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# Deeply Coupled Auto-encoder Networks for
Cross-view Classification
Wen Wang, Zhen Cui, Hong Chang, Shiguang Shan, Xilin Chen
Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
{wen.wang, zhen.cui, hong.chang, shiguang.shan<EMAIL_ADDRESS>
(November 2013)
###### Abstract
The comparison of heterogeneous samples extensively exists in many
applications, especially in the task of image classification. In this paper,
we propose a simple but effective coupled neural network, called Deeply
Coupled Autoencoder Networks (DCAN), which seeks to build two deep neural
networks, coupled with each other in every corresponding layers. In DCAN, each
deep structure is developed via stacking multiple discriminative coupled auto-
encoders, a denoising auto-encoder trained with maximum margin criterion
consisting of intra-class compactness and inter-class penalty. This single
layer component makes our model simultaneously preserve the local consistency
and enhance its discriminative capability. With increasing number of layers,
the coupled networks can gradually narrow the gap between the two views.
Extensive experiments on cross-view image classification tasks demonstrate the
superiority of our method over state-of-the-art methods.
## 1 Introduction
Real-world objects often have different views, which might be endowed with the
same semantic. For example, face images can be captured in different poses,
which reveal the identity of the same object; images of one face can also be
in different modalities, such as pictures under different lighting condition,
pose, or even sketches from artists. In many computer vision applications,
such as image retrieval, interests are taken in comparing two types of
heterogeneous images, which may come from different views or even different
sensors. Since the spanned feature spaces are quite different, it is very
difficult to classify these images across views directly. To decrease the
discrepancy across views, most of previous works endeavored to learn view-
specific linear transforms and to project cross-view samples into a common
latent space, and then employed these newly generated features for
classification.
Though there are lots of approaches used to learn view-specific projections,
they can be divided roughly based on whether the supervised information is
used. Unsupervised methods such as Canonical Correlation Analysis (CCA)[14]
and Partial Least Square (PLS) [26] are employed to the task of cross-view
recognition. Both of them attempt to use two linear mappings to project
samples into a common space where the correlation is maximized, while PLS
considers the variations rather than only the correlation in the target space.
Besides, with use of the mutual information, a Coupled Information-Theoretic
Encoding (CITE) method is developed to narrow the inter-view gap for the
specific photo-sketch recognition task. And in [30], a semi-coupled dictionary
is used to bridge two views. All the methods above consider to reduce the
discrepancy between two views, however, the label information is not
explicitly taken into account. With label information available, many methods
were further developed to learn a discriminant common space For instance,
Discriminative Canonical Correlation Analysis (DCCA) [16] is proposed as an
extension of CCA. And In [22], with an additional local smoothness
constraints, two linear projections are simultaneously learnt for Common
Discriminant Feature Extraction (CDFE). There are also other such methods as
the large margin approach [8] and the Coupled Spectral Regression (CSR) [20].
Recently, multi-view analysis [27, 15] is further developed to jointly learn
multiple specific-view transforms when multiple views (usually more than 2
views) can be available.
Although the above methods have been extensively applied in the cross-view
problem, and have got encouraging performances, they all employed linear
transforms to capture the shared features of samples from two views. However,
these linear discriminant analysis methods usually depend on the assumption
that the data of each class agrees with a Gaussian distribution, while data in
real world usually has a much more complex distribution [33]. It indicates
that linear transforms are insufficient to extract the common features of
cross-view images. So it’s natural to consider about learning nonlinear
features.
A recent topic of interest in nonlinear learning is the research in deep
learning. Deep learning attempts to learn nonlinear representations
hierarchically via deep structures, and has been applied successfully in many
computer vision problems. Classical deep learning methods often stack or
compose multiple basic building blocks to yield a deeper structure. See [5]
for a recent review of Deep Learning algorithms. Lots of such basic building
blocks have been proposed, including sparse coding [19], restricted Boltzmann
machine (RBM) [12], auto-encoder [13, 6], etc. Specifically, the (stacked)
auto-encoder has shown its effectiveness in image denoising [32], domain
adaptation [7], audio-visual speech classification [23], etc.
As we all known, the kernel method, such as Kernel Canonical Correlation
Analysis(Kernel CCA) [1], is also a widely used approach to learn nonlinear
representations. Compared with the kernel method, deep learning is much more
flexible and time-saving because the transform is learned rather than fixed
and the time needed for training and inference process is beyond the limit of
the size of training set.
Inspired by the deep learning works above, we intend to solve the cross-view
classification task via deep networks. It’s natural to build one single deep
neural network with samples from both views, but this kind of network can’t
handle complex data from totally different modalities and may suffer from
inadequate representation capacity. Another way is to learn two different deep
neural networks with samples of the different views. However, the two
independent networks project samples from different views into different
spaces, which makes comparison infeasible. Hence, building two neural networks
coupled with each other seems to be a better solution.
In this work, we propose a Deeply Coupled Auto-encoder Networks(DCAN) method
that learns the common representations to conduct cross-view classification by
building two neural networks deeply coupled respectively, each for one view.
We build the DCAN by stacking multiple discriminative coupled auto-encoders, a
denoising auto-encoder with maximum margin criterion. The discriminative
coupled auto-encoder has a similar input corrupted and reconstructive error
minimized mechanism with the denoising auto-encoder proposed in [28], but is
modified by adding a maximum margin criterion. This kind of criterion has been
used in previous works, like [21, 29, 35], etc. Note that the counterparts
from two views are added into the maximum margin criterion simultaneously
since they both come from the same class, which naturally couples the
corresponding layer in two deep networks. A schematic illustration can be seen
in Fig.1.
The proposed DCAN is related to Multimodal Auto-encoders [23], Multimodal
Restricted Boltzmann Machines and Deep Canonical Correlation Analysis [3]. The
first two methods tend to learn a single network with one or more layers
connected to both views and to predict one view from the other view, and the
Deep Canonical Correlation Analysis build two deep networks, each for one
view, and only representations of the highest layer are constrained to be
correlated. Therefore, the key difference is that we learn two deep networks
coupled with each other in representations in each layer, which is of great
benefits because the DCAN not only learn two separate deep encodings but also
makes better use of data from the both two views. What’s more, these
differences allow for our model to handle the recognition task even when data
is impure and insufficient.
The rest of this paper is organized as follows. Section 2 details the
formulation and solution to the proposed Deeply Coupled Auto-encoder Networks.
Experimental results in Section 3 demonstrate the efficacy of the DCAN. In
section 4 a conclusion is given.
## 2 Deeply Coupled Auto-encoder Networks
In this section, we first present the basic idea. The second part gives a
detailed description of the discriminative coupled auto-encoder. Then, we
describe how to stack multiple layers to build a deep network. Finally, we
briefly describe the optimization of the model.
### 2.1 Basic Idea
Figure 1: An illustration of our proposed DCAN. The left-most and right-most
schematic show the structure of the two coupled network respectively. And the
schematic in the middle illustrates how the whole network gradually enhances
the separability with increasing layers, where pictures with solid line border
denote samples from view 1, those with dotted line border denote samples from
view 2, and different colors imply different subjects.
As shown in Fig.1, the Deeply Coupled Auto-encoder Networks(DCAN) consists of
two deep networks coupled with each other, and each one is for one view. The
network structures of the two deep networks are just like the left-most and
the right-most parts in Fig.1, where circles means the units in each layers
(pixels in a input image for the input layer and hidden representation in
higher layers), and arrows denote the full connections between adjacent
layers. And the middle part of Fig.1 illustrates how the whole network
projects samples in different views into a common space and gradually enhances
the separability with increasing layers.
The two deep networks are both built through stacking multiple similar coupled
single layer blocks because a single coupled layer might be insufficient, and
the method of stacking multiple layers and training each layer greedily has be
proved efficient in lots of previous works, such as those in [13, 6]. With the
number of layers increased, the whole network can compactly represent a
significantly larger set of transforms than shallow networks , and gradually
narrow the gap with the discriminative capacity enhanced.
We use a discriminative coupled auto-encoders trained with maximum margin
criterion as a single layer component. Concretely, we incorporate the
additional noises in the training process while maximizing the margin
criterion, which makes the learnt mapping more stable as well as discriminant.
Note that the maximum margin criterion also works in coupling two
corresponding layers. Formally, the discriminative coupled auto-encoder can be
written as follows:
$\displaystyle\quad\min_{f_{x},f_{y}}\quad L(X,f_{x})+L(Y,f_{y})$ (1)
$\displaystyle s.t.\quad
G_{1}(H_{x},H_{y})-G_{2}(H_{x},H_{y})\leq\varepsilon,$ (2)
where $X,Y$ denote inputs from the two views, and $H_{x},H_{y}$ denote hidden
representations of the two views respectively. $f_{x}:X\longrightarrow
H_{x},f_{y}:Y\longrightarrow H_{y}$ are the transforms we intend to learn, and
we denote the reconstructive error as $L(\cdot)$, and maximum margin criterion
as $G_{1}(\cdot)-G_{2}(\cdot)$, which are described detailedly in the next
subsection.$\varepsilon$ is the threshold of the maximum margin criterion.
### 2.2 Discriminative coupled auto-encoder
In the problem of cross-view, there are two types of heterogenous samples.
Without loss of generality, we denote samples from one view as
$X=[x_{1},\cdots,x_{n}]$ , and those from the other view as
$Y=[y_{1},\cdots,y_{n}]$, in which $n$ is the sample sizes. Noted that the
corresponding labels are known, and $H_{x},H_{y}$ denote hidden
representations of the two views we want to learn.
The DCAN attempts to learn two nonlinear transforms $f_{x}:X\longrightarrow
H_{x}$ and $f_{y}:Y\longrightarrow H_{y}$ that can project the samples from
two views to one discriminant common space respectively, in which the local
neighborhood relationship as well as class separability should be well
preserved for each view. The auto-encoder like structure stands out in
preserving the local consistency, and the denoising form enhances the
robustness of learnt representations. However, the discrimination isn’t taken
into consideration. Therefore, we modify the denoising auto-encoder by adding
a maximum margin criterion consisting of intra-class compactness and inter-
class penalty. And the best nonlinear transformation is a trade-off between
local consistency preserving and separability enhancing.
Just like the one in denoising auto-encoder, the reconstructive error
$L(\cdot)$ in Eq.(1) is formulated as follows:
$\displaystyle
L(X,\Theta)=\sum_{x\in{X^{p}}}{\mathbb{E}_{\tilde{x}\sim{P(\tilde{x}|x)}}}\|\hat{x}-x\|$
(3) $\displaystyle
L(Y,\Theta)=\sum_{y\in{Y^{p}}}{\mathbb{E}_{\tilde{y}\sim{P(\tilde{y}|y)}}}\|\hat{y}-y\|$
(4)
where $\mathbb{E}$ calculates the expectation over corrupted versions
$\tilde{X},\tilde{Y}$ of examples $X,Y$ obtained from a corruption process
$P(\tilde{x}|x),P(\tilde{y}|y)$.
$\Theta=\\{W_{x},W_{y},b_{x},b_{y},c_{x},c_{y}\\}$ specifies the two nonlinear
transforms $f_{x},f_{y}$ , where $W_{x},W_{y}$ is the weight matrix, and
$b_{x},b_{y},c_{x},c_{y}$ are the bias of encoder and decoder respectively,
and $\hat{X},\hat{Y}$ are calculated through the decoder process :
$\begin{split}\hat{X}=s(W_{x}^{T}H_{x}+c_{x})\\\
\hat{Y}=s(W_{y}^{T}H_{y}+c_{y})\end{split}$ (5)
And hidden representations $H_{x},H_{y}$ are obtained from the encoder that is
a similar mapping with the decoder,
$\begin{split}H_{x}=s(W_{x}\tilde{X}+b_{x})\\\
H_{y}=s(W_{y}\tilde{Y}+b_{y})\end{split}$ (6)
where $s$ is the nonlinear activation function, such as the point-wise
hyperbolic tangent operation on linear projected features, i.e.,
$s(x)=\frac{e^{ax}-e^{-ax}}{e^{ax}+e^{-ax}}$ (7)
in which $a$ is the gain parameter.
Moreover, for the maximum margin criterion consisting of intra-class
compactness and inter-class penalty, the constraint term
$G_{1}(\cdot)-G_{2}(\cdot)$ in Eq.(1) is used to realize coupling since
samples of the same class are treated similarly no matter which view they are
from.
Assuming $S$ is the set of sample pairs from the same class, and $D$ is the
set of sample pairs from different classes. Note that the counterparts from
two views are naturally added into $S,D$ since it’s the class rather than the
view that are considered.
Then, we characterize the compactness as follows,
$\displaystyle
G_{1}(H)=\frac{1}{2N_{1}}\sum\limits_{I_{i},I_{j}\in{S}}\|h_{i}-h_{j}\|^{2},$
(8)
where $h_{i}$ denotes the corresponding hidden representation of an input
$I_{i}\in{X\bigcap{Y}}$ and is a sample from either view 1 or view 2, and
$N_{1}$ is the size of $S$.
Meanwhile, the goal of the inter-class separability is to push the adjacent
samples from different classes far away, which can be formulated as follows,
$\displaystyle
G_{2}(H)=\frac{1}{2N_{2}}\sum\limits_{\tiny\begin{subarray}{c}I_{i},I_{j}\in{D}\\\
I_{j}\in{KNN(I_{i})}\end{subarray}}\|h_{i}-h_{j}\|^{2},$ (9)
where $I_{j}$ belongs to the $k$ nearest neighbors of $I_{i}$ with different
class labels, and $N_{2}$ is the number of all pairs satisfying the condition.
And the function of $G_{1}(H),G_{2}(H)$ is illustrated in the middel part of
Fig.1. In the projected common space denoted by $S$, the compactness term
$G_{1}(\cdot)$ shown by red ellipse works by pulling intra-class samples
together while the penalty term $G_{2}(\cdot)$ shown by black ellipse tend to
push adjacent inter-class samples away.
Finally, by solving the optimization problem Eq.(1), we can learn a couple of
nonlinear transforms $f_{x},f_{y}$ to transform the original samples from both
views into a common space.
### 2.3 Stacking coupled auto-encoder
Through the training process above, we model the map between original sample
space and a preliminary discriminant subspace with gap eliminated, and build a
hidden representation $H$ which is a trade-off between approximate
preservation on local consistency and the distinction of the projected data.
But since real-world data is highly complicated, using a single coupled layer
to model the vast and complex real scenes might be insufficient. So we choose
to stack multiple such coupled network layers described in subsection 2.2.
With the number of layers increased, the whole network can compactly represent
a significantly larger set of transforms than shallow networks, and gradually
narrow the gap with the discriminative ability enhanced.
Training a deep network with coupled nonlinear transforms can be achieved by
the canonical greedy layer-wise approach [12, 6]. Or to be more precise, after
training a single layer coupled network, one can compute a new feature $H$ by
the encoder in Eq.(6) and then feed it into the next layer network as the
input feature. In practice, we find that stacking multiple such layers can
gradually reduce the gap and improve the recognition performance (see Fig.1
and Section 3).
### 2.4 Optimization
We adopt the Lagrangian multiplier method to solve the objective function
Eq.(1) with the constraints Eq.(2) as follows:
$\begin{split}\min_{\Theta}\quad&\lambda(L(X,\Theta)+L(Y,\Theta))+(G_{1}(H)-G_{2}(H))+\\\
&\gamma(\frac{1}{2}\|W_{x}\|_{F}^{2}+\frac{1}{2}\|W_{y}\|_{F}^{2})\end{split}$
(10)
where the first term is the the reconstruction error, the second term is the
maximum margin criterion, and the last term is the shrinkage constraints
called the Tikhonov regularizers in [11], which is utilized to decrease the
magnitude of the weights and further to help prevent over-fitting. $\lambda$
is the balance parameter between the local consistency and empirical
separability. And $\gamma$ is called the weight decay parameter and is usually
set to a small value, e.g., 1.0e-4.
To optimize the objective function (10), we use back-propagation to calculate
the gradient and then employ the limited-memory BFGS (L-BFGS) method [24, 17],
which is often used to solve nonlinear optimization problems without any
constraints. L-BFGS is particularly suitable for problems with a large amount
of variables under the moderate memory requirement. To utilize L-BFGS, we need
to calculate the gradients of the object function. Obviously, the object
function in (10) is differential to these parameters $\Theta$, and we use
Back-propagation [18] method to derive the derivative of the overall cost
function. In our setting, we find the objective function can achieve as fast
convergence as described in [17].
## 3 Experiments
In this section, the proposed DCAN is evaluated on two datasets, Multi-PIE [9]
and CUHK Face Sketch FERET (CUFSF) [34, 31].
### 3.1 Databases
Multi-PIE dataset [9] is employed to evaluate face recognition across pose.
Here a subset from the 337 subjects in 7 poses
($-45^{\circ},-30^{\circ},-15^{\circ},0^{\circ},15^{\circ},30^{\circ},45^{\circ}$),
3 expression (Neutral,Smile, Disgust), no flush illumination from 4 sessions
are selected to validate our method. We randomly choose 4 images for each pose
of each subject, then randomly partition the data into two parts: the training
set with 231 subjects (i.e., $231\times 7\times 4=6468$ images) and the
testing set with the rest subjects.
CUHK Face Sketch FERET (CUFSF) dataset [34, 31] contains two types of face
images: photo and sketch. Total 1,194 images (one image per subject) were
collected with lighting variations from FERET dataset [25]. For each subject,
a sketch is drawn with shape exaggeration. According to the configuration of
[15], we use the first 700 subjects as the training data and the rest subjects
as the testing data.
### 3.2 Settings
All images from Multi-PIE and CUFSF are cropped into 64$\times$80 pixels
without any preprocess. We compare the proposed DCAN method with several
baselines and state-of-the-art methods, including CCA [14], Kernel CCA [1],
Deep CCA [3], FDA [4], CDFE [22], CSR [20], PLS [26] and MvDA [15]. The first
seven methods are pairwise methods for cross-view classification. MvDA jointly
learns all transforms when multiple views can be utilized, and has achieved
the state-of-the-art results in their reports [15].
The Principal Component Analysis (PCA) [4] is used for dimension reduction. In
our experiments, we set the default dimensionality as 100 with preservation of
most energy except Deep CCA, PLS, CSR and CDFE, where the dimensionality are
tuned in [50,1000] for the best performance. For all these methods, we report
the best performance by tuning the related parameters according to their
papers. Firstly, for Kernel CCA, we experiment with Gaussian kernel and
polynomial kernel and adjust the parameters to get the best performance. Then
for Deep CCA [3], we strictly follow their algorithms and tune all possible
parameters, but the performance is inferior to CCA. One possible reason is
that Deep CCA only considers the correlations on training data (as reported in
their paper) so that the learnt mode overly fits the training data, which thus
leads to the poor generality on the testing set. Besides, the parameter
$\alpha$ and $\beta$ are respectively traversed in [0.2,2] and [0.0001,1] for
CDFE, the parameter $\lambda$ and $\eta$ are searched in [0.001,1] for CSR,
and the reduced dimensionality is tuned for CCA, PLS, FDA and MvDA.
As for our proposed DCAN, the performance on CUFSF database of varied
parameters, $\lambda,k$, is shown in Fig.3. In following experiments, we set
$\lambda=0.2,\gamma=1.0e-4$, $k=10$ and $a=1$. With increasing layers, the
number of hidden neurons are gradually reduced by $10$, _i.e.,_ $90,80,70,60$
if four layers.
Method | Accuracy
---|---
CCA[14] | 0.698
KernelCCA[10] | 0.840
DeepCCA[3] | 0.599
FDA[4] | 0.814
CDFE[22] | 0.773
CSR[20] | 0.580
PLS[26] | 0.574
MvDA[15] | 0.867
DCAN-1 | 0.830
DCAN-2 | 0.877
DCAN-3 | 0.884
DCAN-4 | 0.879
Table 1: Evaluation on Multi-PIE database in terms of mean accuracy. DCAN-k
means a stacked k-layer network.
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.816 | 0.588 | 0.473 | 0.473 | 0.515 | 0.511
$-30^{\circ}$ | 0.816 | 1.000 | 0.858 | 0.611 | 0.664 | 0.553 | 0.553
$-15^{\circ}$ | 0.588 | 0.858 | 1.000 | 0.894 | 0.807 | 0.602 | 0.447
$0^{\circ}$ | 0.473 | 0.611 | 0.894 | 1.000 | 0.909 | 0.604 | 0.484
$15^{\circ}$ | 0.473 | 0.664 | 0.807 | 0.909 | 1.000 | 0.874 | 0.602
$30^{\circ}$ | 0.515 | 0.553 | 0.602 | 0.604 | 0.874 | 1.000 | 0.768
$45^{\circ}$ | 0.511 | 0.553 | 0.447 | 0.484 | 0.602 | 0.768 | 1.000
(a) CCA, $Ave=0.698$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.878 | 0.810 | 0.756 | 0.706 | 0.726 | 0.737
$-30^{\circ}$ | 0.878 | 1.000 | 0.892 | 0.858 | 0.808 | 0.801 | 0.757
$-15^{\circ}$ | 0.810 | 0.892 | 1.000 | 0.911 | 0.880 | 0.861 | 0.765
$0^{\circ}$ | 0.756 | 0.858 | 0.911 | 1.000 | 0.938 | 0.759 | 0.759
$15^{\circ}$ | 0.706 | 0.808 | 0.880 | 0.938 | 1.000 | 0.922 | 0.845
$30^{\circ}$ | 0.726 | 0.801 | 0.861 | 0.759 | 0.922 | 1.000 | 0.912
$45^{\circ}$ | 0.737 | 0.757 | 0.765 | 0.759 | 0.845 | 0.912 | 1.000
(b) KernelCCA, $Ave=0.840$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.854 | 0.598 | 0.425 | 0.473 | 0.522 | 0.523
$-30^{\circ}$ | 0.854 | 1.000 | 0.844 | 0.578 | 0.676 | 0.576 | 0.566
$-15^{\circ}$ | 0.598 | 0.844 | 1.000 | 0.806 | 0.807 | 0.602 | 0.424
$0^{\circ}$ | 0.425 | 0.578 | 0.806 | 1.000 | 0.911 | 0.599 | 0.444
$15^{\circ}$ | 0.473 | 0.676 | 0.807 | 0.911 | 1.000 | 0.866 | 0.624
$30^{\circ}$ | 0.522 | 0.576 | 0.602 | 0.599 | 0.866 | 1.000 | 0.756
$45^{\circ}$ | 0.523 | 0.566 | 0.424 | 0.444 | 0.624 | 0.756 | 1.000
(c) DeepCCA, $Ave=0.599$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.847 | 0.754 | 0.686 | 0.573 | 0.610 | 0.664
$-30^{\circ}$ | 0.847 | 1.000 | 0.911 | 0.847 | 0.807 | 0.766 | 0.635
$-15^{\circ}$ | 0.754 | 0.911 | 1.000 | 0.925 | 0.896 | 0.821 | 0.602
$0^{\circ}$ | 0.686 | 0.847 | 0.925 | 1.000 | 0.964 | 0.872 | 0.684
$15^{\circ}$ | 0.573 | 0.807 | 0.896 | 0.964 | 1.000 | 0.929 | 0.768
$30^{\circ}$ | 0.610 | 0.766 | 0.821 | 0.872 | 0.929 | 1.000 | 0.878
$45^{\circ}$ | 0.664 | 0.635 | 0.602 | 0.684 | 0.768 | 0.878 | 1.000
(d) FDA, $Ave=0.814$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.854 | 0.714 | 0.595 | 0.557 | 0.633 | 0.608
$-30^{\circ}$ | 0.854 | 1.000 | 0.867 | 0.746 | 0.688 | 0.697 | 0.606
$-15^{\circ}$ | 0.714 | 0.867 | 1.000 | 0.887 | 0.808 | 0.704 | 0.579
$0^{\circ}$ | 0.595 | 0.746 | 0.887 | 1.000 | 0.916 | 0.819 | 0.651
$15^{\circ}$ | 0.557 | 0.688 | 0.808 | 0.916 | 1.000 | 0.912 | 0.754
$30^{\circ}$ | 0.633 | 0.697 | 0.704 | 0.819 | 0.912 | 1.000 | 0.850
$45^{\circ}$ | 0.608 | 0.606 | 0.579 | 0.651 | 0.754 | 0.850 | 1.000
(e) CDFE, $Ave=0.773$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.914 | 0.854 | 0.763 | 0.710 | 0.770 | 0.759
$-30^{\circ}$ | 0.914 | 1.000 | 0.947 | 0.858 | 0.812 | 0.861 | 0.766
$-15^{\circ}$ | 0.854 | 0.947 | 1.000 | 0.923 | 0.880 | 0.894 | 0.775
$0^{\circ}$ | 0.763 | 0.858 | 0.923 | 1.000 | 0.938 | 0.900 | 0.750
$15^{\circ}$ | 0.710 | 0.812 | 0.880 | 0.938 | 1.000 | 0.923 | 0.807
$30^{\circ}$ | 0.770 | 0.861 | 0.894 | 0.900 | 0.923 | 1.000 | 0.934
$45^{\circ}$ | 0.759 | 0.766 | 0.775 | 0.750 | 0.807 | 0.934 | 1.000
(f) MvDA, $Ave=0.867$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.872 | 0.819 | 0.730 | 0.655 | 0.708 | 0.686
$-30^{\circ}$ | 0.856 | 1.000 | 0.881 | 0.825 | 0.754 | 0.737 | 0.650
$-15^{\circ}$ | 0.807 | 0.874 | 1.000 | 0.869 | 0.865 | 0.781 | 0.681
$0^{\circ}$ | 0.757 | 0.854 | 0.896 | 1.000 | 0.938 | 0.858 | 0.790
$15^{\circ}$ | 0.688 | 0.777 | 0.854 | 0.916 | 1.000 | 0.900 | 0.823
$30^{\circ}$ | 0.708 | 0.735 | 0.788 | 0.834 | 0.918 | 1.000 | 0.916
$45^{\circ}$ | 0.719 | 0.715 | 0.697 | 0.752 | 0.832 | 0.909 | 1.000
(g) DCAN-1, $Ave=0.830$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.905 | 0.876 | 0.783 | 0.714 | 0.779 | 0.796
$-30^{\circ}$ | 0.927 | 1.000 | 0.954 | 0.896 | 0.850 | 0.825 | 0.730
$-15^{\circ}$ | 0.867 | 0.929 | 1.000 | 0.905 | 0.905 | 0.867 | 0.757
$0^{\circ}$ | 0.832 | 0.876 | 0.925 | 1.000 | 0.958 | 0.896 | 0.808
$15^{\circ}$ | 0.765 | 0.865 | 0.907 | 0.951 | 1.000 | 0.929 | 0.874
$30^{\circ}$ | 0.779 | 0.832 | 0.870 | 0.916 | 0.945 | 1.000 | 0.949
$45^{\circ}$ | 0.794 | 0.777 | 0.785 | 0.812 | 0.876 | 0.938 | 1.000
(h) DCAN-3, $Ave=0.884$
Table 2: Results of CCA, FDA [4], CDFE [22], MvDA [15] and DCAN on MultiPIE
dataset in terms of rank-1 recognition rate. DCAN-k means a stacked k-layer
network. Due to space limitation, the results of other methods cannot be
reported here, but their mean accuracies are shown in Table 1.
### 3.3 Face Recognition across Pose
First, to explicitly illustrate the learnt mapping, we conduct an experiment
on Multi-PIE dataset by projecting the learnt common features into a 2-D space
with Principal Component Analysis (PCA). As shown in Fig.2. The classical
method CCA can only roughly align the data in the principal directions and the
state-of-the-art method MvDA [15] attempts to merge two types of data but
seems to fail. Thus, we argue that linear transforms are a little stiff to
convert data from two views into an ideal common space. The three diagrams
below shows that DCAN can gradually separate samples from different classes
with the increase of layers, which is just as we described in the above
analysis.
Figure 2: After learning common features by the cross-view methods, we project
the features into 2-D space by using the principal two components in PCA. The
depicted samples are randomly chosen form Multi-PIE [9] dataset. The “$\circ$”
and “$+$” points come from two views respectively. Different color points
belong to different classes. DCAN-k is our proposed method with a stacked
k-layer neural network.
Next, we compare our methods with several state-of-the-art methods for the
cross-view face recognition task on Multi-PIE data set. Since the images are
acquired over seven poses on Multi-PIE data set, in total $7\times 6=42$
comparison experiments need to be conducted. The detailed results are shown in
Table 2,where two poses are used as the gallery and probe set to each other
and the rank-1 recognition rate is reported. Further, the mean accuracy of all
pairwise results for each methods is also reported in Table 1.
From Table 1, we can find the supervised methods except CSR are significantly
superior to CCA due to the use of the label information. And nonlinear methods
except Deep CCA are significantly superior to the nonlinear methods due to the
use of nonlinear transforms. Compared with FDA, the proposed DCAN with only
one layer network can perform better with 1.6% improvement. With increasing
layers, the accuracy of DCAN reaches a climax via stacking three layer
networks. The reason of the degradation in DCAN with four layers is mainly the
effect of reduced dimensionality, where 10 dimensions are cut out from the
above layer network. Obviously, compared with two-view based methods, the
proposed DCAN with three layers improves the performance greatly (88.4% vs.
81.4%). Besides, MvDA also achieves a considerably good performance by using
all samples from all poses. It is unfair to compare these two-view based
methods (containing DCAN) with MvDA, because the latter implicitly uses
additional five views information except current compared two views. But our
method performs better than MvDA, 88.4% vs. 86.7%. As observed in Table 2,
three-layer DCAN achieves a largely improvement compared with CCA,FDA,CDFE for
all cross-view cases and MvDA for most of cross-view cases. The results are
shown in Table 2 and Table 1.
### 3.4 Photo-Sketch Recognition
Method | Photo-Sketch | Sketch-Photo
---|---|---
CCA[14] | 0.387 | 0.475
KernelCCA[10] | 0.466 | 0.570
DeepCCA[3] | 0.364 | 0.434
CDFE[22] | 0.456 | 0.476
CSR[20] | 0.502 | 0.590
PLS[26] | 0.486 | 0.510
FDA[4] | 0.468 | 0.534
MvDA[15] | 0.534 | 0.555
DCAN-1 | 0.535 | 0.555
DCAN-2 | 0.603 | 0.613
DCAN-3 | 0.601 | 0.652
Table 3: Evluation on CUFSF database in terms of mean accuracy. DCAN-k means a
stacked k-layer network.
(a)
(b)
Figure 3: The performance with varied parameter values for our proposed DCAN.
The sketch and photo images in CUFSF [34, 31] are respectively used for the
gallery and probe set. (a) Varied $\lambda$ with fixed $k=10$. (b) Varied $k$
with fixed $\lambda=0.2$.
Photo-Sketch recognition is conducted on CUFSF dataset. The samples come from
only two views, photo and sketch. The comparison results are provided in Table
3. As shown in this table, since only two views can be utilized in this case,
MvDA degrades to a comparable performance with those previous two-view based
methods. Our proposed DCAN with three layer networks can achieve even better
with more than 6% improvement, which further indicates DCAN benefits from the
nonlinear and multi-layer structure.
Discussion and analysis: The above experiments demonstrate that our methods
can work very well even on a small sample size. The reasons lie in three
folds:
1. (1)
The maximum margin criterion makes the learnt mapping more discriminative,
which is a straightforward strategy in the supervised classification task.
2. (2)
Auto-encoder approximately preserves the local neighborhood structures.
For this, Alain et al. [2] theoretically prove that the learnt representation
by auto-encoder can recover local properties from the view of manifold. To
further validate that, we employ the first 700 photo images from CUFSF
database to perform the nonlinear self-reconstruction with auto-encoder. With
the hidden presentations, we find the local neighbors with 1,2,3,4,5 neighbors
can be preserved with the probability of 99.43%, 99.00%, 98.57%, 98.00% and
97.42% respectively. Thus, the use of auto-encoder intrinsically reduces the
complexity of the discriminant model, which further makes the learnt model
better generality on the testing set.
3. (3)
The deep structure generates a gradual model, which makes the learnt transform
more robust. With only one layer, the model can’t represent the complex data
very well. But with layers goes deeper, the coupled networks can learn
transforms much more flexible and hence can be allowed to handle more complex
data.
## 4 Conclusion
In this paper, we propose a deep learning method, the Deeply Coupled Auto-
encoder Networks(DCAN), which can gradually generate a coupled discriminant
common representation for cross-view object classification. In each layer we
take both local consistency and discrimination of projected data into
consideration. By stacking multiple such coupled network layers, DCAN can
gradually improve the learnt shared features in the common space. Moreover,
experiments in the cross-view classification tasks demonstrate the superior of
our method over other state-of-the-art methods.
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|
$\displaystyle=\mathbf{Q}_{n,ww}^{-1}\mathbf{q}_{n,wy}-\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)^{-1}\mathrm{E}\left(\mathbf{q}_{n,wy}\right)$
$\displaystyle=\left[\mathbf{Q}_{n,ww}^{-1}-E\left(\mathbf{Q}_{n,ww}\right)^{-1}+E\left(\mathbf{Q}_{n,ww}\right)^{-1}\right]\left[\mathbf{q}_{n,wy}-\mathrm{E}\left(\mathbf{q}_{n,wy}\right)+\mathrm{E}\left(\mathbf{q}_{n,wy}\right)\right]-\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)^{-1}\mathrm{E}\left(\mathbf{q}_{n,wy}\right)$
$\displaystyle=\left[\mathbf{Q}_{n,ww}^{-1}-E\left(\mathbf{Q}_{n,ww}\right)^{-1}\right]\left[\mathbf{q}_{n,wy}-\mathrm{E}\left(\mathbf{q}_{n,wy}\right)\right]+\left[\mathbf{Q}_{n,ww}^{-1}-\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)^{-1}\right]\mathrm{E}\left(\mathbf{q}_{n,wy}\right)$
$\displaystyle\quad\quad\quad+\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)^{-1}\left[\mathbf{q}_{n,wy}-\mathrm{E}\left(\mathbf{q}_{n,wy}\right)\right].$
Then,
$\displaystyle\left\|\hat{\mathbf{\phi}}-\mathbf{\phi}\right\|$
$\displaystyle\leq\left\|\mathbf{Q}_{n,ww}^{-1}-\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)^{-1}\right\|\left\|\mathbf{q}_{n,wy}-\mathrm{E}\left(\mathbf{q}_{n,wy}\right)\right\|+\left\|\mathbf{Q}_{n,ww}^{-1}-\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)^{-1}\right\|\left\|\mathrm{E}\left(\mathbf{q}_{n,wy}\right)\right\|$
$\displaystyle\quad\quad+\left\|\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)^{-1}\right\|\left\|\mathbf{q}_{n,wy}-\mathrm{E}\left(\mathbf{q}_{n,wy}\right)\right\|.$
By Assumption 1(c), we have
$\left\|\mathbf{Q}_{n,ww}^{-1}-\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)^{-1}\right\|=O_{p}\left(n^{-1/2}\right)$,
$\left\|\mathbf{q}_{n,wy}-\mathrm{E}\left(\mathbf{q}_{n,wy}\right)\right\|=O_{p}\left(n^{-1/2}\right)$,
and by Assumption 1(b),
$\left\|\mathrm{E}\left(\mathbf{q}_{n,wy}\right)\right\|$ and
$\left\|\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)^{-1}\right\|$ are bounded.
Thus,
$\left\|\hat{\mathbf{\phi}}-\mathbf{\phi}\right\|=O_{p}\left(n^{-1/2}\right).$
(S.2.1)
To establish the asymptotic distribution of $\hat{\mathbf{\phi}}$, we first
note that
$\sqrt{n}\left(\hat{\mathbf{\phi}}-\mathbf{\phi}\right)=\mathbf{Q}_{n,ww}^{-1}\left(n^{-1/2}\sum_{i=1}^{n}\mathbf{w}_{i}\xi_{i}\right).$
By Assumption 3, we have
$\mathrm{var}\left(n^{-1/2}\sum_{i=1}^{n}\mathbf{w}_{i}\xi_{i}\right)=\frac{1}{n}\sum_{i=1}^{n}\mathrm{var}\left(\mathbf{w}_{i}\xi_{i}\right)=\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\xi_{i}^{2}\right)\rightarrow\mathbf{V}_{w\xi}\succ
0.$
Note that $\xi_{i}=u_{i}+x_{i}v_{i}$, and $\mathbf{w}_{i}$ is distributed
independently of $u_{i}$ and $v_{i}$. Then
$\mathbf{w}_{i}\xi_{i}=\mathbf{w}_{i}\left(u_{i}+x_{i}v_{i}\right)=\mathbf{w}_{i}u_{i}+\left(\mathbf{w}_{i}x_{i}\right)v_{i},$
and by Minkowski’s inequality, for $r=2+\delta$ with $0<\delta<1$,
$\left[E\left\|\mathbf{w}_{i}\xi_{i}\right\|^{r}\right]^{1/r}\leq\left[E\left\|\mathbf{w}_{i}u_{i}\right\|^{r}\right]^{1/r}+\left[E\left\|\left(\mathbf{w}_{i}x_{i}\right)v_{i}\right\|^{r}\right]^{1/r}.$
Due to the independence of $u_{i}$ and $v_{i}$ from $\mathbf{w}_{i}$, we have
$\mathrm{E}\left(\left\|\mathbf{w}_{i}u_{i}\right\|^{r}\right)\leq
E\left\|\mathbf{w}_{i}\right\|^{r}E\left\|u_{i}\right\|^{r},\text{ and
}E\left\|\left(\mathbf{w}_{i}x_{i}^{\prime}\right)v_{i}\right\|^{r}\leq
E\left\|\mathbf{w}_{i}x_{i}\right\|^{r}E\left\|v_{i}\right\|^{r}.$
Also, $E\left\|\mathbf{w}_{i}x_{i}\right\|^{r}\leq
E\left\|\left(x_{i}^{2},x_{i}\mathbf{z}_{i}^{\prime}\right)^{\prime}\right\|^{r}\leq
E\left\|\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\right\|^{r}\leq
E\left\|\mathbf{w}_{i}\right\|^{2r}$, where $2<r<3$, and hence $2r<6$. By
Assumptions 1(a.ii) and 1(b.ii), we have
$\sup_{i}\mathrm{E}\left(\left\|\mathbf{w}_{i}\right\|^{6}\right)<C$,
$\sup_{i}\mathrm{E}\left(\left\|u_{i}\right\|^{3}\right)<C$, and
$\mathrm{E}\left(\left\|v_{i}\right\|^{3}\right)\leq\max_{1\leq k\leq
K}\left|b_{k}-\mathrm{E}\left(\beta_{i}\right)\right|^{3}<C.$ Then, we
verified that
$\sup_{i}\mathrm{E}\left(\left\|\mathbf{w}_{i}u_{i}\right\|^{r}\right)<C$, and
$E\left\|\left(\mathbf{w}_{i}x_{i}^{\prime}\right)v_{i}\right\|^{r}<C$, and
hence the Lyapunov condition that
$\sup_{i}\mathrm{E}\left(\left\|\mathbf{w}_{i}\xi_{i}\right\|^{r}\right)<C$,
where $r=2+\delta\in(2,3)$. By the central limit theorem for independent but
not necessarily identically distributed random vectors (see Pesaran (2015,
Theorem 18) or Hansen (2022, Theorem 6.5)), we have
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\mathbf{w}_{i}\xi_{i}\rightarrow_{d}N(\mathbf{0},\mathbf{V}_{w\xi}),$
as $n\rightarrow\infty$, and by Assumption 1 and continuous mapping theorem,
$\sqrt{n}(\hat{\mathbf{\phi}}-\mathbf{\phi})\rightarrow_{d}N\left(\mathbf{0},\mathbf{Q}_{ww}^{-1}\mathbf{V}_{w\xi}\mathbf{Q}_{ww}^{-1}\right).$
We then turn to the consistent estimation of the variance matrix. By
Assumption 3, we have
$\displaystyle\left\|\hat{\mathbf{V}}_{w\xi}-\mathbf{V}_{w\xi}\right\|$
$\displaystyle=\left\|\frac{1}{n}\sum_{i=1}^{n}\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\hat{\xi}_{i}^{2}-\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\mathbf{w}_{i}\mathbf{w}_{i}\xi_{i}^{2}\right)+\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\mathbf{w}_{i}\mathbf{w}_{i}\xi_{i}^{2}\right)-\mathbf{V}_{w\xi}\right\|$
$\displaystyle\leq\left\|\frac{1}{n}\sum_{i=1}^{n}\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\xi_{i}^{2}-\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\mathbf{w}_{i}\mathbf{w}_{i}\xi_{i}^{2}\right)\right\|+\left\|\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\mathbf{w}_{i}\mathbf{w}_{i}\xi_{i}^{2}\right)-\mathbf{V}_{w\xi}\right\|$
$\displaystyle\quad\quad+\left\|\frac{1}{n}\sum_{i=1}^{n}\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\left(\hat{\xi}_{i}^{2}-\xi_{i}^{2}\right)\right\|$
$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\left\|\mathbf{w}_{i}\right\|^{2}\left|\hat{\xi}_{i}^{2}-\xi_{i}^{2}\right|+O_{p}(n^{-1/2}).$
(S.2.2)
Note that
$\hat{\xi}_{i}=\xi_{i}-\left(\hat{\mathbf{\phi}}-\mathbf{\phi}\right)^{\prime}\mathbf{w}_{i}$,
then
$\displaystyle\left|\hat{\xi}_{i}^{2}-\xi_{i}^{2}\right|$ $\displaystyle\leq
2\left|\xi_{i}\mathbf{w}_{i}^{\prime}\left(\hat{\mathbf{\phi}}-\mathbf{\phi}\right)\right|+\left(\hat{\mathbf{\phi}}-\mathbf{\phi}\right)^{\prime}\left(\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\right)\left(\hat{\mathbf{\phi}}-\mathbf{\phi}\right)$
$\displaystyle\leq
2\left|\xi_{i}\right|\left\|\mathbf{w}_{i}\right\|\left\|\hat{\mathbf{\phi}}-\mathbf{\phi}\right\|+\left\|\mathbf{w}_{i}\right\|^{2}\left\|\hat{\mathbf{\phi}}-\mathbf{\phi}\right\|^{2}.$
(S.2.3)
Combine (S.2.2) and (S.2.3), we have
$\left\|\hat{\mathbf{V}}_{w\xi}-\mathbf{V}_{w\xi}\right\|\leq
2\left(\frac{1}{n}\sum_{i=1}^{n}\left\|\mathbf{w}_{i}\right\|^{3}\left|\xi_{i}\right|\right)\left\|\hat{\mathbf{\phi}}-\mathbf{\phi}\right\|+\left(\frac{1}{n}\sum_{i=1}^{n}\left\|\mathbf{w}_{i}\right\|^{4}\right)\left\|\hat{\mathbf{\phi}}-\mathbf{\phi}\right\|^{2}+O_{p}\left(n^{-1/2}\right).$
(S.2.4)
By Hölder’s inequality,
$\frac{1}{n}\sum_{i=1}^{n}\left\|\mathbf{w}_{i}\right\|^{3}\left|\xi_{i}\right|\leq\left(\frac{1}{n}\sum_{i=1}^{n}\left\|\mathbf{w}_{i}\right\|^{4}\right)^{3/4}\left(\frac{1}{n}\sum_{i=1}^{n}\xi_{i}^{4}\right)^{1/4}.$
(S.2.5)
By Assumption 1(b.iii),
$n^{-1}\sum_{i=1}^{n}\left\|\mathbf{w}_{i}\right\|^{4}=O_{p}(1)$. By Minkowski
inequality,
$\displaystyle\left(\frac{1}{n}\sum_{i=1}^{n}\xi_{i}^{4}\right)^{1/4}$
$\displaystyle=\left(\frac{1}{n}\sum_{i=1}^{n}\left(u_{i}+x_{i}v_{i}\right)^{4}\right)^{1/4}\leq\left(\frac{1}{n}\sum_{i=1}^{n}u_{i}^{4}\right)^{1/4}+\left(\frac{1}{n}\sum_{i=1}^{n}x_{i}^{4}v_{i}^{4}\right)^{1/4}$
$\displaystyle\leq\left(\frac{1}{n}\sum_{i=1}^{n}u_{i}^{4}\right)^{1/4}+\max_{k}\left\\{\left|b_{k}-\mathrm{E}\left(\beta_{i}\right)\right|\right\\}\left(\frac{1}{n}\sum_{i=1}^{n}x_{i}^{4}\right)^{1/4}$
$\displaystyle=O_{p}(1),$
where the last inequality is from Assumptions 1(a.iii) and (b.iii) that
$n^{-1}\sum_{i=1}^{n}u_{i}^{4}=O_{p}\left(1\right)$, and
$n^{-1}\sum_{i=1}^{n}x_{i}^{4}\leq
n^{-1}\sum_{i=1}^{n}\left\|\mathbf{w}_{i}\right\|^{4}=O_{p}(1)$. Then we
verified in (S.2.5) that
$n^{-1}\sum_{i=1}^{n}\left\|\mathbf{w}_{i}\right\|^{3}\left|\xi_{i}\right|=O_{p}(1)$.
Then using the above results in (S.2.4), and noting from (S.2.1) that
$\left\|\hat{\mathbf{\phi}}-\mathbf{\phi}\right\|=O_{p}\left(n^{-1/2}\right)$,
we have
$\left\|\hat{\mathbf{V}}_{w\xi}-\mathbf{V}_{w\xi}\right\|=O_{p}\left(n^{-1/2}\right)$,
as required.
## Appendix S.3 Monte Carlo Simulation
### S.3.1 Results with $S=5$ and $S=6$
Tables S.1 and S.2 present the summary results corresponding to $S=5$ and
$S=6$, for the data generating processes described in Section 5.1. These
results show that adding more moments does not necessarily improve the
estimation accuracy but could be counter-productive.
Table S.1: Bias, RMSE and size of the GMM estimator for distributional
parameters of $\beta$ with $S=5$
DGP | Baseline | Categorical $x$ | Categorical $u$
---|---|---|---
Sample size $n$ | Bias | RMSE | Size | Bias | RMSE | Size | Bias | RMSE | Size
high variance: $\mathrm{var}\left(\beta_{i}\right)=0.25$
$\pi=0.5$ | 100 | 0.0308 | 0.1869 | 0.1021 | 0.0259 | 0.1986 | 0.1276 | 0.0106 | 0.1944 | 0.1050
1,000 | 0.0048 | 0.1235 | 0.1950 | 0.0054 | 0.1334 | 0.2112 | -0.0364 | 0.1638 | 0.2239
2,000 | -0.0006 | 0.0875 | 0.1641 | -0.0009 | 0.0962 | 0.1887 | -0.0238 | 0.1172 | 0.2059
5,000 | -0.0005 | 0.0484 | 0.1339 | -0.0001 | 0.0591 | 0.1602 | -0.0125 | 0.0740 | 0.1667
10,000 | -0.0002 | 0.0334 | 0.1152 | -0.0005 | 0.0373 | 0.1246 | -0.0080 | 0.0519 | 0.1386
100,000 | -0.0002 | 0.0096 | 0.0636 | 0.0001 | 0.0116 | 0.0738 | -0.0008 | 0.0174 | 0.0766
$\beta_{L}=1$ | 100 | 0.2234 | 0.4541 | 0.3205 | 0.1992 | 0.4777 | 0.2843 | 0.1780 | 0.5090 | 0.2519
1,000 | 0.0503 | 0.1609 | 0.3060 | 0.0475 | 0.1812 | 0.2963 | 0.0100 | 0.2024 | 0.2141
2,000 | 0.0265 | 0.1148 | 0.2501 | 0.0257 | 0.1262 | 0.2501 | 0.0088 | 0.1337 | 0.1905
5,000 | 0.0108 | 0.0606 | 0.1926 | 0.0130 | 0.0702 | 0.2042 | 0.0031 | 0.0803 | 0.1641
10,000 | 0.0054 | 0.0409 | 0.1408 | 0.0061 | 0.0456 | 0.1510 | 0.0008 | 0.0527 | 0.1338
100,000 | 0.0004 | 0.0114 | 0.0716 | 0.0006 | 0.0134 | 0.0790 | 0.0002 | 0.0184 | 0.0834
$\beta_{H}=2$ | 100 | -0.1956 | 0.5486 | 0.2448 | -0.1941 | 0.5638 | 0.2386 | -0.2029 | 0.5801 | 0.2269
1,000 | -0.0418 | 0.2080 | 0.3299 | -0.0414 | 0.2300 | 0.3384 | -0.0752 | 0.2583 | 0.3620
2,000 | -0.0264 | 0.1379 | 0.2799 | -0.0286 | 0.1554 | 0.2860 | -0.0529 | 0.1789 | 0.3048
5,000 | -0.0113 | 0.0696 | 0.2008 | -0.0116 | 0.0883 | 0.2170 | -0.0254 | 0.1038 | 0.2411
10,000 | -0.0053 | 0.0432 | 0.1502 | -0.0064 | 0.0520 | 0.1642 | -0.0156 | 0.0690 | 0.2002
100,000 | -0.0007 | 0.0113 | 0.0662 | -0.0004 | 0.0135 | 0.0764 | -0.0016 | 0.0209 | 0.0818
low variance: $\mathrm{var}\left(\beta_{i}\right)=0.15$
$\pi=0.3$ | 100 | 0.2214 | 0.2820 | 0.1063 | 0.2291 | 0.2942 | 0.1328 | 0.2212 | 0.2876 | 0.1221
1,000 | 0.0477 | 0.1746 | 0.2235 | 0.0605 | 0.1928 | 0.2430 | 0.0348 | 0.2039 | 0.2900
2,000 | 0.0217 | 0.1198 | 0.2020 | 0.0262 | 0.1331 | 0.2246 | -0.0080 | 0.1608 | 0.2822
5,000 | 0.0112 | 0.0709 | 0.1732 | 0.0154 | 0.0828 | 0.1956 | -0.0115 | 0.1072 | 0.2289
10,000 | 0.0063 | 0.0465 | 0.1588 | 0.0106 | 0.0576 | 0.1649 | -0.0075 | 0.0761 | 0.1890
100,000 | 0.0001 | 0.0130 | 0.0810 | 0.0014 | 0.0158 | 0.0882 | 0.0040 | 0.0280 | 0.0978
$\beta_{L}=0.5$ | 100 | 0.4245 | 0.5722 | 0.2938 | 0.4048 | 0.5818 | 0.2612 | 0.3827 | 0.6052 | 0.2278
1,000 | 0.1300 | 0.2692 | 0.3058 | 0.1300 | 0.2890 | 0.3057 | 0.0882 | 0.3673 | 0.1970
2,000 | 0.0763 | 0.1746 | 0.3147 | 0.0735 | 0.1903 | 0.2820 | 0.0149 | 0.2523 | 0.1964
5,000 | 0.0378 | 0.1018 | 0.2690 | 0.0410 | 0.1155 | 0.2695 | 0.0034 | 0.1417 | 0.1905
10,000 | 0.0202 | 0.0674 | 0.2344 | 0.0257 | 0.0822 | 0.2404 | 0.0013 | 0.0961 | 0.1690
100,000 | 0.0013 | 0.0184 | 0.0952 | 0.0026 | 0.0221 | 0.1042 | 0.0060 | 0.0347 | 0.1112
$\beta_{H}=1.345$ | 100 | -0.0646 | 0.3773 | 0.1781 | -0.0616 | 0.4058 | 0.1668 | -0.0564 | 0.4357 | 0.1688
1,000 | -0.0180 | 0.1523 | 0.2496 | -0.0119 | 0.1804 | 0.2615 | -0.0476 | 0.2022 | 0.2721
2,000 | -0.0104 | 0.1021 | 0.2375 | -0.0101 | 0.1147 | 0.2414 | -0.0381 | 0.1448 | 0.2830
5,000 | -0.0027 | 0.0549 | 0.1680 | -0.0016 | 0.0680 | 0.1936 | -0.0193 | 0.0927 | 0.2369
10,000 | -0.0001 | 0.0368 | 0.1458 | 0.0007 | 0.0438 | 0.1458 | -0.0115 | 0.0634 | 0.1976
100,000 | -0.0002 | 0.0102 | 0.0726 | 0.0005 | 0.0120 | 0.0688 | 0.0021 | 0.0214 | 0.0902
Notes: The data generating process is (5.1). high variance and low variance
parametrization are described in (5.2). “Baseline”, “Categorical $x$” and
“Categorical $u$” refer to DGP 1 to 3 as in Section 5.1. Generically, bias,
RMSE and size are calculated by
$R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)$,
$\sqrt{R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)^{2}}$,
and
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{0}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
respectively, for true parameter $\theta_{0}$, its estimate
$\hat{\theta}^{(r)}$, the estimated standard error of $\hat{\theta}^{(r)}$,
$\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical value
$\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.
Table S.2: Bias, RMSE and size of the GMM estimator for distributional
parameters of $\beta$ with $S=6$
DGP | Baseline | Categorical $x$ | Categorical $u$
---|---|---|---
Sample size $n$ | Bias | RMSE | Size | Bias | RMSE | Size | Bias | RMSE | Size
high variance: $\mathrm{var}\left(\beta_{i}\right)=0.25$
$\pi=0.5$ | 100 | 0.0337 | 0.1472 | 0.0456 | 0.0293 | 0.1645 | 0.0695 | 0.0227 | 0.1498 | 0.0469
1,000 | 0.0021 | 0.1405 | 0.2545 | 0.0015 | 0.1469 | 0.2543 | -0.0265 | 0.1635 | 0.2551
2,000 | 0.0008 | 0.1071 | 0.2614 | 0.0006 | 0.1185 | 0.2789 | -0.0201 | 0.1281 | 0.2732
5,000 | -0.0020 | 0.0661 | 0.2261 | -0.0016 | 0.0765 | 0.2518 | -0.0142 | 0.0836 | 0.2510
10,000 | -0.0005 | 0.0444 | 0.1844 | -0.0011 | 0.0505 | 0.2155 | -0.0093 | 0.0587 | 0.2323
100,000 | 0.0000 | 0.0097 | 0.0732 | 0.0000 | 0.0118 | 0.0912 | -0.0020 | 0.0178 | 0.1162
$\beta_{L}=1$ | 100 | 0.2226 | 0.4373 | 0.3341 | 0.2151 | 0.4658 | 0.3237 | 0.1879 | 0.4841 | 0.2896
1,000 | 0.0721 | 0.2081 | 0.4485 | 0.0780 | 0.2197 | 0.4318 | 0.0531 | 0.2283 | 0.3576
2,000 | 0.0443 | 0.1464 | 0.4056 | 0.0455 | 0.1609 | 0.4157 | 0.0342 | 0.1536 | 0.3271
5,000 | 0.0175 | 0.0806 | 0.3035 | 0.0203 | 0.0923 | 0.3341 | 0.0150 | 0.0933 | 0.2770
10,000 | 0.0092 | 0.0510 | 0.2350 | 0.0098 | 0.0594 | 0.2723 | 0.0081 | 0.0629 | 0.2403
100,000 | 0.0010 | 0.0114 | 0.0850 | 0.0013 | 0.0136 | 0.0982 | 0.0002 | 0.0186 | 0.1116
$\beta_{H}=2$ | 100 | -0.2495 | 0.5629 | 0.2563 | -0.2580 | 0.5681 | 0.2608 | -0.2589 | 0.5782 | 0.2248
1,000 | -0.0618 | 0.2530 | 0.4938 | -0.0686 | 0.2733 | 0.4867 | -0.0962 | 0.2814 | 0.4874
2,000 | -0.0334 | 0.1729 | 0.4454 | -0.0365 | 0.1951 | 0.4461 | -0.0625 | 0.2017 | 0.4643
5,000 | -0.0189 | 0.1010 | 0.3457 | -0.0203 | 0.1178 | 0.3638 | -0.0383 | 0.1223 | 0.3946
10,000 | -0.0080 | 0.0634 | 0.2670 | -0.0109 | 0.0732 | 0.3011 | -0.0246 | 0.0830 | 0.3347
100,000 | -0.0013 | 0.0114 | 0.0842 | -0.0012 | 0.0141 | 0.1070 | -0.0043 | 0.0220 | 0.1396
low variance: $\mathrm{var}\left(\beta_{i}\right)=0.15$
$\pi=0.3$ | 100 | 0.2374 | 0.2757 | 0.0591 | 0.2352 | 0.2816 | 0.0829 | 0.2330 | 0.2771 | 0.0801
1,000 | 0.1071 | 0.2107 | 0.2608 | 0.1114 | 0.2244 | 0.2775 | 0.0764 | 0.2158 | 0.2772
2,000 | 0.0702 | 0.1661 | 0.2994 | 0.0786 | 0.1815 | 0.3258 | 0.0242 | 0.1806 | 0.3291
5,000 | 0.0452 | 0.1101 | 0.3217 | 0.0519 | 0.1260 | 0.3466 | 0.0092 | 0.1263 | 0.3329
10,000 | 0.0300 | 0.0816 | 0.3060 | 0.0390 | 0.0933 | 0.3389 | 0.0108 | 0.0954 | 0.3161
100,000 | 0.0018 | 0.0164 | 0.1128 | 0.0041 | 0.0234 | 0.1482 | 0.0055 | 0.0298 | 0.1688
$\beta_{L}=0.5$ | 100 | 0.4146 | 0.5479 | 0.3137 | 0.4191 | 0.5636 | 0.2965 | 0.3844 | 0.5678 | 0.2532
1,000 | 0.2445 | 0.3459 | 0.4601 | 0.2436 | 0.3579 | 0.4561 | 0.2080 | 0.3872 | 0.3187
2,000 | 0.1663 | 0.2539 | 0.4809 | 0.1684 | 0.2620 | 0.4797 | 0.1108 | 0.2830 | 0.3203
5,000 | 0.0977 | 0.1648 | 0.4800 | 0.1051 | 0.1788 | 0.4938 | 0.0590 | 0.1731 | 0.3606
10,000 | 0.0613 | 0.1182 | 0.4230 | 0.0730 | 0.1315 | 0.4717 | 0.0417 | 0.1251 | 0.3667
100,000 | 0.0050 | 0.0242 | 0.1420 | 0.0086 | 0.0333 | 0.1808 | 0.0101 | 0.0386 | 0.1906
$\beta_{H}=1.345$ | 100 | -0.0817 | 0.3703 | 0.1601 | -0.0883 | 0.3842 | 0.1687 | -0.0806 | 0.4136 | 0.1614
1,000 | -0.0086 | 0.1726 | 0.3174 | -0.0144 | 0.1907 | 0.3295 | -0.0560 | 0.2029 | 0.3239
2,000 | 0.0022 | 0.1194 | 0.3267 | 0.0029 | 0.1368 | 0.3401 | -0.0395 | 0.1582 | 0.3736
5,000 | 0.0093 | 0.0722 | 0.2899 | 0.0099 | 0.0876 | 0.3254 | -0.0189 | 0.0998 | 0.3570
10,000 | 0.0092 | 0.0535 | 0.2642 | 0.0117 | 0.0601 | 0.2889 | -0.0076 | 0.0733 | 0.3141
100,000 | -0.0002 | 0.0116 | 0.0972 | 0.0012 | 0.0157 | 0.1326 | 0.0019 | 0.0220 | 0.1454
Notes: The data generating process is (5.1). high variance and low variance
parametrization are described in (5.2). “Baseline”, “Categorical $x$” and
“Categorical $u$” refer to DGP 1 to 3 as in Section 5.1. Generically, bias,
RMSE and size are calculated by
$R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)$,
$\sqrt{R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)^{2}}$,
and
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{0}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
respectively, for true parameter $\theta_{0}$, its estimate
$\hat{\theta}^{(r)}$, the estimated standard error of $\hat{\theta}^{(r)}$,
$\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical value
$\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.
### S.3.2 GMM Estimation of Moments of $\beta_{i}$
With the data generating processes described in Section 5.1, we report the
bias, RMSE and size of the GMM estimator for moments of $\beta_{i}$ in Table
S.3. The GMM estimator for moments of $\beta_{i}$ achieve better small sample
performance as compared to those for the distributional parameters
$\pi,\beta_{L}$ and $\beta_{H}$.
Table S.3: Bias, RMSE and size of the GMM estimator for moments of $\beta$
DGP | Baseline | Categorical $x$ | Categorical $u$
---|---|---|---
Sample size $n$ | Bias | RMSE | Size | Bias | RMSE | Size | Bias | RMSE | Size
high variance: $\mathrm{var}\left(\beta_{i}\right)=0.25$
$\mathrm{E}\left(\beta_{i}\right)=1.5$ | 100 | -0.0080 | 0.2262 | 0.1922 | -0.0117 | 0.2297 | 0.1940 | -0.0030 | 0.2418 | 0.1800
1,000 | -0.0029 | 0.0663 | 0.0936 | -0.0015 | 0.0673 | 0.0848 | -0.0037 | 0.0725 | 0.0804
2,000 | -0.0012 | 0.0431 | 0.0688 | -0.0015 | 0.0463 | 0.0700 | -0.0021 | 0.0494 | 0.0656
5,000 | -0.0003 | 0.0263 | 0.0566 | -0.0009 | 0.0276 | 0.0588 | -0.0013 | 0.0303 | 0.0622
10,000 | 0.0004 | 0.0183 | 0.0530 | -0.0001 | 0.0186 | 0.0498 | -0.0003 | 0.0206 | 0.0492
100,000 | 0.0000 | 0.0056 | 0.0434 | 0.0000 | 0.0058 | 0.0472 | 0.0000 | 0.0066 | 0.0514
$\mathrm{E}\left(\beta_{i}^{2}\right)=2.5$ | 100 | -0.0627 | 0.9082 | 0.3464 | -0.0826 | 0.8821 | 0.3166 | -0.0629 | 0.9459 | 0.3122
1,000 | -0.0300 | 0.2909 | 0.1518 | -0.0275 | 0.2837 | 0.1382 | -0.0362 | 0.3112 | 0.1512
2,000 | -0.0160 | 0.1751 | 0.0976 | -0.0188 | 0.1868 | 0.1074 | -0.0255 | 0.1900 | 0.1048
5,000 | -0.0067 | 0.0916 | 0.0658 | -0.0090 | 0.0993 | 0.0710 | -0.0124 | 0.1091 | 0.0754
10,000 | -0.0015 | 0.0580 | 0.0506 | -0.0036 | 0.0609 | 0.0530 | -0.0061 | 0.0704 | 0.0566
100,000 | -0.0005 | 0.0179 | 0.0462 | -0.0005 | 0.0185 | 0.0498 | -0.0011 | 0.0219 | 0.0542
$\mathrm{E}\left(\beta_{i}^{3}\right)=4.5$ | 100 | -0.2511 | 2.3755 | 0.3698 | -0.2990 | 2.3416 | 0.3424 | -0.2940 | 2.6179 | 0.3522
1,000 | -0.1155 | 0.7641 | 0.1734 | -0.1092 | 0.7613 | 0.1606 | -0.1478 | 0.8856 | 0.1904
2,000 | -0.0667 | 0.4683 | 0.1166 | -0.0745 | 0.5058 | 0.1234 | -0.1066 | 0.5485 | 0.1378
5,000 | -0.0290 | 0.2475 | 0.0800 | -0.0365 | 0.2696 | 0.0788 | -0.0507 | 0.3178 | 0.0942
10,000 | -0.0099 | 0.1559 | 0.0516 | -0.0163 | 0.1699 | 0.0602 | -0.0282 | 0.2088 | 0.0660
100,000 | -0.0020 | 0.0488 | 0.0462 | -0.0023 | 0.0515 | 0.0526 | -0.0052 | 0.0653 | 0.0520
low variance: $\mathrm{var}\left(\beta_{i}\right)=0.15$
$\mathrm{E}\left(\beta_{i}\right)=1.0915$ | 100 | 0.0165 | 0.1943 | 0.1618 | 0.0089 | 0.1983 | 0.1514 | 0.0169 | 0.2112 | 0.1416
1,000 | 0.0045 | 0.0577 | 0.0800 | 0.0042 | 0.0584 | 0.0702 | 0.0033 | 0.0655 | 0.0734
2,000 | 0.0019 | 0.0384 | 0.0594 | 0.0016 | 0.0410 | 0.0698 | 0.0010 | 0.0452 | 0.0632
5,000 | 0.0008 | 0.0243 | 0.0562 | 0.0003 | 0.0250 | 0.0540 | -0.0003 | 0.0283 | 0.0574
10,000 | 0.0007 | 0.0171 | 0.0502 | 0.0001 | 0.0175 | 0.0476 | 0.0000 | 0.0194 | 0.0442
100,000 | 0.0000 | 0.0052 | 0.0430 | 0.0000 | 0.0054 | 0.0476 | 0.0000 | 0.0062 | 0.0472
$\mathrm{E}\left(\beta_{i}^{2}\right)=1.3413$ | 100 | -0.0121 | 0.5119 | 0.2440 | -0.0280 | 0.5095 | 0.2330 | -0.0236 | 0.5724 | 0.2340
1,000 | -0.0061 | 0.1528 | 0.1232 | -0.0084 | 0.1566 | 0.1126 | -0.0163 | 0.1776 | 0.1246
2,000 | -0.0072 | 0.0973 | 0.0836 | -0.0080 | 0.1053 | 0.0922 | -0.0143 | 0.1154 | 0.0964
5,000 | -0.0037 | 0.0565 | 0.0658 | -0.0044 | 0.0603 | 0.0698 | -0.0088 | 0.0699 | 0.0720
10,000 | -0.0018 | 0.0381 | 0.0582 | -0.0027 | 0.0401 | 0.0590 | -0.0054 | 0.0476 | 0.0618
100,000 | -0.0004 | 0.0119 | 0.0496 | -0.0005 | 0.0125 | 0.0538 | -0.0009 | 0.0152 | 0.0506
$\mathrm{E}\left(\beta_{i}^{3}\right)=1.7407$ | 100 | -0.0759 | 0.9761 | 0.2806 | -0.0995 | 1.0052 | 0.2672 | -0.1277 | 1.2814 | 0.2718
1,000 | -0.0364 | 0.2925 | 0.1486 | -0.0396 | 0.3112 | 0.1456 | -0.0687 | 0.3973 | 0.1720
2,000 | -0.0297 | 0.1927 | 0.1040 | -0.0310 | 0.2126 | 0.1178 | -0.0526 | 0.2650 | 0.1324
5,000 | -0.0148 | 0.1141 | 0.0798 | -0.0168 | 0.1252 | 0.0860 | -0.0301 | 0.1619 | 0.0964
10,000 | -0.0078 | 0.0771 | 0.0654 | -0.0097 | 0.0846 | 0.0722 | -0.0188 | 0.1126 | 0.0828
100,000 | -0.0013 | 0.0242 | 0.0478 | -0.0016 | 0.0262 | 0.0554 | -0.0031 | 0.0360 | 0.0566
Notes: The data generating process is (5.1). $S=4$ is used. high variance and
low variance parametrization are described in (5.2). “Baseline”, “Categorical
$x$” and “Categorical $u$” refer to DGP 1 to 3 as in Section 5.1. Generically,
bias, RMSE and size are calculated by
$R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)$,
$\sqrt{R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)^{2}}$,
and
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{0}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
respectively, for true parameter $\theta_{0}$, its estimate
$\hat{\theta}^{(r)}$, the estimated standard error of $\hat{\theta}^{(r)}$,
$\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical value
$\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.
### S.3.3 Three Estimators of $\mathrm{E}\left(\beta_{i}\right)$
Table S.4 compares the finite sample performance of three estimators of
$\mathrm{E}\left(\beta_{i}\right)$ with the data generating processes
described in Section 5.1.
* •
The OLS estimator $\hat{\mathbf{\phi}}$ studied in Section 3.1
* •
The GMM estimator of $\mathrm{E}\left(\beta_{i}\right)$ with moment conditions
given by (3.7).
* •
$\widehat{\mathrm{E}\left(\beta_{i}\right)}=\hat{\pi}\hat{\beta}_{L}+\left(1-\hat{\pi}\right)\hat{\beta}_{H}$,
where $\hat{\pi},\hat{\beta}_{L},\hat{\beta}_{H}$ are the GMM estimators of
$\pi,\beta_{L},$ and $\beta_{H}$.
According to Table S.4, three estimators perform comparably well in terms of
bias and RMSE, whereas the OLS estimator, along with the standard error from
Theorem 3, controls size well when $n$ is small.
Table S.4: Bias, RMSE and size of three estimators for
$\mathrm{E}\left(\beta_{i}\right)$
DGP | Baseline | Categorical $x$ | Categorical $u$
---|---|---|---
Sample size $n$ | Bias | RMSE | Size | Bias | RMSE | Size | Bias | RMSE | Size
high variance: $\mathrm{E}\left(\beta_{i}\right)=1.5$,
$\mathrm{var}\left(\beta_{i}\right)=0.25$
OLS | 100 | -0.0024 | 0.2035 | 0.0966 | -0.0037 | 0.2035 | 0.0858 | -0.0042 | 0.2268 | 0.0920
1,000 | -0.0017 | 0.0669 | 0.0568 | -0.0002 | 0.0657 | 0.0540 | -0.0019 | 0.0738 | 0.0540
2,000 | -0.0008 | 0.0463 | 0.0512 | -0.0015 | 0.0475 | 0.0534 | -0.0010 | 0.0523 | 0.0522
5,000 | -0.0004 | 0.0301 | 0.0540 | -0.0008 | 0.0300 | 0.0546 | -0.0007 | 0.0335 | 0.0560
10,000 | 0.0002 | 0.0214 | 0.0508 | 0.0000 | 0.0212 | 0.0510 | 0.0000 | 0.0229 | 0.0456
100,000 | -0.0001 | 0.0066 | 0.0472 | 0.0000 | 0.0066 | 0.0460 | 0.0000 | 0.0075 | 0.0506
GMM | 100 | -0.0080 | 0.2262 | 0.1922 | -0.0117 | 0.2297 | 0.1940 | -0.0030 | 0.2418 | 0.1800
1,000 | -0.0029 | 0.0663 | 0.0936 | -0.0015 | 0.0673 | 0.0848 | -0.0037 | 0.0725 | 0.0804
2,000 | -0.0012 | 0.0431 | 0.0688 | -0.0015 | 0.0463 | 0.0700 | -0.0021 | 0.0494 | 0.0656
5,000 | -0.0003 | 0.0263 | 0.0566 | -0.0009 | 0.0276 | 0.0588 | -0.0013 | 0.0303 | 0.0622
10,000 | 0.0004 | 0.0183 | 0.0530 | -0.0001 | 0.0186 | 0.0498 | -0.0003 | 0.0206 | 0.0492
100,000 | 0.0000 | 0.0056 | 0.0434 | 0.0000 | 0.0058 | 0.0472 | 0.0000 | 0.0066 | 0.0514
$\hat{\pi}\hat{\beta}_{L}+(1-\hat{\pi})\hat{\beta}_{H}$ | 100 | -0.0087 | 0.2922 | 0.1961 | -0.1232 | 0.2347 | 0.1809 | -0.0037 | 0.2947 | 0.1894
1,000 | -0.0012 | 0.0648 | 0.0709 | -0.0237 | 0.0783 | 0.0665 | -0.0023 | 0.0713 | 0.0652
2,000 | -0.0004 | 0.0410 | 0.0556 | -0.0140 | 0.0537 | 0.0597 | -0.0015 | 0.0479 | 0.0558
5,000 | 0.0000 | 0.0259 | 0.0536 | -0.0063 | 0.0296 | 0.0546 | -0.0011 | 0.0299 | 0.0590
10,000 | 0.0004 | 0.0183 | 0.0526 | -0.0035 | 0.0205 | 0.0496 | -0.0003 | 0.0205 | 0.0488
100,000 | 0.0000 | 0.0056 | 0.0436 | -0.0006 | 0.0062 | 0.0472 | 0.0000 | 0.0066 | 0.0514
low variance: $\mathrm{E}\left(\beta_{i}\right)=1.0915$,
$\mathrm{var}\left(\beta_{i}\right)=0.15$
OLS | 100 | -0.0006 | 0.1829 | 0.0810 | -0.0023 | 0.1855 | 0.0766 | -0.0025 | 0.2094 | 0.0828
1,000 | -0.0005 | 0.0597 | 0.0610 | 0.0005 | 0.0590 | 0.0478 | -0.0006 | 0.0670 | 0.0542
2,000 | -0.0002 | 0.0408 | 0.0516 | -0.0007 | 0.0427 | 0.0606 | -0.0004 | 0.0475 | 0.0544
5,000 | -0.0002 | 0.0264 | 0.0530 | -0.0006 | 0.0266 | 0.0480 | -0.0005 | 0.0302 | 0.0538
10,000 | 0.0000 | 0.0189 | 0.0546 | -0.0002 | 0.0188 | 0.0486 | -0.0002 | 0.0208 | 0.0482
100,000 | -0.0001 | 0.0059 | 0.0474 | 0.0000 | 0.0059 | 0.0494 | 0.0000 | 0.0068 | 0.0508
GMM | 100 | -0.0121 | 0.5119 | 0.2440 | -0.0280 | 0.5095 | 0.2330 | -0.0236 | 0.5724 | 0.2340
1,000 | -0.0061 | 0.1528 | 0.1232 | -0.0084 | 0.1566 | 0.1126 | -0.0163 | 0.1776 | 0.1246
2,000 | -0.0072 | 0.0973 | 0.0836 | -0.0080 | 0.1053 | 0.0922 | -0.0143 | 0.1154 | 0.0964
5,000 | -0.0037 | 0.0565 | 0.0658 | -0.0044 | 0.0603 | 0.0698 | -0.0088 | 0.0699 | 0.0720
10,000 | -0.0018 | 0.0381 | 0.0582 | -0.0027 | 0.0401 | 0.0590 | -0.0054 | 0.0476 | 0.0618
100,000 | -0.0004 | 0.0119 | 0.0496 | -0.0005 | 0.0125 | 0.0538 | -0.0009 | 0.0152 | 0.0506
$\hat{\pi}\hat{\beta}_{L}+(1-\hat{\pi})\hat{\beta}_{H}$ | 100 | 0.0166 | 0.2392 | 0.1496 | 0.0063 | 0.2342 | 0.1412 | 0.0182 | 0.2432 | 0.1586
1,000 | 0.0078 | 0.0621 | 0.0827 | 0.0068 | 0.0615 | 0.0677 | 0.0064 | 0.0674 | 0.0693
2,000 | 0.0024 | 0.0388 | 0.0559 | 0.0021 | 0.0414 | 0.0672 | 0.0019 | 0.0454 | 0.0627
5,000 | 0.0009 | 0.0241 | 0.0554 | 0.0003 | 0.0247 | 0.0524 | 0.0001 | 0.0282 | 0.0548
10,000 | 0.0007 | 0.0170 | 0.0502 | 0.0002 | 0.0174 | 0.0478 | 0.0003 | 0.0193 | 0.0438
100,000 | 0.0000 | 0.0052 | 0.0430 | 0.0000 | 0.0054 | 0.0480 | 0.0004 | 0.0063 | 0.0494
Notes: The data generating process is (5.1). high variance and low variance
parametrization are described in (5.2). “Baseline”, “Categorical $x$” and
“Categorical $u$” refer to DGP 1 to 3 as in Section 5.1. Generically, bias,
RMSE and size are calculated by
$R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)$,
$\sqrt{R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)^{2}}$,
and
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{0}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
respectively, for true parameter $\theta_{0}$, its estimate
$\hat{\theta}^{(r)}$, the estimated standard error of $\hat{\theta}^{(r)}$,
$\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical value
$\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.
### S.3.4 Experiments with higher $\mathrm{var}\left(\beta_{i}\right)$
Following the data generating processes in Section 5.1, we increase the
variance of $\beta_{i}$ by considering the following two parametrizations:
$\left(\pi,\beta_{L},\beta_{H},\mathrm{E}\left(\beta_{i}\right),\mathop{\mathrm{v}ar}\left(\beta_{i}\right)\right)=\begin{cases}(0.3,0.5,6,4.35,6.3525),\\\
(0.3,0.5,10,7.15,18.9525).\end{cases}$ (S.3.1)
Table S.5 presents the results, which show that using larger values of
$\mathrm{var}\left(\beta_{i}\right)$ improves the small sample performance of
the GMM estimators.
Table S.5: Bias, RMSE and size of the GMM estimator for distributional
parameters of $\beta$
DGP | Baseline | Categorical $x$ | Categorical $u$
---|---|---|---
Sample size $n$ | Bias | RMSE | Size | Bias | RMSE | Size | Bias | RMSE | Size
$\mathrm{var}\left(\beta_{i}\right)=6.35$
$\pi=0.3$ | 100 | 0.0755 | 0.3014 | 0.1885 | 0.0628 | 0.2829 | 0.1601 | 0.0760 | 0.2967 | 0.1795
1,000 | -0.0113 | 0.1058 | 0.1485 | -0.0002 | 0.0882 | 0.1406 | -0.0092 | 0.1043 | 0.1509
2,000 | -0.0103 | 0.0646 | 0.1025 | -0.0016 | 0.0495 | 0.1072 | -0.0077 | 0.0598 | 0.1104
5,000 | -0.0026 | 0.0276 | 0.0718 | -0.0009 | 0.0197 | 0.0726 | -0.0021 | 0.0245 | 0.0742
10,000 | -0.0008 | 0.0095 | 0.0576 | -0.0005 | 0.0093 | 0.0608 | -0.0010 | 0.0099 | 0.0588
100,000 | -0.0002 | 0.0027 | 0.0490 | -0.0001 | 0.0026 | 0.0518 | -0.0002 | 0.0028 | 0.0504
$\beta_{L}=0.5$ | 100 | 2.7277 | 3.5109 | 0.2385 | 2.3640 | 3.2861 | 0.2207 | 2.6810 | 3.4783 | 0.2292
1,000 | 0.2951 | 1.1688 | 0.2743 | 0.1539 | 0.9017 | 0.2521 | 0.2473 | 1.1016 | 0.2725
2,000 | 0.0933 | 0.6394 | 0.1916 | 0.0460 | 0.5158 | 0.1988 | 0.0698 | 0.5904 | 0.1951
5,000 | 0.0159 | 0.2570 | 0.1236 | -0.0005 | 0.1786 | 0.1306 | 0.0066 | 0.2080 | 0.1225
10,000 | 0.0009 | 0.0607 | 0.0884 | -0.0005 | 0.0504 | 0.0998 | -0.0014 | 0.0585 | 0.0830
100,000 | 0.0000 | 0.0130 | 0.0572 | 0.0005 | 0.0135 | 0.0630 | -0.0003 | 0.0148 | 0.0622
$\beta_{H}=6$ | 100 | 0.1286 | 1.1700 | 0.0978 | 0.0482 | 1.1467 | 0.1057 | 0.1395 | 1.3662 | 0.0970
1,000 | 0.0031 | 0.2840 | 0.1320 | 0.0062 | 0.2695 | 0.1200 | 0.0043 | 0.3197 | 0.1382
2,000 | -0.0108 | 0.1392 | 0.0982 | 0.0007 | 0.1552 | 0.1094 | -0.0108 | 0.1519 | 0.1088
5,000 | -0.0041 | 0.0621 | 0.0746 | -0.0024 | 0.0608 | 0.0736 | -0.0054 | 0.0652 | 0.0794
10,000 | -0.0018 | 0.0340 | 0.0550 | -0.0012 | 0.0347 | 0.0678 | -0.0034 | 0.0386 | 0.0642
100,000 | -0.0003 | 0.0109 | 0.0530 | 0.0001 | 0.0107 | 0.0518 | -0.0006 | 0.0125 | 0.0588
$\mathrm{var}\left(\beta_{i}\right)=18.95$
$\pi=0.3$ | 100 | 0.0575 | 0.2896 | 0.1761 | 0.0530 | 0.2762 | 0.1524 | 0.0554 | 0.2889 | 0.1646
1,000 | -0.0136 | 0.1070 | 0.1217 | -0.0025 | 0.0892 | 0.1306 | -0.0110 | 0.1024 | 0.1369
2,000 | -0.0101 | 0.0650 | 0.0850 | -0.0032 | 0.0488 | 0.0969 | -0.0077 | 0.0610 | 0.0957
5,000 | -0.0027 | 0.0291 | 0.0668 | -0.0010 | 0.0217 | 0.0625 | -0.0023 | 0.0247 | 0.0713
10,000 | -0.0009 | 0.0122 | 0.0549 | -0.0005 | 0.0097 | 0.0600 | -0.0009 | 0.0100 | 0.0570
100,000 | -0.0002 | 0.0025 | 0.0480 | -0.0001 | 0.0024 | 0.0514 | -0.0002 | 0.0025 | 0.0484
$\beta_{L}=0.5$ | 100 | 4.5691 | 5.9597 | 0.2001 | 4.0139 | 5.6053 | 0.1750 | 4.4575 | 5.8827 | 0.1991
1,000 | 0.5104 | 1.8908 | 0.2327 | 0.2907 | 1.5133 | 0.2146 | 0.4062 | 1.7517 | 0.2522
2,000 | 0.1678 | 1.0260 | 0.1683 | 0.0929 | 0.8581 | 0.1714 | 0.1178 | 0.9144 | 0.1736
5,000 | 0.0292 | 0.3901 | 0.1069 | 0.0073 | 0.3040 | 0.1095 | 0.0186 | 0.3400 | 0.1036
10,000 | 0.0058 | 0.1638 | 0.0719 | 0.0014 | 0.0899 | 0.0834 | 0.0000 | 0.0919 | 0.0740
100,000 | 0.0000 | 0.0171 | 0.0572 | 0.0006 | 0.0171 | 0.0614 | -0.0004 | 0.0185 | 0.0576
$\beta_{H}=10$ | 100 | 0.0520 | 1.5471 | 0.0926 | -0.0530 | 1.4858 | 0.0944 | 0.0460 | 1.6879 | 0.0888
1,000 | -0.0078 | 0.4047 | 0.1185 | -0.0108 | 0.4158 | 0.1020 | -0.0100 | 0.4178 | 0.1195
2,000 | -0.0093 | 0.2058 | 0.0936 | -0.0005 | 0.2067 | 0.0975 | -0.0129 | 0.2546 | 0.0933
5,000 | -0.0037 | 0.0944 | 0.0727 | -0.0034 | 0.0922 | 0.0709 | -0.0052 | 0.0871 | 0.0709
10,000 | -0.0023 | 0.0512 | 0.0555 | -0.0010 | 0.0504 | 0.0684 | -0.0034 | 0.0529 | 0.0580
100,000 | -0.0005 | 0.0160 | 0.0522 | 0.0002 | 0.0154 | 0.0526 | -0.0007 | 0.0171 | 0.0560
Notes: The data generating process is (5.1). Parametrization are described in
(S.3.1). $S=4$ is used. “Baseline”, “Categorical $x$” and “Categorical $u$”
refer to DGP 1 to 3 as in Section 5.1. Generically, bias, RMSE and size are
calculated by
$R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)$,
$\sqrt{R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)^{2}}$,
and
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{0}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
respectively, for true parameter $\theta_{0}$, its estimate
$\hat{\theta}^{(r)}$, the estimated standard error of $\hat{\theta}^{(r)}$,
$\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical value
$\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.
### S.3.5 Experiments with three categories ($K=3)$
#### S.3.5.1 Data generating processes
We generate $y_{i}$ as
$y_{i}=\alpha+x_{i}\beta_{i}+z_{i1}\gamma_{1}+z_{i2}\gamma_{2}+u_{i},\text{
for }i=1,2,...,n,$ (S.3.2)
with $\beta_{i}$ distributed as in (2.2) with $K=3$,
$\beta_{i}=\begin{cases}\beta_{L},&\text{ w.p. }\pi_{L}\\\ \beta_{M},&\text{
w.p. }\pi_{M}\\\ \beta_{L},&\text{ w.p. }1-\pi_{L}-\pi_{M},\end{cases}$
where w.p. denotes “with probability”. The parameters take values
$\left(\pi_{L},\pi_{M},\beta_{L},\beta_{M},\beta_{H}\right)=\left(0.3,0.3,1,2,3\right)$.
Corresponding, the moments of $\beta_{i}$ are
$\left(\mathrm{E}\left(\beta_{i}\right),\mathrm{E}\left(\beta_{i}^{2}\right),\mathrm{E}\left(\beta_{i}^{3}\right),\mathrm{E}\left(\beta_{i}^{4}\right),\mathrm{E}\left(\beta_{i}^{5}\right)\right)=\left(2.1,5.1,13.5,37.5,107.1\right)$.
The remaining parameters are set as $\alpha=0.25$, and
$\mathbf{\gamma}=\left(1,1\right)^{\prime}$.
We first generate $\tilde{x}_{i}\sim\text{IID}\chi^{2}(2)$, and then set
$x_{i}=(\tilde{x}_{i}-2)/2$ so that $x_{i}$ has $0$ mean and unit variance.
The additional regressors, $z_{ij}$, for $j=1,2$ with homogeneous slopes are
generated as
$z_{i1}=x_{i}+v_{i1}\text{ and }z_{i2}=z_{i1}+v_{i2},$
with $v_{ij}\sim\text{IID }N\left(0,1\right)$, for $j=1,2$. The error term,
$u_{i}$, is generated as $u_{i}=\sigma_{i}\varepsilon_{i}$, where
$\sigma_{i}^{2}$ are generated as $0.5(1+\text{IID}\chi^{2}(1))$, and
$\varepsilon_{i}\sim\text{IID}N(0,1)$.
#### S.3.5.2 Results
Table S.6 reports the bias, RMSE and size of the GMM estimator for
distributional parameters and moments of $\beta_{i}$. The results are based on
$5,000$ replications and $S=6$. The results show that even larger sample sizes
are needed for the GMM estimators (both the moments of $\beta_{i}$ and its
distributional parameters) to achieve reasonable finite sample performance,
since higher order of moments are involved.
In additional to the results of jointly estimating distributional parameters
and moments of $\beta_{i}$ by GMM, Table S.7 reports the results of GMM
estimation of moments of $\beta_{i}$ up to order 3 using the moment conditions
as in the $K=2$ case where $S=4$ in the left panel, and the results of OLS
estimation of $\mathbf{\phi}$ in the right panel. These results show that we
are still able to obtain accurate estimation of lower order moments of
$\beta_{i}$ when the fourth and fifth moments of $\beta_{i}$ are not used,
confirming the lower information content of the higher order moments for
estimation of the lower order moments of $\beta_{i}$.
Table S.6: Bias, RMSE and size of the GMM estimator for distributional
parameters and moments of $\beta$ with $K=3$
| | Distribution of $\beta_{i}$ | | Moments of $\beta_{i}$
---|---|---|---|---
Sample size $n$ | | Bias | RMSE | Size | | Bias | RMSE | Size
100 | $\pi_{L}=0.3$ | -0.0405 | 0.1910 | 0.1319 | $\mathrm{E}(\beta_{i})=2.1$ | 0.1484 | 0.7471 | 0.6451
1,000 | -0.0417 | 0.1633 | 0.1915 | -0.0711 | 0.5415 | 0.6128
2,000 | -0.0383 | 0.1474 | 0.2354 | -0.1112 | 0.4408 | 0.5264
5,000 | -0.0299 | 0.1186 | 0.3098 | -0.0904 | 0.3712 | 0.4034
10,000 | -0.0209 | 0.0949 | 0.3371 | -0.0523 | 0.2740 | 0.2910
100,000 | -0.0074 | 0.0314 | 0.2295 | -0.0026 | 0.0400 | 0.0678
200,000 | -0.0050 | 0.0208 | 0.1917 | -0.0004 | 0.0202 | 0.0568
100 | $\beta_{M}=0.3$ | 0.2166 | 0.2995 | 0.0492 | $\mathrm{E}(\beta_{i}^{2})=5.1$ | 0.2841 | 2.8452 | 0.7223
1,000 | 0.1404 | 0.2378 | 0.1364 | -0.6374 | 1.9507 | 0.6456
2,000 | 0.1035 | 0.2117 | 0.1901 | -0.7163 | 1.7408 | 0.5472
5,000 | 0.0615 | 0.1645 | 0.2381 | -0.5478 | 1.4628 | 0.4472
10,000 | 0.0364 | 0.1292 | 0.2477 | -0.3391 | 1.1394 | 0.3432
100,000 | 0.0013 | 0.0322 | 0.1305 | -0.0209 | 0.2300 | 0.0932
200,000 | 0.0006 | 0.0185 | 0.1033 | -0.0046 | 0.1128 | 0.0620
100 | $\beta_{L}=1$ | 0.6881 | 1.1994 | 0.1110 | $\mathrm{E}(\beta_{i}^{3})=13.5$ | 0.4897 | 10.0757 | 0.7189
1,000 | 0.2588 | 0.7438 | 0.1994 | -2.7735 | 7.0573 | 0.6718
2,000 | 0.1096 | 0.5372 | 0.2607 | -2.9100 | 6.3988 | 0.5894
5,000 | 0.0205 | 0.4184 | 0.3426 | -2.1889 | 5.4307 | 0.5078
10,000 | 0.0070 | 0.2733 | 0.3360 | -1.3454 | 4.3382 | 0.4042
100,000 | -0.0064 | 0.0556 | 0.2213 | -0.0942 | 1.0263 | 0.1132
200,000 | -0.0047 | 0.0320 | 0.1775 | -0.0236 | 0.5035 | 0.0738
100 | $\beta_{M}=2$ | 0.1249 | 0.7256 | 0.0642 | $\mathrm{E}(\beta_{i}^{4})=37.5$ | 0.9092 | 35.1538 | 0.7235
1,000 | -0.1190 | 0.6298 | 0.1531 | -10.1071 | 24.1521 | 0.6944
2,000 | -0.1935 | 0.5762 | 0.2303 | -10.7108 | 21.5751 | 0.6268
5,000 | -0.1662 | 0.4777 | 0.3670 | -8.2675 | 18.7735 | 0.5464
10,000 | -0.1261 | 0.3703 | 0.4414 | -5.5310 | 15.4382 | 0.4406
100,000 | -0.0326 | 0.1175 | 0.2681 | -0.4433 | 3.5927 | 0.1240
200,000 | -0.0193 | 0.0682 | 0.2203 | -0.1114 | 1.6644 | 0.0810
100 | $\beta_{H}=3$ | 0.8514 | 3.1645 | 0.1064 | $\mathrm{E}(\beta_{i}^{5})=107.1$ | 2.4059 | 121.1286 | 0.6989
1,000 | 1.6632 | 4.5208 | 0.3124 | -34.0298 | 77.5508 | 0.7012
2,000 | 1.7929 | 4.6701 | 0.4000 | -35.4018 | 69.5876 | 0.6424
5,000 | 1.3425 | 4.0152 | 0.4539 | -27.3828 | 60.4373 | 0.5638
10,000 | 0.9637 | 3.3831 | 0.4333 | -18.1022 | 50.3990 | 0.4590
100,000 | 0.0474 | 0.8321 | 0.2046 | -1.5330 | 11.7796 | 0.1314
200,000 | 0.0033 | 0.3237 | 0.1573 | -0.4226 | 5.9529 | 0.0812
Notes: The data generating process is (S.3.2). Generically, bias, RMSE and
size are calculated by
$R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)$,
$\sqrt{R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)^{2}}$,
and
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{0}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
respectively, for true parameter $\theta_{0}$, its estimate
$\hat{\theta}^{(r)}$, the estimated standard error of $\hat{\theta}^{(r)}$,
$\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical value
$\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.
Table S.7: Bias, RMSE and size of estimation of $\phi$ and moments of
$\beta_{i}$ (using $S=4$) with $K=3$
| | Moments of $\beta_{i}$ ($S=4$) | | OLS Estimate $hat{\phi}i$
---|---|---|---|---
$n$ | | Bias | RMSE | Size | | Bias | RMSE | Size
100 | $\mathrm{E}(\beta_{i})=2.1$ | 0.0025 | 0.2867 | 0.2088 | $\mathrm{E}(\beta_{i})=2.1$ | -0.0031 | 0.2768 | 0.1042
1,000 | -0.0006 | 0.0821 | 0.1008 | -0.0008 | 0.0939 | 0.0588
2,000 | 0.0004 | 0.0537 | 0.0734 | 0.0000 | 0.0653 | 0.0550
5,000 | 0.0004 | 0.0323 | 0.0610 | -0.0008 | 0.0422 | 0.0506
10,000 | 0.0007 | 0.0224 | 0.0572 | -0.0001 | 0.0299 | 0.0510
100,000 | 0.0000 | 0.0069 | 0.0454 | -0.0001 | 0.0093 | 0.0462
200,000 | 0.0000 | 0.0050 | 0.0550 | 0.0000 | 0.0067 | 0.0498
100 | $\mathrm{E}(\beta_{i}^{2})=5.1$ | -0.1195 | 1.8290 | 0.3948 | $\gamma_{1}=1$ | -0.0020 | 0.1817 | 0.0604
1,000 | -0.0455 | 0.5965 | 0.1602 | 0.0000 | 0.0581 | 0.0474
2,000 | -0.0196 | 0.3454 | 0.0902 | 0.0001 | 0.0409 | 0.0474
5,000 | -0.0073 | 0.1630 | 0.0608 | -0.0001 | 0.0259 | 0.0494
10,000 | -0.0004 | 0.1028 | 0.0544 | -0.0004 | 0.0183 | 0.0518
100,000 | 0.0001 | 0.0311 | 0.0488 | -0.0001 | 0.0058 | 0.0490
200,000 | -0.0002 | 0.0217 | 0.0492 | -0.0001 | 0.0041 | 0.0490
100 | $\mathrm{E}(\beta_{i}^{3})=13.5$ | -0.7404 | 6.7772 | 0.4396 | $\gamma_{2}=1$ | 0.0011 | 0.1296 | 0.0672
1,000 | -0.3116 | 2.2732 | 0.1964 | 0.0000 | 0.0414 | 0.0570
2,000 | -0.1433 | 1.3285 | 0.1110 | 0.0000 | 0.0291 | 0.0478
5,000 | -0.0524 | 0.6468 | 0.0702 | -0.0001 | 0.0183 | 0.0506
10,000 | -0.0117 | 0.4052 | 0.0568 | 0.0002 | 0.0130 | 0.0526
100,000 | 0.0001 | 0.1236 | 0.0528 | 0.0001 | 0.0041 | 0.0494
200,000 | -0.0009 | 0.0850 | 0.0462 | 0.0000 | 0.0029 | 0.0542
Notes: The data generating process is (S.3.2). Generically, bias, RMSE and
size are calculated by
$R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)$,
$\sqrt{R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)^{2}}$,
and
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{0}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
respectively, for true parameter $\theta_{0}$, its estimate
$\hat{\theta}^{(r)}$, the estimated standard error of $\hat{\theta}^{(r)}$,
$\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical value
$\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.
### S.3.6 Experiments with idiosyncratic heterogeneity
In addition to the existing results, the following Monte Carlo experiment is
designed to examine the finite sample performance of the estimator under
different degrees of idiosyncratic heterogeneity. Following DGP 1 in Section
5.1, we generate $\tilde{x}_{i}\sim\text{IID}\chi^{2}(2)$, and then set
$x_{i}=(\tilde{x}_{i}-2)/2$. The additional regressors, $z_{ij}$, for $j=1,2$
with homogeneous slopes are generated as
$z_{i1}=x_{i}+v_{i1}\text{ and }z_{i2}=z_{i1}+v_{i2},$
with $v_{ij}\sim\text{IID }N\left(0,1\right)$, for $j=1,2$. The error term,
$u_{i}$, is generated as
$u_{i}=\begin{cases}\sigma_{i}\varepsilon_{i}+e_{i}&\text{if
}i=1,2,\cdots,\lfloor n^{\alpha}\rfloor\\\ \sigma_{i}\varepsilon_{i}&\text{if
}i=\lfloor n^{\alpha}\rfloor+1,\cdots,n\end{cases}$
where $\sigma_{i}^{2}$ are generated as $0.5(1+\text{IID}\chi^{2}(1))$,
$\varepsilon_{i}\sim\text{IID}N(0,1)$, and $e_{i}$ is the idiosyncratic
heterogeneity that is generated from the standard normal distribution and then
set to be fixed across Monte Carlo replications. Then in this case we have
$\left|n^{-1}\sum_{i=1}^{n}\mathrm{E}\left(u_{i}^{2}\right)-1\right|=\left|n^{-1}\sum_{i=1}^{\lfloor
n^{\alpha}\rfloor}e_{i}^{2}\right|\leq n^{-1}\sum_{i=1}^{\lfloor
n^{\alpha}\rfloor}\left|e_{i}^{2}\right|\leq\left(\max_{1\leq i\leq\lfloor
n^{\alpha}\rfloor}\left|e_{i}^{2}\right|\right)n^{\alpha-1}.$
Similar arguments can be made for $r=3$.
Following the same parametrization as in Section 5, we consider the degree of
heterogeneity $\alpha=0.25$, $0.4$, and $0.5$. The estimation results are
reported in Table S.8. The results are similar to that of the Baseline DGP as
reported in Table 3, which suggests that the GMM estimator is robust to
limited degrees of idiosyncratic heterogeneity.
Table S.8: Bias, RMSE and size of the GMM estimator for distributional
parameters of $\beta$
$\alpha$ | 0.25 | 0.40 | 0.50
---|---|---|---
Sample size $n$ | Bias | RMSE | Size | Bias | RMSE | Size | Bias | RMSE | Size
high variance: $\mathrm{var}\left(\beta_{i}\right)=0.25$
$\pi=0.5$ | 100 | 0.0292 | 0.2201 | 0.1957 | 0.0293 | 0.2177 | 0.1859 | 0.0297 | 0.2160 | 0.1609
1,000 | 0.0020 | 0.1273 | 0.1943 | 0.0039 | 0.1293 | 0.2047 | 0.0037 | 0.1356 | 0.2150
2,000 | 0.0014 | 0.0879 | 0.1585 | 0.0003 | 0.0812 | 0.1421 | 0.0020 | 0.0851 | 0.1455
5,000 | 0.0002 | 0.0440 | 0.0980 | 0.0010 | 0.0457 | 0.0982 | -0.0003 | 0.0445 | 0.0946
10,000 | -0.0007 | 0.0301 | 0.0764 | 0.0003 | 0.0304 | 0.0824 | -0.0001 | 0.0311 | 0.0910
100,000 | 0.0000 | 0.0098 | 0.0610 | 0.0000 | 0.0097 | 0.0536 | -0.0002 | 0.0096 | 0.0556
$\beta_{L}=1$ | 100 | 0.2027 | 0.5686 | 0.1807 | 0.1993 | 0.5706 | 0.1738 | 0.2007 | 0.5662 | 0.1712
1,000 | 0.0104 | 0.1711 | 0.2115 | 0.0136 | 0.1750 | 0.2156 | 0.0079 | 0.1827 | 0.2132
2,000 | 0.0094 | 0.1121 | 0.1741 | 0.0069 | 0.1025 | 0.1529 | 0.0087 | 0.1109 | 0.1593
5,000 | 0.0040 | 0.0543 | 0.1090 | 0.0052 | 0.0557 | 0.1136 | 0.0050 | 0.0546 | 0.1112
10,000 | 0.0023 | 0.0365 | 0.0856 | 0.0024 | 0.0365 | 0.0882 | 0.0025 | 0.0367 | 0.0922
100,000 | 0.0004 | 0.0116 | 0.0602 | 0.0005 | 0.0115 | 0.0604 | 0.0004 | 0.0115 | 0.0584
$\beta_{H}=2$ | 100 | -0.1947 | 0.5616 | 0.1307 | -0.1983 | 0.5545 | 0.1421 | -0.2094 | 0.5510 | 0.1358
1,000 | -0.0096 | 0.1720 | 0.1682 | -0.0078 | 0.1729 | 0.1710 | -0.0066 | 0.1802 | 0.1751
2,000 | -0.0060 | 0.1142 | 0.1445 | -0.0068 | 0.1066 | 0.1523 | -0.0070 | 0.1060 | 0.1405
5,000 | -0.0047 | 0.0530 | 0.1130 | -0.0037 | 0.0545 | 0.1110 | -0.0054 | 0.0559 | 0.1088
10,000 | -0.0031 | 0.0360 | 0.0922 | -0.0023 | 0.0370 | 0.0826 | -0.0024 | 0.0372 | 0.0896
100,000 | -0.0004 | 0.0116 | 0.0592 | -0.0003 | 0.0115 | 0.0546 | -0.0005 | 0.0114 | 0.0600
low variance: $\mathrm{var}\left(\beta_{i}\right)=0.15$
$\pi=0.3$ | 100 | 0.2132 | 0.2951 | 0.1851 | 0.2133 | 0.2912 | 0.1797 | 0.2132 | 0.2945 | 0.1716
1,000 | 0.0133 | 0.1591 | 0.1894 | 0.0125 | 0.1613 | 0.1872 | 0.0163 | 0.1637 | 0.1840
2,000 | -0.0051 | 0.1103 | 0.1619 | -0.0055 | 0.1048 | 0.1553 | -0.0027 | 0.1083 | 0.1559
5,000 | -0.0046 | 0.0599 | 0.1198 | -0.0029 | 0.0607 | 0.1070 | -0.0046 | 0.0620 | 0.1208
10,000 | -0.0038 | 0.0418 | 0.0900 | -0.0023 | 0.0418 | 0.0932 | -0.0022 | 0.0423 | 0.0930
100,000 | -0.0003 | 0.0132 | 0.0622 | -0.0003 | 0.0130 | 0.0576 | -0.0004 | 0.0127 | 0.0532
$\beta_{L}=0.5$ | 100 | 0.3935 | 0.6293 | 0.1959 | 0.3900 | 0.6353 | 0.1853 | 0.3917 | 0.6236 | 0.1811
1,000 | 0.0310 | 0.2598 | 0.1590 | 0.0357 | 0.2634 | 0.1589 | 0.0298 | 0.2653 | 0.1609
2,000 | 0.0025 | 0.1590 | 0.1539 | 0.0004 | 0.1478 | 0.1274 | 0.0025 | 0.1565 | 0.1459
5,000 | -0.0008 | 0.0849 | 0.1100 | 0.0018 | 0.0849 | 0.1122 | 0.0003 | 0.0854 | 0.1078
10,000 | -0.0001 | 0.0586 | 0.0922 | 0.0004 | 0.0586 | 0.0958 | 0.0012 | 0.0576 | 0.0918
100,000 | 0.0005 | 0.0183 | 0.0596 | 0.0002 | 0.0181 | 0.0582 | 0.0003 | 0.0177 | 0.0558
$\beta_{H}=1.345$ | 100 | -0.0463 | 0.4194 | 0.1128 | -0.0509 | 0.4224 | 0.1147 | -0.0489 | 0.4386 | 0.1239
1,000 | -0.0097 | 0.1428 | 0.1498 | -0.0106 | 0.1427 | 0.1523 | -0.0094 | 0.1486 | 0.1467
2,000 | -0.0107 | 0.0920 | 0.1443 | -0.0106 | 0.0917 | 0.1439 | -0.0093 | 0.0915 | 0.1389
5,000 | -0.0065 | 0.0492 | 0.1166 | -0.0056 | 0.0500 | 0.1092 | -0.0063 | 0.0532 | 0.1134
10,000 | -0.0045 | 0.0345 | 0.0910 | -0.0037 | 0.0344 | 0.0902 | -0.0035 | 0.0344 | 0.0900
100,000 | -0.0006 | 0.0108 | 0.0602 | -0.0004 | 0.0107 | 0.0572 | -0.0005 | 0.0105 | 0.0560
Notes: The data generating process is (S.3.2). high variance and low variance
parametrization are described in (5.2). $\alpha$ is the degree of
heterogeneity as in Remark 6. Generically, bias, RMSE and size are calculated
by $R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)$,
$\sqrt{R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)^{2}}$,
and
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{0}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
respectively, for true parameter $\theta_{0}$, its estimate
$\hat{\theta}^{(r)}$, the estimated standard error of $\hat{\theta}^{(r)}$,
$\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical value
$\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.
## Appendix S.4 Additional empirical results
In this section, we provide additional results for the empirical application.
In addition to the quadratic in experience in Section 6, we further consider
the following quartic in experience specification,
$\log\text{wage}_{i}=\alpha+\beta_{i}\text{edu}_{i}+\rho_{1}\text{exper}_{i}+\rho_{2}\text{exper}_{i}^{2}+\rho_{3}\text{exper}_{i}^{3}+\rho_{4}\text{exper}_{i}^{4}+\tilde{\mathbf{z}}_{i}^{\prime}\tilde{\mathbf{\gamma}}+u_{i},$
(S.4.1)
where
$\beta_{i}=\begin{cases}b_{L}&\text{w.p. }\pi,\\\ b_{H}&\text{w.p.
}1-\pi.\end{cases}$
Table S.9 and S.10 report the estimates of the distributional parameters of
$\beta_{i}$ and the estimates of $\mathbf{\gamma}$ with the specification
(S.4.1).
The estimates of parameter of interests with specification (S.4.1) are almost
the same as that with quadratic in experience specification (6.3), reported in
Table 5. The qualitative analysis and conclusion discussed in Section 6 remain
robust to adding third and fourth order powers of experi in the regressions.
Table S.9: Estimates of the distribution of the return to education with
specification (S.4.1) across two periods, 1973 - 75 and 2001 - 03, by years of
education and gender
| High School or Less | | Postsecondary Edu. | | All
---|---|---|---|---|---
| 1973 - 75 | 2001 - 03 | | 1973 - 75 | 2001 - 03 | | 1973 - 75 | 2001 - 03
| Both Male and Female
$\pi$ | 0.4841 | 0.5081 | | 0.4281 | 0.3576 | | 0.4689 | 0.3559
| (5274.3) | (0.0267) | | (0.0495) | (0.0089) | | (0.0534) | (0.0046)
$\beta_{L}$ | 0.0617 | 0.0392 | | 0.0627 | 0.0859 | | 0.0567 | 0.0658
| (5.9252) | (0.0013) | | (0.0035) | (0.0009) | | (0.0022) | (0.0004)
$\beta_{H}$ | 0.0628 | 0.0928 | | 0.1108 | 0.1397 | | 0.0938 | 0.1270
| (5.5919) | (0.0019) | | (0.0031) | (0.0007) | | (0.0023) | (0.0004)
$\beta_{H}/\beta_{L}$ | 1.0177 | 2.3645 | | 1.7675 | 1.6267 | | 1.6533 | 1.9299
| (7.1413) | (0.0400) | | (0.0629) | (0.0111) | | (0.0305) | (0.0076)
$\mathrm{E}\left(\beta_{i}\right)$ | 0.0623 | 0.0656 | | 0.0902 | 0.1205 | | 0.0764 | 0.1053
$\mathrm{var}\left(\beta_{i}\right)$ | 0.0005 | 0.0268 | | 0.0238 | 0.0258 | | 0.0185 | 0.0293
$n$ | 77,899 | 216,136 | | 33,733 | 295,683 | | 111,632 | 511,819
| Male
$\pi$ | 0.4835 | 0.4968 | | 0.4478 | 0.3007 | | 0.4856 | 0.3550
| n/a | (0.0394) | | (0.0676) | (0.0095) | | (0.0936) | (0.0052)
$\beta_{L}$ | 0.0648 | 0.0419 | | 0.0520 | 0.0733 | | 0.0553 | 0.0581
| n/a | (0.0019) | | (0.0047) | (0.0012) | | (0.0033) | (0.0005)
$\beta_{H}$ | 0.0651 | 0.0927 | | 0.0988 | 0.1321 | | 0.0875 | 0.1220
| n/a | (0.0026) | | (0.0041) | (0.0008) | | (0.0034) | (0.0005)
$\beta_{H}/\beta_{L}$ | 1.0048 | 2.2143 | | 1.9002 | 1.8015 | | 1.5816 | 2.1003
| n/a | (0.0495) | | (0.1124) | (0.0210) | | (0.0456) | (0.0124)
$\mathrm{E}\left(\beta_{i}\right)$ | 0.0649 | 0.0675 | | 0.0778 | 0.1144 | | 0.0719 | 0.0993
$\mathrm{var}\left(\beta_{i}\right)$ | 0.0002 | 0.0254 | | 0.0233 | 0.0269 | | 0.0161 | 0.0306
$n$ | 44,299 | 116,129 | | 20,851 | 144,138 | | 65,150 | 260,267
| Female
$\pi$ | 0.5000 | 0.5210 | | 0.4512 | 0.3849 | | 0.4733 | 0.3773
| (0.5611) | (0.0281) | | (0.0739) | (0.0167) | | (0.0870) | (0.0083)
$\beta_{L}$ | 0.0453 | 0.0352 | | 0.0804 | 0.0956 | | 0.0644 | 0.0762
| (0.0143) | (0.0016) | | (0.0050) | (0.0013) | | (0.0034) | (0.0006)
$\beta_{H}$ | 0.0724 | 0.0969 | | 0.1307 | 0.1449 | | 0.1032 | 0.1338
| (0.0169) | (0.0025) | | (0.0052) | (0.0011) | | (0.0040) | (0.0007)
$\beta_{H}/\beta_{L}$ | 1.5994 | 2.7540 | | 1.6252 | 1.5154 | | 1.6012 | 1.7564
| (0.1537) | (0.0666) | | (0.0551) | (0.0125) | | (0.0323) | (0.0084)
$\mathrm{E}\left(\beta_{i}\right)$ | 0.0588 | 0.0648 | | 0.1080 | 0.1260 | | 0.0848 | 0.1121
$\mathrm{var}\left(\beta_{i}\right)$ | 0.0136 | 0.0308 | | 0.0250 | 0.0240 | | 0.0193 | 0.0279
$n$ | 33,600 | 100,007 | | 12,882 | 151,545 | | 46,482 | 251,552
Notes: This table reports the estimates of the distribution of $\beta_{i}$
with the quartic in experience specification (S.4.1), using $S=4$ order
moments of $\text{edu}_{i}$. “Postsecondary Edu.” stands for the sub-sample
with years of education higher than 12 and “High School or Less” stands for
those with years of education less than or equal to 12.
$\mathrm{s.d.}\left(\beta_{i}\right)$ corresponds to the square root of
estimated $\mathrm{var}\left(\beta_{i}\right)$. $n$ is the sample size. “n/a”
is inserted when the estimates show homogeneity of $\beta_{i}$ and $\pi$ is
not identified and cannot be estimated.
Table S.10: Estimates of $\mathbf{\gamma}$ associated with control variables
$\mathbf{z}_{i}$ with specification (S.4.1) across two periods, 1973 - 75 and
2001 - 03, by years of education and gender, which complements Table S.9
| High School or Less | | Postsecondary Edu. | | All
---|---|---|---|---|---
| 1973 - 75 | 2001 - 03 | | 1973 - 75 | 2001 - 03 | | 1973 - 75 | 2001 - 03
| Both male and female
exper. | 0.0769 | 0.0526 | | 0.0817 | 0.0763 | | 0.0757 | 0.0603
| (0.0015) | (0.0009) | | (0.0029) | (0.0012) | | (0.0013) | (0.0007)
$\mathtt{exper.}^{2}$ | -0.0040 | -0.0020 | | -0.0045 | -0.0039 | | -0.0038 | -0.0024
| (0.0001) | (0.0001) | | (0.0003) | (0.0001) | | (0.0001) | (0.0001)
$\mathtt{exper.}^{3}$ ($\times 10^{5}$) | 9.2470 | 3.4329 | | 11.2100 | 8.9370 | | 8.3625 | 3.6521
| (0.4146) | (0.2882) | | (1.2538) | (0.4460) | | (0.3677) | (0.2412)
$\mathtt{exper.}^{4}$ ($\times 10^{5}$) | -0.0768 | -0.0236 | | -0.1074 | -0.0777 | | -0.0654 | -0.0169
| (0.0043) | (0.0031) | | (0.0158) | (0.0054) | | (0.0039) | (0.0027)
marriage | 0.0819 | 0.0700 | | 0.0728 | 0.0674 | | 0.0799 | 0.0718
| (0.0037) | (0.0020) | | (0.0060) | (0.0020) | | (0.0031) | (0.0014)
nonwhite | -0.1052 | -0.0808 | | -0.0486 | -0.0613 | | -0.0855 | -0.0719
| (0.0046) | (0.0024) | | (0.0088) | (0.0025) | | (0.0041) | (0.0018)
gender | 0.4146 | 0.2272 | | 0.2933 | 0.2008 | | 0.3854 | 0.2150
| (0.0029) | (0.0017) | | (0.0049) | (0.0018) | | (0.0025) | (0.0013)
$n$ | 77,899 | 216,136 | | 33,733 | 295,683 | | 111,632 | 511,819
| Male
exper. | 0.0823 | 0.0620 | | 0.0859 | 0.0780 | | 0.0825 | 0.0664
| (0.0020) | (0.0012) | | (0.0040) | (0.0018) | | (0.0017) | (0.0010)
$\mathtt{exper.}^{2}$ ($\times 10^{2}$) | -0.0039 | -0.0024 | | -0.0041 | -0.0036 | | -0.0037 | -0.0025
| (0.0002) | (0.0001) | | (0.0004) | (0.0002) | | (0.0001) | (0.0001)
$\mathtt{exper.}^{3}$ ($\times 10^{5}$) | 8.2014 | 4.3686 | | 9.2747 | 7.3170 | | 7.4306 | 3.6749
| (0.5321) | (0.3864) | | (1.7422) | (0.6709) | | (0.4700) | (0.3241)
$\mathtt{exper.}^{4}$ ($\times 10^{5}$) | -0.0650 | -0.0314 | | -0.0880 | -0.0582 | | -0.0552 | -0.0161
| (0.0054) | (0.0042) | | (0.0223) | (0.0081) | | (0.0049) | (0.0036)
marriage | 0.1493 | 0.1052 | | 0.1310 | 0.1234 | | 0.1421 | 0.1192
| (0.0056) | (0.0029) | | (0.0088) | (0.0031) | | (0.0048) | (0.0021)
nonwhite | -0.1362 | -0.1191 | | -0.1214 | -0.1040 | | -0.1309 | -0.1136
| (0.0064) | (0.0035) | | (0.0126) | (0.0039) | | (0.0057) | (0.0027)
$n$ | 44,299 | 116,129 | | 20,851 | 144,138 | | 65,150 | 260,267
| Female
exper. | 0.0713 | 0.0455 | | 0.0911 | 0.0782 | | 0.0729 | 0.0568
| (0.0022) | (0.0013) | | (0.0040) | (0.0016) | | (0.0019) | (0.0011)
$\mathtt{exper.}^{2}$ ($\times 10^{2}$) | -0.0044 | -0.0018 | | -0.0067 | -0.0045 | | -0.0045 | -0.0025
| (0.0002) | (0.0001) | | (0.0004) | (0.0002) | | (0.0002) | (0.0001)
$\mathtt{exper.}^{3}$ ($\times 10^{5}$) | 11.0325 | 3.4767 | | 19.6859 | 11.2858 | | 11.3406 | 4.4944
| (0.6649) | (0.4360) | | (1.7412) | (0.5915) | | (0.6095) | (0.3682)
$\mathtt{exper.}^{4}$ ($\times 10^{5}$) | -0.0974 | -0.0264 | | -0.1979 | -0.1046 | | -0.0969 | -0.0272
| (0.0071) | (0.0048) | | (0.0216) | (0.0071) | | (0.0066) | (0.0042)
marriage | -0.0078 | 0.0278 | | -0.0175 | 0.0168 | | -0.0082 | 0.0234
| (0.0048) | (0.0028) | | (0.0080) | (0.0026) | | (0.0041) | (0.0020)
nonwhite | -0.0714 | -0.0479 | | 0.0276 | -0.0291 | | -0.0356 | -0.0375
| (0.0065) | (0.0033) | | (0.0117) | (0.0033) | | (0.0057) | (0.0024)
$n$ | 33,600 | 100,007 | | 12,882 | 151,545 | | 46,482 | 251,552
Notes: This table reports the estimates of $\mathbf{\ \gamma}$ in (S.4.1).
“Postsecondary Edu.” stands for the sub-sample with years of education higher
than 12 and “High School or Less” stands for those with years of education
less than or equal to 12. The standard error of estimates of coefficients
associated with control variables are estimated based on Theorem 3 and
reported in parentheses. $n$ is the sample size.
## Appendix S.5 Computational algorithm
In this section, we describe the computational procedure used for estimation
of $\mathbf{\gamma}$, moments of $\beta_{i}$, and distributional parameters of
$\beta_{i}$.
1. 1.
Denote $\mathbf{w}_{i}=\left(x_{i},\mathbf{z}_{i}^{\prime}\right)^{\prime}$.
Compute the OLS estimator
$\left(\widehat{\mathrm{E}\left(\beta_{i}\right)}^{(0)},\widehat{\mathbf{\gamma}}^{\prime}\right)^{\prime}=\left(\frac{1}{n}\sum_{i=1}^{n}\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\right)^{-1}\left(\frac{1}{n}\sum_{i=1}^{n}\mathbf{w}_{i}^{\prime}y_{i}\right),$
and $\hat{\tilde{y}}_{i}=y_{i}-\mathbf{z}_{i}^{\prime}\widehat{\gamma}$.
2. 2.
For $r=2,3,\cdots,2K-1$, compute the sample version of the moment conditions
(2.8) and (2.9) in the main paper by replacing $\rho_{r,s}$ by
$n^{-1}\sum_{i=1}^{n}\hat{\tilde{y}}_{i}^{r}x_{i}^{s}$, and solving for
$\widehat{\mathrm{E}\left(\beta_{i}^{r}\right)}^{(0)}$ and
$\widehat{\sigma_{r}}^{(0)}$, recursively.
3. 3.
Use the initial estimates
$\left\\{\widehat{\mathrm{E}\left(\beta_{i}^{r}\right)}^{(0)}\right\\}_{r=1}^{2K-1}$
and $\left\\{\widehat{\sigma_{r}}^{(0)}\right\\}_{r=2}^{2K-1}$ to construct
the weighting matrix $\hat{\mathbf{A}}_{n}$ in (3.10) and compute the GMM
estimators
$\left\\{\widehat{\mathrm{E}\left(\beta_{i}^{r}\right)}^{(1)}\right\\}_{r=1}^{2K-1}$
and $\left\\{\widehat{\sigma_{r}}^{(1)}\right\\}_{r=2}^{2K-1}$ to compute the
moments of $\beta_{i}$ and $\sigma_{r}$. Iterate the GMM estimation one more
time with
$\left\\{\widehat{\mathrm{E}\left(\beta_{i}^{r}\right)}^{(1)}\right\\}_{r=1}^{2K-1}$
and $\left\\{\widehat{\sigma_{r}}^{(1)}\right\\}_{r=2}^{2K-1}$ as initial
estimates to obtain
$\left\\{\widehat{\mathrm{E}\left(\beta_{i}^{r}\right)}\right\\}_{r=1}^{2K-1}$
and $\left\\{\widehat{\sigma_{r}}\right\\}_{r=2}^{2K-1}$.
4. 4.
Solve
$\min_{\pi_{k},b_{k}}\left\\{\sum_{j=1}^{r}\left(\sum_{k=1}^{K}\pi_{k}b_{k}^{r}-\widehat{\mathrm{E}\left(\beta_{i}^{r}\right)}\right)^{2}\right\\}$
to get the initial estimates,
$\widehat{\mathbf{\theta}}^{(0)}=\left(\widehat{\mathbf{\pi}}^{(0)\prime},\widehat{\mathbf{b}}^{(0)\prime}\right)^{\prime}$.
5. 5.
Using
$\widehat{\mathbf{\theta}}^{(0)}=\left(\widehat{\mathbf{\pi}}^{(0)\prime},\widehat{\mathbf{b}}^{(0)\prime}\right)^{\prime}$
construct the weighting matrix $\hat{\mathbf{A}}_{n}$ and compute the GMM
estimator as
$\widehat{\mathbf{\theta}}^{(1)}=\left(\widehat{\mathbf{\pi}}^{(1)\prime},\widehat{\mathbf{b}}^{(1)\prime}\right)^{\prime}$
for $\mathbf{\theta}$. Iterate the GMM estimation one more time with
$\widehat{\mathbf{\theta}}^{(1)}=\left(\widehat{\mathbf{\pi}}^{(1)\prime},\widehat{\mathbf{b}}^{(1)\prime}\right)^{\prime}$
as initial estimates to obtain
$\widehat{\mathbf{\theta}}=\left(\widehat{\mathbf{\pi}}^{\prime},\widehat{\mathbf{b}}^{\prime}\right)^{\prime}$.
In the setup of the optimization problem for the optimization solver, imposing
the constraint $b_{1}<b_{2}<\cdots<b_{K}$ is important to improve the
numerical performance, particularly when $n$ is not sufficiently large (less
than $5,000$).
## References
* Hansen (2022) Hansen, E. B. (2022). Econometrics. Princeton University Press, Princeton.
* Pesaran (2015) Pesaran, M. H. (2015). Time Series and Panel Data Econometrics. Oxford University Press, Oxford.
|
# A Multi-Fluid Dust Module in Athena++: Algorithms and Numerical Tests
Pinghui Huang Institute for Advanced Study, Tsinghua University, Beijing
100084, People’s Republic of China Xue-Ning Bai Institute for Advanced Study,
Tsinghua University, Beijing 100084, People’s Republic of China Center for
Astrophysics, Department of Astronomy, Tsinghua University, Beijing 100084,
People’s Republic of China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
We describe the algorithm, implementation and numerical tests of a multifluid
dust module in the Athena++ magnetohydrodynamic (MHD) code. The module can
accommodate an arbitrary number of dust species interacting with the gas via
aerodynamic drag (characterized by the stopping time), with a number of
numerical solvers. In particular, we describe two second-order accurate, two-
stage, fully-implicit solvers that are stable in stiff regimes including short
stopping time and high dust mass loading, and they are paired with the second-
order explicit van-Leer and Runge-Kutta gas dynamics solvers in Athena++,
respectively. Moreover, we formulate a consistent treatment of dust
concentration diffusion with dust back-reaction, which incorporates momentum
diffusion and ensures Galilean invariance. The new formulation and stiff drag
solvers are implemented to be compatible with most existing features of
Athena++, including different coordinate systems, mesh refinement, shearing-
box and orbital advection. We present a large suite of test problems,
including the streaming instability in linear and nonlinear regimes, as well
as local and global setting, which demonstrate that the code achieves the
desired performance. This module will be particularly useful for studies of
dust dynamics and planet formation in protoplanetary disks.
††software: astropy (Astropy Collaboration et al., 2013), Athena++ (Stone et
al., 2020), Mathematica (Wolfram, 1991)
## 1 Introduction
Protoplanetary disks (PPDs) are composed of gas and dust. Although sharing
only about 1% in mass, dust represents the fundamental building blocks of
planets, and it is primarily the thermal radiation from dust that makes PPDs
observable in continuum emission from infrared to millimeter wavelengths. Dust
is coupled with gas via aerodynamic drag, characterized by the stopping time.
Dust particles of small sizes have small stopping time and are strongly
coupled to the gas, while larger dust particles are more loosely coupled, and
hence do not necessarily trace the gas. This fact is not only important many
processes of planet formation, but also crucial for interpreting disk
observations.
The initial stage of planet formation involves dust growth and transport, both
of which are sensitive to disk structure and level of turbulence (Ormel &
Cuzzi, 2007; Birnstiel et al., 2010). In particular, disk turbulence leads to
dust diffusion (Cuzzi et al., 1993; Youdin & Lithwick, 2007; Carballido et
al., 2010; Zhu et al., 2015), which determines the thickness of the dust layer
in the vertical direction, as well as the mixing in the radial direction.
Additional “psudo-diffusion” can result from complex radial gas flow
structures due to, e.g., wind-driven accretion (Hu & Bai, 2021). Upon growing
to larger sizes, back-reaction from dust-to-gas leads to dust clumping due to
the streaming instability (SI, Goodman & Pindor, 2000; Youdin & Goodman,
2005), and subsequently planetesimal formation (Johansen et al., 2007). While
there has been a large number of further studies (e.g., Bai & Stone, 2010a, b;
Carrera et al., 2015; Simon et al., 2017; Yang et al., 2017; Li & Youdin,
2021), it is less clear how the SI interplay with more realistic gas dynamics
(see Johansen et al., 2011; Schäfer et al., 2020; Xu & Bai, 2022). Finally,
instead of planetesimal accretion, the growth of planetary cores by pebble
accretion has been identified to be more efficient towards higher core mass
(Ormel & Klahr, 2010; Lambrechts & Johansen, 2012). The efficiency of pebble
accretion again depends on disk structure and level of turbulence (e.g.
Morbidelli et al., 2015; Xu et al., 2017), and back-reaction from dust-to-gas
may destabilize the feeding zone (Fu et al., 2014; Pierens et al., 2019; Yang
& Zhu, 2020; Huang et al., 2020; Hsieh & Lin, 2020; Surville et al., 2020),
which requires careful study considering realistic gas dynamics in 3D.
Over the past decade, thanks to the advent of the Atacama Large
Millimeter/submillimeter Array (ALMA), as well as high-contrast imaging
techniques equipped in ground-based telescopes, the dramatically improved
resolution and sensitivity have led to the discovery of disk substructures
prevalent in PPDs, particularly in the form of rings and gaps, as well as
various forms of asymmetries (see Andrews, 2020, for a review). These features
are commonly interpreted as a consequence of planet-disk interaction, which
can open gaps (Bae et al., 2017; Dong et al., 2017, 2018), create vortices
(van der Marel et al., 2013; Zhu et al., 2014; Flock et al., 2015), drive
spirals (Dong et al., 2011b, a; Bae & Zhu, 2018a, b), etc. At millimeter/sub-
millimeter wavelength, the observed substructures reflect the distribution of
mm-sized dust particles, which likely substantially amplify substructures in
the gaseous disk because these particles are not strongly tied to gas and tend
to drift towards pressure maxima (Whipple, 1972; Weidenschilling, 1977).
Alternatively, a number of non-planet mechanisms have been identified which
lead to substructure formation, such as processes involving snow lines (Zhang
et al., 2015; Okuzumi et al., 2016; Owen, 2020) and MHD effects (Suriano et
al., 2018; Riols et al., 2020; Cui & Bai, 2021). Some of the mechanisms
requires active participation from dust itself due to its back-reaction
(Takahashi & Inutsuka, 2014, 2016; Tominaga et al., 2019, 2020). In all these
scenarios, it is crucial to co-evolve gas and dust in a self-consistent manner
to help constrain the physical mechanisms behind the observations.
Computationally, dust is commonly treated either as Lagrangian (super-)
particles, or as pressureless fluids. The particle methods have been
implemented in several MHD codes including Pencil (Johansen et al., 2007),
Athena (Bai & Stone, 2010c), FARGO-ADSG (Baruteau & Zhu, 2016) and PLUTO
(Mignone et al., 2019). It has also been naturally employed in smoothed
particle hydrodynamic (SPH) codes including PHANTOM (Price et al., 2018). One
major advantage of the Lagrangian treatment is being able to properly handle
particle crossing, more relevant for particles that are marginally or loosely
coupled to the gas, which is important for studying planetesimal formation by
the SI. On the other hand, it is generally difficult to handle the highly
stiff regime of extreme particle concentration (Bai & Stone, 2010c, but see
Yang & Johansen, 2016; Moseley et al., 2022), and achieving good load
balancing can be challenging for very large simulations (but see Johansen et
al., 2011). Moreover, it is common to treat the unspecified source of disk
turbulence as an effective viscosity in gas dynamic simulations. Doing so for
particles can be involved, especially if one were to further consider dust
back-reaction.
The alternative fluid treatment of dust is gaining popularity, such as in
PIERNIK (Hanasz et al., 2010a, b), MPI-AMRVAC (Porth et al., 2014; Xia et al.,
2018), LA-COMPASS (Li et al., 2005, 2009; Fu et al., 2014) and FARGO3D
(Benítez-Llambay & Masset, 2016; Benítez-Llambay et al., 2019). This approach
is more appropriate for relatively strongly coupled dust, as they quickly
respond to fluid motion to minimize particle crossing. As separate fluids are
co-located with gas in the computational domain, stiffness issues can be
overcome by designing fully-implicit schemes for the drag source term
simultaneously on gas and dust, and load balancing can be trivially satisfied.
Dust diffusion can be easily handled by incorporating a concentration
diffusion source term (Cuzzi et al., 1993; Youdin & Lithwick, 2007). Finally,
this approach is generalizable to further incorporate dust coagulation (Li et
al., 2019; Drazkowska et al., 2019; Li et al., 2020), so that one can self-
consistently compute the dust size distribution at every simulation cell.
In this paper, we describe the algorithm, implementation and numerical tests
of a multifluid dust module in the Athena++ MHD code (Stone et al., 2020). Our
development features a set of dust integrators, particularly two fully-
implicit integrators that can handle all stiff regimes while maintaining 2nd-
order accuracy, which improve upon previous works which were either explicit,
such as MPI-AMRVAC (Porth et al., 2014), FARGO-ADSG (Baruteau & Zhu, 2016) and
PHANTOM (Price et al., 2018), or implicit but only 1st-order accurate
(FARGO3D, Benítez-Llambay et al., 2019). With Athena++ being a Godunov MHD
code, our implementation naturally conserves total momentum and energy.
Moreover, we provide a consistent formulation of dust concentration diffusion,
and show that additional correction terms in the momentum equations of dust
are necessary to properly conserve total momentum and maintain Galilean
invariance. Implementing these terms yield physically sensible results in a
number of test problems.
The outline of this paper is as follows. In Section 2, we describe the
equations, our numerical schemes and implementations. In Section 3, we present
the benchmark tests, including collisions between gas and dust, dust diffusion
with or without momentum correction, linear and non-linear tests of the SI,
global curvilinear simulations of the SI, as well as static/adaptive mesh
refinement tests. Finally, we summarize and discuss our results in Section 4.
## 2 Numerical Scheme
In this section, we describe the basic equations including consistent
formulation of dust concentration diffusion, as well as the numerical schemes
and implementation of the multifluid dust module in Athena++.
### 2.1 General Equations (Conservative Form)
We start by presenting the full set of equations of gas and multifluid dust.
We use subscripts “d” and “g” to denote “dust” and “gas”. Let there be
$N_{\text{d}}$ dust species, each characterized by a stopping time
$T_{\text{s},n}$ representing the timescale they respond to gas drag, where we
use a label “$n$” for the $n$-th dust species. In conservative form, the
equations read:
$\frac{\partial\rho_{\text{g}}}{\partial
t}+\nabla\cdot\left(\rho_{\text{g}}\boldsymbol{v}_{\text{g}}\right)=0\ ,$ (1)
$\displaystyle\frac{\partial\left(\rho_{\text{g}}\boldsymbol{v}_{\text{g}}\right)}{\partial
t}$
$\displaystyle+\nabla\cdot\left(\rho_{\text{g}}\boldsymbol{v}_{\text{g}}\boldsymbol{v}_{\text{g}}+P_{\text{g}}\mathsf{I}+\boldsymbol{\Pi}_{\nu}\right)=$
(2)
$\displaystyle\rho_{\text{g}}\boldsymbol{f}_{\text{g,src}}+\sum^{N_{\text{d}}}_{n=1}\rho_{\text{d},n}\frac{\boldsymbol{v}_{\text{d},n}-\boldsymbol{v}_{\text{g}}}{T_{\text{s},n}}\
,$
$\displaystyle\frac{\partial E_{\text{g}}}{\partial
t}+\nabla\cdot\left[\left(E_{\text{g}}+P_{\text{g}}\right)\boldsymbol{v}_{\text{g}}+\boldsymbol{\Pi}_{\nu}\cdot\boldsymbol{v}_{\text{g}}\right]=\rho_{\text{g}}\boldsymbol{f}_{\text{g,src}}\cdot\boldsymbol{v}_{\text{g}}$
(3)
$\displaystyle+\sum^{N_{\text{d}}}_{n=1}\rho_{\text{d},n}\frac{\boldsymbol{v}_{\text{d},n}-\boldsymbol{v}_{\text{g}}}{T_{\text{s},n}}\cdot\boldsymbol{v}_{\text{g}}+\omega\sum^{N_{\text{d}}}_{n=1}\rho_{\text{d},n}\frac{\left(\boldsymbol{v}_{\text{d},n}-\boldsymbol{v}_{\text{g}}\right)^{2}}{T_{\text{s},n}}\
,$
$\frac{\partial\rho_{\text{d},n}}{\partial
t}+\nabla\cdot\left(\rho_{\text{d},n}\boldsymbol{v}_{\text{d},n}+\boldsymbol{\mathscr{F}}_{\text{dif},n}\right)=0\
,$ (4)
$\displaystyle\frac{\partial\rho_{\text{d},n}\left(\boldsymbol{v}_{\text{d},n}+\boldsymbol{v}_{\text{d,dif},n}\right)}{\partial
t}$
$\displaystyle+\nabla\cdot(\rho_{\text{d},n}\boldsymbol{v}_{\text{d},n}\boldsymbol{v}_{\text{d},n}+\boldsymbol{\Pi}_{\text{dif},n})=$
(5) $\displaystyle\rho_{\text{d},n}$
$\displaystyle\boldsymbol{f}_{\text{d,src},n}+\rho_{\text{d},n}\frac{\boldsymbol{v}_{\text{g}}-\boldsymbol{v}_{\text{d},n}}{T_{\text{s},n}}\
.$
There are $4N_{\text{d}}+5$ equations in total, where Equations (1) to (3) are
the gas continuity, momentum and energy equations, and Equations (4), (5) are
the continuity and momentum equations for the dust species, which are treated
as pressureless fluids (Garaud et al., 2004). In the above, $\rho$ is the
density, $\boldsymbol{v}$ is the velocity, $P_{\text{g}}$ is the gas pressure,
$\mathsf{I}$ is the identity tensor, and
$E_{\text{g}}=(\rho_{\text{g}}\boldsymbol{v}_{\text{g}}^{2})/2+P_{\text{g}}/(\gamma-1)$
is the total energy density of gas, with $\gamma$ being the adiabatic index.
Here we neglect magnetic fields, thermal conduction, etc., as they do not
directly couple to dust, and our dust fluid module is fully compatible with
these existing features.111We treat dust fluids as neutral, but future
extensions may incorporate dust charge, e.g., Hopkins & Squire, 2018.
We incorporate gas viscosity which mimics the presence of external turbulence,
described by the viscous stress tensor $\boldsymbol{\Pi}_{\nu}$:
$\Pi_{\nu,ij}=\rho_{\text{g}}\nu_{\text{g}}\left(\frac{\partial
v_{\text{g},i}}{\partial x_{\text{g},j}}+\frac{\partial
v_{\text{g},j}}{\partial
x_{\text{g},i}}-\frac{2}{3}\delta_{ij}\nabla\cdot\boldsymbol{v_{\text{g}}}\right)\
,$ (6)
where $\nu_{\text{g}}$ is the kinematic viscosity. Closely related to the gas
viscosity is a dust diffusivity $D_{\text{d},n}$, which leads to concentration
diffusion. We treat the diffusivity for each dust species as a free parameter
to be specified by the user, while they are usually prescribed as (Youdin &
Lithwick, 2007):
$D_{\text{d},n}=\frac{\nu_{\text{g}}}{1+\left(T_{\text{s},n}/T_{\text{g,eddy}}\right)^{2}}\
,$ (7)
where $T_{\rm g,eddy}$ is the turbulent eddy time of the external turbulence.
With this, the dust concentration diffusion flux, acting on the dust
continuity equation, is given by
$\boldsymbol{\mathscr{F}}_{\text{dif},n}\equiv-\rho_{\text{g}}D_{\text{d},n}\nabla\left(\frac{\rho_{\text{d},n}}{\rho_{\text{g}}}\right)=\rho_{\text{d},n}\boldsymbol{v}_{\text{d,dif},n}\
,$ (8)
which also gives the definition of the effective dust drift speed
$\boldsymbol{v}_{\text{d,dif},n}$ due to concentration diffusion. Associated
with this concentration diffusion, correction terms must be incorporated to
the dust momentum equation to ensure consistent momentum diffusion flux
(Tominaga et al., 2019) and Galilean invariance. The individual components of
the momentum diffusion flux tensor are given by
$\Pi_{\text{dif},n,ij}=v_{\text{d},n,j}\mathscr{F}_{\text{dif},n,i}+v_{\text{d},n,i}\mathscr{F}_{\text{dif},n,j}\
.$ (9)
Full derivations of the concentration diffusion terms will be presented in
Section 2.2.
The last terms of the right hand sides of the momentum equations (2) and (5)
correspond to the aerodynamic drag between gas and dust. Here we assume linear
drag law, where $T_{{\rm s},n}$ is independent of velocity. Additional two
source terms are added to the energy equation (3), which correspond to the
work done by the drag, and frictional heating. We have included a parameter
$\omega$ to control the level of frictional heating, being $0$ to be turned
off, and $1$ when all dissipation is deposited to the gas.222In reality, some
of the dissipation must lead to heating of the dust. If assuming gas and dust
should maintain the same temperature, one should assign
$\omega=c_{V,\text{g}}\rho_{\text{g}}/(c_{V,\text{g}}\rho_{\text{g}}+c_{V,\text{d}}\rho_{\text{d}})$,
where $c_{V,\text{g}}$ and $c_{V,\text{d}}$ are the heat capacity of gas and
dust, respectively, and
$\rho_{\text{d}}=\sum_{n=1}^{N_{\text{d}}}\rho_{\text{d},n}$.
Other external source terms are denoted by $\boldsymbol{f}_{\text{src}}$,
which may include stellar and/or planetary gravity in disk problems depending
on applications. They are implemented as explicit source terms added on the
momentum equations (2) and (5) following the standard in Athena++ (Stone et
al., 2020). Associated with them is a source term
$W\equiv\boldsymbol{f}_{\text{g,src}}\cdot\boldsymbol{v}_{\text{g}}$ in the
energy equation accounting for the work done by the source terms.
Note that our formulation does not contain an energy equation for dust, thus
does not ensure global energy conservation of the composite dust-gas system.
While this is not of overwhelming concern in typical applications, future
generalization to incorporate a dust energy equation is possible. At
algorithmic level, we thus aim at full momentum conservation, and implement
energy source terms to match the overall accuracy of the algorithm.
### 2.2 Consistent Formulation of Dust Concentration Diffusion
Here we derive dust fluid equations in the presence of turbulent diffusion,
following the procedures of Cuzzi et al. (1993) and Tominaga et al. (2019). We
use the Reynolds averaging technique with approximate closure relations to
properly account for the role of turbulence at sub-grid level while preserving
global conservation laws. In doing so, we are interested in the physics on
time (and potentially length) scales above those for the turbulence, and hence
any physical variable $A$ is decomposed into a time-averaged part
$\overline{A}$ and a fluctuating part $\Delta A$, i.e., $A=\overline{A}+\Delta
A$.
Without loss of generality, we focus on a single dust species and drop its
label $n$. We start from the standard dust fluid equations in conservation
form:
$\displaystyle\frac{\partial\rho_{\text{d}}}{\partial
t}+\nabla\cdot\left(\rho_{\text{d}}\boldsymbol{v}_{\text{d}}\right)$
$\displaystyle=0\ ,$ (10)
$\displaystyle\frac{\partial\rho_{\text{d}}\boldsymbol{v}_{\text{d}}}{\partial
t}+\nabla\cdot\left(\rho_{\text{d}}\boldsymbol{v}_{\text{d}}\boldsymbol{v}_{\text{d}}\right)$
$\displaystyle=\rho_{\text{d}}\frac{\boldsymbol{v}_{\text{g}}-\boldsymbol{v}_{\text{d}}}{T_{\text{s}}}\
.$
Taking averages to the continuity equation, we obtain
$\displaystyle\frac{\partial\overline{\rho_{\text{d}}}}{\partial
t}+\nabla\cdot\left(\overline{\rho}_{\text{d}}\overline{\boldsymbol{v}_{\text{d}}}+\overline{\Delta\rho_{\text{d}}\Delta\boldsymbol{v}_{\text{d}}}\right)=0\
.$ (11)
The extra term
$\overline{\Delta\rho_{\text{d}}\Delta\boldsymbol{v}_{\text{d}}}$, by
definition, corresponds to dust concentration diffusion flux (8)
$\overline{\Delta\rho_{\text{d}}\Delta\boldsymbol{v}_{\text{d}}}\equiv\boldsymbol{\mathscr{F}}_{\text{dif}}=\overline{\rho_{\text{d}}}\boldsymbol{v}_{\text{d},\text{dif}}\
.$ (12)
Next, by taking averages to the momentum equation, we obtain
$\displaystyle\frac{\partial\overline{\rho}_{\text{d}}\overline{\boldsymbol{v}_{\text{d},j}}}{\partial
t}+\frac{\partial\overline{\Delta\rho_{\text{d}}\Delta\boldsymbol{v}_{\text{d},j}}}{\partial
t}+\frac{\partial}{\partial
x_{i}}(\overline{\rho_{\text{d}}}\overline{\boldsymbol{v}_{\text{d},i}}\overline{\boldsymbol{v}_{\text{d},j}}$
(13)
$\displaystyle+\overline{\Delta\rho_{\text{d}}\Delta\boldsymbol{v}_{\text{d},i}}\overline{\boldsymbol{v}_{\text{d},j}}+\overline{\boldsymbol{v}_{\text{d},i}}\overline{\Delta\rho_{\text{d}}\Delta\boldsymbol{v}_{\text{d},j}}+\overline{\rho_{\text{d}}}\overline{\Delta\boldsymbol{v}_{\text{d},i}\Delta\boldsymbol{v}_{\text{d},j}})$
$\displaystyle=\overline{\rho_{\text{d}}}\frac{\overline{\boldsymbol{v}_{\text{g},j}}-\overline{\boldsymbol{v}_{\text{d},j}}}{T_{\text{s}}}+\frac{\overline{\Delta\rho_{\text{d}}\Delta\boldsymbol{v}_{\text{g},j}}-\overline{\Delta\rho_{\text{d}}\Delta\boldsymbol{v}_{\text{d},j}}}{T_{\text{s}}}\
.$
At this stage, it is often argued that one can drop the second term on the
left assuming the time-dependent diffusion flux is small compared to that of
the bulk flow (Cuzzi et al., 1993; Tominaga et al., 2019). However, our
analysis shows that this would violate Galilean invariance (see Appendix A,
and also numerical tests in Section 2.4.2), and hence it must be kept. The
second and third terms in the momentum flux can be reduced using the effective
dust drift velocity $\boldsymbol{v}_{\text{d,dif}}$, which leads to the
expression of momentum diffusion flux (9). We note that momentum conservation
does not necessarily requires the inclusion of momentum diffusion flux, but
this flux is important when considering angular momentum conservation in disk
problems (Tominaga et al., 2019), as well as for ensuring Galilean invariance.
For the last term in the momentum flux
$\overline{\rho}_{\text{d}}\overline{\Delta\boldsymbol{v}_{\text{d},i}\Delta\boldsymbol{v}_{\text{d},j}}$,
we may use the simple closure relation by Shariff & Cuzzi (2011) and Tominaga
et al. (2019) as
$\overline{\Delta\boldsymbol{v}_{\text{d},i}\Delta\boldsymbol{v}_{\text{d},j}}=\delta_{ij}c_{\text{s,d}}^{2}\
.$ (14)
where $c_{\text{s,d}}$ is the effective dust sound speed. This term can be
neglected in the multifluid approach (Garaud et al., 2004). The second term on
the right hand side is also neglected with the expectation that the standard
drag term dominates, as in Tominaga et al. (2019).
With all these considerations, we recover the dust momentum equation shown in
Section 2.1, here rewritten as
$\displaystyle\frac{\partial\rho_{\text{d}}\left(v_{\text{d},j}+v_{\text{d,dif},j}\right)}{\partial
t}$ $\displaystyle+\frac{\partial}{\partial
x_{i}}(\rho_{\text{d}}v_{\text{d},i}v_{\text{d},j}+\rho_{\text{d}}v_{\text{d,dif},i}v_{\text{d},j}$
(15) $\displaystyle+\rho_{\text{d}}v_{\text{d},i}v_{\text{d,dif},j})$
$\displaystyle=\rho_{\text{d}}\frac{v_{\text{g},j}-v_{\text{d},j}}{T_{\text{s}}}\
.$
where for notational convenience, we can drop the overline and interpret the
dust fluid quantities in the averaged sense. The presence of time-derivative
on $\rho_{d}\boldsymbol{v}_{\text{d,dif}}$ in the momentum equation is the
inevitable consequence of this averaging procedure. Missing this term would
lead to unphysical behaviors as we demonstrate in Section 3.2. Implementing
this term also requires special care, as will be discussed in Section 2.4.2.
### 2.3 Dust-Gas Drag Integrators
The drag term involves interactions between gas and all dust species. As a
special source term to both gas and dust, the drag integrator aims to solve
the following equation
$\displaystyle\frac{\partial\boldsymbol{M}}{\partial
t}=\begin{bmatrix}\sum^{N_{\text{d}}}_{n=1}\alpha_{n}\left(\boldsymbol{M}_{\text{d},n}-\epsilon_{n}\boldsymbol{M}_{\text{g}}\right)\\\
\alpha_{1}(\epsilon_{1}\boldsymbol{M}_{\text{g}}-\boldsymbol{M}_{\text{d},1})\\\
\alpha_{2}(\epsilon_{2}\boldsymbol{M}_{\text{g}}-\boldsymbol{M}_{\text{d},2})\\\
\vdots\\\
\alpha_{n}(\epsilon_{n}\boldsymbol{M}_{\text{g}}-\boldsymbol{M}_{\text{d},n})\\\
\end{bmatrix}\equiv\boldsymbol{f}_{\text{drag}}\left(\boldsymbol{M},\boldsymbol{W}\right)\
,$ (16)
with $\boldsymbol{f}_{\text{drag}}$ being the mutual drag force,
$\boldsymbol{M}\equiv[\boldsymbol{M}_{\text{g}},$
$\boldsymbol{M}_{\text{d,1}},\ \dots,\
\boldsymbol{M}_{\text{d},n}]^{\top}=[\rho_{\text{g}}\boldsymbol{v}_{\text{g}},\
\rho_{\text{d},1}\boldsymbol{v}_{\text{d,1}},\ \dots,\
\rho_{\text{d},n}\boldsymbol{v}_{\text{d},n}]^{\top}$ is the momentum vector
of gas and dust. The remaining variables are denoted as $\boldsymbol{W}$ given
by ($\boldsymbol{\epsilon},\boldsymbol{\alpha})$, where
$\boldsymbol{\epsilon}=\left[\epsilon_{1},\dots,\epsilon_{n}\right]\equiv[\rho_{\text{d},\
1}/\rho_{\text{g}},\ \dots,\ \rho_{\text{d},n}/\rho_{\text{g}}]$, and
$\boldsymbol{\alpha}=\left[\alpha_{1},\dots\alpha_{n}\right]\equiv[T_{\text{s},1}^{-1},\dots
T_{\text{s},n}^{-1}]$. They are treated as constant parameters in the
integrator.
The drag term is potentially stiff in two regimes. First, when the dust
stopping time $T_{s}$ is very small, and stiffness arises when $T_{S}<\Delta
t\equiv h$, the hydrodynamic time step. Second, when $\sum_{n}\epsilon_{n}\gg
1$, which arises when dust is strongly concentrated. The stiff regimes should
be handled by fully-implicit integrators for stability. We note that for
particle-based methods, handling the first regime of stiffness is relatively
straightforward (Bai & Stone, 2010c; Fung & Muley, 2019; Mignone et al.,
2019), whereas handling the second regime requires extra care, where one
either artificially reduces particle back-reaction (Bai & Stone, 2010c), or
sacrifice the time step (Li & Youdin, 2021), and more rigorous treatment
demands substantially more computational cost (Yang & Johansen, 2016). With
the fluid-treatment of dust, one can directly solve the above equation
implicitly, which automatically handles both stiffness regimes (Benítez-
Llambay et al., 2019). In doing so, we need to evaluate the Jacobian of
$\boldsymbol{f}_{\text{drag}}$, and for brevity we drop the subscript “drag”:
$\displaystyle\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}=\begin{bmatrix}-\sum^{N_{\text{d}}}_{n}\epsilon_{i}\alpha_{i}&\alpha_{1}&\alpha_{2}&\cdots&\alpha_{n}\\\
\epsilon_{1}\alpha_{1}&-\alpha_{1}&0&\cdots&0\\\
\epsilon_{2}\alpha_{2}&0&-\alpha_{2}&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
\epsilon_{n}\alpha_{n}&0&0&\cdots&-\alpha_{n}\\\ \end{bmatrix}\ .$ (17)
Note that for linear drag law, this Jacobian applies to individual dimensions
(which are independent of each other).
We have implemented a number of different drag integrators. Here we describe
the algorithms of fully-implicit integrators that we develop for achieving
numerical stability and towards higher-order accuracy. The implementation of
other simpler integrators, which are explicit or semi-implicit that are useful
for non-stiff problems, are described in Appendix C.
#### 2.3.1 First Order Fully-Implicit Method
We start from the standard backward Euler method, which is a single-stage
integrator to be combined with the Runge-Kutta 1 (RK1, or forward Euler) time
integrator in Athena++. Integrating from step $n$ to $n+1$, the format is
given by
$\displaystyle\boldsymbol{M}^{(n+1)}$
$\displaystyle=\boldsymbol{M}^{(n)}+h\boldsymbol{f}\left(\boldsymbol{M}^{(n+1)},\boldsymbol{W}^{(n)}\right)\
.$ (18)
Note that this numerical format guarantees momentum conservation. Substituting
$\displaystyle\boldsymbol{f}(\boldsymbol{M}^{(n+1)},\boldsymbol{W}^{(n)})$
$\displaystyle=\boldsymbol{f}(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n)})$ (19)
$\displaystyle+\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n)}(\boldsymbol{M}^{(n+1)}-\boldsymbol{M}^{(n)})\
,$
we can update the momentum
$\Delta\boldsymbol{M}\equiv\boldsymbol{M}^{(n+1)}-\boldsymbol{M}^{(n)}$ by
$\Delta\boldsymbol{M}=\left(\mathsf{I}-h\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n)}\right)^{-1}h\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n)}\right)\
,\\\ $ (20)
where $\mathsf{I}$ is the identity matrix and evaluating
$\Delta\boldsymbol{M}$ involves matrix inversion. This is the main integrator
implemented in FARGO3D (Benítez-Llambay et al., 2019), which makes the mutual
drag interaction unconditional stable, despite of being only 1st-order
accurate in time. With the simple form of the Jacobian (17), the matrix in the
backward Euler method can be solved efficiently on order $\sim O(N_{d})$
instead of $\sim O(N_{d}^{3})$ as in standard LU decomposition (Krapp &
Benítez-Llambay, 2020).
The energy source term on the gas has two parts. The first arises from the
work done by the drag force. To better preserve energy conservation, this term
should be implemented as the change in the gas kinetic energy due to gas drag:
$\Delta
E_{\text{g},1}=\Delta\boldsymbol{M}_{\text{g}}\cdot(\boldsymbol{v}^{(n)}_{\text{g}}+\boldsymbol{v}^{(n+1)}_{\text{g}})/2\
,$ (21)
The second part is from frictional heating (Marble, 1970; Laibe & Price, 2014;
Mignone et al., 2019), which is associated with the reduction of total kinetic
energy in the gas-dust system. This can be calculated by
$\Delta
E_{\text{g},2}=\Delta\boldsymbol{M}_{\text{g}}\cdot\frac{\boldsymbol{v}^{(n)}_{\text{g}}+\boldsymbol{v}^{(n+1)}_{\text{g}}}{2}+\sum_{n=1}^{N_{\text{d}}}\Delta\boldsymbol{M}_{\text{d},n}\cdot\frac{\boldsymbol{v}^{(n)}_{\text{d},n}+\boldsymbol{v}^{(n+1)}_{\text{d},n}}{2}\
,$ (22)
The source terms for the energy equation should thus be
$E_{\text{g}}^{(n+1)}=E_{\text{g}}^{(n)}+\Delta E_{\text{g},1}-\omega\Delta
E_{\text{g},2}\ .$ (23)
#### 2.3.2 Second Order Fully-Implicit Methods
Next we build two fully-implicit drag integrators to be combined with the van-
Leer 2 (VL2) and the Runge-Kutta 2 (RK2) time integrators in Athena++. We
refer to them as the “VL2-Implicit” and “RK2-Implicit” integrators,
respectively. Both integrators involve two stages. Here we describe their
implementation, while the derivation of the algorithm can be found in Appendix
B.
#### VL2-Implicit
Stage I: We apply the backward Euler method to update the system momenta from
step $n$ for half a time step $h/2$, denoted by a prime ′:
$\Delta\boldsymbol{M}^{\prime}=\left(\mathsf{I}-\frac{h}{2}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n)}\right)^{-1}\frac{h}{2}\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n)}\right)\
,\\\ $ (24)
Matrix inversion in this stage can be similarly achieved on order $O(N_{d})$.
The update in gas energy at this stage is exactly analogous to that in the
backward Euler method, which we do not repeat.
Stage II: the momentum is updated from step $n$ to $n+1$ using the following
$\displaystyle\Delta\boldsymbol{M}$
$\displaystyle=\boldsymbol{\Lambda}^{-1}\left(\mathsf{I}-\frac{h}{2}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}\bigg{|}^{\prime}\right)h\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{\prime}\right)\
,$ (25)
where
$\boldsymbol{\Lambda}\equiv\mathsf{I}-\left(\mathsf{I}-\frac{h}{2}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}\bigg{|}^{\prime}\right)h\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n)}\
.$ (26)
Note that this matrix is more complex and should be inverted by LU
decomposition (Press et al., 1986). The update in gas energy has exactly the
same form as Equations (21) to (23), which we do not repeat.
#### RK2-Implicit
Stage I: We use the backward Euler method with time step $h$ to calculate the
momentum at step $n+1$, which is exactly the same as described in Section
2.3.1. We still denote the quantities at the end of this stage using a prime
′.
Stage II: The momentum at stage $n+1$ is:
$\displaystyle\Delta\boldsymbol{M}=\boldsymbol{\Lambda}^{-1}$
$\displaystyle\bigg{[}h\boldsymbol{f}(\boldsymbol{M}^{(n)},\boldsymbol{W}^{\prime})$
(27)
$\displaystyle+\bigg{(}\mathsf{I}-h\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}\bigg{|}^{\prime}\bigg{)}h\boldsymbol{f}(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n)})\bigg{]}\
,$
where
$\displaystyle\boldsymbol{\Lambda}$
$\displaystyle\equiv\mathsf{I}-h\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}\bigg{|}^{(n)}+\frac{h^{2}}{2}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}\bigg{|}^{\prime}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}\bigg{|}^{(n)}\
.$ (28)
Similarly, matrix inversion is solved by LU decomposition. The update in gas
energy also has exactly the same form as Equations (21) to (23), which we do
not repeat.
Note that this integration scheme is in essence the same as the fully-implicit
particle integrator in Bai & Stone (2010c).
#### 2.3.3 Coupling Explicit Hydrodynamic Integrators with Implicit Drag
Integrators
Special care must be taken when combining implicit integrators with the
explicit hydrodynamic integrators and source terms. When they are treated
separately, the combined algorithm would only be 1st-order accurate,333The
implicit-explicit Runge-Kutta schemes are viable choices (Pareschi & Russo,
2005), they usually involve more stages of integration than the order of
accuracy achieved, and do not necessarily match the existing hydrodynamic
integrators in Athena++. and the implicit drag integrator cannot maintain
exact equilibrium solutions.
To overcome these issues, we may consider the advection, diffusion and other
hydrodynamic source terms as an “add-on” to $\boldsymbol{f}$. In other words,
$\boldsymbol{f}$ in the drag algorithms above represents the combination of
the drag force ($\boldsymbol{f}_{\rm drag}$, treated implicitly), as well as
other explicit terms including advection and other source terms
$\boldsymbol{G}_{M}$ acting on the gas and dust momenta
$\boldsymbol{f}\equiv\boldsymbol{f}_{\rm
drag}(\boldsymbol{M},\boldsymbol{W})+\boldsymbol{G}_{M}(\boldsymbol{\overline{U}})\
,$ (29)
where $\boldsymbol{G}_{M}$ is expressed in terms of conserved variables
$\boldsymbol{U}$. By adding an overline on $\boldsymbol{U}$, we treat these
other explicit terms as known constant, readily obtained in the hydrodynamic
integrator. In the hydrodynamic integration from step 1 to step 2 over time
interval $\Delta t$, we estimate $\boldsymbol{G}_{M}$ to be the momentum
update from explicit terms
$\boldsymbol{G}_{M}(\boldsymbol{\overline{U}})=\frac{\boldsymbol{M}^{(2)}-\boldsymbol{M}^{(1)}}{\Delta
t}\ ,$ (30)
where $\boldsymbol{M}^{(2),(1)}$ represents the momenta before and after
explicit integration steps (advection, diffusion and other explicit source
terms). By treating this term as a constant, the Jacobian and the $\Lambda$
matrices described in the previous subsection remain unchanged.
Implementing the above requires extra storage to store the momentum updates,
and that we must finish all explicit steps in the hydrodynamic integration
before entering the drag integrator. We will show that our approach
successfully achieves 2nd-order accuracy when using VL2-Implicit and
RK2-Implicit integrators, and it also allows us to achieve exact equilibrium
solutions involving the drag force.
### 2.4 Integration of Multifluid Dust Equations
The integration of dust fluid is divided into several parts (advection,
diffusion, source terms and drag). Except for the drag term (described in the
previous subsection), the other terms are treated independently and
explicitly, and we describe their implementation in this subsection.
#### 2.4.1 General Procedures
Following the standard routine in Athena++, each integration time step is
divided into a number of stages depending on the time integrator employed (see
detailed descriptions in Stone et al., 2020). In each stage, the integration
procedures involve updating conserved variables based on primitive variables
by evolving the fluid equations by $dt$. Our multifluid dust module supports
Athena++ time integrators up to second order, including 1st-order Runge-Kutta
(RK1), 2nd-order Runge-Kutta (RK2) and van-Leer integrator (VL2).
For each dust species, the primitive ($\mathbf{W}_{\text{d}}$) and conserved
($\mathbf{U}_{\text{d}}$) variables are
$\mathbf{W}_{\text{d}}=\begin{bmatrix}\rho_{\text{d}}\\\
\boldsymbol{v}_{\text{d}}\\\ \end{bmatrix}\
,\quad\mathbf{U}_{\text{d}}=\begin{bmatrix}\rho_{\text{d}}\\\
\rho_{\text{d}}(\boldsymbol{v}_{\text{d}}+\boldsymbol{v}_{\text{d,dif}})\\\
\end{bmatrix}\ .$ (31)
Note that the presence of time derivative on
$\rho_{\text{d}}\boldsymbol{v}_{\text{d,dif}}$ in the momentum equation
suggests that the concentration diffusion flux should be considered as part of
the conserved dust momentum. The total momentum is thus
$\rho\boldsymbol{v}_{\text{g}}+\sum_{i}\rho_{\text{d},i}(\boldsymbol{v}_{\text{d},i}+\boldsymbol{v}_{\text{d,dif},i})$.
Integrating the bulk part of the dust fluid is very similar that of
hydrodynamics in Athena++. The main procedures involves the reconstruction of
primitive variables at cell interfaces, followed by solving a Riemann problem
to obtain the mass and momentum fluxes, after which we update the dust fluid
quantities from flux gradients. Same as in Athena++, the multifluid dust
module supports spatial reconstructions up to third order.
As pressureless fluids, the Riemann problem for dust fluids is greatly
simplified. In one-dimension along the $x$-direction, given the left/right
states $\mathbf{W}_{\text{d}}^{L/R}$, we provide the Riemann flux for
conserved variables as follows. The density flux reads
$F_{x}(\rho_{d})=\begin{cases}\rho_{\text{d}}^{L}v_{\text{d},x}^{L}&v_{\text{d},x}^{L}>0\
,\ v_{\text{d},x}^{R}>0\ ,\\\
\rho_{\text{d}}^{R}v_{\text{d},x}^{R}&v_{\text{d},x}^{L}<0\ ,\
v_{\text{d},x}^{R}<0\ ,\\\ 0&v_{\text{d},x}^{L}<0\ ,\ v_{\text{d},x}^{R}>0\
,\\\
\rho_{\text{d}}^{L}v_{\text{d},x}^{L}+\rho_{\text{d}}^{R}v_{\text{d},x}^{R}&v_{\text{d},x}^{L}>0\
,\ v_{\text{d},x}^{R}<0\ .\\\ \end{cases}$ (32)
Similar expressions hold for the momentum flux for all three directions.
Essentially, we use the upwind flux when the normal velocity in the L/R states
are the same, set the flux to be zero when the L/R normal velocities diverge,
and sum up the fluxes from the two sides when L/R normal velocities converge.
The last treatment reflects that as pressureless fluids, the flows on the two
sides can penetrate each other, just as particles.444Note that penetration is
still prohibited within each cell, where dust fluid velocities get well mixed.
Alternatively, one may set the flux to zero in this case. We do not find much
practical differences in test problems by using different Riemann solvers for
dust.
The implementation of other source terms on dust, such as stellar gravity and
source terms in shearing-box, as well as geometric source terms in cylindrical
and spherical coordinates, are same as that of gas, which are treated
explicitly.
#### 2.4.2 Dust Diffusion
The implementation of dust concentration diffusion starts by computing the
concentration diffusion flux according to Equation (8). The fluxes are
computed by standard finite differencing, and are located at cell interfaces.
Next, we calculate the momentum diffusion flux according to Equation (9). This
term contains two parts. The first part,
$v_{\text{d},n,j}\mathscr{F}_{\text{dif},n,i}$, describes the diffusion of the
$j$-momentum in the $i$-direction. At the implementation level, its value is
obtained by averaging from the upwind side based on the sign of the
concentration diffusion flux $\mathscr{F}_{\text{dif},n,i}$. The second part,
$v_{\text{d},n,i}\mathscr{F}_{\text{dif},n,j}$, represents the advection of
the $j$-diffusion flux in the $i$-direction. Its value is obtained by
averaging from the upwind side based on the sign of the advection velocity
$v_{\text{d},n,i}$. In addition, we note that in cylindrical/spherical
coordinates, we need to add extra diffusive geometric sources terms on the
momentum and energy equations (Skinner & Ostriker, 2010).
Finally, we compute the concentration diffusion momenta and compare to the
original concentration diffusion flux, from which we can estimate the
contribution from the
$\partial(\rho_{\text{d},n}\boldsymbol{v}_{\text{d,dif},n})/\partial t$ term.
The concentration diffusion momenta are stored in the cell center and are
averaged by the nearby face-centered concentration diffusion fluxes. We note
that although our formulation is Galilean invariant, it is not invariant to
machine precision at implementation level, but the incorporation of this term
is important to ensure approximate Galilean invariance in simulations.
### 2.5 Flow chart
Figure 1: Flow chart of a single integration stage of the multifluid dust
module in Athena++.
Figure 1 shows the flow chart of our multifluid dust module in Athena++, and
we summarize the main steps over one integration stage below.
Step 1: Backup the primitive variables for both gas and dust and calculate the
dust stopping time. The backed up primitive variables are used in the semi-
implicit and fully-implicit drag integrators to ensure higher-order accuracy
of the combined algorithm, as discussed in Section 2.3.3.
Step 2: Calculate the diffusion processes of gas and dust when applicable,
including viscosity, thermal conduction, and resistivity on the gas, and
concentration diffusion and momentum correction on dust fluids.
Step 3: Calculate the Riemann fluxes of both gas and dust, and integrate gas
and dust fluids by applying flux divergence. Send and receive flux corrections
when necessary for mesh refinement.
Step 4: Add explicit source terms on gas and dust, including geometric source
terms for curvilinear coordinates.
Step 5: Apply any of the drag integrators, and use the backed-up variables to
enhance the accuracy in implicit schemes.
Step 6: Do orbital advection when necessary (for disk problems).
Step 7: Send and receive boundary data, set boundary conditions, and do
prolongation/restriction for mesh refinement.
Step 8: Convert conserved variables to primitive variables. When the dust
momentum correction is turned on, the concentration diffusion flux calculated
by Step 2 will be subtracted from the dust momenta.
After finishing all stages of an integration cycle, we calculate the new time
step based on the Courant–Friedrichs–Lewy (CFL) condition for both gas and
dust. The dust CFL condition is set according to maximum dust velocity and
dust diffusion coefficient $D_{\text{d}}$ in the same way as gas velocity and
viscosity.
As a dust fluid module, it has fixed amount of floating point operations per
meshblock per integration cycle, as opposed to particle-based approaches.
Taking the advantage of the task-based execution model with excellent
scalability of Athena++, our dust fluid module primarily adds a fixed fraction
of computational cost. Such cost increases with $N_{d}$ non-linearly when
using higher-order fully implicit drag solvers due to the matrix inversion
whose cost scales as $O[(N_{d}+1)^{3}]$. In practice, we find that linear
scaling approximately applies for $N_{d}\lesssim 5$ and the cost of the drag
solver is no more than the cost from rest of the dust integration scheme for
$N_{d}\lesssim 10$. Further details about code performance are provided in
Appendix D.
## 3 Code Tests
In this section, we show benchmark numerical tests of our multifluid dust
module. They include the collisions between gas and dust, dust diffusion with
momentum correction, linear/non-linear streaming instability and
(static/adaptive) mesh refinement. We also follow the same dusty sound wave
and dusty shock tests in Section 3.2 and 3.3 of Benítez-Llambay et al. (2019).
To avoid repetitions, we show the test results of dusty sound wave and dusty
shock in Appendix F.1 and F.2. They demonstrate that our multi-fluid dust code
achieves full second-order accuracy when coupled with hydrodynamics, and it is
excellent at shock capturing.
### 3.1 Collisions
We start by conducting the 1D dust-gas collision test as a benchmark, similar
to Section 3.1 of Benítez-Llambay et al. (2019). We consider two dust species
with constant stopping time $T_{s,1},T_{s,2}$, and set three collision tests
named A, B and C. The gas and all dust species are homogeneous, each having
its own density ($\rho_{\text{g}},\rho_{\text{d},1},\rho_{\text{d},2}$) and
velocity ($v_{\text{g}},v_{\text{d},1},v_{\text{d},2}$). The system then
evolves under the mutual aerodynamic drag forces, characterized by two
eigenvalues $\lambda_{1}$, $\lambda_{2}$, in the form of
$v=v_{\text{COM}}+c_{1}\exp{(\lambda_{1}\;t)}+c_{2}\exp{(\lambda_{2}\;t)}\ ,$
(33)
where $v_{\text{COM}}$ is the center-of-mass velocity of the system. Their
initial conditions, as well as the associated coefficients and eigenvalues are
given by shown in Table 1, and we provide the calculation procedures in
Appendix E. The three tests are designed to test the non-stiff case (Test A),
the stiff case with small stopping time (Test B) and the stiff case with large
dust-to-gas ratios (Test C). These tests are conducted in 1D Cartesian
coordinates with a periodic boundary condition. We use the adiabatic equation
of state with the adiabatic index being $\gamma=1.4$ and an initial gas sound
speed is set as $c_{\text{s}}^{2}\equiv\gamma\frac{P}{\rho_{\text{g}}}=1.4$
for all three tests. We include the work and friction heating from drags in
the energy equation. We have tested eight drag integrators (Explicit:
“RK1-Explicit”, “RK2-Explicit”, “VL2-Explicit”; Semi-Implicit: “Trapezoid”,
“TrBDF2”; Fully-Implicit: “RK1-Implicit”, “RK2-Implicit” and “VL2-Implicit”),
and the main results are discussed below.
Table 1: The Initial Conditions, Eigenvalues and Coefficients of the Analytic
Solutions in the Collision Tests
Test | A | B | C
---|---|---|---
$\rho_{\text{g}}$ | 1 | 1 | 1
$v_{\text{g}}$ | 1 | 1 | 1
$\rho_{\text{d,1}}$ | 1 | 1 | 10
$v_{\text{d,1}}$ | 2 | 2 | 2
$T_{\text{s,1}}$ | 2 | 0.01 | 2
$\rho_{\text{d,2}}$ | 1 | 1 | 100
$v_{\text{d,2}}$ | 0.5 | 0.5 | 0.5
$T_{\text{s,2}}$ | 1 | 0.002 | 1
Coefficients1 | | |
$v_{\text{COM}}$ | 1.16666666666667 | 1.16666666666667 | 0.63963963963963
$\lambda_{1}$ | -0.63397459621556 | -141.742430504416 | -0.52370200744224
$\lambda_{2}$ | -2.36602540378444 | -1058.25756949558 | -105.976297992557
$c_{\text{g,1}}$ | -0.22767090063074 | -0.35610569612832 | -0.06458203330249
$c_{\text{g,2}}$ | 0.06100423396407 | 0.18943902946166 | 0.42494239366285
$c_{\text{d,1,1}}$ | 0.84967936855889 | 0.85310244713865 | 1.36237475791577
$c_{\text{d,1,2}}$ | -0.01634603522555 | -0.01976911380532 | -0.00201439755542
$c_{\text{d,2,1}}$ | -0.62200846792815 | -0.49699675101033 | -0.13559165545855
$c_{\text{d,2,2}}$ | -0.04465819873852 | -0.16966991565634 | -0.00404798418109
* 1
The analytic solutions of velocities of gas and dust are in the form of
Equation (33). The analytic solutions of gas energy can be obtained by
integrating the right hand side of the Equation (3) with $\omega=1$.
Figure 2: The top eight panels are the velocities and energy of gas and dust
in the collision Test B and C with the numerical time step $\Delta t=0.005$
and $\Delta t=0.05$. Solutions obtained by semi-implicit and fully-implicit
different drag integrators are shown. The black dashed lines represent the
analytic solutions. Note that explicit methods are unstable for our choice of
$\Delta t>1/|\lambda_{2}|$ (see Table 1), and the results are not shown. The
bottom three panels are the scaling of total relative error $\Delta
E_{\text{total}}$ with numerical time step $\Delta t$ for different drag
integrators (see legends) in the collision Tests A (left), B (middle) and C
(right).
The top eight panels of Figure 2 shows the temporal evolution of velocities
and energy of both gas and dust in Test B and C with numerical time steps
$\Delta t=0.005$ and $0.05$, respectively. The results are to be compared with
the analytic solution. Table 1 shows the largest eigenvalue
$|\lambda_{2}|\simeq 1058$ and $106$ for Test B and C. Therefore, the drag
terms become stiff when $\Delta t>1/|\lambda_{2}|\simeq 0.001$ and $0.01$ for
B and C. The explicit integrators are unstable in Test B and C with $\Delta
t=0.005$ and $0.05$. The semi-implicit methods are stable in the stiff drags,
however, the numerical updates oscillate around the analytic solutions
artificially, which is not unexpected as similar behavior was observed in Bai
& Stone (2010c). Our fully-implicit methods “RK1-Implicit”, “VL2-Implicit” and
“RK2-Implicit” handle these two stiff regimes (small stopping times and large
dust-to-gas ratios) very well, and the two second-order integrators are
clearly seen to be more accurate.
To test numerical convergence, we calculate the relative error $\Delta E$ as a
function of the numerical time step $\Delta t$ with different drag
integrators. The $\Delta E$ is calculate by:
$\Delta E\left(\Delta
t\right)=\frac{1}{t_{\text{max}}-t_{\text{min}}}\sum\frac{|U_{\text{num}}\left(\Delta
t\right)-U_{\text{ana}}|}{U_{\text{ana}}}\Delta t$ (34)
where $U$ represents momentum or the gas energy, subscripts “num” and “ana”
represent the numerical and analytic solutions, respectively. The scaling of
the total relative error $\Delta E_{\text{total}}=\Delta E_{\text{mom}}+\Delta
E_{\text{erg}}$ with different time steps $\Delta t$ is shown in the bottom
three panels of Figure 2 for different drag integrators. We vary $\Delta t$
from $10^{-4}$ to $10^{-1}$ which are non-stiff for Test A, but get
increasingly stiff for Test B and C. The total relative errors are calculated
between $t_{\text{min}}=0$ and $t_{\text{max}}=10$.
We see that in the non-stiff regime of Test A, all drag integrators achieve
first or second-order accuracy in time as desired, and there is no significant
difference between integrators of the same order. In the stiff regime of Test
B and C, we see that the error in explicit and semi-implicit integrators
diverge when $\Delta t\gtrsim(1/|\lambda_{2}|)$. The threshold for error
divergence is only slightly higher for semi-implicit integrators. The fully-
implicit integrators, on the other hand, achieve the desired level of accuracy
at small $\Delta t$, while remain stable at large $\Delta t$ for both tests.
Among them, the 2nd-order fully-implicit integrators show error levels that
are at least one order of magnitude smaller, and are the preferred choices
that we generally recommend.
Our drag integrators conserve total momentum of the dust-gas system by
construction. While not shown in the figures, we have verified that in all
three tests, total momentum is conserved to the fractional level of $\lesssim
10^{-14}$ (i.e., approaching machine precision) within the duration of the
simulations.
### 3.2 Momentum Correction in Dust Diffusion
Novel additions in our dust concentration diffusion formulation include the
time derivative of the $\rho_{\text{d}}\boldsymbol{v}_{\text{d,dif}}$ term and
the $\boldsymbol{\Pi}_{\text{d,dif}}$ in the dust momentum equation (5). We
refer to them as “momentum correction”. They reflect the fact that the
concentration diffusion flux carries momentum that back-react to the gas, and
that the new formulation is Galilean invariant. To test these aspects of our
implementation, we consider the following simple test problems without/with
momentum correction in 1D, 1.5D and 2D Cartesian coordinates, where $1.5D$
means 1D test in the presence of a transverse velocity.
#### 3.2.1 Initial Setups
We give the parameters of our tests in Table 2, with more details below. The
1D and 1.5D tests are carried in Cartesian coordinates, with 256 uniform
grids, covering $x\in[0,20]$. The gas has uniform initial density
$\rho_{\text{g0}}=1$ and a single dust species with constant stopping time
$T_{\text{s}}=10^{-2}$ is included with an initial Gaussian density profile:
$\rho_{\text{d0}}=A\exp{\left[-\frac{\left(x-x_{0}\right)^{2}}{2\sigma_{x}^{2}}\right]}+\rho_{\text{g0}}\
.$ (35)
where $A=5$, $x_{0}=10$ and $\sigma_{x}=2$ in both cases. We include gas
viscosity and dust diffusion with coefficients $\nu=D_{\text{d}}=1$. In the 1D
test, initial gas and dust velocities are zero, whereas in the 1.5D test, gas
and dust have constant transverse velocities
$v_{\text{g},y}=v_{\text{d},y}=1$.
In the 2D test, we use 2D Cartesian coordinates in the domain $x,y\in[0,20]$
with $256^{2}$ cells, and set the initial 2D Gaussian dust density profile in
the center of the box:
$\rho_{\text{d0,2D}}=A\exp{\left[-\frac{\left(x-x_{0}\right)^{2}}{2\sigma_{x}^{2}}-\frac{\left(y-y_{0}\right)^{2}}{2\sigma_{y}^{2}}\right]}+\rho_{\text{g0}}\
.$ (36)
where $y_{0}=10$ and $\sigma_{y}=2$, and the rest of parameters are same as in
the 1.5D test.
We note that when including momentum correction, there is an additional
contribution $v_{\text{d,dif}}$ to the conserved dust momentum. This leads to
two possible initial settings. One is to make the initial conserved momentum
to be zero. By the conversion relation (31), the primitive velocity thus
equals to $-v_{\text{d,dif}}$. Alternatively, one can choose the primitive
velocity to be zero (i.e., zero mean dust velocity), so that the conserved
momentum becomes $\rho v_{\text{d,dif}}$ (i.e., non-zero mean dust momentum).
This ambiguity reflects that the initial condition itself is physically
unrealistic to build up without involving additional source terms. As a test
problem, we choose the latter, which we consider to be physically more natural
and intuitive (the alternative choice would lead to different interpretable
outcomes that we omit here for brevity). Note that without momentum
correction, we set $v_{\text{d}}=0$ so that the two setups share the same
initial condition in primitive variables.
Table 2: Problem setup for 1D, 1.5D and 2D dust diffusion tests | Correction555The correction terms by default include both $\boldsymbol{\Pi}_{\text{dif}}$ and $\partial{(\rho_{\text{d}}\boldsymbol{v}_{\text{d,dif}})}/\partial{t}$. | $v_{\text{g},x}$ | $v_{\text{g},y}$ | $v_{\text{d},x}$ | $v_{\text{d},y}$
---|---|---|---|---|---
1D | No | 0 | 0 | 0 | 0
Yes | 0 | 0 | $v_{\text{d,dif},x}$666The diffusion velocity $v_{\text{d,dif}}$ is calculated by Equation (8). | 0
1.5D | No | 0 | 1 | 0 | 1
Yes | 0 | 1 | $v_{\text{d,dif},x}$ | 1
2D | No | 1 | 1 | 1 | 1
Yes777In the 2D cases, we have two corrections tests, one is with only $\boldsymbol{\Pi}_{\text{dif}}$, and the other is with both $\boldsymbol{\Pi}_{\text{dif}}$ and $\partial{(\rho_{\text{d}}\boldsymbol{v}_{\text{d,dif}})}/\partial{t}$. | 1 | 1 | 1+$v_{\text{d,dif},x,\text{2D}}$ | 1+$v_{\text{d,dif},y,\text{2D}}$
In all these tests, dust back-reaction is included. We use an isothermal
equation of state with sound speed $c_{\text{s,iso}}=1$, applying periodic
boundary conditions. We use the Piecewise Linear Method (PLM) spatial
reconstruction for both gas and dust, and a CFL number of 0.3. The
“VL2-Implicit” drag integrator is used to calculate the mutual drags.
#### 3.2.2 Results
Figure 3: 1D (top six panels) and 1.5D (bottom six panels) dust diffusion
tests without (“No Correction, NoCo”, dashed lines)/with (“Correction, Co”,
solid lines) momentum corrections. The first and the third rows show gas
density ($\rho_{\text{g}}$) and velocities ($v_{\text{g},x},\;v_{\text{g},y}$)
at different times ($t=0.0,\;1.0,\;3.0$ and $5.0$). Note that the gas density
and $x$-velocity in the first and third panels are identically one and zero in
the “NoCo” cases with these dashed lines embedded below the black solid line.
The second and the fourth rows show dust concentration
($\rho_{\text{d}}/\rho_{\text{g}}$) and dust velocities
($v_{\text{d},x},\;v_{\text{d},y}$) at different times. Similarly, the dust
$x$-velocity in the “NoCo” case is zero with the corresponding dashed lines
embedded below the black solid line. Figure 4: 2D dust diffusion tests without
correction (top eight panels), with only $\boldsymbol{\Pi}_{\text{dif}}$
correction (middle eight panels), and with both
$\boldsymbol{\Pi}_{\text{dif}}$ and
$\partial(\rho_{\text{d}}\boldsymbol{v}_{\text{d,dif}})/\partial t$
corrections (bottom eight panels). From left to right, the panels are at time
$t=0.0,\;1.0,\;3.0$ and $5.0$. The first, the third and the fifth rows are for
gas density ($\rho_{\text{g}}$), while the second, the fourth and the sixth
rows are for dust concentration ($\rho_{\text{d}}/\rho_{\text{g}}$). The black
(white) lines represent the velocity streamlines of gas (dust).
In the 1D tests shown in the top eight panels of Figure 3, we note that the
correction term $\boldsymbol{\Pi}_{\text{dif}}$ is zero, thus only the time
derivative term
$\frac{\partial{(\rho_{\text{d}}\boldsymbol{v}_{\text{d,dif}})}}{\partial{t}}$
is effective. When the correction is not included, dust proceeds as normal
concentration diffusion, whereas the gas is totally intact. This is unphysical
because turbulent mixing is a two-way process that not only mixes dust with
gas, but should also mix gas with dust. When the momentum correction is
included, we see that gas density exhibits a central deficit and two outside
bumps. This is essentially the outcome of gas being dragged by the outward
diffusion flow of dust, as can be seen in the central region of the middle
panel. The additional structures in the gas act to slow down dust
concentration diffusion, and we can see that without incorporating momentum
correction, dust diffuses more than 2 times more rapidly.
When adding a transverse velocity in the 1.5D test, one should expect
identical results as in 1D except for a velocity shift in the $y$-direction.
However, without momentum correction, we see in the two right panels in the
bottom of Figure 3 that the system develops artificial variations in $v_{y}$.
This is because concentration diffusion changes the dust density profile, but
without properly altering the momentum profile. When momentum corrections are
included, we see that dust and gas momenta are properly advected to ensure
Galilean invariance.
In the 2D tests shown in Figure 4, similar to the 1D case, incorporating
momentum correction leads to the physical behavior where the gas density
develops a dip in the center surrounded by an outside ring, which would be
absent without the correction. To test Galilean invariance, we add a
background advection velocity along the diagonal direction. We run two cases
with momentum correction, one includes only the
$\boldsymbol{\Pi}_{\text{dif}}$ term, and the other includes both the
$\boldsymbol{\Pi}_{\text{dif}}$ and
$\partial(\rho_{\text{d}}\boldsymbol{v}_{\text{d,dif}})/\partial t$ terms. We
see that the circular gas and dust density patterns are well preserved when
both terms are included, whereas artificial density structures show up without
the correction or when including only the $\Pi_{\text{dif}}$ term. These
artificial density patterns are due to artifacts in velocity variations
resulting from the violation of Galilean invariance. Therefore, both
correction terms are essential to ensure Galilean invariance. Overall, these
tests demonstrate the importance of properly incorporating the momentum
correction terms to for a consistent treatment of dust concentration
diffusion, especially when dust feedback is taken into account.
### 3.3 Streaming Instability
The streaming instability (SI) is a stringent test for two-way gas drag, and
its nonlinear evolution with strong dust clumping represents a further test of
code capability to handle sharp discontinuities. We adopt the linear tests
given by Youdin & Johansen (2007) and non-linear runs from Johansen & Youdin
(2007), which has become the standard test problem for codes with particle-
based treatment of dust (Bai & Stone, 2010c), and more recently for multifluid
dust as well (Benítez-Llambay et al., 2019). We will further extend the non-
linear tests to cylindrical coordinates (Section 3.3.4), and to incorporate
mesh refinement (Section 3.4.1).
#### 3.3.1 Shearing Box Equations and Equilibrium State
Most of our SI tests are carried out in the local shearing box framework,
which follows a local patch of a disk at some fiducial radius $r_{0}$ in the
corotating frame with orbital frequency $\Omega_{0}=\Omega\left(r_{0}\right)$
(Goldreich & Lynden-Bell, 1965; Hawley et al., 1995). The equations of gas and
dust are written in a Cartesian coordinate system $\left(x,y,z\right)$ for
radial, azimuthal and vertical directions with Coriolis and centrifugal source
terms (Stone & Gardiner, 2010). The unit vectors along this three directions
are denoted as $\left(\hat{i},\hat{j},\hat{k}\right)$, respectively. We do not
consider viscosity, diffusion, magnetic field and self-/vertical gravity.
Adopting an isothermal equation of state with isothermal sound speed
$c_{\text{s}}$, the continuity and momentum equations of gas and dust are:
$\displaystyle\frac{\partial\rho_{\text{g}}}{\partial
t}+\nabla\cdot\left(\rho_{\text{g}}\boldsymbol{v}_{\text{g}}\right)=0\ ,$ (37)
$\displaystyle\frac{\partial\left(\rho_{\text{g}}\boldsymbol{v}_{\text{g}}\right)}{\partial
t}+\nabla\cdot\left(\rho_{\text{g}}\boldsymbol{v}_{\text{g}}\boldsymbol{v}_{\text{g}}+P_{\text{g}}\mathsf{I}\right)=$
(38) $\displaystyle 2\rho_{\text{g}}q\Omega_{0}^{2}x\hat{\mathbf{i}}$
$\displaystyle-2\rho_{\text{g}}\Omega_{0}\hat{\mathbf{k}}\times\boldsymbol{v}_{\text{g}}-\sum^{n}_{k=1}\rho_{\text{d},k}\frac{\boldsymbol{v}_{\text{g}}-\boldsymbol{v}_{\text{d},k}}{T_{\text{s},k}}\
,$
$\displaystyle\frac{\partial\rho_{\text{d},k}}{\partial
t}+\nabla\cdot\left(\rho_{\text{d},k}\boldsymbol{v}_{\text{d},k}\right)=0\ ,$
(39)
$\displaystyle\frac{\partial\left(\rho_{\text{d},k}\boldsymbol{v}_{\text{d},k}\right)}{\partial
t}+\nabla\cdot$
$\displaystyle\left(\rho_{\text{d},k}\boldsymbol{v}_{\text{d},k}\boldsymbol{v}_{\text{d},k}\right)=$
(40) $\displaystyle
2\rho_{\text{d},k}q\Omega_{0}^{2}x\hat{\mathbf{i}}-2\rho_{\text{d},k}\Omega_{0}\hat{\mathbf{k}}$
$\displaystyle\times\boldsymbol{v}_{\text{d},k}+\rho_{\text{d},k}\frac{\boldsymbol{v}_{\text{g}}-\boldsymbol{v}_{\text{d},k}}{T_{\text{s},k}}\
.$
where $q=-d\ln{\Omega}/dr$ is the shear rate and $q=3/2$ for Keplerian disks
as we adopt.
Ignoring vertical gravity (i.e., unstratified disk), dust and gas can achieve
the so-called Nakagawa-Sekiya-Hayashi (NSH) equilibrium Nakagawa et al.
(1986). The equilibrium is a force balance between background pressure
gradient, centrifugal force, Coriolis force and mutual aerodynamic drags
between gas and dust in the horizontal plane. Here we consider a Keplerian
disk with angular speed $\Omega_{0}=\Omega_{\text{K}}$. In the absent of dust,
gas rotates slower than the Keplerian speed by a small amount $\eta
v_{K}\equiv\Pi c_{\text{s}}$ due to the background pressure gradient, and
$\Pi$ is $\lesssim 0.1$ under typical disk conditions. $\eta$ represents the
strength of radial gas pressure gradient:
$\eta\equiv\frac{1}{2}\frac{\ln P_{\text{g}}}{\ln
r}\left(\frac{h}{r}\right)^{2}=\frac{1}{2}\frac{\ln P_{\text{g}}}{\ln
r}\left(\frac{c_{\text{s}}}{v_{\text{k}}}\right)^{2}\ ,$ (41)
The original NSH solution considered a single dust species. Generalized to
multiple dust species with different stopping times, the velocities of gas and
dust in the multi-species equilibrium are given by (Benítez-Llambay et al.,
2019), and also see Tanaka et al. (2005):
$\displaystyle v_{\text{g0},x}$ $\displaystyle=2\eta
v_{\text{K}}\frac{\mathcal{A}}{\mathcal{A}+\mathcal{B}}\ ,\quad
v_{\text{d0},k,x}=\frac{v_{\text{g0},x}+2St_{k}v^{\prime}_{\text{g0},y}}{1+St_{k}^{2}}\
,$ (42) $\displaystyle v^{\prime}_{\text{g0},y}$ $\displaystyle=-\eta
v_{\text{K}}\frac{\mathcal{B}}{\mathcal{A}+\mathcal{B}}\ ,\quad
v^{\prime}_{\text{d0},k,y}=\frac{v^{\prime}_{\text{g0},y}-St_{k}v_{\text{g0},x}/2}{1+St_{k}^{2}}\
,$
where $St_{\text{k}}\equiv\Omega_{\text{K}}T_{\text{s},k}$ is the
dimensionless stopping time of k-th dust species, the $y-$ velocities with a
prime have Keplerian shear subtracted
$v^{\prime}_{\text{g}0,y}=v_{\text{g}0,y}+(3/2)\Omega_{K}x$,
$v^{\prime}_{\text{d}0,k,y}=v_{\text{d}0,k,y}+(3/2)\Omega_{K}x$, and
$\mathcal{A}=\sum_{k=1}^{n}\frac{\epsilon_{k}St_{k}}{1+St_{k}^{2}}\
,\quad\mathcal{B}=1+\sum_{k=1}^{n}\frac{\epsilon_{k}}{1+St_{k}^{2}}\ .$ (43)
In our numerical setup, we add an additional outward force
$\boldsymbol{f}\equiv 2\rho_{\text{g}}\eta v_{\text{K}}\Omega_{0}\hat{i}$ on
the gas to mimic the radial pressure gradient. Note that it differs from Bai &
Stone (2010c) and Benítez-Llambay et al. (2019), who add this force on the
dust component. The two approaches are equivalent except for a constant
velocity shift. Realize the exact analytic solution of the multi-species NSH
equilibrium is straightforward when using explicit integrators, and our
numerical implementation in Section 2.3.3 ensures that such exact solution can
be realized using the semi-implicit and implicit drag integrators as well.
#### 3.3.2 Linear SI Modes and Growth Rates
The NSH equilibrium is subject to the SI (Youdin & Goodman, 2005). The growth
rate $s$ of the SI is a function of initial dust-to-gas ratio (or metallicity)
$\epsilon_{0}\equiv\rho_{\text{d,0}}/\rho_{\text{g,0}}$, dimensionless
stopping time $St$ and two dimensionless wavenumbers $K_{x}\equiv k_{x}\eta
r_{0}=k_{x}\eta c_{\text{s}}/\Omega_{\text{K}}$ and $K_{z}\equiv k_{z}\eta
r_{0}=k_{z}\eta c_{\text{s}}/\Omega_{\text{K}}$:
$s=s(\epsilon_{0},St,K_{x},K_{z})$. In this work, we choose the Lin-A and
Lin-B tests in Table 1 of Youdin & Johansen (2007), which consist of gas and
one dust species, and the Lin-3 test in Section 3.5 of Benítez-Llambay et al.
(2019), which consists of gas and two dust species.
The numerical setups of the linear tests are similar to those of Youdin &
Johansen (2007), Bai & Stone (2010c) and Benítez-Llambay et al. (2019). The
numerical domain is a square box along $x$ and $z$ directions with
$-L_{x}/2\leq x\leq L_{x}/2$, $-L_{z}/2\leq z\leq L_{z}/2$, and
$L_{x}=L_{z}=1$. We choose $\Omega_{0}=1.0$ and $\eta
v_{\text{K}}=0.05\;c_{\text{s}}$. On top of the multi-species NSH equilibrium
from Equation (42), we add perturbations on densities and velocities of both
gas and dust of the following form, following Youdin & Johansen (2007):
$\displaystyle\delta\rho$
$\displaystyle=[\Re(\tilde{\rho})\cos\phi-\Im(\tilde{\rho})\sin\phi]\cos(k_{z}z)\
,$ (44) $\displaystyle\delta v_{x}$
$\displaystyle=[\Re(\tilde{v}_{x})\cos\phi-\Im(\tilde{v}_{x})\sin\phi]\cos(k_{z}z)\
,$ $\displaystyle\delta v_{y}$
$\displaystyle=[\Re(\tilde{v}_{y})\cos\phi-\Im(\tilde{v}_{y})\sin\phi]\cos(k_{z}z)\
,$ $\displaystyle\delta v_{z}$
$\displaystyle=[\Re(\tilde{v}_{z})\sin\phi+\Im(\tilde{v}_{z})\cos\phi]\sin(k_{z}z)\
.$
where $\phi\equiv k_{x}x-\omega t$, and density and velocity perturbations,
denoted by $\tilde{\rho}$, $\tilde{\boldsymbol{v}}$ for gas and individual
dust species, are given by the respective eigenvectors of the specific linear
modes, which are listed in Table 1 of Youdin & Johansen (2007) and Table 4 of
Benítez-Llambay et al. (2019).
In these tests, we fit one eigenmode in the simulation box. As our box size is
fixed, the sound speed $c_{\text{s}}$ is no longer a free parameter and it
depends on the dimensionless wavenumbers ($K_{x}$ and $K_{z}$). Because of
$K_{z}=K_{x}=k_{x}\eta r_{0}=2\pi\times 0.05c_{\text{s}}/\Omega_{0}$, we
obtain $c_{\text{s}}=K_{x}\Omega_{0}/(0.05\times 2\pi)$. In the Lin-A test,
$K_{x}=K_{z}=30$, thus we have $c_{\text{s,A}}=95.49296585$. Similar setup can
be done in Lin-B and Lin-3 tests, with $c_{\text{s,B}}=19.09859317$ in Lin-B
and $c_{\text{s,3}}=159.1549431$ in Lin-3. We use all 2nd-order drag
integrators, with PLM and Piecewise Parabolic Method (PPM) for spatial
reconstructions on both gas and dust, and the HLLE Riemann solver. Numerical
resolution spans from 8 to 256 cells per wavelength in all each directions,
with the CFL number being 0.3.
Figure 5: Fitted averaging growth rates (normalized as $s/\Omega_{0}$) by
measuring the mean kinetic energy $E_{\text{kinetic}}$ of gas and dust in
Lin-A, Lin-B and Lin-3 tests of SI. Results are shown as a function of the
number of grids per wavelength ($N/\lambda$). The analytic growth rates are
marked with blacked dotted lines. Different drag integrators are marked with
different colors. The solid (dashed) lines represent the runs with PPM (PLM)
spatial reconstruction.
We measure the growth rate of the kinetic energy $E_{\text{kinetic}}$ of both
gas and dust in the linear tests. Because the initial kinetic energy is
dominated by radial drift, we expect the temporal variation of kinetic energy
to grow as $\delta E_{\text{kinetic}}\propto\exp(s\;t)$. We fit the spatial
standard deviation $<\delta E_{\text{kinetic}}^{2}>^{\frac{1}{2}}$ over the
whole mesh as a function of time, and show the fitted growth rate
$s/\Omega_{0}$ in Figure 5 as a function of resolution in Lin-A, Lin-B and
Lin-3 tests. As these test problems are non-stiff, different drag integrators
generally show similar results. With PPM reconstruction, 16 cells per
wavelength is generally sufficient to accurately capture the growth in all
three tests. When using the PLM reconstructions, on the other hand, about
128-256 cells per wavelength is needed for similar accuracy, and the
requirement for Lin-B is the most stringent. When compared with the measured
growth rates in Table 5 of Benítez-Llambay et al. (2019), it appears that we
need more grid points to achieve similar accuracy in FARGO3D. On the other
hand, the level of accuracy that we achieve is similar to those obtained in
other finite volume method code, Athena (Bai & Stone, 2010c) and PLUTO
(Mignone et al., 2019), with PPM spatial reconstructions. We thus attribute
the difference primarily to the different nature of the base code. On the
other hand, we will show that our code show similar outcomes in the nonlinear
regime at a given resolution.
#### 3.3.3 The SI in the Non-Linear Regime
Figure 6: Dust densities of AB test (top) and BA test (bottom) with different
spatial resolutions at $t=40\;\Omega_{0}^{-1}$ for AB test and
$t=800\;\Omega_{0}^{-1}$ for BA test. Figure 7: The dust CDFs of AB test
(top) and BA test (bottom), similar to Figure 10 of Benítez-Llambay et al.
(2019). The left panels are the CDFs calculated by counting the number of
cells whose dust density exceeds certain threshold. The right panels are the
CDFs calculated by additional weighting by dust density. The color shaded
regions are the temporal standard deviations based on many snapshots (from
$30\;\Omega_{0}^{-1}$ to $40\;\Omega_{0}^{-1}$ for AB test, and from
$600\;\Omega_{0}^{-1}$ to $800\;\Omega_{0}^{-1}$ for BA test). The initial
dust densities are $\rho_{\text{d},0}=1.0$ and $\rho_{\text{d},0}=0.2$ for AB
and BA test, respectively. Table 3: Turbulence Properties of AB and BA test
with different resolutions
Run | $Ma_{x}$ | $Ma_{y}$ | $Ma_{z}$ | $\mathscr{Re}/\mathscr{Re}_{\text{NSH}}$ | $v_{\text{d,drift}}/v_{\text{d,drift,NSH}}$
---|---|---|---|---|---
AB-$128^{2}$ | $1.39(06)\times 10^{-2}$ | $8.97(55)\times 10^{-3}$ | $1.18(08)\times 10^{-2}$ | $2.15(22)$ | $1.74(07)$
AB-$256^{2}$ | $1.46(05)\times 10^{-2}$ | $9.30(49)\times 10^{-3}$ | $1.10(03)\times 10^{-2}$ | $2.56(10)$ | $2.03(08)$
AB-$512^{2}$ | $1.37(05)\times 10^{-2}$ | $7.32(39)\times 10^{-3}$ | $1.07(06)\times 10^{-2}$ | $2.66(12)$ | $2.19(07)$
AB-$1024^{2}$ | $1.24(02)\times 10^{-2}$ | $6.01(24)\times 10^{-3}$ | $8.95(28)\times 10^{-3}$ | $2.55(07)$ | $2.15(05)$
AB-$2048^{2}$ | $1.10(02)\times 10^{-2}$ | $4.96(17)\times 10^{-3}$ | $7.61(15)\times 10^{-3}$ | $2.38(02)$ | $2.07(02)$
AB-$4096^{2}$ | $1.07(01)\times 10^{-2}$ | $4.93(14)\times 10^{-3}$ | $7.26(14)\times 10^{-3}$ | $2.34(04)$ | $2.03(02)$
BA-$64^{2}$ | $1.03(15)\times 10^{-2}$ | $1.69(14)\times 10^{-2}$ | $3.92(33)\times 10^{-2}$ | $0.74(08)$ | $0.74(08)$
BA-$128^{2}$ | $1.46(23)\times 10^{-2}$ | $2.03(20)\times 10^{-2}$ | $4.92(26)\times 10^{-2}$ | $0.61(07)$ | $0.63(06)$
BA-$256^{2}$ | $1.78(11)\times 10^{-2}$ | $2.09(15)\times 10^{-2}$ | $4.91(45)\times 10^{-2}$ | $0.54(08)$ | $0.58(07)$
BA-$512^{2}$ | $1.59(21)\times 10^{-2}$ | $2.08(14)\times 10^{-2}$ | $5.02(16)\times 10^{-2}$ | $0.62(06)$ | $0.65(05)$
BA-$1024^{2}$ | $1.57(15)\times 10^{-2}$ | $2.07(07)\times 10^{-2}$ | $4.83(13)\times 10^{-2}$ | $0.58(04)$ | $0.62(04)$
BA-$2048^{2}$ | $1.61(15)\times 10^{-2}$ | $2.12(20)\times 10^{-2}$ | $4.95(24)\times 10^{-2}$ | $0.61(05)$ | $0.64(05)$
* 1
The number in parenthesis quotes the $1\sigma$ uncertainty of the last two
digits.
The non-linear SI runs are also carried in the 2D shearing box in $x-z$. We
select two non-linear tests of the SI, namely, AB and BA test from Johansen &
Youdin (2007). In AB test, the domain is $-\eta r_{0}\leq x\leq\eta r_{0}$,
$-\eta r_{0}\leq z\leq\eta r_{0}$, with parameters being $\epsilon_{0}=1.0$,
$St=0.1$. In BA test, the domain is $-20\;\eta r_{0}\leq x\leq 20\;\eta
r_{0}$, $-20\;\eta r_{0}\leq z\leq 20\;\eta r_{0}$, with parameters being
$\epsilon_{0}=0.2$ and $St=1.0$. The parameters and simulation setups are
similar to Table 1 of Johansen & Youdin (2007), Section 5 of Bai & Stone
(2010c) and Section 3.5.6 of Benítez-Llambay et al. (2019). We use the
“VL2-Implicit” drag integrator, the Roe Riemann solver, with PPM
reconstruction for the gas, and PLM reconstructions for the dust, in order to
more robustly follow the dramatic variation of dust density over space in the
non-linear stage with strong particle clumping. We use an isothermal equation
of state with sound speed is $c_{\text{s}}=1.0$, and the initial gas density
is $\rho_{\text{g}}=1.0$. We also set a density floor
$\rho_{\text{d,floor}}=10^{-6}$ on dust. Gas viscosity and dust diffusion are
not included. The simulations are initiated from the NSH equilibrium, on top
of which we add white-noise velocity perturbations with an amplitude of
$<A>\sim 0.02c_{\text{s}}$ on both gas and dust.
The saturated state of AB and BA tests, following the evolution of
$40\;\Omega_{0}^{-1}$ and $800\;\Omega_{0}^{-1}$ are shown in Figure 6, for
simulations with different resolutions. They are to be compared with with
Figure 8 and 9 of Benítez-Llambay et al. (2019). AB test is characterized by
the development of thin filaments and cavitation towards smaller scales at
higher resolution, well consistent with results in previous works (Johansen &
Youdin, 2007; Bai & Stone, 2010c; Benítez-Llambay et al., 2019). In BA test,
the system develops long dusty stripes and valleys nearly aligned with the $z$
direction and tilted towards the $x$ direction. Different from AB test, these
general features are similar at all resolutions, again consistent with
previous studies.
We further investigate the convergence of dust clumping by calculating the
cumulative dust density distributions (CDFs). Following the same procedures
described in Section 3.5.6 of Benítez-Llambay et al. (2019), there are two
ways of calculating the CDFs. One is based on the probability that local dust
density exceeds certain threshold, obtained by counting the number of cells
whose dust density exceeds the threshold. The other reflects the probability
where a particle resides in regions whose particle density exceeds a certain
threshold, obtained by weighting the first probability by local dust density.
We refer to the two CDFs as obtained by counting cell numbers, and counting
dust density, respectively. The overall results are shown in Figure 7.
The CDFs obtained by counting cell numbers are very similar to those obtained
by Benítez-Llambay et al. (2019) for both AB and BA. The CDFs of AB test
systematically vary with resolution at both the low-density and high-density
ends, in line with the non-convergent behaviors revealed in Figure 6. The CDFs
of BA test show convergence at the low-density end up to $P\sim 10^{-3}$,
while show more significant clumping at higher resolution (BA-$1024^{2}$ and
BA-$2048^{2}$).
Our CDFs by counting dust density, on the other hand, show more clumping than
that in FARGO3D. The clumping is also more significant than that obtained in
the particle module of Athena (see Figure 6 of Bai & Stone, 2010c). This might
be related to the higher-order drag integrators adopted here compared to
FARGO3D. We also note that in the original Athena code, some artificial
reduction of dust feedback was applied in strong dust clumps to alleviate the
stiffness in the system that is circumvented in our approach. We thus leave
this as an open issue. We also anticipate that our dust fluid module generally
finds most of applications in regimes with $St<1$, instead of $St\gtrsim 1$ as
in BA test.
We also examine the properties of gas turbulence triggered by the SI. We
calculate the turbulent Mach numbers along 3 directions ($x$, $y$ and $z$),
the Reynolds stress and mean radial drift velocities of dust fluids. The mach
number is calculated by
$Ma\equiv\sqrt{\langle(v_{\text{g}}-\overline{v_{\text{g}}})^{2}\rangle}/c_{\text{s}}$.
The Reynolds stress is calculated by:
$\mathscr{Re}\equiv\langle\rho_{\text{g}}v_{\text{g},x}(v_{\text{g},y}-v_{\text{K}})\rangle$.
The mean radial drift velocity is computed by dividing the mean dust momentum
over mean density. These properties are all spatially and time averaged,
indicated by angle brackets, based on many snapshots ($30\;\Omega_{0}^{-1}\sim
40\;\Omega_{0}^{-1}$ in AB test, $600\;\Omega_{0}^{-1}\sim
800\;\Omega_{0}^{-1}$ in BA test), and the results are shown in Table 3. These
diagnostic quantities are in broad agreement with the values obtained in
Johansen & Youdin (2007); Bai & Stone (2010c), all saturated into highly
subsonic, anisotropic turbulence, with enhanced radial drift and Reynolds
stress in AB test and reduced radial drift and Reynolds stress in BA test.
#### 3.3.4 Global Curvilinear Run of BA test
In order to test our multifluid module in curvilinear coordinates, we run a
global unstratified SI test in cylindrical coordinates $(r,\phi,z)$, similar
to earlier investigations (Kowalik et al., 2013; Mignone et al., 2019). We
choose to adopt the parameters close to BA test here, which is less demanding
in resolution and has better convergence properties. The computational domains
in three directions are $r\in[0.2,2.6]$, $\phi\in[0,2\pi]$ and
$z\in[-0.15,0.15]$. The numerical resolutions are $4096\times 4\times 512$
cells along $r$, $\phi$ and $z$, where we use a reduced $\phi-$resolution to
preserve axisymmetry.
Figure 8: Different snapshots of dust densities (scaled by $r^{-0.5}$) in the
global unstratified BA test in the cylindrical coordinate. From top to bottom,
the panels are at $t=100\;\Omega_{0}^{-1}$, $200\;\Omega_{0}^{-1}$,
$400\;\Omega_{0}^{-1}$ and $800\;\Omega_{0}^{-1}$, where $\Omega_{0}$ is the
orbital frequency at $r_{0}\equiv 1$. The strength of gas pressure gradient is
$\eta=0.0075$. The top and right axis are scaled by $\eta r_{0}$ for
reference. The effective resolution is the same as the BA-$512^{2}$ in Figure
6.
We set the central star mass $GM=1$, and set the gas radial density profile to
be $\rho_{\text{g}}(r)=\rho_{0}(r/r_{0})^{-0.5}$ with $\rho_{0}\equiv 1$ at
$r_{0}\equiv 1$. We adopt a vertically-isothermal equation of state with
$P(r)=c_{\text{s}}(r)^{2}\rho_{\text{g}}(r)$. The sound speed is chosen so
that the disk aspect ratio is $h/r=c_{s}/v_{K}=0.1$ at all radii, which gives
$c_{\text{s}}=0.1(r/r_{0})^{-0.5}$. From Equation (41), we obtain
$\eta=0.0075$ ($\Pi=0.075$). The effective resolutions along $r$ and $z$
directions are $12.8/\eta r_{0}$, same as BA-$512^{2}$. The initial dust
density follows that of the gas with a uniform dust-to-gas density ratio
$\epsilon_{0}=0.2$. The Stokes number of dust is
$St=T_{\text{s}}\Omega_{\text{K}}=1.0$. The initial velocities of both gas and
dust are set by the NSH equilibrium. Periodic boundary conditions are applied
along $\phi$ and $z$ directions. The radial boundary condition is fixed by the
NSH solution. To minimize of the unphysical wave reflection, we apply wave
damping zones near the inner and outer radial boundaries (de Val-Borro et al.,
2006, 2007), located at $0.20\leq r\leq 0.32$ and $2.44\leq r\leq 2.60$, where
gas and dust density and velocities (represented by $x$) are relaxed according
to:
$\frac{\mathrm{d}x}{\mathrm{~{}d}t}=-\frac{x-x_{\text{init}}}{\tau}R(r)\ ,$
(45)
where $x_{\text{init}}$ is the initial value and $\tau$ is damping rate. We
adopt $\tau=1$ in our simulation, and $R(r)$ is a smoothing parabolic function
that transiting from $0$ in the active zones to $1$ in the ghost zones. We
also use the orbital advection algorithm (Masset, 2000, 2002, FARGO algorithm)
to reduce truncation error. At the beginning, we add small velocity
perturbations in white noise with an amplitude of $0.02c_{\text{s}}$ to seed
the instability.
Figure 8 shows different snapshots of dust density in the global simulations
of this global BA test. We can clearly see the progressive development of the
SI from small radii to large radii, as the inner region has shorter dynamical
time. The simulation reaches saturated state over the entire domain after
about $t=600\;\Omega_{0}^{-1}$, where $\Omega_{0}=\Omega_{\text{K}}$ at
$r=r_{0}$, and we see the characteristic long dusty stripes and valleys with
the maximum of $\rho_{\text{d}}r^{0.5}$ significantly amplified by a factor of
$\gtrsim 100$.
More quantitatively, we have also examined the dust-to-gas ratio $\epsilon$
and the dust CDFs of the global run, and compared them with those from the
local BA-$512^{2}$ shown in Figure 9. The maximum $\epsilon$ in the global
simulation is higher than that in the local simulation by a small margin. Note
that due to the reduction of mean radial drift speed in BA test as the SI
saturates, and that the SI develops faster in the inner region than the outer
region, the mean dust-to-gas ratio $\epsilon$ in the global run increases with
time and reaches about $0.3$ (rather than the initial value of $0.2$) at
$t=800\;\Omega_{0}^{-1}$. This is likely the main cause of the stronger dust
clumping found in the global run. On the other hand, from the dust CDFs, we
see that the dust density distribution in the global and local runs converge
within the error bars at relatively high densities large
$\rho_{\text{threshold}}$ ($\gtrsim 5\;\rho_{\text{d}}$ both by counting
numbers and counting density), suggesting that dust clumping is well captured
in both local and global simulations. However, there are some deviations at
relatively small dust density. The cause of this deviation is unknown and may
require further investigations beyond the scope of this work, but we speculate
it may be related to a combination of higher pressure gradient $\Pi=0.075$
instead of $0.05$, higher mean $\epsilon$, and the global nature of the
simulation.
Figure 9: Left: The evolution of maximum and mean dust-to-gas ratio $\epsilon$
over time in the global and the local BA-$512^{2}$. The calculation of
$\epsilon$ in the global run are conducted over $r\in[0.4,2.4]$. Right: Dust
CDFs of the global and the local BA-$512^{2}$, similar to Figure 7. The dust
CDFs of the global run is calculated by the normalized dust density
$\rho_{\text{d}}r^{0.5}$ on $r\in[0.4,2.4]$. The color shaded regions are the
temporal standard deviations from $600\;\Omega_{0}^{-1}$ and
$800\;\Omega_{0}^{-1}$.
### 3.4 Mesh Refinement
In this subsection, we present additional tests to demonstrate the
compatibility of our multifluid dust module with static and adaptive mesh
refinement (SMR/AMR). Following the convention, here the root mesh is called
level 0, and each level of refinement doubles the resolution and is called
level 1, 2, etc.
#### 3.4.1 SMR Run of AB test
Figure 10: Snapshots of dust densities in the SMR test of AB test with two
levels of refinement. From top to bottom, from left to right, the panels are
at $t=5\;\Omega_{0}^{-1}$, $10\;\Omega_{0}^{-1}$, $15\;\Omega_{0}^{-1}$ and
$40\;\Omega_{0}^{-1}$. The edges of each meshblock are marked by black solid
lines, and meshblocks with different levels are labeled in the top-left panel.
Each meshblock contains $64^{2}$ cells, and there are 8, 16 and 64 meshblocks
in levels 0, 1 and 2.
We first rerun the AB test of the SI (see Section 3.3.3), but with two levels
of SMR. The resolution of the root mesh is $256^{2}$, and each level of
refinement doubles the resolution in the central region along the $z$
direction. Because AB test is sensitive to the amplitude of initial
perturbations, we use perturbations ten times smaller
($~{}0.002\;c_{\text{s}}$) than those in uniform runs. The results are shown
in Figure 10. As noted earlier, the outcome of AB test depends on resolutions.
Indeed, we see that the SI is first developed in the finest level and quickly
becomes nonlinear well within one orbital time, while the SI is developed more
slowly in coarser meshblocks, and it is not until after about $t\simeq
15\;\Omega_{0}^{-1}$ that the SI is fully developed in the entire domain. The
overall pattern in each refinement level closely resembles those shown in
Figure 6 with the same resolution ($256^{2}$ to $1024^{2}$), and there are no
abrupt features seen along coarse-fine meshblock boundaries, which testifies
the compatibility of our multifluid dust module with SMR in shearing box. We
have also examined the CDFs of this SMR run, and found that the CDFs at a
given level are largely consistent with the CDFs in the corresponding uniform-
level runs discussed earlier within the $1\sigma$ uncertainties.
#### 3.4.2 AMR Test of Kelvin-Helmholtz Instability
Figure 11: Snapshots of the KHI tests with 2 dust species with AMR. The top
(bottom) six panels are the cases without (with) dust feedback at $t=1.2$
($t=1.6$). The abbreviations of “nofb” and “fb” are for the cases without and
with feedback, respectively. From left to right, the panels are for gas, dust
species 1 and 2. The stopping times of dust are $T_{\text{s,1}}=10^{-2}$ and
$T_{\text{s,2}}=10^{-8}$. The first and the third rows are the gas and dust
densities from AMR runs with up to 3 refinement levels. The second and the
fourth rows are the relative differences between the AMR runs and the uniform
grid runs at a resolution matching the finest AMR refinement level of 3. The
edges of meshblocks in the AMR runs are also indicated in black solid lines.
We next conduct the standard test problem of the Kelvin-Helmholtz instability
(KHI) in Athena++ with AMR, exactly following the problem setup described in
Section 3.4.3 of Stone et al. (2020), but adding two dust species. The
resolution of the root mesh is $256^{2}$ with each meshblock size being
$8^{2}$, and the refinement condition is determined by
$g=\max{(|\partial_{x}v_{\text{g},y}|,|\partial_{y}v_{\text{g},x}|,|\partial_{x}v_{\text{d},n,y}|,|\partial_{y}v_{\text{d},n,x}|)}\
.\\\ $ (46)
which represents the maximum spatial velocity gradients in gas and all dust
fluids. Meshblocks with $g>0.01$ will be refined, and with $g<0.005$ will be
de-refined. We set a maximum of 3 refinement levels. We add two dust species
with stopping times $T_{\text{s,1}}=10^{-2}$, $T_{\text{s,2}}=10^{-8}$ but no
dust diffusion. We consider the cases without and with dust feedback, and the
dust-to-gas mass ratio for each species is set to unity. We use the
“RK2-Implicit” drag integrator, PLM reconstruction for both gas and dust, the
HLLC Riemann solver on gas, and a CFL number of 0.4. For comparison,
simulations with a uniform resolution of $2048^{2}$ (matching the finest
level) are also conducted.
Figure 11 shows the results of our dusty KHI tests with AMR. The first and the
third rows show gas and dust density patterns, while the second and the fourth
rows show the relative differences from the runs with uniform resolution. We
see that the strongly coupled dust with $T_{\text{s,2}}=10^{-8}$ shares
exactly the same density pattern as gas, as expected. The more marginally
coupled dust with $T_{\text{s,1}}=10^{-2}$, on the other hand, are depleted in
vortex centers, as they are relatively slow in response to the rapid vortical
motion to fill in the vortex eyes. We also see that the size of the vortices
are larger when feedback is included, as the inertia from more dust loading
would require more space for the KHI patterns to roll over.
We find there are no distinguishable differences between the AMR and uniform
grid runs, with relative differences of at most a few percent at vortex
centers in the first dust species with $T_{\text{s,1}}=10^{-2}$, due in part
to the low dust densities in there. The time step in our KHI tests is around
$5\times 10^{-5}$ in code units, which is much larger than the stopping time
of the second dust species ($T_{\text{s,2}}=10^{-8}$), making the drag
interaction highly stiff. The results again testify that our fully-implicit
methods are thus accurate and robust in these extremely stiff regimes with
AMR.
## 4 Summary and Discussion
In this paper, we describe the algorithm and implementation of a multifluid
dust module in Athena++, together with a suite of benchmark numerical tests.
The dust is treated as an arbitrary number of separate pressureless fluids,
each interacting with the gas via the aerodynamic gas drag, characterized by a
stopping time. Our development features two major advances.
First, we have provided a consistent formulation of dust concentration
diffusion. Dust concentration diffusion is commonly implemented as a diffusion
term in the dust continuity equation, which mimics the response to background
turbulence without explicitly simulating turbulence. This approach has been
shown to not conserve angular momentum in disk problems (Tominaga et al.,
2019). We further derive from a Reynolds averaging procedure the proper terms
that should be included in the momentum equation to ensure not only proper
momentum diffusion flux, but also Galilean invariance. The physically
meaningful behavior of dust concentration diffusion including dust feedback is
then illustrated from a simple test problem.
Second, we have developed two fully-implicit, second-order accurate drag
integrators, which naturally combine with the existing VL2 and RK2 integrators
in Athena++ to ensure 2nd-order accuracy in time for the composite system,
together with momentum conservation to machine precision. The integrators are
stable to highly stiff regimes not only in small dust stopping time, but also
in regimes of high dust mass loading. Our code is, to our knowledge, the first
to achieve the combination of these features. We have also implemented a
number of explicit and semi-implicit drag integrators for non-stiff
applications. In addition, we have incorporated frictional heating that can be
applied to any of the drag integrators.
The development of the multifluid dust module in Athena++, a higher-order
Godunov code, compliments the multifluid dust module in the widely used
FARGO3D code (Benítez-Llambay et al., 2019), which is Zeus-like (Stone &
Norman, 1992a, b). We conducted a large suite of code tests demonstrating code
performance, many in parallel to those done in (Benítez-Llambay et al., 2019).
In particular, we studied the SI from linear to nonlinear regimes, and the
results are generally in good agreement. We anticipate that the aforementioned
new features in our implementation represent more benefits, in addition to
better shock-capturing capabilities inherent to Godunov codes.
One of the main advantages in our multifluid dust module is its compatibility
with many of the existing functionalities and physics modules in Athena++. In
particular, our dust fluid module is compatible with static and adaptive mesh
refinement, curvilinear coordinate system including cylindrical and spherical
coordinates, shearing box and orbital advection, magnetic fields, diffusion
physics (viscosity, thermal conduction, non-ideal MHD), etc. The
implementation of the multifluid dust module thus enables a wide range of
applications involving dust dynamics, particularly related to the study of
physics, gas dynamics and observational signatures of protoplanetary disks and
planet formation, especially relevant to current and future disk observations
by ALMA, James Webb Space Telescope (JWST), Chinese Space Station Telescope
(CSST), the Next Generation Very Large Array (ngVLA) and the Square Kilometer
Array (SKA). We will also make this module publicly available in the near
future to benefit the broader astrophysical community.
There is still substantial room for further extensions of the multifluid dust
module, including dust coagulation/fragmentation (Drazkowska et al., 2019),
coupling with non-equilibrium radiative heating and cooling (Kamp & Dullemond,
2004), and self-gravity. Additionally, the drag term is exactly the same as
coupling among charged and neutral species in weakly ionized plasmas, e.g.,
O’Sullivan & Downes (2006), and the coupling term can be extremely stiff in
the strong coupling regime. Thus, our code can also be potentially extended to
accurately handle weakly ionized plasmas from multifluid to strong coupling
regimes. These directions will be considered in future works.
We thank Pablo Benítez-Llambay, Leonardo Krapp, Hui Li, Shengtai Li, Ruobing
Dong, Rixin Li, and Shangfei Liu for helpful discussions. This work is
supported by the National Key R&D Program of China No. 2019YFA0405100, and the
China Manned Space Project with NO. CMS-CSST-2021-B09.
## Appendix A Galilean invariance in momentum diffusion
In this Appendix, we prove the Galilean invariance of our dust concentration
diffusion formulation. In doing so, it would be much easier to recast the dust
momentum equation into an Euler-like equation. By applying the dust continuity
equation (4) and after some algebra, we arrive at
$\displaystyle\frac{\partial}{\partial
t}(\boldsymbol{v}_{\text{d}}+\boldsymbol{v}_{\text{d,dif}})+[(\boldsymbol{v}_{\text{d}}+\boldsymbol{v}_{\text{d,dif}})\cdot\nabla](\boldsymbol{v}_{\text{d}}+\boldsymbol{v}_{\text{d,dif}})=\frac{1}{\rho_{\text{d}}}\nabla\cdot(\rho_{\text{d}}\boldsymbol{v}_{\text{d,dif}}\boldsymbol{v}_{\text{d,dif}})+\frac{\boldsymbol{v}_{\text{g}}-\boldsymbol{v}_{\text{d}}}{T_{s}}\
,$ (A1)
Let us consider a different frame which moves at a constant speed
$\boldsymbol{v}_{0}$, where physical quantities are denoted with a prime ′.
Obviously, we have $\rho^{\prime}=\rho$,
$\boldsymbol{v}^{\prime}=\boldsymbol{v}-\boldsymbol{v}_{0}$, and
$\boldsymbol{v}^{\prime}_{\text{d,dif}}=\boldsymbol{v}_{\text{d,dif}}$. To
prove that the equation is Galilean invariant, it suffices to show that the
equations written in the new frame is exactly the same when expressed with
primed quantities.
With Equation (A1), the proof becomes quite straightforward. We first replace
$\boldsymbol{v}_{\text{d}}$ by
$\boldsymbol{v}^{\prime}_{\text{d}}+\boldsymbol{v}_{0}$,
$\boldsymbol{v}_{\text{g}}$ by
$\boldsymbol{v}^{\prime}_{\text{g}}+\boldsymbol{v}_{0}$, $\rho$ by
$\rho^{\prime}$, and $\boldsymbol{v}_{\text{d,dif}}$ by
$\boldsymbol{v}^{\prime}_{\text{d,dif}}$. We note that the right hand side
remains unchanged. Next, note that $\partial/\partial
t^{\prime}=\partial/\partial t+\boldsymbol{v}_{0}\cdot\nabla$, we can see that
terms proportional to $\boldsymbol{v}_{0}$ all cancel out, thus the form of
equation remain exactly the same as (A1) except all expressed in primed
quantities.
In the above discussion, we emphasize that if drop off the time derivative
term on $\rho_{\text{d}}\boldsymbol{v}_{\text{d,dif}}$, the Euler-like
equation would become
$\displaystyle\frac{\partial\boldsymbol{v}_{\text{d}}}{\partial
t}+[(\boldsymbol{v}_{\text{d}}+2\boldsymbol{v}_{\text{d,dif}})\cdot\nabla]\boldsymbol{v}_{\text{d}}+\frac{\boldsymbol{v}_{\text{d}}}{\rho_{\text{d}}}\nabla\cdot(\rho_{\text{d}}\boldsymbol{v}_{\text{d,dif}})=\frac{\boldsymbol{v}_{\text{g}}-\boldsymbol{v}_{\text{d}}}{T_{s}}\
.$ (A2)
which is very different from that of Equation (A1). In particular, going
through the same procedures, the presence of the third term on the left hand
side makes this equation violate the Galilean invariance.
## Appendix B Design of second order fully-implicit drag integrators
The VL2 implicit integrator is designed from the following format:
$\displaystyle\boldsymbol{M}^{(n+1)}-\boldsymbol{M}^{(n)}$
$\displaystyle=h\boldsymbol{f}\left(\boldsymbol{M}^{(n+\frac{1}{2})},\boldsymbol{W}^{(n+\frac{1}{2})}\right)$
(B1)
$\displaystyle=h\boldsymbol{f}\left[\boldsymbol{M}^{(n+1)}-\frac{h}{2}\boldsymbol{f}\left(\boldsymbol{M}^{(n+1)},\boldsymbol{W}^{(n+\frac{1}{2})}\right),\boldsymbol{W}^{(n+\frac{1}{2})}\right]$
$\displaystyle=h\boldsymbol{f}\left(\boldsymbol{M}^{(n+1)},\boldsymbol{W}^{(n+\frac{1}{2})}\right)-\frac{h^{2}}{2}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n+\frac{1}{2})}\boldsymbol{f}\left(\boldsymbol{M}^{(n+1)},\boldsymbol{W}^{(n+\frac{1}{2})}\right)$
$\displaystyle=h\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n+\frac{1}{2})}\right)+h\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n)}\left(\boldsymbol{M}^{(n+1)}-\boldsymbol{M}^{(n)}\right)$
$\displaystyle\quad-\frac{h^{2}}{2}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n+\frac{1}{2})}\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n+\frac{1}{2})}\right)-\frac{h^{2}}{2}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n+\frac{1}{2})}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n)}\left(\boldsymbol{M}^{(n+1)}-\boldsymbol{M}^{(n)}\right)\
.$
This leads to the integration scheme (25) and (26). The quantities at
$n+\frac{1}{2}$ time step (denoted by a prime in Section 2.3.2) are obtained
from the first stage of the algorithm, and any first-order implicit
integration suffices (we use the backward Euler method).
The “RK2-Implicit” integrator is designed from the following format:
$\displaystyle\boldsymbol{M}^{(n+1)}-\boldsymbol{M}^{(n)}$
$\displaystyle=\frac{h}{2}\boldsymbol{f}\left(\boldsymbol{M}^{(n+1)},\boldsymbol{W}^{\prime}\right)+\frac{h}{2}\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n)}\right)$
(B2)
$\displaystyle=\frac{h}{2}\boldsymbol{f}\left(\boldsymbol{M}^{(n+1)},\boldsymbol{W}^{\prime}\right)+\frac{h}{2}\boldsymbol{f}\left[\boldsymbol{M}^{(n+1)}-h\boldsymbol{f}\left(\boldsymbol{M}^{(n+1)},\boldsymbol{W}^{(n)}\right),\boldsymbol{W}^{(n)}\right]$
$\displaystyle=\frac{h}{2}\boldsymbol{f}\left(\boldsymbol{M}^{(n+1)},\boldsymbol{W}^{\prime}\right)+\frac{h}{2}\left(\mathsf{I}-h\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}\bigg{|}^{\prime}\right)\boldsymbol{f}\left(\boldsymbol{M}^{(n+1)},\boldsymbol{W}^{(n)}\right)$
$\displaystyle=\frac{h}{2}\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{\prime}\right)+\frac{h}{2}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}\bigg{|}^{(n)}\left(\boldsymbol{M}^{(n+1)}-\boldsymbol{M}^{(n)}\right)$
$\displaystyle\quad+\frac{h}{2}\left(\mathsf{I}-h\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}\bigg{|}^{\prime}\right)\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n)}\right)+\frac{h}{2}\left(\mathsf{I}-h\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}\bigg{|}^{\prime}\right)\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}\bigg{|}^{(n)}\left(\boldsymbol{M}^{(n+1)}-\boldsymbol{M}^{(n)}\right)\
.$
where the prime ′ on $W$ indicate the quantity is evaluated at the end of the
first stage (integrating to time step $n+1$ with a first-order implicit
scheme, i.e., backward Euler). This leads to the integration scheme (27) and
(28).
## Appendix C Explicit and Semi-implicit Drag Integrators
When the stopping time of dust is much larger than the numerical time step,
$T_{\text{s}}\gtrsim\delta t\equiv h$ and there is no strong dust mass
loading, the drag is in non-stiff regime. Here we have also implemented a
number of standard explicit and semi-implicit drag integrators in Athena++. We
omit the trivial implementation of the forward Euler method (RK1), and 2nd-
order methods are described below. For all methods, the energy equation is
updated in each stage in a way analogous to Equations (21) to (23), which we
again omit here.
### C.1 Second Order Explicit Methods
Here we document the two explicit integrators following the VL2 and RK2
integrators in Athena++, termed “VL2-Explicit” and “RK2-Explicit” in this
paper. Note that the explicit integrators usually requires the time step
$h<T_{\text{s}}$ for all dust species. The momentum update is as follows.
VL2-Explicit: We first update the system for half a time step, followed by a
full update using the midpoint values:
$\displaystyle\boldsymbol{M}^{(n+\frac{1}{2})}$
$\displaystyle=\boldsymbol{M}^{(n)}+\frac{h}{2}\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n)}\right)\
,$ (C1) $\displaystyle\boldsymbol{M}^{(n+1)}$
$\displaystyle=\boldsymbol{M}^{(n)}+h\boldsymbol{f}\left(\boldsymbol{M}^{(n+\frac{1}{2})},\boldsymbol{W}^{(n+\frac{1}{2})}\right)\
.$
RK2-Explicit: it first provides an estimate after a time step $h$ denoted by
′, followed by a correction:
$\displaystyle\boldsymbol{M}^{\prime}$
$\displaystyle=\boldsymbol{M}^{(n)}+h\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n)}\right)\
,$ (C2) $\displaystyle\boldsymbol{M}^{(n+1)}$
$\displaystyle=\frac{1}{2}\left(\boldsymbol{M}^{(n)}+\boldsymbol{M}^{\prime}\right)+\frac{1}{2}h\boldsymbol{f}\left(\boldsymbol{M}^{\prime},\boldsymbol{W}^{\prime}\right)\
.$
### C.2 Second Order Semi-implicit Methods
Here we present two semi-implicit methods, which are more robust than the
explicit methods.
Trapezoid (Crank-Nicholson Method)
The trapezoid method is derived from
$\displaystyle\boldsymbol{M}^{(n+1)}$
$\displaystyle=\boldsymbol{M}^{(n)}+\frac{1}{2}\left[h\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n)}\right)+h\boldsymbol{f}\left(\boldsymbol{M}^{(n+1)},\boldsymbol{W}^{(n)}\right)\right],$
(C3)
$\displaystyle=\boldsymbol{M}^{(n)}+h\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n)}\right)+\frac{h}{2}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n)}\left(\boldsymbol{M}^{(n+1)}-\boldsymbol{M}^{(n)}\right),$
$\displaystyle\Rightarrow\boldsymbol{M}^{(n+1)}$
$\displaystyle=\boldsymbol{M}^{(n)}+\left(\mathsf{I}-\frac{h}{2}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n)}\right)^{-1}h\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n)}\right).$
The first stage ′ of “Trapezoid” is updated by backward Euler method with $h$,
so as to be compatible with “RK2-Explicit”.
Trapezoid Backward Differentiation Formula 2 (TrBDF2)
In the TrBDF2, the momentum at the middle stage $n+\frac{1}{2}$ is calculated
by the trapezoid method with time step $h/2$, so as to be compatible with
“VL2-Explicit”. Then $\boldsymbol{M}^{(n+1)}$ is updated by backward
Differentiation Formula 2 (BDF2) method at the stage $n+1$.
$\displaystyle\boldsymbol{M}^{(n+\frac{1}{2})}$
$\displaystyle=\boldsymbol{M}^{(n)}+\left(\mathsf{I}-\frac{h}{4}\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n)}\right)^{-1}\frac{h}{2}\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{W}^{(n)}\right),$
(C4) $\displaystyle\boldsymbol{M}^{(n+1)}$
$\displaystyle=\frac{4}{3}\boldsymbol{M}^{(n+\frac{1}{2})}-\frac{1}{3}\boldsymbol{M}^{(n)}+\frac{1}{3}\left(\mathsf{I}-h\frac{\partial\boldsymbol{f}}{\partial\boldsymbol{M}}|^{(n)}\right)^{-1}h\boldsymbol{f}\left(\boldsymbol{M}^{(n)},\boldsymbol{V}^{(n)}\right).$
## Appendix D Performance
Figure 12: Code performance measured by CPU time per cell in micron second,
shown as a function of total number of species (gas and $\mathbf{N_{d}}$
species of dust).
The two higher-order fully-implicit drag integrators VL2-Implicit and
RK2-Implicit involve solving the inverse of an $(N_{d}+1)\times(N_{d}+1)$
matrix. The cost of matrix inversion is $O[(N_{d}+1)^{3}]$. Moreover, it also
takes $O[(N_{d}+1)^{3}]$ to handle matrix multiplications, such as in
computing $\boldsymbol{\Lambda}$ in Equations (26) and (28). Such operations
would make the calculations increasingly expensive as $N_{d}$ increases, and
can become a bottleneck at sufficiently large $N_{d}$.
Here we measure code performance as a function of $N_{d}$ from the NSH
equilibrium test in shearing box. The test is run on a Intel Xeon Gold 6132
CPU with 28 cores. We use all the cores with 28 meshblocks, so that we occupy
the entire CPU (and hence its cache) to mimic more realistic situation in
large-scale simulations (note that communications in Athena++ are mostly
hidden thanks its use of tasklist and performance is more sensitive to catch
use). We use the HLLE Riemann solver for gas and the PLM reconstruction for
both gas and dust. We measure the performance in terms of the time spent to
update a single cell by an individual core. The results for different drag
integrators are shown in Figure 12, as a function of total number of species
($N_{d}+1$), which are further compared to results with gas drag turned off.
For explicit and semi-implicit integrators, we see that the drag integrators
add to very limited computational cost relative to the no-drag case. In
particular, the semi-implicit integrators that involve two matrix inversion
operations remain computationally efficient thanks to the fast analytical
solver (Krapp & Benítez-Llambay, 2020) that reduces the cost to $O(N_{d}+1)$.
The total cost increases linearly with $N_{d}$ for $N_{d}\lesssim 12$, but
gets slightly nonlinear at larger $N_{d}$. We speculate it is likely due to
heavier memory use that reduces cache performance.
For the two fully implicit solvers, we manage to improve the performance by
using the fast matrix inversion at the first integration stage, yet the more
complex matrix computation and inversion at the second stage substantially
increases the computational cost. This cost increases non-linearly with
$N_{d}$. It is relatively negligible for $N_{d}\lesssim 5$, and remains to be
minor compared to the rest of the dust integrator for $N_{d}\lesssim 10$, but
becomes rather significant for larger $N_{d}$.
## Appendix E Solutions to the Collision Tests
The mutual drags between gas and $n$ dust species can be written in the matrix
form:
$\displaystyle\frac{\partial\boldsymbol{M}}{\partial
t}=\boldsymbol{A}\boldsymbol{M}=\begin{bmatrix}-\sum^{n}_{i}\epsilon_{i}\alpha_{i}&\alpha_{1}&\alpha_{2}&\cdots&\alpha_{n}\\\
\epsilon_{1}\alpha_{1}&-\alpha_{1}&0&\cdots&0\\\
\epsilon_{2}\alpha_{2}&0&-\alpha_{2}&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
\epsilon_{n}\alpha_{n}&0&0&\cdots&-\alpha_{n}\\\
\end{bmatrix}\begin{bmatrix}\boldsymbol{M}_{\text{g}}\\\
\boldsymbol{M}_{\text{d},1}\\\ \boldsymbol{M}_{\text{d},2}\\\ \vdots\\\
\boldsymbol{M}_{\text{d},n}\\\ \end{bmatrix}\ .$ (E1)
The analytic solution of the momentum vector must take the form
$\boldsymbol{M(t)}=\sum_{0}^{n}\boldsymbol{c}_{i}\exp{(\lambda_{i}\;t)}$,
where $\boldsymbol{c}_{i}$ are the coefficient for each momentum component
determined by initial condition and $\lambda_{i}$ are the eigenvalue. The key
is to solve the eigensystem
$\boldsymbol{A}\boldsymbol{M}=\lambda\boldsymbol{M}$, yielding eigenvalues
$\lambda_{i}$ and eigenvectors $\boldsymbol{M}_{i}$. The coefficients
$\boldsymbol{c}_{i}$ are obtained by decomposing the initial condition into
the eigenvectors.
Because of momentum conservation, i.e., the drag force acted on the gas equals
to the sum of drag forces acted on dust, the matrix $A$ has an eigenvalue
$\lambda_{0}=0$ corresponding to bulk motion, and hence the coefficient
$c_{0}=v_{\text{COM}}$ is the velocity of center of mass (COM). When multiple
dust species share the same stopping time, the eigensystem can be greatly
simplified (see Table 1 of Benítez-Llambay et al. (2019)), but this is no
longer true in the general case. We use Mathematica (Wolfram, 1991) to
calculate the eigenvalues and the rest of the coefficients in Table 1.
## Appendix F Additional Numerical Tests
In this Appendix, we present additional code tests that largely reproduce the
dusty sound wave test and the dusty shock test in Benítez-Llambay et al.
(2019) to demonstrate our code performance.
### F.1 Dusty Sound Wave
To demonstrate that our multifluid dust module achieves second order accuracy
when combined with the hydrodynamic solver, we conduct the dusty sound wave
test by exactly following Section 3.2 of Benítez-Llambay et al. (2019),
originally proposed by Laibe & Price (2011, 2012). We use the PLM spatial
reconstruction, isothermal equation of state with the HLLE gas Riemann solver,
and consider both the “VL2-Implicit” and “RK2-Implict” drag integrators. The
tests are conducted in 1D, starting from a resolution of $N=64$ cells, and we
double the resolution at a time until reaching a resolution of $N=512$ cells.
We have conducted simulations for both single-species and multi-species cases,
and found excellent agreement with analytical theory. For brevity, we show in
Figure 13 the time evolution of the normalized dust and gas velocities for the
single species dust case, showing that our numerical solution perfectly
matches the analytical solution. Moreover, we measure the root mean square of
the L1 norms $[\sum_{n}(\sum|U_{n}-U_{n,{\rm ana}}|/N)^{2}]^{1/2}$ after one
wave period, similar to the approach in the linear wave test in Athena++ Stone
et al. (2020), where $U_{n}$ and $U_{n,{\rm ana}}$ are the numerical and
analytical solutions of the $n$-th variable. The variables include gas density
and velocity, and dust density and velocity, all in normalized units. We see
in the right panel of Figure 13 that our code clearly achieves second order
convergence.
Figure 13: Results from the dusty sound wave test. The left and the middle
panels are the time evolution of normalized density and normalized velocities
of gas and dust located at $x=0$, to be compared with Figure 3 of Benítez-
Llambay et al. (2019). The solid lines represent the analytical solutions, and
the triangle and circle markers are the numerical results from test runs with
256 cells. The right panel demonstrates the numerical convergence, where we
calculate the mean L1-error after one wave period, and show how it varies with
the number of grid cells using the “VL2-Implicit” and “RK2-Implicit” drag
integrators.
### F.2 Dusty Shock
Being a Godunov code, Athena++ has excellent shock capturing properties that
we demonstrate using the generalized dusty shock test presented in Section 3.3
of Benítez-Llambay et al. (2019), which is generalized from Lehmann & Wardle
(2018). We follow the same procedures and adopt identical parameters as in
Benítez-Llambay et al. (2019) to conduct two simulations with one and three
dust species (two and four species in total) on 400 grid points. PLM spatial
reconstruction and “VL2-Implicit” drag integrator are used for these tests.
The results are shown in Figure 14, which is to be compared with Figure 5 of
Benítez-Llambay et al. (2019) side by side. Note that the shocks are in steady
state thus we focus on the overall shock profile rather than the specific
shock locations. Because of dust drag, the dust profile near the shock is
connected by a precursor that is accurately reproduced, similar to that in
FARGO3D. On the other hand, Athena++ very captures the discontinuity in the
gas within neighboring cells, as opposed to $\sim 4$ cells in FARGO3D.
Figure 14: Normalized velocities (top) and densities (bottom) of dusty wave
tests with 1 (left) and 3 (right) dust species, to be directly compared with
Figure 5 of Benítez-Llambay et al. (2019). The gas profile is shown in red,
while the other colors correspond to the dust species, with the insets showing
the zoomed in profiles across the shock. Analytical solutions are shown in
solid lines, while numerical results are shown in filled circles.
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|
# Learning to Navigate Intersections
with Unsupervised Driver Trait Inference
Shuijing Liu, Peixin Chang, Haonan Chen,
Neeloy Chakraborty, and Katherine Driggs-Campbell S. Liu, P. Chang, H. Chen,
N. Chakraborty and K. Driggs-Campbell are with the Department of Electrical
and Computer Engineering at the University of Illinois at Urbana-Champaign.
emails<EMAIL_ADDRESS>material is
based upon work supported by the National Science Foundation under Grant No.
2143435.
###### Abstract
Navigation through uncontrolled intersections is one of the key challenges for
autonomous vehicles. Identifying the subtle differences in hidden traits of
other drivers can bring significant benefits when navigating in such
environments. We propose an unsupervised method for inferring driver traits
such as driving styles from observed vehicle trajectories. We use a
variational autoencoder with recurrent neural networks to learn a latent
representation of traits without any ground truth trait labels. Then, we use
this trait representation to learn a policy for an autonomous vehicle to
navigate through a T-intersection with deep reinforcement learning. Our
pipeline enables the autonomous vehicle to adjust its actions when dealing
with drivers of different traits to ensure safety and efficiency. Our method
demonstrates promising performance and outperforms state-of-the-art baselines
in the T-intersection scenario. For code implementation and videos, please
visit https://github.com/Shuijing725/VAE_trait_inference.
## I Introduction
To successfully navigate through uncontrolled intersections, autonomous
vehicles must carefully reason about how to interact with different types of
human drivers [1]. It is important for the vehicle to infer the traits of
human drivers, such as the propensity for aggression or cooperation, and
adjust its strategies accordingly [2, 3]. Inspired by recent advancements of
unsupervised learning and deep reinforcement learning, we propose a novel
pipeline to learn a representation of traits of other drivers, which is used
for autonomous navigation in uncontrolled intersections.
Trait inference is challenging yet essential for the navigation of autonomous
vehicles for two reasons. First, the environment is not fully observable to
the ego vehicle, since each traffic participant runs its own policy
individually and has its own internal states. The ego vehicle needs to
interpret the hidden states such as driving styles and intents of other agents
to understand future behaviors that may influence planning [2, 3]. Second,
uncontrolled intersections are less structured since traffic lights and stop
signs are not present to coordinate agent behaviors. The traffic participants
implicitly interact and negotiate with each other, making the environment
complex and potentially dangerous [4, 5, 6, 7]. By inferring the traits of
other drivers, the ego vehicle can be cautious when other drivers are
aggressive or irrational and bold when they are passive or cooperative,
improving both the safety and efficiency of interactive navigation.
To address the above problems, Morton et al learns a latent representation of
driver traits, which is fed into a feedforward policy to produce multimodal
behaviors [8]. However, the feedforward policy only considers current states
and actions which are not sufficient to fully express long-term properties of
drivers such as traits. As a result, this representation fails to distinguish
between different traits. Ma et al classifies driver traits with supervised
learning to aid navigation in intersections [3] but has the following two
problems. First, the trait labels are expensive to obtain and usually do not
exist in most real driving datasets [9, 1]. Second, the navigation policy is
trained with ground truth trait labels instead of predicted traits. When the
trait classifier and the policy are combined in testing, intermediate and
cascading errors cause severe performance degradation.
Figure 1: The T-intersection scenario in left-handed traffic. The goal of the
ego car (yellow) is to take a right turn and merge into the upper lane without
colliding with other cars. The conservative car (blue) yields to the ego car
while the aggressive cars (red) do not yield.
In this paper, we study the same uncontrolled T-intersection navigation
problem as in [3], which is shown in Fig. 1. Before entering the
T-intersection, the ego vehicle needs to infer the latent driving styles of
other drivers to determine whether they are willing to yield to the ego
vehicle. Based on the inferred driving style, the ego vehicle must intercept
the drivers who will yield to achieve the goal.
We seek to create a pipeline that first learns a representation of driver
traits in an unsupervised way, and then uses the trait representation to
improve navigation in the T-intersection. In the first stage, we encode the
trajectories of other drivers to a latent representation using a variational
autoencoder (VAE) with recurrent neural networks (RNN). Since trajectory
sequences reveal richer information about driver traits than single states,
the VAE+RNN network effectively learns to distinguish between different traits
without any explicit trait labels. In the second stage, we use the latent
representations of driver traits as inputs to the ego vehicle’s navigation
policy, which is trained with model-free reinforcement learning (RL). With the
inferred traits, the ego vehicle is able to adjust its decisions when dealing
with different drivers, which leads to improved performance. Since the RL
policy is trained with the learned representations instead of ground truth
labels, our pipeline is much less sensitive to cascading errors from the trait
inference network.
We present the following contributions: (1) We propose a novel unsupervised
approach to learn a representation of driver traits with a VAE+RNN network;
(2) We use the learned representation to improve the navigation of an
autonomous vehicle through an uncontrolled T-intersection; (3) Our trait
representation and navigation policy exhibit promising results and outperform
previous works.
This paper is organized as follows: We review related works in Section II. We
formalize the problem and propose our method in Section III. Experiments and
results are discussed in Section IV. We conclude the paper in Section V.
## II Related Works
### II-A Driver internal state estimation
Driver internal state estimation can be divided into intent estimation and
trait estimation [10]. Intent estimation often uses methods such as
probabilistic graph model and unparameterized belief tracker to predict the
future maneuvers of other drivers [6, 11, 12], which can then inform
downstream planning for the ego driver [13, 6, 12]. Trait estimation infers
the properties of drivers such as driving styles, preferences, fatigue, and
level of distraction [10]. Some works distinguish between distracted and
attentive drivers for behavior prediction and cooperative planning [14, 15].
Driving style recognition has been addressed with both unsupervised and
supervised learning methods, which we will discuss in detail below [16, 17, 8,
3].
Morton et al propose a method that first encodes driving trajectories with
different driving styles to a latent space. Then, the latent encodings and the
current driver states are fed into a feedforward policy that produces
multimodal actions [8]. The encoder and the policy are optimized jointly.
However, since the feedforward policy only considers the relations between
current states and actions, the joint optimization encourages the encoder to
encode short-term information of the trajectories such as accelerations while
ignoring the persistent properties such as traits. In contrast, since our
method uses an RNN decoder to reconstruct the trajectories, the encoder must
encode the trait information. Ma et al propose a graph neural network that
classifies the driving styles with supervised learning, which requires large
amounts of labeled data and is difficult to scale in reality [3]. In addition,
since the definitions of trait properties vary by individuals and cultures,
manually labeled trait information will likely be noisy and inconsistent. In
contrast, our method is unsupervised and does not need any trait labels.
### II-B Representation learning for sequential data
Contrastive learning is widely used to learn representations from sequential
data such as videos and pedestrian trajectories [18, 19, 20]. However, the
performance of contrastive learning is sensitive to the quality of negative
samples, and finding efficient negative sampling strategies remains an open
challenge [21].
Another category of representation learning methods is VAE and its variants
[22, 23]. Bowman et al introduce an RNN-based VAE to model the latent
properties of sentences [24], which inspires us to learn the traits of drivers
from their trajectories. Conditional VAEs (CVAE) are widely used in pedestrian
and vehicle trajectory predictions since discrete latent states can represent
different behavior modes such as braking and turning [25, 26, 27, 28]. While
these behavior modes change frequently, the driver traits that we aim to learn
are persistent with each driver [10]. Also, the goal of these CVAEs is to
generate multimodal trajectory predictions while, to the best of our
knowledge, our work is the first to infer driver traits with VAE for
autonomous driving.
### II-C Autonomous driving in uncontrolled intersections
Autonomous navigation through uncontrolled intersections is well studied and
has had many successful demonstrations [29, 6, 30, 3]. Some heuristic methods
use a time-to-collision (TTC) threshold to decide when to cross [31, 30].
However, the TTC models assume constant velocity for the surrounding vehicles,
which ignores the interactions and internal states of drivers. Also, the TTC
models can be overly cautious and cause unnecessary delays [29]. Another line
of works formulates the problem as a partially observable Markov decision
process (POMDP), which accounts for the uncertainties and partial
observability in the intersection scenario but is computationally expensive to
solve [6, 32, 33]. RL-based methods use neural networks as function
approximators to learn driving policies. Isele et al learns to navigate in
occluded intersections using deep Q-learning [29]. Ma et al focuses on
navigation in a T-Intersection where the drivers exhibit different traits [3].
However, the navigation policy is trained with ground truth traits and only
uses the inferred traits in testing. Thus, the performance of the RL policy is
sensitive to the intermediate errors from the trait inference module. In
contrast, our method trains the RL policy directly with inferred trait
representations, which eliminates the problems caused by the intermediate
errors.
## III Methodology
In this section, we first present our unsupervised approach to represent
driver traits from unlabeled driving trajectories with a VAE+RNN network.
Then, we discuss how we use the inferred traits to improve the navigation
policy.
Figure 2: The network architectures. (a) The VAE+RNN network for trait
representation learning. The fully connected layers before and after GRUs are
eliminated for clarity. The dice represents the reparameterization trick. The
start-of-sequence state is denoted by $\langle SOS\rangle$. (b) The navigation
policy network. The weights of the encoder (blue) is fixed and only the yellow
part is trained with RL. We use $[\bullet\bullet]$ to denote concatenation.
For illustration purposes, we assume that the inferred trait $z_{1},...,z_{n}$
is updated at time $t$.
### III-A Preliminaries
Consider a T-intersection environment with an ego vehicle and $n$ surrounding
vehicles. The number of surrounding vehicles $n$ may change at any timestep
$t$. Suppose that all vehicles move in a 2D Euclidean space. Let ${o}_{0}^{t}$
be the state of the ego vehicle and ${o}_{i}^{t}$ be the observable state of
the $i$-th surrounding vehicle at time $t$, where $i\in\\{1,...,n\\}$. The
state of the ego vehicle ${o}_{0}^{t}$ consists of the position
$(p_{x},p_{y})$ and the velocity $(v_{x},v_{y})$ of the vehicle. The
observable state of each surrounding vehicle ${o}_{i}^{t}$ consists of its
position $(p_{x}^{i},p_{y}^{i})$. In contrast to the previous work [3],
${o}_{i}^{t}$ does not include other vehicles’ velocities because they are
hard to accurately measure without special facilities and algorithms in the
real world [34, 35]. In addition, each surrounding vehicle has a latent state
$z_{i}$ that indicates the aggressiveness (i.e., the trait) of the $i$-th
driver. We assume that the driving style of each driver $z_{i}$ does not
change throughout an episode. The positions of surrounding vehicles
${o}_{1}^{t},...,{o}_{n}^{t}$ are observable to the ego vehicle, while the
latent states $z_{1},...,z_{n}$ are not.
### III-B Trait representation learning
#### III-B1 Data collection
For each vehicle in the T-Intersection, the vehicle in front of it has the
most direct influence on its behaviors. For this reason, in the trait
representation learning stage, we define the observable state of each vehicle
to be $x=(\Delta p_{x},\Delta p_{x,f})$, where $\Delta p_{x}$ is the
horizontal offset from the vehicle’s starting position in the trajectory, and
$\Delta p_{x,f}$ is the horizontal displacement of the vehicle from its front
vehicle. Then, the trajectory of each driver is a sequence of states
$\mathbf{x}=[x^{1},...,x^{l}]$, where $l$ is the length of the trajectory.
We only keep the longitudinal state information because all vehicles move in
horizontal lanes and the lateral states are not insightful in this setting,
except indicating which lane the vehicle is in. We rotate the trajectories so
that all trajectories in the dataset are aligned in the same direction. Thus,
the lane and the directional information is indistinguishable within the
trajectory data, allowing the network to focus on learning the trait
difference instead of other differences between vehicles.
To collect the trajectory data, we run a simulation of the T-intersection
scenario without the presence of the ego car and record the trajectories of
all surrounding vehicles, which are controlled by the Intelligent Driver Model
(IDM) [36]. Learning trait representations from this dataset allows the ego
car to infer the traits from the trajectories of other drivers before deciding
to intercept or wait. The dataset is denoted as
$\\{\mathbf{x}_{j}\\}_{j=1}^{N}$, where $N$ is the total number of
trajectories.
#### III-B2 Network architecture
We use the collected dataset to train the VAE+RNN network to learn a
representation of traits, as shown in Fig. 2a. The VAE network consists of an
encoder, which compresses a trajectory $\mathbf{x}$ to a distribution of a
latent trait vector $z$, and a decoder, which reconstructs the trajectory from
the latent vector $z$. Both the encoder and the decoder are gated recurrent
unit (GRU) networks since GRUs are more computationally efficient than long
short-term memory networks (LSTM).
Given a trajectory $\mathbf{x}=[x^{1},...,x^{l}]$, the encoder GRU first
applies a non-linear embedding layer $f_{\textrm{encoder}}$ to each state
$x^{t}$ and then feeds the embedded features to the GRU cell:
$h^{t}_{e}=\mathrm{GRU}\left(h^{t-1}_{e},f_{\textrm{encoder}}(x^{t})\right)$
(1)
where $h^{t}_{e}$ is the hidden state of the encoder GRU at time
$t\in\\{1,...,l\\}$. After the entire trajectory is fed through the encoder
GRU, we take the last hidden state $h^{l}_{e}$ as the encoded latent feature
of the trajectory $\mathbf{x}$ and apply fully connected layers $f_{\mu}$ and
$f_{\sigma}$ to get the Gaussian parameters of the latent driving style
$z\in\mathbb{R}^{2}$ in a two-dimensional latent space:
$\mu=f_{\mu}(h^{t}_{e}),\quad\sigma_{i}=f_{\sigma}(h^{t}_{e}).$ (2)
Finally, we use the reparameterization trick to sample $z$ from
$\mathcal{N}(\mu,\sigma)$ for efficient learning:
$z=\mu+\epsilon\sigma,\epsilon\sim\mathcal{N}(0,I)$.
In the decoding stage, since the driving style of each driver does not change
over time, we treat the latent state $z$ as part of the vehicle state instead
of the initial hidden state of decoder GRU. To this end, at each timestep $t$,
we concatenate the reconstructed state $\hat{x}^{t-1}$ from the last timestep
and the latent state $z$ from the encoder. Then, we embed the joint states
using $f_{\textrm{decoder}}$, feed the embeddings into the next decoder GRU
cell, and apply another embedding $g_{\textrm{decoder}}$ to the next hidden
state $h^{t}_{d}$, which outputs the next reconstructed state $\hat{x}^{t}$:
$\displaystyle h^{t}_{d}$
$\displaystyle=\mathrm{GRU}\left(h^{t-1}_{d},f_{\textrm{decoder}}([\hat{x}^{t-1},z])\right)$
(3) $\displaystyle\hat{x}^{t}$
$\displaystyle=g_{\textrm{decoder}}(h^{t}_{d}).$
In the first timestep, we use a special start-of-sequence (SOS) state, which
is similar to the start-of-sequence symbol in natural language processing to
reconstruct $\hat{x}^{1}$ [37]. The process in Eq. 3 repeats until we
reconstruct the whole trajectory
$\hat{\mathbf{x}}=[\hat{x}^{1},...,\hat{x}^{l}]$.
The objective for training our VAE+RNN network is
$\mathcal{L}=\beta
D_{KL}\left(\mathcal{N}(\mu,\sigma)||\mathcal{N}(0,I)\right)+||\mathbf{x}-\hat{\mathbf{x}}||_{2}$
(4)
where $D_{KL}$ is the Kullback–Leibler (KL) divergence. The first term
regularizes the distribution of the latent trait $z$ to be close to a prior
with a standard normal distribution. The second term is the reconstruction
loss and measures the $L2$ error of the reconstructed trajectories from the
original trajectories. The two terms are summed with a weight $\beta$.
By optimizing Eq. 4, our network learns latent encodings that represent the
trait of each trajectory without any ground truth trait labels. Note that we
also make no assumptions on the number of trait classes or the semantic
meanings of the trait classes. Thus, our network has the potential to
generalize to real trajectory datasets with more complex traits.
### III-C Navigation policy learning
We model the T-intersection scenario as a POMDP, defined by the tuple
$\langle\mathcal{S},\mathcal{A},\mathcal{T},\mathcal{R},\mathcal{O},\mathcal{P},\gamma\rangle$.
Suppose that there are $n$ surrounding vehicles at timestep $t$. We use
${o}_{t}=[{o}_{0}^{t},{o}_{1}^{t},...,{o}_{n}^{t}]\in\mathcal{O}$ to denote
the observations of the ego vehicle, where $\mathcal{O}$ is the observation
space. Let ${u}^{t}_{i}=[{o}^{t}_{i},z_{i}]\in\mathbb{R}^{4}$ be the state of
the $i$-th surrounding vehicle. Since the ego vehicle is influenced by all
surrounding vehicles, we use
$s_{t}=[{o}^{t}_{0},{u}^{t}_{1},...,{u}^{t}_{n}]\in\mathcal{S}$ to denote the
state of the POMDP, where $\mathcal{S}$ is the state space. And
$\mathcal{P}:\mathcal{S}\rightarrow\mathcal{O}$ is the set of conditional
observation probabilities.
At each timestep $t$, the ego vehicle chooses the desired speed for the low-
level controller $a_{t}\in\mathcal{A}$ according to its policy
$\pi(a_{t}|s_{t})$, where $\mathcal{A}=\\{0,0.5,3\\}\mathrm{m/s}$ is the
action space. In return, the agent receives a reward $r_{t}$ and transits to
the next state $s_{t+1}$ according to an unknown state transition
$\mathcal{T}(\cdot|s_{t},a_{t})$. Meanwhile, all other vehicles also take
actions according to their policies and move to the next states with unknown
state transition probabilities. The episode continues until $t$ exceeds the
maximum episode length $T$, the ego vehicle reaches its goal, or the ego
vehicle collides with any other vehicle. The goal of the agent is to maximize
the expected return, $R_{t}=\mathbb{E}[\sum^{T}_{i=t}\gamma^{i-t}r_{i}]$,
where $\gamma$ is a discount factor. The value function $V^{\pi}(s)$ is
defined as the expected return starting from $s$, and successively following
policy $\pi$.
To handle the unknown transitions and environment complexity, we train our
policy network illustrated in Fig. 2b using model-free deep RL. During RL
training, we fix the trainable weights of the encoder. For every $l$
timesteps, we feed the trajectories of the surrounding vehicles over the past
$l$ timesteps to the encoder and infer the driving style $z_{i}^{t}$ of each
driver $i$. To improve efficiency, we only update the latent states of the
drivers in the lanes that the ego car has not passed yet, as shown in Fig. 5.
Since the ego car is not allowed to move backward, the latent states of the
drivers that the ego car has already passed or the drivers that have already
yielded to the ego car have no effect on the ego car’s decisions and thus are
not updated anymore.
Our policy network is a GRU with an attention module. We concatenate the state
of each driver ${u}^{t}_{i}$ with the state of the ego vehicle ${o}_{0}$ to
obtain ${q}_{i}^{t}=[{u}^{t}_{i},{o}_{0}^{t}]$, where
${q}_{i}^{t}\in\mathbb{R}^{8}$ and $i\in\\{1,...,n\\}$. We feed each
concatenated state ${q}_{i}^{t}$ into an attention module which assigns
attention weights to each surrounding vehicle. Specifically, we embed
${q}_{i}^{t}$ using a multi-layer perceptron (MLP) denoted as $f_{emb}$ and
calculate the mean of the embeddings of each surrounding vehicle:
$m^{t}=\frac{1}{n}\sum^{n}_{i=1}{e}_{i}^{t},\quad{e}_{i}^{t}=f_{emb}({q}_{i}^{t})$
(5)
where ${m}^{t}$ is the resulting mean. The weighted feature of each
surrounding vehicle ${c}_{i}^{t}$ is calculated by
${c}_{i}^{t}=\alpha_{i}^{t}\cdot{e}_{i}^{t},\quad\alpha_{i}=f_{attn}([{e}_{i}^{t},{m}^{t}])$
(6)
where $\alpha_{i}^{t}\in\mathbb{R}$ is the attention score for the $i$-th
vehicle, and $f_{attn}$ is another MLP. We then concatenate the sum of
$c_{1}^{t},...,c_{n}^{t}$ with the state of the ego vehicle ${o}_{0}^{t}$ and
feed the result to a GRU:
$h^{t}_{\pi}=\mathrm{GRU}\left(h^{t-1}_{\pi},\left[\sum^{n}_{i=1}c_{i}^{t},o_{0}^{t}\right]\right)$
(7)
Finally, the hidden state of the GRU $h^{t}_{\pi}$ is fed to a fully connected
layer to obtain the value $V(s_{t})$ and the policy $\pi(a_{t}|s_{t})$. We use
Proximal Policy Optimization (PPO), a model-free policy gradient algorithm,
for policy and value function learning [38].
Figure 3: Visualizations of the latent representations of two driver traits.
(a) Our method. (b) The method by Morton et al. (c) The original processed
trajectories corresponding to the labeled latent vectors in (a) and (b). The
$x$-axis is the horizontal displacement in meters. Each triangle marker
indicates the position of a car at each timestep and the markers become darker
over time. Denser triangles indicate smaller velocities. The blue and red
trajectories indicate conservative and aggressive cars respectively. The brown
trajectories indicate the front cars. If the front car goes out of the
boundary before the trajectory ends, the brown trajectories will be shorter
than the red or blue trajectories, such as #1 and #2.
## IV Experiments and results
In this section, we first describe the simulation environment for training. We
then present our experiments and results for trait representation and
navigation policy.
### IV-A Simulation environment
Fig. 5 shows our simulation environment adapted from [3]. At the beginning of
an episode, the surrounding vehicles are randomly placed in a two-way street
with opposite lanes and we assume that they never take turns or change lanes.
The number of surrounding vehicles varies as vehicles enter into or exit from
the T-intersection. We assume that all cars can always be detected and
tracked. The surrounding vehicles are controlled by IDM [36]. Conservative
drivers vary their front gaps from the preceding vehicles between $0.5m$ and
$0.7m$ and have the desired speed of $2.4m/s$. Aggressive drivers vary their
front gaps between $0.3m$ and $0.5m$ and have the desired speed of $3m/s$. If
the ego car moves in front of other cars, conservative drivers will yield to
the ego car, while aggressive drivers will ignore and collide with the ego
car.
The ego car starts at the bottom of the T-intersection. The goal of the ego
car is to take a right turn to merge into the upper lane without colliding
with other cars. The ego car is controlled by a longitudinal PD controller
whose desired speed is set by the RL policy network. The right-turn path of
the ego car is fixed to follow the traffic rule. The ego car also has a safety
checker that clips the magnitude of its acceleration if it gets dangerously
close to other cars.
Let $S_{goal}$ be the set of goal states, where the ego vehicle successfully
makes a full right-turn, and $S_{fail}$ be the set of failure states, where
the ego vehicle collides with other vehicles. Let
$r_{speed}(s)=0.05\times\lVert v_{ego}\rVert_{2}$ be a small reward on the
speed of the ego vehicle and $r_{step}=-0.0013$ be a constant penalty that
encourages the ego vehicle to reach the goal as soon as possible. The reward
function is defined as
$\begin{split}\begin{gathered}r(s,a)=\begin{cases}2.5,&\text{if }s\in
S_{goal}\\\ -2,&\text{if }s\in S_{fail}\\\
r_{speed}(s)+r_{step},&\text{otherwise}.\end{cases}\end{gathered}\end{split}$
(8)
### IV-B Trait inference
#### IV-B1 Baseline
We compare the latent representation of our method with the method proposed by
Morton et al [8].
TABLE I: Testing results of simple supervised classifiers with learned latent representations Method | Accuracy
---|---
Ours | 98.08%
Morton et al. | 60.22%
The encoder of the baseline is the same as our encoder except that the
baseline takes the longitudinal acceleration at each timestep as an extra
input. The policy network of the baseline is a 4-layer multilayer perceptron
(MLP) network that imitates the IDM policy from the trajectory dataset.
#### IV-B2 Training and evaluation
Our dataset contains approximately $696,000$ trajectories from both classes,
and the train/test split ratio is $2$:$1$. We train both methods for $1000$
epochs with a decaying learning rate $5\times 10^{-4}$. The weight of the KL
divergence loss $\beta$ is $5\times 10^{-8}$ for both methods.
To better understand the learned latent representation and how it provides
trait information to the navigation policy, we fix the trainable weights of
the encoders, train linear support vector classifiers using the inferred
latent states as inputs, and record the testing accuracy of the classifiers.
To visualize the learned representations, we feed a set of testing
trajectories into the encoder and visualize their latent representations.
#### IV-B3 Results and interpretability
In Table I, the supervised classifier with our latent representation achieves
much higher classification accuracy than that of Morton et al. Together with
Fig. 3a, we show that our representation successfully separates the vehicle
trajectories with different traits in most cases. For example, in Fig. 3c, #1,
#5, and #6 are in the aggressive cluster, while #2, #7, and #8 are in the
conservative cluster. The overlapped region encodes the trajectories with
ambiguous traits, such as very short trajectories (#3) and the trajectories
with ambiguous front gaps. For example, the average front gap of #4 is between
the aggressive #6 and the conservative #8. Besides the binary traits, our
latent representation also captures other meaningful information. First, the
trajectories of the cars whose front cars go out of the border, such as #1 and
#2, are mapped into the fan-shaped peripherals of the two clusters
respectively. Second, in the central regions of the two clusters, the
trajectories with larger average speeds are mapped in the lower left part (#5
and #7), while those with smaller average speeds are in the upper right part
(#6 and #8).
From Table I and Fig. 3b, the baseline suffers from severe model collapse and
the representation fails to separate the two traits. The reason is that the
MLP policy only considers current state-action pairs, which encourages the
encoder to only encode features within short time windows such as
accelerations while ignoring the properties of the trajectories such as
traits. Despite the model collapse, the baseline still learns some meaningful
representations. For example, the trajectories with uniform speeds together
such as #5 and #7 and forms separate clusters for the decelerating
trajectories such as #6. Therefore, we conclude that compared with single
states, trajectories exhibit richer information about traits and are better
suited for trait representation learning.
### IV-C Navigation with inferred traits
#### IV-C1 Baselines and ablations
We use the following two baselines: (1) The pipeline proposed by Ma et al,
which trains a supervised trait predictor and an RL policy with binary ground
truth trait labels separately and combines them at test time [3]; (2) We use
the latent representation by Morton et al to train a policy network using the
same method described in Sec. III-C. In addition, we use an RL policy trained
with ground truth labels as an oracle, and another RL policy trained without
any trait information as a naïve model. To examine the effectiveness of the
attention mechanism, we train an ablated model of our method without the
attention module. For a fair comparison, the architectures of all policy
networks are kept the same.
#### IV-C2 Training
We run three experiments with different proportions of two types of drivers,
as shown in Fig. 4. We train the policy networks for all methods and ablations
for $1\times 10^{7}$ timesteps with a decaying learning rate $1\times
10^{-4}$. To accelerate and stabilize training, we run twelve instances of the
simulation environment in parallel for collecting the ego car’s experiences.
At each policy update, 30 steps of 6 episodes are used. For Ma et al, we
pretrain a trait classification network with a $96\%$ testing accuracy and use
the classifier to infer traits for the RL policy.
#### IV-C3 Evaluation
We test all models with $500$ random unseen test cases. We measure the
percentage of success, collision, and timeout episodes as the evaluation
criteria.
Figure 4: Success, timeout, and collision rates w.r.t. different driver trait
distributions. $P(\textrm{conservative})$ is the probability for each
surrounding driver to be conservative. The task difficulty increases as
$P(\textrm{conservative})$ decreases. The numbers on the bars indicate the
percentages of the corresponding bars. Figure 5: Qualitative result of our
method. The ego car is in yellow, the conservative cars are in blue, and the
aggressive cars are in red. As mentioned in Sec. III-C, the latent traits of
the light blue and light red cars are updated periodically, while those of the
dark blue and dark red cars are not updated and stay the same as before.
#### IV-C4 Results
As shown in Fig. 4, the success rates of our method are closest to the oracle
who has access to true trait labels in all experiments, with an average
difference of $2\%$. We believe the main reason is that our trait
representation effectively captures the trait differences of the surrounding
drivers and aids the decision-making in RL. The performance gap between the
oracle and our method is caused by the drivers with ambiguous traits, as well
as the fact that the learned representation is noisier than the true labels.
Although the model with no labels can implicitly infer traits to some extent
in RL training, our policy is able to utilize the existing trait
representation and focuses more on the decision-making of the ego vehicle,
which leads to better navigation performance. Fig. 5 shows a successful
episode of our method, where the ego car waits until a conservative car
appears, cuts in the front of the conservative cars when passing both lanes,
and completes the right turn.
Compared with our model, the model by Morton et al has a lower success rate
especially in more challenging experiment settings when
$P(\textrm{conservative})$ is smaller. The reason is that the latent
representation does not distinguish between different traits and only provides
very limited useful information to RL. This implies that the policy still
needs to implicitly infer the traits while being distracted by the latent
representation, which negatively affects the performance.
For Ma et al, the trait classifier and the RL policy both have good
performance when tested separately. However, when the two modules are combined
together, intermediate and cascading errors significantly lower the success
rates. The performance drop is due to the distribution shift between the true
traits and inferred traits. Since the policy is trained with true traits, it
fails easily whenever the trait classifier makes a small mistake. Moreover, Ma
et al requires trait labels, but our trait representation is learned without
any labels and still outperforms Ma et al in navigation.
### IV-D Attention vs. no attention
The second from the rightmost graph in Fig. 4 shows that the removal of the
attention module results in $3\%\sim 14\%$ lower success rates. Since the
attention module assigns different weights to the cars with different traits
and in different positions, the policy is able to focus on the cars which have
a bigger influence on the ego car, which validates the necessity of the
attention mechanism.
## V Conclusions and future work
We propose a novel pipeline that encodes the trajectories of drivers to a
latent trait representation with a VAE+RNN network. Then, we use the trait
representation to improve the navigation of an autonomous vehicle through an
uncontrolled T-intersection. The trait representation is learned without any
explicit supervision. Our method outperforms baselines in the navigation task
and the trait representation shows interpretability. Possible directions to
explore in future work include (1) validating our method with real driving
trajectory data, (2) generalizing our model to more sophisticated driver
internal states and more navigation tasks, and (3) incorporating occlusions
and limited sensor range of the ego vehicle to close the gap between the
simulation and the real world.
## Acknowledgements
We thank Xiaobai Ma for thoughtful discussions and for providing the baseline
code. We thank Zhe Huang and Tianchen Ji for feedback on paper drafts.
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|
Miyachi’s Theorem For the Quaternion Fourier Transform
Youssef El Haoui1,111Corresponding author., Said Fahlaoui1
1Department of Mathematics and Computer Sciences, Faculty of Sciences, Equipe
d’Analyse Harmonique et Probabilités, University Moulay Ismail, BP 1120,
Zitoune, Meknes, Morocco
E-MAIL<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
The quaternion Fourier transform (QFT) satisfies some uncertainty principles
similar to the Euclidean Fourier transform. In this paper, we establish
Miyachi’s theorem for this transform and consequently generalize and prove the
analogue of Hardy’s theorem and Cowling-Price uncertainty principle in the QFT
domain.
Key words: Quaternion Fourier transform, Miyachi’s theorem.
## 1 Introduction
Uncertainty principle (UP) is an important tool in harmonic anlysis; it states
that a nonzero function and its Fourier transform cannot both be very rapidly
decreased. UP has implications in different areas like quantum physics,
information processing, signal analysis, etc.
In signal analysis, it gives in general a lower bound for the simultaneous
localization of signals in phase and frequency spaces. There are many
advantageous ways to get the statement about localization precise; examples
include theorems of Hardy theorem [11], Beurling [13], and Miyachi [17] which
interpreted differently the localisation, as sharp pointwise estimates of a
signal and its Fourier transform. More precisely Miyachi’s theorem asserts
that if $f$ is a measurable function on $\mathbb{R}$ satisfying :
$\displaystyle e^{{\alpha x}^{2}}f\in
L^{1}({{\mathbb{R}}})+L^{\infty}({{\mathbb{R}}}),$
and
$\displaystyle\int_{{\mathbb{R}}}{{\log}^{+}}\left(\frac{\left|{\hat{f}}(y)e^{{\frac{{\pi}^{2}}{\alpha}y^{2}}}\right|}{\rho}\right)\
{\hbox{d}}y<\infty,$
for some positive constants $\alpha$ and $\rho$, where
$\displaystyle\log^{+}(x)={\left\\{\begin{array}[]{ll}\log(x),&{}\text{if
}x>1.\\\ 0&{}\text{otherwise}.\end{array}\right.}$
and $\hat{f}$ stands for the classical Fourier transform of $f$,
then $f$ is a constant multiple of the Gaussian $e^{{-\alpha x}^{2}}$.
Miyachi’s theorem has been extended in several different directions in recent
years, including extensions to Dunkl transform [5], Clifford–Fourier
transform[9], and much more generally, to nilpotent lie groups [1] and
Heisenberg motion groups [2].
The quaternion Fourier transform (QFT) is a non-trivial extension of the real
and complex classical Fourier transform to the algebra of the quaternions.
Since the quaternion multiplication is non-commutative, there are three types
of the QFT depending on which side multiplication of the kernel is done, that
is the so-called left-sided, right-sided and the two-sided QFT.
The QFT was introduced at first by Ell [10] for the analysis linear time-
invariant partial differential systems and then applied in color image
processing.
Later, Bülow [3] investigated the important properties of the two-sided QFT
for real signals and applied it to signal and image processing. Furthermore,
several uncertainty principles have been formulated for the quaternion Fourier
transform.
In [15], the authors generalized a component-wise UP for the right-sided QFT.
The directional UP related to the two-sided QFT was proposed in [12].
Recently, in [4], the authors established logarithmic UP associated with the
QFT. Meanwhile, Mawardi [16] obtained the connection between the QFT and
quantum mechanics and then established the modified UP (full UP) for the two-
sided QFT.
Our contribution to these developments is that we propose a new UP for the
QFT, namely Miyachi’s UP.
So far, no such uncertainty principle for the QFT (one-sided or two-sided) had
been established. In our previous works, other UPs: Heisenberg, Hardy[7], and
Beurling[8], have been extended for the two-sided QFT. Also, we derived in [8]
the UPs of Cowling-Price and Hardy using the extension of Beurling theorem in
a quaternion framework. Here, we will obtain, in a different way, by the main
result of Miyachi, the same UPs of Cowling-Price and Hardy in QFT domain. The
techniques used here are also applicable for the left-sided and the right-
sided QFT as well.
Our paper is organized as follows. In Sect. 2, we review some basic notions
and notations related to the quaternion algebra. In Sect. 3, we recall the
definition and some results for the quaternion Fourier transform useful in the
sequel. In Sect. 4, we prove Miyachi’s theorem for the quaternion Fourier
transform, and provide an extension of certain UPs to the quaternion Fourier
transform domain. In Sect. 5, we conclude the paper.
## 2 The Algebra of Quaternions
In order to extend complex numbers to a four-dimensional algebra, the Irish W.
R. Hamilton invented in 1843 the quaternion algebra $\mathbb{H}$.
Any quaternion $q\in\mathbb{H}$ can be expressed by
$q=q_{0}+\bm{i}q_{1}+\bm{j}q_{2}+\bm{k}q_{3};\
q_{0},q_{1},q_{2},q_{3}\in\mathbb{R},$
where $\bm{i},\bm{j},\bm{k}$ satisfy Hamilton’s rules
$\displaystyle\bm{i}^{2}=\bm{j}^{2}=\bm{k}^{2}=-1,\
\bm{i}\bm{j}=-\bm{j}\bm{i}=\bm{k},$ $\bm{j}\bm{k}=-\bm{k}\bm{j}=\bm{i};\
\bm{k}\bm{i}=-\bm{i}\bm{k}=\bm{j}.$
Quaternions are isomorphic to the Clifford algebra ${Cl}_{(0,2)}$ of
${\mathbb{R}}^{(0,2)}$:
$\mathbb{H}\cong Cl_{(0,2)}.$ (2.1)
We define the conjugation of $q\in\mathbb{H}$ by:
$\overline{q}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}q_{0}-\bm{i}q_{1}-\bm{j}q_{2}-\bm{k}q_{3}$
and its modulus $|q|_{Q}$ is defined by
${|q|}_{Q}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}\sqrt{q\overline{q}}=\sqrt{q^{2}_{0}+q^{2}_{1}+q^{2}_{2}+q^{2}_{3}}.$
Particularly, when $q=q_{0}$ is a real number, the module ${|q|}_{Q}$ reduces
to the ordinary Euclidean module $\left|q\right|=\sqrt{q^{2}_{0}}$. Also, we
observe that for $\bm{x}\in{\mathbb{R}}^{2},\
{\left|\bm{x}\right|}_{Q}=\left|\bm{x}\right|,$ where $\left|.\right|\ $is the
Euclidean norm $\left|(x_{1},x_{2})\right|^{2}=x^{2}_{1}+x^{2}_{2}$
Moreover, for arbitrary $p,q\in\mathbb{H}$ the following identity holds
${|pq|}_{Q}={|p|}_{Q}{|q|}_{Q}.$
Clearly, the inverse of $0\neq q\in\mathbb{H}$ is defined by :
$q^{-1}=\frac{\overline{q}}{|q|^{2}_{Q}}.$
which shows that $\mathbb{H}$ is a normed division algebra.
Due to (2.1), we recall the following properties:
if $\bm{x}$ is a vector in ${Cl}_{(0,2)}$, then
${\left|\bm{x}\right|}^{2}=-\bm{x}^{2}.$ (2.2)
Let (,) be the inner product on ${\mathbb{R}}^{(0,2)}$; for 2.3rs $\bm{x}$ and
$\bm{y}$ we have
$(\bm{x},\bm{y})~{}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}\sum^{2}_{l=1}{x_{l}}y_{l}=-\frac{1}{2}(\bm{x}\bm{y}+\bm{y}\bm{x}).$
(2.3)
In this paper, we will study the quaternion-valued signal
$f:\mathbb{R}^{2}\to\mathbb{H}$, which can be written in this form
$f=f_{0}+\bm{i}f_{1}+\bm{j}f_{2}+\bm{k}f_{3},$
with $f_{m}~{}:{\mathbb{R}^{2}}\to\ {\mathbb{R}}\ for\ m=0,1,2,3.$
We introduce the Banach spaces
$L^{p}\left({\mathbb{R}}^{2},\mathbb{H}\right)$, $1\leq p\leq\infty$,
$L^{p}\left({\mathbb{R}}^{2},\mathbb{H}\right)\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}\
\\{f|f:\mathbb{R}^{2}\rightarrow\mathbb{H},{\left|f\right|}_{p,Q}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}({\int_{{\mathbb{R}}^{2}}{{\left|f(\bm{x})\right|}^{p}_{Q}\
d\bm{x}}})^{\frac{1}{p}}<\infty\\},1\leq p<\infty,$
$L^{\infty}\left({\mathbb{R}}^{2},\mathbb{H}\right)\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}\
\\{f|f:\mathbb{R}^{2}\rightarrow\mathbb{H},{\left|f\right|}_{\infty,Q}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}ess\
{sup}_{\bm{x}\in\mathbb{R}^{2}}{\left|f(\bm{x})\right|}_{Q}<\infty\\}.$
where $d\bm{x}=dx_{1}dx_{2},$ refers to the usual Lebesgue measure in
$\mathbb{R}^{2}.$
If $f\ {\in L}^{\infty}\left({\mathbb{R}}^{2},\mathbb{H}\right)$ is
continuous, then
${\left|f\right|}_{\infty,Q}={sup}_{{\bm{x}\in{\mathbb{R}}}^{2}}{{\left|f(\bm{x})\right|}_{Q}}$
Furthermore, we define naturally the two following Banach spaces
$L^{1}(\mathbb{R}^{2},\mathbb{H})\cap
L^{\infty}(\mathbb{R}^{2},\mathbb{H})=\\{f|f\in
L^{1}(\mathbb{R}^{2},\mathbb{H})\ \textnormal{and}\ f\in
L^{\infty}(\mathbb{R}^{2},\mathbb{H})\\},$
$L^{1}(\mathbb{R}^{2},\mathbb{H})+L^{\infty}(\mathbb{R}^{2},\mathbb{H})=\\{f=f_{1}+f_{2},f_{1}\in
L^{1}(\mathbb{R}^{2},\mathbb{H}),\ f_{2}\in
L^{\infty}(\mathbb{R}^{2},\mathbb{H})\\}.$
We denote by ${\mathcal{S}}(\mathbb{R}^{2},\mathbb{H})$ the quaternion
Schwartz test function space, i.e., the set $C^{\infty}$ of smooth functions
$f$, from ${{\mathbb{R}}}^{2}$ to $\mathbb{H}$, given by
$\mathcal{S}(\mathbb{R}^{2},\mathbb{H})\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}\\{f\in
C^{\infty}(\mathbb{R}^{2},\mathbb{H}):{sup}_{\bm{x}\in\mathbb{R}^{2},\
{|\alpha|\leq
n}}{({\left(1+\left|\bm{x}\right|^{m}\right)}\partial^{\mathbf{\alpha}}{\left|f(\bm{x})\right|}_{Q})}<\infty,\
m,n\in{\mathbb{N}}\\},$
where $\partial^{\mathbf{\alpha}}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}\frac{\partial^{|\alpha|}}{\partial^{\alpha_{1}}_{x_{1}}\partial^{\alpha_{2}}_{x_{2}}}$,
$|\alpha|=\alpha_{1}+\alpha_{2}$ for a multi-index
$\alpha=({\alpha}_{1},{\alpha}_{2})\in\mathbb{N}^{2}$.
## 3 Quaternion Fourier Transform
In this section, we review the definition and some properties of the two-sided
QFT.
###### Definition 3.1.
Let$\ f$ in $L^{1}\left({\mathbb{R}}^{2},\mathbb{H}\right)$. Then, the two-
sided quaternion Fourier transform of the function $f$ is given by
$\mathcal{F}\\{f\\}(\bm{\xi})\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}\int_{{\mathbb{R}}^{2}}{e^{-\bm{i}{2\pi\xi}_{1}x_{1}}}f(\bm{x})e^{-\bm{j}{2\pi\xi}_{2}x_{2}}d\bm{x},$
(3.1)
where $\bm{\xi},\bm{x}\in{\mathbb{R}}^{2}.$
###### Lemma 3.2.
Inverse QFT [3, Thm. 2.5]
If $f\in L^{2}\left(\mathbb{R}^{2},\mathbb{H}\right),and\
\mathcal{F}\\{f\\}\in L^{1}\left(\mathbb{R}^{2},\mathbb{H}\right)$, then the
two-sided QFT is an invertible transform and its inverse is given by
$f(\bm{x})=\int_{\mathbb{R}^{2}}{e^{\bm{i}2\pi{\xi}_{1}x_{1}}}\mathcal{F}\\{f\\}(\bm{\xi})e^{\bm{j}{2\pi\xi}_{2}x_{2}}d\bm{\xi},\
\ d\bm{\xi}=d\xi_{1}d\xi_{2}.$
###### Lemma 3.3.
Scaling property
Let $\alpha$ be a positive scalar constant; then, the two-sided QFT of
${f}_{\alpha}\left({\bm{x}}\right)={f}(\alpha{\bm{x}})$ becomes
$\mathcal{F}\left\\{f_{\alpha}\right\\}(\bm{\xi})={\left(\frac{1}{\alpha}\right)}^{2}\mathcal{F}\left\\{f\right\\}(\frac{1}{\alpha}\bm{\xi}).$
(3.2)
Proof. Equation (3.1) gives
$\displaystyle\mathcal{F}\\{{{f}}_{\alpha}\\}(\bm{\xi})=\int_{{\mathbb{R}}^{2}}{e^{-\bm{i}{2\pi\xi}_{1}x_{1}}}\
f(\alpha\bm{x})\ e^{-\bm{j}{2\pi\xi}_{2}x_{2}}d\bm{x}.$
We substitute $\bm{y}$ for $\alpha\bm{x}$ and get
$\displaystyle\mathcal{F}\\{{{f}}_{\alpha}\\}(\bm{\xi})$ $\displaystyle=$
$\displaystyle{\left(\frac{1}{\alpha}\right)}^{2}\int_{{\mathbb{R}}^{2}}{e^{-\bm{i}2\pi(\frac{1}{\alpha}{\xi}_{1})y_{1}}}f(\bm{y})e^{-\bm{j}2\pi(\frac{1}{\alpha}{\xi}_{2})y_{2}}d\bm{y}$
$\displaystyle=$
$\displaystyle{\left(\frac{1}{\alpha}\right)}^{2}\mathcal{F}\left\\{f\right\\}(\frac{1}{\alpha}\bm{\xi}).$
The next lemma states that the QFT of a Gaussian quaternion function is
another quaternion Gaussian quaternion function.
###### Lemma 3.4.
QFT of a Gaussian quaternion function.
Consider a two-dimensional Gaussian quaternion function $f$ given by
$f(\bm{x})=qe^{-(\alpha_{1}x_{1}^{2}+\alpha_{2}x_{2}^{2})},$
where $q=q_{0}+iq_{1}+jq_{2}+kq_{3}$ is a constant quaternion, and
$\alpha_{1},\alpha_{2}$ are positive real constants.
Then
$\mathcal{F}\\{f\\}(\bm{\xi})=q\frac{\pi}{\sqrt{\alpha_{1}\alpha_{2}}}e^{-\frac{\pi^{2}}{\alpha_{1}}\xi_{1}^{2}-\frac{\pi^{2}}{\alpha_{2}}\xi_{2}^{2}}.$
(3.3)
where $\bm{x},\bm{\xi}\in{\mathbb{R}}^{2}.$
Proof. Let $g$ be defined by
$g(\bm{x})\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}e^{-(\alpha_{1}x_{1}^{2}+\alpha_{2}x_{2}^{2})},$ we have
$\mathcal{F}\\{f\\}(\bm{\xi})=q_{0}\mathcal{F}\\{g\\}(\bm{\xi})+\bm{i}q_{1}\mathcal{F}\\{g\\}(\bm{\xi})+q_{2}\mathcal{F}\\{g\\}(\bm{\xi})\bm{j}+q_{3}\bm{i}\mathcal{F}\\{g\\}(\bm{\xi})\bm{j},$
where we used the $\mathbb{R}$-linearity of the QFT and the properties
$\mathcal{F}\\{\bm{i}h\\}=\bm{i}\mathcal{F}\\{h\\},\mathcal{F}\\{\bm{j}h\\}=\mathcal{F}\\{h\\}\bm{j},$
and $\mathcal{F}\\{\bm{k}h\\}=\bm{i}\mathcal{F}\\{h\\}\bm{j}$ for $h$ real-
valued function.
On the other hand, we have
$\displaystyle\mathcal{F}\\{g\\}(\bm{\xi})$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{2}}{e^{-\bm{i}{2\pi\xi}_{1}x_{1}}}g(\bm{x})e^{-\bm{j}{2\pi\xi}_{2}x_{2}}d\bm{x}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}e^{-\alpha_{1}x_{1}^{2}}{e^{-\bm{i}{2\pi\xi}_{1}x_{1}}}dx_{1}\int_{\mathbb{R}}e^{-\alpha_{2}x_{2}^{2}}e^{-\bm{j}{2\pi\xi}_{2}x_{2}}dx_{2}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{\pi}{\alpha_{1}}}e^{-\frac{\pi^{2}}{\alpha_{1}}\xi_{1}^{2}}\sqrt{\frac{\pi}{\alpha_{2}}}e^{-\frac{\pi^{2}}{\alpha_{2}}\xi_{2}^{2}}$
$\displaystyle=$
$\displaystyle\frac{\pi}{\sqrt{\alpha_{1}\alpha_{2}}}e^{-\frac{\pi^{2}}{\alpha_{1}}\xi_{1}^{2}-\frac{\pi^{2}}{\alpha_{2}}\xi_{2}^{2}}.$
This completes the proof of Lemma 3.4.∎
###### Lemma 3.5.
[8, Lemma 3.11]
Let $f:\mathbb{R}^{2}\to\mathbb{R}$ be of the form
$f\left(\bm{x}\right)=P(\bm{x})e^{-\pi\alpha{\left|\bm{x}\right|}^{2}},$
where $P$ is a polynomial and $\alpha>0,$
Then
$\mathcal{F}\\{f\\}(\bm{\xi}{\rm)}=\ Q(\bm{\xi})\
e^{-\frac{\pi}{\alpha}{\left|\bm{\xi}\right|}^{2}},$
where $Q$ is a quaternion polynomial with $degP=degQ.$
## 4 Miyachi’s theorem
In this section, we prove Miyachi’s theorem for the quaternion Fourier
transform. For this, we need the following technical lemma of the complex
analysis.
###### Lemma 4.1.
[5, Lemma 1]
Let $h$ be an entire function on ${\mathbb{C}}^{2}$ such that
${\left|h\left(\bm{z}\right)\right|}\leq{Ae}^{B{\left|Re(\bm{z})\right|}^{2}}\
for\ all\ \bm{z}\in{\mathbb{C}}^{2}$
and
$\int_{{\mathbb{R}}^{2}}{{\log}^{+}}\left({\left|h\left(\bm{y}\right)\right|}\right)d\bm{y}<\infty,$
for some positive constants $A$ and $B$.
Then $h$ is a constant function.
###### Theorem 4.2.
(Miyachi’s Theorem).
Let $\alpha$, $\beta>0$. Suppose that $f$ is a measurable function such that
$e^{{\alpha\left|\bm{x}\right|}^{2}}f\in
L^{1}({\mathbb{R}}^{2},\mathbb{H})+L^{\infty}({\mathbb{R}}^{2},\mathbb{H}),$
(4.1)
and
$\int_{{\mathbb{R}}^{2}}{{\log}^{+}}(\frac{{\left|\mathcal{F}\left\\{f\right\\}(\bm{y})e^{{\beta\left|\bm{y}\right|}^{2}}\right|}_{Q}}{\rho})\
d\bm{y}<\infty,$ (4.2)
for some $\rho,\ 0<\rho<+\infty$
Then, three cases can occur:
1. (i)
If $\alpha\beta>{\pi}^{2},\ $ then $\ f=0$ almost everywhere.
2. (ii)
If $\ \alpha\beta={\pi}^{2},\ $then
$f\left(\bm{x}\right)=Ce^{{-\alpha\left|\bm{x}\right|}^{2}},$ where $C$ is a
constant quaternion.
3. (iii)
$If\ \alpha\beta<{\pi}^{2},\ $then there exist infinitely many functions
satisfying (4.1) and (4.2).
Proof.
1. (i)
We first prove the result for the case $\alpha\beta={\pi}^{2}.$
By scaling, we can assume that $\alpha=\beta=\pi.$
Indeed, let $g\left(\bm{x}\right)=f(\sqrt{\frac{\pi}{\alpha}}\ \bm{x})$; then,
by Lemma 3.3 we obtain
$\mathcal{F}\left\\{g\right\\}\left(\bm{t}\right)=\frac{\alpha}{\pi}\
\mathcal{F}\left\\{f\right\\}(\sqrt{\frac{\alpha}{\pi}}\bm{t}),\ \ \
\bm{t}\in\mathbb{R}^{2}.$ (4.3)
In addition, we get by (4.1)
$\displaystyle
e^{{\pi\left|\bm{x}\right|}^{2}}g(\bm{x})=e^{{\pi\left|\bm{x}\right|}^{2}}f(\sqrt{\frac{\pi}{\alpha}}\
\bm{x})\in
L^{1}({\mathbb{R}}^{2},\mathbb{H})+L^{\infty}({\mathbb{R}}^{2},\mathbb{H}).$
Also, we have
$\displaystyle\int_{\mathbb{R}^{2}}{{\log}^{+}}\left(\frac{{\left|\mathcal{F}\left\\{f\right\\}\left(\bm{y}\right)e^{{\beta\left|\bm{y}\right|}^{2}}\right|}_{Q}}{\rho}\right)d\bm{y}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{\pi}\int_{{\mathbb{R}}^{2}}{{\log}^{+}}\left(\frac{{\left|\mathcal{F}\left\\{f\right\\}\left(\sqrt{\frac{\alpha}{\pi}}\bm{t}\right)e^{{\frac{\alpha\beta}{\pi}\left|\bm{t}\right|}^{2}}\right|}_{Q}}{\rho}\right)d\bm{t}$
$\displaystyle\overset{\makebox{\mbox{\eqref{revision}}}}{=}$
$\displaystyle\frac{\alpha}{\pi}\int_{{\mathbb{R}}^{2}}{{\log}^{+}}\left(\frac{{\left|\mathcal{F}\left\\{g\right\\}\left(\bm{t}\right)e^{{\pi\left|\bm{t}\right|}^{2}}\right|}_{Q}}{\rho^{{}^{\prime}}}\right)d\bm{t}<\infty,$
where ${\rho}^{{}^{\prime}}=\frac{\alpha}{\pi}\rho.$
If the result is shown for $\alpha=\beta=\pi,$
then $g\left(\bm{x}\right)=Ce^{{-\pi\left|\bm{x}\right|}^{2}}$, and thus,
$f\left(\bm{x}\right)=\ g\left(\sqrt{\frac{\alpha}{\pi}}\bm{x}\right)=C\
e^{{-\alpha\left|\bm{x}\right|}^{2}}.$
Now, we assume that $\alpha=\beta=\pi:$
Applying the same method as in [7, Thm. 5.3], by complexifying the variable
$\bm{z}=\bm{a}+i_{{\mathbb{C}}}\bm{b},$
where $\bm{a}=(a_{1},a_{2}),\bm{b}=(b_{1},b_{2})\in\mathbb{R}^{2}$, and we
note by $i_{{\mathbb{C}}}$ the complex number which satisfies
$i^{2}_{\mathbb{C}}=-1.$
We have
${\left|\bm{z}\right|}^{2}_{Q}={\left|\bm{a}\right|}^{2}+{\left|\bm{b}\right|}^{2}={\left|\bm{z}\right|}^{2},$
where $\left|.\right|\ $is the Euclidean norm in ${\mathbb{C}}^{2}.$
Let
$\displaystyle w\left({\bm{x}}\right)$ $\displaystyle=$
$\displaystyle{{e}}^{\pi({\left|{{a}}_{1}\right|^{2}{+}\left|{{b}}_{1}\right|^{2})}}{{e}}^{\pi({\left|{{a}}_{2}\right|^{2}{+}\left|{{b}}_{2}\right|^{2})}}{{e}}^{{-}\pi({\left|x_{1}\right|-(\left|{{a}}_{1}\right|{+}\left|{{b}}_{1}\right|))}^{2}}{{e}}^{{-}\pi({\left|x_{2}\right|-(\left|{{a}}_{2}\right|{+}\left|{{b}}_{2}\right|))}^{2}}$
$\displaystyle=$ $\displaystyle
e^{\pi{\left|{z}\right|}^{2}}{{e}}^{{-}\pi({\left|x_{1}\right|-(\left|{{a}}_{1}\right|{+}\left|{{b}}_{1}\right|))}^{2}}{{e}}^{{-}\pi({\left|x_{2}\right|-(\left|{{a}}_{2}\right|{+}\left|{{b}}_{2}\right|))}^{2}}.$
Clearly, $w$ belongs to $L^{1}(\mathbb{R}^{2},\mathbb{H})\cap
L^{\infty}(\mathbb{R}^{2},\mathbb{H}).$
By assumption, $e^{{\pi\left|\bm{x}\right|}^{2}}f$ belongs to
$L^{1}$(${{\mathbb{R}}}^{2}{,\mathbb{H})+}L^{\infty}$(${{\mathbb{R}}}^{2}{,\mathbb{H})}.$
As
$\mathcal{F}\left\\{f\right\\}\left(\bm{z}\right)=\int_{{{\mathbb{R}}}^{2}}{e^{-2\pi\bm{i}x_{1}{(}a_{1}{+}i_{\mathbb{C}}b_{1})}f\left(\bm{x}\right)e^{-2\pi\bm{j}{\bm{x}}_{2}(a_{2}+i_{\mathbb{C}}b_{2})}d\bm{x}},$
we have
$\displaystyle\left|\mathcal{F}\left\\{f\right\\}\left(\bm{z}\right)\right|_{Q}$
$\displaystyle\leq$
$\displaystyle\int_{{{\mathbb{R}}}^{2}}{{\left|f\left(\bm{x}\right)\right|}_{Q}}e^{2\pi(\left|x_{1}a_{1}\right|+\left|x_{1}b_{1}\right|{+}\left|x_{2}a_{2}\right|+\left|x_{2}b_{2}\right|)}d\bm{x}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2}}{{\left|e^{{\pi\left|\bm{x}\right|}^{2}}f\left({\bm{x}}\right)\right|}_{Q}\
}{w}\left({\bm{x}}\right)d\bm{x}.$
Hence, $\mathcal{F}\left\\{{f}\right\\}\left(\bm{z}\right)$ is well defined,
and is an entire function on ${\mathbb{C}}^{2}.$
Furthermore, by (4.1) there exists $u\in
L^{1}({{\mathbb{R}}}^{2}{,\mathbb{H})}$ and $v\in
L^{\infty}({{\mathbb{R}}}^{2}{,\mathbb{H})}$ such that
$e^{{\pi\left|\bm{x}\right|}^{2}}f\left({\bm{x}}\right)=u\left({\bm{x}}\right)+v\left({\bm{x}}\right),$
Using the triangle inequality and the linearity of the integral we get
${\left|\mathcal{F}\left\\{f\right\\}\left(\bm{z}\right)\right|}_{Q}\leq\int_{\mathbb{R}^{2}}{{\left|u\left({\bm{x}}\right)\right|}_{Q}}{w}\left({\bm{x}}\right)d{\bm{x}}+\int_{\mathbb{R}^{2}}{{\left|v\left({\bm{x}}\right)\right|}_{Q}\
}{w}\left({\bm{x}}\right)d{\bm{x}}.$
Then according to the Hôlder’s inequality, we have
${\left|\mathcal{F}\left\\{f\right\\}\left(z\right)\right|}_{Q}\leq{\left|u\right|}_{1,Q}{\left|w\right|}_{\infty,Q}+{\left|v\right|}_{\infty,Q}{\left|w\right|}_{1,Q},$
Since
$\int_{\mathbb{R}}{e^{-\pi({\left|t\right|+m)}^{2}}}dt\leq 2,\ \
\textnormal{where}\ m\in{\mathbb{R}},$
we obtain
${\left|w\right|}_{1,Q}\leq 4e^{\pi{\left|\bm{z}\right|}^{2}},\
\textnormal{and}\ {\left|w\right|}_{\infty,Q}\leq
e^{\pi{\left|\bm{z}\right|}^{2}}.$
Then
$\displaystyle{\left|\mathcal{F}\left\\{f\right\\}\left(\bm{z}\right)\right|}_{Q}$
$\displaystyle\leq$
$\displaystyle{e^{\pi{\left|\bm{z}\right|}^{2}}\left|u\right|}_{1,Q}+4\
e^{\pi{\left|\bm{z}\right|}^{2}}{\left|v\right|}_{\infty,Q}$
$\displaystyle\leq$ $\displaystyle K\ e^{\pi{\left|\bm{z}\right|}^{2}},$
where $K$ is a positive constant independent of $z.$
Now, let
$h(\bm{z})=e^{-\pi
z^{2}}|\mathcal{F}\left\\{f\right\\}\left(\bm{z}\right)|_{Q},\
\textnormal{for}\ \bm{z}\in{\mathbb{C}}^{2},$
then, $h$ is an entire function.
By (2.2) and (2.3), we have
$\bm{z}^{2}=\left(\bm{a}+i_{\mathbb{C}}\bm{b}\right)\left(\bm{a}+i_{\mathbb{C}}\bm{b}\right)=-{\left|\bm{a}\right|}^{2}+{\left|\bm{b}\right|}^{2}-2i_{\mathbb{C}}(\bm{a},\bm{b}),$
then
${\left|e^{-\pi\bm{z}^{2}}\right|}_{Q}\leq
e^{\pi{\left|\bm{a}\right|}^{2}}e^{-\pi{\left|\bm{b}\right|}^{2}},$
where we used ${\left|e^{2\pi i_{\mathbb{C}}(\bm{a},\bm{b})}\right|}_{Q}=1$.
As a result
$\left|h\left(z\right)\right|\leq{Ke}^{\pi{\left|a\right|}^{2}}e^{-\pi{\left|b\right|}^{2}}e^{\pi{\left|a\right|}^{2}}e^{\pi{\left|b\right|}^{2}},$
$=Ke^{2\pi{\left|a\right|}^{2}}.$ (4.4)
On the other hand, by (2.2) and by assumption
$\int_{{\mathbb{R}}^{2}}{{\log}^{+}}\left(\frac{\left|h\left(\bm{y}\right)\right|}{\rho}\right)d\bm{y}=\int_{{\mathbb{R}}^{2}}{{\log}^{+}}\left(\frac{{\left|e^{\pi{\left|\bm{y}\right|}^{2}}\
\mathcal{F}\left\\{f\right\\}\left(\bm{y}\right)\right|}_{Q}}{\rho}\right)d\bm{y}<\infty.$
(4.5)
Then by (4.4),(4.5) and by applying Lemma 4.1 to the function
$h\left(\bm{y}\right){/}\rho$, we deduce that $h\left(\bm{y}\right){=const.}$
i.e
$|\frac{\mathcal{F}\left\\{f\right\\}\left(\bm{y}\right)}{e^{-\pi{\left|\bm{y}\right|}^{2}}}|_{Q}=\
const.$
Therefore
$\mathcal{F}\left\\{f\right\\}\left(\bm{y}\right)=Ce^{-\pi{\left|\bm{y}\right|}^{2}},$
where $C$ is a constant quaternion.
Then by Lemmas 3.2 and 3.4, we have
$f\left(\bm{x}\right)=C\ e^{-\pi{\left|\bm{x}\right|}^{2}}.$
2. (ii)
If $\alpha\beta>{\pi}^{2}.$
Let $g\left(\bm{x}\right)=f(\sqrt{\frac{\pi}{\alpha}}\ \bm{x})$, a simple
calculation shows that
$\displaystyle\int_{\mathbb{R}^{2}}{{\log}^{+}}\left(\frac{{\left|\mathcal{F}\left\\{g\right\\}\left(t\right)e^{{\pi\left|t\right|}^{2}}\right|}_{Q}}{\rho^{{}^{\prime}}}\right)dt$
$\displaystyle<$
$\displaystyle\frac{\pi}{\alpha}\int_{{\mathbb{R}}^{2}}{{\log}^{+}}\left(\frac{{\left|\mathcal{F}\left\\{f\right\\}\left(\bm{y}\right)e^{{\beta\left|\bm{y}\right|}^{2}}\right|}_{Q}}{\rho}\right)d\bm{y}$
$\displaystyle<$ $\displaystyle\infty,\ \ \ \ \ \ (\textnormal{by}\
\eqref{4.2})$
where ${\rho}^{{}^{\prime}}=\frac{\alpha}{\pi}\ \rho.$
Then, according to the first case
$g\left(\bm{x}\right)=C\ e^{-\pi{\left|\bm{x}\right|}^{2}},$
where $C$ is a constant quaternion.
Consequently
${f}\left({\bm{x}}\right)=C\ e^{-\alpha{\left|\bm{x}\right|}^{2}}.$
Hence, by Lemma 3.4 we get
$\mathcal{F}\left\\{f\right\\}\left(\bm{y}\right)=Ce^{-\frac{{\pi}^{2}}{\alpha}{\left|\bm{y}\right|}^{2}}.$
Refering to (4.2), $C$ must be zero.
3. (iii)
For the final case, $\alpha\beta<{\pi}^{2}$
Let ${f\left(\bm{x}\right)=\varphi}_{k,l}(\bm{x})\
e^{-\pi\gamma{\left|\bm{x}\right|}^{2}}$ with
$\frac{\alpha}{\pi}$$<$$\gamma$$<$$\frac{\pi}{\beta},$
where ${{\\{\varphi}_{k,l}\\}}_{k,l\in{\mathbb{N}}}$ is a basis of
${\mathcal{S}}({\mathbb{R}}^{2},\mathbb{H})$, which is defined by
${\varphi}_{k,l}\left(x_{1},x_{2}\right)\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}{\varphi}_{k}\left(x_{1}\right){\varphi}_{l}\left(x_{2}\right),$
for $\left(x_{1},x_{2}\right)\in\mathbb{R}^{2},$
and
${\varphi}_{k}\left(x\right)=\frac{{(-1)}^{k}}{k!}e^{\pi
x^{2}}\frac{d^{k}}{{dx}^{k}}(e^{-{2}\pi\bm{x}^{2}}),x\in\mathbb{R}.$
It is important to see that${{\\{\varphi}_{k,l}\\}}_{k,l\in\mathbb{N}}$ is a
basis of ${\mathcal{S}}(\mathbb{R}^{2},\mathbb{H})$ (see [6, 7]).
We have
$\displaystyle e^{\alpha{\left|\bm{x}\right|}^{2}}f$ $\displaystyle=$
$\displaystyle
e^{\alpha{\left|\bm{x}\right|}^{2}}{\varphi}_{k,l}\left(\bm{x}\right)e^{-\pi\gamma{\left|\bm{x}\right|}^{2}}$
$\displaystyle=$
$\displaystyle{\varphi}_{k,l}\left(\bm{x}\right)e^{(\alpha-\pi\gamma){\left|\bm{x}\right|}^{2}}\in
L^{1}(\mathbb{R}^{2}{,\mathbb{H})+}L^{\infty}(\mathbb{R}^{2},\mathbb{H}).$
Lemma 3.5 implies
$\mathcal{F}\left\\{f\right\\}\left(\bm{y}\right)=Q(\bm{y})\
e^{-\frac{\pi}{\gamma}{\left|\bm{y}\right|}^{2}},$
where $Q$ is a quaternion polynomial.
Then, since $\beta<\frac{\pi}{\alpha}$,
$\displaystyle\int_{\mathbb{R}^{2}}{{\log}^{+}}\left(\frac{{\left|\mathcal{F}\left\\{f\right\\}\left(\bm{y}\right)e^{{\beta\left|\bm{y}\right|}^{2}}\right|}_{Q}}{\rho}\right)d\bm{y}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2}}{{\log}^{+}}\left(\frac{{\left|Q(\bm{y})\
e^{{(\beta-\frac{\pi}{\gamma})\left|\bm{y}\right|}^{2}}\right|}_{Q}}{\rho}\right)d\bm{y}$
$\displaystyle<$ $\displaystyle\infty.$
This completes the proof of Theorem 4.2.∎
In the following, we illustrate the effectiveness of Theorem 4.2, by giving an
example, and derive two generalizations of uncertainty principle associated
with the QFT.
###### Example 4.3.
Consider $\alpha,\beta$ two positive numbers with $\alpha\beta=\pi^{2},$ and
the quaternion Gaussian function
$f(\bm{x})=qe^{-\alpha|\bm{x}|^{2}},$
where $q=q_{0}+\bm{i}q_{1}+\bm{j}q_{2}+\bm{k}q_{3}$ is a constant quaternion.
Obviously, we have
$e^{\alpha|\bm{x}|^{2}}f=q\in L^{\infty}(\mathbb{R}^{2},\mathbb{H})\subset
L^{1}(\mathbb{R}^{2},\mathbb{H})+L^{\infty}(\mathbb{R}^{2},\mathbb{H}).$
Thus, $f$ satisfies condition (4.1).
By lemma 3.4, we get
$\mathcal{F}\\{f\\}(\bm{y})=q\frac{\pi}{\alpha}e^{-\beta|\bm{y}|^{2}}.$
Then
$\int_{{\mathbb{R}}^{2}}\log^{+}(\frac{|\mathcal{F}\\{f\\}(\bm{y})|_{Q}e^{\beta|\bm{y}|^{2}}}{\rho})d\bm{y}=\int_{{\mathbb{R}}^{2}}\log^{+}(\frac{|q|_{Q}\frac{\pi}{\alpha}}{\rho})d\bm{y}<\infty,$
whenever $\rho>|q|_{Q}\ \frac{\pi}{\alpha}.$
###### Corollary 4.4.
Hardy’s uncertainty principle for the QFT.
Let $\alpha$ and $\beta$ be both positive constants. Suppose $f$ be in
$L^{1}(\mathbb{R}^{2},\mathbb{H})$ with
1. (i)
$|f(\bm{x})|^{2}<Ce^{-\alpha|\bm{x}|^{2}}.$
2. (ii)
$|\mathcal{F}\\{f\\}(\bm{y})|^{2}<C^{{}^{\prime}}e^{-\beta|\bm{y}|^{2}}.$
for some constants $C>0$ and $C^{\prime}>0$. Then, three cases can occur :
1. 1.
If $\alpha\beta>\pi^{2},$ $f=0$ almost everywhere on $\mathbb{R}^{2}.$
2. 2.
If $\alpha\beta=\pi^{2},$ then $f$ is a constant quaternion multiple of
$e^{-\alpha|\bm{x}|^{2}}$.
3. 3.
If $\alpha\beta<\pi^{2},$ there are infinitely many linearly independent
functions satisfying both conditions $(i)$ and $(ii).$
Proof. Immediately using the decay condition $(i)$ one has
$fe^{\alpha|\bm{x}|^{2}}\in L^{\infty}(\mathbb{R}^{2},\mathbb{H}).$ Hence $f$
verifies condition (4.1) of Theorem 4.2.
Moreover, for $\rho>0$ we have
$\frac{|\mathcal{F}\\{f\\}(\bm{y})|_{Q}e^{\beta|\bm{y}|^{2}}}{\rho}\leq\frac{C^{{}^{\prime}}}{\rho}.$
Thus
$\log^{+}(\frac{|\mathcal{F}\\{f\\}(\bm{y})|_{Q}e^{\beta|\bm{y}|^{2}}}{\rho})\leq\underbrace{\log^{+}(\frac{C^{{}^{\prime}}}{\rho})}_{=0},$
whenver $\rho>C^{{}^{\prime}}.$
So condition (4.2) of Theorem 4.2 is verified. Then, direct application of
Theorem 4.2 enables us to achieve the proof.
###### Corollary 4.5.
Cowling-Price’s uncertainty principle for the QFT.
Let $\alpha$ and $\beta$ be positive real numbers, $1\leq p,q\leq\infty$ such
that $min(p,q)$ is finite, and let $f$ are a square integrable quaternion-
valued function satisfying the following decay conditions Suppose $f$ be in
$L^{1}(\mathbb{R}^{2},\mathbb{H})$
1. (i)
$\int_{\mathbb{R}^{2}}(|f(\bm{x})|_{Q}e^{\alpha|\bm{x}|^{2}})^{p}d\bm{x}<\infty.$
2. (ii)
$\int_{\mathbb{R}^{2}}(|\mathcal{F}\\{f\\}(\bm{y})|_{Q}e^{\beta|\bm{y}|^{2}})^{q}d\bm{y}<\infty.$
Then the three following conclusions hold:
1. 1.
$f=0$ almost everywhere whenever $\alpha\beta>\pi^{2}.$
2. 2.
If $\alpha\beta=\pi^{2},$ then $f$ is a constant quaternion multiple of
$e^{-\alpha|\bm{x}|^{2}}$.
3. 3.
If $\alpha\beta<\pi^{2},$ there are infinitely many linearly independent
functions satisfying both conditions $(i)$ and $(ii).$
Proof. According to (i), we get $fe^{\alpha|\bm{x}|^{2}}\in
L^{p}(\mathbb{R}^{2},\mathbb{H})$, and using the fact that $L^{p}\subset
L^{1}+L^{\infty}$ we obtain that $f$ fulfills condition (4.1) of Theorem 4.2.
Furthermore, based on the inequality
$\displaystyle\log^{+}(x)\leq x\ \ \textnormal{for}\ x\in\mathbb{R}_{+},$
we can easily see that $f$ satisfies the second condition (4.2).
Therefore, by applying Theorem 4.2 we conclude the proof.
## 5 Conclusions
In this paper, based on some obtained results of the two-sided QFT and one
technical lemma of the complex analysis, a generalization of Miyachi’s
uncertainty principle associated with the QFT was proposed. Consequently, two
variants of this UP were provided, namely the theorems of Hardy and Cowling-
Price. The extension of these qualitative UPs to the quaternionic algebra
framework shows that a quaternionic 2D signal $f$ and its QFT cannot both
simultaneously decrease very rapidly.
The QFT has proved to be very significant tool for applications in color image
processing[14], quantum mechanics, engineering, signal processing, optics,
etc. Apart from their importance to pure mathematics, our results are also
relevant to applied mathematics and signal processing.
## References
* [1] A. Baklouti, S. Thangavelu, Variants of Miyachi’s theorem for nilpotent lie groups. J. Aust. Math. Soc. 88, 1–17 (2010). https://doi.org/10.1017/S144678870900038X
* [2] A. Baklouti and S. Thangavelu, Hardy and Miyachi theorems for Heisenberg motion groups, Nagoya Math. J. 229, 1–20 (2016). https://doi.org/10.1017/nmj.2016.58
* [3] T. Bülow, Hypercomplex spectral signal representations for the processing and analysis of images, Ph.D. Thesis, Institut für Informatik und Praktische Mathematik, University of Kiel, Germany (1999)
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* [6] H. De Bie, New techniques for two-sided quaternion Fourier transform. In: Procedings of AGACSE (2012).
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* [8] Y. El Haoui, S. Fahlaoui, Beurling’s theorem for the quaternion Fourier transform. J. Pseudo-Differ. Oper. Appl. (2019). https://doi.org/10.1007/s11868-019-00281-7
* [9] J. El Kamel, R. Jday, Uncertainty Principles for the Clifford–Fourier Transform, Adv. Appl. Clifford Algebras (2017) https://doi.org/10.1007/s00006-017- 0791-1
* [10] T.A. Ell, Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems. In: Proceeding of the 32nd Conference on Decision and Control, San Antonio, Texas, pp. 1830–1841 (1993).
* [11] G.H. Hardy, A theorem concerning Fourier transform, J. Lond. Math. Soc. 8, 227–231 (1933)
* [12] E. Hitzer, Directional uncertainty principle for quaternion Fourier transform. Adv. Appl. Clifford Algebras 20, 271–284 (2010)
* [13] L. Hörmander, A uniqueness theorem of Beurling for Fourier transform pairs. Ark. Math. 2, 237–240 (1991)
* [14] K.M. Hosny, Y.M. Khedr, W.I. Khedr et al., Robust color image hashing using quaternion polar complex exponential transform for image authentication. Circuits Syst. Signal Process. 37, 5441 (2018). https://doi.org/10.1007/s00034-018-0822-8
* [15] B. Mawardi, E. Hitzer, A. Hayashi, R. Ashino, An uncertainty principle for quaternion fourier transform. Comput. Math. Appl. 56, 2398–2410 (2008)
* [16] B. Mawardi, A modified uncertainty principle for two-sided quaternion Fourier transform. Adv. Appl. Clifford Algebras 26(2), 513–527 (2016)
* [17] A. Miyachi, A generalization of theorem of Hardy, Harmonic AnalysisSeminar held at Izuna-gaoka, Shizuoka-Ken, Japon, pp. 44–51 (1997)
|
# On The Temporal Evolution of Particle Production in $f(T)$ Gravity
Sanjay Mandal 0000-0003-2570-2335 Department of Mathematics,
Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India
<EMAIL_ADDRESS>P.K. Sahoo 0000-0003-2130-8832 Department of
Mathematics,
Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India
<EMAIL_ADDRESS>
###### Abstract
The thermodynamical study of the universe allow particle production in
modified $f(T)$ ($T$ is the torsion scalar) theory of gravity within a flat
FLRW framework for line element. The torsion scalar $T$ plays the same role as
the Ricci scalar $R$ in the modified theories of gravity. We derived the
$f(T)$ gravity models by taking $f(T)$ as the sum of $T$ and an arbitrary
function of $T$ with three different arbitrary function. We observe that the
particle production describes the accelerated expansion of the universe
without a cosmological constant or any unknown “quintessence” component. Also,
we discussed the supplementary pressure, particle number density and particle
production rate for three cases.
###### keywords:
Modified $f(T)$ gravity; Particle creation; Thermodynamics
Received (22 July 2020)Accepted (05 October 2020)
PACS Nos.: 04.50.kd
## 1 Introduction
At the beginning of the 19th century, the General Theory of Relativity brought
the revolution to the modern cosmology proposed by Albert Einstein. The
Riemannian space-time formulates this theory based on the Levi-Civita
connection, a torsion-free, and metric compatibility connection. It also helps
us to understand the geodesic structure of the Universe. Later on, it faced
problems like fine-tuning, cosmic co-incidence, initial singularity,
cosmological constant, and flatness [1], since modern cosmology is growing by
a prominent number of accurate observations. Besides, the cosmological
observation, such as type Ia supernovae [2, 3], cosmic microwave background
(CMB) radiation [4, 5], large scale structure [6, 7], baryon acoustic
oscillations [8], and weak lensing [9] confirms that currently, our Universe
is going through an accelerated expansion phase that happens because of the
highly negative pressure produced by the unknown form of matter and energy,
called dark energy and dark matter. To overcome the above issues, researchers
started to modify the Einstein’s theory of relativity, and they ended up with
several modified theories of gravity such as $f(R)$ gravity [10], $f(R,T)$
gravity [11], $f(T)$ gravity [12], $f(Q)$ gravity [13], $f(Q,T)$ gravity [14],
etc. As a result, cosmologists found many interesting results such as Yousaf
et al., studied the self-gravitating structures [15], gavastars [16], Sahoo et
al. [17] studied the wormhole geometry, bouncing cosmology, and accelerated
expansion of the universe using modified theories of gravity. The main
advantage of the modified theories of gravity is that it successfully
describes the late-time cosmic acceleration and the early time inflation. In
the early stage of the Universe, there is a possibility of particle creation.
In this study, we are focusing on particle production in the teleparallel
gravity.
The $f(T)$ theories of gravity are the generalization of the Teleparallel
Equivalent of General Relativity (TEGR), where $T$ is the torsion scalar [18].
TEGR was first presented by Einstein. In the context of $f(T)$ theories, the
Reimann-Cartan space-time requires the torsional curvature to vanish.
Furthermore, this type of space-time is constructed by the Weitzenböck
connection [19, 20]. TEGR is equivalent to GR; the reason is that both cases’
action is the same except the surface term in TEGR. But, the physical
interpretation is different from each other. The construction of gravitational
Lagrangian in TEGR formulation was done in [21, 22]. TEGR is Lorentz invariant
theories, whereas the modified $f(T)$ theories are not Lorentz invariant.
Also, the motion equations in $f(T)$ gravity are not necessarily Lorentz
invariant, because the certainty is that this theory’s property explains the
recent interests [23, 24, 25, 26]. Moreover, the modification of TEGR was
motivated by the $f(R)$ gravity theory. In the teleparallel gravity theory, we
use the Weitzenböck connection instead of the Levi-Civita connection, which
uses $f(R)$ gravity to vanish the non-zero torsion curvature. Here, we would
like to mention that the $f(T)$ gravity does not need the Equivalence
principle because the Weitzenböck connection describes its gravitational
interaction. It is a simple modified theory compared to other modified
theories because the torsion scalar $T$ contains only the first-order
derivatives of the vierbeins. In contrast, Ricci scalar $R$ contains the
second-order derivatives of the metric tensors. Recently, Mandal et al.,
studied the acceleration expansion of the Universe using the parametrization
technique with presuming exponential and logarithmic form of $f(T)$ [27] in
$f(T)$ gravity. They also studied a complete cosmological scenario of the
Universe in $f(T)$ gravity, where they discussed the difference between
General Relativity and Teleparallel gravity [28] in $f(T)$ gravity. M. Sharif
and S. Rani studied the dynamical instability ranges in Newtonian as well as
post-Newtonian regimes considering power-law $f(T)$ model with anisotropic
fluid in $f(T)$ gravity [29]. Cai et al., studied the matter bounce cosmology
using perturbation technique and they found a scale-invariant power spectrum,
which is consistent with cosmological observations in $f(T)$ gravity [30]. In
[31], the wormhole solutions with non-commutative geometry have been studied
assuming power-law $f(T)$ model and a particular shape function in
teleparallel gravity. Inflationary universe studied using power-law $f(T)$
function and logamediate scale factor in [32], and constant-roll inflation
studied in [33].
In the early stage of the universe, the possibility of particle creation has
been discussed for curved space-time by Schrodinger [34], Dewitt [35], Imamura
[36]. Later, the first ever particle creation was treated by an external
gravitational field by Parkar [37, 38]. In flat space-time, the unique vacuum
state is identifying by the guidance of Lorentz invariance. Moreover, we do
not have Lorentz symmetry in curved space-time. In general, there are more
than one vacuum state exists in a curved space-time. Therefore, the particle
creation idea becomes open to discuss, but it’s physical interpretation
becomes more difficult [39, 40]. The interaction between the dynamical
external gradients causes the particle creation from the vacuum. The particle
creation produces negative pressure, so it is considered to explain the
accelerated expansion of the universe and got some unexpected outcomes. Also,
it might play the role of unknown gradients of the universe. In [41, 42]
studied the particle creation with SNe Ia data. Singh [43], and Singh and
Beesham [44, 45] studied the particle creation with some kinematical tests in
FLRW cosmology. The continuous creation of particle predicts the assumptions
of standard Big Bang cosmology.
The thermodynamical study of black hole gives the fundamental relation between
thermodynamics and gravitation [46, 47, 48, 49, 50]. In GR, the relation
between the entropy and the horizon area with the Einstein equation derives
from the Clausius relation in thermodynamics [51, 52]. This idea is also used
for other theories, mainly, the generalized thermodynamics laws and modified
theories of gravity which are derived from the GR [53, 54]. Among the modified
theories of gravity, $f(R)$ gravity got more attention on this framework.
Thereby, one can obtained the gravitational field equation through the non-
equilibrium feature of thermodynamics by using the Clausius approach. There
are some work have been done in the thermodynamics of particle creation in
$f(T)$ gravity theory [56, 57, 55, 58, 50].
In this work, we study the theoretical significance of particle creation in
$f(T)$ gravity theory considering a flat FRW model. Assuming $f(T)$ as the sum
of torsion scalar $T$ and an arbitrary function of torsion scalar $T$, we
studied the thermodynamics of particle creation with $f(T)=0$ is a simple
teleparallel gravity, $f(T)=A(-T)^{q}$ as power law gravity and
$f(T)=A(1-e^{-qT})$ as exponential gravity. After that we discussed the
behaviour of supplementary pressure $p_{c}$, particle number density $n$, and
the particle creation rate $\psi$ for three models. Also we compared the
effect of the cosmological pressure $p_{m}$ with the supplementary pressure
$p_{c}$ for different values of equation of state parameter $\omega$ on
particle creation.
This work is organised as follows. In Sec. 2, we discussed the thermodynamics
of particle creation, which is followed by the overview of $f(T)$ gravity and
it’s field equations in Sec. 3. In Sec. 4, we discussed three $f(T)$ gravity
models. Finally, the results are summarized in Sec. 5
## 2 Thermodynamics of particle creation
If we assume the total number of particles in the universe to be conserved,
the laws of thermodynamics can be expressed as
$dQ=d(\rho_{m}V)+p_{m}dV$ (1)
and
$TdS=p_{m}dV+d(\rho_{m}V)$ (2)
where $p_{m}$, $\rho_{m}$, $V$, $T$ and $S$ denote respectively the
cosmological pressure, density, volume, temperature and entropy. Also, $dQ$
represent the heat exchange in the time interval $dt$. From (1) and (2), we
further obtain,
$dQ=TdS$ (3)
Eq. (3) reflects the fact that the entropy is a conserved quantity, since for
lose adiabatic system $dQ=0$. We now consider a scenario in which the total
number of particles in the universe is not constant. Under this condition, Eq
(1) gets modified to [59]
$dQ=d(\rho_{m}V)+p_{m}dV+(h/n)d(nV)$ (4)
where $N=nV$, $n$ being the number density of the particles and
$h=(p_{m}+\rho_{m})$ the enthalpy per unit volume of the system. For an
adiabatic system where $dQ=0$, (4) reads [59]
$d(\rho_{m}V)+p_{m}dV=(h/n)d(nV)$ (5)
In [59], the authors stated that in cosmology this change in the total number
of particles in the universe can be understood as a transformation of
gravitational field energy to the matter.
For an open thermodynamic system, Eq. (5) can be expressed as [59]
$d(\rho_{m}V)=-\left(p_{m}+p_{c}\right)dV$ (6)
where
$p_{c}=-(h/n)(dN/dV)$ (7)
represents supplementary pressure associated with the creation of particles
[59]. Note that negative $p_{c}$ indicate production of particles whereas
positive $p_{c}$ implies particle annihilation and finally for $p_{c}=0$ the
total number of particles is constant. Using Eq (2) and (5), it can also be
shown that [59]
$S=S_{0}\left(\frac{N}{N_{0}}\right)$ (8)
where $S_{0}$ and $N_{0}$ represent current values of these quantities.
Additionally, we assume the particles follow a barotropic equation of state
and therefore can be written as
$p_{m}=(\omega)\rho_{m}$ (9)
where $-1\leq\omega\leq 1$ is the EoS parameter. The number density of
particles is related to the density $\rho_{{}_{m}}$ as [45]
$n=n_{0}\left(\frac{\rho_{m}}{\rho_{0}}\right)^{\frac{1}{1+\omega}}$ (10)
where $\rho_{0}\geq 0$ and $n_{0}\geq 0$ are the present values of density and
particle number density respectively.
We now consider the matter creation rate to be defined as [60]
$\psi(t)=3\beta nH$ (11)
where $0\leq\beta\leq 1$ is assumed to be a constant and $\psi(t)$ represent
the rate of particle creation and has a dimension of $t^{-1}$. $\psi$ can
either be positive or negative depending on the creation or annihilation of
particles. $\psi=0$ indicate particle number being conserved in the universe.
For cosmological matter following barotropic equation of state (Eq. 9), the
supplementary pressure $p_{c}$ can be expressed as [59]
$p_{c}=-\beta(\omega+1)\rho_{m}$ (12)
## 3 Overview of $f(T)$ Gravity
Let us consider the extension of Einstein-Hilbert Lagrangian of $f(T)$ theory
of gravity (which is similar to $f(R)$ gravity extension from the Ricci scalar
$R$ to $R+f(R)$ in the action), namely the teleparallel gravity term $T$ to
$T+f(T)$, where $f(T)$ is an arbitrary function of $T$ as
$S=\frac{1}{16\pi G}\int[T+f(T)]ed^{4}x,$ (13)
where $e=det(e^{i}_{\mu})=\sqrt{-g}$ and $G$ is the gravitational constant.
Assume $k^{2}=8\pi G=M_{p}^{-1}$, where $M_{p}$ is the Planck mass.The
gravitational field is defined by the torsion one as
$T^{\gamma}_{\mu\nu}\equiv
e^{\gamma}_{i}(\partial_{\mu}e^{i}_{\nu}-\partial_{\nu}e^{i}_{\mu}).$ (14)
The contracted form of the above torsion tensor is
$T\equiv\frac{1}{4}T^{\gamma\mu\nu}T_{\gamma\mu\nu}+\frac{1}{2}T^{\gamma\mu\nu}T_{\nu\mu\gamma}-T^{\gamma}_{\gamma\mu}T^{\nu\mu}_{\nu}.$
(15)
By the variation of the total action $S+L_{m}$, here $L_{m}$ is the matter
Lagrangian gives us the field equation for $f(T)$ gravity as
$e^{-1}\partial_{\mu}(ee^{\gamma}_{i}S^{\mu\nu}_{\gamma})(1+f_{T})-(1+f_{T})e^{\lambda}_{i}T^{\gamma}_{\mu\lambda}S^{\nu\mu}_{\gamma}\\\
+e^{\gamma}_{i}S^{\mu\nu}_{\gamma}\partial_{\mu}(T)f_{TT}+\frac{1}{4}e^{\nu}_{i}[T+f(T)]=\frac{k^{2}}{2}e^{\gamma}_{i}T^{(M)\nu}_{\gamma},$
(16)
where $f_{T}=df(T)/dT$, $f_{TT}=d^{2}f(T)/dT^{2}$, the ”superpotential “
tensor $S^{\mu\nu}_{\gamma}$ written in terms of cotorsion
$K^{\mu\nu}_{\gamma}=-\frac{1}{2}(T^{\mu\nu}_{\gamma}-T^{\nu\mu}_{\alpha}-T^{\mu\nu}_{\alpha})$
as
$S^{\mu\nu}_{\gamma}=\frac{1}{2}(K^{\mu\nu}_{\gamma}+\delta^{\mu}_{\gamma}T^{\alpha\nu}_{\alpha}-\delta^{\nu}_{\gamma}T^{\alpha\mu}_{\alpha})$
and $T^{(M)\nu}_{\gamma}$ represents the energy-momentum tensor to the matter
Lagrangian $L_{m}$. Now we consider a flat FLRW universe with the metric as
$ds^{2}=dt^{2}-a^{2}(t)dx^{\mu}dx^{\nu},$ (17)
where $a(t)$ is the scale factor, which gives us
$e^{i}_{\mu}=diag(1,a,a,a).$ (18)
Using equation (18) into the field equation (16), we get the modified field
equation as follows
$H^{2}=\frac{8\pi G}{3}\rho_{m}-\frac{f}{6}+\frac{Tf_{T}}{3},$ (19)
$\dot{H}=-\left[\frac{4\pi G(\rho_{m}+p_{m}+p_{c})}{1+f_{T}+2Tf_{TT}}\right],$
(20)
where $H\equiv\dot{a}/a$ be the Hubble parameter and ”dot“ represents the
derivative with respect to $t$. Here, $\rho_{m}$ and $p_{m}$ be the energy
density and pressure of the matter content, $p_{c}$ be the supplementary
pressure. Also, we have used
$T=-6H^{2},$ (21)
which holds for a FLRW Universe according to equation (15).
## 4 $f(T)$ gravity models
In this section we shall investigate the temporal evolution of particle
production in radiation ($\omega=1/3$) and dust universe ($\omega=0$) for
various $f(T)$ gravity models with model parameters constrained from
cosmological observations related to gravitational baryogenesis.
For the purpose of analysis, we shall assume a power law evolution of scale
factor of the form
$a(t)=a_{0}t^{\left[\frac{2}{3(1+\omega)}\right]}$ (22)
where $a_{0}>0$ is a constant.
### 4.1 Simple Teleparallel gravity
In simple teleparallel equivalent of general relativity [61], where $f(T)=0$ ,
for a universe composed of perfect fluid, the field equations (19) and (20)
becomes
$H^{2}=\frac{8\pi G}{3}\rho_{m}$ (23) $\dot{H}=-4\pi G(\rho_{m}+p_{m}+p_{c})$
(24)
Substituting (22) in (23), we obtain the expression of density $\rho_{m}$ as
$\rho_{m}=\frac{4}{3}\left(\frac{1}{t^{2}(1+\omega)^{2}}\right)$ (25)
The expression of supplementary pressure $p_{c}$, particle number density $n$
and particle creation rate $\psi$ are obtained respectively as
$p_{c}=\frac{4}{3}\left(\frac{\beta(1+\omega)}{t^{2}(1+\omega)^{2}}\right)$
(26)
$n=\left[\frac{4}{3}\left(\frac{1}{t^{2}(1+\omega)^{2}}\right)\right]^{\frac{1}{(1+\omega)}}$
(27)
$\psi=3\beta\left[\frac{2}{3t(1+\omega)}\right]\times\left[\frac{4}{3}\left(\frac{1}{t^{2}(1+\omega)^{2}}\right)\right]^{\frac{1}{(1+\omega)}}$
(28)
Figure 1: The behaviour of supplementary pressure $p_{c}$ with respect to
cosmic time $t$ for $\omega=\frac{1}{3},\omega=0$ and $\beta=1$. Figure 2: The
behaviour of particle number density $n$ with respect to cosmic time $t$ for
$\omega=\frac{1}{3},\omega=0$ and $\beta=1$. Figure 3: The behaviour of
particle creation rate $\psi$ with respect to cosmic time $t$ for
$\omega=\frac{1}{3},\omega=0$ and $\beta=1$.
### 4.2 Power Law Gravity
The power law model of Bengochea and Ferraro [62] reads
$f(T)=A(-T)^{q}$ (29)
where $A$ is a constant and $q>1$. In [63], the authors reported viable
baryon-to-entropy ratio for $A=-10^{-7}\texttt{or}-10^{-6}$ and $q\gtrsim
4.8$. However, other values of the model parameters could also yield viable
estimates of baryon-to-entropy ratio. Nonetheless, we restrict ourselves to
the values $A=-10^{-7}$ and $q=5$ for the present analysis. Substituting (22)
and (29) in (19) and (20), the expression of density $\rho_{m}$ reads
$\rho_{m}=A6^{(q-1)}(1-18q)t^{-2q}\left[\frac{2}{3(1+\omega)}\right]^{2q}$
(30)
The expression of supplementary pressure $p_{c}$, particle number density $n$
and particle creation rate $\psi$ for the power law gravity are obtained
respectively as
$p_{c}=-\beta(1+\omega)A6^{(q-1)}(1-18q)t^{-2q}\left[\frac{2}{3(1+\omega)}\right]^{2q}$
(31)
$n=\left[A6^{(q-1)}(1-18q)t^{-2q}\left[\frac{2}{3(1+\omega)}\right]^{2q}\right]^{1/(1+\omega)}$
(32)
$\psi=3\beta\left[\frac{2}{3t(1+\omega)}\right]\times\left[A6^{(q-1)}(1-18q)t^{-2q}\left[\frac{2}{3(1+\omega)}\right]^{2q}\right]^{1/(1+\omega)}$
(33)
Figure 4: The behaviour of supplementary pressure $p_{c}$ with respect to
cosmic time $t$ for $\omega=\frac{1}{3},\omega=0$ and $\beta=1,q=5,A=-10^{-7}$
Figure 5: The behaviour of particle number density $p_{c}$ with respect to
cosmic time $t$ for $\omega=\frac{1}{3},\omega=0$ and
$\beta=1,q=5,A=-10^{-7}$. Figure 6: The behaviour of particle creation rate
$\psi$ with respect to cosmic time $t$ for $\omega=\frac{1}{3},\omega=0$ and
$\beta=1,q=5,A=-10^{-7}$.
### 4.3 Exponential Gravity
The exponential $f(T)$ model given in [64] reads
$f(T)=A(1-e^{-qT})$ (34)
where $A$ and $q$ are model parameters. In [63] the authors reported a wide
range of values of $A$ and $q$ for which a viable baryon-to-entropy ratio
could be realized. However, we shall work with $A=1$ and $q=10^{-10}$ as these
values were used in [63] to fit the baryon-to-entropy ratio with observations.
Substituting (22) and (34) in (19) and (20), the expression of density
$\rho_{m}$ reads
$\rho_{m}=\left[\frac{18A\left[\frac{2}{3(1+\omega)}\right]^{2}qe^{\frac{6\left[\frac{2}{3(1+\omega)}\right]^{2}q}{t^{2}}}}{t^{2}}\right]$
(35)
The expression of supplementary pressure $p_{c}$, particle number density $n$
and particle creation rate $\psi$ for the exponential gravity are obtained
respectively as
$p_{c}=-\beta(1+\omega)\left[\frac{18A\left[\frac{2}{3(1+\omega)}\right]^{2}qe^{\frac{6\left[\frac{2}{3(1+\omega)}\right]^{2}q}{t^{2}}}}{t^{2}}\right]$
(36)
$n=\left[\frac{18A\left[\frac{2}{3(1+\omega)}\right]^{2}qe^{\frac{6\left[\frac{2}{3(1+\omega)}\right]^{2}q}{t^{2}}}}{t^{2}}\right]^{1/(1+\omega)}$
(37)
$\psi=3\beta\left[\frac{2}{3t(1+\omega)}\right]\times\left[\left(\frac{18A\left[\frac{2}{3(1+\omega)}\right]^{2}qe^{\frac{6\left[\frac{2}{3(1+\omega)}\right]^{2}q}{t^{2}}}}{t^{2}}\right)^{1/(1+\omega)}\right]$
(38)
Figure 7: The behaviour of supplementary pressure $p_{c}$ with respect to
cosmic time $t$ for $\omega=\frac{1}{3},\omega=0$ and
$\beta=1,q=10^{-10},A=1$. Figure 8: The behaviour of particle number density
$n$ with respect to cosmic time $t$ for $\omega=\frac{1}{3},\omega=0$ and
$\beta=1,q=10^{-10},A=1$. Figure 9: The behaviour of particle creation rate
$\psi$ with respect to cosmic time $t$ for $\omega=\frac{1}{3},\omega=0$ and
$\beta=1,q=10^{-10},A=1$.
## 5 Discussion of Outcomes and Conclusions
In this article, we have studied the thermodynamics of an open system with
particle creation of a flat FLRW universe in $f(T)$ theory of gravity. We have
constructed three cosmological models by assuming suitable functions for
$f(T)$ as $f(T)=0,f(T)=A(-T)^{q},f(T)=A(1-e^{-qT})$ and the particle creation
rate $\psi$. To analyze our models, we have considered the power law evolution
of the scale factor and studied the behaviour of physical quantities (i.e. the
supplementary pressure $p_{c}$, particle number density $n$, and the particle
creation rate $\psi$) through their graphical representations with respect to
cosmic time $t$ and some fixed values of $\beta$ in various phase of the
evolution of the universe. And, details of our cosmological models discussed
in the following.
In our simple Teleparallel gravity model, we have considered the minimal
coupling between matter and geometry. In Fig. 3, profiles of $\psi$ have been
shown. From Fig. 3, one can easily observe that the rate of particle creation
is high in the early time and tends to zero when $t$ tends to infinity. But,
the number of particle in the universe increases with cosmic time $t$ shown in
Fig. 2. The supplementary pressure $p_{c}$ has higher negative which shows
that the particle production is high during the early stage and tends to zero
when $t$ tends infinity, in Fig. 1. From this model we have concluded that the
evolution of the universe depends on the contribution of the particle
production.
In power law gravity and exponential gravity models, we have considered the
non-minimal coupling between matters. The profiles of $\psi,n$ and $p_{c}$
have been shown for the corresponding models. In Fig. 6,9, the particle
creation rate $\psi$ is high in the early stage and it tends to zero as cosmic
time $t$ tends to infinity. Also, the particle number density n in Fig. 5,8
goes to zero as cosmic time $t$ goes to infinity which concluded that the
expansion rate overcomes the rate particle creation as the supplementary
pressure in Fig. 4,7 is negative throughout the evolution of the universe in
different phases. The density parameters in three models shows that the
universe is open in the presence of particle creation in $f(T)$ theory of
gravity.
In summary, we have studied the cosmological models with particle production
in $f(T)$ theory of gravity to explore the current accelerated phenomenon of
the universe. We have found that the particle creation produces negative
pressure which may derive the accelerated expansion of the universe and play
the role of unknown matter called “dark energy” in $f(T)$ theory of gravity.
We may expect that the particle creation process be a constraint for the
unexpected observational outcomes. The new fact about this article is that the
particle creation is studied by the thermodynamics approach in $f(T)$ theory
of gravity.
Acknowledgements S.M. acknowledges Department of Science & Technology (DST),
Govt. of India, New Delhi, for awarding Junior Research Fellowship (File No.
DST/INSPIRE Fellowship/2018/IF180676). PKS acknowledges DST, New Delhi, India
for providing facilities through DST-FIST lab, Department of Mathematics,
BITS-Pilani, Hyderabad Campus where a part of this work was done. The authors
thank S. Bhattacharjee for stimulating discussions. We are very much grateful
to the honorable referee and the editor for the illuminating suggestions that
have significantly improved our work in terms of research quality and
presentation.
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|
# Self-Supervised Path Consistency Learning for HOI Detection
Jihwan Park1,2 SeungJun Lee1 Hwan Heo1 Hyeong Kyu Choi1 Hyunwoo J. Kim1,
1Department of Computer Science and Engineering, Korea University 2Kakao Brain
{jseven7071, lapal0413, gjghks950, imhgchoi<EMAIL_ADDRESS>
<EMAIL_ADDRESS>corresponding author.
# DP-Aug: Decoding Path Augmentation for Transformers in HOI Detection
Jihwan Park1,2 SeungJun Lee1 Hwan Heo1 Hyeong Kyu Choi1 Hyunwoo J. Kim1,
1Department of Computer Science and Engineering, Korea University 2Kakao Brain
{jseven7071, lapal0413, gjghks950, imhgchoi<EMAIL_ADDRESS>
<EMAIL_ADDRESS>corresponding author.
# Consistency Learning via Decoding Path Augmentation
for Transformers in Human Object Interaction Detection
Jihwan Park1,2 SeungJun Lee1 Hwan Heo1 Hyeong Kyu Choi1 Hyunwoo J. Kim1,
1Department of Computer Science and Engineering, Korea University 2Kakao Brain
{jseven7071, lapal0413, gjghks950, imhgchoi<EMAIL_ADDRESS>
<EMAIL_ADDRESS>corresponding author.
###### Abstract
Human-Object Interaction detection is a holistic visual recognition task that
entails object detection as well as interaction classification. Previous works
of HOI detection has been addressed by the various compositions of subset
predictions, _e.g_., Image $\rightarrow$ HO $\rightarrow$ I, Image
$\rightarrow$ HI $\rightarrow$ O. Recently, transformer based architecture for
HOI has emerged, which directly predicts the HOI triplets in an end-to-end
fashion (Image $\rightarrow$ HOI). Motivated by various inference paths for
HOI detection, we propose cross-path consistency learning (CPC), which is a
novel end-to-end learning strategy to improve HOI detection for transformers
by leveraging augmented decoding paths. CPC learning enforces all the possible
predictions from permuted inference sequences to be consistent. This simple
scheme makes the model learn consistent representations, thereby improving
generalization without increasing model capacity. Our experiments demonstrate
the effectiveness of our method, and we achieved significant improvement on
V-COCO and HICO-DET compared to the baseline models. Our code is available at
https://github.com/mlvlab/CPChoi.
## 1 Introduction
Human-Object Interaction (HOI) detection is a holistic visual recognition task
that includes detecting individual objects as <human, object>, while properly
classifying the type of <interaction>. Previous HOI detectors [15, 52, 31, 49]
were mainly built on object detection models. They commonly extend CNN-based
object detectors [45, 34, 42] with an additional head for interaction
classification, _e.g_., humans and objects are detected first, and their
interaction is associated subsequently.
To alleviate the high computation cost of such two-stage HOI detection
methods, one-stage models [52, 22, 33] have been proposed for faster
detection. These models perform interaction prediction and object detection in
parallel. They compensate for their lower performance with auxiliary
predictions for the HOI subsets, _i.e_., auxiliary predictions for subset
<human, interaction> or <object, interaction> may help HOI prediction through
post-processing. However, these works demand different network architectures
for each auxiliary prediction, due to strict disciplines for each network’s
input. Hence, to introduce flexibility, transformer-based architectures [23,
46, 6, 48] have recently been adopted for HOI detection. They reformulate the
HOI detection problem as a direct set prediction building on DETR [4].
(a) cycle consistency
(b) cross-task consistency
(c) cross-path consistency
Figure 1: Comparison on the variants of consistencies. The black line refers
to the main task function $f_{\mathcal{X}\mathcal{Y}}$, and the red, blue
lines refer to the pair of tasks trained to be consistent with each other. (a)
Cycle consistency enforces the composite function of
$f_{\mathcal{Y}\mathcal{X}}\circ f_{\mathcal{X}\mathcal{Y}}$ to be consistent
with identity function $f_{\mathcal{X}\mathcal{X}}$. (b) Cross-task
consistency requires an auxiliary pretrained network
$f_{\mathcal{Y}\mathcal{Y_{\text{a}}}}$, represented in dashed lines, to give
consistent outputs across tasks. (c) Cross-path consistency does not require
task-specific pretrained networks. The output of main task function
$f_{\mathcal{X}\mathcal{Y}}$ should be consistent with the composition of the
outputs from sub-task functions $f_{\mathcal{X}\mathcal{Y_{\text{s}}}}$ and
$f_{\mathcal{Y}_{\text{s}}\mathcal{Y}}\circ
f_{\mathcal{X}\mathcal{Y}_{\text{s}}}$.
Motivated by various inference paths in HOI detectors, we propose a simple yet
effective method to train HOI transformers. We augment the decoding paths with
respect to the possible prediction sequences of HOI triplets. Then, with the
cascade structure of transformers, an input query is sequentially decoded into
auxiliary sub-task outputs and the final output. The stage of each augmented
paths stage shares a decoder, in a multi-task learning fashion. We further
improve our method to leverage the augmented decoding paths by enforcing the
outputs from the various paths to be consistent. Accordingly, we propose
Cross-Path Consistency (CPC) Learning, which aims to predict HOI triplets
regardless of inference sequences.
Similar to cross-task consistency [55], cross-path consistency retains
inference path invariance. However, cross-path consistency learning does not
require additional pre-trained networks. In contrast to cross-task
consistency, which demands an auxiliary network to train the main task
$\mathcal{X}\rightarrow\mathcal{Y}$ (Figure 1-(b)), cross-path consistency
defines an auxiliary domain $\mathcal{Y}_{s}$ in between $\mathcal{X}$ and
$\mathcal{Y}$ (Figure 1-(c)). In other words, the main task
$\mathcal{X}\rightarrow\mathcal{Y}$ (_i.e_., Image$\rightarrow$HOI) is divided
into subtasks $\mathcal{X}\rightarrow\mathcal{Y}_{s}$ and
$\mathcal{Y}_{s}\rightarrow\mathcal{Y}$ (_e.g_.,
Image$\rightarrow$HO$\rightarrow$I). The main task function
$f_{\mathcal{X}\mathcal{Y}}$ is then trained by enforcing its output and the
composition of sub-task predictions to be consistent. Moreover, cross-path
consistency learning is temporarily adopted for training only.
Our training strategy can be generalized to any transformer based
architecture, and can be applied in an end-to-end method. Extensive
experiments show that HOI transformers trained with CPC learning strategy
achieves substantial improvements in two popular HOI detection benchmarks:
V-COCO and HICO-DET. The contribution of this work can be summarized as the
followings:
* •
We propose Cross-Path Consistency (CPC) learning, which is a novel end-to-end
learning strategy to improve transformers for HOI detection leveraging various
inference paths. In this learning scheme, we use Decoding-Path Augmentation to
generate various inference paths which are compositions of subtasks with a
shared decoder for effective training.
* •
Our training scheme achieves substantial improvements on V-COCO and HICO-DET
without increasing model capacity and inference time.
## 2 Related Works
### 2.1 Human Object Interaction Detection
Human-Object Interaction (HOI) detection has been proposed in [16]. Later,
human-object detectors have been improved using human or instance appearance
and their spatial relationship [15, 25, 12]. On the other hand, graph-based
approaches [44, 49, 11, 51] have been proposed to clarify the action between
the <human, object> pair.
HOI detection models based on only visual cues often suffer from the lack of
contextual information. Thus, recent works utilize external knowledge to
improve the quality of HOI detection. Human pose information extracted from
external models [3, 7, 19, 28] or linguistic priors and knowledge graph models
show meaningful improvement in performance [58, 36, 43, 54, 18, 14, 31, 57,
37].
Since the majority of the previous works are based on two-stage methods with
slower inference time, attempts for faster HOI detection by introducing simple
end-to-end multi-layer perceptrons [17], or directly detecting interaction
points [52, 33], or union regions [22, 20, 30] have been suggested.
### 2.2 Transformers in Computer Vision
Transformer has become the state-of-the-art method in many computer vision
tasks. In image classification, [9] has shown competitive performance on
ImageNet without any convolution layers. DeiT [48] applied knowledge
distillation to data-efficiently train the vision transformer. To extract
multi-scale image features, Swin Transformer [38] proposed shifted window
based self-attention modules that effectively aggregate small patches to
increase the receptive field. In the object detection task, DETR [4] has
proposed an end-to-end framework eliminating the need for hand-designed
components. DETR’s bipartite matching loss between the predicted set and the
ground truth labels enables direct set prediction at inference. Recently,
DETR’s late convergence problem has been tackled in [62, 40, 13].
Inspired by DETR, transformer-based HOI (Human-Object Interaction) detectors
[23, 46, 6, 63, 8] have been recently proposed. HOI transformer models have
two types of structure, one decoder model and the two decoder model. The one-
decoder model which follows the structure of DETR [4] predicts triplets from
the output of a single decoder. QPIC [46] and HoiT [63] are one-decoder models
that output <human, object, interaction> triplets directly with multiple
interaction detection heads. Two-decoder models use two transformer decoders
to output distinctive targets. For instance, HOTR [23] and AS-NET [6] are
composed of an instance decoder that outputs object and an interaction decoder
that outputs interaction. In contrast to previous works that are trained with
a single inference path, our model learns with the augmented decoding paths.
Also, our framework can be applied to any transformer-based model. More
explanation of HOI transformers are in Section 3.1.
### 2.3 Consistency Learning in Vision
Consistency constraints applied to many computer vision topics have been
extensively studied. In semi-supervised learning, consistency regularization
is widely used to train the model to be invariant to input noise. Label
consistency methods [27, 53, 41, 47] augment or perturb an input image and
apply consistency loss between model predictions. CDS [21] explored object
detection in a semi-supervised setting with classification and localization
consistency regularization. Also, consistency regularization in cyclic form is
commonly used in generative models [61], image matching [60, 59], temporal
correspondence [10], and in many other domains.
#### Comparison with Consistency Learning
Our consistency training scheme is relevant to cross-task consistency learning
[55]. Cross-task consistency learning is based on inference-path invariance,
where the predictions should be consistent regardless of the inference paths.
As shown in Figure 1 (b), cross-task consistency learning uses an auxiliary
task $\mathcal{Y}\rightarrow\mathcal{Y}_{a}$ to train the main task function
$f_{\mathcal{X}\mathcal{Y}}$, _i.e_., given $x$ from the query domain, and $y$
from target domain $\mathcal{Y}$, predictions of
$f_{\mathcal{Y}\mathcal{Y}_{a}}\circ f_{\mathcal{X}\mathcal{Y}}(x)$ and
$f_{\mathcal{Y}\mathcal{Y}_{a}}(y)$ are expected to be consistent. Different
from cross-task consistency, our cross-path consistency learning (Figure 1
(c)) trains the main task function $f_{\mathcal{X}\mathcal{Y}}$ by enforcing
the prediction of target domain $\mathcal{Y}$ of $f_{\mathcal{X}\mathcal{Y}}$
and $f_{\mathcal{Y}_{s}\mathcal{Y}}\circ f_{\mathcal{X}\mathcal{Y}_{s}}$,
where auxiliary domain $\mathcal{Y}_{s}$ is decomposed from the target domain
$\mathcal{Y}$, to be consistent. Also, while cross-task consistency learning
requires the mapping function $f_{\mathcal{Y}\mathcal{Y}_{a}}$ to be
pretrained to avoid suboptimal training with the noisy estimator, cross-path
consistency learning does not demand any task-specific pre-trained networks
since the auxiliary domain $\mathcal{Y}_{s}$ is part of the target domain
$\mathcal{Y}$. Details for our framework is described in section 3.2.
## 3 Method
In this section, we present our novel end-to-end training strategy for
Transformers with cross-path consistency in Human-Object Interaction
Detection. The training strategy includes 1) augmenting the decoding path and
2) consistency regularization between predictions of multiple decoding paths.
Before discussing our training strategy, we briefly summarize transformers in
Human-Object Interaction detection.
### 3.1 Transformer in HOI detection
HOI transformers are commonly extended upon DETR [4], which is composed of a
CNN backbone followed by the encoder-decoder architecture of Transformer [1].
The CNN backbone first extracts a locally aggregated feature map
$f\in\mathbb{R}^{H^{\prime}\times W^{\prime}\times D}$ from input image
$x\in\mathbb{R}^{H\times W\times 3}$. Then, the feature map $f$ is passed into
the encoder to globally aggregate features via the self-attention mechanism,
resulting in the encoded feature map $X\in\mathbb{R}^{H^{\prime}\times
W^{\prime}\times D}.$ At a decoding stage, a decoder takes learnable query
embeddings ${\color[rgb]{0,0,0}q}\in\mathbb{R}^{N\times D}$ and outputs
$e\in\mathbb{R}^{N\times D}$ by interacting with encoded feature map $X$
through cross-attention. The outputs are converted to final HOI predictions
(_i.e_., human, object, interaction) by read-out functions, which are
generally feed-forward networks.
Training Transformers for detection entails matching between predictions and
ground truth labels since Transformers provide detections as set predictions.
To compute losses, the Hungarian algorithm [26] is used to associate
detections with ground truth labels. The predictions unmatched with ground
truth labels are considered as no object or no interactions. In general, HOI
transformers can be categorized into two groups based on human/object
localization schemes. [46, 63] directly predict the box coordinates of human
and object from an HOI prediction. But this causes problems that human or
object can be redundantly predicted by multiple query embeddings and the
localizations of the same object often differ across HOI triplet predictions.
To address these problems, [23, 6] propose parallel architectures to perform
interaction detection separately from object detection.
### 3.2 Decoding-Path Augmentation
(a)
(b)
Figure 2: Cross-path consistency for HOI detection. (a) Main task path
${\color[rgb]{1,0,0}\mathcal{P}_{1}}$ should be consistent with each augmented
path. _e.g_. path ${\color[rgb]{0,0,1}\mathcal{P}_{2}}$. (b) Augmented paths
should be consistent with one another. _e.g_. path
${\color[rgb]{0,0,1}\mathcal{P}_{2}}$ and
${\color[rgb]{1,0,0}\mathcal{P}_{3}}$. Figure 3: The overall process of
Cross-Path Consistency Learning. The encoded image features are passed into
the shared decoder with multiple inference paths
$\\{\mathcal{P}_{1},...,\mathcal{P}_{k-1},\mathcal{P}_{k}\\}$. Each path is
augmented based on the decoding-path augmentation to generate various
sequences of inference paths (see Section 3.2). To avoid clutter, we visualize
only the main path $\mathcal{P}_{1}$ and an augmented path $\mathcal{P}_{k}$.
The main path $\mathcal{P}_{1}$ consists of a single decoding stage, and the
augmented path $\mathcal{P}_{k}$ is a composition of decoding stages; all $f$
blocks share parameters. Given queries $q$ a learnable position embeddings,
each decoder extracts output embeddings denoted as $e_{1,1}$, $e_{k,1}$, and
$e_{k,2}$. Then, each of the output embeddings is fed into the readout
function FFN to predict each HOI element _i.e_. <human, object, interaction>.
With Cross-Path Consistency Learning (Section 3.3), all the outputs supervised
with the same ground truth label are trained to be consistent regardless of
their inference paths. Cross-Matching is used to match the queries that are
considered to be consistent by leveraging ground truth label. Along with the
supervision loss $\mathcal{L}_{\mathbf{sup}}^{k}$ for all paths
$\mathcal{P}_{k}$, cross-path consistency loss $\mathcal{L}_{\textbf{CPC}}$ is
added to our final loss.
We observe that HOI detection can be achieved by various sequences of
predictions. For instance, CNN-based HOI detection models [15, 5, 11, 17]
first detect instances (human and object) and then predict interactions
between the instances, _i.e_., $x\rightarrow\text{HO}\rightarrow\text{I}$,
where $x$ is an input image and H, O, I are predictions for human, object,
interaction, respectively. On the other hand, the HOI Transformers by [23, 46,
6, 63] directly predict HOI triplets, _i.e_., $x\rightarrow\text{HOI}$.
Inspired by Cross-Task Consistency [56] and this observation, we propose
decoding-path augmentation to generate various decoding paths (or prediction
paths) and impose consistency regularization. Decoding-path augmentation for
Transformers in HOI detection can be easily achieved by partially decoded HOI
predictions. Furthermore, sharing decoders across paths is beneficial in terms
of knowledge sharing.
In our experiments, we consider four decoding paths as follows:
$\begin{split}&\;\mathcal{P}_{1}=x\rightarrow\text{HOI}\\\
&\begin{rcases*}\mathcal{P}_{2}=x\rightarrow\text{HO}\rightarrow\text{I}\\\
\mathcal{P}_{3}=x\rightarrow\text{HI}\ \rightarrow\text{O}\\\
\mathcal{P}_{4}=x\rightarrow\text{OI}\
\rightarrow\text{H}\end{rcases*}Augmented.\end{split}$ (1)
Each decoding stage of path $\mathcal{P}_{k}$ can be written as:
$\begin{split}e_{{k,1}}&=f(e_{k,0}+q_{k,1},\ X),\\\
e_{{k,2}}&=f(e_{k,1}+q_{k,2},\ X),\end{split}$ (2)
where ${q}_{k,j},\ e_{k,j}$ denote learnable query and output embeddings on
$k^{\text{th}}$ path at $j^{\text{th}}$ decoding stage. The decoder $f$ is
shared across all paths and stages. The $e_{k,0}$ above is dummy output
embeddings set to zeros since there is no 0-th stage, see Figure 3. Each
decoding stage and path use a separate readout function FFN to translate the
output embeddings into HOI instance predictions. For example, on
$\mathcal{P}_{2}:x\rightarrow\text{HO}\rightarrow\text{I}$, at stage 1
$e_{2,1}$ is read out by $\text{FFN}^{\mathcal{P}_{2}}_{h}$ and
$\text{FFN}_{o}^{\mathcal{P}_{2}}$ to predict bounding boxes of human and
object respectively. Prediction for HOI element $m\in\\{h,o,act\\}$ in each
$k^{th}$ path at $j^{th}$ decoding stage can be written as
$\hat{y}_{k}^{m}=\text{FFN}^{\mathcal{P}_{k}}_{m}\left(e_{k,j}\right)$.
### 3.3 Cross-Path Consistency Learning
We now present our Cross-Path Consistency Learning framework (CPC) that
imposes consistency regularization between predictions from different decoding
paths as shown in Figure 2. Learning with CPC leads better generalization
without any additional data or labels.
#### Cross-Path Consistency.
We explain our consistency learning scheme with an exemplary case of main path
$\mathcal{P}_{1}$ and augmented path $\mathcal{P}_{2}$ given as
$\begin{split}&\mathcal{P}_{1}:x\rightarrow\text{HOI}\\\
&\mathcal{P}_{2}:x\rightarrow\text{HO}\rightarrow\text{I}.\end{split}$ (3)
Here, the main path $\mathcal{P}_{1}$ is the HOI transformers’ original
inference path. In path $\mathcal{P}_{2}$, human and object detection logits
$\hat{y}^{h}_{2}$ and $\hat{y}^{o}_{2}$ are obtained reading out $e_{2,1}$,
which is the output embeddings on path 2 at stage 1. Then, the interaction
logit $\hat{y}^{act}_{2}$ is obtained after another subsequent decoder pass
defined as $f_{2,2}$. The corresponding inference scheme of $\mathcal{P}_{2}$
can be written in more formal terms:
$\begin{split}\hat{y}_{2}^{h}&=\text{FFN}_{h}^{\mathcal{P}_{2}}(f_{2,1}(X))\\\
\hat{y}_{2}^{o}&=\text{FFN}_{o}^{\mathcal{P}_{2}}(f_{2,1}(X))\\\
\hat{y}_{2}^{act}&=\text{FFN}_{act}^{\mathcal{P}_{2}}(f_{2,2}\circ
f_{2,1}(X))\\\ \end{split}$ (4)
In (4), input arrays for $f$ other than feature map $X$ were omitted for
simplicity.
With the predictions, we impose regularization to make the outputs from path
$\mathcal{P}_{1}$ and path $\mathcal{P}_{2}$ consistent. Note that HOI
detections from $\mathcal{P}_{2}$ consist of both final and intermediate
decoder outputs. To this end, we define the loss function
$\mathcal{L}_{\mathcal{P}_{1}\mathcal{P}_{2}}$ by aggregating losses from
multiple augmented paths to enforce consistency. The loss function is given
as:
$\begin{split}\mathcal{L}_{\mathcal{P}_{1}\mathcal{P}_{2}}=\lambda_{h}\cdot\mathcal{L}_{h}\big{(}\hat{y}_{1}^{h},\hat{y}_{2}^{h}\big{)}+\lambda_{o}\cdot\mathcal{L}_{o}\big{(}\hat{y}_{1}^{o},\hat{y}_{2}^{o}\big{)}\\\
+\lambda_{act}\cdot\mathcal{L}_{act}\big{(}\hat{y}_{1}^{act},\hat{y}_{2}^{act}\big{)},\end{split}$
(5)
where $\hat{y}^{h}_{1}$, $\hat{y}^{o}_{1}$ and $\hat{y}^{act}_{1}$ are the
output from the main path $\mathcal{P}_{1}$ and $\lambda$ are the loss
weights. In our experiments, softmax-type outputs use Jensen-Shannon
divergence (JSD) for consistency loss to give loss to each path symmetrically,
while outputs followed by sigmoid, _e.g_., box regression, multi-label action
classes, take the Mean-Squared Error loss. More details on type-specific loss
functions are in the supplement.
In the case of other path pairs, loss is computed in the same manner. The
final loss should thus incorporate all possible pairs. Then, the cross-path
consistency (CPC) loss can be written as:
$\mathcal{L}_{\textbf{CPC}}=\frac{1}{S}\sum_{(k,k^{\prime})\in\mathcal{K}}\mathcal{L}_{\mathcal{P}_{k}\mathcal{P}_{k^{\prime}}}$
(6)
where $\mathcal{K}$ denotes the set of all possible path pairs, and $S$ refers
to the size of set $\mathcal{K}$, _i.e_. the number of path combinations.
#### Cross Matching.
Cross-path consistency learning encourages outputs from different paths to be
consistent. However, since the outputs from a path are given as a set, we
first need to resolve correspondence to specify the pairs of predictions to
enforce consistency. We present cross matching, a simple method that tags each
instance with its corresponding ground truth label. The instances tagged with
the same label are paired to compute consistency loss. On the other hand, if
an instance is not matched with any of the paths’ output, we simply exclude
the instance from consistency learning treating it as no object or no
interaction. Our cross-path consistency loss is introduced below.
Let $\sigma_{k}(i)$ denote the index of the ground truth label that matches
the $i^{th}$ query in the $k^{th}$ path. We define
$\sigma_{k}^{-1}\left(n\right)$ as the query index of path $\mathcal{P}_{k}$
which is matched with the ground truth index $n$. To avoid clutter, we use
$\tilde{\sigma}_{k,n}$ as a shorthand notation for
$\sigma_{k}^{-1}\left(n\right)$. The outputs from different paths with the
same ground-truth label should be consistent. For example,
$\hat{y}^{m}_{k,{\tilde{\sigma}_{k,n}}}$ and
$\hat{y}^{m}_{k,{\tilde{\sigma}_{k^{\prime},n}}}$ which are predictions for
$m$ from $\mathcal{P}_{k}$ and $\mathcal{P}_{k^{\prime}}$ with the same
ground-truth index $n$ should be consistent.
Cross-path consistency loss between output predictions from $\mathcal{P}_{k}$
and $\mathcal{P}_{k^{\prime}}$ with the same ground-truth with index $n$ is
defined as follows:
$\begin{split}\mathcal{L}_{\mathcal{P}_{k}\mathcal{P}_{k^{\prime}}}^{n}=&\hskip
5.69054pt\lambda_{h}\cdot\mathcal{L}_{h}\big{(}\hat{y}^{h}_{k,\tilde{\sigma}_{k,n}},\hat{y}^{h}_{k^{\prime},\tilde{\sigma}_{k^{\prime},n}}\big{)}\\\
&+\lambda_{o}\cdot\mathcal{L}_{o}\big{(}\hat{y}^{o}_{k,\tilde{\sigma}_{k,n}},\hat{y}^{o}_{k^{\prime},\tilde{\sigma}_{k^{\prime},n}}\big{)}\\\
&+\lambda_{act}\cdot\mathcal{L}_{act}\big{(}\hat{y}^{act}_{k,\tilde{\sigma}_{k,n}},\hat{y}^{act}_{k^{\prime},\tilde{\sigma}_{k^{\prime},n}}\big{)}\end{split}$
(7)
#### Final Loss
The final cross-path consistency loss for all $\mathcal{P}_{k}$ is derived as,
$\mathcal{L}_{\textbf{CPC}}=\frac{1}{{S}\cdot\mathcal{N}}\sum_{n=1}^{\mathcal{N}}\sum_{(k,k^{\prime})\in\mathcal{K}}\mathcal{L}_{\mathcal{P}_{k}\mathcal{P}_{k^{\prime}}}^{n}$
(8)
where $\mathcal{N}$ is the number of ground truth labels. Then, the final form
of our training loss $\mathcal{L}$ is defined by
$\mathcal{L}=\sum_{k}\mathcal{L}_{\mathbf{sup}}^{k}+w(t)\cdot\mathcal{L}_{\textbf{CPC}},$
(9)
where $\mathcal{L}_{\mathbf{sup}}^{k}$ is the supervision loss for each path
$\mathcal{P}_{k}$ and $w(t)$ is a ramp-up function [27, 2, 47] for stable
training. Our overall framework is illustrated in Figure 3.
## 4 Experiments
In this section, we empirically evaluate the effectiveness of our cross-path
consistency learning with HOI transformers. Our experiments are conducted on
public HOI detection benchmark datasets: V-COCO and HICO-DET. We first briefly
introduce the datasets and provide implementation details. Our extensive
experiments demonstrate that our training strategy renders significant
improvement on the baseline models without additional parameters or inference
time.
### 4.1 Dataset
#### V-COCO
[16] is a subset of the COCO dataset [35] which contains 5,400 trainval images
and 4,946 test images. V-COCO is annotated with 29 common action classes. For
evaluation of the V-COCO dataset, we report the mAP metric over 25
interactions for two scenarios, The first scenario includes a prediction of
occluded objects and is evaluated with respect to AP${}_{\text{role1}}$. On
the other hand, the second scenario does not contain such cases, and
performance is measured in AP${}_{\text{role2}}$.
#### HICO-DET
[5] is a subset of the HICO dataset which has more than 150K annotated
instances of human-object pairs in 47,051 images (37,536 for training and
9,515 for testing). It is annotated with 600 <interaction, object> instances.
There are 80 unique object types, identical to the COCO object categories, and
117 unique interaction verbs. For evaluation of the HICO-DET, we report the
mAP over three different set categories: (1) all 600 HOI categories in HICO
(Full), (2) 138 HOI categories with less than 10 training samples (Rare), and
(3) 462 HOI categories with more than 10 training samples (Non-Rare).
### 4.2 Implementation Details
#### Training
In our experiment, QPIC [46] and HOTR [23] were used as the baseline for the
HOI transformer respectively. During training, we initialize the network with
pretrained DETR [4] on MS-COCO with a Resnet-50 backbone. For all decoding
paths, parameters of the model are shared except for stage-wise queries and
feedforward networks.
All our experiments using consistency regularization are trained for 90 epochs
and the learning rate is decayed at the 60-th epoch by a factor of 0.1. As an
exception, HOTR is trained up to 50 epochs and the learning rate is decayed at
epoch 30 by a factor of 0.1 for HICO-DET. Following the original training
schemes in QPIC and HOTR, we freeze the encoder and backbone for HOTR, whereas
unfreeze those for QPIC. We use the AdamW [39] optimizer with a batch size of
16, and the initial learning rates for the transformer and backbone parameters
are set to $10^{-4}$ and $10^{-5}$ respectively, and weight decay is set to
$10^{-4}$. All experiments are trained on 8 V100 GPUs.
We re-implement the result of QPIC and HOTR on V-COCO [16] since our
reproduction results are quite different from the official ones in the paper.
For a fair comparison, all the loss coefficients overlapping between baselines
and our training strategy are identical to the ones reported in the paper [23,
46]. Details for hyperparameters relevant to our training strategy are
reported in the supplementary material.
#### Inference
We mainly use $\mathcal{P}_{1}$ ( $x\rightarrow\text{HOI}$ ) for inference to
compare with the baseline models without increasing the number of parameters.
Also, we report the results of other inference paths in our ablation studies.
### 4.3 Comparison with HOI transformer
We evaluate the effectiveness of our method compared to the existing HOI
transformers. All experiments are reported with the main path
$\mathcal{P}_{1}$ that infers HOI triplets by a single decoding stage
($x\rightarrow\text{HOI}$) which is identical to the original HOI transformer.
As shown in Table 1, our CPC training strategy significantly outperforms on
two baselines, HOTR [23] and QPIC [46]. In the V-COCO dataset, the experiment
shows improvement in performance by a considerable margin of 0.9 mAP for QPIC
in AP${}_{\text{role1}}$, and 1.8 mAP for HOTR. For AP${}_{\text{role2}}$,
QPIC and HOTR gain improvement by 0.9 mAP and 1.9 mAP respectively, similar to
that of AP${}_{\text{role1}}$.
In the HICO-DET dataset, our CPC learning with HOTR and QPIC outperforms all
the evaluation categories of HICO-DET, except negligible degradation in the
Non-Rare category on HOTR. Results on rare class on the HICO-DET are improved
by a significant margin of 1.29 mAP and 5.5 mAP for QPIC and HOTR
respectively. In both models, we observe a more prominent performance
improvement in the Rare category. This supports that our training strategy
performs well on rarely seen examples. Our strategy improves the conventional
HOI transformer models.
| V-COCO | HICO-DET
---|---|---
Method | AP${}_{\text{role1}}$ | AP${}_{\text{role2}}$ | Full | Rare | Non-Rare
QPIC | 62.2* | 64.5* | 29.07 | 21.85 | 31.23
QPIC + ours | 63.1 | 65.4 | 29.63 | 23.14 | 31.57
HOTR | 59.8* | 64.9* | 25.10 | 17.34 | 27.42
HOTR + ours | 61.6 | 66.8 | 26.16 | 22.84 | 27.15
Table 1: Comparison of our training strategy with vanilla HOI transformers on V-COCO and HICO-DET. * signifies our results reproduced with the official implementation codes of QPIC and HOTR. Method | Backbone | AP${}_{\text{role1}}$ | AP${}_{\text{role2}}$
---|---|---|---
CNN-based HOI Detection Model
InteractNet [15] | R50-FPN | 40.0 | 48.0
iCAN [12] | R50 | 45.3 | 52.4
TIN [32] | R50 | 47.8 | -
RPNN [58] | R50 | - | 47.5
Verb Embd. [54] | R50 | 45.9 | -
PMFNet [50] | R50-FPN | 52.0 | -
PastaNet [31] | R50-FPN | 51.0 | 57.5
VCL [20] | R50 L | 48.3 | -
UniDet [22] | R50-FPN | 47.5 | 56.2
DRG [11] | R50-FPN | 51.4 | -
FCMNet [36] | R50 | 53.1 | -
ConsNet [37] | R50-FPN | 53.2 | -
PDNet [57] | R50-FPN | 53.3 | -
IDN [30] | R50 | 53.3 | 60.3
GPNN [44] | R152 | 44.0 | -
IPNet [52] | H.G.104 | 51.0 | -
VSGNet [49] | R152 | 51.8 | 57.0
PDNet [57] | Res152 | 52.2 | -
ACP [24] | Res152 | 53.0 | -
Transformer-based HOI Detection Model
HoiT [63] | R101 | 52.9 | -
AS-Net [6] | R50 | 53.9 | -
HOTR [23] | R50 | 55.2 | 64.4
HOTR+ Ours | R50 | 61.6 | 66.8
QPIC [46] | R50 | 58.8 | 61.0
QPIC+ Ours | R50 | 63.1 | 65.4
Table 2: Comparison of performances on the V-COCO test set. AP${}_{\text{role1}}$ and AP${}_{\text{role2}}$ denotes performances under Scenario 1 and Scenario 2 in V-COCO respectively. | Default
---|---
Method | Detector | Backbone | Extra | Full | Rare | Non Rare
CNN-based HOI Detection Model | | | |
InteractNet [15] | COCO | R50-FPN | ✗ | 9.94 | 7.16 | 10.77
iCAN [12] | COCO | R50 | S | 14.84 | 10.45 | 16.15
TIN [32] | COCO | R50 | S+P | 17.03 | 13.42 | 18.11
RPNN [58] | COCO | R50 | P | 17.35 | 12.78 | 18.71
PMFNet [50] | COCO | R50-FPN | S+P | 17.46 | 15.65 | 18.00
No-Frills HOI [17] | COCO | R152 | S+P | 17.18 | 12.17 | 18.68
UnionDet [22] | COCO | R50-FPN | ✗ | 14.25 | 10.23 | 15.46
DRG [11] | COCO | R50-FPN | S+L | 19.26 | 17.74 | 19.71
VCL [20] | COCO | R50 | S | 19.43 | 16.55 | 20.29
FCMNet [36] | COCO | R50 | S+P | 20.41 | 17.34 | 21.56
ACP [24] | COCO | R152 | S+P | 20.59 | 15.92 | 21.98
DJ-RN [29] | COCO | R50 | S+V | 21.34 | 18.53 | 22.18
ConsNet [37] | COCO | R50-FPN | S+L | 22.15 | 17.12 | 23.65
PastaNet [31] | COCO | R50 | S+P+L | 22.65 | 21.17 | 23.09
IDN [30] | COCO | R50 | S | 23.36 | 22.47 | 23.63
GPNN [44] | COCO | R152 | ✗ | 13.11 | 9.41 | 14.23
IPNet [52] | COCO | HourGlass104 | ✗ | 19.56 | 12.79 | 21.58
VSGNet [49] | COCO | R152 | S | 19.80 | 16.05 | 20.91
PD-Net [57] | COCO | R152 | S+P+L | 20.81 | 15.90 | 22.28
Transformer-based HOI Detection Model | | | |
HoiT [63] | HICO-DET | R50 | ✗ | 23.46 | 16.91 | 25.41
AS-Net [6] | HICO-DET | R50 | ✗ | 28.87 | 24.25 | 30.25
HOTR [23] | HICO-DET | R50 | ✗ | 25.10 | 17.34 | 27.42
HOTR+ Ours | HICO-DET | R50 | ✗ | 26.16 | 22.84 | 27.15
QPIC [46] | HICO-DET | R50 | ✗ | 29.07 | 21.85 | 31.23
QPIC+ Ours | HICO-DET | R50 | ✗ | 29.63 | 23.14 | 31.57
Table 3: Performance comparison in HICO-DET. For the Detector, COCO means that the detector is trained on COCO, while HICO-DET means that the detector is first trained on COCO and then fine-tuned on HICO-DET. The each letter in Extra column stands for S: Interaction Patterns (Spatial Correlations), P: Pose, L: Linguistic Priors, V: Volume. Method | Share Dec. | CPC | $\mathcal{P}_{1}$ | $\mathcal{P}_{2}$ | $\mathcal{P}_{3}$ | $\mathcal{P}_{4}$ | Average
---|---|---|---|---|---|---|---
QPIC | ✓ | ✓ | 63.1 | 63.3 | 63.1 | 63.0 | 63.13 $\pm$ 0.05†
| ✓ | 62.4 | 62.9 | 60.8 | 59.4 | 61.38 $\pm$ 1.38
✓ | | 60.7 | 60.7 | 59.9 | 58.1 | 59.85 $\pm$ 1.06
HOTR | ✓ | ✓ | 61.6 | 61.5 | 61.6 | 61.6 | 61.58 $\pm$ 0.02†
| ✓ | 61.2 | 61.6 | 61.1 | 60.6 | 61.13 $\pm$ 0.36
✓ | | 60.6 | 60.6 | 61.2 | 60.6 | 60.75 $\pm$ 0.13
Table 4: Ablation Study on our learning strategies. Ablation results on
shared decoder (Share Dec.), and Cross-Path Consistency (CPC) are
demonstrated. For main path $\mathcal{P}_{1}$, and each augmented path
$\mathcal{P}_{2}$, $\mathcal{P}_{3}$, $\mathcal{P}_{4}$, their performances
are reported measured in mAP. They are evaluated on the V-COCO test set with
respect to Scenario 1. The best performances for each path are highlighted in
bold, and $\dagger$ refers to the case where the least standard deviation is
observed.
### 4.4 Comparison with State-of-the-Art Methods
In Table 2 and Table 3, we compare previous HOI detection methods with ours.
As demonstrated in the tables, our training strategy achieves the best
performance among its peers. Table 2 shows the result on V-COCO dataset in
both AP${}_{\text{role1}}$ and AP${}_{\text{role2}}$. In the V-COCO dataset,
our method achieves outstanding performance of 63.1 mAP in
AP${}_{\text{role1}}$ and 66.8 mAP in AP${}_{\text{role2}}$. Also, the results
on the HICO-DET dataset in Table 3 show that our CPC further improves the
state-of-the-art models (_e.g_., HOTR, and QPIC) in the default setting
achieving 26.16 mAP and 29.63 mAP, respectively.
### 4.5 Ablation Study
We further discuss the effectiveness of our framework through a series of
ablation studies. We first provide a path-wise analysis for our cross-path
consistency learning method. The effect of our training technique components
was tested on each path to validate our method. Subsequently, we analyze the
impact of the number of augmented paths on the main task performance. We
experimentally prove the validity of our method by demonstrating the
correlation between the number of paths and performance.
#### Efficiency of CPC.
Table 4 presents ablation experiment results for all inference paths,
$\mathcal{P}_{1}$, $\mathcal{P}_{2}$, $\mathcal{P}_{3}$, and
$\mathcal{P}_{4}$. Path $\mathcal{P}_{1}$ is the main path, which we aim to
boost performance with the rest of the augmented paths. We try ablating
decoder sharing or cross-path consistency regularization one at a time to
confirm each component’s contribution to our training strategy. Note that all
of our experiments are conducted with the encoder block shared across paths.
When our CPC training strategies are applied, QPIC and HOTR achieve an mAP of
63.1, and 61.6 on main path $\mathcal{P}_{1}$. When the decoder parameters are
not shared, performance degradation in path $\mathcal{P}_{1}$ was observed for
both baselines; a 0.7 mAP drop for QPIC, and 0.4 mAP drop for HOTR. On the
other hand, when CPC regularization is left out while decoder parameters are
shared, performance of QPIC and HOTR decreased by a large margin of 2.4 mAP
and 1.0 mAP each. In terms of overall performance across all paths, the
average mAP showed a similar trend for each experiment condition. The overall
results support that our learning strategy improves generalization of base
architectures, and boosts performance by sharing knowledge throughout paths
and stages.
Interestingly, the standard deviation of all performances dramatically
increases without both components. With unshared decoders, deviation increases
by 1.33 for QPIC and 0.35 for HOTR. Also, when CPC regularization is removed,
deviation increases by 1.01 for QPIC and 0.11 for HOTR. This implies that our
training strategy with shared decoder and CPC leads to more stable training as
well as consistent representations.
#### Impact of Augmented Paths.
We explore how the number of augmented paths affects the performance of the
main path $\mathcal{P}_{1}$ in V-COCO benchmark. Starting from
$\mathcal{P}_{1}$, the augmented paths are gradually added with respect to mAP
of Scenario 1 from Table 5, where each path is independently trained with
default settings with no training techniques applied. We leverage the
augmented path with better performance first, as performance of each model
will serve as a lower bound for the ensemble of paths. Specifically, as shown
in Table 5, both HOTR and QPIC showed better performance in the order of
$\mathcal{P}_{1}$, $\mathcal{P}_{2}$, $\mathcal{P}_{3}$, and
$\mathcal{P}_{4}$, when trained independently.
We compare the four cases where the augmented paths are gradually added in the
corresponding order; _i.e_., $\mathcal{P}_{1}$,
$\mathcal{P}_{1}+\mathcal{P}_{2}$,
$\mathcal{P}_{1}+\mathcal{P}_{2}+\mathcal{P}_{3}$, and
$\mathcal{P}_{1}+\mathcal{P}_{2}+\mathcal{P}_{3}+\mathcal{P}_{4}$. As shown in
Figure 4, performance is gradually improved as augmented paths are added. The
ablation study evidences that regardless of each path performance, taking
advantage of more paths bolsters the learning capability of our main task, and
its performance builds up as the number of augmented paths increases.
Method | $\mathcal{P}_{1}$ | $\mathcal{P}_{2}$ | $\mathcal{P}_{3}$ | $\mathcal{P}_{4}$ | Average
---|---|---|---|---|---
QPIC | 62.2 | 61.9 | 61.7 | 60.4 | 61.55 $\pm$ 0.69
HOTR | 59.8 | 59.5 | 59.0 | 58.9 | 59.3 $\pm$ 0.37
Table 5: Path-wise results on V-COCO.
(a) QPIC
(b) HOTR
Figure 4: Ablation on the number of augmented paths. As the number of
augmented paths increases, main task performance increases accordingly.
## 5 Conclusion
We propose end-to-end Cross-Path Consistency learning for Human-Object
Interaction detection. Through decoding-path augmentation, various decoder
paths are generated which predict HOI triplets in permuted sequences. Then,
consistency regularization is applied across paths to enforce the predictions
to be consistent. Parameter sharing and cross-matching were introduced as well
to enhance learning.
Our method is conceptually simple, and can be applied to a wide range of
transformer architectures. Also, it does not require additional model capacity
nor inference time. The substantial improvements on V-COCO and HICO-DET
support our method’s efficacy in various HOI detection tasks. Through further
empirical studies, its capabilities to improve generalization and to encourage
consistent representations are approved.
#### Acknowledgements
This work was partly supported by Institute of Information & communications
Technology Planning & Evaluation (IITP) grant funded by the Korea government
(MSIT) (No.2021-0-02312, Efficient Meta-learning Based Training Method and
Multipurpose Multi-modal Artificial Neural Networks for Drone AI),
(IITP-2022-2020-0-01819, the ICT Creative Consilience program); ETRI grant
(22ZS1200, Fundamental Technology Research for Human-Centric Autonomous
Intelligent System); and KakaoBrain corporation.
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|
\geq \mu_o \left( B \cap \mathcal O^+_o(g_no,r) \right)
\geq \mu_o(B)
\end{equation*}
By <ref>, $\mathcal O^+_o(g_no,r) \cap \Lambda$ is contained in $\mathcal O_o(g_no,r+20\alpha)$.
Since $\nu_o$ gives full measure to $\Lambda$, we get
\begin{equation*}
< \mu_o(B)
\leq \nu_o \left(\mathcal O^+_o(g_no,r) \right)
\leq \nu_o \left(\mathcal O^+_o(g_no,r) \cap \Lambda\right)
\leq \nu_o \left(\mathcal O_o(g_no,r+20\alpha)\right).
\end{equation*}
Recall that $\dist o{g_no}$ diverges to infinity.
Hence the above inequality contradicts the Shadow Lemma (<ref>).
Let us prove <ref>.
It follows from the construction that $\mu$ is $G$-invariant, $\norm{\mu_o} = 1$, and $\mu_x \ll \mu_y$, for every $x,y \in X$.
Hence we are only left to prove that $\mu$ is quasi-conformal.
Let $x,y \in X$.
We define two auxiliary maps as follows.
\begin{equation*}
\begin{array}{ccccccc}
\bar X & \to & \R & \quad \text{and} \quad\ & \bar X & \to & \R \\
c & \mapsto &\displaystyle \inf_{c' \sim c} c'(x,y) && c & \mapsto &\displaystyle \sup_{c' \sim c} c'(x,y)
\end{array}
\end{equation*}
We denote them by $c \mapsto \beta^-_c(x,y)$ and $c \mapsto \beta^+_c(x,y)$ respectively.
As $\bar X$ is separable, one checks that these maps are $\mathfrak R$-measurable.
Let $B$ be a saturated Borel subset.
Using the conformality of $\nu$ we have
\begin{equation*}
\nu_x(B)
\leq \int \mathbb 1_B(c) e^{-\omega_Gc(x,y)} d\nu_y(c)
\leq \int \mathbb 1_B(c) e^{-\omega_G\beta^-_c(x,y)} d\nu_y(c).
\end{equation*}
Since $B$ is saturated and $c \mapsto \beta^-_c(x,y)$ is $\mathfrak R$-measurable, we get
\begin{equation*}
\mu_x(B)
\leq \int \mathbb 1_B(c) e^{-\omega_G\beta^-_c(x,y)} d\mu_y(c).
\end{equation*}
This inequality holds for every $B \in \mathfrak R$.
\begin{equation*}
\frac {d\mu_x}{d\mu_y} (c) \leq e^{-\omega_G \beta^-_c(x,y)}, \quad \mu\text{-a.e.}
\end{equation*}
In the same way, we obtain a lower bound for the Radon-Nikodym derivative with $\beta^+_c(x,y)$ in place of $\beta^-_c(x,y)$.
By <ref>, for $\mu$-almost every $c \in \bar X$, we have
\begin{equation*}
c(x,y) -20\alpha \leq \beta^-_c(x,y) \leq \beta^+_c(x,y) \leq c(x,y) + 20\alpha.
\end{equation*}
Hence $\mu$ is quasi-conformal.
Point <ref> now follows from <ref> and the quasi-conformality.
§.§ More applications
Assume that the action of $G$ on $X$ is divergent.
For every infinite normal subgroup of $G$ we have
\begin{equation*}
\omega_N > \frac 12 \omega_G.
\end{equation*}
Let $Q = G/N$ and $\omega_Q$ be the growth rate of $Q$ on $X/N$.
According to <ref> we have
\begin{equation*}
\omega_N + \frac 12 \omega_Q \geq \omega_G.
\end{equation*}
Since the map $X \to X/N$ is $1$-Lipschitz, $\omega_Q \leq \omega_G$.
\begin{equation*}
\omega_N \geq \frac 12 \omega_G.
\end{equation*}
Suppose now that, contrary to our claim, $\omega_G = 2 \omega_N$.
We choose
* a $G$-invariant, $\omega_G$-conformal density $\nu = (\nu_x)$ and
* an $N$-invariant, $\omega_N$-conformal density $\nu' = (\nu'_x)$ such that the action of $N$ on $(\bar X, \mathfrak B, \nu'_o)$ is ergodic.
We write $\mu$ and $\mu'$ for their respective restrictions to the reduced horocompactification $(\bar X, \mathfrak R)$.
In particular, the action of $N$ on $(\bar X, \mathfrak R, \mu'_o)$ is ergodic.
We claim that $\mu_0$ is absolutely continuous with respect to $\mu'_0$.
According to <ref>, $(G, \nu')$ satisfies the Shadow Lemma for some parameters $(\epsilon, r_0) \in \R_+^* \times \R_+$.
By <ref>, there exists $\alpha,r \in \R^*_+$ such that $\nu_o$ gives full measure to $\Lambda = \Lambda_{\rm ctg} (G, o, \alpha,r)$.
Without loss of generality, we can assume that $r \geq r_0$.
For simplicity we set $r' = r + 16\alpha$ and write $\Lambda'$ for $\Lambda_{\rm ctg} (G, o, \alpha,r)$.
Let $B$ be a saturated subset contained in $\Lambda_{\rm ctg} (G, o,\alpha,r)$.
Let $V$ be an open set containing $B$.
Observe that $(B \cap \Lambda)^+$ is contained in $\Lambda'$ (<ref>)
Fix $L > r' +16\alpha$.
Using <ref> with $(B \cap \Lambda)^+$, we build a subset $S \subset \mathcal T(\alpha, L)$ such that
\begin{equation*}
B \cap \Lambda \subset (B \cap \Lambda)^+ \subset \bigcup_{g \in S}\mathcal O_o(go,r') \subset V.
\end{equation*}
According to <ref>, there is a subset $S^*$ of $S$ such that
* the collection $\left( \mathcal O_o(go,r')\right)_{g \in S^*}$ is pairwise disjoint, and
* $B\cap \Lambda$ is covered by $\left( \mathcal O_o(go,r'+42\alpha)\right)_{g \in S^*}$
Since $\nu$ gives full measure to $\Lambda$, we have $\nu_o(B) = \nu_o(B \cap \Lambda)$.
Using <ref> with the density $\nu$ we get
\begin{equation*}
\nu_o(B)
\leq \sum_{g \in S^*} \nu_o(\mathcal O_o(go,r'+42\alpha))
\leq e^{2\omega_G(r'+42\alpha)} \sum_{g \in S^*} e^{-\omega_G \dist o{go}}.
\end{equation*}
Recall that for every $g \in G$, we have $\norm{\nu'_{go}} \geq e^{-\omega_N \dist o{go}}$.
Since $\omega_G = 2 \omega_N$, we obtain
\begin{equation*}
\nu_o(B)
\leq e^{2\omega_G(r'+42\alpha)} \sum_{g \in S^*} \norm{\nu'_{go}} e^{-\omega_N \dist o{go}}.
\end{equation*}
Using now the Shadow Principle with the density $\nu'$ we obtain
\begin{equation*}
\nu_o(B)
\leq C \sum_{g \in S^*} \nu'_o(\mathcal O_o(go,r'))
\leq C \nu'_o(V),
\quad \text{where} \quad
C = \frac 1\epsilon e^{2\omega_G(r'+42\alpha)}
\end{equation*}
does not depend on $B$.
This inequality holds for every open subset $V$ containing $B$, hence $\nu_o(B) \leq C\nu'_o(B)$, i.e. $\mu_o(B) \leq C \mu'_o(B)$.
This completes the proof of our claim.
Denote by $f$ the Radon-Nikodym derivative $f = d\mu_o / d \mu'_o$.
Both $\mu$ and $\mu'$ are $N$-invariant.
Hence the set $A = \set{c \in \bar X}{f(c) > 0}$ is $N$-invariant.
Note that $\mu'_o(A) > 0$.
Indeed otherwise $\mu_o$ would be the zero measure.
Since the action of $N$ on $(\bar X, \mathfrak R, \mu'_o)$ is ergodic, we get $\mu'_o(A) = 1$.
Hence $\mu_o$ and $\mu'_o$ are in the same class of measures.
Since $\mu_o$ is $G$-invariant, $\mu'_o$ is $G$-quasi-invariant.
We assumed that $\mu'_o$ is ergodic for the action of $N$.
It follows from <ref> that $\mu'$ is almost fixed by $G$.
Thus $\omega_N \geq \omega_G$ by <ref>.
This contradicts our assumption and completes the proof.
Let $H \subset G$ be a subgroup which is not virtually cyclic and contains a contracting element.
Let $\nu = (\nu_x)$ be an $H$-invariant, $\omega_H$-conformal density and $\mu = (\mu_x)$ its restriction to the reduced horocompactification $(\bar X, \mathfrak R)$.
Assume that the action of $H$ is divergent.
If $\mu$ is almost fixed by $G$, then $(G, \nu)$ satisfies the Shadow Principle.
According to <ref>, there are $\alpha, r_0 \in \R^*_+$ such that $\nu$ gives full measure to $\Lambda_{\rm ctg}(H,o,\alpha,r_0)$.
Proceeding as in the proof of <ref>, we show that for every $g \in G$ and $r \in \R_+$,
\begin{equation}
\label{eqn: shadow lemma almost-fixed reduced}
\nu_o\left(\mathcal O_o(go,r)\right)
\geq \norm{\nu_{go}} e^{-\omega_H \dist o{go}}\nu^g_o\left(\mathcal O_{g^{-1}o}(o,r)\right).
\end{equation}
Choose now $r \geq r_0$ and $g \in G$.
We denote by $\mathcal O^+_{g^{-1}o}(o,r)$ the saturation of the shadow $\mathcal O_{g^{-1}o}(o,r)$, which is mesurable by <ref>.
According to <ref>,
\begin{equation*}
\mathcal O^+_{g^{-1}o}(o,r) \cap \Lambda_{\rm ctg}(H,o,\alpha, r) \subset \mathcal O_{g^{-1}o}(o,r+20\alpha).
\end{equation*}
Recall that $\nu$ gives full measure to $\Lambda_{\rm ctg}(H,o,\alpha, r_0)$, thus to $\Lambda_{\rm ctg}(H,o,\alpha, r)$ as well.
Since $\mu$ is almost fixed by $G$ we have
\begin{align*}
\nu^g_o\left(\mathcal O_{g^{-1}o}(o,r+20\alpha)\right)
\geq \mu^g_o\left(\mathcal O^+_{g^{-1}o}(o,r) \right)
& \geq \epsilon \mu_o\left(\mathcal O^+_{g^{-1}o}(o,r) \right) \\
& \geq \epsilon \nu_o\left(\mathcal O_{g^{-1}o}(o,r) \right),
\end{align*}
where $\epsilon \in \R_+^*$ does not depend on $g$ and $r$.
Combined with (<ref>) it shows that for every $r \geq r_0$, for every $g \in G$, we have
\begin{equation*}
\nu_o\left(\mathcal O_o(go,r +20\alpha)\right)
\geq \epsilon \norm{\nu_{go}} e^{-\omega_H \dist o{go}}\nu_o\left(\mathcal O_{g^{-1}o}(o,r)\right).
\end{equation*}
According to our assumption, $H$ is not virtually cyclic and contains a contracting element.
The conclusion now follows from <ref> applied with the group $H$ and the set $\mathcal D_0 = \{ \nu\}$..
Let $H$ be a commensurated subgroup of $G$.
If the action of $H$ on $X$ is divergent, then the following holds.
Any $H$-invariant, $\omega_H$-conformal density is $G$-almost invariant when restricted to the reduced horocompactification $(\bar X, \mathfrak R)$.
$\omega_H = \omega_G$.
The action of $G$ on $X$ is divergent.
Let $\nu = (\nu_x)$ be an $H$-invariant, $\omega_H$-conformal density.
We denote by $\mu = (\mu_x)$ its restriction to the reduced horocompactification $(\bar X, \mathfrak R)$.
Let $g \in G$.
By definition of commensurability the intersection $H_0 = H^g \cap H$ has finite index in $H$.
In particular, $H_0$ is divergent and $\omega_{H_0} = \omega_H$.
Recall that $\nu^g$ is the image of $\nu$ under the right action of $g \in G$.
It is an $H^g$-invariant, $\omega_H$-conformal density, thus an $H_0$-invariant, $\omega_{H_0}$-conformal density.
Similarly $\nu$ is an $H_0$-invariant, $\omega_{H_0}$-conformal density.
Since $H_0$ is divergent, <ref> tells us that
* there is $C \in \R_+$ such that $\mu^g \leq C \mu$,
* the action of $H^g \cap H$ on $(\bar X, \mathfrak R, \mu_o)$ is ergodic.
Note that $C$ depends a priori on $H_o$ and thus on $g$.
Nevertheless, it still proves that $\mu$ is $G$-quasi-invariant.
Consequently $\mu$ is $C_0$-almost fixed by $G$, for some $C_0 \in \R_+^*$ (<ref>).
We deduce from <ref> that $(G, \nu)$ satisfies the Shadow Principle.
Point <ref> now follows from <ref>.
Recall that $\mathcal P_H(s) \leq \mathcal P_G(s)$, for every $s \in \R_+$.
Since the action of $H$ on $X$ is divergent, $\mathcal P_G(s)$ diverges at $s = \omega_H = \omega_G$.
Hence the action of $G$ on $X$ is divergent as well, which proves <ref>.
We already know that $\mu$ is almost-fixed by $G$, so that the map $\chi \colon G \to \R$ sending $g$ to $\ln \norm{\mu_{go}}$ is a quasi-morphism (<ref>).
We are left to prove that $\mu$ is actually $G$-almost invariant, i.e. $\chi$ is bounded.
Recall that $(G, \nu)$ satisfies the Shadow Principle.
It follows from <ref> that the critical exponent of the series
\begin{equation*}
\sum_{g \in G} e^{\chi(g)}e^{-s\dist o{go}} = \sum_{g \in G} \norm{\nu_{go}} e^{-s\dist o{go}}
\end{equation*}
is exactly $\omega_H$.
Hence $\omega_{-\chi} = \omega_\chi = \omega_H$.
Note also that, since $\nu$ is $H$-invariant, $\chi(hg) = \chi(g)$ for every $h \in H$ and $g \in G$.
Using <ref>, with the quasi-morphism $- \chi$, we produce an $H$-invariant, $\omega_H$-conformal density $\nu^* = (\nu^*_x)$ satisfying the following additional property:
there is $C_1 \in \R_+^*$ such that for every $g \in G$, for every $x \in X$, we have
\begin{equation*}
\frac 1{C_1} \nu_x \leq e^{\chi(g)} {g^{-1}}_\ast \nu_{gx} \leq C_1 \nu_x.
\end{equation*}
Denote by $\mu^*$ its restriction to the reduced horocompactification $(\bar X, \mathfrak R)$.
According to <ref><ref>, there is $C_2 \in \R_+$ such that $\mu \leq C_2 \mu^*$.
Recall that $\mu$ is $C_0$-almost fixed by $G$.
Consequently for every $g \in G$, we have
\begin{equation*}
\leq C_0 \left( {g^{-1}}_\ast \mu_{go}\right)
\leq C_0C_2 \left( {g^{-1}}_\ast\mu^*_{go} \right)
\leq C_0C_1C_2 \left(e^{-\chi(g)}\mu^*_o\right).
\end{equation*}
Since $\mu_o$ and $\mu^*_o$ are probability measures, $\chi$ is bounded, whence the result.
As we noticed in the introduction, every finite index and every normal subgroup of $G$ is commensurated.
More generally, consider a subgroup $H$ of $G$ and $N$ a normal subgroup of $G$.
If $H$ and $N$ are commensurable (i.e. $H\cap N$ has finite index in both $H$ and $N$) then $H$ is commensurated.
However there are plenty of other examples.
Here is a construction suggested by Uri Bader [4].
Consider the free group $\free 2$ and morphism $\phi \colon \free 2 \to M$ where $M$ is a topological group.
Let $K$ be an open compact subgroup of $M$.
Then $H = \phi^{-1}(K)$ is commensurated.
Now if $\phi$ has dense image, then $H$ is commensurable with a normal subgroup of $\free 2$ if and only if $K$ is commensurable with a normal subgroup of $M$.
Consider now for instance a prime $p$ and a morphism $\phi \colon \free 2 \to {\rm PSL}_2(\Q_p)$ with dense image.
Then $\phi^{-1}({\rm PSL}_2(\Z_p))$ provides an example of a commensurated subgroup of $\free 2$ that is not commensurable with a normal subgroup of $\free 2$.
§ STRONGLY POSITIVELY RECURRENT ACTIONS
§.§ Definition
Let $G$ be a group acting properly, by isometries on a proper, geodesic, metric space $X$.
Given a compact subset $K \subset X$, we define a subset $G_K \subset G$ as follows:
an element $g \in G$ belongs to $G_K$, if there exist $x,y \in K$ and a geodesic $\gamma$ joining $x$ to $gy$ such that the intersection $\gamma \cap G K$ is contained in $K \cup gK$.
Although $G_K$ is not a subgroup of $G$, its exponential growth rate $\omega(G_K, X)$ is defined in the same way as for the one of $G$.
The entropy at infinity of the action of $G$ on $X$ is
\begin{equation*}
\omega_\infty(G, X) = \inf_K \omega(G_K, X),
\end{equation*}
where $K$ runs over all compact subsets of $X$.
The action of $G$ on $X$ is strongly positively recurrent (or statistically convex co-compact) if $\omega_\infty(G, X) < \omega(G,X)$.
We refer the reader to [45, 19] for examples of strongly positively recurrent actions in the context of hyperbolic geometry.
Arzhantseva, Cashen and Tao <cit.> also observed that the work of Eskin, Mirzakani, and Rafi <cit.> implies that the action of the mapping class group on the Teichmüller space endowed with the Teichmüller metric is strongly positively recurrent.
§.§ Divergence
If the action of $G$ on $X$ is strongly positively recurrent, then it is divergent.
The statement was proved by Yang [51].
We give here an alternative approach in the spirit of Schapira-Tapie [45].
The idea is to build a $G$-invariant, $\omega_G$-conformal density which gives positive measure to the radial limit set.
Indeed, according to <ref>, this will imply that the action of $G$ on $X$ is divergent.
As we explained in <ref> this part of <ref> does not require that $G$ contains a contracting element.
First, we give a description of the complement of the radial limit set.
To that end we introduce some notations.
Given a compact subset $K \subset X$ and $\epsilon \in \R_+^*$, we denote by $A_{K, \epsilon}$ the set of all cocycles $c \in \partial X$ with the following property:
there is a point $x \in K$ such that for every $\epsilon$-quasi-gradient ray $\gamma \colon \R_+ \to X$ for $c$ starting at $x$, for every $u \in G$, if the intersection $\gamma \cap uK$ is non-empty, then $d(K, uK) \leq 1$.
The radial set of $G$ satisfies the following inclusion
\begin{equation*}
\partial X \setminus \Lambda_{\rm rad} (G)
\subset \bigcap_{K \subset X} G \left( \bigcup_{\epsilon > 0} A_{K, \epsilon}\right),
\end{equation*}
where $K$ runs over all compact subsets of $X$.
The proof is by contraposition.
Consider a cocycle $c \in \partial X$ that is not in the set
\begin{equation*}
\bigcap_{K \subset X} G \left( \bigcup_{\epsilon > 0} A_{K, \epsilon}\right).
\end{equation*}
There is a compact subset $K \subset X$ such that for every $g \in G$ and $\epsilon > 0$, the cocycle $c$ does not belong to $gA_{K, \epsilon}$.
Fix $\epsilon \in (0, 1)$ and $x_0 \in K$.
In addition we let $g_0 = 1$.
We are going to build by induction, a sequence of points $x_1, x_2\dots$ in $X$, a sequence of elements $g_1, g_2,\dots$ in $G$, and a sequence of rays $\gamma_1, \gamma_2,\dots$, such that for every $i \in \N\setminus\{0\}$ the following holds.
* $x_i$ belongs to $g_iK$.
* $c(x_0,x_i) \geq i /2$.
* For every $i \in \N\setminus \{0\}$, the path $\gamma_i$ is a $2^{-i}\epsilon$-quasi-gradient ray of $c$ starting at $x_{i-1}$ and passing through $x_i$.
Let $i \in \N$.
Assume that $x_i \in X$, $g_i \in G$ have been defined.
By assumption $c$ does not belong to the set
\begin{equation*}
g_iA_{K, 2^{-(i+1)}\epsilon}.
\end{equation*}
Hence there exists a $2^{-(i+1)}\epsilon$-quasi-gradient ray $\gamma_{i+1} \colon \R_+ \to X$ for $c$ starting at $x_i$ and an element $u_i \in G$ such that $\gamma_{i+1} \cap g_iu_iK$ is non-empty and $d(g_iK, g_iu_iK) > 1$.
We let $g_{i+1} = g_iu_i$ and denote by $x_{i+1}$ a point in $\gamma_{i+1} \cap g_iu_iK$.
Since $x_i \in g_iK$ and $x_{i+1} \in g_iu_iK$, we have $\dist {x_i}{x_{i+1}} > 1$.
However $\gamma_{i+1}$ is a quasi-gradient line.
\begin{equation*}
c(x_i, x_{i+1}) \geq \dist {x_i}{x_{i+1}} - 2^{-(i+1)}\epsilon \geq 1/2.
\end{equation*}
Using the induction hypothesis, we get
\begin{equation*}
c(x_0, x_{i+1}) \geq c(x_0, x_i) + c(x_i, x_{i+1}) \geq (i+1)/2.
\end{equation*}
Consequently $x_{i+1}$, $g_{i+1}$, and $\gamma_{i+1}$ satisfy the announced properties.
Note that the sequence $(x_i)$ is unbounded.
Indeed otherwise $c(x_0, x_i)$ should be bounded as well.
Thus we can build an infinite path $\gamma$ by concatenating the restriction of each $\gamma_i$ between $x_{i-1}$ and $x_i$.
It follows from the construction that $\gamma$ is an $\epsilon$-quasi-gradient line for $c$, see <ref>.
Moreover $\gamma$ intersects $g_iK$ for every $i \in \N$.
One proves using the triangle inequality that $c$ belongs to the radial limit set.
Let $K \subset X$ be a compact subset and $\epsilon \in \R_+^*$.
For every compact subset $F \subset X$, we define $U_{K, \epsilon}(F)$ to be the set of cocycles $b \in \bar X$ for which there is a cocycle $c \in A_{K, \epsilon}$ satisfying $\norm[F]{b - c} < \epsilon$.
Observe that $U_{K,\epsilon}(F)$ is an open subset of $\bar X$ containing $A_{K, \epsilon}$.
Let $K \subset X$ be a compact set and $\epsilon \in \R_+^*$.
Fix a base point $o \in K$.
There exist $r \in \R_+$ and a finite subset $S \subset G$, such that for every $T \geq \epsilon$, if $F$ stands for the closed ball of radius $T + 2r$ centered at $o$, then
\begin{equation*}
U_{K, \epsilon}(F) \cap Go \subset S \left(\bigcup_{\substack{k \in G_K \\ \dist o{ko} \geq T}} \mathcal O_o(ko,r)\right).
\end{equation*}
Since the action of $G$ on $X$ is proper, the set
\begin{equation*}
S = \set{u \in G}{d(K, uK) \leq 1}
\end{equation*}
if finite.
We fix $r > 2 \diam K + 1$.
Let $T \geq \epsilon$ and $F$ be the closed ball of radius $R = T+2r$ centered at $o$.
Let $g \in G$ such that $go$ belongs to $U_{K, \epsilon}(F)$.
We write $b = \iota(go)$ for the corresponding cocycle.
By definition, there is $c \in A_{K, \epsilon}$ such that $\norm[F]{b-c} < \epsilon$.
Observe first that $\dist o{go} > R - \epsilon$.
Indeed the map $x \mapsto b(x,go)$ admits a global minimum at $go$, while there exists a $c$-gradient line starting at $go$.
We cannot have at the same time $\dist o{go} \leq R - \epsilon$ and $\norm[F]{b-c} < \epsilon$.
In particular, $g \notin S$.
Since $c \in A_{K, \epsilon}$, there exists $x \in K$, such that for every $\epsilon$-quasi-gradient ray $\gamma \colon \R_+ \to X$ for $c$, starting at $x$, if $\gamma$ intersects $uK$ for some $u \in G$, then $u \in S$.
Consider now a geodesic $\alpha\colon \intval 0\ell \to X$ from $x$ to $go$.
We denote by $s \in \intval 0\ell$, the largest time such that the point $y = \alpha(s)$ belongs to $SK$.
We now denote by $t \in \intval s\ell$, the smallest time such that the point $z = \alpha(t)$ lies in $hK$ for some $h \in G\setminus S$ (such a time $t$ exists since $\alpha(\ell)$ belongs to $gK$).
It follows from the construction that $h$ can be written $h = uk$ with $u \in S$ and $k \in G_K$.
Moreover $y \in uK$.
Observe that $\gro y{go}z = 0$, while $\dist y{uo} \leq r/2$ and $\dist z{uko}\leq r/2$.
The triangle inequality yields $\gro {uo}{go}{uko} \leq r$, i.e. $go$ belongs to $u\mathcal O_o(ko, r)$.
We are left to prove that $\dist o{ko} \geq T$.
By construction $\dist o{uo} \leq r$.
Thanks to the triangle inequality, it suffices to show that $\dist oz \geq R$.
Assume on the contrary that $\dist oz < R$.
In particular, both $x$ and $z$ belong to $F$.
Since $b$ and $c$ differ by at most $\epsilon$ on $F$, we get that $c(x,z) \geq \dist xz - \epsilon$.
Hence any geodesic from $x$ to $z$ is an $\epsilon$-quasi-gradient arc for $c$.
If we concatenate this path with a gradient ray for $c$ starting at $z$, we obtain an $\epsilon$-quasi-gradient ray for $c$ starting at $x$ and intersecting $hK$ with $d(K, hK) > 1$.
This contradicts the fact that $c$ belongs to $A_{K, \epsilon}$, and completes the proof.
If the action of $G$ on $X$ is strongly positively recurrent, then there is a $G$-invariant, $\omega_G$-conformal density which gives full measure to the radial limit set $\Lambda_{\rm rad}(G)$.
By definition, there is a compact subset $K \subset X$ such that $\omega_{G_K} < \omega_G$.
We fix once and for all a base point $o \in K$.
The argument relies on Patterson's construction recalled in the proof of <ref> with $H = G$ and $\chi$ the trivial morphism.
In particular, $\mathcal Q(s)$ stands for the weighted Poincaré series defined in (<ref>).
For every $s > \omega_G$, we consider the density $\nu^s = (\nu_x^s)$ defined as in (<ref>).
As we explained there is a sequence $(s_n)$ converging to $\omega_G$ from above such that $\nu^{s_n}$ converges to a $G$-invariant, $\omega_G$-conformal density $\nu$ supported on $\partial X$.
Let $\eta > 0$ such that $\omega_{G_K} < \omega_G - \eta$.
The weight $\theta$ used to construct $\nu$ is slowing increasing.
More precisely, according to <ref> there exists $t_0$ such that for every $t \geq t_0$ and $u \in \R_+$ we have $\theta(t + u) \leq e^{\eta u}\theta(t)$.
Let $\epsilon > 0$.
Let $r \in \R_+$ and $S \subset G$ be the data provided by <ref> applied with $K$ and $\epsilon$.
For every $T \in \R_+$, we write $F_T$ for the closed ball of radius $R =T+2r$ centered at $o$.
Let $s > \omega_G$ and $T \geq \max\{t_0, \epsilon\}$.
In view of <ref>, we have
\begin{equation*}
\nu^s_o\left(U_{K,\epsilon}(F_T)\right)
\leq \card S \sum_{\substack{k \in G_K \\ \dist o{ko} \geq T}} \nu^s_o\left(\mathcal O_o(ko, r) \right).
\end{equation*}
Let us estimate the measures of the shadows in the sum.
Let $k \in G_K$, such that $\dist o{ko} \geq T$.
Any element $g \in G$ such that $go \in \mathcal O_o(ko,r)$ can be written $g = ku$ with $u \in G$ and
\begin{equation*}
\dist o{ko} + \dist o{uo} - 2r \leq \dist o{go} \leq \dist o{ko} + \dist o{uo}.
\end{equation*}
Unfolding the definition of $\nu^s$, we get
\begin{equation}
\label{eqn: spr gives full measure}
\nu^s_o\left(\mathcal O_o(ko, r) \right)
\leq \frac{e^{2sr}e^{-s \dist o{ko}}}{\mathcal Q(s)}\sum_{u \in G} \theta(\dist o{go}) e^{-s \dist o{uo}}.
\end{equation}
Observe that if $\dist o{uo} \geq t_0$, then it follows from our choice of $t_0$ that
\begin{equation*}
\theta\left(\dist o{go}\right)
\leq \theta\left( \dist o{ko} + \dist o{uo}\right)
\leq e^{\eta\dist o{ko}} \theta\left(\dist o{uo}\right).
\end{equation*}
Otherwise, since $\dist o{ko}\geq T \geq t_0$, we have
\begin{equation*}
\theta\left(\dist o{go}\right)
\leq \theta\left(t_0 + \dist o{uo}+ \dist o{ko} -t_0 \right)
\leq e^{\eta\dist o{ko}} \theta\left(t_0\right).
\end{equation*}
We break the sum in (<ref>) according to the length of $u$ and get
\begin{equation*}
\nu^s_o\left(\mathcal O_o(ko, r) \right)
\leq \frac{e^{2sr}e^{-(s-\eta) \dist o{ko}}}{\mathcal Q(s)} \left[\theta(t_0)\Sigma_1(s) + \Sigma_2(s)\right],
\end{equation*}
\begin{align*}
\Sigma_1(s) & = \sum_{\substack{u \in G \\ \dist o{uo} \leq t_0}} e^{-s \dist o{uo}}, \quad \text{and}\quad \\
\Sigma_2(s) & = \sum_{\substack{u \in G \\ \dist o{uo} > t_0}} \theta(\dist o{uo}) e^{-s \dist o{uo}}.
\end{align*}
Note that $\Sigma_1(s)$ is a finite sum that does not depend on $k$, while $\Sigma_2(s)$ is the remainder of the series $\mathcal Q(s)$.
\begin{equation*}
\nu^s_o\left(\mathcal O_o(ko, r) \right)
\leq e^{2sr}
\left[ \frac {\theta(t_0)}{\mathcal Q(s)} \Sigma_1(s) + 1\right] e^{-(s-\eta) \dist o{ko}},
\end{equation*}
Summing over all long elements $k \in G_K$, we get
\begin{equation*}
\nu^s_o\left(U_{K,\epsilon}(F_T)\right)
\leq \card S e^{2sr}
\left[ \frac {\theta(t_0)}{\mathcal Q(s)} \Sigma_1(s) + 1\right]
\sum_{\substack{k \in G_K \\ \dist o{ko} \geq T}} e^{-(s-\eta) \dist o{ko}}.
\end{equation*}
Note that $\Sigma_1(s)$ is bounded, while $\mathcal Q(s)$ diverges to infinity.
Since $U_{K,\epsilon}(F_T)$ is an open subset of $\bar X$, we can pass to the limit and get
\begin{equation*}
\nu_o\left(U_{K,\epsilon}(F_T)\right)
\leq \card S e^{2\omega_Gr}
\sum_{\substack{k \in G_K \\ \dist o{ko} \geq T}} e^{-(\omega_G-\eta) \dist o{ko}}.
\end{equation*}
The sum corresponds to the remainder of the Poincaré series of $G_K$ at $s = \omega_G - \eta$.
However $\omega_G - \eta > \omega_{G_K}$.
Hence this series converges, and its reminder tends to zero when $T$ approaches infinity.
Consequently, for every $\epsilon > 0$,
\begin{equation*}
\nu_o\left(\bigcap_{T \geq 0}U_{K,\epsilon}(F_T)\right) = 0.
\end{equation*}
By construction the set $A_{K, \epsilon}$ is contained in $U_{K, \epsilon}(F_T)$ for every $T \in \R_+$.
It follows from <ref> that
\begin{equation*}
\partial X \setminus \Lambda_{\rm rad} (G)
\subset G \left( \bigcup_{\epsilon > 0} \bigcap_{T \geq 0}U_{K, \epsilon}(F_T)\right).
\end{equation*}
Since $G$ is countable we conclude that $\nu_o(\partial X \setminus \Lambda_{\rm rad}(G)) = 0$.
Recall that $\nu_o$ is supported on $\partial X$, thus $\nu_o(\Lambda_{\rm rad}(G)) = 1$.
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|
# Luminous Late-time Radio Emission from Supernovae Detected by the Karl G.
Jansky Very Large Array Sky Survey (VLASS)
Michael C. Stroh Center for Interdisciplinary Exploration and Research in
Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern
University, Evanston, IL 60201, USA Giacomo Terreran Center for
Interdisciplinary Exploration and Research in Astrophysics (CIERA) and
Department of Physics and Astronomy, Northwestern University, Evanston, IL
60201, USA Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA
93117-5575, USA Deanne L. Coppejans Center for Interdisciplinary Exploration
and Research in Astrophysics (CIERA) and Department of Physics and Astronomy,
Northwestern University, Evanston, IL 60201, USA Department of Physics,
University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK Joe S. Bright
Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA)
and Department of Physics and Astronomy, Northwestern University, Evanston, IL
60201, USA Department of Astronomy, University of California, Berkeley, CA
94720, USA Raffaella Margutti Center for Interdisciplinary Exploration and
Research in Astrophysics (CIERA) and Department of Physics and Astronomy,
Northwestern University, Evanston, IL 60201, USA Department of Astronomy,
University of California, Berkeley, CA 94720, USA Michael F. Bietenholz
Department of Physics and Astronomy, York University, Toronto, M3J 1P3,
Ontario, Canada Hartebeesthoek Radio Observatory, P.O. Box 443, Krugersdorp,
1740, South Africa Fabio De Colle Instituto de Ciencias Nucleares,
Universidad Nacional Autónoma de México, A. P. 70-543 04510 D.F. Mexico
Lindsay DeMarchi Center for Interdisciplinary Exploration and Research in
Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern
University, Evanston, IL 60201, USA Rodolfo Barniol Duran Department of
Physics and Astronomy, California State University, Sacramento, 6000 J Street,
Sacramento, CA 95819, USA Danny Milisavljevic Department of Physics and
Astronomy, Purdue University, 525 Northwestern Avenue, West Lafayette, IN
47907, USA Kohta Murase Department of Physics, The Pennsylvania State
University, University Park, PA 16802, USA Department of Astronomy &
Astrophysics, The Pennsylvania State University, University Park, PA 16802,
USA Center for Mulitmessenger Astrophysics, Institute for Gravitation and the
Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
Center for Gravitational Physics, Yukawa Institute for Theoretical Physics,
Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan Kerry Paterson Center for
Interdisciplinary Exploration and Research in Astrophysics (CIERA) and
Department of Physics and Astronomy, Northwestern University, Evanston, IL
60201, USA Wendy L. Williams Leiden Observatory, Leiden University, PO Box
9513, NL-2300 RA Leiden, the Netherlands
###### Abstract
We present a population of 19 radio-luminous supernovae (SNe) with emission
reaching $L_{\nu}{\sim}10^{26}-10^{29}\,\rm{erg\,s^{-1}Hz^{-1}}$ in the first
epoch of the Very Large Array Sky Survey (VLASS) at $2-4$ GHz. Our sample
includes one long Gamma-Ray Burst, SN 2017iuk/GRB171205A, and 18 core-collapse
SNe detected at $\approx(1-60)$ years after explosion. No thermonuclear
explosion shows evidence for bright radio emission, and hydrogen-poor
progenitors dominate the sub-sample of core-collapse events with spectroscopic
classification at the time of explosion (79%). We interpret these findings
into the context of the expected radio emission from the forward shock
interaction with the circumstellar medium (CSM). We conclude that these
observations require a departure from the single wind-like density profile
(i.e., $\rho_{\rm{CSM}}\propto r^{-2}$) that is expected around massive stars
and/or a departure from a spherical Newtonian shock. Viable alternatives
include the shock interaction with a detached, dense shell of CSM formed by a
large effective progenitor mass-loss rate $\dot{M}\sim(10^{-4}-10^{-1})$ M⊙
yr-1 (for an assumed wind velocity of $1000\,\rm{km\,s^{-1}}$); emission from
an off-axis relativistic jet entering our line of sight; or the emergence of
emission from a newly-born pulsar-wind nebula. The relativistic SN 2012ap that
is detected 5.7 and 8.5 years after explosion with $L_{\nu}{\sim}10^{28}$ erg
s-1 Hz-1 might constitute the first detections of an off-axis jet+cocoon
system in a massive star. However, none of the VLASS-SNe with archival data
points are consistent with our model off-axis jet light curves. Future multi-
wavelength observations will distinguish among these scenarios. Our VLASS
source catalogs, which were used to perform the VLASS cross matching, are
publicly available athttps://doi.org/10.5281/zenodo.4895112 (catalog
10.5281/zenodo.4895112).
Core-collapse supernovae, Radio transient sources, Sky surveys, Very Large
Array
††software: APLpy (Robitaille & Bressert, 2012; Robitaille, 2019), Astropy
(Astropy Collaboration et al., 2013, 2018), BOXFIT (van Eerten et al., 2012),
Pandas (Wes McKinney, 2010), PyBDSF (Mohan & Rafferty, 2015), Q3C (Koposov &
Bartunov, 2006), SExtractor (Bertin & Arnouts, 1996)
## 1 Introduction
Radio observations of stellar explosions in the years-to-decades after stellar
demise constitute a probe of the physical properties of the fastest ejecta in
the explosion (i.e. their velocity and energy), and of the environment at
large distances of $r\geq 10^{17}\,\rm{cm}$ (i.e., the density of the
circumstellar medium, CSM), e.g., Chevalier & Fransson (2017). There are three
main sources of bright non-thermal synchrotron radio emission in SNe at $t\geq
1$ yr: (i) the deceleration of the forward shock in a dense environment (e.g.,
Chevalier, 1998; Chevalier & Fransson, 2006); (ii) emission from an off-axis
relativistic jet entering our line of sight (e.g., Granot et al., 2002); (iii)
emergence of emission from a newly-formed pulsar-wind nebula (PWN; Slane,
2017). Late-time radio observations of cosmic explosions can thus reveal a
complex mass-loss history of the stellar progenitors in the years leading up
to core-collapse; they can reveal jet-driven explosions similar to long Gamma-
Ray Bursts (GRBs) that launched a jet that was misaligned with our line of
sight; or they can reveal the energetics and properties of the compact object
remnant. However, most SNe are not observed at radio wavelengths at very late
times. For example, in the sample of 294 SNe observed at $\sim 5-8$ GHz
compiled by Bietenholz et al. (2021), only 87 were observed at more than 1000
days post explosion and of these only 28 were detected111This sample is one of
the largest compilations of radio observations of SNe. We note that it is not
a complete sample (see Bietenholz et al. 2021), so these numbers are likely
underestimates. One important selection effect is that the peak frequency of
the radio emission from SNe declines with time, so lower frequency
observations would yield a higher detection rate.. As a result, the late-time
radio emission from SNe constitutes a poorly explored region of the phase
space (Figure 2 in Bietenholz et al. 2021).
Here we present a sample of 19 radio-luminous SNe detected in the first epoch
of the Very Large Array Sky Survey (VLASS, Lacy et al., 2020) carried out by
the Karl G. Jansky Very Large Array (VLA). VLASS is a successor and
complementary survey to the National Radio Astronomy Observatory (NRAO) VLA
Sky Survey (NVSS, Condon et al., 1998) and Faint Images of the Radio Sky at
Twenty centimeters (FIRST, Becker et al., 1995) surveys. The survey is
conducted at $2$ to $4$ GHz and is split into three distinct epochs, each
scanning the full survey region (declination $\delta{>}{-}40^{\circ}$) with an
${\approx}\,32$ month observing cadence reaching an RMS noise of
$120\mu$Jy/beam per epoch. Exploring ‘Hidden Explosions and Transient Events’
is a key VLASS science theme, and Hallinan et al. (2020) demonstrated the
synergy between VLASS and newly discovered transients with the detection of
the type II SN 2019xhb in the 2nd VLASS observing epoch 202 days after
discovery. Importantly, by scanning the northern sky, VLASS offers the
opportunity to perform a systematic and unbiased survey of the late-time radio
emission from the tens of thousands of previously reported SNe. With the
exception of SN 2017iuk/GRB171205A, the sample of radio luminous SNe presented
in this paper was imaged at an epoch corresponding to $\approx(1-60)$ years
post explosion.
This paper is organized as follows. In Section 2, we describe our methodology
for identifying optically detected SNe in the VLASS sample, how we filtered
out potential spurious detections, and present our list of VLASS detected SNe.
Section 3 discusses physical processes that could produce the bright radio
emission associated with the SNe. Finally, in Section 4 we summarize the
conclusions.
## 2 A Sample of SNe with luminous late-time radio emission
We created a catalog consisting of all publicly-announced, optical SNe by
combining the Bright Supernovae222http://rochesterastronomy.org/snimages, Open
SNe333https://sne.space, and Transient Name Server444https://www.wis-tns.org
catalogs. We included all optical SNe detected prior to 2020-01-01 leading to
an initial sample of $\approx$70,000 unique SNe. Two independently generated
VLASS detection catalogs were produced using the ${\approx}35,000$ VLASS epoch
1 quick look images provided by NRAO.
### 2.1 PyBDSF detections
The complete VLASS epoch 1 was processed using the Python Blob Detector and
Source Finder version 1.9.1 (PyBDSF, Mohan & Rafferty, 2015). We created a
local catalog consisting of all detections using a source detection threshold
of 5$\sigma$ (thresh_pix=5.0), threshold for islands of 3$\sigma$
(thresh_isl=3.0), and fixing source components to be Gaussian with major axis,
minor axis, and position angle equal to the synthesised beam shape from the
respective VLASS observations (fix_to_beam=True). The values of thresh_pix and
thresh_isl are used to calculate pixel islands of significant emission. The
thresh_pix parameter is used to identify significant pixels (where the pixel
value is greater than mean + $\texttt{thresh\\_pix}\times\sigma$, where
$\sigma$ is the rms noise of the image) used for fitting, with the mean map
calculated using a box of pixel size and pixel step size either calculated
within PyBDSF or set manually by the user. The fitting region is then extended
based on the value of thresh_isl such that all pixels greater than mean +
$\texttt{thresh\\_isl}\times\sigma$ which are adjacent (including diagonally)
to a significant pixel identified using thresh_pix are included in the fitting
region or ‘island’ (if this process causes islands to overlap they are
combined). Multiple Gaussians are then fit to the island in order to best
describe the source (the collection of Gaussians within an island) and we
fixed the Gaussians to the shape of the synthesized beam.
This VLASS-PyBDSF catalog resulted in 3 752 214 sources, similar to the 3 381
277 VLASS sources cataloged by the Canadian Initiative for Radio Astronomy
Data Analysis (CIRADA, Gordon et al., 2020), and more than the 2 232 726
sources in the VLASS Quicklook Catalog (Bruzewski et al., 2021). Our VLASS-
PyBDSF catalog is publicly available athttps://doi.org/10.5281/zenodo.4895112
(catalog 10.5281/zenodo.4895112).
Figure 1: Field of SN 2012ap in the Pan-STARRS1 r-band (left) and at 3 GHz
with VLASS (right). The red lines indicate the optical SN position, and each
image notes the date of each observation. The optical image clearly shows the
host galaxy of SN 2012ap, but no emission from the host is found in the
optical image near the optical SN position. No radio emission is found in the
field near SN 2012ap except near the optical SN position. The VLASS image was
taken 2065 days since explosion (Milisavljevic et al., 2014).
### 2.2 Source extractor detections
We created an initial list of possible VLASS source detections using Source
Extractor version 2.25.0 (SExtractor, Bertin & Arnouts, 1996). SExtractor is
not optimized for radio imaging analysis, but through trial and error with
fields where transients were known to exist, we settled on a requirement of at
least 5 contiguous pixels (detect_minarea) to be above the low
detect_threshold=5$\sigma$. The VLASS-SExtractor catalog contained 9 652 665
possible source detections. Most of these detections are likely spurious, as
evidenced by the number of sources detected by the previous section. We note
that SExtractor runs at least an order of magnitude faster through VLASS quick
look images than PyBDSF, so it may be preferred for studies where a large
number of possible bogus detections are acceptable. Our VLASS-SExtractor
catalog is publicly available athttps://doi.org/10.5281/zenodo.4895112
(catalog 10.5281/zenodo.4895112).
### 2.3 Identifying supernovae for cross matching
We chose low detection thresholds in order to minimize the chance that we may
miss potential SNe cross matches on our initial pass. In order to reduce the
number of spurious matches, we required that the potential SNe cross matches
must be detected by both VLASS-SExtractor and VLASS-PyBDSF catalogs. We cross-
matched the locations of the optical SNe with the VLASS-PyBDSF and VLASS-
SExtractor catalogs using a $5^{\prime\prime}$ angular separation. This
separation is ${\approx}2$ times the average VLASS beam size, and helps
account for a lack of positional precision in SN discovery reports. Using our
initial list of ${\approx}70,000$ SNe, ${\approx}1600$, and ${\approx}1400$
SNe have potential PyBDSF, and SExtractor cross-matches, respectively. By
requiring that a source must have cross-matches in both the VLASS-PyBDSF and
VLASS-SExtractor catalogs, we have ${\approx}1300$ potential cross-matches in
VLASS. We further required that the VLASS observation must have occurred
_after_ the SN discovery date.
The possible VLASS-SNe were visually inspected to ensure that the VLASS-SNe
detections are real and are not due to radio imaging artifacts (see for
example Figure 1). We also rejected SNe when the location of the radio source
broadly overlapped with that of the galactic nucleus. After visual inspection,
only ${\approx}100$ potential VLASS-SNe detections remained.
### 2.4 Multiwavelength cross matching
In order to filter out known radio sources, we rejected associations that had
counterparts in the NVSS or FIRST catalogs prior to their explosion date. We
removed VLASS-SNe near active galactic nuclei (AGN) identified by Assef et al.
(2018), who cataloged probable AGN in the Wide-field Infrared Survey Explorer
(WISE, Wright et al., 2010) AllWISE data release (Cutri et al., 2021).
D’Abrusco et al. (2013) examined the chance of probability for spurious
associations between a sample of NVSS detected blazars and AllWISE. They
calculated the number of additional cross-matches between their NVSS blazar
catalog and AllWISE $\Delta N_{t}$ as the cross-matching radius increases.
Similarly, the number of additional spurious cross-matches per increasing
cross-matching radius, $\Delta N_{r}$, was calculated by adding a random
offset to the NVSS blazar positions. D’Abrusco et al. (2013) found that
$\Delta N_{r}>\Delta N_{t}$ for cross-matching radii above
$3.3^{\prime\prime}$, thus a cross-matching radius of $3.3^{\prime\prime}$ can
be considered a cross-matching between point-like VLA sources and AllWISE. For
the infrared AGN cross matching, we adopted the $3.3^{\prime\prime}$ angular
search radius suggested by D’Abrusco et al. (2013).
Possibly misidentified AGN were also removed by cross matching our VLASS-SNe
candidates against the Chandra Source Catalog v2.0 (CSC 2.0, Evans et al.,
2020a), the most recent XMM-Newton X-ray source data release (4XMM-DR10, Webb
et al., 2020), and the 2nd Swift-XRT point source catalog (2SXPS, Evans et
al., 2020b). The error in the X-ray position is generally greater than the
astrometric uncertainties in the VLASS positions (see the Appendix of
Bruzewski et al., 2021), thus for X-ray catalog cross-matching, we removed
sources within the 1$\sigma$ X-ray error region in the respective X-ray
catalog. We ensured that no VLASS-SNe candidates were rejected by targeted SNe
follow-up observations.
We further inspected the VLASS-SNe candidates within the 424 square degrees
covered so far by the LOFAR Two-metre Sky Survey (LoTSS Data Release 1,
Shimwell et al., 2019), to ensure that none were classified as AGN based on
any of their $150$ MHz radio morphology or luminosity, or based on their
multi-wavelength cross-identifications (Williams et al., 2019). While only two
of the candidates lie within this area, both were detected as star-forming
galaxies. Further releases of LoTSS over the Northern sky will enable further
such comparisons.
This multiwavelength filtering procedure leads to a sample of 19 core-collapse
SNe with associated VLASS emission. The final VLASS-SNe sample is listed in
Table 1. All VLASS-SNe also have counterparts in the CIRADA VLASS catalogue,
and SN 2017hcb is the only source without a cross-match in the VLASS Quicklook
Catalog. Interestingly, we note that no thermonuclear explosion (i.e. Ia-like)
passed the criteria above, in spite of it largely dominating the initial
optical SN sample. The lack of type Ia SNe in the sample is consistent with
the lack of radio emission associated with type Ia (e.g., Chomiuk et al.,
2016), but the lack chance coincidence matches may be evidence of the strength
of our multiwavelength filtering described above. Of the 14 VLASS detected SNe
with early-time spectroscopic classification, 13 (93%) and 12 (86%) are
detected at ${>}10^{2}$ and ${>}10^{3}$ days post explosion, respectively. SNe
with hydrogen poor progenitors at the time of explosion make-up the majority
11 (79%) of the sources with early-time spectroscopic classification.
Remarkably, we find that SN 2012ap, one of the only two known SNe with
relativistic ejecta without a GRB (Margutti et al., 2014a; Chakraborti et al.,
2015; Milisavljevic et al., 2015a), shows evidence for bright radio emission
years after explosion, and is a member of the sample. The very nearby SN
2017iuk/GRB171205A, at redshift $z{=}0.0368$ (de Ugarte Postigo et al., 2017),
was detected in the first VLASS epoch less than 60 days following the Swift
trigger.
### 2.5 Final flux densities
In addition to a complete and consistent processing of VLASS epoch 1 (as
described in Section 2.1) we performed an optimized manual analysis on each of
the fields containing the sources listed in Table 1. The default significance
parameters used for this analysis were thresh_isl=5.0 and thresh_pix=5.0, with
an adaptive region used to calculate the RMS and mean maps
(adaptive_rms_box=True), and components fixed to be the same shape as the
synthesised beam (fix_to_beam=True). The island and pixel threshold values
were adjusted on a per-field basis depending on e.g. bright source artifacts
or extended host structure; however, we require
$\texttt{thresh\\_pix}\geq\texttt{thresh\\_isl}$. Adaptive RMS calculation
ensures that region size near bright sources is reduced compared to regions
devoid of bright emission, properly accounting for elevated RMS levels
resulting from imaging artifacts (an issue in a number of the VLASS fields).
For fields with particularly strong artifacts we set the RMS box size and step
size manually such that the noise map captured the variation caused by the
artifacts. In the cases where the SN emission formed part of an extended
emission complex from the host galaxy we set fix_to_beam=False in order to
better model the emitting region. We ran PyBDSF in interactive mode
(interactive=True) and manually inspected the result of the island and source
detection, adjusting our significance threshold and the size of the region
used to calculate the RMS noise to improve the source fitting.
For fields with extreme imaging artifacts around bright sources, we manually
set the RMS box size and disabled adaptive region sizing. In a handful of
cases, emission from the transient was part of a larger emission complex
(radio emission from the host galaxies) and the emission island was better
described using Gaussians with unconstrained shapes. We list the results of
our fitting in Table 1, and note any deviations from the default parameters.
Additionally we analyzed each of the target fields and list the flux densities
and position of the sources in the first half of the second VLASS epoch (i.e.
epoch 2.1). In the cases of SN 2017iuk/GRB171205A and SDSS-II SN 8524, the
source is no longer detected in the second epoch, so instead we list a
$3\sigma$ upper limit. We compare the luminosities and timescales of the VLASS
detected SNe to historical radio light curves in Figure 2.
Figure 2: VLASS-SNe detections in the context of H-rich SNe (red), H-poor SNe
(blue) SNe and long GRBs (gray). A number of the VLASS observations were taken
at a later stage than SNe are typically observed and detected at radio
wavelengths, and show brighter emission than would be expected at this epoch.
Notably, the H-poor VLASS-SNe were observed years after the radio emission (at
$\gtrsim$1 GHz) from this class typically fades. Archival radio light curves
for VLASS detected SNe are included: SN 1986J ($3-5$ GHz; Bietenholz & Bartel,
2017), SN 2003bg (4.86 GHz; Soderberg et al., 2006a), SN 2004C (4.9 GHz;
DeMarchi in prep.), SN 2004dk ($3-5$ GHz; Wellons et al., 2012;
Balasubramanian et al., 2021), PTF11qcj ($3-4$ GHz; Palliyaguru et al., 2019a;
Corsi et al., 2014), SN 2012ap ($3$ GHz extrapolation based on radio SED
modeling; Chakraborti et al., 2015), SN 2012au ($3-4$ GHz; Kamble et al.,
2014a, Terreran et al. in prep.), SN 2014C (7.1 GHz; Margutti et al., 2017),
and SN 2016coi (3 GHz; Terreran et al., 2019). The archival radio observations
of SNe are from Bietenholz et al. (2021), and the archival long GRBs are from
Chandra & Frail (2012). Most archival historical light curves are at 8.6 GHz,
as the 3 GHz light curves are not well sampled. From the VLASS detected
sample, SDSS-II SN 8524 is not included since it has neither a known host
galaxy nor redshift, thus a luminosity cannot be calculated. The H-rich and
H-poor designations are inferred from the spectrum near the time of explosion.
The upper x-axis provides a reference distance scale for a fiducial normal SN
shock velocity of $0.05$c with no deceleration. This figure highlights the
presence of two groups of H-rich SNe in the radio phase space, with IIn SNe
belonging to the group with luminous radio emission years after explosion (see
e.g. Bietenholz et al., 2021).
## 3 Powering luminous late-time radio emission
Winds from massive stars enhance and shape the density of their immediate
surroundings (e.g., Smith, 2014). Radio emission from stellar explosions is
normally associated with an interaction between the fastest SN ejecta (i.e.
the blastwave) and the wind-shaped CSM. As the forward shock propagates
through the CSM, the electrons are accelerated, creating a bell-shaped non-
thermal synchrotron spectrum (Chevalier & Fransson, 2017). The radio spectrum
is characterized by a peak frequency, $\nu_{pk}$, that cascades to lower
frequencies as the blastwave expands (e.g., Chevalier 1998; Chevalier &
Fransson 2006). For synchrotron self-absorption (SSA) dominated spectra,
Chevalier (1998) suggests $\nu_{pk}\propto R^{2/7}B^{9/7}$, and the spectral
peak flux $F_{pk}\propto R^{19/7}B^{19/7}$, where $R$ is the forward shock
radius and $B$ is the post-shock magnetic field. The optically thin flux
density at $\nu>\nu_{pk}$ scales as $F_{\nu,thin}{\propto}\,\nu^{-(p-1)/2}$
(where $p$ is the power-law index of the electron distribution,
$N_{e}(\gamma_{e}){\propto}\,\gamma_{e}^{-p}$, and $\gamma_{e}$ is the
electron Lorentz factor) and the optically thick spectrum at $\nu<\nu_{pk}$ is
described as $F_{\nu,thick}\propto\nu^{5/2}$.
During the SN interaction phase with a “wind density profile” environment
expected around massive stars ($\rho_{\rm{CSM}}{\propto}r^{-2}$), the self-
similar solutions by Chevalier (1982) apply and the shock radius evolves with
time as $R\propto t^{\frac{n-3}{n-2}}$, where the density in the outer layers
of the stellar progenitor has been parametrized as $\rho_{\rm SN}\propto
r^{-n}$. In the limit of no evolution of the shock microphysical parameters
(e.g., the fraction of post-shock energy in magnetic fields and relativistic
electrons, $\epsilon_{B}$ and $\epsilon_{e}$), and using $n\approx 10$ as
appropriate for compact massive stars (Chevalier & Fransson, 2006; Matzner &
McKee, 1999), the equations above imply $\nu_{pk}\approx t^{-1}$ and
$F_{pk}\approx$ constant. Since radio SNe typically show $p\approx 3$ (or
$F_{\nu,thin}\propto\nu^{-1}$), $L_{\nu,pk}\leq
10^{28}\,\rm{erg\,s^{-1}Hz^{-1}}$ and $\nu_{pk}\leq 10$ GHz at ${<}0.1$ year
after explosion, the prediction of this single wind model is $v_{pk}{\ll}1$
GHz and a luminosity $<10^{27}\,\rm{erg\,s^{-1}Hz^{-1}}$ in the VLASS bandpass
at the current epoch (which corresponds to ${>}10^{3}$ days since explosion,
Figure 2). We conclude that our sample of VLASS SNe require a deviation from a
single-wind model.
In the remainder of this section, we discuss three alternative explanations:
(i) interaction of the SN shock with dense shells of CSM (Section 3.1); (ii)
emission from an off-axis relativistic jet entering our line of sight (Section
3.2); (iii) emergence of emission from a PWN (Section 3.3).
### 3.1 Dense detached CSM shells in the local SN environment
VLASS SNe show a level of radio emission comparable to the most luminous type
IIn SNe (Fig. 2). We place the VLASS-SNe into the phase space of radio
observables $\nu_{pk}$, $L_{\nu,pk}$ and peak time $t_{pk}$ in Figure 3, where
we calculated lines of constant shock velocity $v_{sh}$ and mass-loss
$\dot{M}$ rate following the standard formulation of SSA radio emission from a
blast wave during the interaction phase with a wind-like environment (e.g.,
Chevalier, 1998; Chevalier & Fransson, 2006; Soderberg et al., 2005, 2012).
Equipartition of energy between the relativistic electrons, protons and
magnetic field, i.e. $\epsilon_{e}=\epsilon_{B}=1/3$, where $\epsilon_{e}$ is
the fraction of thermal energy stored in electrons, and $\epsilon_{B}$ is the
fraction of magnetic energy relative to the thermal energy leads to a solid
lower limit on the mass-loss rate parameter $\dot{M}$ for a given wind
velocity ($v_{w}$), where $\rho_{CSM}=\frac{\dot{M}}{4\pi v_{w}r^{2}}$. We
present our results for both $\epsilon_{e}=\epsilon_{B}=1/3$ and for
$\epsilon_{e}=0.1$ and $\epsilon_{B}=0.01$ Our discussion below focuses on our
fiducial case of $\epsilon_{e}=0.1$ and $\epsilon_{B}=0.01$. All $\dot{M}$
values quoted are for a wind velocity $v_{w}=10^{3}\,\rm{km\,s^{-1}}$. A few
considerations follow from Figure 3:
Figure 3: Location of the VLASS SNe (filled red squares) in the phase space
of radio observables. A blue outline marks the VLASS SNe with an H-poor
spectrum at the time of explosion. Black filled circles: GRB-SNe. Grey filled
circles: H-stripped SNe from radio observations typically acquired at
$\lesssim 100$ days since explosion. We assume $p=3$ and the shock
microphysics indicated in the title of each plot. Black dashed lines: lines of
constant shock velocity assuming SSA only. Orange dashed lines: lines of
constant mass-loss rate, here calculated for an assumed wind velocity of
$10^{3}$ km/s. The red filled squares show what the properties of the VLASS
SNe would be in the case that the emission peaked in the VLASS band at the
time of the observations (see Section 3.1). Red open squares: location of the
VLASS SNe for an optically thin spectrum $L_{\nu}\propto\nu^{-1}$, assuming
that the $\nu_{pk}$ of the SSA spectrum is below the VLASS band at $\approx
0.3$ GHz. The VLASS object that crosses the $v=c$ line is SN
2017iuk/GRB171205A. VLASS Memo 13 report an ${\sim}10\%$ overestimate in flux
densities from the VLASS epoch 1 QuickLook data. No appreciable difference is
found in this analysis when applying a 10% correction to these figures.
References: Soderberg et al. (2012) and references therein.
* •
If $\nu_{pk}\gtrsim\nu_{obs}$ (where $\nu_{obs}$ is the frequency of the VLASS
observations), then VLASS-SNe require very dense environments with an
effective $\dot{M}\gtrsim 0.1\,\rm{M_{\odot}year^{-1}}$, which is
significantly larger than the typical $\dot{M}$ inferred for non-type-IIn SN
progenitors that comprise the majority of our sample (Smith 2014). In absolute
terms, the inferred $\dot{M}$ would compete with the most extreme mass-loss
rates invoked for evolved massive stars.
* •
A lower $\nu_{pk}<\nu_{obs}$ would bring the VLASS-SNe in line with the lower
$\dot{M}\sim 10^{-4}-10^{-3}\,\rm{M_{\odot}year^{-1}}$ that are typical of
massive stars. The empty squares of Figure 3 show the location of VLASS-SNe
for $\nu_{pk}=0.3$ GHz as an example. However, the lower $\nu_{pk}$ would also
lead to shock velocities $v_{sh}$ $\geq 0.1$c and
$L_{\nu,pk}>10^{28}\,\rm{erg\,s^{-1}Hz^{-1}}$, implying that VLASS-SNe would
constitute a class of radio SNe as luminous as long GRBs and with mildly
relativistic shocks at ${>}10^{3}$ days (and likely faster at earlier times).
Since earlier-time radio follow up of some VLASS-SNe indicated “normal” SN
shock speeds of $\sim 0.1-0.2c$ at a few months post-explosion (e.g., SN
2012au in Kamble et al. 2014b), it is clear that the relativistic ejecta
scenario cannot explain the entire VLASS-SNe sample unless the relativistic
ejecta is highly collimated (i.e. a jet) and pointing away from our line of
sight at early times (i.e. off-axis). We further explore the relativistic
ejecta scenario in Section 3.2.
Mass-loss rates $\dot{M}{\gtrsim}0.1\,\rm{M_{\odot}year^{-1}}$, _if_ sustained
until the time of explosion, would lead to very prominent type-IIn like
spectroscopic features at earlier times for all the VLASS-SNe, which were not
observed for the majority of the sample. Earlier radio observations of some
targets also pointed to significantly lower $\dot{M}\approx
10^{-5}\,\rm{M_{\odot}year^{-1}}$ (e.g., SNe 2004dk, 2012au and 2012ap;
Wellons et al., 2012; Kamble et al., 2014b; Chakraborti et al., 2015) at the
smaller radii probed at those epochs $r{\lesssim}5\times 10^{16}\,\rm{cm}$.
The emerging picture is that at least some VLASS-SNe exploded in a low-density
bubble surrounded by a shell of dense material at $r{\sim}v_{sh}\delta
t=(v_{sh}/10^{4}\,\rm{km\,s^{-1}})(\delta t/8000\,\rm{days}){\approx}0.5$ pc,
consistent with the findings from the multi-wavelength monitoring of SNe
2003bg, 2004C, 2004dk, 2014C, and PTF11qcj (Soderberg et al. 2006b, Margutti
et al. 2017, Pooley et al. 2019, Corsi et al. 2014, Palliyaguru et al. 2019b,
Murase et al. 2019, Brethauer et al. 2020, Balasubramanian et al. 2021,
DeMarchi in prep.). For VLASS-SNe from H-poor stellar progenitors (which
interestingly dominate the sample), these overdensities might represent the
shedding of their H-rich envelope in the centuries before core-collapse.
Optical spectroscopy at the time of the radio re-brightenings of SN 2003bg, SN
2004dk, SN 2014C and PTF11qcj confirmed the later appearance of H features in
the spectra (Soderberg et al., 2006b; Pooley et al., 2019; Milisavljevic et
al., 2015b; Palliyaguru et al., 2019b), consistent with this scenario.
Potential theoretical explanations of this phenomenology include the
interaction of faster Wolf-Rayet winds with the slower winds of the red
supergiant phase coupled with a shorter-than expected Wolf-Rayet phase;
envelope ejection due to binary interaction; or mass shedding due to gravity-
wave powered mass loss (e.g., Smith 2014, Zhao & Fuller 2020, Wu & Fuller
2021).
### 3.2 Off-axis relativistic jets
Figure 4: VLASS-SNe 3 GHz light curves for SN 2005ha, SN 2012ap, and SDSS-II
SN 12882. The 3 GHz SN 2012ap archival light curve is included using the model
from Chakraborti et al. (2015). The SN 2012ap light curve is shown along with
the 10 top-hat jet models in our grid that best fit the VLASS only light curve
(gray dashed lines) and the 10 models that best fit the combined VLASS and
archival light curve (gray dotted lines). SN 2005ha and SDSS-II SN 12882 are
the only multi-epoch detected VLASS-SNe that are consistent with off-axis jet
models from our grid (i.e. $\chi^{2}<1$). The solid black lines represent the
off-axis jet models that are consistent with SN 2005ha and SDSS-II SN 12882
light curves. SN 2005ha and SDSS-II SN 12882 had limited spectroscopic follow-
up and were not classified as H-rich or H-poor (e.g., Marsden 2005). We show
this figure as a proof of concept, but we note that the models that best fit
the SN 2005ha and SDSS-II SN 12882 light curves have high isotropic kinetic
energies of $E_{iso}=10^{55}$ erg, corresponding to beaming-corrected energies
of $E=3\times 10^{53}$ and $1\times 10^{54}$ erg for SN 2005ha and SDSS-II SN
12882, respectively, and are likely unrealistic.
Off-axis jets can result in bright synchrotron emission that peaks years after
explosion (e.g., Granot et al., 2002, 2018). The emission from off-axis jets
enters our line of sight as the jet decelerates in the ambient medium and
relativistic beaming becomes less severe (Rhoads, 1997; Sari et al., 1999).
The fraction of stellar explosions that are jet driven is still unclear (e.g.,
Corsi & Lazzati 2021; Corsi et al. 2016; Bietenholz et al. 2014; Soderberg et
al. 2006c). Successful relativistic jet have been so far associated with
broad-line type-Ic SNe accompanying cosmological GRBs, while partially
successful and partially failed jets have been proposed to be powering low-
luminosity GRBs and relativistic SNe, respectively (e.g., for observations see
Margutti et al. 2014a, and for theory see Morsony et al. 2007; Lazzati et al.
2012). While observations of energetic H-stripped SNe point at a continuum of
jet properties from normal Ibc SNe to GRB/SNe (e.g., Xu & Wei, 2008; Mazzali
et al., 2008; Margutti et al., 2014b; Corsi & Lazzati, 2021), no bona fide
off-axis jet has ever been associated with a SN without a GRB detection. In
this context it is particularly interesting to note that the relativistic SN
2012ap, which is one of only two known relativistic SNe without a GRB
counterpart (Chakraborti et al., 2015; Soderberg et al., 2010), is detected by
VLASS observations 5.7 and 8.5 years after explosion. Thus SN 2012ap is a
clear candidate for an off-axis jet driving late-time emission.
To determine whether the detected VLASS emission is associated with off-axis
jets, we generated a set of synthetic 3 GHz jet afterglow light-curves with
Boxfit v2 (van Eerten et al., 2012). Boxfit assumes a top-hat jet structure,
i.e. a jet with energy uniformly distributed within $\theta\leq\theta_{jet}$
and $E=0$ for $\theta>\theta_{jet}$. We assumed a wind-like CSM density
profile. We explored the parameter space with a grid of parameter values
defined as follows: isotropic-equivalent jet kinetic energies of
$E_{iso}=[10^{50},10^{51},10^{52},10^{53},10^{54},10^{55}]\,\rm{erg}$; jet
opening angles of $\theta_{jet}=[5^{\circ},15^{\circ},30^{\circ}]$; off-axis
angle $\theta_{obs}=[30^{\circ},60^{\circ},90^{\circ}]$ from our line of
sight; mass-loss rates of
$\dot{M}=[10^{-8},10^{-7},10^{-6},10^{-5},10^{-4}10^{-3}]$ $M_{\odot}$ year-1
for $v_{w}=1000\,\rm{km\,s^{-1}}$; shock micro-physical parameters
$\epsilon_{e}=0.1$, $\epsilon_{B}=[0.001,0.01]$, $p=[2,2.5,3]$. Finally we
used the $\chi^{2}$ as a metrics to evaluate the agreement between the models
and the VLASS data of Figure 2.
None of the VLASS-SNe with archival (i.e. pre-VLASS) data points are
consistent with our model off-axis jet light curves. We find that the
synthetic models that best approximate the VLASS data of SDSS-II SN 12882, SN
2002hi, SN 2005ha, SN 2009fi, and SN 2012cc with $\chi^{2}\lesssim 1$ have
large off-axis angles $\theta_{obs}\geq 60^{\circ}$, large densities
corresponding to $\dot{M}>10^{-5}$ $M_{\odot}$ year-1, $p\sim 3$,
$\epsilon_{B}=0.01$ coupled with large $E_{iso}\geq 10^{54}\,\rm{erg}$ and
large jet angle $\theta_{jet}\geq 15^{\circ}$. The values of these parameters
is driven by the large radio luminosities of the sample at late times, and
imply extremely large beaming-corrected jet energies $3\times
10^{52}-10^{54}\,\rm{erg}$. While we show some examples of top-hat off-axis
jet light-curve consistent with the VLASS data in Figure 4, we consider this
top-hat jet scenario unlikely because of the large jet energies needed and the
fact that only SNe with a sparse data set can be fitted. We consider
alternative jet models and environments below.
We start by noting that in SN 1965G, SN 2004C, SN 2005ha, SN 2012ap, SN
2012at, and SDSS-II SN 12882, the radio flux density remains nearly constant
over ${\sim}2-3$ years between two VLASS epochs. SN 1986J also has nearly
constant radio flux densities (see Figure 2); however, it has only been
observed in a single VLASS epoch. Numerical simulations of GRB jets
propagating through a stratified media show that nearly flat, wide peaks are
obtained only if the jet propagates through a wind-profile medium with
$\rho\propto r^{-2}$ (see Figure 1 in Granot et al., 2018). Jets propagating
through a uniform density environment have a much more narrow peak (e.g., as
seen in the GRB 170817A afterglow Margutti & Chornock 2020) and are ruled out
by our observations. GRB 170817A also clearly showed that relativistic jets
can have angular structure (i.e. the jet is not necessarily top-hat; see e.g.,
Nakar 2020 and references therein).
The propagation of relativistic GRB jets through a massive Wolf-Rayet
progenitor star leads to the production of extended wide-angle outflows known
as cocoons, with masses ${\approx}10^{-2}-10^{-1}$ M⊙ and energies
${\approx}10^{50}-10^{51}$ ergs (see, e.g., Lazzati & Begelman, 2005; Nakar &
Piran, 2017; De Colle et al., 2021) possibly observationally identified in SN
2017iuk/GRB171205A (Izzo et al., 2019). Once the jet breaks out of the star,
the cocoon engulfs the star and expands nearly spherically into the
environment (see e.g. Figure 3 of De Colle et al. 2021). The cocoon initially
expands with relativistic velocities (corresponding to Lorentz factors
${\sim}2-10$), but later decelerates to mildly relativistic velocities at
${\sim}10^{16}$ cm (see Figure 2 in De Colle et al., 2018). Particle
acceleration through the shock cocoon itself will lead to a bright afterglow.
While GRB jets are collimated and enter into the observer line of sight only
at late times, the cocoon radio emission should be detectable at early times
by observers located at nearly all angles (as beaming effects are much less
important in the slower moving cocoon material) and would thus be able to
explain the larger radio fluxes of pre-VLASS observations. The predicted
early-time radio emission (De Colle et al., 2018, 2021) is similar to that
observed in relativistic SNe 2009bb and 2012ap (Soderberg et al., 2010;
Margutti et al., 2014a; Chakraborti et al., 2015). Several SNe in our sample,
including SN 2012ap, have been observed at early times (${\sim}$days to a
month after explosion), but only SN 2012ap showed mildly relativistic material
consistent with the expectations from the cocoon model. SN 2012ap is the only
VLASS-SN for which a cocoon and off-axis relativistic jet is a viable
explanation. The largely uncollimated, mildly relativistic cocoon would be
responsible for the early emission. The late-time VLASS emission would be
powered by the off-axis relativistic jet. In this case, SN 2012ap would
represent the first evidence of a cocoon and jet system from a massive stellar
explosion. Future multifrequency observations will test this scenario. Thus,
with the exception of SN 2012ap, we find that the late-time VLASS emission is
unlikely to be caused by relativistic jets.
### 3.3 Emergence of emission from a pulsar wind nebula
Another candidate for the cause of late-time radio emission from SNe is the
presence of a PWN (e.g., Gaensler & Slane 2006; Slane 2017). Core-collapse
SNe, which comprise the totality of our sample, are expected to leave a
compact remnant. If a fast rotating NS is left behind, it can feed a steady
highly-energetic wind of relativistic particles into the SN ejecta, and this
“bubble” of relativistic particles is referred to as a PWN. As this wind
interacts with the slower SN ejecta, a termination shock forms and high-energy
photon emission heats and ionizes the surrounding SN ejecta. Shortly after the
explosion, the emission is absorbed by the dense ejecta (e.g., Metzger et al.,
2014; Murase et al., 2015, 2016, 2021). Over time, as the ejecta expands and
the optical depth decreases, the PWN emission becomes observable. No SN has
unambiguously shown the transition from ejecta-dominated emission to PWN-
dominated emission. Recently, there have been hints towards the detection of a
PWN associated with SN 1987A. This suggestion is due to non-thermal emission
in the hard X-rays (Greco et al., 2021), and from the radio detection of a
warm dust concentration at the center of the remnant (Cigan et al., 2019);
however, alternative mechanisms to explain the emission cannot be ruled out.
Beyond SN 1987A, two young SNe have been suggested to harbor PWNe (SN 1986J
and SN 2012au), and both are in our VLASS-SNe sample.
The presence of a PWN energizing the ejecta in a young SN has been proposed to
explain the anomalous state of high ionization inferred from optical
spectroscopy of the H-stripped energetic SN 2012au ${\approx}6$ years after
explosion by Milisavljevic et al. (2018). The spectra of this transient
acquired $\approx$7 years after explosion were dominated by forbidden oxygen
lines with velocities of ${\approx}2300$ km s-1. Oxygen resides in the inner
part of the SN ejecta, thus one explanation for this emission is the presence
of a pulsar that ionizes the internal material (Milisavljevic et al., 2018).
The lack of narrow hydrogen in the early spectra of SN 2012au suggests a
different powering mechanism than CSM-ejecta interaction, and supports the
scenario of ionization by a pulsar as the origin of the emission.
Bietenholz et al. (2002) suggested the late-time radio emission from SN 1986J
is evidence of a PWN. SN 1986J showed a broad radio SED $7-16$ years after
explosion with a spectrum at $\nu>10$ GHz which evolved from thin to thick
(i.e. an inverted radio spectrum). However, observed SEDs of evolved PWNe are
relatively flat, with typical spectral indices between $-0.3$ and $0.0$. In
contrast, SN 1986J has an SED that peaked at ${\approx}20$ GHz, with an
absorbed optically thick region, and an optically thin spectral index of
$-0.76$. SN 2012au has a similarly shaped SED at $8$ years post-explosion
(Terreran et al. in prep.). The bell-like synchrotron SEDs produced by CSM-
interaction of the SN shock wave peak below GHz frequencies on these time-
scales. The observed radio spectrum of SN 1986J and SN 2012au is also unusual
for evolved PWNe, but we emphasize that the spectral properties of nascent
PWNe that are a few years old are not observationally well constrained. From a
theoretical perspective, we expect the radiative electrons to be in the “fast-
cooling” regime, which can lead to radio spectra similar to those observed
(e.g., Murase et al., 2016; Omand et al., 2018; Murase et al., 2021). A young
PWN is expected to be smaller in size than the SN ejecta, thus one can
distinguish between the shock interaction and a PWN with very long baseline
interferometry.
Interestingly, both SN 1986J and SN 2012au are in the VLASS-SNe sample. If the
bright radio emission is confirmed to be powered by PWNe, the associated PWNe
would be the two youngest discovered to date. No forming PWN has been observed
in the Milky Way or the Magellanic Clouds. The youngest known galactic PWN,
Kes 75, has an estimated age of $480\pm 40$ years (Reynolds et al., 2018), and
thus little is known about PWN properties in the years to decades after the SN
explosion.
## 4 Conclusions
We present evidence for a population of 19 radio luminous SNe ($L_{\nu}\sim
10^{26}-10^{29}$ erg s-1 Hz-1 at 3 GHz) ${\approx}1-60$ years after explosion
found in the first epoch of the VLASS. This is part of the radio phase space
of stellar explosions that has not been systematically explored so far. Our
filtering procedure leveraged multiwavelength catalogs to remove potential AGN
contaminants, and other known radio sources leading to a sample that is
entirely comprised of core-collapse SNe and surprisingly dominated by stellar
explosions with hydrogen stripped progenitors at the time of collapse. Our
main result is that the large radio luminosities at these late stages of
evolution require deviation from the traditional single wind mass-loss
scenario and/or spherical shock assumption. Potential alternatives include the
following:
1. 1.
Initial expansion of the SN shock into a lower-density bubble, followed by
strong shock interaction with a sharp density increase (i.e. a “bubble” plus
detached shell CSM structure). This dense shell might be connected to the
shedding of the H-rich stellar envelope in the centuries before core-collapse
through mass-loss mechanisms that have yet to be observationally identified.
VLASS SNe are as luminous as the most luminous radio SNe IIn few yrs post
explosion, which indicates CSM densities at large radii from the progenitors
that are comparable to those inferred for SNe IIn.
2. 2.
While top-hat relativistic jets viewed off-axis are unlikely to provide an
adequate explanation due to the under-prediction of the pre-VLASS radio
observations of most elements of the sample, relativistic jets with structure
are not ruled out. SN 2012ap, which showed evidence for an uncollimated mildly
relativistic outflow at $\delta t<40$ days, is the primary candidate for being
the first jet+cocoon system in a massive star observed off-axis, which may
signal that relativistic SNe are “cocoons” observed early on.
3. 3.
The final alternative is the emergence of a PWN. The VLASS-SNe sample includes
SN 1986J and SN 2012au, the two young SNe that have previously been suggested
to have PWNe powered late-time radio emission.
The VLA Sky Survey provides an unprecedented and unbiased window into the
variable radio sky, combining the large survey area of the Northern VLA Sky
Survey with the depth and angular resolution of the Faint Images of the Radio
Sky survey. These features, and the planned multiple field visits, are
particularly useful for the discovery and study of extragalaxtic transients,
where the angular resolution (and higher frequency, 3 GHz vs. 1 GHz) minimizes
confusion by the host galaxies of transients of interest. Planned
interferometers such as the Next Generation VLA (Carilli et al., 2015) and the
Square Kilometer Array (Dewdney et al., 2009) will expand our ability to study
the variable radio sky with increased depth. The VLASS is complimented by
other surveys and serendipitous transient discovery programs being carried out
with SKA pathfinder instruments such as ASKAP (VAST; Murphy et al. 2013),
Westerbork (Apertif; Adams & van Leeuwen 2019), MeerKAT (ThunderKAT; Fender et
al. 2016), LOFAR (LoTSS; Shimwell et al. 2017), which encompass a range of
frequencies and angular resolutions while providing access to the southern
sky.
Follow-up with multiwavelength observations including radio spectral energy
distributions, and optical spectroscopy will help constrain the mechanisms
responsible for the bright radio emission of our VLASS-SN sample. We will
present the multiwavelength follow-up of the VLASS detected SNe sample in
future papers.
We thank the referee for providing constructive comments. We also thank Seth
Bruzewski for providing astrometric corrections to the VLASS Quicklook epoch 1
data. This work is supported by the Heising-Simons Foundation under grant
#2018-0911 (PI: Margutti). R.M. acknowledges support by the NSF under grants
AST-1909796 and AST-1944985. F.D.C. acknowledges support from the UNAM-PAPIIT
grant AG100820. R.B.D. acknowledges support from National Science Foundation
(NSF) under grant 1816694. The National Radio Astronomy Observatory is a
facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc. This publication makes use of data
products from the Wide-field Infrared Survey Explorer, which is a joint
project of the University of California, Los Angeles, and the Jet Propulsion
Laboratory/California Institute of Technology, and NEOWISE, which is a project
of the Jet Propulsion Laboratory/California Institute of Technology. WISE and
NEOWISE are funded by the National Aeronautics and Space Administration. This
research has made use of data obtained from the Chandra Source Catalog,
provided by the Chandra X-ray Center (CXC) as part of the Chandra Data
Archive. This research has made use of data obtained from the 4XMM XMM-Newton
serendipitous source catalogue compiled by the 10 institutes of the XMM-Newton
Survey Science Centre selected by ESA. This work made use of data supplied by
the UK Swift Science Data Centre at the University of Leicester. This research
was supported in part through the computational resources and staff
contributions provided for the Quest high performance computing facility at
Northwestern University which is jointly supported by the Office of the
Provost, the Office for Research, and Northwestern University Information
Technology. Development of the BOXFIT code was supported in part by NASA
through grant NNX10AF62G issued through the Astrophysics Theory Program and by
the NSF through grant AST-1009863. This research made use of APLpy, an open-
source plotting package for Python (Robitaille & Bressert, 2012; Robitaille,
2019).
## Appendix A VLASS detected SNe
We present in Table 1 the sample of SNe detected in the VLASS data set. The
second column lists whether the progenitor was hydrogen rich or hydrogen poor
at the time of explosion. Flux densities and positions were derived as
described in Section 2.5, with any deviations from the default procedure given
in the ‘Notes’ column. The list of VLASS SNe is ordered by increasing right
ascension. Bruzewski et al. (2021) calculated the astrometric corrections
required to align the VLASS Quicklook epoch 1 data with the Gaia catalog, and
we list the coordinates that include these astrometric corrections. The
positional errors include the uncertainties from PyBDSF fits and the
astrometric corrections (Bruzewski private communication) added in quadrature.
We applied the Bruzewski et al. (2021) derived corrections and astrometric
uncertainties for the VLASS epoch 2.1 observations. We note that applying the
VLASS epoch 1 uncertainties may overstate the positional uncertainties in the
second epoch observations, since the second VLASS epoch will have likely
benefited from studying the systematic uncertainties in the first epoch (see
e.g. VLASS Memo 13555The VLASS Project Memo Series is listed at
https://go.nrao.edu/vlass-memos.).
We report the PyBDSF flux density errors which are purely statistical. There
are known flux density offsets in the VLASS Quicklook images, as detailed in
VLASS Memo 13 and the CIRADA Catalogue User Guide. The detection type defines
the nature of the source structure, where “S” indicates a single Gaussian that
is the only source in the island, “C” indicates a single source in an island
with other sources, and “M” indicates multiple Gaussian source. The angular
separation lists the distance between the listed VLASS position and the
optical position. For the SN classifications, “Pec” and “BL” stand for
peculiar and broad-lined, respectively. Four SNe (20% of the sample) had
limited follow-up leaving the classification unknown, but they are believed to
be core-collapse SNe (i.e. SN 1965G, SN 2005ha, SDSS-II SN 8524, and SDSS-II
SN 12882).
Table 1: Supernovae detected in VLASS epoch 1
| | VLASS | | Angular |
---|---|---|---|---|---
Name | Progenitor | R.A. | Decl. | Flux Density | Detection | Obs. Date | Luminosity | Separation | Classification
| [H-rich/poor] | [hh:mm:ss.ss] | [dd:mm:ss.ss] | [mJy] | [S, M or C] | [MJD] | [erg/s/Hz] | [′′] |
SN 1986J | ? | 02:22:31.293(15) | $+$42:19:57.5(3) | $1.3\pm 0.2$ | C | 58588 | $(1.6\pm 0.2)\times 10^{26}$ | $0.56$ | IIn
SN 2017hcb | H-poor | 02:36:23.756(14) | $+$31:42:36.2(3) | $0.5\pm 0.2$ | M | 58569 | $(3.0\pm 1.2)\times 10^{27}$ | $0.99$ | Ib
SDSS-II SN 12882 | ? | 03:03:49.977(9) | $-$00:12:14.3(3) | $1.7\pm 0.4$ a | S | 58103 | $(2.8\pm 0.7)\times 10^{28}$ | $0.32$ | ?
| | 03:03:49.975(10) | $-$00:12:14.2(3) | $1.6\pm 0.3$ a | S | 59078 | $(2.6\pm 0.5)\times 10^{28}$ | $0.41$ |
SN 2003bg | H-poor | 04:10:59.436(6) | $-$31:24:50.2(3) | $4.1\pm 0.2$ | S | 58663 | $(2.83\pm 0.14)\times 10^{27}$ | $0.61$ | IcPecBL
SN 2012at | H-poor | 04:54:52.783(7) | $-$37:19:16.9(3) | $2.0\pm 0.2$ | S | 58153 | $(2.5\pm 0.2)\times 10^{27}$ | $0.41$ | Ic
| | 04:54:52.786(8) | $-$37:19:17.4(3) | $1.9\pm 0.2$ | S | 59155 | $(2.3\pm 0.2)\times 10^{27}$ | $0.70$ |
SN 2012ap | H-poor | 05:00:13.734(5) | $-$03:20:51.4(3) | $4.0\pm 0.3$ | S | 58027 | $(8.9\pm 0.7)\times 10^{27}$ | $0.25$ | IcBL
| | 05:00:13.738(5) | $-$03:20:51.6(3) | $4.5\pm 0.3$ | S | 59078 | $(1.00\pm 0.07)\times 10^{28}$ | $0.23$ |
SN 2005ha | ? | 06:21:49.110(6) | $+$00:21:56.2(3) | $2.2\pm 0.2$ | S | 58123 | $(3.7\pm 0.3)\times 10^{27}$ | $1.23$ | ?
| | 06:21:49.106(6) | $+$00:21:56.0(3) | $1.9\pm 0.3$ | S | 59048 | $(3.2\pm 0.5)\times 10^{27}$ | $1.09$ |
SN 2002hi | H-rich | 07:19:54.127(9) | $+$17:58:18.5(3) | $2.1\pm 0.3$ | S | 58572 | $(1.7\pm 0.2)\times 10^{29}$ | $0.72$ | IIn
SN 2017iuk | H-poor | 11:09:39.519(5) | $-$12:35:18.5(3) | $4.8\pm 0.2$ | S | 58150 | $(1.39\pm 0.06)\times 10^{29}$ | $0.24$ | IcBL
| | $\cdots$ | $\cdots$ | $<0.45$ | $\cdots$ | 59133 | $<1.3\times 10^{28}$ | $\cdots$ |
SN 2004C | H-poor | 11:27:29.80(2) | $+$56:52:47.9(3) | $4.2\pm 0.6$ b | S | 58020 | $(2.8\pm 0.4)\times 10^{27}$ | $0.57$ | Ic
| | 11:27:29.77(2) | $+$56:52:47.9(3) | $5.3\pm 0.9$ | S | 59064 | $(3.5\pm 0.6)\times 10^{27}$ | $0.79$ |
SN 1965G | ? | 12:11:54.049(5) | $+$24:06:58.5(3) | $7.7\pm 0.4$ | S | 58082 | $(1.15\pm 0.06)\times 10^{28}$ | $2.59$ | ?
| | 12:11:54.045(5) | $+$24:06:58.4(3) | $7.8\pm 0.3$ | S | 59099 | $(1.17\pm 0.05)\times 10^{28}$ | $2.52$ |
SN 2012cc | H-rich | 12:26:56.829(9) | $+$15:02:45.6(3) | $2.3\pm 0.4$ | S | 58590 | $(1.10\pm 0.19)\times 10^{27}$ | $0.36$ | II
SN 2012au | H-poor | 12:54:52.257(5) | $-$10:14:50.5(3) | $4.5\pm 0.3$ | S | 58553 | $(3.0\pm 0.2)\times 10^{27}$ | $1.16$ | Ib
PTF11qcj | H-poor | 13:13:41.480(9) | $+$47:17:56.8(3) | $6.8\pm 0.2$ | S | 58561 | $(1.12\pm 0.03)\times 10^{29}$ | $0.44$ | IcBL
SN 2009fi | H-rich | 14:06:05.757(6) | $+$11:47:13.6(3) | $2.2\pm 0.2$ | S | 58611 | $(1.17\pm 0.11)\times 10^{28}$ | $0.88$ | IIb
SN 2004dk | H-poor | 16:21:48.872(4) | $-$02:16:17.6(3) | $6.3\pm 0.2$ | S | 58624 | $(3.34\pm 0.11)\times 10^{27}$ | $0.75$ | Ib
SDSS-II SN 8524 | ? | 21:29:23.354(6) | $+$00:56:42.9(3) | $1.7\pm 0.2$ | S | 58023 | $\cdots$ c | $1.18$ | ?
| | $\cdots$ | $\cdots$ | $<0.5$ | $\cdots$ | 59049 | $\cdots$ c | $\cdots$ |
SN 2016coi | H-poor | 21:59:04.127(8) | $+$18:11:10.8(3) | $1.8\pm 0.2$ | S | 58604 | $(4.7\pm 0.5)\times 10^{26}$ | $0.35$ | IcBL
SN 2014C | H-poor | 22:37:05.601(6) | $+$34:24:31.5(3) | $29.0\pm 0.3$ | S | 58642 | $(7.91\pm 0.08)\times 10^{27}$ | $0.49$ | Ibd
Note. — a) The region surrounding SDSS-II SN 12882 is contaminated by radial
artifacts from quasar PB 6989, so flux density could be less reliable.
b) The region surrounding SN 2004C is contaminated by radial artifacts from
NVSS J112731$+$565240, so the flux density could be less reliable. SN 2004C is
clearly part of an extended emission complex, which is not detected with
PyBDSF unless the island and detection threshold are both lowered.
There is a clear point source at the location of SN 2004C which we associate
with the transient, but the flux density here is likely unreliable.
c) SDSS-II SN 8524 has no known redshift, thus a luminosity cannot be
calculated.
d) SN 2014C was initially classified as a type Ib but was later classified as
a type IIn.
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|
# HEIGHTS AND SINGULAR MODULI OF DRINFELD MODULES
Zhenlin Ran
###### Abstract
We prove that there are only finitely many singular moduli of rank 2 Drinfeld
modules that are units. In particular, we develop some techniques of heights
of Drinfeld modules to approach it.
###### Contents
1. 1 Introduction
2. 2 Minimal models of Drinfeld $A$-modules
3. 3 Heights
4. 4 Arithmetic on quadratic fundamental domain
5. 5 Bounding $h(J)$
6. A More on Complex Multiplication
## 1 Introduction
A singular modulus is the $j$-invariant of a CM elliptic curve over the
complex numbers. It is well-known that a singular modulus is an algebraic
integer. In 2015, Habegger proved the following theorem [10]:
###### Theorem 1.1.
At most finitely many singular moduli are algebraic units.
His proof employs many classical tools from Diophantine geometry. The main
idea of his proof is to bound the Weil height of a unitary singular modulus.
In particular, assuming the singular modulus is a unit, an upper bound of its
Weil height could be obtained by applying an equidistribution theorem from
Clozel and Ullmo [8, Section 2.3]. This is the biggest difficulty of the
entire proof as the number of singular moduli with big Galois orbits is hard
to control.
In this paper, we consider a function field analogue and prove the same result
for Drinfeld $\mathbb{F}_{q}[t]$-modules.
Let $C$ be a smooth, projective and geometrically irreducible curve over a
finite field $\mathbb{F}_{q}$. Fix a closed point $\infty\in C$ and let $A$ be
the ring of functions regular outside $\infty$. A Drinfeld $A$-module over
some scheme $S$ over $A$ is a pair $(\mathbb{G}_{a,\mathcal{L}},\phi)$ such
that $\mathbb{G}_{a,\mathcal{L}}$ is a line bundle over $S$ and $\phi$ is a
ring homomorphism from $A$ to $\text{End}(\mathbb{G}_{a,\mathcal{L}})$ with
some extra conditions (cf. Definition 2.1). It is well-known that Drinfeld
$A$-modules of rank 2 are the analogue of elliptic curves. Most of the
concepts and results of elliptic curves could be found for Drinfeld
$A$-modules of rank 2. For example, we can define singular modulus of Drinfeld
$\mathbb{F}_{q}[t]$-modules in the same way as elliptic curves. The main
theorem of this paper is:
###### Theorem 1.2.
Let $q$ be odd. There are only finitely many singular moduli of rank 2
Drinfeld $\mathbb{F}_{q}[t]$-modules that are algebraic units.
The strategy of proving our main theorem follows that of Habbeger. As we
pointed out earlier, the key tool that Habegger uses to control the number of
Galois orbits of quadratic imaginary numbers close to the roots of the
$j$-function is an equidistribution theorem from Clozel and Ullmo, which
enables him to obtain an upper bound for the Weil height of a unitary singular
modulus. However, this idea does not work well for our case since to the best
of our knowledge, there are not any equidistribution results for Drinfeld
$A$-modules like the one of Clozel and Ullmo. Instead, our idea to address
this issue is to study the arithmetic of quadratic imaginary points. Though
our method for the case of Drinfeld $\mathbb{F}_{q}[t]$-modules is somehow
elementary, we can show that there is at most one quadratic imaginary point in
a certain small neighbourhood of a root of the $j$-function (cf. Proposition
4.5). Thus, we could also obtain an upper bound for the Weil height of unitary
singular moduli of Drinfeld $\mathbb{F}_{q}[t]$-modules.
On the other hand, Habegger also gives a lower bound for the Weil height of
singular moduli that grows faster than the upper bound he obtained. Many tools
for the case of elliptic curves were already known while the analogues for
Drinfeld $A$-modules are not available. We follow Habegger’s strategy and
prove some analogous results for the case of Drinfeld modules, which will lead
us to a lower bound for the Weil height of singular moduli for Drinfeld
$\mathbb{F}_{q}[t]$-modules. In Drinfeld $A$-modules, the analogue of Faltings
height is Taguchi height which was introduced by Taguchi in [20] for the case
of finite characteristic and in [21] for the case of generic characteristic.
In particular, we obtain an analogous result of Nakkajima and Taguchi for
Drinfeld $\mathbb{F}_{q}[t]$-modules, which gives an explicit description for
the variation of the Taguchi heights of rank 2 Drinfeld
$\mathbb{F}_{q}[t]$-modules under isogeny, where one Drinfeld
$\mathbb{F}_{q}[t]$-module of rank $2$ has CM by arbitrary order and the other
one has CM by the maximal order. From this point on, we can obtain the
variation of the graded heights of the same Drinfeld
$\mathbb{F}_{q}[t]$-modules. As the graded height of Drinfeld $A$-modules is
the generalization of the Weil height of $j$-invariants for Drinfeld
$A$-modules, we thus obtain a lower bound for the Weil height of singular
moduli by applying a theorem of Wei where he proves the Colmez conjecture for
Drinfeld $A$-modules.
More recently Bilu, Habegger and Kühne proved the stronger result that there
are no singular moduli that are algebraic units in the case of elliptic curves
[7]. Their approach is the same as Habegger’s while the method in [7] is
effective. What makes Habegger’s proof ineffective is the equidistribution
theorem he applied. Compared to this work, a totally different approach using
Gross-Zagier, Gross-Kohnen-Zagier and their generalizations was given by Li
[15]. Li could deduce the same result of [7] as a special case of [15, Theorem
1.1].
This paper is organized as follows:
* •
In Section 2, we recall the concept of Drinfeld $A$-modules over arbitrary
$A$-schemes and the minimal models of Drinfeld $A$-modules. In particular, we
study some properties of the minimal models.
* •
In Section 3, three different types of heights are introduced and their
relationships are discussed. In particular, we study the variation of heights
of Drinfeld $A$-modules under isogenies.
* •
In Section 4, we study the number of quadratic imaginary points near the root
of the $j$-function of Drinfeld $A$-modules.
* •
In Section 5, we bound the Weil height of $j$-invariants and prove the main
theorem.
* •
The appendix is written to give some results related to CM Drinfeld
$A$-modules that could not be found in some common literature. In particular,
this section is devoted to the proof of Proposition 3.14.
Acknowledgements The author is very grateful to his advisor Florian Breuer for
advising him such an interesting project and for being supportive in all
aspects throughout his PhD study. He would also thank Marc Hindry and Urs
Hartl for inviting him to visit Paris and Münster respectively where the
author had delightful and helpful discussions with them and learnt mathematics
from them. He also appreciate Fabien Pazuki and Fu-Tsun Wei for helpful
discussions. Many thanks go to Philipp Habegger for his suggestion of working
on Lemma 5.7.
## 2 Minimal models of Drinfeld $A$-modules
In this section, we review the definition of Drinfeld $A$-modules and the
associated minimal models. In particular, we prove an analogue of a result of
Néron models of abelian varieties.
Let $C$ be a smooth, projective and geometrically irreducible curve over
$\mathbb{F}_{q}$, where $\mathbb{F}_{q}$ is a finite field with $q$ elements.
Let $\infty\in C$ be a closed point. We set
$A:=\Gamma(C\backslash\\{\infty\\},\mathcal{O}_{C})$ to be the ring of
functions regular outside $\infty$. We fix $k$ to be the field of fractions of
$A$. Let $M_{k}$ denote the set of all places of $k$. To each place $v\in
M_{k}$ we associate an absolute value $|\cdot|_{v}$ as follows:
$|x|_{v}=|q|^{-\deg(v)v(x)},\ \forall x\in k.$
Let $k_{\infty}$ denote the completion of $k$ with respect to $\infty$ and
$\mathbb{C}_{\infty}$ denote the completion of an algebraic closure of
$k_{\infty}.$
Throughout this paper, we denote by $\log$ the logarithm function of base $q$
and assume that $q$ is odd.
###### Definition 2.1.
Let $S$ be a scheme over $\text{Spec}(A)$ with structure morphism
$i:S\rightarrow\text{Spec}(A)$. A Drinfeld $A$-module over $S$ is a pair
$\textbf{E}=(\mathbb{G}_{a,\mathcal{L}},\phi)$, where $\mathcal{L}$ is an
invertible sheaf over $S$ and $\phi$ is a homomorphism from $A$ to
$\text{End}(\mathbb{G}_{a,\mathcal{L}})$ such that
1. (1)
$\partial\circ\phi=i^{\\#}$, where $i^{\\#}:A\rightarrow\mathcal{O}_{S}(S)$
and $\partial$ is the natural homomorphism of taking the constant term of
$\text{End}(\mathbb{G}_{a,\mathcal{L}})$.
2. (2)
for any $0\neq a\in A$, the morphism $\phi(a)$ is finite, and at any point of
$S$ its degree is $>1$ for some $a\in A$.
###### Remark 2.2.
1. (1)
If $\varphi\in\text{End}(\mathbb{G}_{a,\mathcal{L}})$ then
$\varphi=\sum_{n\geq 0}a_{n}\tau_{p}^{n}$, where
$a_{n}\in\Gamma(S,\mathcal{L}^{1-p^{n}})$ and $\tau_{p}$ is the relative
$p$-Frobenius. So $D\circ\phi$ lands in $\mathcal{O}_{S}(S)$.
2. (2)
We denote the subalgebra of $\mathbb{F}_{q}$-linear endomorphisms of
$\text{End}(\mathbb{G}_{a,\mathcal{L}})$ by
$\text{End}_{q}(\mathbb{G}_{a,\mathcal{L}})$. The elements of
$\text{End}_{q}(\mathbb{G}_{a,\mathcal{L}})$ are of the form
$\sum_{n}a_{n}\tau_{q}^{n}$ for $a_{n}\in\Gamma(S,\mathcal{L}^{1-q^{n}})$.
3. (3)
For any $\varphi=\sum_{n\geq
0}a_{n}\tau_{p}^{n}\in\text{End}(\mathbb{G}_{a,\mathcal{L}})$, the sum is
locally finite [14, Remark 1.2.4].
4. (4)
If $S$ is connected, then there exists an integer $r>0$ such that
$\deg(\phi(a))=|a|_{\infty}^{r}$ [14, Proposition 2.2.2]. The integer $r$ is
called the rank of the Drinfeld module E.
5. (5)
A homomorphism (resp. isogeny) from $(\mathbb{G}_{a,\mathcal{L}},\phi)$ to
$(\mathbb{G}_{a,\mathcal{M}},\psi)$ is a (resp. finite) homomorphism
$f:\mathbb{G}_{a,\mathcal{L}}\rightarrow\mathbb{G}_{a,\mathcal{M}}$ such that
$f\circ\phi(a)=\psi(a)\circ f$ for all $a\in A$.
We abbreviate $\phi(a)$ as $\phi_{a}$ for $a\in A$. If $S$ is the spectrum of
a field, then the line bundle on $S$ is unique in which case we specify a
Drinfeld module over $S$ only by $\phi$. In this paper, we assume for any
Drinfeld module $(\mathbb{G}_{a,\mathcal{L}},\phi)$, $\phi$ is $q$-linear and
$\tau:=\tau_{q}$.
###### Definition 2.3.
(Taguchi) Let $S$ be an integral normal scheme of finite type over $A$ with
function field $F$. Let $\phi$ be a Drinfeld $A$-module over $F$. A model
$\mathscr{M}=(\mathbb{G}_{a,\mathcal{L}},\varphi,f)$ of $\phi$ over $S$ is an
$A$-module scheme $\textbf{E}=(\mathbb{G}_{a,\mathcal{L}},\varphi)$ over $S$
such that $f:\textbf{E}\times_{S}\text{Spec}(F)\rightarrow\phi$ is an
isomorphism of Drinfeld modules over $F$. A model $\mathscr{M}$ of $\phi$ over
$S$ is minimal if given any other model
$\mathscr{N}=(\mathbb{G}_{a,\mathcal{L}^{\prime}},\varphi^{\prime},f^{\prime})$,
there exists a unique homomorphism $\mathscr{N}\rightarrow\mathscr{M}$ which
induces an isomorphism on the generic fibre compatible with $f$ and
$f^{\prime}$.
###### Proposition 2.4.
([21, Proposition 2.2]) Let $S$ and $F$ be as in Definition 2.3, and we
further assume $S$ is a scheme on which the two concepts of Weil divisors and
Cartier divisors coincide. Then there exists a minimal model over $S$ of
$\phi$.
###### Remark 2.5.
1. (1)
If $\phi$ has a minimal model, then the minimal model is unique up to
isomophism;
2. (2)
By checking on fibres we see that every model of $\phi$ over $S$ is smooth
over $S$.
3. (3)
By [21, Remark (4), p. 299], each model is isomorphic to a standard one. To
avoid confusion with the term standard Drinfeld module, we call the standard
model the normalized model. That is, a normalized model is a model whose
generic fibre is exactly the given Drinfeld module over $F$.
Let $\phi$ be a Drinfeld $A$-module of rank $r$ over $F$ which is a field of
finite degree over $k$, and let $R$ be the integral closure of $A$ in $F$.
Thus we can identify the invertible sheaves over $\text{Spec}(R)$ with the
fractional ideals of $R$ in $F$. For any $a\in A$, we write:
$\phi_{a}=a\tau^{0}+\cdots+\Delta_{a}\tau^{r\deg(a)}.$
Let $\mathscr{M}=(\mathbb{G}_{a,\mathcal{L}},\varphi)$ be the normalized
minimal model of $\phi$ over $R$. Then
$\Delta_{a}\in\mathcal{L}^{1-q^{r\deg(a)}}$. Moreover, if $\phi$ has good
reduction at a place $v$ of $R$, then
$v(\Delta_{a})=v(\mathcal{L}^{1-q^{r\deg(a)}})$ [21, Example, p. 301], where
$v(\mathcal{I})$ is the ramification index of a fractional ideal $\mathcal{I}$
at the prime corresponding to $v$.
###### Remark 2.6.
In this paper we do not distinguish invertible sheaves over $\text{Spec}(R)$
and fractional ideals of $R$ in $F$. We use them interchangeably without
causing any ambiguity since we only work on $\text{Spec}(R).$ Moreover, We do
not distinguish prime ideals of $R$ and valuations on $F$. That is, if we take
a prime ideal $v\in\text{Spec}(R)$, by $v(I)$ we mean the ramification index
of the fractional ideal $I$ at $v$.
###### Proposition 2.7.
Let $\phi$ be as above.
1. 1.
If $\phi$ has everywhere good reduction on $R$, then the minimal model
$\mathscr{M}$ of $\phi$ over $R$ is a Drinfeld module.
2. 2.
If $\phi$ has everywhere good reduction on $R$, $\phi^{\prime}$ is another
Drinfeld module over $F$ and $f:\phi\rightarrow\phi^{\prime}$ is an isogeny of
Drinfeld modules over $F$, then $f$ is an isogeny of their normalized minimal
models.
###### Proof.
To prove the first statement, we are left to show that $\phi_{a}$ is finite
for each $0\neq a\in A$. The above argument implies that
$v(\Delta_{a})=v(\mathcal{L}^{1-q^{r\deg(a)}})$ for every place $v$ on $R$.
This means that $\Delta_{a}$ corresponds to an isomorphism in
$\text{End}(\mathbb{G}_{a,\mathcal{L}})$. By [14, Proposition 1.2.6] we see
$\phi_{a}$ is finite for any $a\in A$. So the first statement is true.
To prove the second statement, we first write:
$\phi^{\prime}_{a}=a\tau^{0}+\cdots+\Delta_{a}^{\prime}\tau^{r\deg(a)},\
\forall a\in A;\quad f=f_{0}\tau^{0}+\cdots+f_{n}\tau^{n}.$
For any $a\in A$ we have $f\phi_{a}=\phi_{a}^{\prime}f$. By comparing the
coefficients we get
$f_{n}\Delta_{a}^{q^{n}}=\Delta_{a}^{\prime}f_{n}^{q^{r\deg(a)}}.$ (1)
Let
$\mathscr{M}^{\prime}=(\mathbb{G}_{a,\mathcal{L}^{\prime}},\varphi^{\prime})$
be the normalized minimal model of $\phi^{\prime}$. Then $f$ extends uniquely
to a morphism from $\mathscr{M}$ to $\mathscr{M}^{\prime}$ whose generic fibre
is $f$ [21, Proposition 2.5]. We therefore denote the two morphisms by $f$
interchangeably without causing any ambiguity. Since $\phi$ has good reduction
everywhere, so does $\phi^{\prime}$. The above argument again implies
$v(\Delta_{a}^{\prime})=v(\mathcal{L}^{\prime 1-r\deg(a)})$. From (1) we see
for every place $v$ on $R$
$v(f_{n})=v(\mathcal{L}^{\prime})-q^{n}v(\mathcal{L})=v(\mathcal{L}^{\prime}\mathcal{L}^{-q^{n}}).$
Thus by the same argument in the proof of the first statement, $f$ is finite,
hence an isogeny. ∎
###### Remark 2.8.
This proposition actually indicates an analogue of a well-known result that if
an abelian variety over a number field has good reduction everywhere then its
Néron model is an abelian scheme, and moreover, an isogeny between abelian
varieties with everywhere good reductions extends to a finite flat
homomorphism between their Néron models. In our case, the flatness of $f$ is
in consequence of the finiteness since a homomorphism between two line bundles
is quasi-finite if and only if it is flat [14, Proposition 1.2.5].
## 3 Heights
In this section, we study Taguchi heights, Weil heights and graded heights. In
particular, we calculate the variation of Taguchi heights and graded heights
of rank 2 Drinfeld $\mathbb{F}_{q}[t]$-modules under an isogeny.
Taguchi heights
Let $A,k,k_{\infty}$ and $\mathbb{C}_{\infty}$ be as in Section 2. In [21],
Taguchi introduced his so-called differential heights of Drinfeld $A$-modules
which are now called Taguchi heights. He defines this concept in the case when
the lattices associated to Drinfeld $A$-modules are free. Wei generalizes
Taguchi’s definition for arbitrary Drinfeld $A$-modules [23, Section 5]. We
copy the following definition from Wei. For more details, the reader could
refer to [22, Section 4] or [23, Remark 2.10].
###### Definition 3.1.
Let $\Lambda$ be an $A$-lattice of rank $r$ in $\mathbb{C}_{\infty}$, and let
$\mathcal{O}_{\infty}$ be the ring of $\infty$-adic integers in $k_{\infty}$.
Choose an orthogonal $k_{\infty}$-basis $\\{\lambda_{i}\\}_{i=1}^{r}$ of
$k_{\infty}\otimes\Lambda$ such that:
1. (1)
$\lambda_{i}\in\Lambda$ for $1\leq i\leq r$;
2. (2)
$|a_{1}\lambda_{1}+\cdots+a_{r}\lambda_{r}|_{\infty}=\max\\{|a_{i}\lambda_{i}|_{\infty}:1\leq
i\leq r\\}$ for all $a_{1},...,a_{r}\in k_{\infty}$;
3. (3)
$k_{\infty}\otimes\Lambda=\Lambda+(\mathcal{O}_{\infty}\lambda_{1}+\cdots+\mathcal{O}_{\infty}\lambda_{r})$.
The covolume $D_{A}(\Lambda)$ of the $A$-lattice $\Lambda$ is defined as
follows:
$D_{A}(\Lambda):=q^{1-g_{k}}\cdot\left(\frac{\prod_{i=1}^{r}|\lambda_{i}|_{\infty}}{\\#\left(\Lambda\cap\left(\mathcal{O}_{\infty}\lambda_{1}+\cdots+\mathcal{O}_{\infty}\lambda_{r}\right)\right)}\right)^{\frac{1}{r}}=\left(\frac{\prod_{i=1}^{r}|\lambda_{i}|_{\infty}}{\\#(\Lambda/(A\lambda_{1}+\cdots+A\lambda_{r}))}\right)^{\frac{1}{r}},$
where $g_{k}$ is the genus of the field $k$.
Let $F/k$ be a finite field extension and $R$ be the integral closure of $A$
in $F$. We may assume that $F\subset\mathbb{C}_{\infty}$ and for any infinite
place $w$ of $F$ we fix $F_{w}\subset\mathbb{C}_{\infty}$. A metrized line
bundle $(\mathcal{L},\|\cdot\|)$ on $\text{Spec}(R)$ is a projective
$R$-module $\mathcal{L}$ of rank $1$, together with norms
$\|\cdot\|_{w}:\mathcal{L}\otimes_{R}F_{w}\rightarrow\mathbb{R}$
for all infinite places $w$ of $F$. The degree $\deg(\mathcal{L},\|\cdot\|)$
of a metrized line bundle $(\mathcal{L},\|\cdot\|)$ on $R$ is
$\deg(\mathcal{L},\|\cdot\|):=\log\\#(\mathcal{L}/lR)-\sum_{w|\infty}\epsilon_{w}\log\|l\|_{w}$
for some $l\in\mathcal{L}$, and $\epsilon_{w}$ is the local degree at $w$. It
is independent of the choice of $l$ by the product formula. We note that by
taking a norm we implicitly indicate an extension of the absolute value
$|\cdot|_{\infty}$ on $k$. In this above equation, the extension of absolute
value is taken as the one remaining unchanged on $k$ while our normalization
below is different. The reader should note only in this definition we take the
infinite absolute value on $F$ such that it remains the same on $k$.
Let $\phi$ be a Drinfeld $A$-module of rank $r$ over $F$ and
$\mathscr{M}=(\mathbb{G}_{a,\mathcal{L}},\varphi,f)$ the minimal model of
$\phi$ over $R$, where $\textbf{E}=(\mathbb{G}_{a,\mathcal{L}},\varphi)$ is an
$A$-module scheme and $\mathcal{L}$ is an invertible sheaf over
$\text{Spec}(R)$. Since $\mathbb{G}_{a,\mathcal{L}}\rightarrow\text{Spec}(R)$
is smooth of finite type and relative dimension $1$, we see
$\Omega_{\mathbb{G}_{a,\mathcal{L}}/R}^{1}$ is locally free of rank $1$. Let
$e:\text{Spec}(R)\rightarrow\mathbb{G}_{a,\mathcal{L}}$ be the unit section,
then we set
$\omega_{\textbf{E}/R}:=e^{*}(\Omega_{\mathbb{G}_{a,\mathcal{L}}/R}^{1}).$
Thus $\omega_{\textbf{E}/R}\cong\mathcal{L}^{-1}$ which is the inverse of
$\mathcal{L}$ in $\text{Pic}(R)$. Without causing any ambiguity, we treat
$\omega_{\textbf{E}/R}$ as a rank $1$ projective $R$-module. Let $w$ be an
infinite place of $F$ and $\textbf{E}_{w}$ be the Drinfeld module over $F_{w}$
by extension of scalars $R\rightarrow F_{w}$. Let $\Lambda_{w}$ be the
corresponding $A$-lattice of rank $d$ in $\mathbb{C}_{\infty}$ ([9, Theorem
4.6.9] or [14, Proposition 2.1.5]). If $x$ is the coordinate function of
$\mathbb{G}_{a}/F_{w}$, then $dx$ is a generator of
$\omega_{\textbf{E}_{w}/F_{w}}(\cong\mathcal{L}^{-1}\otimes_{R}F_{w})$. We put
a metric $\|\cdot\|_{w}$ on $\omega_{\textbf{E}_{w}/F_{w}}$ by
$\|dx\|:=D_{A}(\Lambda_{w}).$
###### Definition 3.2.
Let $\phi$ be a Drinfeld $A$-module of rank $r$ over $F$ and
$\mathscr{M}=(\mathbb{G}_{a,\mathcal{L}},\varphi,f)$ be its minimal model over
$R$, where $\textbf{E}=(\mathbb{G}_{a,\mathcal{L}},\varphi)$ is an $A$-module
scheme over $R$. The Taguchi height of $\phi$ over $F$ is
$h_{\text{Tag}}(\phi/F):=\frac{1}{[F:k]}\deg(\omega_{\textbf{E}/R},\|\cdot\|),$
where the metric $\|\cdot\|$ is given as above.
It is obvious the Taguchi height of a Drinfeld $A$-module $\phi$ depends on
the choice of the field $F$. However, it will remain unchanged when taking a
finite field extension of $F$ if $\phi$ has everywhere stable reduction over
$F$. Since every Drinfeld $A$-modules has everywhere potential stable
reduction, we can define the stable Taguchi height of $\phi$ to be
$h_{\text{Tag}}^{\text{st}}(\phi):=h_{\text{Tag}}(\phi/F^{\prime}),$
where $F^{\prime}$ is a field of finite degree over $F$ on which $\phi$ has
everywhere stable reduction. The following isogeny lemma is standard.
###### Lemma 3.3.
Let $f:\phi_{1}\rightarrow\phi_{2}$ be an isogeny of Drinfeld $A$-modules over
$F$ with everywhere stable reduction. Then we have:
$h_{\text{Tag}}^{\text{st}}(\phi_{2})-h_{\text{Tag}}^{\text{st}}(\phi_{1})=\frac{1}{r}\log|\deg(f)|-\frac{1}{[F:k]}\log\\#(R/D_{f}),$
where $D_{f}$ is the different of $f$ (cf. [21, section 5.4] or [18, section
1.3]).
###### Remark 3.4.
Let $G$ be the kernel of the induced homomorphism of the minimal models of
$\phi_{1}$ and $\phi_{2}$. If $f:\phi_{1}\rightarrow\phi_{2}$ is an isogeny of
Drinfeld $A$-modules over $F$ with everywhere good reduction, then by
Proposition 2.7 we see $f$ induce an isogeny on the minimal models. In this
case, according to [13, equation 4.9.6], $D_{f}$ is the absolute different of
$G$ [17, Apendice, Définition 8].
If $\phi$ is a Drinfeld $A$-module of rank $r$ over $F$, we set:
$v(\phi):=-\min_{a\in A-\\{0\\}}\min_{i}\left\\{\frac{v(a_{i})}{q^{i}-1}:1\leq
i\leq r\deg(a)\right\\},$
where the $a_{i}$’s are coefficients in
$\phi_{a}=a\tau^{0}+\sum_{i=1}^{r\deg(a)}a_{i}\tau^{i}.$
###### Lemma 3.5.
Let $f:\phi_{1}\rightarrow\phi_{2}$ be an isogeny of Drinfeld $A$-modules over
$F$ with everywhere good reduction on $R$, and let $\mathscr{M}_{1}$ and
$\mathscr{M}_{2}$ be the normalized minimal models over $R$ of $\phi_{1}$ and
$\phi_{2}$ respectively. If $v\in\text{Spec}(R)$ is a finite place, then
$v(D_{f})=v(f_{0})+v(\phi_{1})-v(\phi_{2}),$
where $f_{0}=\partial(f)$ is the coefficient of the linear term.
###### Proof.
It will suffice to prove for the local cases, i.e. we may assume that $R$ is a
discrete valuation ring. Suppose $v$ is the valuation on $R$. Let us first
look at the case when the normalized minimal models
$\mathscr{M}_{1}=(\text{Spec}(R[X]),\varphi_{1},\text{Id})$ and
$\mathscr{M}_{2}=(\text{Spec}(R[Y]),\varphi_{2},\text{Id})$. We use $f$ to
denote the isogeny between $\text{Spec}(R[X])$ and $\text{Spec}(R[Y])$. We
denote $f^{\\#}:R[Y]\rightarrow R[X]$ the corresponding homomorphism of rings.
Thus we have
$f^{\\#}(Y)=f_{0}X+f_{1}X^{q}+\cdots+f_{n}X^{q^{n}}\in R[X].$
The kernel $G$ of $f$ is then given by $\text{Spec}(R[X]/(f^{\\#}(Y))$. By
[13, Equations 4.9.5, 4.9.6], the absolute different of $G$ is $(f_{0})\subset
R$. Thus $D_{f}=(f_{0})$ (Remark 3.4). By the construction of minimal models
[21, Proposition 2.2], we note that in this case $v(\phi_{1})=v(\phi_{2})=0$.
Hence our claim is true in this case.
To prove the general cases, let $\mathbb{G}_{a,\mathcal{L}_{i}}$ be the line
bundle of the normalized minimal model of $\phi_{i}$ for $i=1,2$. We assume
$\mathcal{L}_{i}=(a_{i})$ to be a fractional ideal for some $a_{i}\in F$. Thus
we have
$f_{j}\in(a_{2}a_{1}^{-q^{j}}).$
In particular, $f_{0}=a_{2}a_{1}^{-1}b$ for some $b\in R$. Now apply the same
argument above with the variables $X$ replaced by $a_{1}^{-1}X$ and $Y$
replaced by $a_{2}^{-1}Y$, we see $D_{f}=(b)=(f_{0}a_{1}a_{2}^{-1})$.
Therefore we have
$v(D_{f})=v(f_{0})+v(a_{1})-v(a_{2}).$
This proves our claim. ∎
Logarithmic heights
The absolute value $|\cdot|_{\infty}$ on $k$ naturally extends to a unique
absolute value on $\mathbb{C}_{\infty}$, which we denote $|\cdot|$. If $F/k$
is a finite field extension and $w$ is a place of $F$ lying over $v\in M_{k}$,
we normalize the absolute value associated to $w$ as
$|y|_{w}=|\text{N}_{F_{w}/k_{v}}(y)|_{v}^{\frac{1}{[F:k]}},\ \forall y\in F.$
Since $k$ has degree of imperfection 1 (see the Remark 3.6 below), for any
place $v\in M_{k}$ we have:
$F\otimes_{k}k_{v}\cong\prod_{w|v}F_{w}.$ (2)
Let $M_{F}$ be the set of places $w$ normalized as above. By $(2)$ we have
following two properties:
* •
Product formula: For every $y\in F$, $\sum_{w\in M_{F}}\log|y|_{w}=0.$
* •
Extension formula: $[F:k]=\sum_{w|v}[F_{w}:k_{v}].$
###### Remark 3.6.
If $F$ is a field of characteristic $p\neq 0$, by degree of imperfection of
$F$ we mean the number $n$ such that $[F:F^{p}]=p^{n}$. The isomorphism in
$(2)$ holds in a more general case of simple extension. A theorem from Becker
and MacLane [1, Theorem 6] tells us any finite extension $L/F$ can be
generated by at most $\max\\{1,n\\}$ elements.
Let $\overline{k}$ be the algebraic closure of $k$ in $\mathbb{C}_{\infty}$
and we denote $\mathbb{P}^{n}(\overline{k})$ the $n$-dimensional projective
space over $\overline{k}$. If
$\textbf{x}=(x_{0}:\cdots:x_{n})\in\mathbb{P}^{n}(\overline{k})$ and $F$ is a
finite extension of $k$ containing these coordinates, then the Weil height of
x is:
$h(\textbf{x}):=\sum_{w\in M_{F}}\max_{j}\log|x_{j}|_{w}.$
As in the number field case, this definition is independent of the choice of
both the field $F$ and the coordinates. The definition of Weil heights of
points in affine space is naturally obtained by embedding the affine space to
a projective space. In particular, for $x\in\overline{k}$ and $F$ a finite
extension of $k$ containing $x$ we have
$h(x)=\sum_{w\in M_{F}}\log\max\\{1,|x|_{w}\\}=\sum_{w\in
M_{F}}\log^{+}|x|_{w}.$
The proof of the following lemma is similar to [2, Proposition 1.6.6].
###### Lemma 3.7.
Let $\alpha\in\overline{k}$ of degree $d$ and $f(X)$ be the minimal polynomial
of $\alpha$ over $A$ with leading coefficient $a_{d}$ and roots $\alpha_{j}$,
$j=1,...,d$. Then
$dh(\alpha)=\log|a_{d}|+\sum_{j=1}^{d}\log^{+}|\alpha_{j}|.$
###### Definition 3.8.
Let $\phi$ be a Drinfeld $A$-module of rank $r$ over $F$. The global graded
degree $h_{G}(\phi)$ (resp. local graded degree $h_{G}^{w}(\phi/F)$ at a place
$w$ over $F$) of $\phi$ is
$h_{G}(\phi):=\frac{1}{[F:k]}\sum_{w\in M_{F}}\deg(w)w(\phi)\
\left(\text{resp}.\ h_{G}^{w}(\phi/F):=\frac{\deg(w)w(\phi)}{[F:k]}\right).$
###### Remark 3.9.
1. (1)
It is obvious the global graded height of $\phi$ does not depend on the choice
of the field $F$ and it is invariant under isomorphisms, while a local graded
height of $\phi$ will not satisfy such properties.
2. (2)
The global graded height is a direct interpretation of “finite” Taguchi height
(cf. [20, Definition 2.3]).
3. (3)
Let $\phi$ be a Drinfeld $\mathbb{F}_{q}[t]$-module of rank $r$ over $F$. Then
it is characterised by:
$\phi_{t}=t\tau^{0}+g_{1}\tau+\cdots+g_{r}\tau^{r},\ g_{i}\in F,g_{r}\neq 0.$
Let $m=\text{lcm}\\{q-1,...,q^{r}-1\\}$. We set
$J:=(j_{1}:\cdots:j_{r})\in\mathbb{P}^{r-1}(\overline{k})$ where
$j_{i}=g_{i}^{m/(q^{i}-1)},\ \text{for}\ i=1,...,r.$
If $r=2$, then $j_{\phi}:=j_{1}/j_{2}$ is the $j$-invariant of the Drinfeld
$A$-module $\phi$. It plays the same role as the $j$-invariant of elliptic
curves. The global graded height of $\phi$ is then given by:
$h_{G}(\phi)=\sum_{w\in M_{F}}\max_{1\leq i\leq
r}\log|g_{i}|_{w}^{1/(q^{i}-1)}.$
Thus the global graded height coincides the one in [5, equation 6]. It is
obvious that $mh_{G}(\phi)=h(J)$.
###### Proposition 3.10.
Let $\phi$ be a Drinfeld $A$-module of rank $r$ over $F$ such that $F/k$ is a
separable extension. For any $\sigma\in\text{Gal}(k^{\text{sep}}/k)$, we
denote by $\sigma(\phi)$ the Drinfeld $A$-module obtained by acting $\sigma$
on the coefficients of a Drinfeld $A$-module $\phi$. Then we have
$h_{G}(\phi)=h_{G}(\sigma(\phi)).$
###### Proof.
By Remark 3.9 (1) we may assume $F/k$ is a Galois extension so that for any
$\sigma\in\text{Gal}(F/k)$ the Drinfeld $A$-module $\sigma(\phi)$ is defined
over $F$. For any places $v\in M_{k}$ and $w\in M_{F}$ such that $w$ lies over
$v$, we see $w\circ\sigma$ is again a place lying over $v$. Thus $\sigma$
permutes the places lying over $v$. By a result of algebraic number theory,
$w$ and $w\circ\sigma$ have the same degree. Therefore
$h_{G}(\sigma(\phi))=\sum_{v\in M_{k}}\sum\limits_{\begin{subarray}{c}w\in
M_{F}\\\ w|v\end{subarray}}\deg(w\circ\sigma)w\circ\sigma(\phi)=\sum_{v\in
M_{k}}\sum\limits_{\begin{subarray}{c}w\in M_{F}\\\
w|v\end{subarray}}\deg(w)w(\phi)=h_{G}(\phi).$
∎
###### Theorem 3.11.
Let $f:\phi_{1}\rightarrow\phi_{2}$ be an isogeny of Drinfeld $A$-modules over
$F$ with everywhere good reduction on $\text{Spec}(R)$. Then we have:
$h_{\text{Tag}}^{\text{st}}(\phi_{2})-h_{\text{Tag}}^{\text{st}}(\phi_{1})=\frac{1}{r}\log|\deg(f)|-\sum\limits_{\begin{subarray}{c}w|\infty\\\
w\in
M_{F}\end{subarray}}\log|f_{0}|_{w}+h_{G}^{\text{fin}}(\phi_{2})-h_{G}^{\text{fin}}(\phi_{1}),$
where $f_{0}$ is the linear coefficient of $f$ and for $i=1,2$,
$h_{G}^{\text{fin}}(\phi_{i})$ is the sum of the local graded heights running
over all finite places of $F$.
###### Proof.
By applying Lemma 3.3 and Lemma 3.5 we obtain:
$h_{\text{Tag}}^{\text{st}}(\phi_{2})-h_{\text{Tag}}^{\text{st}}(\phi_{1})=\frac{1}{r}\log|\deg(f)|-\frac{1}{[F:k]}\sum\limits_{\begin{subarray}{c}w\in
M_{F}\\\
w\nmid\infty\end{subarray}}\deg(w)(w(f_{0})+w(\phi_{1})-w(\phi_{2})).$ (3)
By applying the product formula, we get:
$\sum\limits_{\begin{subarray}{c}w\in M_{F}\\\
w\nmid\infty\end{subarray}}\deg(w)w(f_{0})=-\sum\limits_{\begin{subarray}{c}w\in
M_{F}\\\ w|\infty\end{subarray}}\deg(w)w(f_{0}).$ (4)
Under our normalization, we have
$\log|f_{0}|_{w}=\frac{-\deg(w)w(f_{0})}{[F:k]}.$ (5)
Now substitute (5) and (4) to (3) we obtain our formula. ∎
Heights of Drinfeld $A$-modules with complex multiplication
We first recall the CM theory for Drinfeld modules. Let $\phi$ be a Drinfeld
$A$-module of rank $r$ over $\mathbb{C}_{\infty}$. We say $\phi$ has complex
multiplication if the ring of endomorphisms $\mathcal{O}:=\text{End}(\phi)$ is
a projective $A$-module of rank $r$, and $K:=\mathcal{O}\otimes_{A}k$ is
called the CM field of $\phi$. In this case, $K/k$ is an imaginary extension
of degree $r$. Here by an imaginary extension we mean there is only one place
of $K$ extending the infinite place $\infty$ of $k$. As with the case of
abelian varieties, there is also a standard theory of complex multiplication
for Drinfeld modules:
###### Theorem 3.12.
(Main Theorem of Complex Multiplication) Let $\phi$ be a Drinfeld $A$-module
of rank $r$ over $\mathbb{C}_{\infty}$ with complex multiplication. Let
$\mathcal{O}$ be the ring of endomorphisms of $\phi$ and $K$ be the CM field.
The following statements are true:
1. 1.
There is a finite extension $H_{\mathcal{O}}/K$ such that
$\text{Gal}(H_{\mathcal{O}}/K)\cong\text{Pic}(\mathcal{O})$ via the Artin map.
The field $H_{\mathcal{O}}$ is the ring class field of $\mathcal{O}$. The
prime $\infty$ of $K$ splits completely in $H_{\mathcal{O}}$, and
$\mathcal{O}$ is unramified outside $\mathcal{C}$ which is the conductor of
$\mathcal{O}$, i.e. the largest common ideal of $\mathcal{O}$ and
$\mathcal{O}_{K}$.
2. 2.
$\phi$ has good reduction at every finite place of $H_{\mathcal{O}}$.
3. 3.
If $r=2$ and $A=\mathbb{F}_{q}[t]$, then the $j$-invariant $j_{\phi}$ of
$\phi$ is integral over $A$ and $H_{\mathcal{O}}=K(j_{\phi})$.
The reader could find a proof to the above statements in [11], as well as a
complete treatment of theory of complex multiplications for Drinfeld
$A$-modules.
Suppose $\phi$ is a Drinfeld $A$-module of rank $r$ with CM by an order
$\mathcal{O}$ in a CM field $K$. We denote by $\text{Pr}(\mathcal{O})$ the
monoid of proper fractional ideals of $\mathcal{O}$ quotient by principal
ideals. It is then obvious
$\text{Pic}(\mathcal{O})\subset\text{Pr}(\mathcal{O})$ and
$\text{Pic}(\mathcal{O})$ has a natural action on $\text{Pr}(\mathcal{O})$.
Since $\phi$ has CM by $\mathcal{O}$, its associated lattice is isomorphic to
a proper ideal $I_{\phi}$ of $\mathcal{O}$.
###### Lemma 3.13.
Let $\phi_{1}$ and $\phi_{2}$ be two Drinfeld $A$-modules of rank $r$ with CM
by the same order $\mathcal{O}$, and $I_{1}$ (resp. $I_{2}$) be a proper ideal
of $\mathcal{O}$ such that the associated lattice of $\phi_{1}$ (resp.
$\phi_{2}$) is isomorphic to $I_{1}$ (resp. $I_{2}$). If $I_{1}$ and $I_{2}$
are in the same orbit of $\text{Pr}(\mathcal{O})$ under the action of
$\text{Pic}(\mathcal{O})$, then $h_{G}(\phi_{1})=h_{G}(\phi_{2})$. In
particular, if $\text{Pr}(\mathcal{O})=\text{Pic}(\mathcal{O})$ then all the
Drinfeld $A$-modules with CM by $\mathcal{O}$ have the same graded height.
###### Proof.
Without loss of generality, we assume $\phi_{i}$ has associated lattice
$I_{i}$ where $i=1,2$. We choose an invertible ideal
$J\in\text{Pic}(\mathcal{O})$ such that $I_{1}=J^{-1}\cdot I_{2}$. Thus
$I_{1}$ is homothetic to the lattice associated to $J*\phi_{2}$ (cf. [11,
Proposition 5.10 and Equation (5.18)]). On the other hand, $J*\phi_{2}$ is
isomorphic to a Drinfeld $A$-module $\phi_{2}^{\prime}$ obtained by a Galois
action on the coefficients of $\phi_{2}$ [23, Theorem A.1 (2)]. We note that
we can always choose suitable $I_{1}$ and $I_{2}$ to make $J$ integral so that
our argument makes sense. Now by Remark 3.9 (1) and Proposition 3.10 we
complete our proof. ∎
From now on, we always fix $A=\mathbb{F}_{q}[t]$. A proof of the following
result for elliptic curves is due to Nakkajima and Taguchi [16]. We only make
a few arguments here to adapt their formula to Drinfeld $A$-modules.
###### Proposition 3.14.
Let $\phi_{1}$ and $\phi_{2}$ be two Drinfeld $A$-modules of rank 2 with CM by
$\mathcal{O}_{K}$ and $\mathcal{O}$ respectively, where $K$ is an imaginary
quadratic field and $\mathcal{O}_{K}$ (resp. $\mathcal{O}$) is a maximal
(resp. arbitrary) order. We write $\mathcal{O}=A+f_{0}\mathcal{O}_{K}$ for
some $f_{0}\in A$. If $F/K$ is a finite field extension such that both
$\phi_{1}$ and $\phi_{2}$ are defined over $F$ with everywhere good reduction
then
$h_{\text{Tag}}^{\text{st}}(\phi_{2})-h_{\text{Tag}}^{\text{st}}(\phi_{1})=\frac{1}{2}\log|f_{0}|-\frac{1}{2}\sum_{v|f_{0}}\deg(v)e_{f_{0}}(v),$
where $v$ runs over all monic prime factors of $f_{0}$ and for
$l:=q^{\deg(v)}$
$e_{f_{0}}(v)=\frac{(1-\chi(v))(1-l^{-v(f_{0})})}{(l-\chi(v))(1-l^{-1})},$
and $\chi(v)=1$ if $v$ splits in $K$; $\chi(v)=0$ if $v$ ramifies in $K$;
$\chi(v)=-1$ if $v$ is inert in $K$.
###### Proof.
First we note that the Taguchi height of rank 2 Drinfeld $A$-modules with CM
does not depend on the choice of lattice that analytically generates the
corresponding Drinfeld $A$-module. So we may assume that $\phi_{1}$ is given
by $\mathcal{O}_{K}$ and $\phi_{2}$ is given by $\mathcal{O}$, and an isogeny
$f:\phi_{1}\rightarrow\phi_{2}$ given by
$f_{0}\mathcal{O}_{K}\subset\mathcal{O}$.
For the case when $\chi(v)=1$, from Theorem A.2 we see $\phi_{1}$ has ordinary
reduction at any place lying over $v$. Applying the same argument from [16,
Proposition 4] we get $e_{f_{0}}(v)=0$. The reduction process in the argument
for Drinfeld $A$-modules is given by Theorem A.4. For the case of
supersingular reduction, the argument for our case is exactly the same as [16,
Section 2.2] with only one modification that we take $l=q^{\deg(v)}$. ∎
###### Corollary 3.15.
Assume the same conditions as in Proposition 3.14. The following formula is
true
$h_{G}(\phi_{2})-h_{G}(\phi_{1})=\log|f_{0}|-\frac{1}{2}\sum_{v|f_{0}}\deg(v)e_{f_{0}}(v)+h_{G}^{\infty}(\phi_{2}^{\prime})-h_{G}^{\infty}(\phi_{1}^{\prime}),$
where $\phi_{1}^{\prime}$ is the Drinfeld $A$-module given by the lattice
$\mathcal{O}_{K}$ and $\phi_{2}^{\prime}$ is given by $\mathcal{O}$, and
$h_{G}^{\infty}(\phi_{i}^{\prime})$ is the sum of local graded heights of
$\phi_{i}^{\prime}$ at infinite places for $i=1,2$.
###### Proof.
By Lemma 3.13 we can choose $\phi_{i}$ to be $\phi_{i}^{\prime}$, $i=1,2.$ It
is then a trivial consequence of Theorem 3.11 and Proposition 3.14. ∎
## 4 Arithmetic on quadratic fundamental domain
We assume our Drinfeld $A$-module $\phi$ has rank 2 with CM for the rest this
paper. Let $\Omega:=\mathbb{C}_{\infty}\backslash k_{\infty}$ be the Drinfeld
upper-half plane so that $\text{PGL}_{2}(A)\backslash\Omega$ are the
$\mathbb{C}_{\infty}$-points of the coarse moduli space of rank 2 Drinfeld
$A$-modules over $\mathbb{C}_{\infty}$. As in the case of elliptic curves,
there are bijections:
$\text{PGL}_{2}(A)\backslash\Omega\xrightleftharpoons{\quad\quad}\\{\text{Lattices
of rank 2 in }\mathbb{C}_{\infty}/\cong\\}\xrightleftharpoons{\quad\
\quad}\mathbb{C}_{\infty}.$
Thus we obtain a natural $j$-function$:\Omega\rightarrow\mathbb{C}_{\infty}$.
The set of $j$-invariants of rank 2 Drinfeld $A$-modules with CM is precisely
the image of the $j$-function at imaginary quadratic arguments.
Unfortunately, the Drinfeld upper-half plane doesn’t have a good geometry as
the Poincaré upper-half plane of complex numbers does. However, we can still
define the quadratic fundamental domain.
###### Definition 4.1.
(cf. [4, Definition 3.4]) The quadratic fundamental domain is
$\displaystyle\mathcal{D}=\\{z\in\Omega$ $\displaystyle:z\text{ satisfies an
equation of the form }az^{2}+bz+c=0,$ (6) $\displaystyle\quad\text{where
}a,b,c\in A,\ a\text{ is monic, }|b|<|a|\leq|c|\text{ and}$
$\displaystyle\quad\text{gcd}(a,b,c)=1\\}.$
Any rank 2 lattice corresponding to a CM Drinfeld module is homothetic to
$\Lambda_{z}$ for some $z\in\mathcal{D}_{K}$, where $\Lambda_{z}$ denotes the
lattice generated by $z$ and 1, and $\mathcal{D}_{K}:=\mathcal{D}\cap K$ for
some quadratic imaginary extension $K/k$ in $\mathbb{C}_{\infty}$. Unlike the
case of elliptic curves, such $z$ is not necessarily unique.
###### Proposition 4.2.
([3, Proposition 1.2.1]) Let $q$ be odd and $K$ be a quadratic extension of
$k$. Then $K$ is a Kummer extension and can be written in the form
$K=k(\sqrt{\delta})$ for some square-free $\delta\in A$. Let $m=\deg(\delta)$.
Then we have
1. 1.
The place $\infty$ ramifies in $K$ if and only if $m$ is odd;
2. 2.
The place $\infty$ is inert in $K$ if and only if $m$ is even and the leading
coefficient of $\delta$ is not a square in $\mathbb{F}_{q}$;
3. 3.
The place $\infty$ splits in $K$ if and only if $m$ is even and the leading
coefficient of $\delta$ is a square in $\mathbb{F}_{q}$.
Let $\delta\in A$ be a polynomial of odd degree or even degree with the
leading coefficient not being a square in $\mathbb{F}_{q}$, and let
$\sqrt{\delta}\in\overline{k}$ be a root of $X^{2}-\delta$. The field
$K:=k(\sqrt{\delta})$ is a quadratic imaginary field by Proposition 4.2. Let
$\mathcal{O}_{K}$ be its maximal order. If $\mathcal{O}\subset\mathcal{O}_{K}$
is a suborder of discriminant $\delta$, then there exsits some $f\in A$ such
that $\mathcal{O}=A+f\mathcal{O}_{K}$ and such $f$ is called the conductor of
$\mathcal{O}$. The discriminant $\delta_{0}$ of $\mathcal{O}_{K}$ is called
the fundamental discriminant of $K$ and $K=k(\sqrt{\delta_{0}})$. The
discriminant of $\mathcal{O}$ is $\delta=4f^{2}\delta_{0}$.
We denote $T_{\delta}$ the set of triples $(a,b,c)$ with $a,b,c\in A$ such
that $b^{2}-4ac=\delta$ and satisfying (6). For $(a,b,c)\in T_{\delta}$ we set
$z(a,b,c)=\frac{-b+\sqrt{\delta}}{2a}\in K=k(\sqrt{\delta}).$
The map $(a,b,c)\mapsto j(z(a,b,c))$ is a bijection from $T_{\delta}$ to the
Galois conjugates of $j(z)$.
###### Lemma 4.3.
If $\infty$ ramifies in $K$, then there doesn’t exist any
$z\in\mathcal{D}_{K}$ such that $z$ is in the open ball of radius 1 of the
points $u$ for any $u\in\mathbb{F}_{q^{2}}\backslash\mathbb{F}_{q}$.
###### Proof.
For some $u\in\mathbb{F}_{q^{2}}\backslash\mathbb{F}_{q}$, we assume there
exists $z\in\mathcal{D}_{K}$ such that $|z-u|<1$. In this case, we have
$|z|=1$ as $|u|=1$. Besides, there exists a triple $(a,b,c)$ satisfying (6)
such that $b^{2}-4ac=\delta$ and $z$ is root of the equation:
$aX^{2}+bX+c=0.$
We write $K=k(\sqrt{\delta_{0}})$, where $\delta_{0}\in A$ is square-free. By
Proposition 4.2 and since $K$ is a quadratic imaginary extension of $k$, we
see $\delta_{0}$ is either of odd degree or of even degree with leading
coefficients not a square in $\mathbb{F}_{q}$. Also we have
$|az^{2}+bz|=|c|.$
Since $|bz|=|b|<|a|=|az^{2}|$, we have $|a|=|az^{2}+bz|=|c|$, which implies
$a$ and $c$ have the same degree and $b$ has degree less than $a$ and $c$.
Therefore $\delta$ has even degree. Since $\delta=4f^{2}\delta_{0}$, we see
$\delta_{0}$ has even degree, hence with leading coefficients not a square in
$\mathbb{F}_{q}$. This is equivalant to saying that $\infty$ is inert in $K$.
∎
###### Remark 4.4.
If $\delta$ has even degree with leading coefficient not a square in
$\mathbb{F}_{q}$, then $\sqrt{\delta}\in\mathbb{F}_{q^{2}}((\frac{1}{t})).$
Moreover, if there exists some $z\in\mathcal{D}_{K}$ with discriminant
$\delta$ such that $|z-u|<1$, then $\delta$ has leading coefficient $4u^{2}$.
This also suggests such $z$ can be close only to those
$u\in\mathbb{F}_{q^{2}}\backslash\mathbb{F}_{q}$ such that
$u^{2}\in\mathbb{F}_{q}$.
###### Proposition 4.5.
Let $u\in\mathbb{F}_{q^{2}}\backslash\mathbb{F}_{q}$. The number of
$(a,b,c)\in T_{\delta}$ such that $|z(a,b,c)-u|<\sqrt{|\delta|}^{-1}$ is at
most 1.
###### Proof.
We first notice that $\delta$ has degree $>0$. If not, then
$z\in\mathbb{F}_{q^{2}}\backslash\mathbb{F}_{q}$. This contradicts the fact
that $\mathcal{D}_{K}\cap\mathbb{F}_{q^{2}}=\emptyset$. So $\delta$ has
positive degree, which implies $\sqrt{|\delta|}\geq 1$. Thus we have
$|z-u|<\sqrt{|\delta|}^{-1}\leq 1$. By the proof of Lemma 4.3, we have
$\delta=\alpha_{2e}t^{2e}+\cdots+\alpha_{0},\ \alpha_{i}\in\mathbb{F}_{q}\
\text{for }i=0,...,2e\ \text{and $e$ is a positive integer.}$
Thus we have
$\sqrt{\delta}=\lambda_{e}t^{e}+\cdots+\lambda_{0}+\lambda_{-1}t^{-1}+\lambda_{-2}t^{-2}\cdots$
with coefficients in $\mathbb{F}_{q^{2}}$. By identifying
$(\sqrt{\delta})^{2}=\delta$ we obtain:
$\displaystyle\alpha_{2e}$ $\displaystyle=\lambda_{e}^{2};$
$\displaystyle\alpha_{2e-1}$
$\displaystyle=\lambda_{e}\lambda_{e-1}+\lambda_{e-1}\lambda_{e};$
$\displaystyle\vdots$ $\displaystyle\alpha_{e}$
$\displaystyle=\lambda_{e}\lambda_{0}+\lambda_{e-1}\lambda_{1}+\cdots+\lambda_{1}\lambda_{e-1}+\lambda_{0}\lambda_{e};$
$\displaystyle\vdots$ $\displaystyle\alpha_{0}$
$\displaystyle=\lambda_{e}\lambda_{-e}+\lambda_{e-1}\lambda_{-(e-1)}+\cdots+\lambda_{-(e-1)}\lambda_{e-1}+\lambda_{-e}\lambda_{e}.$
First notice $\lambda_{e}=2u\neq 0$ because $|z-u|<1$. We first claim that
$(2u)^{-1}\lambda_{i}\in\mathbb{F}_{q}$ when $i=0,1,...,e$.
Our claim is trivial when $i=e$. We proceed by induction and suppose it’s true
for $\lambda_{e},...,\lambda_{n}$ when $0<n\leq e$. From the equations above
we have
$\alpha_{e+n-1}=\lambda_{e}\lambda_{n-1}+\lambda_{e-1}\lambda_{n}+\cdots+\lambda_{n}\lambda_{e-1}+\lambda_{n-1}\lambda_{e}.$
By multiplying $(2u)^{-2}$ on both sides we obtain
$(2u)^{-2}\alpha_{e+n-1}=(2u)^{-1}\lambda_{e}(2u)^{-1}\lambda_{n-1}+\cdots+(2u)^{-1}\lambda_{n-1}(2u)^{-1}\lambda_{e}.$
Since all terms other than $(2u)^{-1}\lambda_{e}(2u)^{-1}\lambda_{n-1}$ are in
$\mathbb{F}_{q}$ and $\lambda_{e}=2u$, we have
$(2u)^{-1}\lambda_{n-1}\in\mathbb{F}_{q}$.
Now recall $z=\frac{-b+\sqrt{\delta}}{2a}$ with triple $(a,b,c)\in T_{\delta}$
and $\delta=b^{2}-4ac$. Then $|z-u|<\sqrt{|\delta|}^{-1}$ is equivalent to
$\deg(a)-\deg(\sqrt{\delta}-b-2au)>\deg(\sqrt{\delta})=\deg(a).$
So we have $\deg(\sqrt{\delta}-b-2au)<0$. Suppose
$a=\sum_{i=0}^{e}a_{i}t^{i},\ b=\sum_{i=0}^{e}b_{i}t^{i},\ \text{with all
$a_{i},b_{i}\in\mathbb{F}_{q}$}.$
Thus for all $i=0,...,e$ we have $\lambda_{i}-2a_{i}u=b_{i}$. Since
$(2u)^{-1}\lambda_{i}\in\mathbb{F}_{q}$, we see $b_{i}=0$ and
$a_{i}=(2u)^{-1}\lambda_{i}$ for all $i=0,...,e$. Thus, $a,b,c$ are completely
determined by $|z(a,b,c)-u|<\sqrt{|\delta|}^{-1}.$ ∎
## 5 Bounding $h(J)$
We prove our main theorem in this section.
Upper bound on $h(J)$
Let $\phi$ be a CM Drinfeld $A$-module of rank 2 over $\mathbb{C}_{\infty}$
and $J$ be its $j$-invariant of degree $d$ over $A$. Let
$\mathcal{O}=\text{End}(\phi)$ and $K=\mathcal{O}\otimes_{A}k$. We denote
$J=J_{1},...,J_{d}$ all the Galois conjugates of $J$ and $z_{1},...,z_{d}$ the
corresponding points in $\mathcal{D}_{K}$. Then for each $i$, we have $z_{i}$
satisfying the equation:
$a_{i}X^{2}+b_{i}X+c_{i}=0,\ (a_{i},b_{i},c_{i})\in T_{\delta}$
and $b_{i}^{2}-4a_{i}c_{i}=\delta$ for some $\delta\in A$ that is the
discriminant of $\mathcal{O}$. In this subsection we prove:
###### Proposition 5.1.
Assuming the notations above and $J$ is an algebraic unit, we have
$h(J)\leq\frac{2}{d}(q+1)\log\sqrt{|\delta|}+O(q),$
where $O(q)$ is some constant depending on $q$.
We first fix some notations. Let $u$ be a point such that
$u\in\mathbb{F}_{q^{2}}\backslash\mathbb{F}_{q}$ and
$|z|_{A}=\inf_{a\in A}|z-a|,$ $|z|_{i}=\inf_{x\in k_{\infty}}|z-x|.$
###### Lemma 5.2.
([3, Proposition 3.2.5]) If $z\in\mathcal{D}_{K},$ then
$|z|_{i}=|z|_{A}=|z|\geq 1$.
###### Lemma 5.3.
For each $z_{i}$, we have $h(z_{i})\leq\log\sqrt{|\delta|}.$
###### Proof.
Proof is similar to [10, Lemma 5]. ∎
###### Lemma 5.4.
([6, Lemma 2.6.9]) Suppose $z\in\Omega$ such that $|z|_{A}>q^{-1}$. If
$u\in\mathbb{F}_{q^{2}}\backslash\mathbb{F}_{q}$ and $|z-u|<q^{-1}$, then
there exists $\zeta\in\mathbb{C}_{\infty}$ with $|\zeta|<1$ such that
$\displaystyle j(z)$
$\displaystyle=t^{q}u^{-2}(1-u^{q-1})^{-2}(z-u)^{q+1}(1+\zeta),$
$\displaystyle|j(z)|$ $\displaystyle=q^{q}|z-u|^{q+1}.$
Let $h(J)$ be the Weil height of $J$. Therefore, by Lemma 3.7 we have
$dh(J)=\sum_{i=1}^{d}\log^{+}|J_{i}|=\sum_{i=1}^{d}\log^{+}|j(z_{i})|.$
If we assume $J$ is an algebraic unit, then pick a
$u\in\mathbb{F}_{q^{2}}\backslash\mathbb{F}_{q}$ we have:
$\displaystyle dh(J)=dh(J^{-1})$
$\displaystyle=\sum_{i=1}^{d}\log^{+}|j(z_{i})|^{-1}$
$\displaystyle=\sum_{|z_{i}-u|<\sqrt{|\delta|}^{-1}}\log^{+}|j(z_{i})|^{-1}+\sum_{\sqrt{|\delta|}^{-1}\leq|z_{i}-u|\leq
q^{-1}}\log^{+}|j(z_{i})|^{-1}+dO(q).$ (7)
###### Lemma 5.5.
Assuming the notations above, we have
$\sum_{|z_{i}-u|<\sqrt{|\delta|}^{-1}}\log^{+}|j(z_{i})|^{-1}\leq
2(q+1)\log\sqrt{|\delta|}-q.$
###### Proof.
Since $z_{i}\in\mathcal{D}_{K}$, we have $|z|_{A}\geq 1$ by Lemma 5.2. Thus
according to Lemma 5.4 we have
$\log|j(z_{i})|^{-1}=-q-(q+1)\log|z_{i}-u|.$ (8)
Let $L=k(z_{i},u)$, then $[L:k]=2$ if
$z_{i}\in\mathbb{F}_{q}(t)(u)=\mathbb{F}_{q^{2}}(t)$ and $[L:k]=4$ otherwise.
In the first case, there is only one $w\in M_{L}$ lying over $\infty\in
M_{k}$. Therefore we have $|z_{i}-u|_{w}=|z_{i}-u|$ under our normalization.
In the second case, there are two places in $M_{L}$ lying over $\infty\in
M_{k}$, and we take $w$ to be one of the two. Thus we have
$|z_{i}-u|=|z_{i}-u|_{w}^{2}$. Either way, we find
$\log|z_{i}-u|\leq 2\sum_{w}\log^{+}|z_{i}-w|_{w}=2h(z_{i}-u).$
Note the fact that $h(\alpha)=h(1/\alpha)$ for any $\alpha\in\overline{k}$ and
we have
$\log|z_{i}-u|\geq-2h(z_{i}-u)\geq-2(h(z_{i})+h(u))=-2h(z_{i}).$
Now substitute this inequality to (8) and apply Lemma 5.3 we obtain
$\log|j(z_{i})|^{-1}\leq 2(q+1)\log\sqrt{|\delta|}-q.$
Since the number of such $z_{i}$ is at most one by Proposition 4.5, we
conclude. ∎
Now we are ready to prove Proposition 5.1.
###### Proof.
We are left to estimate
$\sum_{\sqrt{|\delta|}^{-1}\leq|z_{i}-u|\leq q^{-1}}\log^{+}|j(z_{i})|^{-1}.$
From (8) and $\sqrt{|\delta|}^{-1}\leq|z_{i}-u|$ we have
$\sum_{\sqrt{|\delta|}^{-1}\leq|z_{i}-u|\leq
q^{-1}}\log^{+}|j(z_{i})|^{-1}\leq\left(-q+(q+1)\log\sqrt{|\delta|}\right)\cdot\sum_{\sqrt{|\delta|}^{-1}\leq|z_{i}-u|\leq
q^{-1}}1$
* •
If there exists some $z_{i}$ such that $|z_{i}-u|<\sqrt{|\delta|}^{-1}$, then
we have
$\sum_{\sqrt{|\delta|}^{-1}\leq|z_{i}-u|\leq q^{-1}}1\leq d-1;$
* •
If there doesn’t exist $z_{i}$ such that $|z_{i}-u|<\sqrt{|\delta|}^{-1}$,
then
$\sum_{\sqrt{|\delta|}^{-1}\leq|z_{i}-u|\leq q^{-1}}1\leq d.$
Combining with Lemma 5.5, we are done. ∎
Lower bound on $h(J)$
###### Lemma 5.6.
Let $\phi^{0}$ be a Drinfeld $A$-module of rank 2 with CM by the maximal order
$\mathcal{O}_{K}$ in an imaginary quadratic field $K$. We denote the genus of
$K$ by $g_{K}$. Then we have:
$h_{\text{Tag}}^{\text{st}}(\phi^{0})\geq\left(\frac{1}{2}-\frac{1}{\sqrt{q}+1}\right)g_{K}-\frac{5q-3}{4(q-1)}.$
###### Proof.
By the equation of stable Taguchi height from [23, Section 5.2.1], we get
$h_{\text{Tag}}^{\text{st}}(\phi^{0})=\frac{g_{K}}{2}\log
q_{K}+\frac{1}{2}\left(\log q_{\infty}-\log
q_{K}\right)-1+\frac{\gamma_{K}}{2\ln q},$
where $q_{K}$ is the cardinality of the field of constants in $K$,
$q_{\infty}$ is the cardinality of the residue field of $K_{\infty}$, and
$\gamma_{K}$ is the Euler-Kronecker constant of $K$ [12, Equation (0.2)]. We
remind the reader that the definition of stable Taguchi height in [23,
Equation 5.1] is a multiple of our stable Taguchi height by the constant $\ln
q$. From [12, Equation 1.4.6] we get
$\frac{\gamma_{K}}{2\ln q}\geq\frac{-g_{K}}{\sqrt{q}+1}+\frac{q-3}{4(q-1)}.$
Now using the fact $q\leq q_{K},q_{\infty}\leq q^{2}$ we get the lower bound.
∎
###### Lemma 5.7.
Let $e_{f_{0}}(v)$ be as in Proposition 3.14 and $f_{0}\in A$. Then we have
$\frac{1}{2}\sum_{v|f_{0}}\deg(v)e_{f_{0}}(v)\leq\frac{9}{4}\log\log|f_{0}|+C_{q},$
where $v$ runs through all the monic prime factors of $f_{0}$, and $C_{q}$ is
a computable constant depending on $q$.
###### Proof.
First we need a Mertens-type formula for function field, i.e. the following
inequality:
$\sum_{|v|\leq x}\frac{\log|v|}{|v|}\leq\log x+O(1),\text{where $v$'s are
monic prime polynomials}.$
To see this, we notice that
$\sum_{|v|\leq x}\frac{\log|v|}{|v|}=\sum_{i=1}^{n:=\lfloor\log
x\rfloor}\frac{i}{q^{i}}\cdot a_{i},$
where $a_{i}$ is the number of monic prime polynomials of degree $i$. By [19,
Theorem 2.2] we obtain
$\sum_{i=1}^{n}\frac{i}{q^{i}}\cdot
a_{i}=\sum_{i=1}^{n}\left(1+O({q^{-i/2}})\right)\leq\log x+O(1).$
Recall that
$e_{f_{0}}(v)=\frac{(1-\chi(v))(1-l^{-v(f_{0})})}{(l-\chi(v))(1-l^{-1})},\text{
where }l=|v|.$
Note that $|l|\geq 3$ and $\chi(v)\in\\{-1,0,1\\}$. Thus we get
$e_{f_{0}}(v)\leq\frac{2}{l-l^{-1}}\leq\frac{9}{4l}.$
Thus we have
$\sum_{v|f_{0}}\deg(v)e_{f_{0}}(v)\leq\frac{9}{4}\sum_{v|f_{0}}\frac{\log
l}{l}=\frac{9}{4}\left(\sum\limits_{\begin{subarray}{c}v|f_{0}\\\
|v|\leq\log|f_{0}|\end{subarray}}\frac{\log
l}{l}+\sum\limits_{\begin{subarray}{c}v|f_{0}\\\
|v|>\log|f_{0}|\end{subarray}}\frac{\log l}{l}\right).$
We have proven
$\sum\limits_{\begin{subarray}{c}v|f_{0}\\\
|v|\leq\log|f_{0}|\end{subarray}}\frac{\log l}{l}\leq\log\log|f_{0}|+O(1).$
For the other term, we have
$\sum\limits_{\begin{subarray}{c}v|f_{0}\\\
|v|>\log|f_{0}|\end{subarray}}\frac{\log
l}{l}\leq\frac{\log\log|f_{0}|}{\log|f_{0}|}\cdot\sum\limits_{\begin{subarray}{c}v|f_{0}\\\
|v|>\log|f_{0}|\end{subarray}}1.$
Using the product formula we see there are at most $\log|f_{0}|$ monic prime
factors of $f_{0}$. As [19, Theorem 2.2] is effective, all the constant terms
are summed up to a computable constant $C_{q}$. ∎
###### Proposition 5.8.
Let $J$ be a singular modulus of rank 2 Drinfeld $A$-module with corresponding
discriminant $\delta$ with conductor $f_{0}$. There exists some computable
constant $C_{q}$ with respect to $q$ such that
$h(J)\geq(q^{2}-1)\left(\frac{1}{2}-\frac{1}{\sqrt{q}+1}\right)\log\sqrt{|\delta|}+\left(\frac{1}{2}+\frac{1}{\sqrt{q}+1}\right)\log|f_{0}|-\frac{9}{4}\log\log|f_{0}|-C_{q}.$
###### Proof.
We recall that for $K=k(\sqrt{\delta_{0}})$ with $\delta_{0}$ square free, the
genus $g_{K}$ of $K$ is given by (cf. [4, Section 3])
$g_{K}=\begin{cases}\frac{\log|\delta_{0}|-1}{2}&\text{if $\deg(\delta_{0})$
is odd,}\\\ \frac{\log|\delta_{0}|-2}{2}&\text{if $\deg(\delta_{0})$ is
even.}\end{cases}$
We input a result from [5, equation (23)], which says that
$|h_{G}^{\infty}(\phi^{\prime})-h_{G}^{\infty}(\phi)|\leq\frac{q}{q-1}-\frac{q^{r}}{q^{r}-1},$
where $\phi$ and $\phi^{\prime}$ are two isogenous Drinfeld $A$-modules of
rank $r$. Using Corollary 3.15, Lemma 5.6, Lemma 5.7 and the facts that
$h_{G}(\phi)\geq h_{\text{Tag}}^{\text{st}}(\phi)$ and
$h(J)=(q^{2}-1)h_{G}(\phi)$ we complete our proof. ∎
###### Remark 5.9.
1. (1)
We note that [5, Equation (23)] holds true only for reduced Drinfeld modules
[5, definition before Lemma 4.2]. However, Lemma 3.13 ensures in the rank 2
case we can always choose the graded height of the Drinfeld module with
associated lattice being the CM order, hence reduced.
2. (2)
One can also take $q\geq 3$ to make our statement independent of $q$, i.e.
$h(J)\geq
4(2-\sqrt{3})\log\sqrt{|\delta|}+\frac{1}{2}\log|f_{0}|-\frac{9}{4}\log\log|f_{0}|-C_{3}.$
Now we are ready to prove our main theorem. We restate our theorem:
###### Theorem 5.10.
Let $q$ be odd. There are only finitely many singular moduli of rank 2
Drinfeld $A$-modules that are algebraic units.
###### Proof.
Assume $J$ is a unitary singular modulus. There are finitely many singular
moduli with bounded degree. So we can choose $d$ large enough in Proposition
5.1 such that
$\frac{2}{d}(q+1)\log\sqrt{|\delta|}<(q^{2}-1)\left(\frac{1}{2}-\frac{1}{\sqrt{q}+1}\right)\log\sqrt{|\delta|}.$
We note that
$\left(\frac{1}{2}+\frac{1}{\sqrt{q}+1}\right)\log|f_{0}|-\frac{9}{4}\log\log|f_{0}|\geq
0$. Therefore, by Proposition 5.1 and Proposition 5.8 we find a constant upper
bound for $\log\sqrt{|\delta|}$. Lemma 5.3 implies that this is also a
constant upper bound for $h(z_{i})$ for $z_{i}$. We note that $z_{i}$ has
degree 2. Thus the Northcott theorem implies our theorem. ∎
## Appendix A More on Complex Multiplication
We use notations as in section 2. This appendix is mainly devoted for the
proof of Proposition 3.14. Actually the results stated here are already known
for elliptic curves. The results below may be already known to many experts.
Because of a dearth of literature for Drinfeld modules, the details are worked
out here for the convenience of the reader.
Let $\phi_{1}$ and $\phi_{2}$ both be rank $r$ Drinfeld $A$-module over
$\mathbb{C}_{\infty}$ with complex multiplication. Let $F/k$ be a finite field
extension such that both $\phi_{1}$ and $\phi_{2}$ are defined over $F$ with
everywhere good reduction. Let $R$ be the integral closure of $A$ in $F$ and
denote $\mathscr{M}_{1}$, $\mathscr{M}_{2}$ the minimal model over $R$ of
$\phi_{1}$, $\phi_{2}$ respectively. If
$f:\mathscr{M}_{1}\rightarrow\mathscr{M}_{2}$ is an isogeny, then it induces
an isogeny of Drinfeld modules after taking reduction at a prime
$v\in\text{Spec}(R)$.
###### Lemma A.1.
Let $\phi_{1}^{v}$ and $\phi_{2}^{v}$ be Drinfeld modules over $\mathbf{k}(v)$
obtained by taking reduction on $\mathscr{M}_{1}$ and $\mathscr{M}_{2}$
respectively at $v$, where $\mathbf{k}(v)$ is the residue field at $v$. Let
$\text{Hom}_{F}(\phi_{1},\phi_{2})$ denote the set of isogenies over $F$
between $\phi_{1}$ and $\phi_{2}$, similarly for
$\text{Hom}_{\mathbf{k}(v)}(\phi_{1}^{v},\phi_{2}^{v})$. Then there is a
canonical injection of $A$-modules:
$\text{Hom}_{F}(\phi_{1},\phi_{2})\xhookrightarrow{}\text{Hom}_{\mathbf{k}(v)}(\phi_{1}^{v},\phi_{2}^{v}).$
###### Proof.
It is easy to check the map is a morphism of $A$-modules. By Proposition 2.7
and [21, Proposition 2.5] we obtain
$\text{Hom}_{F}(\phi_{1},\phi_{2})=\text{Hom}_{R}(\mathscr{M}_{1},\mathscr{M}_{2}).$
Let $f\in\text{Hom}_{R}(\mathscr{M}_{1},\mathscr{M}_{2})$ be an isogeny. Then
it is finite, which implies it has leading coefficient that is non-zero after
reduction at $v$. This proves the injectivity. ∎
From now on, we assume $A=\mathbb{F}_{q}[t]$ and $\phi$ is a rank $2$ Drinfeld
$A$-module over $\mathbb{C}_{\infty}$ with CM by the maximal order
$\mathcal{O}_{K}$, where $K\subset\mathbb{C}_{\infty}$ is a quadratic
imaginary field over $k$. Further we assume $\phi$ is obtained through the
$A$-lattice $\mathcal{O}_{K}$ and $\phi$ is defined over $F$. Let $P\in A$ be
a prime element.
###### Theorem A.2.
For any place $v\in\text{Spec}(R)$, the reduction of $\phi$ at $v$ is ordinary
if and only if $v\cap k$ splits in $\mathcal{O}_{K}$.
###### Proof.
Let $P=v\cap k$ and $\bar{\pi}$ be the Frobenius morphism of $\phi^{v}$. From
Lemma A.1 we see
$K=\text{End}_{F}(\phi_{1})\otimes_{A}k\xhookrightarrow{}\text{End}_{\mathbf{k}(v)}(\phi^{v})\otimes_{A}k:=D.$
By [9, Proposition 4.12.17] we deduce that $\phi$ is ordinary at $v$ if and
only if $K=\text{End}_{\mathbf{k}(v)}(\phi^{v})\otimes_{A}k$. This is
equivalent to saying that
$\mathcal{O}_{K}=\text{End}_{\mathbf{k}(v)}(\phi^{v})$. We can embed $A$ into
$\text{End}_{\mathbf{k}(v)}(\phi^{v})$ via the Drinfeld module $\phi^{v}$. Let
$E:=k(\bar{\pi})$. Then there is only one prime $\mathscr{P}$ of $E$
containing $\bar{\pi}$ and $\mathscr{P}$ lies over $P$ [9, Theorem 4.12.8].
If $\phi$ has ordinary reduction at $v$, then $E\cong K$ as $\bar{\pi}\notin
k$. Again by [9, Proposition 4.12.17] there are more than one primes of $E$
lying over $P$. Thus $P$ splits in $\mathcal{O}_{K}$. Next we show the other
way around. First, we write
$P\mathcal{O}_{K}=\mathcal{P}\mathcal{P}^{\prime}$. Assume the reduction of
$\phi$ at $v$ is supersingular. So it is a consequence that
$\dim_{k}D=r^{2}=4$. Since $\phi$ has good reduction at $v$, $\phi^{v}$ has
rank 2 over $\mathbf{k}(v)$. Thus we have
$2=\text{rank}(\phi^{v})=t\cdot[E:k]$, where $t$ is an integer such that
$t^{2}=\dim_{E}D$. As $\dim_{k}D=4=\dim_{E}D\cdot[E:k]$, we have $t=2$.
Therefore, $E=k$. In particular, $\bar{\pi}\in A$. In this case, it is clear
that $\mathscr{P}=(P)\subset A$. On the other hand, we can obtain a Drinfeld
$\mathcal{O}_{K}$-module $\psi$ over $F$ by extending $\phi$ to
$\text{End}_{F}(\phi)$. By taking reduction at $v$ again, we obtain a Drinfeld
$\mathcal{O}_{K}$-module $\psi^{v}$ over $\mathbf{k}(v)$. It is trivial
$\bar{\pi}$ is the Frobenius element of $\psi^{v}$. As $\bar{\pi}\in A\subset
K$, there is only one prime ideal of $K$ containing $\bar{\pi}$. However,
$\bar{\pi}\in P\mathcal{O}_{K}=\mathcal{P}\mathcal{P}^{\prime}$. This is a
contradiction. ∎
###### Corollary A.3.
If $P\mathcal{O}_{K}=\mathscr{P}\mathscr{P}^{\prime}$ where $\mathscr{P}$ and
$\mathscr{P}^{\prime}$ are prime ideals of $\mathcal{O}_{K}$ both lying over
$P$, then for any place $v\in\text{Spec}(R)$ over $P$ the natural morphism
$\text{End}_{F}(\phi)\rightarrow\text{End}_{\mathbf{k}(v)}(\phi^{v})$ is an
isomorphism.
Reduction process
Let $\mathscr{M}$ be the minimal model of $\phi$. We set
$\mathscr{M}[P]:=\text{Ker}(\phi_{P}:\phi\rightarrow\phi)$. We suppose
moreover that $P\mathcal{O}_{K}=\mathscr{P}\mathscr{P}^{\prime}$. Then it is
easy to see:
$\mathscr{M}[P](\bar{A})=\mathscr{M}[P](\mathbb{C}_{\infty})\cong\mathcal{O}_{K}/P\mathcal{O}_{K}=\mathscr{P}/P\mathcal{O}_{K}\oplus\mathscr{P}^{\prime}/P\mathcal{O}_{K}\cong\mathcal{O}_{K}/\mathscr{P}\oplus\mathcal{O}_{K}/\mathscr{P}^{\prime}.$
There is a natural morphism
$\theta:\mathscr{M}[P](\bar{A})\rightarrow\phi^{v}[P](\overline{\mathbb{F}_{q}})$
by taking reduction at $v\in\text{Spec}(R)$ such that $v$ lies over $P$.
###### Theorem A.4.
If we assume further that
$\pi\in\mathscr{P}\subset\mathcal{O}_{K}=\text{End}_{F}(\phi)=\text{End}_{R}(\mathscr{M})$
is the lifting of the Frobenius element $\bar{\pi}$, then $\theta$ is a
surjection, and the kernel of $\theta$ is isomorphic to
$\mathcal{O}_{K}/\mathscr{P}$.
###### Proof.
Since $P\mathcal{O}_{K}=\mathscr{P}\mathscr{P}^{\prime}$, by Theorem A.2
$\phi^{v}[P](\overline{\mathbb{F}_{q}})$ is non-trivial and finite. We embed
$A$ into $\text{End}_{\mathbf{k}(v)}(\phi^{v})=\mathcal{O}_{K}$ via
$\phi^{v}$. As an $\mathcal{O}_{K}$-module, we have
$\phi^{v}[P](\overline{\mathbb{F}_{q}})\cong\mathcal{O}_{K}/I$ for some proper
ideal $I\subset\mathcal{O}_{K}$. Therefore, we have
$\phi^{v}_{P}\cdot\mathcal{O}_{K}/I=0$, which implies $P\in I$. It is clear
$\\#\\{\phi^{v}[P](\overline{\mathbb{F}_{q}})\\}<\\#\\{\mathscr{M}[P](\mathbb{C}_{\infty})\\}.$
So either $I=\mathscr{P}$ or $I=\mathscr{P}^{\prime}$. Since $\bar{\pi}$ acts
on $\phi^{v}[P](\overline{\mathbb{F}_{q}})$ non-trivially, we see
$I=\mathscr{P}^{\prime}$. Therefore, the kernel of $\theta$ is
$\mathscr{P}^{\prime}/P\mathcal{O}_{K}$ that is isomorphic to
$\mathcal{O}_{K}/\mathscr{P}$. ∎
###### Remark A.5.
1. (1)
If we identify
$\mathcal{O}_{K}=\text{End}_{F}(\phi)=\text{End}_{R}(\mathscr{M})$, then
$\mathscr{P}$ is the collection of isogenies whose reduction has linear
coefficient 0.
2. (2)
Another approach to Theorem A.4 using canonical subgroup of Drinfeld modules
has be shown to the author by Urs Hartl. The two approaches essentially have
the same core.
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Zhenlin Ran, University of Newcastle, Australia
E-mail address<EMAIL_ADDRESS>
|
=18pt plus1pt
CHAPTER: INTRODUCTION
This work is about the geometry of moduli spaces of vortices and antivortices
on a Riemann surface $\surface$. We are interested mostly in the
gauged $O(3)$ Sigma
where the fields are represented by a connection
$\gp$ and a section $\hf$ of a fibre bundle
with fibres
diffeomorphic to $\mathbb{P}^1$, the Riemann sphere.
We say $\hf$ is a Higgs field with target the Riemann sphere.
Static solutions of the field equations modulo gauge equivalence
form the moduli space of vortices and antivortices, each solution is
determined by the cores of the fields:
the preimages of the north pole (vortex points) and the south pole (antivortex
points). It can be proved the total number of the cores is enumerable and if
$\surface$ is compact, it is finite. We will assume without
loss of generality this is the
case, even though $\surface$ can be the complex plane. The dynamics
of slowly varying fields can be described by geodesic motion of curves
on the moduli space [Manton, 1982] with
a metric called the $\Lsp^2$ metric. This metric is Kähler and well
understood for the moduli space of vortices of the Ginzburg-Landau
functional, in which case it is known that the moduli space is a complete
metric space and if the ambient surface is compact the moduli space is also
compact, hence of finite volume.
The $O(3)$ Sigma model we will study is
asymmetric, vortices and antivortices have different
effective mass, moreover,
the existence of two types of cores
means vortices and
antivortices cannot coalesce, therefore, a natural question is if
the moduli space is still complete.
Another question we address is how the asymmetry
affects the volume of the moduli space. These questions were addressed for the
symmetric case in the
reference [Romão and Speight, 2020]. The techniques used in the reference however
do not apply in general, we developed analytical tools to
extend the results to the asymmetric case.
Later, we add a Chern-Simons deformation to the model and
describe the change in the dynamics of the fields on the moduli space.
The deformation is tuned by means of a deformation constant
$\kappa$ which we assume small.
It turns out that the dynamics of the theory is described by
geodesic motion perturbed with a connection term proportional to
$\kappa$, i.e. a term
dependent on the velocity of the cores. Our model resembles the model of Kim and
Lee [Kim and Lee, 2002] with the difference that the target is the
sphere and there are two types of cores to consider.
It is well known for several related models with Chern-Simons deformations
that multiple solutions of the field equations occur.
We study the problem of existence and multiplicity of
solutions to the field equations of the deformed
$O(3)$ Sigma model, the main result is that even though
multiple solutions of the equations can exist,
there is a minimal deformation, such that
no matter which configuration of vortices and antivortices on the
moduli space we choose, we can find exactly one
solution close to the undeformed solution of the
$O(3)$ Sigma model.
We conclude with a description of the chapters of the thesis.
In chapter <ref> we describe the ideas of localization in
abstract terms. Our approach is general and suits equally well
Ginzburg-Landau vortices as well as the $O(3)$ Sigma model, with the benefit
that it
makes clear what we mean by adding a Chern-Simons term.
We also present
analytical results that are common to other parts of the next
In chapter <ref> we focus on the $O(3)$ Sigma
model on the
euclidean plane. We study asymmetric vortex-antivortex pairs,
supporting our
analysis with numerical evidence of the behaviour of colliding
vortex-antivortex pairs.
We compute the metric on the moduli space of vortex-antivortex
pairs numerically
and use this computations to study the scattering of
approaching cores.
The main result is theorem <ref>
says that the moduli space is incomplete.
In chapter <ref> we move to a compact ambient
The main results are the incompleteness of the moduli space of
vortex-antivortex pairs,
theorem <ref>,
and the computation of the volume of
the moduli space for the round sphere and for flat tori in
theorem <ref>, confirming
a general conjecture by Romão-Speight [Romão and Speight, 2020]
in these cases.
Chapter <ref> is devoted to the study of Chern-Simons
deformations on compact surfaces.
We prove the existence of multiple solutions for small
deformations of the $O(3)$ Sigma model if the number of vortices and
antivortices is different and find bounds for
the deformation constant.
We also solve the
field equations numerically on the sphere for two configurations of
vortices and antivortices at antipodal positions.
The main result is theorem <ref>,
describing the behaviour of the solutions to the field
We finalise the chapter
applying the localization technique to vortices of the Ginzburg-Landau
model and vortices/antivortices of the $O(3)$ Sigma model, both with a
Chern-Simons deformation. We found that dynamics is deviated from
geodesic motion by a connection term consistent with
previous results of
Kim-Lee [Kim and Lee, 2002] and Collie-Tong [Collie and Tong, 2008],
and compared our result with theirs.
CHAPTER: PRELIMINARIES
This chapter is for basic definitions and results of field theory
that we will use in the successive. To study the geometry
of the moduli space of vortices we need several analytical tools, this
chapter is intended to be a bridge between field theory and analysis.
In section <ref> we introduce the $O(3)$ Sigma model, which
will play a central role all along the thesis.
In section <ref> we discuss a localization formula
for the $O(3)$ Sigma model, we compute a metric for the moduli space
of vortices and antivortices, the $\spL^2$ metric,
prove that
it is Kähler.
Section <ref> is about the analytic properties of
the Taubes equation, this is the elliptic PDE that guarantees the
existence of the moduli space of vortices and antivortices. Several
theorems of analysis are introduced in this section to keep them
collected in the same place for further reference.
In subsection <ref> we prove that the
solution to the Taubes equation depends differentiably on the
position of the vortices and antivortices.
In section <ref> we state less known theorems of functional
analysis about compact non-linear operators that we will need later.
ASYMMETRIC VORTEX-ANTIVORTEX SYSTEMS IN THE EUCLIDEAN PLANE
CHAPTER: ASYMMETRIC VORTEX-ANTIVORTEX PAIRS ON A COMPACT SURFACE
In this chapter we study vortex-antivortex systems on a compact surface.
We aim to prove that the moduli space is
incomplete and to compute the volume of the moduli space for the
round sphere and flat tori. On a general compact domain, the problem
of the statistical mechanics of Ginzburg-Landau vortices was
addressed by Manton [Manton, 1993] and
by Manton-Nasir [Manton and Nasir, 1999].
As shown
in [Manton, 1993], it can be described if we know the volume
of the moduli space. For
the abelian $O(3)$ Sigma model however, the problem of the volume of the moduli
space is constrained by the fact that vortices and antivortices cannot
coalesce, however, computing the volume is necessary for the partition function
of a gas of BPS vortices
[Romão and Speight, 2020, Manton and Nasir, 1999, Manton, 1993]. There is a
conjectured formula for the volume by Speight and Romão that depends on
topological data, the
volume of the domain,
$\tau$ and the size of the sets $\vset$, $\avset$ of
core positions [Romão and Speight, 2020]. The content of the chapter is as follows.
In section <ref>, we prove that the Taubes equation has
exactly one solution for any $\tau \in (-1, 1)$.
The main result of section <ref>
is theorem <ref>
which asserts that the moduli
space of vortex-antivortex pairs is incomplete. We prove the theorem
after proving several lemmas necessary to bound the derivatives of solutions
to the Taubes equation.
In section <ref> we compute the volume of
the moduli space of vortex-antivortex pairs for the round sphere and
flat tori and compare our results with the conjecture.
§ EXISTENCE OF VORTICES
In this section we will prove the existence of solutions to the
Taubes equation
on a compact surface. In [Sibner et al., 2000]
Sibner-Signer-Yang proved existence and
uniqueness of solutions of the gauged $O(3)$ Sigma model on a compact manifold
for $\tau = 0$.
We prove the following generalisation of their results.
On any compact
Riemann surface there exists exactly one
solution $u$ to the
Taubes equation (<ref>), provided the
\begin{equation}
\label{eq:vav-size-constraint}
- \frac{1 + \tau}{2\pi}\abs{\surface}
k_+ - k_-
\frac{1 - \tau}{2\pi}\abs{\surface}
\end{equation}
holds. Moreover,
$u$ is of class $C^2$ except for the core positions.
We prove the theorem at the end of the section. The inequality
(<ref>) is a Bradlow type restriction
[Bradlow, 1990], constraining the relative number of vortices and
antivortices on a compact surface.
It arises naturally from the second Bogomolny
equation (<ref>), since the total magnetic flux is,
\begin{align}
2\pi(k_+ - k_-) &=
\int_{\surface} B \nonumber\\
&= \int_\surface \lproduct{N,\hf}\,\vol -
\tau\,|\surface|,
\end{align}
where $N$ is the north pole section on the target sphere
and hence $\lproduct{N,\phi}\in [-1, 1]$, it follows
that (<ref>) is a necessary condition for a pair
$(\hf, \gp)$ of a field and a connection to be a solution to the Bogomolny
We will define
the function $\Fstable: \reals \to \reals$,
\begin{equation}
\Fstable(t) = 2 \left(
\frac{e^t - 1}{e^t + 1} + \tau
\right),
\end{equation}
and the constant,
\begin{equation}
\label{eq:fstable-pm-infty}
\Fstable^{\pm\infty} = 2(\pm 1 + \tau),
\end{equation}
in order to simplify notation in the proof of theorem <ref>.
Let us define
$\Fstable_0: \reals \to \reals$ as the function,
\begin{equation}
\label{eq:param-fstable-sphere}
\begin{aligned}[b]
\Fstable_0(t) &= 2\brk(\frac{e^t - 1}{e^t + 1} + \tau) + \frac{4\pi(k_+ -
k_-)}{\abs\surface }\\
&= \frac{4 e^t}{e^t + 1} - C_0,
\end{aligned}
\end{equation}
where the constant $C_0$ is,
\begin{align}
C_0 = 2(1 - \tau) - \frac{4\pi}{\abs\surface}(k_+ - k_-).
\end{align}
For a given configuration of non-coalescent vortices, recall the function
$v : \surface \to \reals\cup\set{\pm\infty}$, defined on
equation (<ref>), if $u$ is the solution of the Taubes equation, and
we define $\tilde h = u - v$, then
the regularized Taubes equation on a compact surface,
equation (<ref>), is equivalent to,
\begin{align}
\label{eq:regular-taubes-bps}
-\laplacian \tilde h = \Fstable_0(v + \tilde h).
\end{align}
Equation (<ref>) shows why Bradlow's bound is
necessary: If a smooth solution exists, by the divergence
theorem a necessary condition for $C_0$ is,
\begin{equation}
\label{eq:abstract-c0-cond}
C_0 = \frac{1}{|\surface|}\,\int_\surface\frac{4\,e^{v + \tilde h}}
{e^{v + \tilde h} + 1}\,\vol \in [0, 4],
\end{equation}
Bradlow's bound is equivalent to
(<ref>). Let
\begin{equation}
\label{eq:Xsp-def}
\Xsp = \left\{
u \in \Hsp^1(\surface) \,:\, \int_\surface u \,\vform = 0
\right\}
\end{equation}
be the subspace of Sobolev's space $\Hsp^1(\surface)$ of functions of zero
average. Since $\surface$ is compact, $\Hsp^1(\surface)$ can be
decomposed as
\begin{equation}
\label{eq:cann-decomp}
\Hsp^1(\surface) = \Xsp \oplus \reals.
\end{equation}
Any $h \in\Hsp^1(\surface)$ can be decomposed as a pair
$(u, \ctilde) \in \Xsp\times\reals$, such that $h = u +
\ctilde$. Hence, $u$ is a solution to the equation,
\begin{equation}
\label{eq:htilde}
-\laplacian u = \Fstable_0(v + u + \ctilde).
\end{equation}
We will use
Leray-Schauder theory to prove existence of solutions to the Taubes
equation as in the proof of Sibner et
al. [Sibner et al., 2000] for $\tau = 0$. Given $\htilde \in \Xsp$, the
\begin{equation}
\label{eq:integral-eqn}
c \mapsto \int_{\surface} \Fstable_0(v + \htilde + c) \,\vol,
\end{equation}
is a well defined, monotonous, continuous function. By Bradlow's
bound, there exists a unique number
$\ctilde$ such that
\begin{equation}
\label{eq:ctilde-defs}
\int_{\surface} \Fstable_0(v + \htilde + \ctilde) \,\vol = 0.
\end{equation}
The function $ \mathcal{C}: \Xsp \to \reals$, $\mathcal{C}(\tilde h) = \tilde
sequentially continuous in
We will highlight the steps different from [Sibner et al., 2000] in the
general case. If $\htilde_n \wto \htilde_0$ in $\Xsp$, then
$\htilde_n$ is a bounded sequence in $\Xsp$, and by the Rellich lemma,
after passing to a sub-sequence if necessary, we can assume
$\htilde_n \to \htilde_0$ in $\Lsp^p$ for $p \geq 1$. Let $\ctilde_n =
\ctilde(\htilde_n)$, $\ctilde_0 = \ctilde(\htilde_0)$ and assume
towards a contradiction that $\ctilde_n$ does not converge to
$\ctilde_0$. In this case we can assume the existence of a constant
$\epsilon_0$ such that,
\begin{equation}
\label{eq:cn-conv-contradiction}
\abs{\ctilde_n - \ctilde_0} \geq \epsilon_0,
\end{equation}
for all $n$. We claim the sequence $\left\{\ctilde_n\right\}$
is bounded. Assume the contrary, after passing to a sub-sequence if
necessary, we can assume the limit
$\ctilde_n \to \infty$. Let $K$ be any bound
for $\Fstable_0$. By Egorov's theorem [Lieb, 1997]
and the strong convergence in $\Lsp^p$,
there exists a measurable set $\surface_{\epsilon}$ and a constant
$K_{\epsilon}$, such that $\abs{\surface_{\epsilon}} < \epsilon
K^{-1}$, the sequence $\htilde_n$ converges uniformly to $\htilde_0$
in $\surface \setminus \surface_{\epsilon}$ and $\abs{\htilde_n} \leq
K_{\epsilon}$ in $\surface\setminus\surface_{\epsilon}$.
On the one hand,
the equality
\begin{equation}
\int_{\surface\setminus\surface_{\epsilon}}\Fstable_0(v +
\ctilde_n + \htilde_n) \,\vol =
-\int_{\surface_{\epsilon}}\Fstable_0(v + \ctilde_n +
\htilde_n) \,\vol,
\end{equation}
\begin{equation}
\label{eq:egorov-integral-impl}
\abs*{\int_{\surface\setminus\surface_{\epsilon}}\Fstable_0(v +
\ctilde_n + \htilde_n) \,\vol } \leq \epsilon,
\end{equation}
and on the other hand, by monotony of $\Fstable_0$,
\begin{equation}
\label{eq:monotony-fstable0-integral}
\int_{\surface\setminus\surface_{\epsilon}}\Fstable_0(v + \ctilde_n
- K_{\epsilon}) \,\vol \leq
\int_{\surface\setminus\surface_{\epsilon}}\Fstable_0(v + \ctilde_n
+ \htilde_n) \,\vol.
\end{equation}
Taking the limit as $n \to \infty$, from these two equations
we have,
\begin{equation}
\label{eq:int-finty-surface-minus-eps}
(\Fstable^{\infty} - C_0)(\abs\surface - \abs{\surface_{\epsilon}}) \leq \epsilon.
\end{equation}
\begin{equation}
\label{eq:fstable-inf-bound-contradiction}
(\Fstable^{\infty} - C_0) \abs\surface \leq \epsilon +
K\abs{\surface_{\epsilon}} < 2\epsilon,
\end{equation}
a contradiction since $\epsilon$ is arbitrary. A similar
argument shows $\ctilde_n$ is bounded below. Therefore,
$\ctilde_n$ is a bounded sequence of real numbers. By the
Bolzano-Weierstrass theorem, we can assume towards a contradiction
$\ctilde_n \to \ctilde$, but $\ctilde \neq \ctilde_0$
by (<ref>). Let
\begin{equation}
\label{eq:alpha-def}
\alpha = \abs*{
\int_{\surface}\Fstable_0(v + \htilde_0 + \ctilde) \,\vol
} > 0,
\end{equation}
bearing in mind the definition of $\htilde_n$,
\begin{multline}
\alpha = \abs*{
\int_{\surface}\Fstable_0(v + \htilde_0 + \ctilde)
- \Fstable_0(v + \htilde_n + \ctilde_n)
\,\vol
\leq
\sup_{t \in \reals}{\set{\Fstable'(t)}}\cdot\left(
\abs{\ctilde - \ctilde_n}\cdot\abs{\surface} + C\,\norm{\htilde_0 -
\htilde_n}_0 \cdot \abs{\surface}^{1/2}
\right) \to 0.\label{eq:alpha-bound}
\end{multline}
Hence $\alpha = 0$, a
contradiction. Therefore (<ref>) is
false and $\ctilde_n \to \ctilde_0$. This proves the lemma.
Let us consider the operator $T: \Xsp \to \Xsp$, mapping each
$\htilde\in\Xsp$ to the weak solution $H \in \Xsp$ of the equation
\begin{equation}
\label{eq:T-op}
-\laplacian H = \Fstable_0(v + \ctilde + \htilde).
\end{equation}
Given that $\int_\surface \Fstable_0(v + \ctilde + \htilde)\,\vol = 0$,
existence of a weak $\Hsp^1$ solution to (<ref>) is a well established
analysis fact <cit.>, moreover, any two weak
solutions to the equation differ by a constant, by taking $H \in \Xsp$ we
guarantee it is unique.
Recall a compact operator is an operator that maps bounded sequences
to sequences with convergent subsequences. We aim to use Schäfer's
alternative, theorem <ref>, to prove $T$
has a fixed point.
The operator $T:\Xsp \to \Xsp$ is compact in the strong topology of
$\Xsp$ as a subspace of $\Hsp^1(\surface)$.
Let $\{\tilde h_n\} \subset \Xsp$ be a bounded sequence, after
passing to a subsequence if necessary, we can assume $\htilde_n \wto
\htilde_0$ in $\Xsp$ and strongly in $\Lsp^2$. Let $H_n = T\htilde_n$,
$n \geq 0$, by lemma <ref> $\ctilde_n \to
\ctilde_0$. Moreover,
\begin{align}
\nonumber
\norm{\grad H_n - \grad H_0}_{\Lsp^2}^2
&= \int_{\surface} (H_n - H_0)\, \laplacian (H_n - H_0)\,\vol\\
\nonumber
&= \int_{\surface} (H_n - H_0)\left(\Fstable(v + \ctilde_n + \htilde_n)
- \Fstable(v + \ctilde_0 + \htilde_0)\right) \,\vol\\
\nonumber
&\leq \sup_{t \in \reals} \set{F'(t)}\,\int_{\surface} \left(
\abs{\ctilde_n - \ctilde_0} + \abs{\htilde_n - \htilde_0}\right)
\abs{H_n - H_0}\,\vol\\
&\leq \sup_{t \in \reals} \set{F'(t)}\, \left(
\abs{\ctilde_n - \ctilde_0}\cdot\abs{\surface}^{1/2} + \norm{\htilde_n -
\htilde_0}_{\Lsp^2}
\right) \norm{H_n - H_0}_{\Lsp^2}.
\end{align}
The last inequality is a consequence of the Cauchy-Schwarz
inequality. By the Poincaré inequality, there are constants $C_1$,
$C_2$ such that
\begin{equation}
\label{eq:Hn-H0-X-convergence}
\norm{H_n - H_0}_{\Hsp^1} \leq C_1\abs{\ctilde_n - \ctilde_0}
+ C_2\norm{\htilde_n - \htilde_0}_{\Lsp^2} \to 0.
\end{equation}
This proves compactness of $T$.
Let us consider the set
\begin{equation}
\label{eq:wt-def}
S = \set{\htilde \in \Xsp\, : \,
\exists\, t \in [0, 1]\;s.t.\;
\htilde = t\cdot
\end{equation}
If $\htilde \in S$, then it is a solution of the equation,
\begin{equation}
\label{eq:htilde-t-sol}
\laplacian \htilde = t \Fstable_0(v + \ctilde + \htilde),
\end{equation}
where $\tilde c = \mathcal C(\tilde h)$ was defined on
lemma <ref>.
By the Cauchy-Schwarz inequality,
\begin{equation}
\begin{aligned}
\norm{\grad\htilde_t}_{\Lsp^2}^2 &= \lproduct{\tilde h_t, \laplacian
\tilde h_t}
\leq C
\int_{\surface}\abs{\htilde_t} \,\vol
\leq C\,
\abs{\surface}^{1/2}\,\norm{\htilde_t}_{\Lsp^2}.
\end{aligned}
\end{equation}
By the Poincaré
inequality we conclude the existence of a constant $C$ such that
\begin{equation}
\label{eq:ht-uniform-bound}
\norm{\htilde_t}_{\Hsp^1} \leq C.
\end{equation}
Since $S$ is bounded, by Schäfer's alternative there is a fixed point
$\htilde$ of $T$. Let $h = \htilde + \ctilde$, where $\ctilde = \mathcal
C(\tilde h)$, then $h$ is a weak
solution to the regularised Taubes equation. By the elliptic
estimates $h$ is also a strong solution in $\Hsp^2$. We follow a
bootstrap argument to prove $h \in C^2$: By Sobolev's embedding we
know $h$ is continuous, hence $h \in \Lsp^p$ for any $p \geq
1$. By (<ref>) and the elliptic estimates $h
\in \Wsp^{2,p}$ for some $p > 2$, once more by Sobolev's embedding $h
\in C^1$. Let $u = h + v$, the derivative $d h \in
\Gamma(T^{*}\surface)$ is a weak solution of the linearized equation,
\begin{align}
-\laplacian\,d h = \frac{4\,e^u}{(e^u + 1)^2}\,d h +
\frac{4\,e^u}{(e^u + 1)^2}\,d v.
\end{align}
The potential function $e^u(e^u + 1)^{-2}$ is continuous and with
zeros of the same order than the singularities of $d v$ at the cores,
hence $\laplacian(d h) \in \Lsp^p$, $p > 2$. Since $d h$ is
continuous, it is also an $\Lsp^p$ form. By the elliptic estimates and
Sobolev's embedding we conclude $h \in C^2$. Since $\Fstable$
is monotonous, $h$ is unique by the strong maximum
principle. Finally, $u$ is the necessarily unique solution to the
Taubes equation.
§ INCOMPLETENESS OF THE MODULI SPACE
In [Romão and Speight, 2020] Romão and Speight prove that the moduli space of
vortex-antivortex pairs on the sphere is incomplete. In this section we extend
their result to general $\tau$ on a compact manifold. In order to prove this,
we find bounds for
the derivatives $\partial_{z_j}\nabla h_\epsilon$ on a holomorphic chart,
where the cores are at positions $z_1, z_2$.
Let $\mu = \log\, (1 - \tau) - \log\, (1 + \tau)$, first we prove a pair of
technical lemmas.
Let $\diag$ be the diagonal set of $\surface \times \surface$ and
let $\set{\vb x_n} \subset \moduli^{1,1}(\surface)$ be a
sequence such that $\vb x_n \to \vb x \in \diag$ in the product metric.
Let $\tilde h_n$ be the
solution of the regular Taubes equation corresponding to each
$\vb x_n$, then
$\tilde h_n \wto \mu$ in $\Hsp^1$ and $\tilde h_n \to \mu$
strongly in $\Lsp^2$.
Let $v_n = v_{\vb x_n}$ for
each point $\vb x_n$ in the given sequence. Let us
decompose each
solution to the regular Taubes equation as $\tilde h_n = u_n +
\tilde c_n \in \Xsp \oplus \reals$. We claim the sequence
$\set{ \tilde c_n}$ is bounded. Assume towards a contradiction
$\tilde c_n \to \infty$. Notice that in the vortex-antivortex
case the functions $\fnF$ and $\fnF_0$ coincide. We know that,
\begin{align}
-\laplacian u_n = \fnF( u_n + \tilde c_n +
\fnv_n).\label{eq:lap-h0n}
\end{align}
By the standard elliptic estimates, there is a constant $C$ such
\begin{align}
\norm{u_n}_{\Hsp^2} \leq C \norm{\laplacian u_n}_{\Lsp^2}.
\end{align}
Since $\fnF$ is a
bounded function, $\set{u_n}$ is bounded in $\Hsp^2$ and by
Sobolev's embedding also in $C^0$.
Assume $\vb x = (x_*, x_*)$ and notice that,
\begin{align}
|v_n(x)| = 4\pi |G(x, x_{1n}) - G(x, x_{2n})|,
\end{align}
where $\vb x_n = (x_{1n}, x_{2n})$,
since $G(x,y)$ is continuous away of the diagonal set,
$\fnv_n(x) \to 0$ for $x \neq x_*$, whence,
we also have the convergence,
\begin{align}
\fnF( u_n + \tilde c_n +
\fnv_n) \to 2(1 + \tau),
\end{align}
pointwise almost everywhere. Applying the dominated convergence theorem
equation (<ref>),
\begin{align}
\int_{\surface} \fnF( u_n + \tilde c_n +
\fnv_n) \,\vform = 0 \to 2(1 +
\tau)\abs\surface,
\end{align}
a contradiction. If $\tilde c_n \to -\infty$ a similar argument
holds. Therefore the sequence of averages $\tilde c_n$ is bounded,
$\{\tilde h_n\}$ is bounded in $C^0$.
Hence, the sequence is also bounded in $\Lsp^p$ for any
positive $p$. By the elliptic estimate
\begin{align}
\norm{\tilde h_n}_{\Hsp^2} \leq C
\pbrk{\norm{\laplacian \tilde
h_n}_{\Lsp^2} + \norm{\tilde h_n}_{\Lsp^2}},
\end{align}
$\{\tilde h_n\}$ is also bounded in $\Hsp^1$. By the Alaoglu
and Rellich theorems, after passing to a subsequence if
necessary, we can assume $\tilde h_n \wto h_{*} \in \Hsp^1$ and
strongly in $\Lsp^2$. We claim that $h_*$ is the constant
function $\mu$. To see this, let $\varphi \in \Hsp^1$. From the
regularized Taubes equation we have,
\begin{align}
\lproduct{\tilde h_n, \varphi}_{\Hsp^1}
&= \lproduct{\tilde h_n,
\varphi}_{\Lsp^2} + \lproduct{\grad \tilde h_n, \grad
\varphi}_{\Lsp^2},\nonumber \\
&= \lproduct{\tilde h_n,
\varphi}_{\Lsp^2} + \lproduct{\laplacian \tilde h_n,
\varphi}_{\Lsp^2}\nonumber \\
&= \lproduct{\tilde h_n,
\varphi}_{\Lsp^2} - \lproduct{\fnF(\tilde h_n + \fnv_n),
\varphi}_{\Lsp^2}.
\end{align}
Since $\tilde h_n \to h_{*}$ strongly in $\Lsp^2$, after passing
to a subsequence if necessary, we can assume
$\tilde h_n\to h_{*}$ pointwise almost everywhere. By the weak
convergence of $\tilde h_n$ in $\Hsp^1$, together with the
strong convergence in $\Lsp^2$ and the dominated convergence
\begin{align}
\lproduct{h_{*}, \varphi}_{\Hsp^1} &= \lim\,\lproduct{\tilde h_n,
\varphi}_{\Hsp^1}\nonumber\\
&= \lim \lproduct{\tilde h_n,
\varphi}_{\Lsp^2} - \lim \lproduct{\fnF(\tilde h_n + \fnv_n),
\varphi}_{\Lsp^2}\nonumber\\
&= \lproduct{h_{*},\varphi}_{\Lsp^2} - \lproduct{\fnF(h_{*}),
\varphi}_{\Lsp^2}.
\end{align}
From this equation, we infer
\begin{align}
\lproduct{\grad h_{*}, \grad\varphi}_{\Lsp^2}
= -\lproduct{\fnF(h_{*}), \varphi}_{\Lsp^2}.
\end{align}
Therefore, $h_{*}$ is a weak solution to the equation
\begin{align}
-\laplacian h_{*} = \fnF(h_{*}).
\end{align}
By elliptic regularity, $h_{*}$ is also a strong solution, and
by the maximum principle, $h_{*}$ is constant since $\fnF$ is an
increasing function. Since the only zero of $\fnF$ is at
$t = \mu$, we conclude $h_{*} = \mu$. If $\tilde h_{n_k}$ is any
subsequence of $\tilde h_{n}$, this argument shows it
has a subsequence weakly converging to $\mu$ in $\Hsp^1$ and
strongly in $\Lsp^2$, the
claim of the lemma follows.
$\tilde h_n \to \mu$ strongly in $\Wsp^{2,p}$ for any positive $p$.
We will prove that any subsequence of $\tilde h_n$ has another
subsequence converging to $\mu$ in $\Wsp^{2,p}$, implying the
lemma. To simplify
notation, we denote subsequences of $\tilde h_n$ by the
same symbol.
From the previous lemma, $\tilde h_n \to \mu$ strongly in
$\Lsp^2$. After passing to a subsequence if necessary, we can
assume that $\tilde h_n \to \mu$ pointwise almost
everywhere. We apply the
dominated convergence theorem to deduce the limit,
\begin{align}
\norm{\laplacian \tilde h_n}_{\Lsp^{p}}
= \norm{\fnF(\tilde h_n + \fnv_n)}_{\Lsp^p}
\to \norm{\fnF(\mu)}_{\Lsp^p} = 0.
\end{align}
If $p = 2$, by the standard elliptic estimates, there is a
constant $C$, such that,
\begin{align}
\norm{\tilde h_n - \mu}_{\Hsp^2} \leq C \pbrk{
\norm{\laplacian \tilde h_n}_{\Lsp^2} +
\norm{\tilde h_n - \mu}_{\Lsp^2}} \to 0.
\end{align}
By Sobolev's embedding, $\tilde h_n \to \mu$ uniformly in
$C^0$, hence also in $\Lsp^p$ for any positive $p$. We apply one
more time the elliptic estimate,
\begin{align}
\norm{\tilde h_n - \mu}_{\Wsp^{2,p}} \leq C \pbrk{
\norm{\laplacian \tilde h_n}_{\Lsp^p} +
\norm{\tilde h_n - \mu}_{\Lsp^p}} \to 0.
\end{align}
As a consequence of this lemma and Sobolev's embedding, we have
the convergence,
\begin{align}
\norm{\tilde h_n - \mu}_{C^1} \to 0,
\end{align}
for any arbitrary sequence $\set{\vb x_n} \subset \moduli^{1,1}(\surface)$,
such that $\vb x_n \to \vb x \in \Delta$. This proves the following
The limit,
\begin{align}
\lim_{d(x_1, x_2) \to 0}\norm*{\tilde h(x; x_1, x_2) - \mu
= 0,
\end{align}
holds, where $d(x_1, x_2)$ is the Riemannian distance in $\surface$.
Let $\surface^2_{\diag} = (\surface\times\surface)\setminus\diag$
endowed with the product metric. As differentiable manifolds,
$\moduli^{1,1}$ and
$\surface^2_\diag$ are equivalent. In what follows, we will consider
$\tilde h$ as a
function $\surface \times \surface^2_{\diag} \to \reals$. Let $U \subset
\surface$ be an open and dense subset and let
$\varphi:U \to V \subset \cpx$ be a
holomorphic chart. In what follows we denote points on the surface as $x$
and points on $\cpx$ as $z$, so $z = \varphi(x)$ for $x \in U$. We also
assume vortices
and antivortices are both located in $U$, such that up to a holomorphic
$\tilde h: \surface \times V^2_\diag \to \reals$, where $V^2_\diag
= V^2\setminus \diag_V$ and
$\diag_V \subset \cpx^2$ is the diagonal set. On this chart partial
derivatives $\partial_{z_j}\tilde h$ are well defined functions
\begin{align}
\partial_{z_j}\tilde h : \surface \times V^2_\diag \to \cpx.
\end{align}
We denote the covariant derivative and Laplacian with respect to the first
variable by $\grad$ and $\laplacian$ and emphasize that
the metric on $V^2_\diag$ is the push forward of the metric induced
by the surface. Our aim is
to estimate the rate at which the second
derivatives $\grad \partial_{z_j}\tilde h$ grow as a sequence $\vb z_n \in
V^2_\diag$ diverges to the diagonal set. This will
allow us to prove that the moduli space is incomplete.
Since, $\laplacian$ and $\partial_{z_j}$ commute, $\partial_{z_j}\tilde h$
is the solution to the elliptic problem,
\begin{align}
-\laplacian\partial_{z_j}\tilde h = \fnV(h)
\partial_{z_j} \tilde h + \sign_j\,\fnV(h)\,\partial_{z_j} \fnv_j,
\end{align}
where $v_j(x) = 4\pi \,G(x, \varphi^{-1}(z_j))$.
Let $\dist_j(x) = \dist(x, x_j)$, $x_j = \varphi^{-1}(z_j)$, we know
there is a uniform constant $C$,
such that the derivative of Green's function is
bounded [Aubin, 2013],
\begin{align}
\abs{\grad G(x, x_j)} < \frac{C}{\dist_j}, &&
\abs{\grad_2 G(x, x_j)} < \frac{C}{\dist_j},
\end{align}
where $\grad_2 G$ is the covariant derivative with respect to the
second variable.
Recall in holomorphic coordinates the metric is
$e^{\Lambda(z)}\abs{dz}^2$, hence, if $z_j$ is restricted to a
bounded domain,
\begin{align}
\abs{\del_{z_j}\fnv_j} \leq 4\pi e^{-\Lambda(z_j)}\,\abs{\grad_2
< \frac{C}{d_j}.
\end{align}
For any positive constant
$C_1$, there is another constant $C$, such that, for all
$x, x_1, x_2 \in U$,
\begin{align}
\frac{\dist_{12}^2}{C_1\dist_1^2 + \dist_2^2}
& \leq C,\label{eq:vol-1}
\\
\frac{\dist_j\dist_k^2}{(C_1\dist_1^2 + \dist_2^2)^2}
&\leq \frac{C}{\dist_{12}},\label{eq:vol-2}
\end{align}
where $\set{\dist_j, \dist_k} = \set{\dist_1, \dist_2}$
and $\dist_{12}=\dist(x_1, x_2)$.
By the triangle inequality and Cauchy-Schwarz,
\begin{align}
\dist_{12} \leq \dist_1 + \dist_2 \leq C\,(\dist_1^2 + \dist_2^2)^{1/2},
\end{align}
on the other hand, any two norms in a finite dimensional vector
space are equivalents, hence, there is another constant such
\begin{align}
(\dist_1^2 + \dist_2^2)^{1/2} \leq C\,(C_1\dist_1^2 + \dist_2^2)^{1/2},
\end{align}
from these two inequalities we obtain the first claim of the
lemma. For the second claim, it is enough to prove that the
\begin{align}
\frac{\dist_1\dist_2^2}{(C_1\dist_1^2 + \dist_2^2)^2}
\leq \frac{C}{\dist_{12}},
\end{align}
holds, the remaining case being equivalent to this one after
relabelling $\dist_1$ and $\dist_2$. Let us note that since,
\begin{align}
\dist_1\dist_2 \leq \half (\dist_1^2+\dist_2^2)
\leq C\,(C_1\dist_1^2 + \dist_2^2),
\end{align}
is sufficient to prove that,
\begin{align}
\frac{\dist_2}{C_1\dist_1^2 + \dist_2^2}
\leq \frac{C}{\dist_{12}}.\label{eq:vol-dist2-c1norm-bound}
\end{align}
$\dist_2 \leq \dist_1$, by the triangle inequality we have,
\begin{align}
\dist_2\dist_{12}
&\leq \dist_1\dist_2 + \dist_2^2\nonumber\\
&\leq \dist_1^2 + \dist_2^2\nonumber\\
&\leq C\,(C_1\dist_1^2 + \dist_2^2),
\end{align}
hence (<ref>). On the other hand, if
$\dist_1\leq \dist_2$, repeating the
previous step, we find that
\begin{align}
\dist_1\dist_{12} \leq C\,(C_1\dist_1^2 + \dist_2^2),
\end{align}
this inequality, together with (<ref>) and the triangle
inequality, implies,
\begin{align}
\frac{\dist_2}{C_1\dist_1^2+\dist_2^2}
\leq \frac{\dist_1}{C_1\dist_1^2+\dist_2^2}
+ \frac{\dist_{12}}{C_1\dist_1^2+\dist_2^2}
\leq \frac{C}{\dist_{12}}.
\end{align}
In any case, we conclude that equation (<ref>) holds.
There is a constant $C$ such that for any pair of distinct
points $x_1, x_2 \in \surface$,
\begin{align}
\abs*{G(x_1,x_2) - \frac{1}{2\pi}\log\,\dist(x_1,x_2)} \leq C.
\end{align}
We cover $\surface$ with a finite cover of metric disks
$\disk_{R_j/2}(p_j)$ such that $R_j < \delta$, where $\delta$ is the
injectivity radius of the metric and for each disk there is a
chart $\varphi_j: U_j \to \cpx$, $\disk_{R_j}(p_j) \subset
U_j$. Let $R = \min\set{R_j}$, for any pair of distinct points $x_1,
\in \surface$,
such that $d(x_1, x_2) < R/2$,
there is a disk such that $x_1, x_2 \in \disk_{R_j}(p_j)$.
For any disk in the cover, let $R_j'$ be a positive radius, such that,
\begin{align}
|\varphi_j(x) - \varphi_j(p_j)| < R_j', \qquad
\forall x \in \disk_{R_j}(p_j).
\end{align}
Let $z_j = \varphi_j(p_j)$ and let us denote by $D_{R'_j}(z_j) \subset
\cpx$ the
holomorphic disk of radius $R'_j$ centred at $z_j$. For any small
$\epsilon > 0$ there are continuous functions $\tilde G_j: D_{R'_j +
\epsilon}(z_j) \times D_{R'_j + \epsilon}(z_j) \to \reals$ such that
if $x_1, x_2 \in \disk_{R_j}(p_j)$,
\begin{align}
G(x_1, x_2) = \frac{1}{2\pi} \log\,\abs{\varphi_j(x_1) -
\varphi_j(x_2)}
+ \tilde G_j(\varphi_j(x_1), \varphi_j(x_2)).
\end{align}
If $\exp \Lambda_j(z)$ is the conformal factor of the metric in the
chart $\varphi_j$, let
\begin{equation}
\begin{aligned}
M_j &= \max \set{e^{\Lambda_j(z)/2}\,:\, z \in
\overline{D_{R'_j}(z_j)}},
\\
m_j &= \min \set{e^{\Lambda_j(z)/2}\,:\, z \in
\overline{D_{R'_j}(z_j)}},
\end{aligned}
\end{equation}
and $M = \max_j \set{M_j}$, $m = \min_j \set{m_j}$. Since each
$\disk_{R_j}(p_j)$ is geodesically convex, for any $x_1, x_2 \in
\disk_{R_j}(p_j)$,
\begin{align}
m\,\abs{\varphi_j(x_1) - \varphi_j(x_2)} \leq d(x_1, x_2)
\leq M\,\abs{\varphi_j(x_1) - \varphi_j(x_2)}.
\end{align}
Taking the log of this inequality we find a positive constant such that,
\begin{align}
\abs{\,d(x_1, x_2) - \log\,\abs{\varphi_j(x_1) - \varphi_j(x_2)}\,}
\leq C,
\end{align}
whenever $x_1, x_2 \in \disk_{R_j}(p_j)$. Since each function $\tilde
G_j$ is continuous in the compact set $\overline{D_{R'_j}(z_j)}$, we find
another constant such that,
\begin{multline}
\abs*{G(x_1, x_2) - \frac{1}{2\pi}\log\,d(x_1, x_2)} =\\
\abs*{\frac{1}{2\pi}\pbrk{\log\,\abs{\varphi_j(x_1) - \varphi_j(x_2)} -
\log\,d(x_1,
x_2)} + \tilde G_j(\varphi_j(x_1), \varphi_j(x_2))}
\leq C.
\end{multline}
This proves the inequality whenever $d(x_1, x_2) < R/2$. Since $G$ and
the distance function are continuous on the compact set,
\begin{align}
\set{(x_1, x_2) \in \surface \times \surface \,:\, d(x_1, x_2) \geq
\frac{R}{2}},\
\end{align}
we can find a second constant satisfying the inequality whenever
$d(x_1, x_2) \geq R/2$. Taking the maximum of both constants
concludes the lemma.
Let $D$ be any bounded domain on $\cpx$. For any $p > 0$, there is a
constant $C$, independent of $z_1, z_2 \in D$, $z_1 \neq z_2$ such
that, if $x_j = \varphi^{-1}(z_j)$,
\begin{align}
\norm{\fnV(h)\partial_{z_j} v_j}_{\Lsp^p} \leq \frac{C}{\dist(x_1, x_2)}.
\end{align}
By lemma <ref>, there is
a constant, such that for all $x, y \in \surface$, $x \neq y$,
\begin{align}
\abs*{G(x, y) - \frac{1}{2\pi} \log \dist(x,y)} \leq C.
\end{align}
\begin{align}
\abs{\fnV(h)\partial_{z_j}\fnv_j} = \abs*{
\frac{4e^{\fnv_1}e^{\fnv_2}e^{\tilde h}}{
(e^{\fnv_1}e^{\tilde h} + e^{\fnv_2})^2
} \partial_{z_j}\fnv_j} \leq
C \abs*{\frac{4 \dist_1^2\dist_2^2\,e^{\tilde h}}{
(\dist_1^2e^{\tilde h} + \dist_2^2)^2
} \frac{1}{\dist_j}},
\end{align}
where the constant depends on $D$.
Since $\tilde h$ is uniformly
bounded on $\surface$, there are constants $C$, $C_1$, such that
by lemma <ref>.
\begin{align}
\abs{\fnV(h)\partial_{z_j}\fnv_j} \leq
C \abs*{\frac{\dist_1^2\dist_2^2}{
(\dist_1^2C_1 + \dist_2^2)^2
} \frac{1}{\dist_j}}
\leq \frac{C}{\dist(x_1,x_2)},
\end{align}
inequality implies the claim.
The proof of the lemma depends only on properties of Green's
function, we could repeat the
proof of lemma <ref> using $\grad \fnv_j$
instead of $\partial_{z_j}v_j$ to prove for
any given domain $D\subset \cpx$ the existence of a constant,
independent of $z_1, z_2 \in D$ , such
\begin{align}
\norm{V(h)\grad v_j}_{\Lsp^p} \leq \frac{C}{d(x_1, x_2)}.
\end{align}
In the next lemmas we prove that the bilinear form,
\begin{align}
\Bop:\Hsp^1\times \Hsp^1 \to \reals,
\Bop(\phi,\psi) = \lproduct{\grad\phi, \grad\psi}^2_{\Lsp^2}
+ \lproduct{\fnV(h)\phi, \psi}_{\Lsp^2},
\end{align}
is coercive with a uniform coercivity constant.
If $\fnV_n: \surface \to \reals$ is a sequence of continuous,
uniformly bounded functions converging pointwise to the
continuous function $\fnV_*$, and $\phi_n\to \phi_*$ in $\Lsp^2$,
\begin{align}
\lproduct{\fnV_n, \phi_n^2}_{\Lsp^2} \to \lproduct{\fnV_*,
\phi_*^2}_{\Lsp^2}.
\end{align}
We have,
\begin{align}
\abs{\lproduct{\fnV_n,\phi_n^2}_{\Lsp^2} -
\lproduct{\fnV_*,\phi_*^2}_{\Lsp^2}}
\leq
\abs{\lproduct{\fnV_n,\phi_n^2 - \phi_*^2}_{\Lsp^2}}
\abs{\lproduct{\fnV_n - \fnV_*, \phi_*^2}_{\Lsp^2}}.
\end{align}
Since the functions $\fnV_n$ are uniformly bounded, there is a
constant $C$ such that,
\begin{align}
\abs{\lproduct{\fnV_n,\phi_n^2 - \phi_*^2}_{\Lsp^2}}
&\leq C\,\lproduct{
\abs{\phi_n - \phi_*},
\abs{\phi_n + \phi_*}}_{\Lsp^2}\nonumber\\
&\leq C\,\norm{\phi_n - \phi_*}_{\Lsp^2}\,\norm{\phi_n +
\phi_*}_{\Lsp^2},
\end{align}
by the convergence $\phi_n \to \phi_*$ in $\Lsp^2$, we obtain
the limit
\begin{align}
\abs{\lproduct{\fnV_n,\phi_n^2 - \phi_*^2}_{\Lsp^2}} \to 0.
\end{align}
Since there is a constant $C$ such that the functions
$(\fnV_n - \fnV_*)\phi_*^2$ are bounded by the measurable
function $C\phi_*^2$ and $\fnV_n - \fnV_* \to 0$ pointwise, by
the dominated convergence theorem,
\begin{align}
\abs{\lproduct{\fnV_n - \fnV_*, \phi_*^2}_{\Lsp^2}} \to 0.
\end{align}
\begin{align}
\abs{\lproduct{\fnV_n,\phi_n^2}_{\Lsp^2} -
\lproduct{\fnV_*,\phi_*^2}_{\Lsp^2}} \to 0,
\end{align}
concluding the proof of the lemma.
There is a positive constant $C$, independent of $(x_1, x_2) \in
\surface^2_\diag$, such
that for any $\phi \in \Hsp^1$,
\begin{align}
C\norm{\phi}^2_{\Hsp^1} \leq \Bop(\phi, \phi).
\end{align}
By the bilinearity of $\Bop$, it is sufficient to prove the
lemma assuming $\norm{\phi}_{\Hsp^1} = 1$.
Let us assume towards a contradiction the statement is false, in
this case there is a sequence
$(\phi_n,\vb{x}_n) \subset \Hsp^1\times \surface^2_{\diag}$,
with $\norm{\phi_n}_{\Hsp^1} = 1$, such that,
\begin{align}
\Bop(\phi_n, \phi_n) = \norm{\grad\phi_n}_{\Lsp^2}^2 +
\lproduct{\fnV_n, \phi_n^2}_{\Lsp^2} \to 0,
\end{align}
where $\fnV_n = \fnV(h_n)$ is the potential function determined
by $h_n$, the solution to the Taubes equation with data
$\vb{x}_n$. Since the functions $\fnV_n$ are non-negative,
\begin{align}
\norm{\grad\phi_n}_{\Lsp^2} \to 0,
\lproduct{\fnV_n, \phi_n^2}_{\Lsp^2} \to 0.
\end{align}
Passing to a subsequence if necessary, we can assume
$\phi_n \wto \phi_*$ in $\Hsp^1$ and strongly in $\Lsp^2$ and
$\vb{x}_n \to \vb{x}_{*}$ in $\surface\times \surface$. Since
the functions
\begin{align}
e^{\fnv_j}: \surface\times\surface \to \reals
\end{align}
are continuous and $\tilde h_n$ varies continuously with the
initial data, if $\vb{x}_{*} \not\in \diag$, we have the uniform
convergence $\fnV_n \to \fnV_* = \fnV(h_{*})$, where $h_{*}$ is
the solution to the Taubes equation determined by $\vb{x}_{*}$. On
the other hand, if $\vb{x}_{*} \in \diag$, we know that
$\tilde h_n \to \mu$ in $C^1$, hence, we have pointwise
$\fnV_n \to \fnV_{*} \equiv 4\exp(\mu) (\exp(\mu) + 1)^{-2}$. In
any case, by our previous lemma,
\begin{align}
\lproduct{\fnV_n, \phi_n^2}_{\Lsp^2} \to
\lproduct{\fnV_*, \phi_*^2}_{\Lsp^2},
\end{align}
but this limit is zero, hence $\phi_{*} = 0$ almost everywhere
and $\phi_n \to 0$ in $\Hsp^1$ strongly, a contradiction.
Let $D \subset V$ be any bounded domain. There is a positive constant
$C(D)$, such that
\begin{align}
\norm{\partial_{z_j}\tilde h}_{C^1} \leq \frac{C}{\dist_{12}},
\text{and}
\norm{\grad \tilde h}_{C^1} \leq \frac{C}{\dist_{12}},
\end{align}
for all $z_1, z_2 \in D$
with $z_1 \neq z_2$, where $\tilde h(x) = \tilde h(x; \varphi^{-1}(z_1),
\varphi^{-1}(z_2))$ and
$\dist_{12} = \dist(x_1, x_2)$.
$\partial_{z_j}\tilde h$ is a solution to the equation
\begin{align}
-\laplacian \partial_{z_j}\tilde h
= \fnV(h)\partial_{z_j}\tilde h + \sign_j\fnV(h)\partial_{z_j}\fnv_j.
\end{align}
By lemma <ref>, there is a positive
constant $C_1$
independent of $z_1, z_2$, such that
\begin{align}
C_1\,\norm{\phi}_{\Hsp^1}^2 \leq \norm{\grad\phi}_{\Lsp^2}^2 +
\lproduct{\fnV(h)\,\phi, \phi}_{\Lsp^2},
\end{align}
for all $\phi \in \Hsp^1$. As in the proof of
lemma <ref>, a
second uniform constant, dependent on $D$, can be found such that,
\begin{align}
\norm{\fnV(h)\partial_{z_j}\fnv_j}_{\Lsp^2} \leq \frac{C_2}{\dist_{12}}.
\end{align}
By the Lax-Milgram theorem, we obtain the bound,
\begin{align}
\norm{\partial_{z_j}\tilde h}_{\Hsp^1} \leq \frac{C}{\dist_{12}},
\end{align}
where $C = C_2/C_1$. Now we follow a recursive argument: by
Schauder's estimates,
$\norm{\partial_{z_j}\tilde h}_{\Hsp^2}$ is also bounded
by $C\,d_{12}^{-1}$ for some constant $C$. By Sobolev's
embedding, there is another constant such that
$\norm{\partial_{z_j}\tilde h}_{C^0}$ is also bounded by
$C\,d_{12}^{-1}$. Thence, for any given $p > 2$,
\begin{align}
\norm{\partial_{z_j}\tilde h}_{\Lsp^p} \leq \frac{C}{\dist_{12}}.
\end{align}
By the elliptic estimates,
\begin{align}
\norm{\partial_{z_j}\tilde h}_{\Wsp^{2, p}}
&\leq C\,(\norm{\laplacian \partial_{z_j}\tilde h}_{\Lsp^p}
+ \norm{\tilde h}_{\Lsp^p})\nonumber\\
&\leq C\,(\norm{\fnV(h)\partial_{z_j}\tilde h}_{\Lsp^p}
+ \norm{\fnV(h)\partial_{z_j}\fnv_j}_{\Lsp^p} + \norm{\partial_{z_j}\tilde
&\leq \frac{C}{\dist_{12}},
\end{align}
for the last inequality we have used that the function $\fnV(t)$
is bounded. Sobolev's embedding implies the claimed
\begin{align}
\norm{\partial_{z_j}\tilde h}_{C^1} \leq \frac{C}{\dist_{12}}.
\end{align}
This argument is also valid for $\grad \tilde h$, because it is a
solution to the elliptic problem,
\begin{align}
-\laplacian (\grad \tilde h) = \fnV(h)\grad \tilde h + \fnV(h)(
\grad\fnv_{1} - \grad\fnv_{2}),
\end{align}
and the upper bound
\begin{align}
\norm{\fnV(h)\grad \fnv_{j}} \leq \frac{C}{\dist_{12}}
\end{align}
also holds.
For latter application, we need to translate this estimate to a
holomorphic chart.
Let $\varphi:U \subset \surface \to V \subset \cpx$ be a holomorphic
chart and let $D$ be a geodesically convex neighbourhood such that
$\overline D \subset U$, there is a positive constant
$C$, such that for all $z_1, z_2 \in \varphi(D)$,
\begin{align}
C\,\abs{z_1 - z_2} \leq d(\varphi^{-1}(z_1), \varphi^{-1}(z_2)).
\end{align}
The conformal factor is
continuous positive function on
$V$ and $\varphi(\overline D)$ is compact, hence there is a constant
$C > 0$, such that for all $z \in \varphi(D)$,
\begin{align}
C^2 \leq e^{\Lambda(z)}.
\end{align}
Since $D$ is geodesically convex, for any pair $z_1, z_2 \in
\varphi(D)$, there is a curve $\gamma: [0, 1] \to \varphi(D)$ joining $z_1$
to $z_2$
such that $\varphi^{-1}\circ \gamma$ is a minimizing geodesic joining
$\varphi^{-1}(z_1)$ to $\varphi^{-1}(z_2)$, hence,
\begin{align}
C\int_0^1\abs{\dot \gamma}\,ds \leq \int_0^1 e^{\Lambda/2}\abs{\dot
\gamma}\,ds = \dist(\varphi^{-1}(z_1), \varphi^{-1}(z_2)).
\end{align}
By the triangle inequality,
\begin{align}
\abs{z_1 - z_2} = \abs*{\int_0^1\dot\gamma}\leq \int_0^1 \abs {\dot
\gamma}\,ds,
\end{align}
yielding the result.
The advantage of the holomorphic chart is that it makes computations
possible, on the other hand, the Riemannian distance
is a geometric invariant defined globally on the surface and better suited
to prove analytical properties of the solutions to the Taubes equation.
For the next lemma, notice that if $\surface_1 \times \surface_2$ is a
product of
Riemmann surfaces, for any function $f: \surface_1 \times \surface_2
\to \cpx$ in local coordinates $\varphi_j:U_j \to \cpx$,
$\varphi_j(x_j) = z_j$,
\begin{align}
\del_{x_1}\del_{x_2}f = \partial_{z_1z_2}f\,dz^1\otimes dz^2
\in \Omega^{(2,0)}(\surface_1\times\surface_2).
\end{align}
In the product metric, $dz^1$ and $dz^2$ are orthogonal, hence,
\begin{align}
\abs{\partial_{x_1}\partial_{x_2}f} = \abs{\partial_{z_1, z_2}f}\,
\abs{dz^1}\,\abs{dz^2}.
\end{align}
For any holomorphic chart $\varphi: U\subset \surface \to V \subset \cpx$
and any geodesically convex neighbourhood $D$ such that $\overline D
\subset U$, there is a constant $C > 0$
such that, for all $z_1, z_2 \in \varphi(D)$, $z_1 \neq z_2$,
\begin{align}
\abs{\del_{z_1}b_1(z_1, z_2)} \leq \frac{C}{\abs{z_1 - z_2}},
\end{align}
where the coefficient $b_1$ appearing in the metric
of $\moduli^{1,1}(\surface)$ is defined as in (<ref>).
If $z_1, z_2 \in \varphi(D)$, there is a smooth
function $\tilde v: \varphi(\overline D) \times \varphi(\overline D) \times
\varphi(\overline D) \to \reals$, such that
all triples $z, z_1, z_2$ of points in the domain with $z_1 \neq z_2$,
\begin{align}
v(\varphi^{-1}(z)) = \log\,\abs{z - z_1}^2 - \log\,\abs{z - z_2}^2+ \tilde
v(z, z_1, z_2).
\end{align}
\begin{align}
b_1(z_1, z_2) &= 2\eval{\bar\del}{z=z_1}(h(\varphi^{-1}(z))
- \log\,\abs{z - z_1}^2)\nonumber\\
&= 2\eval{\bar\del}{z = z_1}(\tilde h(\varphi^{-1}(z)) - \log\,\abs{z -
z_2}^2 +
\tilde v(z,z_1,z_2))\nonumber\\
&= 2\,\bar\del_z\tilde h(\varphi^{-1}(z_1); \varphi^{-1}(z_1),
\varphi^{-1}(z_2)) - \frac{2}{\bar z_1 - \bar
z_2} + 2\,\bar\del_z\tilde v(z_1, z_1, z_2),
\end{align}
where $\bar\del_z$ refers to complex derivatives with respect to the first
entry. In the following calculation we denote $\tilde h(\varphi^{-1}(z_1);
\varphi^{-1}(z_1), \varphi^{-1}(z_2))$ by $\tilde h$ and $\tilde v(z_1,
z_1, z_2)$ by $\tilde v$, whence,
\begin{align}
\del_{z_1}b_1 &= 2 \pbrk{
\del_z\bar\del_z\tilde h
+{\del_{z_1}}\bar\del_z\tilde h
+\del_{z}\bar\del_z \tilde v
+\del_{z_1}\bar\del_z \tilde v}\nonumber\\
&= 2 \pbrk{ -\frac{e^{\Lambda(z_1)}}{2}\laplacian_{\surface}\tilde h
+ \bar\del_z \del_{z_1} \tilde h
+\del_{z}\bar\del_z \tilde v
+\del_{z_1}\bar\del_z \tilde v}\nonumber\\
&= 2\pbrk{\frac{e^{\Lambda(z_1)}}{2}\Fstable(h)
+ \bar\del_z \del_{z_1} \tilde h
+\del_{z}\bar\del_z \tilde v
+\del_{z_1}\bar\del_z \tilde v}.
\end{align}
Since $\varphi(\overline D)$ is compact, $\Lambda(z_1)$ and the last two
terms are
bounded functions on $\varphi(D)$ by continuity. Since function $F(t)$ is
bounded, we conclude the same statement for the first term. For the second
term, if $x = \varphi^{-1}(z)$ and $x_j =
\varphi^{-1}(z_j)$, we
have by lemma <ref> and
proposition <ref>,
\begin{align}
\abs*{\bar\del_z \del_{z_1} \tilde h} &=
e^{\Lambda(z_1)/2} \abs*{\bar\del_z \del_{z_1} \tilde
h} \abs*{dz}\nonumber\\
&= \abs{\bar \partial_x\partial_{z_1}\tilde h(x_1, \varphi^{-1}(z_1),
\varphi^{-1}(z_2))}\nonumber\\
&\leq \frac{C}{\dist(x_1,x_2)}\nonumber\\
&\leq \frac{C}{\abs{z_1 - z_2}}.
\end{align}
Therefore the lemma is proved.
The moduli space is incomplete. There is a
Cauchy sequence $\set{\vb x_n} \subset \moduli^{1,1}(\surface)$
such that $\vb x_n \to \vb x \in \diag$ as a sequence in
$\surface\times \surface$.
Let $\varphi: U\subset \surface \to \cpx$ be an holomorphic chart defined on
an open and dense neighbourhood $U$.
Let $z_1 \in \cpx$ be chosen such that $\varphi^{-1}(s z_1)$, $0 \leq s
\leq
1$ is contained in a geodesically convex neighbourhood of
Let us define the curve,
\begin{align}
\gamma: (0, 1] \to \cpx^2_\diag, &&
\gamma(s) = (s\,z_1, 0).
\end{align}
Let $z(s) = s z_1$ and let
$\varphi^{-1}_*\gamma(s) = (\varphi^{-1}(z(s)), \varphi^{-1}(0))$,
be the push forward of the curve $\gamma$ to the moduli space, hence,
\begin{align}
\abs{\varphi^{-1}_*\dot\gamma}_\moduli^2 = (e^{\Lambda(z)}(1-\tau) +
\del_{z_1}b_1)\,\abs{z_1}^2,
\end{align}
where we denote by $\abs{\cdot}_\moduli$
the norm of vectors in $T_{\varphi^{-1}_*\gamma(s)}\moduli^{1,1}$.
By Lemma <ref> there is a constant $C$,
such that,
\begin{align}
\abs{\del_{z_1}b_1} \leq \frac{C}{\abs z} = \frac{C}{s\,\abs {z_1}}.
\end{align}
Since the conformal factor is a continuous positive function
defined on the whole plane, there is another constant, also
denoted $C$, such that,
\begin{align}
\abs {\varphi^{-1}_*\dot \gamma}_\moduli \leq \frac{C}{s^{1/2}}.
\end{align}
Let $\ell[\gamma, a, b]$ be the arc-length of the segment
$\gamma|_{[a, b]}$, $a, b \in (0, 1)$, there is another constant,
also denoted by $C$, such that,
\begin{align}
\ell[\gamma, a, b] = \int_a^b\abs{\varphi^{-1}_*\dot \gamma}_\moduli\,ds
\leq C (b^{1/2} - a^{1/2}),
\end{align}
\begin{align}
\dist(\varphi^{-1}_* \gamma(b), \varphi^{-1}_* \gamma(a)) \leq
C\,(b^{1/2} - a^{1/2}).
\end{align}
This inequality shows if $\set{s_n} \subset (0, 1]$ is any
converging sequence $s_n \to 0$, the new sequence,
\begin{align}
\vb x_n = \varphi^{-1}_*\gamma(s_n) \in \moduli^{1,1}(\surface),
\end{align}
is Cauchy, however $\gamma$ is continuous which
implies $\vb x_n \to (\varphi^{-1}(0), \varphi^{-1}(0)) \in
\diag_\surface$.
Therefore, the moduli space
is incomplete.
§ THE VOLUME OF THE MODULI SPACE
We conclude this chapter computing the volume of the moduli space
$\moduli^{1,1}(\surface)$ for the
round sphere and flat tori. As it will turn out, the existence of a Lie
group of isometries will play an important role in the
calculations. Symmetries were studied for their relation to conservation laws
in a Schrodinger-Chern-Simons model by Manton and Nasir
in [Manton and Nasir, 1999], for the Riemann sphere, symmetries of the coefficients
the $\Lsp^2$ metric for vortices of a non-relativistic Chern-Simons model
were treated by Romão [Romao, 2001]. We follow similar
ideas for asymmetric vortices of the $O(3)$ Sigma model. There is a
general conjecture for the volume of the moduli space by
Romão-Speight [Romão and Speight, 2020], which can be stated as follows,
Given a compact Riemann surface $\surface$ of genus $g$ and total area
$\abs\surface$, let,
\begin{align*}
J_\pm &= 2\pi (1 \mp \tau)\abs\surface - 4\pi^2(k_\pm - k_\mp),\\
K_\pm &= \mp 2\pi^2,
\end{align*}
then the total volume of the moduli space $\moduli^{k_+, k_-}(\surface)$ is,
\begin{equation*}
\vol(\moduli^{k_+, k_-}(\surface)) =
\sum_{l = 0}^g \frac{g!(g - l)!}{(-1)^l l!} \prod_{\sigma = \pm}
\sum_{j_\sigma = l}^g
\frac{
(2\pi)^{2l} J_\sigma^{k_\sigma - j_\sigma} K_\sigma^{j_\sigma - l}
(j_\sigma - l)! (g - j_\sigma)! (k_\sigma - j_\sigma)!
\end{equation*}
For $\surface = \sphere_{round}$, they corroborated
it for a
vortex-antivortex pair and $\tau = 0$. We aim to confirm the
conjecture on the round sphere and flat tori for vortex-antivortex
pairs and general $\tau$.
§.§ The Riemann sphere
On the round sphere, the three dimensional Lie group of orthogonal
transformations, $O(3)$, acts by
isometries. The vortex equations are invariant under
isometric actions on the domain, if $\isometry: \surface \to
\surface$ is an isometry and $u$ is the solution of the Taubes equation
with vortex set $\vset$ and antivortex set $\avset$, then $u \circ
\isometry$ is the solution with data $\isometry^{-1}(\vset)$,
$\isometry^{-1}(\avset)$. We will make use of this symmetry to obtain
conservation laws for the non-trivial coefficients $b_j$ and an explicit
formula in the subspace of vortices and
antivortices located at antipodal positions. This formula will lead us
to the volume formula. We will prove the following theorem,
Recall the conformal factor of the sphere of
radius $R$ in a stereographic projection chart with coordinate $z$ is,
\begin{align}
\cf=\frac{4R^{2}}{\left(1+\abs z^{2}\right)^{2}}.
\end{align}
We can give an explicit description of the coefficients in the metric
in the case of only $k_{+}$ coincident vortices or $k_{-}$ coincident
antivortices. By rotational symmetry,
the function $u$ depends only on the chordal distance to
either the vortex or antivortex [Manton and Nasir, 1999],
the coefficients $b_{\pm}$ in this case are,
\begin{equation}
b_{\pm} = -\frac{ 2 k_{\pm}z_{\pm} }{ 1 +
\abs{z_{\pm}}^{2}}.\label{eq:bpm-antipodal-position}
\end{equation}
The proof relies on the rotational symmetry
of the configuration and is analogous to the proof for $n$ coincident
Ginzburg-Landau vortices on the sphere that can be found in
[Manton, 1993]. With this identity at hand, we prove the
following theorem,
The volume of the moduli space $\moduli^{k_{+},0}\left( \sphere
\right)$ is,
\begin{equation}
\label{eq:vol-moduli-k-0}
\vol \left( \moduli^{k_{+},0}(\sphere) \right)
\end{equation}
and the volume of $\moduli^{0, k_-}(\sphere)$ can be obtained from
equation (<ref>) by changing $\tau$ into
$-\tau$. For a vortex-antivortex pair, the volume of $\moduli^{1,1}\left(
\sphere\right)$ is
\begin{align}
\vol \left( \moduli^{1,1}(\sphere) \right)
\end{align}
For $k_+ = 0$ or $k_- = 0$ we follow ideas of
Manton-Nasir [Manton and Nasir, 1999], as their proof relies on the topology
of the symmetric product $(\sphere)^N/S_N$, $S_N$ being the $N$ symmetric
group, and can be adapted easily to vortices of the $O(3)$ Sigma model of the
same type. For the case $k_+ = k_- = 1$, we extend the proof given
by Romão-Speight <cit.> for the symmetric case. For
general $\tau$ we no longer have the symmetry $(z_1, z_2) \mapsto (z_2, z_1)$,
instead, we complement the symmetries induced by $SO(3)$ in the moduli space
the symmetry $(z_1, z_2) \mapsto (\conj z_1, \conj z_2)$ to deduce a suitable
formula for the volume of a general Kähler metric on $\sphere\!_{\diag}$.
§.§.§ $k_+$ vortices of the same type
If there are $k_+$ vortices on $\sphere$ and no antivortices, the moduli
space is isomorphic to $\pSpace^{n}$, the complex projective space of
dimension $k_+$ [Manton and Nasir, 1999]. The subspace
$\moduli_0^{k_+,0}(\sphere) \subset \moduli^{k_+,0}(\sphere)$ of $k_+$
coincident vortices on the other hand is isomorphic to $\pSpace^1$,
and can be parametrized with the coordinate $z_+$ of the coincident
vortices. By equation (<ref>) we know how
to compute the coefficient $b_{+}$ in $\moduli_0^{k_+,0}(\sphere)$,
\begin{equation}
b_{+}=-\frac{2 k_+ z_+}{1+\abs {z_+}^2}.\label{eq:b-formula-k-0}
\end{equation}
The metric in $\moduli^{k_+,0}_0(\sphere)$ therefore is,
\begin{align}
ds^{2} & =2 k_+\pi\left((1-\tau)\cf + \frac{\del b_{+}}{\del
z_+}\right)\abs{dz_+}^{2}\nonumber \\
& = k_+
\pi\left(2(1-\tau) - \frac{k_+}{R^2}
\right)\Omega\abs{dz_+}^{2},\label{eq:metric-moduli-kp-0}
\end{align}
as can be seen, the metric is a multiple of the round metric,
hence, the volume of $\moduli^{k_+,0}_0(\sphere)$ is,
\begin{equation}
4\pi^{2} R^{2}k_+ \left( 2 (1-\tau)-\frac{k_+}{R^{2}}
\right),\label{eq:vol-moduli-N}
\end{equation}
this volume is $k_+$ times the volume of the generating cycle in
\begin{equation}
4\pi^{2}R^{2}\left( 2 (1-\tau)-\frac{k_+}{R^{2}}
\right).\label{eq:generating-cycle-vol}
\end{equation}
The total volume of the moduli space therefore is,
\begin{equation}
\vol\left(\moduli^{k_+,0}(\sphere)\right) =
\frac{\left(8\pi^{2}R^{2}(1-\tau)-4\pi^{2} k_+
\right)^{k_+}}{k_+!},\label{eq:total-volume-moduli-space}
\end{equation}
the proof of the volume formula in $\moduli^{0,k_-}(\sphere)$ is analogous,
\begin{equation}
\vol\left(\moduli^{0,k_-}(\sphere)\right) =
\frac{\left(8\pi^{2}R^{2}(1+\tau)-4\pi^{2} k_-
\right)^{k_-}}{k_-!}.%\label{eq:total-volume-moduli-space}
\end{equation}
§.§.§ The moduli space of vortex-antivortex pairs
In general, there is no explicit expression for the
coefficients $b_j$ of the metric if the cores are at general position, however,
can deduce from the invariance of
the Taubes equation under the action of $O(3)$ several constraints on the
coefficients due to symmetry. Before doing so, we need a general lemma that
will also be necessary for flat tori in the next section.
Let $\varphi: U \subset \surface \to V\subset \cpx$ be a
holomorphic chart, containing the core set $\mathcal{Z}$ of a
point in the
moduli space $\moduli^{1,1}(\surface)$.
For any bounded domain $D \subset V$, such that $\mathcal{Z}
\subset \varphi^{-1}(D)$, there are continuous functions $\tilde b_j:
D\times D \to
\cpx$,
$j = 1, 2$,
such that:
* If $\varphi(\mathcal{Z}) = \left\{z_1, z_2\right\}$, where
$z_1$ $(z_2)$ is the vortex (antivortex),
\begin{align}
b_j(z_1, z_2) = \frac{-2\,\sign_j}{\bar z_1 - \bar z_2} + \tilde
b_j(z_1, z_2),
\end{align}
where $b_j$, $j = 1, 2$, are the non-trivial coefficients in the
metric, defined in lemma <ref>.
\begin{align}
\lim_{|z_1 - z_2| \to 0} \tilde b_j(z_1, z_2) = 0.
\end{align}
On $\varphi^{-1}(D)$, Green's function
can be written as
\begin{align}
G(x_1, x_2) = \frac{1}{2\pi}\,\log\,\abs{\varphi(x_1) - \varphi(x_2)} +
\tilde G(x_1,
\end{align}
with a smooth regular part $\tilde G: \varphi^{-1}(D)\times
\varphi^{-1}(D) \to \reals$. Therefore, the solution $h$ to the Taubes
equation can be written as
\begin{align}
h(x; x_1, x_2) = \tilde h(x; x_1, x_2) + \log\,\abs{\varphi(x) -
\varphi(x_1)}^2 - \log\,\abs{\varphi(x) - \varphi(x_2)}^2 + \tilde v(x;
x_1, x_2),
\end{align}
\begin{align}\label{eq:tilde-v-vav-sigma}
\tilde v(x; x_1, x_2) = 4\pi\, \tilde G(x, x_1)
- 4\pi\, \tilde G(x, x_2),
\end{align}
and $\tilde h(x; x_1, x_2)$ can be extended in $C^1$ to the coincidence set
$x_1 = x_2$ by corollary <ref>.
Denoting $h(\varphi^{-1}(z); \varphi^{-1}(z_1),
\varphi^{-1}(z_2))$ and $\tilde h(\varphi^{-1}(z); \varphi^{-1}(z_1),
\varphi^{-1}(z_2))$ as $h$, $\tilde h$, etcetera,
\begin{align}
b_j(z_1, z_2) &= 2\,\conj\del|_{z=z_j}
\pbrk{\sign_j\,h - \log\abs{z - z_j}}\nonumber\\
&= 2\,\conj\del_{z=z_j}
\pbrk{\sign_j\tilde h - \log\abs{z - z_k} + \sign_j\tilde v}\nonumber\\
&= \frac{-2}{\conj z_j - \conj z_k}
+ 2\,\sign_j\conj\del|_{z=z_j}(\tilde h + \tilde v)\nonumber\\
&= \frac{-2\,\sign_j}{\conj z_1 - \conj z_2} + \tilde b_j,
\end{align}
where the regular part $\tilde b_j$ is continuous in $D\times D$. This
proves the first statement. The second statement is a consequence of
corollary <ref> and the fact that
by (<ref>),
\begin{align}
\lim_{|z_1 - z_2| \to 0} \conj\del\lvert_{z=z_j}\pbrk{\tilde
\varphi^{-1}(z_1), \varphi^{-1}(z_2))} = 0.
\end{align}
Suppose $\gamma:U_{1} \subset \cpx \to U_{2}
\subset \cpx$ is
a holomorphic change of coordinates in ambient space, such that
$z_{k}\in U_{1}$
for all cores. There are pairs of corresponding
coefficients $b_{s}(z_{1},\ldots,z_{n})$,
in each of the charts. Let $z'=\gamma(z)$,
$z'_{k}=\gamma(z_{k})$, as in [Romao, 2001], we have the
transformation rule
\begin{equation}
b'_j = \frac{1}{\conj{\gamma'_j}}b_j-\frac{\conj{\gamma''_j}} {\left(
\conj{ \gamma'_j}\right)^{2}}.\label{eq:b-coeffs-change-of-coords}
\end{equation}
Manton and Nasir noted in [Manton and Nasir, 1999] that
equation (<ref>) is similar to the transformation
rule for the Levi-Civita connection on $\sphere$ and resembles the
topological nature of the coefficients $b_j$. In the sphere, the group of
isometries is large, in the sense that it is a Lie group, and each of this
isometries induces a holomorphic change of coordinates on the moduli space.
We exploit this remark to prove the following lemmas.
In the projective chart, the coefficients $b_j$ satisfy the identities,
\begin{gather}
\sum_k(2\,\conj z_k + \conj z_k^2\,b_k + \conj
b_k) = C,\label{eq:bs-gen-rel}\\
\sum_k\conj z_kb_k\in\reals,\label{eq:z-axis-symmetry-b}
\end{gather}
for some constant $C$. For a vortex-antivortex pair, $C = 0$.
Romão deduced similar identities for vortices of a modified Chern-Simons
on the sphere in [Romao, 2001], employing the action of $SO(3)$ on
the moduli space.
Let $\mathbb{S}^2_\Delta$ be the diagonal in the product $\sphere
\times \sphere$. The orthogonal group acts diagonally on the moduli space
$\moduli^{1,1}(\sphere) \cong (\sphere \times \sphere)
\setminus \mathbb{S}^2_\Delta$ by isometries. We can always assume
there is a projective chart such that the pair is located with the
vortex at $z_1 = \epsilon$ and the antivortex at $z_2 =
-\epsilon$. From (<ref>) and the fact that
\begin{equation}
b_{j}(\epsilon, -\epsilon) =
\conj{b_j}(\epsilon, -\epsilon),\label{eq:refl-invariance}
\end{equation}
we conclude,
\begin{equation}
b_{1}(\epsilon,-\epsilon) +
\end{equation}
The $\Lsp^2$ metric in $\moduli^{1,1}(\sphere)$ is Kähler and invariant
under the diagonal action of $O(3)$, given any
pair $(z_1, z_2) \in \moduli^{1,1}(\sphere)$, we can always find a
rotation of $\sphere$ such that in south pole stereographic
projection, $z_1 = \epsilon$, $z_2 = -\epsilon$. In this way, we have
a diffeomorphism,
\begin{equation}
(\sphere \times \sphere) \setminus \mathbb{S}^2_\Delta \cong (0, 1]
\times SO(3),
\label{eq:moduli11-so3-param}
\end{equation}
hence, the moduli space can be parametrized as $(0, 1]\times SO(3)$.
Let $g$ be a Kähler metric in $\sphere\times\sphere$ such that if $o
\in O(3)$ and $(z_1, z_2) \in \sphere\times\sphere$, then the action
\begin{equation}
o * (z_1, z_2) = (o * z_1, o * z_2),
\end{equation}
is by isometries. Let $E_0 = \del_\epsilon$ and let $E_j \in
\mathfrak{so}(3)$ be the left invariant vector field corresponding to
rotations with respect to the $j$-th coordinate axis in $\reals^3$.
Then there exists a function
\begin{equation}
A : (0, 1] \to \reals,
\end{equation}
and a real constant $c$ such that in the parametrization
\begin{multline}
\label{eq:so3-diag-inv-metric}
g = A\left(\frac{1-\epsilon^{2}}{1+\epsilon^{2}}\,(\sigma^{1})^{2} +
\frac{1+\epsilon^{2}}{1-\epsilon^{2}}(\sigma^{2})^{2}\right) -
\frac{1}{\epsilon}
\frac{dA}{d\epsilon}
\left((\sigma^{0})^{2}+\epsilon^{2}(\sigma^{3})^{2}\right)\\
+ \frac{c}{1+\epsilon^{2}}\left(\sigma^{0}\sigma^{2}
+ \frac{\epsilon(1-\epsilon^{2})}{1+\epsilon^{2}} \sigma^{1}
\sigma^{3}\right),
\end{multline}
where $\sigma^k\in T^*((0,1]\times SO(3))$ is the co-vector dual to
$E_k$, $k = 0, \ldots, 3$. For this metric, the volume is,
\begin{equation}
\label{eq:vol-formula}
\vol\left(\sphere\times\sphere\right) =
\end{equation}
Applying lemma <ref> to the $\Lsp^2$ metric,
we obtain,
The $\Lsp^2$ metric on $\moduli^{1,1}(\sphere)$ has the structure
provided by Lemma <ref>, with
\begin{gather}
A = 2\pi \left(
\frac{4R^2}{1 + \epsilon^2} - \epsilon\,b_1 - 2R^2 - 1
\right),
\label{eq:A-formula}\\
c = 8\pi R^2\tau.
\label{eq:c-formula}
\end{gather}
To compute the constant $c$, we calculate
$g\left(E_{0},E_{2}\right)$. Tangent vectors $E_{0}$, $E_{2}$ in
projective coordinates $(z_1, z_2) \in \sphere \times \sphere$ with
respect to the south pole are,
\begin{equation}
E_{0} = \frac{\del}{\del x_1} - \frac{\del}{\del x_{2}}, \qquad
E_{2} = \frac{1+\epsilon^2}{2}\,\left( \frac{\del}{\del x_{1}} +
\frac{\del}{\del x_2} \right).\label{eq:E0-E2-proj}
\end{equation}
where $z_k = x_k + i y_k$. Thence,
\begin{align}
& =\frac{1+\epsilon^{2}}{2}g\left(\frac{\del}{\del
x_{1}}-\frac{\del}{\del x_{2}},\frac{\del}{\del
x_{1}}+\frac{\del}{\del x_{2}}\right)\nonumber \\
& =\frac{1+\epsilon^{2}}{2}2\pi\left(\cf(1+\tau)-\cf(1-\tau)+\frac{\del
b_{1}}{\del z_{1}}+\frac{\del b_{2}}{\del z_{1}}-\frac{\del
b_{1}}{\del z_{2}}-\frac{\del b_{2}}{\del
\end{align}
To simplify (<ref>), we use the symmetries of the
coefficients $b_j$, lemma <ref>,
\begin{align}
\sum_j\left(\pdv{z_1}{b_j} - \pdv{z_2}{b_j}\right) &=
\frac{1}{2} \sum_j{\dv\epsilon}b_j(\epsilon,-\epsilon) -
\frac{i}{2}\sum_j
\left(\pdv{y_1}{b_j} - \pdv{y_2}{b_j}\right)\nonumber\\
&=\frac{1}{2}{\dv \epsilon} (b_1 + b_2) + \frac{1}{2\epsilon}
(b_1 + b_2)\nonumber\\
&= 0.
\end{align}
\begin{equation}
\label{eq:gl2-e0-e2}
\moduliMetric(E_0, E_2) = \frac{8\pi R^2\tau}{1 + \epsilon^2}
\end{equation}
and consequently $c = 8\pi R^2\tau$. Let us compute
$\moduliMetric(E_0, E_0)$,
\begin{align}
\nonumber
\moduliMetric(E_0, E_0) &= \moduliMetric \left( \pdv{x_1} -
\pdv{x_2}, \pdv{x_1} -\pdv{x_2} \right)\\
&= 2 \pi \left( \cf (1 + \tau) + \cf (1 - \tau)
+ \pdv{z_1}{b_1} - \pdv{z_1}{b_2}
- \pdv{z_2}{b_1} + \pdv{z_2}{b_2} \right).
\end{align}
Again by symmetry,
\begin{equation}
\label{eq:db1-db2-prop-at-qeps}
\pdv{z_1}{b_j} - \pdv{z_2}{b_j} = \frac{1}{2}
\frac{d b_j}{d\epsilon}
\end{equation}
\begin{equation}
\label{eq:gl2-e0-e0}
\moduliMetric(E_0, E_0) = 2\pi \left( \frac{8R^2}{(1 +
\epsilon^2)^2} +
\frac{d b_1}{d\epsilon}
+ \frac{1}{\epsilon}b_1 \right).
\end{equation}
Comparing (<ref>) and (<ref>),
\begin{equation}
\label{eq:A-diff-eq}
- \frac{1}{\epsilon} \frac{d A}{d\epsilon} = 2\pi \left(
\frac{8R^2}{(1 + \epsilon^2)^2} + \frac{db_1}{d\epsilon}
+ \frac{1}{\epsilon}b_1 \right),
\end{equation}
Solving this equation, we find,
\begin{equation}
\label{eq:A-diff-eq-sol}
A = \frac{8\pi R^2}{1 + \epsilon^2} - 2\pi\epsilon b_1 +
\mathrm{const}.
\end{equation}
From the regularity condition $\lim_{\epsilon\to 1}A(\epsilon) = 0$ used
to compute the formula for the volume of the moduli space and the
explicit formula (<ref>) for $b_1$ in the
antipodal case, the constant is
\begin{equation}
\label{eq:A-sol-const}
\mathrm{const}. = -2\pi (2R^2 + 1).
\end{equation}
\begin{equation}
A = 2\pi \left( \frac{4R^2}{1 + \epsilon^2} - \epsilon b_1 - 2R^2 -
\end{equation}
We claim that
\begin{equation}
\label{eq:lim-eps-b1}
\lim_{\epsilon\to 0} \epsilon b_1 = -1
\end{equation}
as can be seen numerically in figure <ref> for the
symmetric case in the unit sphere.
For a vortex-antivortex pair,
\begin{equation}
b_1(\epsilon, -\epsilon) = 2 \eval{\frac{\del}{\del
x}}{z=\epsilon} h_{\epsilon} - \frac{1}{\epsilon}.
\label{eq:vav-b1}
\end{equation}
Since $h_{\epsilon} \to \mu$ in $C^1$ as $\epsilon\to 0$,
\begin{equation}
\lim_{\epsilon\to 0} \epsilon\,b_1(\epsilon, -\epsilon) = -1.
\label{eq:the-limit-conclusion}
\end{equation}
Applying lemmas <ref>
and <ref>, the volume of the moduli space is
\begin{equation}\label{eq:volume-conjecture-sphere-formula}
\vol \left( \moduli^{1,1}(\sphere) \right) = \left( 8\pi^2
R^2\right)^2 (1 - \tau^2).
\end{equation}
Notice that another way to express the volume is as $4\pi^2(1-\tau^2)\vol
(\sphere)$, which corresponds to the volume of a product of spheres,
each factor weighted by $2\pi (1\pm\tau)$, the effective mass of a core,
hence, it is expected that as $\tau \to \pm 1$, the volume vanishes,
because of the negligible weight of one of the factors.
Three views of the declination data of $\tilde h_{\epsilon}$,
the regular part of the solution to the Taubes equation, for
three different values of the asymmetry parameter $\tau$ on the
unit sphere.
Top. Vortex and
antivortex are symmetric, with the same effective
mass. Middle and bottom. The antivortex becomes more
massive. We solved
from $\epsilon = 1$ down to $0.05$ in steps of
$0.05$, except that for $\tau = 0.5$, the computation
stopped at $\epsilon = .20$ due to algorithm divergence.
As $\epsilon \to 0$ the data shows how $\tilde h_\epsilon$ flattens
as expected.
Top. Real profile of $\epsilon b$ in the symmetric
case. The limit $\lim_{\epsilon\to 0} \epsilon b = -1$ is apparent
in the numerical data. Bottom. Real profile of a
vortex-antivortex pair located at $\pm \epsilon$ on the real axis
of the extended complex plane for several values of
$\epsilon$. In both cases, the domain is the unit sphere, the bottom
plot shows the behaviour of the real profile of
$\tilde h$ as $\epsilon \to 0$ in the south pole of the domain. The dashed
horizontal line is $\log \left( (1 - \tau)(1 +
\tau)^{-1} \right)$. The data shows how the
regular part of the solution to the Taubes equation converges to this
constant value as the pair collides at the north pole.
§.§ Flat tori
In this section we compute the volume of the moduli space for a flat
tori, to
this end, we extend the coefficients $b_q$ in
the $\Lsp^2$ metric to a global object and relate it to the volume of
$\moduli^{1,1}(\mathbb{T}^2)$ in lemma <ref>.
Consider a holomorphic chart $\varphi: U \subset \mathbb{T}^2 \to \cpx$
on an open and dense set $U$, with coordinates $z = \varphi(x)$, $x \in U$. Let
us define,
\begin{align}
b_U = b_j\,d\bar z^j \in \Omega^{(0,1)}((U \times U) \setminus \diag_U).
\end{align}
In general $b_U$ is only well defined on a chart, however, flat tori admit
atlases such that the holomorphic changes of coordinates are
translations. Since
translations have trivial second derivatives, by
(<ref>) $b_U$ extends to a global form $b \in
\Omega^{(0,1)}(\moduli^{1,1}(\mathbb{T}^2))$. By the symmetries of the
coefficients $b_j$, this form is holomorphic, as the following short
calculation shows in coordinates:
\begin{align}
\bar\del b_U &= \sum_{i,j} \bar\del_{z_i}b_j\,d\bar z^i\wedge d\bar
&= -\sum_{i,j} \bar\del_{z_j}b_i\,d\bar z^j\wedge d\bar z^i\nonumber\\
&= -\bar\del b_U,
\end{align}
hence, $\bar \del b_U = 0$.
To compute the volume of flat tori, we will use the $(1,1)$-form $\del b$ to
define another form in the moduli space which is more convenient
for calculations. Let $\proj_j:\mathbb{T}^2\times \mathbb{T}^2\to \mathbb{T}^2$
be the canonical projection map onto the $j$-th factor of the product. Let
us define the form
\begin{align}
\kform_0 = 2\pi\,(1-\tau)\,\proj_1^{*}\,\kform_{\mathbb{T}^2} + 2\pi\,(1 +
\tau)\, \proj_2^{*}\,\kform_{\mathbb{T}^2}.
\end{align}
The Kähler form on the moduli space can be written as,
\begin{align}
\kform &= \kform_0 + \pi i\,\del b \in
\Lambda^{1,1}(\moduli^{1,1}(\mathbb{T}^2)).
\end{align}
Notice that,
\begin{align}
\vform &= \half\,\kform\wedge\kform\nonumber\\
&= \vform_0 + \pi i\,\kform_0\wedge \del b -
\frac{\pi^2}{2} \del b \wedge \del b,
\end{align}
where $\vform_0 = \half \kform_0\wedge\kform_0$ is the restriction of
the volume form in the product $\mathbb{T}^2\times\mathbb{T}^2$ to the moduli
Let $\diag_\epsilon$ be the $\epsilon$-tubular neighbourhood of
the diagonal set of $\mathbb{T}^2\times\mathbb{T}^2$ for small
$\epsilon$. The volume of the moduli space can be computed as,
\begin{multline}
\Volume(\moduli^{1,1}(\mathbb{T}^2)) =
4\pi^2 (1 - \tau^2)\,\Volume(\mathbb{T}^2)^2 \\
+ \lim_{\epsilon\to 0}\int_{\mathbb{T}^2 \times \mathbb{T}^2
\setminus \diag_\epsilon}
\pbrk{\pi i\, \kform_0\wedge \del b
- \frac{\pi^2}{2}
\del b \wedge \del b}.\label{eq:vol-int-lim}
\end{multline}
\begin{align}
\Volume(\moduli^{1,1}(\mathbb{T}^2))
&= \lim_{\epsilon\to 0}
\int_{\mathbb{T}^2 \times \mathbb{T}^2 \setminus \diag_\epsilon}
\vform\nonumber\\
&= \int_{\mathbb{T}^2\times\mathbb{T}^2}\vform_0
+ \lim_{\epsilon\to 0}\int_{\mathbb{T}^2 \times
\mathbb{T}^2 \setminus \diag_\epsilon}
\pbrk{\pi i\,\kform_0\wedge \del b
- \frac{\pi^2}{2} \del b \wedge \del b}.
\end{align}
On the other hand,
\begin{align}
\vform_0 = 4\pi^2(1 - \tau^2)\,\proj_1^{*}\,\kform_{\mathbb{T}^2} \wedge
\proj_2^{*}\,\kform_{\mathbb{T}^2}.
\end{align}
Applying Fubini and the change of variables theorems,
\begin{align}
\int_{\mathbb{T}^2\times\mathbb{T}^2}\vform_0
&= 4\pi^2(1-\tau^2)\pbrk{\int_{\mathbb{T}^2}\kform_{\mathbb{T}^2}}^2
= 4\pi^2(1-\tau^2)\Volume(\mathbb{T}^2)^2.
\end{align}
This concludes the proof of the lemma.
According to lemma <ref>, to compute the volume
of $\moduli^{1,1}(\mathbb{T})$, we must compute the two non-trivial
terms in (<ref>).
Let $\pi: \cpx \to \mathbb{T}^2$ be the canonical covering map and let
$R \subset \cpx$ be an open parallelogram such that $\pi|_R: R \to
\mathbb{T}^2$ is a bi-holomorphism onto its image and $U = \pi|_R(R)$ is
open and dense. On the local coordinates $\pi|_R^{-1}:U \to R$, there is a
$c \in \cpx$ such that for any pair of
different points $z_1, z_2 \in R$,
\begin{align}
b_1(z_1, z_2) + b_2(z_1, z_2) &= c,
\end{align}
If $\mathcal{I} : \mathbb{T}^2 \to \mathbb{T}^2$ is an isometry, the Taubes
equation is invariant under $\mathcal{I}$,
\begin{align}
h(\mathcal{I}(x); \mathcal{I}(x_1), \mathcal{I}(x_2)) = h(x; x_1, x_2),
\end{align}
$x, x_1, x_2 \in \mathbb{T}^2$, $x_1 \neq x_2$.
By construction, there is a $v \in \cpx$ such that
$\mathcal{I}_\varphi = \varphi\circ \mathcal{I} \circ \varphi^{-1}(z) = z +
v$ for $z \in \varphi(\mathcal{I}^{-1}(U)\cap U)$. For small $v$, the
translation $\mathcal{I}_\varphi$ maps a neighbourhood,
not necessarily connected,
$N \subset R$ of $x_1$ and $x_2$ into $R$.
This implies $b_j$ has the symmetries,
\begin{align}
b_j(z_1 + v, z_2 + v) = b_j(z_1, z_2),
\end{align}
$v$ small. Hence,
\begin{align}
\del_{z_1}b_j + \del_{z_2}b_j = \bar\del_{z_1}b_j + \bar\del_{z_2}b_j = 0.
\end{align}
Applying the symmetries of the coefficients $b_j$,
\begin{align}
\del_{z_j}(b_1 + b_2) = \bar\del_{z_1}\bar b_j + \bar\del_{z_2}\bar b_j = 0.
\end{align}
\begin{align}
\bar\del_{z_j}(b_1 + b_2) = 0.
\end{align}
Hence $b_1 + b_2$ is constant on the connected neighbourhood $R$.
In a flat torus $\mathbb{T}^2$, for the $(1,1)$ form $\del b$ we
\begin{align}
\del b \wedge \del b = 0.
\end{align}
We apply the previous lemma to prove the proposition.
By lemma <ref>, there is an open and dense
set $U \subset \mathbb{T}^2$ and a chart $\varphi: U \to R\subset \cpx$,
$R$ an open parallelogram, such that in this local coordinates $b_1 + b_2$ is
a constant. Denoting points in $R$ as $z_j$, a direct calculation shows,
\begin{align}
b_U \wedge \del b_U
(b_2\,\del_{z_1}b_1 - b_1\,\del_{z_1}b_2)\,
dz_1\wedge d\bar z_1 \wedge d\bar z_2\nonumber\\
&\quad + (-b_2\,\del_{z_2}b_1 + b_1\,\del_{z_2} b_2)\,
d\bar z_1 \wedge dz_2 \wedge d\bar z_2\nonumber\\
-c\,\del_{z_1}b_2\, dz_1\wedge d\bar z_1 \wedge d\bar z_2
-c\,\del_{z_2}b_1\,d\bar z_1 \wedge dz_2 \wedge d\bar z_2.
\end{align}
Since $b_1$ and $b_2$ add to a constant,
\begin{align}
\del b_U \wedge \del b_U
-c\,(\del_{z_2}\del_{z_1} b_2 + \del_{z_1}\del_{z_2}b_1)\,
dz_1 \wedge d\bar z_1 \wedge dz_2 \wedge d\bar z_2 = 0.
\end{align}
Since $U$ is dense, we conclude $\del b \wedge \del b \equiv 0$.
By this proposition and lemma <ref>, to compute the volume
of the moduli space, we have to integrate
$\kform_0\wedge \del b$.
For a flat torus $\mathbb{T}^2$, the volume of the moduli space is,
\begin{align}
\Volume(\moduli^{1,1}(\mathbb{T}^2))
4\pi^2(1 - \tau^2)\,\Volume(\mathbb{T}^2)^2 +
\end{align}
Notice that the first term of the formula is
similar to the case of the
sphere (<ref>), however, the second
term is new, bearing in mind
the volume conjecture, <ref>, one can argue the
extra term is related to the genus of the base surface, however, it
is not clear how to relate our computation to this fact and
the relation is open to
future work.
\begin{align}
\mathbb{T}^2({\epsilon})
&= (\mathbb{T}^2 \times \mathbb{T}^2) \setminus \diag_{\epsilon},\\
\kform_j &= \proj^{*}_j\kform_{\mathbb{T}^2}, \qquad j = 1, 2,
\end{align}
and let $k$ be the complementary index of $j$, such that
$\set{j, k} = \set{1, 2}$. By Fubini's theorem,
\begin{align}
\int_{{\mathbb{T}^2}(\epsilon)}\kform_0\wedge \del b
2\pi \sum_j (1 - \sign_j\tau)\int_{\mathbb{T}^2}\pbrk{
\int_{\mathbb{T}^2 \setminus
\disk_{\epsilon}(x_j)} \inc_k^{*}\del b}\, \kform_{\mathbb{T}^2},
\end{align}
where for any given $x_j \in \mathbb{T}^2$, $\inc_k: \mathbb{T}^2
\hookrightarrow
\mathbb{T}^2\times\mathbb{T}^2$ is the
inclusion of the torus as the k-th factor of the product anchored at $x_j$.
$b$ is well defined globally,
\begin{align}
\int_{\mathbb{T}^2\setminus\disk_{\epsilon}(x_j)} \inc_k^{*}\del b
\int_{\del \mathbb{T}^2\setminus\disk_{\epsilon}(x_j)} \inc_k^{*}b
-\int_{\del \disk_{\epsilon}(x_j)} \inc_k^{*}b,
\end{align}
where we always orient a submanifold by the outward pointing
normal. Let $\varphi: U \to \cpx$ be a holomorphic chart defined on an open
and dense set $U$. If $x_j \in U$, for small $\epsilon$, $\disk_\epsilon(x_j)
\subset U$. Assume $j = 1$, $k = 2$, in the chart,
\begin{align}
(\varphi^{-1})^*\inc_2^{*}b = b_2 d\bar z.
\end{align}
If $z_1 = \varphi(x_1)$ and $D(z_1) \subset \cpx$ is a
bounded domain and neighbourhood of $z_1$, by lemma <ref>,
\begin{align}
b_2 = \frac{2}{\conj z_1 - \conj z_2} + \tilde b_2(z_1, z_2),\qquad
z_2 \in D(z_1).
\end{align}
If $D_\epsilon(z_j) = \varphi(\disk_\epsilon(x_j))$, by Cauchy's residue
\begin{align}
\int_{\del \disk_{\epsilon}(x_1)} \inc_2^{*}b =
-2\int_{\del D_\epsilon(z_1)} \frac{d\conj z}{\conj z - \conj z_1}
+ \int_{\del D_\epsilon(z_1)}\tilde b_2(z_1,z)d\conj z
= 4\pi i + \int_{\del D_\epsilon(z_1)}\tilde b_2(z_1,z)d\conj z.
\end{align}
If $j = 2$, $k = 1$, we find a similar result,
\begin{align}
\int_{\del \disk_{\epsilon}(x_2)} \inc_1^{*}b =
4\pi i + \int_{\del D_\epsilon(z_2)} \tilde b_1(z, z_2) d\conj z.
\end{align}
Since $\tilde b_k$ is a continuous function in a neighbourhood of each
$z_j \in \cpx$,
\begin{align}
\lim_{\epsilon \to 0} \int_{\del D_\epsilon(z_j)} \tilde b_k\,d\conj z = 0.
\end{align}
Hence, since $U$ is dense in $\mathbb{T}^2$,
\begin{align}
lim_{\epsilon\to 0}
\int_{\mathbb{T}^2}\pbrk{
\int_{\mathbb{T}^2 \setminus
\disk_{\epsilon}(x_j)} \inc_k^{*}\del b} \kform_{\mathbb{T}^2}
&= -4\pi i \Volume(\mathbb{T}^2)\nonumber\\
&\quad - \frac{i}{2} \int_{\cpx}
lim_{\epsilon\to 0} \pbrk{\int_{\del D_{\epsilon}(z_j)} \tilde b_k}
e^{\Lambda(z_j)} dz_j\wedge d\conj z_j\nonumber\\
&= -4\pi i \Volume(\mathbb{T}^2).
\end{align}
\begin{align}
\int_{\moduli^{1,1}(\mathbb{T}^2)}\kform_0\wedge\del b
&= 2\pi\sum_j(1 - \sign_j\tau)
lim_{\epsilon\to 0}
\int_{\mathbb{T}^2}\pbrk{
\int_{\mathbb{T}^2 \setminus
\disk_{\epsilon}(x_j)} \inc_k^{*}\del b} \kform_{\mathbb{T}^2}.\nonumber\\
&= 2\pi\sum_j(1 - \sign_j\tau)
\left(-4\pi i \Volume(\mathbb{T}^2) \right)\nonumber\\
&= -16\pi^2\,i\,\Volume(\mathbb{T}^2).
\end{align}
By lemma <ref>, we conclude the volume formula.
CHAPTER: CHERN-SIMONS DEFORMATIONS OF VORTICES
§ DYNAMICS OF THE MODULI SPACE OF GINZBURG-LANDAU VORTICES WITH A
CHERN-SIMONS TERM
⟨#2 ⟩
⟨#2 ⟩
CHAPTER: CONCLUSION
In this work we focused on geometric models of vortices and
antivortices of the $O(3)$ Sigma model. We emphasised the geometric nature of
the interaction of a vortex-antivortex pair on the moduli space.
We were able to prove that the $\Lsp^2$ metric in the moduli space is
both on the euclidean plane and on a compact surface. We also analysed the
properties of the interaction on the plane, focusing on
scattering of vortex-antivortex pairs.
We also computed the volume of the moduli space on spheres and flat tori,
corroborating the work of Speight and Rõmao who conjectured a
formula for the volume of the moduli space for a general surface.
The fact that the moduli space is incomplete imposed some technical
difficulties on the proofs, that we overcame by analysing the
behaviour of solutions to the Taubes equations in the collision of a
vortex and an antivortex.
Finally, we added a Chern-Simons interaction term to our model and
applied the geodesic approximation ideas to determine the extra term in the
metric of the moduli space for small perturbations due to the
interaction. Our analysis indicates that the extra term
can be extended to the coincidence set.
Some questions remain opened, representing an opportunity for future
work. The short range
approximation formula for the metric on the space of vortex-antivortex
pairs of the euclidean plane relies on uniform convergence of the
family $\tilde h_{\epsilon}/\epsilon$, as $\epsilon \to 0$, where
$\tilde h_{\epsilon}$ is the regular part of the Taubes equation.
Numerical evidence
suggests this conjecture is true. Should it be the case, we would be
able to prove formally that the Gaussian curvature of
$\moduli^{1,1}_0(\plane)$ diverges as $\epsilon \to 0$
as expected from the numerical evidence and we could
also justify analytically the effective potential of Ricci magnetic
geodesics. The equivalent conjecture for a compact surface would allow
to compute the volume formula for a general surface, where we no
longer have the extra symmetries that we used for the task.
In conclusion, geometric ideas to study field theory originated in the realm of
superconductivity with the Ginzburg-Landau functional at critical
coupling, but they have proved to be fruitful in a broader context. In
particular, for asymmetric vortex-antivortex systems of the $O(3)$ Sigma model,
where with these ideas one can understand dynamics from a geometric point of
[Alqahtani and Speight, 2015]
LS Alqahtani and JM Speight.
Ricci magnetic geodesic motion of vortices and lumps.
Journal of Geometry and Physics, 98:0 556–574, 2015.
[Aubin, 2013]
Thierry Aubin.
Some nonlinear problems in Riemannian geometry.
Springer Science & Business Media, 2013.
[Bradlow, 1990]
Steven B Bradlow.
Vortices in Holomorphic Line Bundles over Closed Kahler Manifolds.
Commun. Math. Phys, 1350 (1):0 1–17, 1990.
URL <https://projecteuclid.org:443/euclid.cmp/1104201917>.
[Carroll, 2004]
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Addison Wesley, San Francisco, 2004.
ISBN 0805387323.
[Chae and Nam, 2001]
D Chae and Hee Seok Nam.
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maxwell-chern-simmons cp1 model.
Annales de l'Institut Henri Poincare, 0 (2):0
887–906, 2001.
ISSN 03779017.
[Chen and Chern, 2019]
Zhi-You Chen and Jann-Long Chern.
The analysis of solutions for Maxwell–Chern–Simons O(3) sigma
Calculus of Variations and Partial Differential Equations,
580 (4):0 147, jul 2019.
ISSN 1432-0835.
URL <https://doi.org/10.1007/s00526-019-1590-4>.
[Chiacchio and Ricciardi, 2007]
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[Deimling, 2010]
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Courier Corporation, 2010.
[Demoulini and Stuart, 2009]
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Chern-Simons-Schrödinger System.
Communications in Mathematical Physics, 2900
(2):0 597–632, sep 2009.
ISSN 0010-3616, 1432-0916.
[Evans, 2010]
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ISBN 9780821849743.
[Flood and Speight, 2018]
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Journal of Geometry and Physics, 133:0 153–167, 2018.
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URL <https://doi.org/10.1016/j.geomphys.2018.07.009>.
[Gilbarg and Trudinger, 2015]
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springer, 2015.
[Goldstein et al., 2002]
Herbert Goldstein, Charles Poole, and John Safko.
Classical mechanics, 2002.
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Existence of topological multivortex solutions in the self-dual gauge
Proceedings of the Royal Society of Edinburgh Section A:
Mathematics, 1300 (6):0 1293–1309, 2000.
[Han and Lin, 2014]
Jongmin Han and Chang-Shou Lin.
Multiplicity for Self-Dual Condensate Solutions in the
Maxwell-Chern-Simons O(3) Sigma Model.
Communications in Partial Differential Equations, 390
(8):0 1424–1450, 2014.
URL <https://doi.org/10.1080/03605302.2014.908909>.
[Han and Nam, 2005]
Jongmin Han and Hee Seok Nam.
On the topological multivortex solutions of the self-dual
Maxwell-Chern-Simons gauged O(3) sigma model.
Letters in Mathematical Physics, 730 (1):0
17–31, 2005.
ISSN 03779017.
[Han and Song, 2011]
Jongmin Han and Kyungwoo Song.
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Maxwell-Chern-Simons O(3) sigma model.
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420 (10):0 3488–3499, 1990.
[Jaffe and Taubes, 1980]
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Birkhäuser, Basel, 1980.
[Kim and Min, 1992]
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Statistical interactions between chern-simons vortices.
Physics Letters B, 2810 (1):0 81 – 85, 1992.
ISSN 0370-2693.
[Kim and Lee, 1994]
Yoonbai Kim and Kimyeong Lee.
Vortex dynamics in self-dual Chern-Simons-Higgs systems.
Physical Review D, 490 (4):0 2041–2054, feb
URL <http://0.link.aps.org/doi/10.1103/PhysRevD.49.2041>.
[Kim and Lee, 2002]
Yoonbai Kim and Kimyeong Lee.
First and second order vortex dynamics.
Physical Review D, 660 (4):0 045016, aug 2002.
URL <http://0.link.aps.org/doi/10.1103/PhysRevD.66.045016>.
[Kimm et al., 1996]
Kyoungtae Kimm, Kimyeong Lee, and Taejin Lee.
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Phys. Rev. D, 530 (8):0 4436–4440, 1996.
URL <https://link.aps.org/doi/10.1103/PhysRevD.53.4436>.
[Krusch and Speight, 2010]
Steffen Krusch and J Martin Speight.
Exact moduli space metrics for hyperbolic vortex polygons.
Journal of Mathematical Physics, 510 (2):0
022304, 2010.
[Lee et al., 1990]
Choonkyu Lee, Kimyeong Lee, and Hyunsoo Min.
Self-dual Maxwell Chern-Simons solitons.
Physics Letters B, 2520 (1):0 79–83, 1990.
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ISBN 0821806327.
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First Order Vortex Dynamics.
Annals of Physics, 2560 (1):0 114–131, may
ISSN 0003-4916.
[Manton and Nasir, 1999]
N S Manton and S M Nasir.
Conservation laws in a first-order dynamical system of vortices.
Nonlinearity, 120 (4):0 851,
URL <http://stacks.iop.org/0951-7715/12/i=4/a=306>.
[Manton and Nasir, 1999]
N S Manton and S M Nasir.
Volume of vortex moduli spaces.
Communications in mathematical physics, 1990
(3):0 591–604, 1999b.
[Manton and Nasir, 1999]
N S Manton and S M Nasir.
Conservation laws in a first-order dynamical system of vortices.
Nonlinearity, 120 (4):0 851,
URL <http://stacks.iop.org/0951-7715/12/i=4/a=306>.
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N.S. Manton.
Statistical mechanics of vortices.
Nuclear Physics B, 4000 (1):0 624 – 632,
ISSN 0550-3213.
[Manton and Nasir, 1999]
NS Manton and SM Nasir.
Volume of vortex moduli spaces.
Communications in mathematical physics, 1990
(3):0 591–604, 1999d.
[Manton and Speight, 2003]
N.S. Manton and J.M. Speight.
Asymptotic interactions of critically coupled vortices.
Communications in Mathematical Physics, 2360
(3):0 535–555, Jun 2003.
ISSN 1432-0916.
URL <https://doi.org/10.1007/s00220-003-0842-4>.
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Boundary layer methods for lipschitz domains in riemannian manifolds.
Journal of Functional Analysis, 1630 (2):0
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Communications in Partial Differential Equations, 250
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On the condensate multivortex solutions of the self-dual
maxwell-chern-simmons cp1 model.
Annales de l'Institut Henri Poincare, 0 (2):0
887–906, 2001.
ISSN 03779017.
[Chen and Chern, 2019]
Zhi-You Chen and Jann-Long Chern.
The analysis of solutions for Maxwell–Chern–Simons O(3) sigma
Calculus of Variations and Partial Differential Equations,
580 (4):0 147, jul 2019.
ISSN 1432-0835.
URL <https://doi.org/10.1007/s00526-019-1590-4>.
[Chiacchio and Ricciardi, 2007]
Francesco Chiacchio and Tonia Ricciardi.
Multiple vortices for a self-dual CP(1) Maxwell-Chern-Simons model.
Nonlinear Differential Equations and Applications, 130
(5-6):0 563–584, 2007.
ISSN 10219722.
[Chipot, 2011]
Michel Chipot.
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differential equations.
Elsevier, 2011.
[Christ, 1991]
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On the $\bar\partial$ equation in weighted $\mathrm{L}^2$ norms in
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193–230, 1991.
[Collie and Tong, 2008]
Benjamin Collie and David Tong.
Dynamics of Chern-Simons vortices.
Physical Review D - Particles, Fields, Gravitation and
Cosmology, 2008.
ISSN 15507998.
[Deimling, 2010]
Klaus Deimling.
Nonlinear functional analysis.
Courier Corporation, 2010.
[Demoulini and Stuart, 2009]
Sophia Demoulini and David Stuart.
Adiabatic Limit and the Slow Motion of Vortices in a
Chern-Simons-Schrödinger System.
Communications in Mathematical Physics, 2900
(2):0 597–632, sep 2009.
ISSN 0010-3616, 1432-0916.
[Evans, 2010]
L.C. Evans.
Partial Differential Equations.
Graduate studies in mathematics. American Mathematical Society, 2010.
ISBN 9780821849743.
[Flood and Speight, 2018]
S. P. Flood and J. M. Speight.
Chern-Simons deformation of vortices on compact domains.
Journal of Geometry and Physics, 133:0 153–167, 2018.
ISSN 03930440.
URL <https://doi.org/10.1016/j.geomphys.2018.07.009>.
[Gilbarg and Trudinger, 2015]
David Gilbarg and Neil S Trudinger.
Elliptic partial differential equations of second order.
springer, 2015.
[Goldstein et al., 2002]
Herbert Goldstein, Charles Poole, and John Safko.
Classical mechanics, 2002.
[Han, 2000]
Jongmin Han.
Existence of topological multivortex solutions in the self-dual gauge
Proceedings of the Royal Society of Edinburgh Section A:
Mathematics, 1300 (6):0 1293–1309, 2000.
[Han and Lin, 2014]
Jongmin Han and Chang-Shou Lin.
Multiplicity for Self-Dual Condensate Solutions in the
Maxwell-Chern-Simons O(3) Sigma Model.
Communications in Partial Differential Equations, 390
(8):0 1424–1450, 2014.
URL <https://doi.org/10.1080/03605302.2014.908909>.
[Han and Nam, 2005]
Jongmin Han and Hee Seok Nam.
On the topological multivortex solutions of the self-dual
Maxwell-Chern-Simons gauged O(3) sigma model.
Letters in Mathematical Physics, 730 (1):0
17–31, 2005.
ISSN 03779017.
[Han and Song, 2011]
Jongmin Han and Kyungwoo Song.
Existence and asymptotics of topological solutions in the self-dual
Maxwell-Chern-Simons O(3) sigma model.
Journal of Differential Equations, 2500 (1):0
204–222, 2011.
ISSN 00220396.
URL <http://dx.doi.org/10.1016/j.jde.2010.08.003>.
[Hwang, 2003]
AD Hwang.
A symplectic look at surfaces of revolution.
Enseignement Mathematique, pages 1–17, 2003.
[Jackiw et al., 1990]
R Jackiw, Kimyeong Lee, and Erick J. Weinberg.
Self-dual Chern-Simons solitons.
420 (10):0 3488–3499, 1990.
[Jaffe and Taubes, 1980]
A Jaffe and CH Taubes.
Monopoles and Vortices.
Birkhäuser, Basel, 1980.
[Kim and Min, 1992]
Sung Ku Kim and Hyunsoo Min.
Statistical interactions between chern-simons vortices.
Physics Letters B, 2810 (1):0 81 – 85, 1992.
ISSN 0370-2693.
[Kim and Lee, 1994]
Yoonbai Kim and Kimyeong Lee.
Vortex dynamics in self-dual Chern-Simons-Higgs systems.
Physical Review D, 490 (4):0 2041–2054, feb
URL <http://0.link.aps.org/doi/10.1103/PhysRevD.49.2041>.
[Kim and Lee, 2002]
Yoonbai Kim and Kimyeong Lee.
First and second order vortex dynamics.
Physical Review D, 660 (4):0 045016, aug 2002.
URL <http://0.link.aps.org/doi/10.1103/PhysRevD.66.045016>.
[Kimm et al., 1996]
Kyoungtae Kimm, Kimyeong Lee, and Taejin Lee.
Anyonic Bogomol'nyi solitons in a gauged O(3) $\sigma$ model.
Phys. Rev. D, 530 (8):0 4436–4440, 1996.
URL <https://link.aps.org/doi/10.1103/PhysRevD.53.4436>.
[Krusch and Speight, 2010]
Steffen Krusch and J Martin Speight.
Exact moduli space metrics for hyperbolic vortex polygons.
Journal of Mathematical Physics, 510 (2):0
022304, 2010.
[Lee et al., 1990]
Choonkyu Lee, Kimyeong Lee, and Hyunsoo Min.
Self-dual Maxwell Chern-Simons solitons.
Physics Letters B, 2520 (1):0 79–83, 1990.
ISSN 0370-2693.
[Lieb, 1997]
Elliott H. Lieb.
Graduate studies in mathematics, v. 14. American Mathematical
Society, Providence, R.I, 1997.
ISBN 0821806327.
[Manton and Sutcliffe, 2004]
N. Manton and P. Sutcliffe.
Topological Solitons.
Cambridge Monographs on Mathematical Physics. Cambridge
University Press, 2004.
ISBN 978-1-139-45469-8.
[Manton, 1982]
N. S. Manton.
A remark on the scattering of BPS monopoles.
Physics Letters B, 1100 (1):0 54 – 56, 1982.
ISSN 0370-2693.
[Manton, 1997]
N. S. Manton.
First Order Vortex Dynamics.
Annals of Physics, 2560 (1):0 114–131, may
ISSN 0003-4916.
[Manton and Nasir, 1999]
N S Manton and S M Nasir.
Conservation laws in a first-order dynamical system of vortices.
Nonlinearity, 120 (4):0 851,
URL <http://stacks.iop.org/0951-7715/12/i=4/a=306>.
[Manton and Nasir, 1999]
N S Manton and S M Nasir.
Volume of vortex moduli spaces.
Communications in mathematical physics, 1990
(3):0 591–604, 1999b.
[Manton and Nasir, 1999]
N S Manton and S M Nasir.
Conservation laws in a first-order dynamical system of vortices.
Nonlinearity, 120 (4):0 851,
URL <http://stacks.iop.org/0951-7715/12/i=4/a=306>.
[Manton, 1993]
N.S. Manton.
Statistical mechanics of vortices.
Nuclear Physics B, 4000 (1):0 624 – 632,
ISSN 0550-3213.
[Manton and Nasir, 1999]
NS Manton and SM Nasir.
Volume of vortex moduli spaces.
Communications in mathematical physics, 1990
(3):0 591–604, 1999d.
[Manton and Speight, 2003]
N.S. Manton and J.M. Speight.
Asymptotic interactions of critically coupled vortices.
Communications in Mathematical Physics, 2360
(3):0 535–555, Jun 2003.
ISSN 1432-0916.
URL <https://doi.org/10.1007/s00220-003-0842-4>.
[Mawhin, 1999]
Jean Mawhin.
Leray-Schauder degree: a half century of extensions and
Topological Methods in Nonlinear Analysis, 140
(2):0 195, 1999.
ISSN 1230-3429.
[Mitrea and Taylor, 1999]
Marius Mitrea and Michael Taylor.
Boundary layer methods for lipschitz domains in riemannian manifolds.
Journal of Functional Analysis, 1630 (2):0
181–251, 1999.
[Mitrea and Taylor, 2000]
Marius Mitrea and Michael Taylor.
Potential theory on lipschitz domains in riemannian manifolds: Holder
continuous metric tensors.
Communications in Partial Differential Equations, 250
(7-8):0 1487–1536, 2000.
[Nagy, 2017]
Ákos Nagy.
The Berry Connection of the Ginzburg-Landau Vortices.
Communications in Mathematical Physics, 3500
(1):0 105–128, 2 2017.
ISSN 0010-3616.
URL <http://link.springer.com/10.1007/s00220-016-2701-0>.
[Payne and Weinberger, 1960]
L. E. Payne and H. F. Weinberger.
An optimal poincaré inequality for convex domains.
Archive for Rational Mechanics and Analysis, 50
(1):0 286–292, 1960.
ISSN 1432-0673.
URL <https://doi.org/10.1007/BF00252910>.
[Romão and Speight, 2020]
N. M. Romão and J. M. Speight.
The Geometry of the Space of BPS Vortex–Antivortex Pairs.
Communications in Mathematical Physics, 3790
(2):0 723–772, 2020.
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|
Semiclassical Trans-Series from the Perturbative Hopf-Algebraic DSEs
Semiclassical Trans-Series from the Perturbative
Hopf-Algebraic Dyson–Schwinger Equations:
$\boldsymbol{\phi^{3}}$ QFT in 6 Dimensions††This paper is a contribution to
the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory
in honor of Dirk Kreimer for his 60th birthday. The full collection is
available at https://www.emis.de/journals/SIGMA/Kreimer.html
Michael BORINSKY a, Gerald V. DUNNE b and Max MEYNIG b M. Borinsky, G.V. Dunne
and M. Meynig
a) Nikhef Theory Group, Amsterdam 1098 XG, The Netherlands
<EMAIL_ADDRESS>
b) Department of Physics, University of Connecticut, Storrs CT 06269-3046, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received April 07, 2021, in final form September 16, 2021; Published online
September 23, 2021
We analyze the asymptotically free massless scalar $\phi^{3}$ quantum field
theory in 6 dimensions, using resurgent asymptotic analysis to find the trans-
series solutions which yield the non-perturbative completion of the divergent
perturbative solutions to the Kreimer–Connes Hopf-algebraic Dyson–Schwinger
equations for the anomalous dimension. This scalar conformal field theory is
asymptotically free and has a real Lipatov instanton. In the Hopf-algebraic
approach we find a trans-series having an intricate Borel singularity
structure, with three distinct but resonant non-perturbative terms, each
repeated in an infinite series. These expansions are in terms of the
renormalized coupling. The resonant structure leads to powers of logarithmic
terms at higher levels of the trans-series, analogous to logarithmic terms
arising from interactions between instantons and anti-instantons, but arising
from a purely perturbative formalism rather than from a semi-classical
analysis.
renormalons; resurgence; non-perturbative corrections; quantum field theory;
renormalization; Hopf algebra; trans-series
81T15; 81Q15; 34E10
## 1 Introduction
The seminal work of Kreimer and Connes showed that there is an underlying
Hopf-algebraic structure to the renormalization of quantum field theory (QFT)
[28, 29, 61]. This new perspective has led to deep insights into QFT, and also
to novel computational methods that have enabled significant progress in
higher order perturbative computations [14, 16, 17, 20, 22, 23, 27, 60, 62,
63, 64, 81, 83, 84, 85, 88, 89, 90, 91]. The Hopf-algebraic formulation of QFT
is inherently perturbative in nature, so an important open question is to
understand how the non-perturbative features of QFT arise naturally within the
perturbative Hopf algebra structure. In a recent paper [18] we showed how this
works for 4 dimensional massless Yukawa theory, using Écalle’s theory of
resurgent trans-series and alien calculus [3, 33, 44, 48, 80, 82]. Here we
extend this analysis to a conformal field theory: massless scalar $\phi^{3}$
theory in six dimensional space-time. This QFT has been studied extensively
from numerous directions, and has many interesting features, both perturbative
and non-perturbative. The theory is asymptotically free for real coupling $g$
[26, 30, 67, 68], and has a Yang–Lee edge singularity when $g$ is imaginary
[50]. The perturbative beta function and anomalous dimensions have been
computed to 4 loop order [53] (and very recently to 5 loop order [19, 20]).
The perturbative Hopf algebra structure of Dyson–Schwinger equations of this
model was formulated in the pioneering papers [22, 23]. On the non-
perturbative side, this QFT has a real Lipatov instanton when $g$ is real, for
which the conventional one-instanton semi-classical analysis [21, 66, 78, 93]
of the fluctuation determinant has been studied [76, 77]. Further extensions
to multi-dimensional cubic interactions have many interesting applications and
implications for conformal quantum field theories in general [11, 19, 38, 39,
49, 52, 53, 54, 55, 56, 57]. For other analyses of resurgence properties of
renormalization group and Dyson–Schwinger equations see [5, 6, 7, 8, 9, 13].
Our technical analysis is based on the fundamental result [22, 23, 62, 63]
that the Dyson–Schwinger equations have a recursive Hopf-algebraic structure
which, when combined with the renormalization group equations describing the
anomalous scaling under re-scaling of parameters and in the absence of vertex
renormalization, reduces the problem to a non-linear ordinary differential
equation (ODE), where the variable is the renormalized coupling. This Hopf-
algebraic approximation goes well beyond the familiar rainbow [42] and chain
[22, 23] approximations to the Dyson–Schwinger equations. These results cast
the Hopf algebra renormalization problem in a form in which very high orders
of perturbation theory become accessible, and as we show here it also enables
direct access to the associated non-perturbative structure. We employ the
trans-series approach to the resurgence properties of non-linear differential
equations, along the lines of [31, 32, 33]. Our main new result is that the
perturbative Hopf algebra formulation encodes a non-perturbative trans-series
that involves powers of all three trans-monomial elements: $x$, ${\rm
e}^{-1/x}$, and $\log(x)$, all expressed in terms of the renormalized
coupling. Moreover, this trans-series has the form of an all-orders multi-
instanton expansion, and the logarithms appear with the characteristic
structure of logarithmic terms arising from the interaction of instantons and
anti-instantons.111This logarithmic structure does not occur for the 4
dimensional Yukawa model studied in [18]. Logarithmic terms are familiar in
semi-classical computations [1, 40, 41, 45, 46, 47, 65, 71, 79, 92, 94, 95],
and have been studied in differential equations where resonant Borel
singularities $\pm A$ interact [4, 37, 51], but here we find a quite different
resonant Borel structure, with three resonant singularities of the same sign
yet in integer multiples. All this non-perturbative information is encoded in
the original perturbative Hopf-algebraic formulation, which at first sight
makes no explicit mention of instantons, let alone interactions between
instantons and anti-instantons.
## 2 Perturbative Hopf-algebraic analysis
of massless $\boldsymbol{\phi^{3}}$ theory in 6 dimensions
In this paper we analyze the massless scalar $\phi^{3}$ theory in 6
dimensional spacetime. This is the critical dimension in which the theory is
asymptotically free [68] and in which it has a Lipatov instanton [66, 76, 77].
We analyze the non-perturbative features arising in the Hopf-algebraic
approach of [22, 23, 62, 63]. The Lagrangian density is
$\displaystyle{\mathcal{L}}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{g}{3!}\phi^{3}.$
As in [22, 23] we consider the renormalized scalar self-energy
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{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@moveto{22.33957pt}{0.0pt}\pgfsys@curveto{22.33957pt}{0.55229pt}{21.89186pt}{1.0pt}{21.33957pt}{1.0pt}\pgfsys@curveto{20.78728pt}{1.0pt}{20.33957pt}{0.55229pt}{20.33957pt}{0.0pt}\pgfsys@curveto{20.33957pt}{-0.55229pt}{20.78728pt}{-1.0pt}{21.33957pt}{-1.0pt}\pgfsys@curveto{21.89186pt}{-1.0pt}{22.33957pt}{-0.55229pt}{22.33957pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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and take all propagator self-insertions into account. This Hopf-algebraic
approach is depicted by the Dyson–Schwinger equation
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{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@moveto{57.90552pt}{0.0pt}\pgfsys@curveto{57.90552pt}{0.55229pt}{57.45781pt}{1.0pt}{56.90552pt}{1.0pt}\pgfsys@curveto{56.35323pt}{1.0pt}{55.90552pt}{0.55229pt}{55.90552pt}{0.0pt}\pgfsys@curveto{55.90552pt}{-0.55229pt}{56.35323pt}{-1.0pt}{56.90552pt}{-1.0pt}\pgfsys@curveto{57.45781pt}{-1.0pt}{57.90552pt}{-0.55229pt}{57.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@lineto{35.56595pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{49.79233pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@lineto{64.0187pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@curveto{7.11319pt}{13.74994pt}{18.25941pt}{24.89616pt}{32.00935pt}{24.89616pt}\pgfsys@curveto{45.7593pt}{24.89616pt}{56.90552pt}{13.74994pt}{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\cdots-\text{subtractions}\end{aligned}$
(2.1)
with the appropriate BPHZ subtractions indicated. Another way to describe the
relevant set of graphs is to start with the one-loop graph and add all
possible iterated and multiple insertions of this graph into one of the
propagators. Figure 1 shows the resulting low order diagrams and compares this
Hopf expansion with two other common approximations to the Dyson–Schwinger
equations: the rainbow approximation [42] and the chain approximation [22,
23]. The Hopf expansion includes a much larger class of diagrams than either
the rainbow or the chain approximation, and leads to a much richer non-
perturbative structure. The differences between these approximations is
discussed further below, in Sections 3 and 4.222The effect of including also
the vertex corrections will be addressed in future work.
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{}{}{}\pgfsys@moveto{71.1319pt}{0.0pt}\pgfsys@lineto{78.24509pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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($b$) Chain approximation
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# Effects of backreaction and exponential nonlinear electrodynamics on the
holographic superconductors
A<EMAIL_ADDRESS>F. Shaker 1 1 Physics Department and
Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran
###### Abstract
We analytically study the properties of a $(2+1)$-dimensional $s$-wave
holographic superconductor in the presence of exponential nonlinear
electrodynamics. We consider the case in which the scalar and gauge fields
back react on the background metric. Employing the analytical Sturm-Liouville
method, we find that in the black hole background, the nonlinear
electrodynamics correction will affect the properties of the holographic
superconductors. We find that with increasing both backreaction and nonlinear
parameters, the scalar hair condensation on the boundary will develop more
difficult. We obtain the relation connecting the critical temperature with the
charge density. Our analytical results support that, even in the presence of
the nonlinear electrodynamics and backreaction, the phase transition for the
holographic superconductor still belongs to the second order and the critical
exponent of the system always takes the mean-field value $1/2$.
## I Introduction
The AdS/CFT correspondence is an equivalence between a conformal field theory
(CFT) in $d$ spacetime dimensions, and a theory of gravity in
$(d+1)$-dimensional anti-de Sitter (AdS) spacetime AdS.CFT1 ; AdS.CFT2 ;
AdS.CFT3 . The $d$-dimensional theory does not have a gravitational force, and
is to be viewed as a hologram of the $(d+1)$-dimensional theory. The AdS/CFT
correspondence is a well-known approach to explore strongly coupled field
theories in which certain questions become computationally smooth and
conceptually more explicit. The AdS/CFT correspondence can be applied to
condensed matter phenomena. In condensed matter physics, there are many
strongly coupled systems such as superconductors. In this regards, it was
recently argued that it is quite possible to shed some light on the problem of
understanding the mechanism of the high temperature superconductors in
condensed matter physics, by studying a classical general relativity in one
higher dimensional spacetime Har1 ; Har2 . The holographic superconductivity
is a phenomenon associated with asymptotic AdS black holes. The studies on the
holographic superconductors have received a lot of attentions HS.HM ; HS.H ;
HS.FGR ; Wang1 ; Wang2 .
Most studies on the holographic superconductors are focused on the cases where
the gauge field is in the form of the linear Maxwell field. But nonlinear
electrodynamics is constructed by the desire to find non-singular field
theories. One may consider nonlinear electrodynamics as a possible mechanism
for avoiding the singularity of the point-like charged particle at the origin.
The nonlinear extension of the original Maxwell electrodynamics in the context
of holographic superconductors have arisen intensive investigations b.Nu.GB.BI
; p.BI&LN.4D ; p.BI.4D ; JC.p.Nu.BI.4D ; P.BGRL ; P.JPC ; b.BI.5D ; Shey2 . In
particular, in order to see what difference will appear for holographic
superconductor in the presence of Born-Infeld (BI) nonlinear electrodynamics,
compared with the case of linear Maxwell electrodynamics, the authors of Ref.
LPJW2015.p.BI have studied condensation and critical phenomena of the
holographic superconductors with BI electrodynamics in $d$-dimensional
spacetime. Their analytical results indicate that the nonlinear BI
electrodynamics decreases the critical temperature of the holographic
superconductor. It was observed that the higher BI corrections make it harder
for the condensation to form but do not affect the critical phenomena of the
system LPJW2015.p.BI .
It is also interesting to investigate the effects of gauge and scalar fields
of the holographic superconductor on the background geometry. Although if we
ignore this backreaction, the problem is simplified, but retains most of the
interesting physics since the nonlinear interactions are retained. Indeed,
considering the holographic superconductor model away from the probe limit may
bring rich physics. Therefore, many authors have tried to study the
holographic superconductors away from the probe limit PJWC2012 ; B1 ; B2 ; B3
; B4 ; B5 ; B6 . Employing the analytical Sturm-Liouville method, the effects
of both backreaction and BI nonlinear parameter on the critical temperature as
well as scalar condensation were explored in Ref. BIBR . Furthermore, the
relation between the critical temperature and charge density was established
BIBR . It was shown that it is more difficult to have scalar condensation in
BI electrodynamics when the backreaction is taken into account BIBR .
In the present work we would like to extend our analytical study on the
backreacting holographic superconductors by considering another form of the
higher order corrections to the gauge field, i.e., the exponential form of
nonlinear electrodynamics. It was shown that when the backreaction is taken
into account, even the uncharged scalar field can form a condensation in the
$(2+1)$-dimensional holographic superconductor model Har2 . Numerical studies
on the holographic superconductors with exponential nonlinear (EN)
electrodynamics are carried out in the probe limit ZPCJ.P.N.N . It was shown
that the higher nonlinear electrodynamics corrections makes the condensation
harder to form ZPCJ.P.N.N . As far as we know, analytical study on the
holographic superconductor in the presence of EN electrodynamics and away from
the probe limit has not been done. Considering exponential form of the higher
corrections to the gauge field, we shall analytically investigate the
properties of the holographic superconductors when the gauge and scalar field
do back react on the metric background. We shall use the analytical Sturm-
Liouville eigenvalue problem. We will also compare our results with those for
the holographic superconductors with BI nonlinear electrodynamics with
backreaction given in BIBR .
This paper is outlined as follows. In section II, we introduce the action and
basic field equations of the $(2+1)$-holographic superconductor with EN
electrodynamics with backreaction. In section III, we compute the critical
temperature in terms of the charge density and disclose its dependence on the
both nonlinear and backreaction parameters. Section IV, includes step-by-step
computations for obtaining the critical exponent and the condensation values
of the holographic superconductor and provides explanations about them.
Section V will help us to collect the obtained results briefly.
## II Basic Equations of Holographic Superconductors with Backreactions
The action of Einstein gravity coupled to a charged complex scalar field in
the presence of nonlinear electrodynamics is described by
$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2\kappa^{2}}(R-2\Lambda)+\mathcal{L}(\mathcal{F})-|\nabla\psi-
iqA\psi|^{2}-m^{2}|\psi|^{2}\right],$ (1)
where $\kappa$ is the usual four dimensional gravitational constant,
$\kappa^{2}=8\pi G_{4}$, $\Lambda=-{3}/{L^{2}}$ is the cosmological constant,
where $L$ is the AdS radius which will be scaled unity in our calculations.
$R$ and $g$ are, respectively, representing the Ricci scalar and the
determinant of the metric. $A$ is the gauge field and $\psi$ represents a
scalar field with charge $q$ and mass $m$. $\mathcal{L}(\mathcal{F})$ is a
simple generalization of Maxwell Lagrangian in a exponential form Hendi
$\mathcal{L}(\mathcal{F})=\frac{1}{4b}\left(e^{-b\mathcal{F}}-1\right),$ (2)
where $b$ is the nonlinear parameter, $\mathcal{F}=F_{\mu\nu}F^{\mu\nu}$ and
$F^{\mu\nu}$ is the electromagnetic field tensor. Expanding this nonlinear
Lagrangian for small $b$, the leading order term is the linear Maxwell theory,
$\mathcal{L}(\mathcal{F})=-\mathcal{F}/{4}+\mathcal{O}(b)$. The plane-
symmetric black hole solution with an asymptotically AdS behavior including
the backreaction is described by the metric,
$\displaystyle ds^{2}$
$\displaystyle=-f(r)e^{-\chi(r)}dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(dx^{2}+dy^{2}).$
(3)
We adopt the following gauge choices for the vector field and the scalar
field,
$\displaystyle A_{\mu}=\left(\phi(r),0,0,0\right),\ \ \ \psi=\psi(r),$ (4)
with these functions being real-valued. Then, we need to establish the
Einstein equations by varying action (1) with respect to the metric. We find
$\displaystyle
R^{\mu\nu}-\frac{g^{\mu\nu}}{2}R-\frac{3}{L^{2}}g^{\mu\nu}=\kappa^{2}T^{\mu\nu},$
(5)
where the energy momentum tensor is given by
$\displaystyle T^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4b}g^{\mu\nu}\left(e^{-b\mathcal{F}}-1\right)+e^{-b\mathcal{F}}F_{\sigma}^{\
\mu}F^{\sigma\nu}-m^{2}g^{\mu\nu}|\psi|^{2}-g^{\mu\nu}|\nabla\psi-
iqA\psi|^{2}$ (6)
$\displaystyle+\left[(\nabla^{\nu}+iqA^{\nu})\psi^{*}(\nabla^{\mu}-iqA^{\mu})\psi+\mu\leftrightarrow\nu\right].$
Variation with respect to the scalar field yields
$\displaystyle(\nabla_{\mu}-iqA_{\mu})(\nabla^{\mu}-iqA^{\mu})\psi-m^{2}\psi=0,$
(7)
while the electrodynamic equation,
$\displaystyle\nabla_{\mu}\left(F^{\mu\nu}e^{-b\mathcal{F}}\right)=iq\Bigg{[}\psi^{*}(\nabla^{\mu}-iqA^{\mu})\psi-\psi(\nabla^{\mu}+iqA^{\mu})\psi^{*}\Bigg{]},$
(8)
is obtained by varying action (1) with respect to the gauge field. These
equations can easily reduce to those of the holographic superconductor in
Maxwell theory Har2 , provided $b\rightarrow 0$. Calculations of the Einstein,
scalar and electrodynamic field equations, with respect to the metric (3),
yield the following expressions,
$\displaystyle\chi^{\prime}$ $\displaystyle+2r\kappa^{2}\left(\psi^{\prime
2}+\frac{q^{2}e^{\chi}\phi^{2}\psi^{2}}{f^{2}}\right)=0,$ (9) $\displaystyle
f^{\prime}-\left(\frac{3r}{L^{2}}-\frac{f}{r}\right)-\frac{\chi^{\prime}}{2}f+r\kappa^{2}\left[m^{2}\psi^{2}+\frac{1}{4b}(1-e^{2b\phi^{\prime
2}e^{\chi}})+\phi^{\prime 2}e^{\chi+2b\phi^{\prime 2}e^{\chi}}\right]=0,$ (10)
$\displaystyle\phi^{\prime\prime}(1+4be^{\chi}\phi^{\prime
2})+\frac{2}{r}\phi^{\prime}(1+rbe^{\chi}\chi^{\prime}\phi^{\prime
2})+\frac{\chi^{\prime}\phi^{\prime}}{2}-\frac{2q^{2}\phi\psi^{2}}{f}e^{-2be^{\chi}\phi^{\prime
2}}=0,$ (11)
$\displaystyle\psi^{\prime\prime}+\left(\frac{f^{\prime}}{f}+\frac{2}{r}-\frac{\chi^{\prime}}{2}\right)\psi^{\prime}+\left(\frac{q^{2}e^{\chi}\phi^{2}}{f^{2}}-\frac{m^{2}}{f}\right)\psi=0,$
(12)
where the prime denotes derivative with respect to $r$. We further assume
there exists an event horizon $r_{+}$ for which $f(r_{+})=0$, and thus the
corresponding Hawking temperature of the black hole reads
$\displaystyle T=\frac{f^{\prime}(r_{+})e^{-\chi(r_{+})/2}}{4\pi}.$ (13)
For the case with $b\rightarrow 0$, Eqs. (9)-(12) coincide with their
corresponding equations presented in PJWC2012 . Also in the probe limit where
$\kappa=0$, Eqs. (11) and (12) go back to the $(2+1)$-dimensional holographic
superconductor model studied in ZPCJ.P.N.N . In this case the solution of Eq.
(10) is
$\displaystyle
f(r)=\frac{r^{2}}{L^{2}}\left(1-\frac{r_{+}^{3}}{r^{3}}\right).$ (14)
It should be noted that we can set the charge parameter, $q$, as unity and
keep $\kappa^{2}$ finite when the backreaction is taken into account by
adopting the scaling symmetry ssym . When the Hawking temperature is above the
critical temperature $T>T_{c}$, the system leads to the well-known exact black
holes as $b\rightarrow 0$ with the metric coefficient and the potential
function given by
$\displaystyle
f(r)=\frac{r^{2}}{L^{2}}-\frac{1}{r}\left(\frac{r_{+}^{3}}{L^{2}}+\frac{\kappa^{2}\rho^{2}}{2r_{+}}\right)+\frac{\kappa^{2}\rho^{2}}{2r^{2}},\
\ \ \phi\approx\mu-\frac{\rho}{r}.$ (15)
On the dual side, $\mu$ and $\rho$ are, respectively, the chemical potential
and charge density of the holographic superconductor. When $\kappa=0$, the
metric coefficient $f(r)$ recovers the case of Schwarzschild AdS black holes
(14). For investigating the properties of dual model in superconducting phase,
i.e., $\psi(r)\neq 0$, we need the suitable boundary conditions. Examining the
behavior of the fields near the horizon, we find the suitable boundary
conditions as
$\displaystyle\phi(r_{+})=0,\ \ \
\psi(r_{+})=\frac{f^{\prime}(r_{+})\psi^{\prime}(r_{+})}{m^{2}},$ (16)
and hence the metric functions $\chi$ and $f(r)$ satisfy
$\displaystyle\chi^{\prime}(r_{+})$
$\displaystyle=-2r_{+}\kappa^{2}\left(\psi^{\prime
2}(r_{+})+\frac{q^{2}e^{\chi(r_{+})}\phi^{\prime
2}(r_{+})\psi^{2}(r_{+})}{f^{\prime 2}(r_{+})}\right),$ (17) $\displaystyle
f^{\prime}(r_{+})=\frac{3r_{+}}{L^{2}}-r_{+}\kappa^{2}\left[m^{2}\psi^{2}(r_{+})+\frac{1}{4b}(1-e^{2b\phi^{\prime
2}(r_{+})e^{\chi(r_{+})}})+\phi^{\prime 2}(r_{+})e^{\chi(r_{+})+2b\phi^{\prime
2}(r_{+})e^{\chi(r_{+})}}\right].$ (18)
The asymptotic behavior of the fields, corresponding to the solution of Eqs.
(11) and (12) in the limit $r\rightarrow\infty$, are given by
$\displaystyle\phi\approx\mu-\frac{\rho}{r},$ (19)
$\displaystyle\psi\approx\frac{\psi_{-}}{r^{\Delta_{-}}}+\frac{\psi_{+}}{r^{\Delta_{+}}},$
(20)
where
$\displaystyle\Delta_{\pm}=\frac{3}{2}\pm\frac{\sqrt{9+4m^{2}}}{2},$ (21)
is the conformal dimension of the dual operator $\mathcal{O_{\pm}}$ in the
boundary field theory. Here $\psi_{+}$ and $\psi_{-}$ can be considered as the
source and the vacuum expectation values of the dual operator. Hereafter, we
set $\psi_{+}=0$ and investigate the condensation of
$\psi_{-}=<\mathcal{O_{-}}>$, analytically. In what follows we choose the
scalar to have $m^{2}=-2$, and hence the corresponding dual operator has mass
dimension $\Delta_{-}=1$.
## III Analytical study and critical tempreture
In this section, we investigate the analytical properties of a
$(2+1)$-holographic superconductor in the framework of EN electrodynamics. We
study the problem by taking the backreaction into account. We find the
critical temperature $T_{c}$ via the Sturm-Liouville variational approach.
Further, we obtain a relationship between the critical temperature and the
charge density and investigate the effects of both backreaction and EN
parameter on the critical temperature. In order to get the solutions in
superconducting phase, we can define a new variable $z={r_{+}}/{r}$. Then, the
equations of motion can be rewritten as
$\displaystyle\chi^{\prime}-2\kappa^{2}\left(z{\psi^{\prime}}^{2}+\frac{r_{+}^{2}}{z^{3}f^{2}}e^{\chi}\phi^{2}\psi^{2}\right)=0,$
(22) $\displaystyle
f^{\prime}-\frac{f}{z}+\frac{3r_{+}^{2}}{z^{3}}-\frac{\chi^{\prime}f}{2}-\frac{\kappa^{2}r_{+}^{2}}{z^{3}}\left[m^{2}\psi^{2}+\frac{1}{4b}(1-e^{\frac{2bz^{4}}{r_{+}^{2}}\phi^{\prime
2}e^{\chi}})+\frac{z^{4}}{r_{+}^{2}}\phi^{\prime
2}e^{\chi+\frac{2bz^{4}}{r_{+}^{2}}\phi^{\prime 2}e^{\chi}}\right]=0,$ (23)
$\displaystyle\phi^{\prime\prime}\Bigg{(}1+\frac{4bz^{4}}{r_{+}^{2}}e^{\chi}\phi^{\prime
2}\Bigg{)}+\frac{8bz^{3}}{r^{2}_{+}}e^{\chi}\phi^{\prime
3}+\frac{2bz^{4}}{r_{+}^{2}}e^{\chi}\phi^{\prime
3}\chi^{\prime}+\frac{\phi^{\prime}\chi^{\prime}}{2}-\frac{2r_{+}^{2}\psi^{2}}{fz^{4}}e^{-\frac{2bz^{4}}{r^{2}_{+}}e^{\chi}\phi^{\prime
2}}\phi=0,$ (24)
$\displaystyle\psi^{\prime\prime}-\left(\frac{\chi^{\prime}}{2}-\frac{f^{\prime}}{f}\right)\psi^{\prime}-\frac{r_{+}^{2}}{z^{4}}\left(\frac{m^{2}}{f}-\frac{e^{\chi}\phi^{2}}{f^{2}}\right)\psi=0,$
(25)
where now the prime denotes derivative with respect to $z$. When $b\rightarrow
0$, the above equations restore the corresponding equations in Ref. PJWC2012 ,
while in the absence of the backreaction, Eqs. (24) and (25) reduce to their
corresponding equations in Ref. ZPCJ.P.N.N . Following the perturbation
scheme, since close to the critical point, the value of the scalar operator is
small, it can be introduced as an expansion parameter
$\displaystyle\epsilon\equiv<\mathcal{O}_{i}>,$ (26)
with $i=+$ or $i=-$. Besides, near the critical point the scalar and gauge
fields are small and therefore we can expand the gauge field $\phi$, the
scalar field $\psi$, and the metric functions $f(z)$, $\chi(z)$ as PJWC2012
$\displaystyle\psi=\epsilon\psi_{1}+\epsilon^{3}\psi_{3}+\epsilon^{5}\psi_{5}+...,$
(27)
$\displaystyle\phi=\phi_{0}+\epsilon^{2}\phi_{2}+\epsilon^{4}\phi_{4}+...,$
(28) $\displaystyle f=f_{0}+\epsilon^{2}f_{2}+\epsilon^{4}f_{4}+...,$ (29)
$\displaystyle\chi=\epsilon^{2}\chi_{2}+\epsilon^{4}\chi_{4}+...,$ (30)
where the metric function $f(z)$ and $\chi(z)$ are expanded around the
Reissner-Nordström AdS spacetime. Also, the chemical potential $\mu$ may be
expanded as PJWC2012
$\displaystyle\mu=\mu_{0}+\epsilon^{2}\delta\mu_{2}+...,$ (31)
where $\delta\mu_{2}$ is positive. Thus, near the phase transition, the order
parameter as a function of the chemical potential has the form
$\displaystyle\epsilon\thickapprox\Bigg{(}\frac{\mu-\mu_{0}}{\delta\mu_{2}}\Bigg{)}^{1/2}.$
(32)
whose critical exponent $\beta=1/2$ is the same as in the Ginzburg-Landau mean
field theory. The phase transition can take place when
$\mu\rightarrow\mu_{0}$. In this case the critical value of the chemical
potential is given by $\mu_{c}=\mu_{0}$.
From Eq. (24) the equation for $\phi$ is obtained at zeroth order as
$\displaystyle\phi^{\prime\prime}(z)\Bigg{(}1+4b\frac{z^{4}}{r_{+c}^{2}}\phi^{\prime
2}\Bigg{)}+\frac{8bz^{3}}{r^{2}_{+c}}\phi^{\prime 3}(z)=0,$ (33)
which admits the following solutions for the gauge field
$\displaystyle\phi(z)=\int_{1}^{z}{dz\frac{r_{+c}}{2z^{2}\sqrt{b}}\sqrt{L_{W}\left(\frac{4bz^{4}\beta^{2}}{r_{+c}^{2}}\right)}}.$
(34)
In the above expression $\beta$ is an integration constant and
$L_{W}(x)={LambertW(x)}$ is the Lambert function which satisfies Lambert
$L_{W}(x)e^{L_{W}(x)}=x,$ (35)
and has the following series expansion
$L_{W}(x)=x-x^{2}+\frac{3}{2}x^{3}-\frac{8}{3}x^{4}+....$ (36)
Obviously, the series (36) converges provided $|x|<1$. If we expand the
solution (34) for small $b$ and keep the only linear terms in $b$, we arrive
at
$\displaystyle\phi(z)=-\beta(1-z)+\frac{2\beta^{3}b}{5r_{+c}^{2}}\left(1-z^{5}\right)+\mathcal{O}(b^{2}).$
(37)
Differentiating Eqs. (19) and (37) with respect to $z$ and equating them at
$z=0$, we find $\beta=-{\rho}/{r_{+c}}$. Rearranging Eq. (37) and using the
relation $\beta$, we arrive at
$\displaystyle\phi_{0}(z)=\lambda
r_{+c}(1-z)\Bigg{\\{}1-\frac{2}{5}b\lambda^{2}(1+z+z^{2}+z^{3}+z^{4})\Bigg{\\}},\
\ \ b\lambda^{2}<1,$ (38)
where
$\displaystyle\lambda=\frac{\rho}{r^{2}_{+c}},$ (39)
and we have neglected $\mathcal{O}(b^{2})$. Thus, to zeroth order the equation
for $f$ is solved as
$\displaystyle
f_{0}(z)=r_{+}^{2}g(z)=r_{+}^{2}\left[\frac{1}{z^{2}}-z-\frac{\kappa^{2}\lambda^{2}}{2}z(1-z)+\frac{b}{10}\kappa^{2}\lambda^{4}z\left(1-z^{5}\right)\right].$
(40)
At the first order, the behavior of $\psi$ at the asymptotic AdS boundary is
given by
$\displaystyle\psi_{1}\approx\frac{\psi_{-}}{r_{+}^{\Delta_{-}}}z^{\Delta_{-}}+\frac{\psi_{+}}{r_{+}^{\Delta_{+}}}z^{\Delta_{+}}.$
(41)
Next, we introduce a variational trial function $F(z)$ near the boundary
$\displaystyle\psi_{1}(z)=\frac{<\mathcal{O}_{i}>}{\sqrt{2}r_{+}^{\triangle_{i}}}z^{\triangle_{i}}F(z),$
(42)
with the boundary condition $F(0)=1$ and $F^{\prime}(0)=0$. Then, we can
obtain the equation of motion for $F(z)$ by substituting (42) into Eq. (25).
We find
$\displaystyle
F^{\prime\prime}(z)+\Bigg{[}\frac{2\Delta}{z}+\frac{g^{\prime}}{g}\Bigg{]}F^{\prime}(z)+\Bigg{[}\frac{\Delta}{z}\Bigg{(}\frac{\Delta-1}{z}+\frac{g^{\prime}}{g}\Bigg{)}-\frac{m^{2}}{z^{4}g}\Bigg{]}F(z)$
$\displaystyle+\frac{\lambda^{2}(1-z)^{2}}{z^{4}g^{2}}\Bigg{[}1-\frac{4}{5}b\lambda^{2}\Bigg{(}1+z+z^{2}+z^{3}+z^{4}\Bigg{)}\Bigg{]}F(z)=0.$
(43)
Defining the new functions
$\displaystyle T(z)$ $\displaystyle=$ $\displaystyle
z^{2\Delta_{i}+1}\Bigg{[}2(z^{-3}-1)-\kappa^{2}\lambda^{2}(1-z)+\frac{b}{5}\kappa^{2}\lambda^{4}(1-z^{5})\Bigg{]},$
(44) $\displaystyle P(z)$ $\displaystyle=$
$\displaystyle\frac{\Delta_{i}}{z}\Bigg{(}\frac{\Delta_{i}-1}{z}+\frac{g^{\prime}}{g}\Bigg{)}-\frac{m^{2}}{z^{4}g},$
(45) $\displaystyle Q(z)$ $\displaystyle=$
$\displaystyle\frac{(1-z)^{2}}{z^{4}g^{2}}\Bigg{[}1-\frac{4}{5}b\lambda^{2}\Bigg{(}1+z+z^{2}+z^{3}+z^{4}\Bigg{)}\Bigg{]}.$
(46)
we can rewrite Eq. (III) as
$\displaystyle TF^{\prime\prime}+T^{\prime}F^{\prime}+PF+\lambda^{2}QF=0.$
(47)
According to the Sturm-Liouville eigenvalue problem Gan , the eigenvalue
$\lambda^{2}$ can be obtained by minimizing the expression
$\displaystyle\lambda^{2}=\frac{\int_{0}^{1}T\left(F^{\prime
2}-PF^{2}\right)dz}{\int_{0}^{1}TQF^{2}dz},$ (48)
where we have chosen the trial function in the form $F(z)=1-\alpha z^{2}$. In
order to simplify our calculations, we express the backreaction parameter as
PJWC2012
$\displaystyle\kappa_{n}=n\Delta\kappa,\ \ \ n=0,1,2,...$ (49)
where $\Delta\kappa=\kappa_{n+1}-\kappa_{n}$ is the step size of our iterative
procedure. The main purpose is to work in the small backreaction approximation
so that all the functions can be expanded by $\kappa^{2}$ and the $\kappa^{4}$
term can be neglected. Furthermore, we retain the terms that are linear in
nonlinear parameter $b$ and keep terms upto $\mathcal{O}(b)$. So we use the
following relations
$\displaystyle\kappa^{2}\lambda^{2}=\kappa_{n}^{2}\lambda^{2}=\kappa_{n}^{2}(\lambda^{2}|_{\kappa_{n-1}})+\mathcal{O}\left[(\Delta\kappa)^{4}\right],$
(50) $\displaystyle
b\lambda^{2}=b\left(\lambda^{2}|_{b=0}\right)+\mathcal{O}(b^{2}),$ (51)
and
$\displaystyle
b\kappa^{2}\lambda^{4}=b\kappa_{n}^{2}(\lambda^{4}|_{\kappa_{n-1},b=0})+\mathcal{O}(b^{2})+\mathcal{O}[(\Delta\kappa)^{4}],$
(52)
where we have assumed $\kappa_{-1}=0$, $\lambda^{2}|_{\kappa_{-1}}=0$ and
$\lambda^{2}|_{b=0}$ is the value of $\lambda^{2}$ for $b=0$. Now we are going
to compute the critical temperature $T_{c}$. First of all, we start with the
following equation
$\displaystyle T_{c}=\frac{f^{\prime}(r_{+c})}{4\pi}.$ (53)
From Eq. (18), $f^{\prime}(r_{+c})$ is expressed as
$\displaystyle
f^{\prime}(r_{+c})=3r_{+c}-\kappa^{2}r_{+c}\left[\frac{{{\phi_{0}}^{\prime}}^{2}(r_{+c})}{2}+\frac{3}{2}b{{\phi_{0}}^{\prime}}^{4}(r_{+c})\right].$
(54)
Substituting Eq. (38) in the above equation, and then inserting the result
back into Eq. (53), we arrive at the following expression for the critical
temperature,
$\displaystyle
T_{c}=\frac{1}{4\pi}\sqrt{\frac{\rho}{\lambda}}\Bigg{[}3-\frac{\kappa_{n}^{2}(\lambda^{2}|_{\kappa_{n-1}})}{2}+\frac{1}{2}b\kappa_{n}^{2}(\lambda^{4}|_{\kappa_{n-1},b=0})\Bigg{]}.$
(55)
With these obtained computations out of the analytical approach at hand, we
are in a position to present the results of the critical temperature $T_{c}$
for a $(2+1)$-dimensional holographic superconductors in the presence of both
EN electrodynamics as well as backreaction. To do this, we assume the
nonlinear parameter $b$ is small, by choosing it as $b=0,0.1,0.2,0.3$. We also
get the values $m^{2}=-2$, $\Delta_{i}=\Delta_{-}=1$ and $\Delta\kappa=0.05$.
As an example, we bring the details of our calculations for the case of $n=4$
and summarize all results in table $1$.
For $b=0$, we obtain $\lambda^{2}$ From Eq. (48) as
$\displaystyle\lambda^{2}=\frac{-8.279205\alpha^{2}+4.924220\alpha-4.957900}{-3.579048+0.878281\alpha-0.153258\alpha^{2}}.$
(56)
From it we get the minimum eigenvalues of $\lambda^{2}$ and the corresponding
value of $\alpha$, as $\lambda^{2}_{\rm min}=1.2593$ at $\alpha=0.2361$. And
thus the critical temperature is obtained from Eq. (55) as
$T_{c}=0.2235\sqrt{\rho}$, which is in good agreement with the result of
PJWC2012 .
For $b=0.1$, we find
$\displaystyle\lambda^{2}=\frac{-4.95286+4.91624\alpha-8.27411\alpha^{2}}{-3.035+0.673477\alpha-0.109325\alpha^{2}},$
(57)
which has a minimum value $\lambda^{2}_{\rm min}=1.4757$ at $\alpha=0.2417$,
and we can get the critical temperature $T_{c}=0.2147\sqrt{\rho}$.
For $b=0.2$, we arrive at
$\displaystyle\lambda^{2}=\frac{-4.94387+4.90127\alpha-8.26415\alpha^{2}}{-2.49088+0.468581\alpha-0.0652154\alpha^{2}},$
(58)
whose minimum is $\lambda^{2}_{\rm min}=1.7811$ at $\alpha=0.24955$ and the
critical temperature becomes $T_{c}=0.2046\sqrt{\rho}$.
For $b=0.3$, we have
$\displaystyle\lambda^{2}=\frac{-4.93051+4.87819\alpha-8.24877\alpha^{2}}{-1.94661+0.263502\alpha-0.021046\alpha^{2}},$
(59)
which attains its minimum $\lambda^{2}_{\rm min}=2.2451$ at $\alpha=0.26133$
and the critical temperature reads $T_{c}=0.1927\sqrt{\rho}$. We summarize our
results for the critical temperature in cases of different values of nonlinear
and backreaction parameters In table $1$. From this table, we see that, for
fixed value of the backreaction parameter, with the nonlinear parameter $b$
getting stronger, the critical temperature decreases. Similarly, for a fixed
value of the nonlinear parameter $b$, the critical temperature drops as the
backreaction parameter increases. Thus, we conclude that the critical
temperature becomes smaller and so, make the condensation harder when we
increase the values of both backreaction and nonlinear parameters. These
features were also observed in the study a $(2+1)$-dimensional holographic
superconductors with backreaction when the gauge field is in the form of BI
nonlinear electrodynamics BIBR . Comparing the results obtained here with
those of BIBR , we observe that the effect of the EN corrections on the
condensation with respect to the BI nonlinear one is stronger when the
backreactions is taken into account in both cases. In other words, the
formation of scalar hair in the presence of EN electrodynamics is harder
compared to the case of BI nonlinear electrodynamics. Obviously, our analytic
results back up the findings in other articles. In the case of $b=\kappa=0$,
we observe that the analytic results for the critical temperature are
consistent with both the analytical results of Ref. P.ZGJZ as well as the
numerical result of Ref. Har2 . Also, we confirm the numerical result found in
Ref. ZPCJ.P.N.N when the backreaction parameter $\kappa$ is equal to zero. On
the other hand, for $b=0$ the data obtained for the critical temperature, is
analogous to those reported for the holographic superconductors with
backreaction in Maxwell theory PJWC2012 .
n | b=0 | | b=0.1 | | b=0.2 | | b=0.3 |
---|---|---|---|---|---|---|---|---
| BI | EN | BI | EN | BI | EN | BI | EN
0 | 0.2250 | 0.2250 | 0.2228 | 0.2161 | 0.2206 | 0.2060 | 0.2184 | 0.1942
1 | 0.2249 | 0.2249 | 0.2227 | 0.2160 | 0.2204 | 0.2059 | 0.2181 | 0.1941
2 | 0.2246 | 0.2246 | 0.2225 | 0.2158 | 0.2203 | 0.2057 | 0.2180 | 0.1938
3 | 0.2241 | 0.2241 | 0.2220 | 0.2153 | 0.2199 | 0.2050 | 0.2176 | 0.1934
4 | 0.2235 | 0.2235 | 0.2214 | 0.2147 | 0.2192 | 0.2046 | 0.2170 | 0.1927
5 | 0.2226 | 0.2226 | 0.2208 | 0.2141 | 0.2184 | 0.2038 | 0.2162 | 0.1919
6 | 0.2216 | 0.2216 | 0.2196 | 0.2130 | 0.2174 | 0.2029 | 0.2152 | 0.1909
Table $1$: The critical temperature $T_{c}/\sqrt{\rho}$ for holographic
superconductors in the presence of BI and EN electrodynamics. Here we have
taken $\kappa_{n}=n\Delta\kappa$ where $\Delta\kappa=0.05$. The results for BI
case are invoked from Ref. BIBR .
## IV CRITICAL EXPONENT AND THE CONDENSATION OF THE SCALAR OPERATOR
We use the Sturm-Liouville method to analytically examine the scalar
condensation and the order of the phase transition with backreactions near the
critical temperature. With the help of Eq. (42), when $T$ is close to $T_{c}$,
the equation of motion (24) can be rewritten as
$\displaystyle\phi^{\prime\prime}\left(1+\frac{4bz^{4}}{r_{+}^{2}}\phi^{\prime
2}\right)+\frac{8bz^{3}}{r_{+}^{2}}{\phi^{\prime}}^{3}=\frac{\langle\mathcal{O}\rangle^{2}}{r_{+}^{2}}\mathcal{B}(z)\phi(z),$
(60)
$\displaystyle\mathcal{B}(z)=\frac{F^{2}(z)}{1-z^{3}}\left(1-\frac{2bz^{4}}{r_{+}^{2}}\phi^{\prime
2}(z)\right)\left[1+\frac{\kappa^{2}z^{3}}{1+z+z^{2}}\left(\frac{\lambda^{2}}{2}-\frac{b\lambda^{4}}{10}\xi(z)\right)\right],$
(61)
where $\zeta(z)=1+z+z^{2}+z^{3}+z^{4}$. Since the parameter
${\langle\mathcal{O}\rangle^{2}}/{r_{+}^{2}}$ is very small, we can expand
$\phi(z)$ as
$\displaystyle\frac{\phi(z)}{r_{+}}=\lambda(1-z)\left(1-\frac{2}{5}b\lambda^{2}\xi(z)\right)+\frac{\langle\mathcal{O}\rangle^{2}}{r_{+}^{2}}\chi(z).$
(62)
Substituting Eq. (62) into Eq. (60), we can obtain the equation of motion for
$\chi(z)$ as
$\displaystyle\Bigg{[}K(z)\chi^{\prime}(z)\Bigg{]}^{\prime}$ $\displaystyle=$
$\displaystyle(1+4b\lambda^{2}z^{4})^{1/2}\frac{\lambda F^{2}}{1+z+z^{2}}$
(63)
$\displaystyle\times\Bigg{\\{}1-\frac{2}{5}b\lambda^{2}(\xi(z)+5z^{4})+\frac{z^{3}}{1+z+z^{2}}\left(\frac{\kappa^{2}\lambda^{2}}{2}-\frac{b\kappa^{2}\lambda^{4}}{10}(3\xi(z)+10z^{4})\right)\Bigg{\\}}.$
with $\chi(1)=0=\chi^{\prime}(1)$ and we have defined
$\displaystyle K(z)=\left(1+4b\lambda^{2}z^{4}\right)^{3/2}.$ (64)
Integrating both sides of Eq. (63) between $z=0$ to $z=1$, we reach
$\displaystyle\chi^{\prime}(0)$ $\displaystyle=$
$\displaystyle-\lambda\int_{0}^{1}dz(1+4b\lambda^{2}z^{4})^{1/2}\frac{F^{2}}{1+z+z^{2}}$
(65)
$\displaystyle\times\Bigg{\\{}1-\frac{2}{5}b\lambda^{2}(\xi(z)+5z^{4})+\frac{z^{3}}{1+z+z^{2}}\left(\frac{\kappa^{2}\lambda^{2}}{2}-\frac{b\kappa^{2}\lambda^{4}}{10}(3\xi(z)+10z^{4})\right)\Bigg{\\}}.$
Equating $\phi(z)$ from Eqs. (19) and (62), we arrive at
$\displaystyle\frac{\mu}{r_{+}}-\frac{\rho}{r_{+}^{2}}z$ $\displaystyle=$
$\displaystyle\lambda(1-z)\Bigg{\\{}1-\frac{2}{5}b\lambda^{2}\xi(z)\Bigg{\\}}+\frac{\langle\mathcal{O}\rangle^{2}}{r_{+}^{2}}\chi(z)$
(66) $\displaystyle=$
$\displaystyle\lambda(1-z)\Bigg{\\{}1-\frac{2}{5}b\lambda^{2}\xi(z)\Bigg{\\}}+\frac{\langle\mathcal{O}\rangle^{2}}{r_{+}^{2}}\left(\chi(0)+z\chi^{\prime}(0)+...\right),$
where in the last step we have expanded $\chi(z)$ around $z=0$. Considering
the coefficients of $z$ term in both sides of Eq. (66), we find that
$\displaystyle\frac{\rho}{r_{+}^{2}}=\lambda-\frac{\langle\mathcal{O}\rangle^{2}}{r_{+}^{2}}\chi^{\prime}(0).$
(67)
Substituting $\chi^{\prime}(0)$ from Eq. (65) in the above relation, we get
$\displaystyle\frac{\rho}{r_{+}^{2}}=\lambda\Bigg{\\{}1+\frac{\langle\mathcal{O}\rangle^{2}}{r_{+}^{2}}\mathcal{A}\Bigg{\\}},$
(68)
where
$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dz(1+4b\lambda^{2}z^{4})^{1/2}\frac{F^{2}}{1+z+z^{2}}$
(69)
$\displaystyle\times\Bigg{\\{}1-\frac{2}{5}b\lambda^{2}(\xi(z)+5z^{4})+\frac{z^{3}}{1+z+z^{2}}\left(\frac{\kappa^{2}\lambda^{2}}{2}-\frac{b\kappa^{2}\lambda^{4}}{10}(3\xi(z)+10z^{4})\right)\Bigg{\\}}.$
Using Eqs. (13), (18) and (38), and taking into account the fact that $T$ is
very close to $T_{c}$, we can deduce
$\displaystyle r_{+}=\frac{4\pi
T}{\left[3-\frac{\kappa^{2}\lambda^{2}}{2}+\frac{b}{2}\kappa^{2}\lambda^{4}\right]}.$
(70)
Eqs. (39) and (70) show that Eq. (68) can be rewritten as
$\displaystyle
T_{c}^{2}-T^{2}=\langle\mathcal{O}\rangle^{2}\frac{\mathcal{A}}{(4\pi)^{2}}\left[3-\frac{\kappa^{2}\lambda^{2}}{2}+\frac{b}{2}\kappa^{2}\lambda^{4}\right]^{2}.$
(71)
Thus, we find the expectation value $\langle\mathcal{O}\rangle$ near the
critical point as
$\displaystyle\langle\mathcal{O}\rangle=\gamma T_{c}\sqrt{1-\frac{T}{T_{c}}},$
(72)
where $\gamma$ is the condensation parameter of the system which is given by
$\displaystyle\gamma=\frac{4\pi\sqrt{2}}{\sqrt{\mathcal{A}}}\left[3-\frac{\kappa^{2}\lambda^{2}}{2}+\frac{b}{2}\kappa^{2}\lambda^{4}\right]^{-1}.$
(73)
The relation obtained in Eq. (72) is valid for small nonlinear coupling and
backreaction parameters and satisfies
$\langle\mathcal{O}\rangle\sim\sqrt{1-\frac{T}{T_{c}}}$. Therefore, the
analytical result supports that the phase transition for the superconductor
belongs to the second order and the critical exponent of the system takes the
mean-field value $1/2$. This implies that considering nonlinear coupling and
backreaction parameters the value of the critical exponent will not be
altered. As we see in table $2$, condensation values $\gamma$ increases with
increasing the nonlinear parameter $b$ for the fixed parameter $\kappa$. Also,
we see the same behavior between the condensation values $\gamma$ and the
backreaction parameter with a fixed value of the nonlinear parameter $b$. This
means that the condensation becomes harder to be formed by considering both
the nonlinear corrections to the gauge field and taking the backreactions into
account. It should be noted that, at a temperature slightly below $T_{c}$ for
the $(2+1)$-dimensional holographic superconductors with backreaction,
condensation values for both BI and EN holographic superconductors have the
same behaviour, as we see in table $2$. Also, because of the larger parameter
$\gamma$, effect of the EN electrodynamics on the condensation of the scalar
operators is bigger than that of BI case. This implies that the scalar hair is
more difficult to be developed in the holographic superconductors with EN
electrodynamics.
n | b=0 | | b=0.1 | | b=0.2 | | b=0.3 |
---|---|---|---|---|---|---|---|---
| BI | EN | BI | EN | BI | EN | BI | EN
0 | 8.07 | 8.07 | 8.1801 | 8.5298 | 8.3094 | 9.1579 | 8.4696 | 10.0355
1 | 8.09 | 8.09 | 8.1869 | 8.5331 | 8.3212 | 9.1616 | 8.4890 | 10.0399
2 | 8.11 | 8.11 | 8.1943 | 8.5443 | 8.3237 | 9.1742 | 8.4893 | 10.0565
3 | 8.115 | 8.115 | 8.2121 | 8.5630 | 8.3417 | 9.1951 | 8.5023 | 10.0818
4 | 8.13 | 8.13 | 8.2370 | 8.5889 | 8.3669 | 9.2241 | 8.5277 | 10.1014
5 | 8.16 | 8.16 | 8.2909 | 8.6425 | 8.3994 | 9.2615 | 8.5606 | 10.1639
6 | 8.20 | 8.20 | 8.3079 | 8.6617 | 8.4391 | 9.3070 | 8.6007 | 10.2193
Table 2: The values of the condensation parameter $\gamma$ for holographic
superconductors in the presence of EN electrodynamics. Here we have taken
$\kappa_{n}=n\Delta\kappa$ where $\Delta\kappa=0.05$. We have also provided
the results for BI holographic superconductor from Ref. BIBR , for comparison.
## V Conclusions
We have introduced a different type of gravity dual models, i.e., the charged
AdS black holes in the context of Einstein-nonlinear electrodynamics with a
scalar field. We have assumed the EN electrodynamics as the gauge field, and
analytically investigated the behavior of the $(2+1)$-dimensional holographic
superconductors. We have worked in a limit in which the scalar and gauge
fields backreact on the background metric. We have employed the Sturm-
Liouville analytic method to explore the problem. We have found the influence
of the nonlinear corrections to the gauge filed as well as the backreaction
effects on the critical temperature and the process of the scalar field
condensation. We observed that the formation of the scalar hair condensation
on the boundary becomes harder in the presence of nonlinear electrodynamics.
This is mainly caused by the decreasing of the critical temperature when the
both nonlinear and backreaction parameters become stronger. This phenomenon
was also obtained in the study of the effect of the BI and backreaction
parameters in the $(2+1)$-dimensional holographic superconductors BIBR .
Comparing these different models show that for a specific $b$ the critical
temperature $T_{c}$ becomes larger for a holographic superconductor with BI
nonlinear electrodynamics comparing to the case with EN electrodynamics. This
implies that the scalar hair is more difficult to develop in the latter case
than the former one. We have also given the critical exponent for the EN
holographic superconductor model with backreaction, which still takes the
mean-field value $1/2$. We found out that the condensation parameter $\gamma$
in Eq. (72) increases with (i) increasing the nonlinear parameter $b$ with a
fixed backreaction parameter $\kappa$, (ii) increasing the backreaction
parameter with a fixed value of the nonlinear parameter $b$. This implies that
both the nonlinear corrections to the gauge field as well as backreaction,
cause the formation of condensation harder.
###### Acknowledgements.
We thank Shiraz University Research Council. This work has been supported
financially by Research Institute for Astronomy and Astrophysics of Maragha
(RIAAM), Iran.
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|
# Transcendence measure of $e^{1/n}$
Marta Dujella Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland.
<EMAIL_ADDRESS>, Anne-Maria Ernvall-Hytönen University of Helsinki,
PL 68, 00014 Helsingin yliopisto, Finland. anne-maria.ernvall-
<EMAIL_ADDRESS>, Linda Frey Universität Göttingen, Bunsenstraße 3-5,
37073 Göttingen, Germany<EMAIL_ADDRESS>and Bidisha Roy Scuola
Normale di Pisa, Piazza dei Cavlieri,7, 56126, Pisa, Italy.
<EMAIL_ADDRESS>
###### Abstract.
For a given transcendental number $\xi$ and for any polynomial
$P(X)=:\lambda_{0}+\cdots+\lambda_{k}X^{k}\in\mathbb{Z}[X]$, we know that
$P(\xi)\neq 0.$ Let $k\geq 1$ and $\omega(k,H)$ be the infimum of the numbers
$r>0$ satisfying the estimate
$\left|\lambda_{0}+\lambda_{1}\xi+\lambda_{2}\xi^{2}+\ldots+\lambda_{k}\xi^{k}\right|>\frac{1}{H^{r}},$
for all
$(\lambda_{0},\ldots,\lambda_{k})^{T}\in\mathbb{Z}^{k+1}\setminus\\{\overline{0}\\}$
with $\max_{1\leq i\leq k}\\{|\lambda_{i}|\\}\leq H$. Any function greater
than or equal to $\omega(k,H)$ is a transcendence measure of $\xi$. In this
article, we find out a transcendence measure of $e^{1/n}$ which improves a
result proved by Mahler([7]) in 1975.
We thank the organizers of the conference Women in Numbers Europe 4,
especially but not exclusively Valentijn Karemaker and Nirvana Coppola. This
conference set the foundation for this article. Furthermore, we thank the
Universiteit Utrecht for granting unlimited coffee.
## 1\. Introduction
Let $\xi$ be a transcendental number. Then $P(\xi)\neq 0$ for any polynomial
$P$ with integer coefficient.
Let $k,H\geq 1$ and $\omega(k,H)$ is the infimum of the numbers $r>0$
satisfying the estimate
(1)
$\left|\lambda_{0}+\lambda_{1}\xi+\lambda_{2}\xi^{2}+\ldots+\lambda_{k}\xi^{k}\right|>\frac{1}{H^{r}},$
for all
$\overline{\lambda}=(\lambda_{0},\ldots,\lambda_{k})^{T}\in\mathbb{Z}^{k+1}\setminus\\{\overline{0}\\}$
with $\max_{1\leq i\leq k}\\{|\lambda_{i}|\\}\leq H$. Any function greater
than or equal to $\omega(k,H)$ is a transcendence measure of $\xi$.
Bounding transcendence measures of different constants is a classical problem
in number theory. It is widely investigated in particular in the context of
the Napier’s constant $e$. In 1873, Hermite proved it to be transcendental
[5]. This work started with Borel in 1899 [1] when proved that
$\omega(k,H)<c\log\log H$ for some constant $c$ depending on $k$. Popken [9,
10] improved the bound to $k+\frac{c}{\log\log H}$, where $c$ depends on $k$
in 1928–1929. Mahler [8] made the dependance on $k$ explicit in 1931 by
deriving the bound $k+\frac{ck^{2}\log(k+1)}{\log\log H}$, where $c$ is an
absolute constant. This result is already of the shape of modern state of art
results. In 1991, Khassa and Srinivasan [6] showed that $c=98$ is valid for
$\log\log Hd(k+1)^{6k}$ for some constant $d>e^{950}$. This result was
improved by Hata in 1995 [4] who showed that one can choose $c=1$, and also
considerably improved the lower bound for $H$. This was further improved by
Ernvall-Hytönen, Matala-aho and Seppälä ion 2018 [3].
In this paper, we concentrate on the expression
$\left|\lambda_{k}e^{k/n}+\ldots+\lambda_{1}e^{1/n}+\lambda_{0}\right|,$
where $k\geq n\geq 2$. This gives a transcendence measure for roots of $e$.
Since $e$ is transcendental, all its roots and powers are also transcendental.
Ernvall-Hytönen, Matala-aho and Seppälä considered also sparse polynomials in
the context of the transcendence measure of $e$. As a corollary, they also
derived a transcendence measure for integer powers of $e$. However, we were
not able to find any transcendence measures tailored for _roots of $e$_ in the
literature. There are some general results in the literature, for instance, by
Mahler [7] which can be used to derive a bound. Also, the generalized
transcendence measure by Ernvall-Hytönen, Leppälä and Matala-aho can be used
to derive a bound. Our bound will be compared to these bounds in Section 2.
In this article, we prove the following bound:
###### Theorem 1.
Assume $k\geq n\geq 2$. We have
(2)
$\left|\lambda_{0}+\lambda_{1}e^{1/n}+\lambda_{2}e^{2/n}+\ldots+\lambda_{k}e^{k/n}\right|>\frac{1}{H^{r}},$
where $r>\omega(k,H)$ and we can choose
$\omega(k,H)=k+\frac{k^{2}\log k}{\log\log H}\left(1+\frac{0.69}{\log
k-1}\right),$
for $k\geq 5$ and $\omega(k,H)=k+\frac{k^{2}\log k}{\log\log H}d(k)$, where
$d(k)=\begin{cases}&3.319\textrm{ for }k=2\\\ &1.145\textrm{ for }k=3\\\
&1.114\textrm{ for }k=4\end{cases}$
and $\log H\geq s(n,k)e^{s(n,k)}$ with $s(n,k)=(k+n)(\log(k+n))^{2}$.
We follow the approach used in [3].
## 2\. Earlier results and comparisons to our bound
The following result can be obtained as a corollary of a much more general
result presented in by Mahler in 1975.
###### Theorem 2 (Theorem 1 in [7]).
Take $a_{i}=i$ for $i=0,\ldots,k$ (with $k\geq 2$) and $a=n$ a positive
integer. Let $\lambda_{0},\ldots,\lambda_{k}$ be integers not all zero (in
Mahler’s paper $x_{0},\ldots,x_{k}$), $C(r)=(k+1)^{2}r\sqrt{(\log(n+k+1)\log
r}$ and $T$ be the product of the non-zero $\lambda_{i}$. Then for $r$ the
smallest integer for which
$\displaystyle\frac{(r-1)!}{e^{2C(r-1)}}\leq\max\\{|x_{i}|\\}<\frac{r!}{e^{2C(r)}}$
we have
(3)
$\displaystyle|T(\lambda_{0}+\lambda_{1}e^{\frac{1}{n}}+\ldots+\lambda_{k}e^{\frac{k}{n}})|>\frac{\max\\{|\lambda_{i}|\\}}{e^{(2(k+1)-\frac{1}{4})C(r)}}.$
The following result is due to Ernvall-Hytönen, Leppälä and Matala-aho, and
can be obtained as a corollary of a much more general result presented in [2].
###### Theorem 3 (Corollary of [2]).
We have
$\left|\lambda_{0}+\lambda_{1}e^{1/n}+\cdots+\lambda_{k}e^{k/n}\right|>\frac{M^{1-\hat{\delta}(M)}}{h_{0}h_{1}\ldots
h_{k}},$
where $M=\max_{0\leq i\leq k}\\{|\lambda_{i}|\\}$,
$\hat{\delta}(M)\leq\frac{\hat{B}(\overline{\alpha})}{\sqrt{\log\log M}}\leq
c_{k}k^{2}\sqrt{\log(g_{1}(\overline{\alpha})(1+g_{3}(\overline{\alpha})))}/\sqrt{\log\log
M}$ and $h_{i}=\max\\{1,|\lambda_{i}|\\}$, for $i=1,\ldots,k$. Moreover,
$c_{k}=13$ if $k<3$ and $12$ otherwise.
In particular, for $\overline{\alpha}=(0,\frac{1}{n},\ldots,\frac{k}{n})$ we
have $g_{1}(\overline{\alpha})=n$ and $g_{3}(\overline{\alpha})=\frac{k}{n}$.
Therefore,
$\left|\lambda_{0}+\lambda_{1}e^{1/n}+\cdots+\lambda_{k}e^{k/n}\right|>\frac{M}{h_{0}\cdots
h_{k}M^{\frac{c_{k}k^{2}\sqrt{\log(n(1+k/n))}}{\sqrt{\log\log M}}}}.$
Let us now compare these results with our bound.
###### Example 4.
Let us look at the family of polynomials with
$\frac{H}{2}\leq|\lambda_{i}|\leq H$ for all coefficients $\lambda_{i}$ when
$1\leq i\leq k$ and compare our result with results in [2] and [7].
Our result gives the bound
$\left|\lambda_{0}+\lambda_{1}e^{1/n}+\cdots+\lambda_{k}e^{k/n}\right|>H^{-k-\frac{k^{2}\log
k}{\log\log H}\left(1+\frac{0.639}{\log k-1}\right)}$
The bound by Ernvall-Hytönen, Matala-aho and Leppälä:
$\displaystyle\left|\lambda_{0}+\lambda_{1}e^{1/n}+\cdots+\lambda_{k}e^{k/n}\right|$
$\displaystyle>\frac{H}{h_{0}\cdots
h_{k}H^{\frac{c_{k}k^{2}\sqrt{\log(n(1+k/n))}}{\sqrt{\log\log H}}}}$
This bound is certainly not better than
$\left(\frac{H}{2}\right)^{-k-\frac{12k^{2}\sqrt{\log(n(1+k/n))}}{\sqrt{\log\log
H}}}=H^{-k+k\frac{\log 2}{\log
H}-\frac{12k^{2}\sqrt{\log(n+k)}}{\sqrt{\log\log H}}},$
which is weaker than ours for large values of $H$ because for large
$\sqrt{\log\log H}$ grows slower than $\log\log H$.
Mahler 1975:
(4)
$\displaystyle|\lambda_{0}+\lambda_{1}e^{\frac{1}{n}}+\ldots+\lambda_{k}e^{\frac{k}{n}}|>\frac{\max\\{|\lambda_{i}|\\}}{Te^{(2(k+1)-\frac{1}{4})C(r)}},$
where $T$ is the product of all non-zero $\lambda_{i}$’s.
(5)
$\displaystyle\frac{\max\\{|\lambda_{i}|\\}}{Te^{(2(k+1)-\frac{1}{4})C(r)}}$
$\displaystyle\leq\frac{1}{\left(\frac{H}{2}\right)^{k}e^{(2(k+1)-1/4)C(r)}}.$
Mahler gives the bound $\frac{\log x}{\log\log x}<r<\frac{6\log x}{\log\log
x}$, where in his notation $x$ is the maximum of the absolute values of the
coefficients of the polynomial. In our setting, this inequality would
approximately translate to
$\frac{\log H}{\log\log H}<r<\frac{6\log H}{\log\log H}.$
We are losing some accuracy here because we only expected the coefficients of
the polynomial to be on the interval $[\frac{H}{2},H]$. However, for the
current purposes, this is not an issue.
The denominator of (5) can now be written as
$\left(\frac{H}{2}\right)^{k}e^{(2(k+1)-1/4)(2(k+1)-1/4)C(r)}=H^{k-\frac{k\log
2-(2(k+1)-1/4)C(r)}{\log H}}=H^{k-\frac{k\log 2}{\log
H}+\frac{(2(k+1)-1/4)C(r)}{\log H}}.$
Let us now look at the expression $\frac{(2(k+1)-1/4)C(r)}{\log H}$. Let us
use the expression for $C(r)$ and the bound for $r$:
$C(r)=(k+1)^{2}r\sqrt{(\log(n+k+1)\log r}\approx(k+1)^{2}\frac{\log
H}{\log\log H}\sqrt{(\log(n+k+1)\log\frac{\log H}{\log\log H}}\\\
\approx(k+1)^{2}\frac{\log H\sqrt{\log k}}{\sqrt{\log\log H}},$
where we used the bound $n\leq k$ to estimate $\log(n+k+1)\approx\log k$, and
that for large H, $\log\frac{\log H}{\log\log H}\approx\log\log H$. Hence,
$\frac{(2(k+1)-1/4)C(r)}{\log H}\approx\frac{(2(k+1)-1/4)}{\log
H}\cdot(k+1)^{2}\frac{\log H\sqrt{\log k}}{\sqrt{\log\log
H}}\approx\frac{2(k+1)^{2}\sqrt{\log k}}{\sqrt{\log\log H}}.$
Hence, our bound is also better than Mahler’s bound, because $\sqrt{\log\log
H}$ grows slower than $\log\log H$, and the numerator is bigger (dependance on
$k^{3}$ instead of $k^{2}$).
## 3\. Preliminaries and the outline of the method
Ernvall-Hytönen, Seppälä and Matala-aho [3] used the following approach:
Assume that there is a sequence of simultaneous approximations
$L_{m,j}(h)=B_{m,0}(h)\Theta_{j}+B_{m,j}(h)$
with $m=0,1,\dots,k$ and $j=1,2,\dots,k$. Further assume
$B_{m,j}(\ell)\in\mathbb{Z}$ for all $m,j\in\\{0,1,\dots,k\\}$. Assume further
that the coefficients $B_{m,j}$ satisfy the following determinant condition:
$\begin{vmatrix}B_{0,0}&B_{0,1}&\cdots&B_{0,k}\\\
B_{1,0}&B_{1,1}&\cdots&B_{1,k}\\\ \vdots&\vdots&\ddots&\vdots\\\
B_{k,0}&B_{m,1}&\cdots&B_{k,k}\\\ \end{vmatrix}\neq 0.$
Pick the functions $Q(h)$, $q(h)$, $R(h)$ and $r(h)$ to be such that they
satisfy the following inequalities:
$B_{m,0}(h)\leq Q(h)=e^{q(h)}$
and
$\sum_{j=1}^{k}|L_{m,j}(h)|\leq R(h)=e^{-r(h)},$
for all $h\geq h_{0}$, where the functions are of the form
$q(h)=ah\log h+bh$
and
$-r(h)=-ch\log h+dh.$
Assume that $z(y)$ is the inverse function of the function $y(z)=z\log z$.
Further, denote
$B=b+\frac{ad}{c},\quad C=a,\quad D=a+b+e^{-s(m)},\quad F^{-1}=2e^{D},\quad
v=c-\frac{d}{s(m)},\quad h_{1}=\max\\{h_{0},e,e^{s(m)}\\}.$
Our choice will be $s(n,k)=(n+k)(\log(n+k))^{2}$, and we will actually have
$h_{1}=e^{s(n,k)}$. Under the assumptions above, they proved the following
lemma:
###### Lemma 5 ([3]).
Let $m\geq 1$ and $\log(2H)\geq vh_{1}\log h_{1}$. Then under the assumptions
above
(6)
$|\lambda_{0}+\lambda_{1}\Theta_{1}+\cdots+\lambda_{m}\Theta_{m}|>F(2H)^{-\frac{a}{c}-\epsilon(H)},$
where
$\epsilon(H)\log(2H)=Bz\left(\frac{\log(2H)}{v}\right)+C\log\left(z\left(\frac{\log(2H)}{v}\right)\right).$
Furthermore, they gave the following construction for the approximations in
the case of $e^{\alpha_{j}}$: Write
$\overline{\alpha}=(\alpha_{0},\ldots,\alpha_{k})$ and set
(7)
$\displaystyle\Omega(x,\overline{\alpha})=\prod_{j=0}^{m}(\alpha_{j}-x)^{\ell_{j}}=\sum_{i=0}^{L}\sigma_{i}x^{i},$
where $L=\ell_{0}+\ell_{1}+\cdots+\ell_{m}$ and
$\sigma_{i}=\sigma_{i}(\overline{\ell},\overline{\alpha})$. Then choosing
$A_{0}(t)=\sum_{i=\ell_{0}}^{L}t^{L-i}i!\sigma_{i},$
we get
(8) $e^{\alpha_{j}}A_{0}(t)-A_{j}(t)=R_{j}(t),$
where $A_{j}(t)$ is a polynomial with integer coefficients and
$\begin{cases}\deg A_{0}(t)=L-\ell_{0}\\\ \deg A_{j}(t)=L-\ell_{j}\\\
\mathrm{ord}_{t=0}R_{j}(t)\geq L+1.\end{cases}$
Notice that the polynomials depend on the values of
$\ell_{0},\ell_{1},\dots,\ell_{m}$ and on $\overline{\alpha}$. We will
explicitly describe $A_{j}$ and $R_{j}$ in the following chapter.
In the following, we will be choosing $\Theta_{j}=e^{j/n}$ for some $n\geq 2$.
We will then proceed in the same fashion as in [3] to construct the explicit
polynomials used in the simultaneous approximations and to bound them.
Finally, we simplify the estimate given by (6).
## 4\. Explicit polynomial construction
We start by constructing the simultaneous approximations of the powers of the
roots of $e$. For estimating the required term, we set
$\overline{\alpha}=(\alpha_{0},\ldots,\alpha_{k})$ with $\alpha_{s}=s/n$, for
$s=0,1,\ldots k$. Let
$\overline{\ell}=(l_{0},\ldots,l_{k})\in\mathbb{Z}^{k+1}_{\geq 1}$ and
$L=l_{0}+\ldots+\l_{k}$. As explained in the previous chapter, we get the
following approximation formulas for $j=1,\ldots,k$
$e^{\alpha_{j}t}A_{0}(t)-A_{j}(t)=L_{j}(t),$
where
$A_{0}(t)=\sum_{i=\ell_{0}}^{L}t^{L-i}i!\sigma_{i}.$
With a direct computation (similarly as in [3]), we obtain
$\displaystyle\sigma_{i}$ $\displaystyle=(-1)^{i}\sum_{\ell_{0}+i_{1}+\ldots
i_{k}=i}\binom{\ell_{1}}{i_{1}}\ldots\binom{\ell_{k}}{i_{k}}\left(\frac{1}{n}\right)^{\ell_{1}-i_{1}}\ldots\left(\frac{k}{n}\right)^{\ell_{k}-i_{k}}$
$\displaystyle=(-1)^{i}\sum_{\ell_{0}+i_{1}+\ldots
i_{k}=i}\binom{\ell_{1}}{i_{1}}\ldots\binom{\ell_{k}}{i_{k}}n^{-L+i}2^{\ell_{2}-i_{2}}\dots
k^{\ell_{k}-i_{k}}$
Furthermore, $\sigma_{i}=0$ when $0\leq i<\ell_{0}$ and so
$A_{0}(t)=\sum_{i=0}^{L}t^{L-i}i!\sigma_{i}.$
We wish to now bound the polynomials. Laplace transform gives us a tool to
switch from sums to integrals, which is helpful in estimates. Since
$\frac{i!\sigma_{i}(\overline{\ell},\overline{\alpha})}{t^{i+1}}=\mathcal{L}(\sigma_{i}(\overline{\ell},\overline{\alpha})x^{i})(t)$
(where $\mathcal{L}$ denotes the Laplace transform), we have
$A_{0}(t)=\sum_{i=0}^{L}t^{L-i}i!\sigma_{i}=t^{L+1}\sum_{i=0}^{L}\mathcal{L}(\sigma_{i}x^{i})(t)=t^{L+1}\int_{0}^{\infty}e^{-xt}\Omega(x)dx,$
where $\Omega(x):=\Omega(x,\overline{\alpha})$ is given by (7). Now for any
$\alpha_{j}$, we have
$e^{\alpha_{j}t}A_{0}(t)=t^{L+1}\int_{0}^{\infty}e^{(\alpha_{j}-x)t}\Omega(x)dx=t^{L+1}\left(\int_{0}^{\alpha_{j}}+\int_{\alpha_{j}}^{\infty}\right)e^{(\alpha_{j}-x)t}\Omega(x)dx.$
Changing the variable in the second integral: $y=x-\alpha$ gives us:
$e^{\alpha_{j}t}A_{0}(t)=t^{L+1}\int_{0}^{\alpha_{i}}e^{(\alpha_{j}-x)t}\Omega(x)dx+t^{L+1}\int_{0}^{\infty}e^{-yt}\Omega(y+\alpha_{j})dy$
Hence we get
$\displaystyle A_{j}(t)=t^{L+1}\int_{0}^{\infty}e^{-yt}\Omega(y+\alpha_{j})dy$
and
$\displaystyle
L_{j}(t)=t^{L+1}\int_{0}^{\alpha_{j}}e^{(\alpha_{j}-x)t}\Omega(x)dx$
for $j=1,\ldots,k$.
We can make the result stronger if the terms in (8) are as small as possible.
But at the same time, we want to keep the coefficients in $A_{j}(t)$ as
integers. Therefore, we try to find as large common factors in coefficients as
possible.
To do that, we proceed as in [3]. We start by picking very specific values of
$\ell_{0},\ell_{1},\dots,\ell_{k}$ in relation to each other.
For any $u$ with $0\leq u\leq k$, we take
$\ell_{s}^{(u)}=\begin{cases}\ell-1&\mbox{if }s=u\\\
\ell&\mbox{otherwise}\end{cases}$ and
$\overline{\ell}^{(u)}=(\ell_{0}^{(u)},\ldots,\ell_{k}^{(u)})$. For these
values of $\overline{\ell}$, we denote $A_{j}(t)=A_{\overline{\ell},j}(t)$ by
$A_{u,j}(t)$ and $L_{j}(t)=L_{\overline{\ell},j}(t)$ by $L_{u,j}(t)$.
$A_{0}(t)=\sum_{i=\ell_{0}}^{L}t^{L-i}i!(-1)^{i}\sum_{\ell_{0}+i_{1}+\ldots
i_{k}=i}\binom{\ell_{1}}{i_{1}}\ldots\binom{\ell_{k}}{i_{k}}n^{-L+i}2^{\ell_{2}-i_{2}}\dots
k^{\ell_{k}-i_{k}}.$
For our chosen $\overline{\ell}$-s we always have
$\ell_{0}\in\\{\ell,\ell-1\\}$, so we can see that
$\displaystyle\frac{n^{L-\ell+1}}{(\ell-1)!}A_{u,0}(t)\in\mathbb{Z}[t].$
Similarly, we can look at polynomials $A_{u,j}(t)$ for $j=1,\ldots,k$. We have
$A_{u,j}(t)=\int_{0}^{\infty}e^{-yt}\Omega(y+\alpha_{j})dy=t^{L+1}\sum_{i=0}^{L}\mathcal{L}(\sigma_{i}(\overline{\ell},\overline{\beta^{(j)}}))=\sum_{i=0}^{L}t^{L-i}i!\sigma_{i}(\overline{\ell},\overline{\beta^{(j)}}),$
where
$\overline{\beta^{(j)}}=(\alpha_{0}-\alpha_{j},\ldots,\alpha_{k}-\alpha_{j})$
for each $j$. Since the coefficients
$\sigma_{i}(\overline{\ell},\overline{\beta^{(j)}})$ are defined using
polynomial $\Omega(\overline{\ell},\overline{\beta^{(j)}})$, and in the
product representation of $\Omega$, the term
$(t-\beta_{j}^{(j)})^{\ell_{j}}=(t-(\alpha_{j}-\alpha_{j}))^{\ell_{j}}=t^{\alpha_{j}}$
will define the lowest degree of terms occuring in
$\Omega(\overline{\ell},\overline{\beta})$, and thereby in $A_{u,j}$, we have
$\sigma_{i}(\overline{\ell},\overline{\beta})=0$ unless $i\geq\ell_{j}$.
Hence, all the coefficients in the representation of $A_{u,j}(t)$ have the
factor $\ell_{j}!$, which is again either $\ell$ or $\ell-1$. On the other
hand, these terms have the denominator $n^{L-i}$, so again
$\frac{n^{L-\ell+1}}{(\ell-1)!}A_{u,j}(t)\in\mathbb{Z}[t].$
## 5\. Estimation of $A_{u,0}(1)$
Next, we would like to estimate the term $A_{u,0}(t)$, for which its
representation as an integral will be useful. We have fixed
$\overline{\alpha}=(1/n,\ldots,k/n)$ and so
$\Omega(x)=\prod_{j=0}^{k}\left(j/n-x\right)$. Furthermore, for the choices of
$\overline{\ell}$ as in the previous section we have $L-1=(k+1)\ell$.
Therefore, $A_{u,0}(t)$ looks like
$\displaystyle A_{u,0}(t)$
$\displaystyle=t^{(k+1)\ell}\int_{0}^{\infty}e^{-yt}\prod_{j=0}^{k}\left(\frac{j}{n}-x\right)^{\ell_{s}}dx$
$\displaystyle=t^{(k+1)\ell}\int_{0}^{\infty}e^{-xt}(-x)^{\ell}\left(\frac{1}{n}-x\right)^{\ell}\cdots\left(\frac{u}{n}-x\right)^{\ell-1}\cdots\left(\frac{k}{n}-x\right)^{\ell}dx.$
Note that $\left|\frac{x^{\ell}(x-1/n)\cdots(x-k/n)}{(x-u/n)}\right|\leq
x^{(k+1)\ell-1}\leq x^{(k+1)\ell}$, for $x>\frac{k}{n}.$ This gives us an idea
of how the function inside the integral behaves: while $0\leq
x\leq\frac{k}{n}$, the function stays relatively small, and it touches zero at
points $0,\frac{1}{n},\dots,\frac{k}{n}$. However, when $x\geq\frac{k}{n}$, it
starts behaving roughly as $x^{(k+1)\ell-1}e^{-x}$.
Therefore, we split the above integral in the following way
(9)
$\int_{0}^{\infty}e^{-xt}\prod_{j=0}^{k}\left(\frac{j}{n}-x\right)^{\ell_{s}}dx=\left(\int_{0}^{k/n}+\int_{k/n}^{2(k+1)\ell}+\int_{2(k+1)\ell}^{\infty}\right)e^{-xt}\prod_{j=0}^{k}\left(\frac{j}{n}-x\right)^{\ell_{s}}dx.$
Now we treat the above integrals with $t=1$. In that case,
(10)
$\displaystyle\left|\left(\int_{k/n}^{2(k+1)\ell}+\int_{2(k+1)\ell}^{\infty}\right)e^{-x}\frac{\prod_{s=0}^{k}\left(\frac{s}{n}-x\right)^{\ell}}{\left(\frac{u}{n}-x\right)}dx\right|\leq\left(\int_{k/n}^{2(k+1)\ell}+\int_{2(k+1)\ell}^{\infty}\right)e^{-x}x^{(k+1)\ell-1}dx:=I_{1}+I_{2}.$
For estimating the above integrals, we first consider $I_{1}$. In that case,
$\displaystyle|I_{1}|=\left|\int_{k/n}^{2(k+1)\ell}e^{-x}x^{(k+1)\ell-1}dx\right|<2(k+1)\ell
e^{-(k+1)\ell+1}((k+1)\ell-1)^{(k+1)\ell-1}$
because the expression $e^{-x}x^{(k+1)\ell-1}$ is maximal when
$x=(k+1)\ell-1$.
Let us now move to $I_{2}$.
###### Lemma 6.
Let $c>1$ be a constant. We have
$\int_{c\ell(k+1)}^{\infty}e^{-x}\frac{\left(\prod_{j=0}^{k}\left|\frac{j}{n}-x\right|\right)^{\ell}}{\left|\frac{k^{\prime}}{n}-x\right|}\leq\frac{c}{c-1}e^{-c(k+1)l}(c(k+1)\ell)^{\ell(k+1)-1}.$
for any $0\leq k^{\prime}\leq k$.
###### Proof.
We can partially integrate:
$\int e^{-x}x^{t}dx=\left[-e^{-x}x^{t}\right]+\int e^{-x}tx^{t-1},$
which gives us the series expansion for the integral above:
$\int_{c\ell(k+1)}^{\infty}e^{-x}x^{\ell(k+1)-1}dx\\\
=e^{-c(k+1)\ell}(c(k+1)\ell)^{\ell(k+1)-1}+(\ell(k+1)-1)e^{-c(k+1)\ell}(c(k+1)\ell)^{\ell(k+1)-2}\\\
+(\ell(k+1)-1)(\ell(k+1)-2)e^{-c(k+1)\ell}(c(k+1)\ell)^{\ell(k+1)-3}+\ldots\\\
\leq
e^{-c(k+1)l}(c(k+1)\ell)^{\ell(k+1)-1}\left(1+\frac{(k+1)\ell-1}{c(k+1)\ell}+\frac{(k+1)\ell-1}{c(k+1)\ell}\cdot\frac{(k+1)\ell-2}{c(k+1)\ell}\ldots\right)\\\
\leq\frac{c}{c-1}e^{-c(k+1)l}(c(k+1)\ell)^{\ell(k+1)-1}.$
∎
Notice that if we pick $c=2$, we get the following corollary:
###### Corollary 7.
We have the following estimate
$|I_{2}|\leq\int_{2\ell(k+1)}^{\infty}e^{-x}x^{\ell(k+1)-1}dx\leq
2e^{-2(k+1)l}(2(k+1)\ell)^{\ell(k+1)-1}.$
Next, it remains to get a bound for
$\displaystyle\int_{0}^{k/n}e^{-y}\frac{\prod_{s=0}^{k}\left(\frac{s}{n}-y\right)^{\ell}}{\left(\frac{u}{n}-y\right)}dy.$
###### Lemma 8.
Assume $k\geq 5$. Now
$\int_{0}^{k/n}e^{-y}\frac{\prod_{s=0}^{k}\left(\frac{s}{n}-y\right)^{\ell}}{\left(\frac{u}{n}-y\right)}dy\leq\frac{(k!)^{\ell}}{120^{\ell-1}n^{k\ell-5(\ell-1)}}c(n)^{\ell-1},$
where $c(n)=\displaystyle\max_{0\leq y\leq
1}\prod_{s=0}^{5}\left|\frac{s}{n}-y\right|^{\ell-1}.$ Furthermore,
$|c(n)|\leq 1$.
###### Proof.
Observe that $\displaystyle\max_{v\leq y\leq
v+1}\prod_{s=0}^{k}\left|\frac{s}{n}-y\right|^{\ell-1}\leq\max_{0\leq y\leq
1}\prod_{s=0}^{k}\left|\frac{s}{n}-y\right|^{\ell-1},$ for positive integer
$v$ with $0\leq v\leq k-1$. Therefore,
$\left|\frac{\prod_{s=0}^{k}\left(\frac{s}{n}-y\right)^{\ell}}{\left(\frac{u}{n}-y\right)}\right|\leq\frac{k!}{n^{k}}\max_{0\leq
y\leq 1}\prod_{s=0}^{k}\left|\frac{s}{n}-y\right|^{\ell-1}.$
Hence
$\displaystyle\left|\int_{0}^{k/n}e^{-y}\frac{\prod_{s=0}^{k}\left(\frac{s}{n}-y\right)^{\ell}}{\left(\frac{u}{n}-y\right)}dy\right|$
$\displaystyle\leq\int_{0}^{k/n}e^{-y}\frac{k!}{n^{k}}\max_{0\leq y\leq
1}\prod_{s=0}^{k}\left|\frac{s}{n}-y\right|^{\ell-1}dy$
$\displaystyle\leq\frac{k!}{n^{k}}\frac{(k!)^{\ell-1}}{(5!)^{\ell-1}n^{(k-5)(\ell-1)}}\int_{0}^{k/n}e^{-y}\max_{0\leq
y\leq 1}\prod_{s=0}^{5}\left|\frac{s}{n}-y\right|^{\ell-1}dy$
$\displaystyle\leq\frac{(k!)^{\ell}}{120^{\ell-1}n^{k\ell-5(\ell-1)}}c(n)^{\ell-1},\mbox{
writing }c(n)=:\displaystyle\max_{0\leq y\leq
1}\prod_{s=0}^{5}\left|\frac{s}{n}-y\right|.$
By checking small values individually, and bounding
$\prod_{s=0}^{5}\left|\frac{s}{n}-y\right|\leq 1$
for $n\geq 5$, we obtain $|c(n)|\leq 1$. ∎
###### Lemma 9.
Assume $k\leq 5$. Now
$\int_{0}^{k/n}e^{-y}\frac{\prod_{s=0}^{k}\left(\frac{s}{n}-y\right)^{\ell}}{\left(\frac{u}{n}-y\right)}dy\leq\frac{k!}{n^{k}}c(n,k)^{\ell-1},$
where
$c(n,k)=\begin{cases}0.049&\textrm{for }(n,k)=(2,2)\\\
\frac{1}{16}&\textrm{for }(n,k)=(2,3)\\\ \frac{1}{81}&\textrm{for
}(n,k)=(3,3)\\\ 0.114&\textrm{for }(n,k)=(2,4)\\\ 0.015&\textrm{for
}(n,k)=(3,4)\\\ 0.004&\textrm{for }(n,k)=(4,4)\par\end{cases}$
###### Proof.
We simply bound
$\int_{0}^{k/n}e^{-y}\frac{\prod_{s=0}^{k}\left(\frac{s}{n}-y\right)^{\ell}}{\left(\frac{u}{n}-y\right)}dy\leq\frac{k!}{n^{k}}\max_{0\leq
y\leq 1}\prod_{s=0}^{k}\left|y-\frac{s}{n}\right|\int_{0}^{k/n}e^{-y}dy,$
where the integral can be bounded to be at most $1$, and the individual maxima
can be determined using WolframAlpha.
∎
###### Lemma 10.
Assume $k\geq 2$. We have
$\int_{0}^{\infty}e^{-y}\frac{\prod_{s=0}^{k}\left(\frac{s}{n}-y\right)^{\ell}}{\left(\frac{u}{n}-y\right)}dy\leq\exp\left(\log
4+(k+1)\ell\log 2+(\ell(k+1)-1)\log((k+1)\ell)-\ell(k+1)+1\right)$
###### Proof.
Assume first $k\geq 5$. Now taking the above three estimations into account,
we obtain
$\displaystyle\left|A_{u,0}(1)\right|=\left|\int_{0}^{\infty}e^{-y}(-y)^{\ell}(\frac{1}{n}-y)^{\ell}\cdots(\frac{u}{n}-y)^{\ell-1}\cdots(\frac{k}{n}-y)^{\ell}dy\right|$
$\displaystyle\leq\frac{(k!)^{\ell}}{120^{\ell-1}n^{k\ell-5(\ell-1)}}c(n)^{\ell-1}+2(k+1)\ell
e^{-(k+1)\ell+1}((k+1)\ell-1)^{(k+1)\ell-1}+2e^{-2(k+1)l}(2(k+1)\ell)^{\ell(k+1)-1}$
$\displaystyle\leq\frac{(k!)^{\ell}}{n^{k\ell-5(\ell-1)}}\left(\frac{c(n)}{120}\right)^{\ell-1}+\left(2(k+1)\ell+2\cdot
2^{\ell(k+1)-1}\right)e^{-(k+1)\ell+1}((k+1)\ell)^{\ell(k+1)-1}$
$\displaystyle\leq\frac{(k!)^{\ell}}{n^{k\ell-5(\ell-1)}}\left(\frac{c(n)}{120}\right)^{\ell-1}+3\cdot
2^{(k+1)\ell}e^{-(k+1)\ell+1}((k+1)\ell)^{\ell(k+1)-1}$ $\displaystyle\leq
4\cdot 2^{(k+1)\ell}e^{-(k+1)\ell+1}((k+1)\ell)^{\ell(k+1)-1},\qquad\mbox{ for
}n\geq 2$ $\displaystyle\leq\exp\left(\log 4+(k+1)\ell\log
2+(\ell(k+1)-1)\log((k+1)\ell)-\ell(k+1)+1\right)$
For $k\in\\{2,3,4\\}$ the only thing that changes is the first term, but
because $c(n,k)<1$ for all choices of $k$ and $n$ that interest us, we have
that
$\frac{k!}{n^{k}}c(n,k)^{\ell-1}\leq
2^{(k+1)\ell}e^{-(k+1)\ell+1}((k+1)\ell)^{\ell(k+1)-1},$
so the final bound is also valid in for these values. ∎
Finally, we actually need to get a bound for $A_{u,0}^{\star}(1)$. The
following corollary gives us the desired bound.
###### Corollary 11.
For $k\geq 3$ and $\ell\geq\exp(s(n,k))$, we have
$\left|\frac{n^{L-\ell+1}}{(\ell-1)!}A_{u,0}(1)\right|\leq\exp\left(\ell
k\log\ell+\ell\left[k\log k+k\log n+0.72k+0.000003\right]\right)$
For $k=2$, we have
$\displaystyle\left|\frac{n^{L-\ell+1}}{(\ell-1)!}A_{u,0}(1)\right|\leq\exp(2\ell\log\ell+3.377257\ell+2\ell\log
2).$
###### Proof.
Applying the previous lemma and Stirling’s formula we have
$\displaystyle\log\left|A_{u,0}(1)\right|-\log\left((l-1)!\right)$
$\displaystyle\leq\ell
k\log\ell+\ell\left((k+1)\log(k+1)+\log\ell-\log(\ell-1)-k+(k+1)\log 2\right)$
$\displaystyle-\log\ell-\log(k+1)+\frac{1}{2}\log(\ell-1)+\log
4-\log\sqrt{2\pi}$
We first deal with the case when $k\geq 3$. With this assumption, because
$\log\ell\geq s(n,k)\geq s(2,3)$ we have that
$\log(\ell)-\log(\ell-1)=\int_{\ell-1}^{\ell}\frac{dx}{x}\leq\frac{1}{\ell-1}<0.000003.$
Furthermore,
$(k+1)\log(k+1)-k\log(k)=k\log(1+1/k)+\log(k+1)\leq 1+\log(k+1).$
and
$-k+(k+1)\log 2+1+\log(k+1)<0.72k.$
Additionally, for all $k\geq 2$
$\frac{-\log\ell-\log(k+1)+\frac{1}{2}\log(\ell-1)+\log
4-\log\sqrt{2\pi}}{\ell}<0$
and goes to $0$ as $\ell$ grows.
Thus, multiplying by $n^{L-\ell+1}=n^{k\ell}$ we get
$\left|\frac{n^{L-\ell+1}}{(\ell-1)!}A_{u,0}(1)\right|\leq\exp\left(\ell
k\log\ell+\ell\left[k\log k+k\log n+0.72k+0.000003\right]\right)$
Let us now move to the case $k=2$. We have
$\log(\ell)-\log(\ell-1)=\int_{\ell-1}^{\ell}\frac{dx}{x}\leq\frac{1}{\ell-1}<0.00046.$
For $k=2$ see that
$\displaystyle\log\left|A_{u,0}(1)\right|-\log\left((l-1)!\right)$
$\displaystyle\leq(\log 4-3\ell+3\ell
log(6\ell))-(\ell-1)\log(\ell-1)+(\ell-1)-\frac{1}{2}\log(\ell-1)-\log\sqrt{2\pi}$
$\displaystyle\leq 3\ell\log\ell+\ell\left(3\log 6-3+\frac{\log
4-1}{\ell}-\frac{\ell\log(\ell-1)}{\ell}+1+\frac{\log(\ell-1)}{2\ell}-\frac{\log\sqrt{2\pi}}{\ell}\right)$
$\displaystyle\leq 2\ell\log\ell+\ell\left(3\log 6-2+\frac{\log
4/\sqrt{2\pi}-1}{\ell}+\log(\frac{\ell}{\ell-1})+\frac{\log(\ell-1)}{2\ell}\right)$
$\displaystyle\leq 2\ell\log\ell+3.377\ell.$
Therefore, in this case,
$\displaystyle\left|\frac{n^{L-\ell+1}}{(\ell-1)!}A_{u,0}(1)\right|\leq\exp(2\ell\log\ell+3.377257\ell+2\ell\log
2)$.
∎
## 6\. Integrals corresponding to terms $L_{u,j}^{\star}$
Next we need to get a suitable bound on terms $L_{u,j}^{\star}$, or more
precisely their sum $\sum_{j=1}^{k}\left|L_{u,j}^{\star}\right|$. The
following lemma will be useful
###### Lemma 12.
Let $k\geq 3$ and $n\geq 2$. Then
$\displaystyle\max_{0<x<k/n}\left|x\left(\frac{1}{n}-x\right)\left(\frac{2}{n}-x\right)\cdot\ldots\cdot\left(\frac{k}{n}-x\right)\right|\leq\frac{k!}{6n^{k+1}}.$
If $k=2$, then we have
$\max_{0<x<2/n}|x(\frac{1}{n}-x)(\frac{2}{n}-x)|\leq\frac{2}{3\sqrt{3}n^{3}}$.
###### Proof.
By doing a change of variable $y=nx$ the expression on the left becomes
$\displaystyle\max_{0<x<k/n}\left|x\left(\frac{1}{n}-x\right)\cdot\ldots\cdot\left(\frac{k}{n}-x\right)\right|=\frac{1}{n^{k+1}}\max_{0<y<k}\left|y(1-y)\cdot\ldots\cdot(k-y)\right|.$
By analyzing the function $\left|y(1-y)(2-y)\cdot\ldots\cdot(k-y)\right|$ we
see that its maximum on the interval $(0,k)$ is attained for the first time
already on the interval $(0,1)$. If $k\geq 3$ we have the following
$\displaystyle\max_{0<y<k}\left|y(1-y)\ldots(k-y)\right|$
$\displaystyle\leq\max_{0<y<1}|(4-y)\ldots(k-y)|\cdot\max_{0<y<1}|y(1-y)(2-y)(3-y)|$
$\displaystyle\leq\frac{k!}{3!}\max_{0<y<1}y(1-y)(2-y)(3-y).$
By taking the derivative we can see that the function $y(y-1)(y-2)(y-3)$
achieves its maximum $1$ for $y=(3\pm\sqrt{5})/2$, which finally implies that
$\displaystyle\max_{0<x<k/n}\left|x\left(\frac{1}{n}-x\right)\cdot\ldots\cdot\left(\frac{k}{n}-x\right)\right|\leq\frac{k!}{6n^{k+1}}.$
Similarly for $k=2$ we need to analyze the function $y(y-1)(y-2)$, whose
maximum $\frac{2}{3\sqrt{3}}$ is achieved for $y=1\pm\frac{1}{\sqrt{3}}$, from
which the claim follows. ∎
###### Lemma 13.
Let $k\geq 2$ and $n\geq 2$. Then
$\displaystyle|L_{u,j}^{*}(1)|\leq
n^{L-\ell+1}\frac{(e^{\frac{j}{n}}-1)(k!)^{\ell}}{(\ell-1)!(c(k)n^{k+1})^{\ell-1}n^{k}},$
where $c(k)=6$ for $k\geq 3$ and $c(2)=3\sqrt{3}$.
###### Proof.
Let $j\in\\{1,\ldots,k\\}$. By the definition of $|L_{u,j}^{*}(1)|$ we have
$\displaystyle|L_{u,j}^{*}(1)|(\ell-1)!$
$\displaystyle=n^{L-\ell+1}e^{\frac{j}{n}}\int_{0}^{\frac{j}{n}}e^{-x}\frac{\prod_{r=0}^{k}|\frac{r}{n}-x|^{\ell}}{|\frac{u}{n}-x|}dx$
$\displaystyle=n^{L-\ell+1}e^{\frac{j}{n}}\int_{0}^{\frac{j}{n}}e^{-x}\prod_{r=0}^{k}|\frac{r}{n}-x|^{\ell-1}\frac{\prod_{r=0}^{k}|\frac{r}{n}-x|}{|\frac{u}{n}-x|}dx$
$\displaystyle\leq
n^{L-\ell+1}e^{\frac{j}{n}}\int_{0}^{\frac{j}{n}}e^{-x}\prod_{r=0}^{k}|\frac{r}{n}-x|^{\ell-1}\frac{k!}{n^{k}}dx.$
Because $j\geq k$, we have that
$\max_{0<x<j/n}\prod_{r=0}^{k}|\frac{r}{n}-x|\leq\max_{0<x<k/n}\prod_{r=0}^{k}|\frac{r}{n}-x|$
which is at most $\frac{k!}{c(k)n^{k+1}}$ due to the previous lemma. So we
further have
$\displaystyle|L_{u,j}^{*}(1)|(\ell-1)!$ $\displaystyle\leq
n^{L-\ell+1}e^{\frac{j}{n}}\int_{0}^{\frac{j}{n}}e^{-x}\left(\frac{k!}{c(k)n^{k+1}}\right)^{\ell-1}\frac{k!}{n^{k}}dx$
$\displaystyle\leq
n^{L-\ell+1}\frac{(k!)^{l}}{(c(k)n^{k+1})^{\ell-1}n^{k}}e^{\frac{j}{n}}\int_{0}^{\frac{j}{n}}e^{-x}dx$
$\displaystyle\leq
n^{L-\ell+1}\frac{(k!)^{\ell}}{(c(k)n^{k+1})^{\ell-1}n^{k}}(e^{\frac{j}{n}}-1)$
∎
###### Lemma 14.
Let $k\geq 2$ and $c(k)$ as in the previous lemma. We have
$\sum_{j=1}^{k}|L_{u,j}^{*}|\leq\frac{(k!)^{\ell}}{c(k)^{\ell-1}(\ell-1)!}n^{2-\ell}e^{(k+1)/n}.$
###### Proof.
We have the following:
$\sum_{j=1}^{k}(e^{j/n}-1)<\sum_{j=1}^{k}e^{j/n}=\frac{e^{(k+1)/n}-e^{1/n}}{e^{1/n}-1}<\frac{e^{(k+1)/n}}{e^{1/n}-1}.$
Since
$e^{1/n}-1=\int_{0}^{1/n}e^{x}dx>\frac{1}{n},$
this can be further estimated to
$\sum_{j=1}^{k}(e^{j/n}-1)<ne^{(k+1)/n}.$
By summing up the above estimation for $j=1,\dots,k$ we get
$\sum_{j=1}^{k}|L_{u,j}^{*}|\leq
n^{L-\ell+1}\frac{(k!)^{\ell}}{(c(k)n^{k+1})^{\ell-1}n^{k}(\ell-1)!}ne^{(k+1)/n}.$
We can further simplify the above expression, by noticing that
$\displaystyle\frac{n^{L-\ell+1}n}{(n^{k+1})^{\ell-1}n^{k}}=n^{2-\ell},$
which follows from $L=k\ell-1$. This finally gives us the bound
$\displaystyle\sum_{j=1}^{k}|L_{u,j}^{*}|\leq\frac{(k!)^{\ell}}{c(k)^{\ell-1}(\ell-1)!}n^{2-\ell}e^{(k+1)/n}.$
∎
To make this bound suitable for application in $r(\ell)=\exp(R(\ell)$ we need
to simplify it further.
###### Lemma 15.
Let $k\geq 3$ and $\ell\geq e^{s(k,n)}=e^{(k+n)(\log(k+n))^{2}}$. We have
$\sum_{j=1}^{k}|L_{u,j}^{*}|\leq\exp\left(-\ell\log\ell+\ell(k\log
k-0.81k-\log n+0.174)\right).$
For $k=2$ we have
$\sum_{j=1}^{2}|L_{u,j}^{*}|\leq\exp\left(-\ell\log\ell-0.64\ell\right).$
###### Proof.
First let $k\geq 3$. We need to simplify the following expression:
$\log\left(\frac{(k!)^{\ell}}{6^{\ell-1}(\ell-1)!}n^{2-\ell}e^{(k+1)/n}\right)=\ell\log(k!)-(\ell-1)\log
6-\log(\ell-1)!+(2-\ell)\log n+\frac{k+1}{n}.$
We have $\log(\ell-1)!=\log\ell!-\log\ell$. Further, we can use Stirling’s
formula to bound the factorials:
$\sqrt{2\pi\ell}\left(\frac{\ell}{e}\right)^{\ell}e^{1/(12\ell+1)}<\ell!<\sqrt{2\pi\ell}\left(\frac{\ell}{e}\right)^{\ell}e^{1/(12\ell)}.$
Hence
$\log\ell!>\frac{1}{2}\log(2\pi)+\frac{1}{2}\log\ell+\ell\log\ell-\ell+\frac{1}{12\ell+1}$
and similarly for $k!$:
$\log k!<\frac{1}{2}\log(2\pi)+\frac{1}{2}\log k+k\log k-k+\frac{1}{12k}.$
Hence, we have
$\displaystyle\log\left(\frac{(k!)^{\ell}}{6^{\ell-1}(\ell-1)!}n^{2-\ell}e^{(k+1)/n}\right)=$
$\displaystyle\ell\left(\frac{1}{2}\log(2\pi)+\frac{1}{2}\log k+k\log
k-k+\frac{1}{12k}\right)-(\ell-1)\log 6$
$\displaystyle-\log\ell!+\log\ell+(2-\ell)\log n+\frac{k+1}{n}$
$\displaystyle=$ $\displaystyle\ell\left(\frac{1}{2}\log(2\pi)+\frac{1}{2}\log
k+k\log k-k+\frac{1}{12k}\right)-(\ell-1)\log 6+\log\ell$
$\displaystyle-\left(\frac{1}{2}\log(2\pi)+\frac{1}{2}\log\ell+\ell\log\ell-\ell+\frac{1}{12\ell+1}\right)+(2-\ell)\log
n+\frac{k+1}{n}$ $\displaystyle=$ $\displaystyle-\ell\log\ell+\ell(k\log
k-0.81k-\log n+0.16)+\frac{1}{2}\log\ell$ $\displaystyle+2\log
n+\frac{k+1}{n}+0.88,$
because $\log 6-\frac{1}{2}\log(2\pi)-\frac{1}{12\ell+1}<0.88$ and
$\frac{1}{2}\log(2\pi)-\log 6+\frac{1}{12k}+1<0.16$ and $\frac{1}{2}\log
k-k<-0.81k$. We can further simplify by using the inequality
$\left(\frac{1}{2}\log\ell+2\log n+\frac{k+1}{n}+0.88\right)<0.00004\ell$
to obtain
$\log\left(\frac{(k!)^{\ell}}{6^{\ell-1}(\ell-1)!}n^{2-\ell}e^{(k+1)/n}\right)<-\ell\log\ell+\ell(k\log
k-0.81k-\log n+0.17).$
Similar calculation for $k=2$ gives us explicitly:
$\displaystyle\log\left(\sum_{j=1}^{2}|L_{u,j}^{*}|\right)\leq$
$\displaystyle-\ell\log\ell+\ell(2\log 2+\frac{1}{2}\log
2-2+\frac{1}{24}-\log(3\sqrt{3})+\frac{1}{2}\log(2\pi)+1-\log n)$
$\displaystyle+\log(3\sqrt{3})-\frac{1}{2}\log(2\pi)+\frac{1}{2}\log\ell-\frac{1}{12\ell+1}+2\log
n+\frac{k+1}{n}$ $\displaystyle\leq$
$\displaystyle-\ell\log\ell+\ell(0.0456-\log n)+0.0035\ell$
$\displaystyle\leq$
$\displaystyle-\ell\log\ell-0.65\ell+0.0035\ell\leq-\ell\log\ell-0.64\ell.$
∎
## 7\. Transcendence measure for $e^{1/n}$
We are now ready to put together the bounds
$q(\ell)=\ell k\log\ell+\ell\left[k\log k+k\log n+0.72k+0.000003\right]$
which is true for $k\geq 3$ and for $k=2$
$q(\ell)=2\ell\log\ell+\ell(3.377257+2\log n).$
Estimating sum of $L_{u,j}^{*}$, for $k\geq 3$, we obtained
$-r(\ell)=-\ell\log\ell+\ell(k\log k-0.81k-\log n+0.17)$
and for $k=2$, it is
$-r(\ell)=-\ell\log\ell-0.64\ell.$
Using notation from Section 3 in [3], equations $(8)$ and $(9)$, we have
$\displaystyle s(n,k)$ $\displaystyle=(k+n)(\log(k+n))^{2}\quad\textrm{this
function is in place of $s(m)$ there}$ $\displaystyle a$ $\displaystyle=k$
$\displaystyle b$ $\displaystyle=k\log k+k\log n+0.72k+0.000003$
$\displaystyle c$ $\displaystyle=1$ $\displaystyle d$ $\displaystyle=k\log
k-0.81k-\log n+0.17.$
Now with the notation from Section 3 in [3], equation $(10)$, we have
$\displaystyle B$ $\displaystyle=b+\frac{ad}{c}=k\log k+k\log
n+0.72k+0.000003+k(k\log k-0.81k-\log n+0.17)$ $\displaystyle=k\log
k+0.89k+0.000003+k^{2}\log k-0.81k^{2}$ $\displaystyle C$ $\displaystyle=a=k$
$\displaystyle D$ $\displaystyle=a+b+ae^{-s(k,n)}=k+k\log k+k\log
n+0.72k+0.000003+\frac{k}{e^{(k+n)(\log(k+n))^{2}}}$ $\displaystyle F^{-1}$
$\displaystyle=2e^{D}$ $\displaystyle v$ $\displaystyle=1-\frac{k\log
k-0.81k-\log n+0.17}{(k+n)(\log(k+n))^{2}}$ $\displaystyle n_{1}$
$\displaystyle=e^{(n+k)(\log(n+k))^{2}}.$
Now we have
$|\lambda_{0}+\lambda_{1}e^{1/n}+\cdots+\lambda_{k}e^{k/n}|>F(2H)^{-a/c-\epsilon(H)},$
where
$\epsilon(H)=\frac{1}{\log(2H)}\left(Bz\left(\frac{\log(2H)}{v}\right)+C\log\left(z\left(\frac{\log(2H)}{v}\right)\right)\right).$
The term $H^{-a/c}=H^{-k}$ will form the main term, and everything else will
be put together into the second term in the exponent (of $H$). This second
term will be formed of the terms
$F2^{-a/c}(2H)^{-\epsilon(H)}$
For large $k$, we have
$1<|\Lambda|2(2H)^{\frac{a}{c}}e^{\epsilon(H)\log(2H)+D}=|\Lambda|H^{\frac{a}{c}+Y}=|\Lambda|H^{k+Y},$
where
$\displaystyle Y:$ $\displaystyle=\frac{1}{\log
H}\left(Bz\left(\frac{\log(2H)}{v}\right)+C\log\left(z\left(\frac{\log(2H)}{v}\right)\right)+D+(k+1)\log
2\right)$ $\displaystyle\leq\frac{1}{\log
H}\left(\frac{uB}{v}\frac{\log(2H)}{\log\log(2H)}+k\log\left(\frac{u}{v}\frac{\log(2H)}{\log\log(2H)}\right)+D+(k+1)\log
2\right)$
and
$u=1+\frac{\log(s(n,k))}{s(n,k)}.$
We use the fact that $\log H\geq s(n,k)e^{s(n,k)}$. Not, we need to estimate
the terms involving $D$ and $B$.
$\displaystyle D+(k+1)\log 2$ $\displaystyle=k+k\log k+k\log
n+0.72k+0.000003+\frac{k}{e^{(k+n)(\log(k+n))^{2}}}+(k+1)\log 2$
$\displaystyle=k+k\log k+k\log
n+k\left(0.72+\frac{0.000003}{k}+\frac{1}{e^{(k+n)(\log(k+n))^{2}}}+\log
2+\frac{\log 2}{k}\right)$ $\displaystyle\leq k\log k+k\log n+3.4k.$
For the next term, we observe
$k\log\left(\frac{u}{v}\frac{\log(2H)}{\log\log(2H)}\right)\leq
k\log(u\log(2H)).$
Therefore, we get
$\displaystyle Y$ $\displaystyle\leq\frac{1}{\log
H}\left(\frac{uB}{v}\frac{\log(2H)}{\log\log(2H)}+k\log(u\log(2H))+k\log
k+k\log n+3.4k\right)$ $\displaystyle\leq\frac{1}{\log
H}\left(\frac{uB}{v}\frac{\log(2H)}{\log\log(2H)}+k\log(2\log H)+k\log(2\log
2)+k\log k+k\log n+3.4k\right)$ $\displaystyle\leq\frac{1}{\log
H}\left(\frac{uB}{v}\frac{\log(2H)}{\log\log(2H)}+k\log(2\log
H)+\frac{k}{2}+k\log k+k\log n+3.4k\right)$ $\displaystyle\leq\frac{1}{\log
H}\left(\frac{uB}{v}\frac{\log(2H)}{\log\log(2H)}+\frac{69}{10}k\log(2\log
H)\right)$ $\displaystyle\leq\frac{1}{\log\log H}\left(\frac{\log(2H)}{\log
H}\cdot\frac{uB}{v}+\frac{6.9\log\log H\cdot k\log(2\log H)}{\log H}\right)$
$\displaystyle=\frac{u}{v\log\log H}\left(B+\frac{1}{\log
H}\left(\log(2)B+\frac{v\log\log H\cdot 6.9k\log(2\log H)}{u}\right)\right)$
We have
$Y\leq\begin{cases}&\frac{u}{v\log\log H}\left(B+0.744754115\right),\mbox{ for
}k=2\\\ &\frac{u}{v\log\log H}\left(B+0.04386773\right),\mbox{ for }k=3\\\
&\frac{u}{v\log\log H}\left(B+0.00075786\right),\mbox{ for }k=4\\\
&\frac{u}{v\log\log H}\left(B+0.00000412\right),\mbox{ for }k=5\\\
&\frac{u}{v\log\log H}\left(B+7.976\times 10^{-9}\right),\mbox{ for }k=6\\\
\end{cases}$
For $k\geq 6,$ we observe
$Y\leq\frac{u}{v\log\log H}\left(B+10^{-8}\right).$
For calculating the small values of $k$, we do it case by case.
For $k=2$, take $n=2$ and recall
$\displaystyle b=3.377257+2\log 2$ $\displaystyle d=-0.64$ $\displaystyle
a=2\mbox{ and }d=1$
which implies
$3.878864\mbox{ and }D=7.159781\qquad\frac{u}{v}\leq 1.151906$
We also have $\log H\geq s(2,2)e^{s(2,2)}=7.69\times e^{7.69}$
$\displaystyle Y$ $\displaystyle\leq\frac{1}{\log
H}\left(\frac{uB}{v}\frac{\log(2H)}{\log\log(2H)}+2\log\left(\frac{u}{v}\frac{\log(2H)}{\log\log(2H)}\right)+D+3\log
2\right)$ $\displaystyle\leq\frac{9.202255}{\log\log H}$
Let us now look at the case $k\geq 3$. For simplicity’s sake, write
$Y\leq\frac{u}{v\log\log H}(B+\theta)$
Define
$f(k,n)=\frac{u}{vk^{2}\log
k}\left(B+\theta\right)=\frac{\left(1+\frac{\log((k+n)(\log(k+n))^{2})}{(k+n)(\log(k+n))^{2}}\right)\left(1+\frac{1}{k}+\frac{0.89}{k\log
k}+\frac{0.000003+\theta}{k^{2}\log k}-\frac{0.81}{\log
k}\right)}{1-\frac{k\log k-0.81k-\log n+0.17}{(k+n)(\log(k+n))^{2}}}$
The expression can be further simplified to
$f(k,n)=\frac{\left(1+\frac{1}{(k+n)\log(k+n)}+\frac{2\log\log(k+n)}{(k+n)(\log(k+n))^{2}}\right)\left(1+\frac{1}{k}+\frac{0.89}{k\log
k}+\frac{0.000003+\theta}{k^{2}\log k}-\frac{0.81}{\log
k}\right)}{1-\frac{k\log k-0.81k-\log n+0.17}{(k+n)(\log(k+n))^{2}}}$
Let us look at the numerator. We have
$\left(1+\frac{1}{(k+n)\log(k+n)}+\frac{2\log\log(k+n)}{(k+n)(\log(k+n))^{2}}\right)\left(1+\frac{1}{k}+\frac{0.89}{k\log
k}+\frac{0.000003+\theta}{k^{2}\log k}-\frac{0.81}{\log k}\right)\\\
=1+\frac{1}{k}+\frac{0.89}{k\log k}+\frac{0.000003+\theta}{k^{2}\log
k}-\frac{0.81k}{\log
k}+\frac{1}{(k+n)\log(k+n)}+\frac{1}{k(k+n)\log(k+n)}+\frac{0.89}{k(k+n)\log
k\log(k+n)}\\\ +\frac{0.000003+\theta}{k^{2}\log
k(k+n)\log(k+n)}-\frac{0.81}{(k+n)\log(k+n)\log
k}+\frac{2\log\log(k+n)}{(k+n)(\log(k+n))^{2}}+\frac{2\log\log(k+n)}{k(k+n)(\log(k+n))^{2}}\\\
+\frac{2\cdot 0.89\log\log(k+n)}{k(k+n)\log
k(\log(k+n))^{2}}+\frac{2\cdot(0.000003+\theta)\log\log(k+n)}{k^{2}(k+n)\log
k(\log(k+n))^{2}}-\frac{2\cdot 0.81\log\log(k+n)}{(k+n)(\log(k+n))^{2}\log
k}.$
We can now verify using WolframAlpha that
$\frac{1}{k(k+n)\log(k+n)}+\frac{0.89}{k(k+n)\log
k\log(k+n)}+\frac{0.000003+\theta}{k^{2}\log
k(k+n)\log(k+n)}-\frac{0.81}{(k+n)\log(k+n)\log k}<0$
and
$\frac{2\log\log(k+n)}{k(k+n)(\log(k+n))^{2}}\\\ +\frac{2\cdot
0.89\log\log(k+n)}{k(k+n)\log
k(\log(k+n))^{2}}+\frac{2\cdot(0.000003+\theta)\log\log(k+n)}{k^{2}(k+n)\log
k(\log(k+n))^{2}}-\frac{2\cdot 0.81\log\log(k+n)}{(k+n)(\log(k+n))^{2}\log
k}<0.$
for $k\geq 3$. Furthermore, the denominator can be written as
$1-\frac{k\log
k}{(k+n)(\log(k+n))^{2}}+\frac{0.81k}{(k+n)(\log(k+n))^{2}}+\frac{\log
n}{(k+n)(\log(k+n))^{2}}-\frac{0.17}{(k+n)(\log(k+n))^{2}}\\\ >1-\frac{k\log
k}{(k+n)(\log(k+n))^{2}}+\frac{0.81k}{(k+n)(\log(k+n))^{2}},$
since $\frac{\log
n}{(k+n)(\log(k+n))^{2}}-\frac{0.17}{(k+n)(\log(k+n))^{2}}>0$. We can thus
estimate
$f(n,k)<\frac{1+\frac{1}{k}+\frac{0.89}{k\log
k}+\frac{0.000003+\theta}{k^{2}\log k}-\frac{0.81}{\log
k}+\frac{1}{(k+n)\log(k+n)}+\frac{2\log\log(k+n)}{(k+n)(\log(k+n))^{2}}}{1-\frac{k\log
k}{(k+n)(\log(k+n))^{2}}+\frac{0.81k}{(k+n)(\log(k+n))^{2}}}.$
If $k=3$, we have $f(2,3)<1.145$ and $f(3,3)<1.08$, so $f(2,3)$ gives the
larger value.
For $k=4$, we have $f(2,4)<1.114$, $f(3,4)<1.05$ and $f(4,4)<1$, so $f(2,4)$
yields the largest bound. Estimating
$\frac{1}{(k+n)\log(k+n)}+\frac{2\log\log(k+n)}{(k+n)(\log(k+n))^{2}}<\frac{0.81k}{(k+n)(\log(k+n))^{2}}$
(which is based on the fact that the second term on the left side is at most
$1+\frac{2}{e}<1.74$ and the term on the right side is at least $>1.75$ if
$k\geq 5$ and $n\leq k$), and
$\frac{k\log k}{(k+n)(\log(k+n))^{2}}<\frac{1}{\log k},$
we can further simplify the expression:
$f(n,k)<\frac{1+\frac{1}{k}+\frac{0.89}{k\log
k}+\frac{0.000003+\theta}{k^{2}\log k}-\frac{0.81}{\log
k}+\frac{0.81k}{(k+n)(\log(k+n))^{2}}}{1-\frac{1}{\log
k}+\frac{0.81k}{(k+n)(\log(k+n))^{2}}}$
Now
$\frac{1}{k}+\frac{0.89}{k\log k}+\frac{0.000003+\theta}{k^{2}\log
k}=\frac{1}{\log k}\left(\frac{\log
k}{k}+\frac{0.89}{k}+\frac{0.000003+\theta}{k^{2}\log k}\right)\\\
<\frac{1}{\log k}\left(0.299+0.149+0.0000003\right)<\frac{0.5}{\log k},$
where the terms are estimated using $k\geq 5$. Hence
$f(n,k)<\frac{1-\frac{0.31}{\log
k}+\frac{0.81k}{(k+n)(\log(k+n))^{2}}}{1-\frac{1}{\log
k}+\frac{0.81k}{(k+n)(\log(k+n))^{2}}}=1+\frac{0.69}{\log k-1+\frac{0.81k\log
k}{(k+n)(\log(k+n))^{2}}}<1+\frac{0.69}{\log k-1}$
Hence,
$Y\leq\frac{k^{2}\log k}{\log\log H}f(n,k)<\frac{k^{2}\log k}{\log\log
H}\left(1+\frac{0.69}{\log k-1}\right).$
.
## References
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* [2] Ernvall-Hytönen, A.-M., Leppälä, K., & Matala-aho, T. (2015). An explicit Baker-type lower bound of exponential values. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145(6), 1153-1182. `doi:10.1017/S0308210515000049`
* [3] Ernvall-Hytönen, A.-M., Matala-aho, T. & Seppälä, L. (2019). On Mahler’s Transcendence Measure for e. Constr Approx 49, 405–444. `https://doi.org/10.1007/s00365-018-9429-3`
* [4] Hata M. Remarks on Mahler’s Transcendence Measure for e, J. Number Theory 54 (1995), 81–92.
* [5] Hermite Ch. Sur la fonction exponentielle, C. R. Acad. Sci. 77 (1873), 18–24, 74–79, 226–233, 285–293.
* [6] Khassa D. S. and Srinivasan S. A transcendence measure for e, J. Indian Math. Soc. 56 (1991), 145–152.
* [7] Mahler, K. (1975). On a paper by A. Baker on the approximation of rational powers of e Tom 27 / 1975 Acta Arithmetica 27, 61-87. `doi:10.4064/aa-27-1-61-87`
* [8] Mahler K. Zur Approximation der Exponentialfunktion und des Logarithmus. Teil I, J. Reine Angew. Math. 166 (1931), 118–136.
* [9] Popken J. Sur la nature arithmétique du nombre e, C. R. Acad. Sci. Paris, 186 (1928), 1505–1507.
* [10] Popken J. Zur Transzendenz von e, Math. Z. 29 (1929), 525–541.
|
# Accurate quantum simulation of molecular ground and excited states with a
transcorrelated Hamiltonian
Ashutosh Kumar<EMAIL_ADDRESS>Theoretical Division, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA Ayush Asthana Department of Chemistry,
Virginia Tech, Blacksburg, VA 24061, USA Conner Masteran Department of
Chemistry, Virginia Tech, Blacksburg, VA 24061, USA Edward F. Valeev
Department of Chemistry, Virginia Tech, Blacksburg, VA 24061, USA Yu Zhang
<EMAIL_ADDRESS>Theoretical Division, Los Alamos National Laboratory, Los Alamos,
NM 87545, USA Lukasz Cincio Theoretical Division, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA Sergei Tretiak Theoretical Division,
Los Alamos National Laboratory, Los Alamos, NM 87545, USA Center for
Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, NM
87545, USA Pavel A. Dub<EMAIL_ADDRESS>Chemistry Division, Los Alamos
National Laboratory, Los Alamos, NM 87545, USA
###### Abstract
NISQ era devices suffer from a number of challenges like limited qubit
connectivity, short coherence times and sizable gate error rates. Thus,
quantum algorithms are desired that require shallow circuit depths and low
qubit counts to take advantage of these devices. We attempt to reduce quantum
resource requirements for molecular simulations on a quantum computer, a
promising application on NISQ devices, while maintaining the desired accuracy
with the help of classical quantum chemical theories of canonical
transformation and explicit correlation. In this work, compact ab initio
Hamiltonians are generated classically through an approximate similarity
transformation of the Hamiltonian with a) an explicitly correlated two-body
unitary operator with generalized pair excitations that remove the Coulombic
electron-electron singularities from the Hamiltonian and b) a unitary one-body
operator to efficiently capture the orbital relaxation effects required for
accurate description of the excited states. The resulting transcorelated
Hamiltonians are able to describe both ground and excited states of molecular
systems in a balanced manner. Using the fermionic-ADAPT-VQE method based on
the unitary coupled cluster with singles and doubles (UCCSD) ansatz and only a
minimal basis set (ANO-RCC-MB), we demonstrate that the transcorrelated
Hamiltonians can produce ground state energies comparable to the much larger
cc-pVTZ basis. This leads to a potential reduction in the number of required
CNOT gates by more than three orders of magnitude for the chemical species
studied in this work. Furthermore, using the qEOM formalism in conjunction
with the transcorrelated Hamiltonian, we reduce the errors in excitation
energies by an order of magnitude. The transcorrelated Hamiltonians developed
here are Hermitian and contain only one- and two-body interaction terms and
thus can be easily combined with any quantum algorithm for accurate electronic
structure simulations.
## I Introduction
In 1982, Feynman envisioned the idea of simulating quantum mechanical
processes occurring in nature through devices that operate on the principles
of quantum mechanics themselves [1]. Since then, a lot of important progress
has been made in the development of such quantum mechanical devices, also
referred to as quantum computers as they promise a near-exponential speed-up
(“quantum advantage”) over classical computers for a wide variety of
computational tasks[2]. Solving the many-body electronic Schrödinger equation
is quite naturally one of the most promising applications for quantum
computers. A number of algorithms based on quantum phase estimation (QPE)[3,
4], adiabatic state preparation[5, 6], variational quantum optimization[7, 8]
have been developed and refined to calculate the ground and low-lying excited
states of the many-body systems with the aim of realizing the promised
“quantum advantage” in the field of electronic structure theory. The
contemporary quantum hardware, however, is still in its infancy and suffers
from a variety of challenges, such as limited qubit connectivity, short
coherence times and sizable gate error rates. Furthermore, the mostly one-to-
one correspondence between spin-orbitals and qubits ensures that the quantum
simulations can only utilize a minimal number of qubits which can, at most,
only give a qualitative description of the desired solution. Thus, a lot of
efforts lately have focused on reducing the quantum resource requirements for
electronic structure simulations on noisy intermediate-scale quantum (NISQ)
devices[9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. A majority of these
efforts utilize the variational quantum eigensolver (VQE) algorithm in
conjunction with unitary coupled-cluster based ansatzes[8, 7, 20, 21] and are
able to produce highly compact quantum circuits through a variational
minimization of the expectation value of the Hamiltonian with respect to the
circuit parameters, and hence by construction, are more suited for
contemporary quantum hardware. Some specific examples include development of
adaptive ansatzes for simulation of ground[9, 10, 11] and excited-states[12,
13], correlation informed permutation of qubits (PermVQE)[14] or qubits
clustering (ClusterVQE)[19] approaches, construction of highly compact
molecular Hamiltonians through a basis-set free formalism[15] utilizing pair-
natural orbital (PNO) based compression [22, 23] in conjunction with multi-
resolution[24] strategies, low rank factorization techniques for approximating
operators[16]. In a similar work, Bauman et al. employed the QPE algorithm and
double unitary coupled-cluster (DUCC) formalism to downfold or embed many-body
correlation effects into active-spaces of effective Hamiltonians for both
ground[17] and excited states[18]. The explicitly correlated theories[25, 26,
27, 28], which are routinely used in classical electronic structure
calculations to accelerate the convergence of electronic energies and other
molecular properties with respect to the size of basis sets, is another
attractive approach, which has the potential to significantly reduce the
computational resources required for accurate quantum simulations. It is well
known that the traditional many-body wavefunctions generated from a
superposition of single slater determinants which are nothing but an
antisymmetrized product of one-electron orbitals fail to capture short-range
dynamic correlation effects efficiently.[26]. The singularity of the Coulombic
electron-electron interactions near the coalescent point introduces cusps[29]
in the wavefunction, which can only be described accurately by using a large
number of one-electron basis functions. The explicitly correlated methods
alleviate this problem through an explicit parametrization of the wavefunction
in terms of inter-electronic distances and are hence referred to as “R12” or
“F12” methods. Other F12-based formalisms have also been developed which focus
on removing the singularities in the Hamiltonian itself[30, 31, 32]. In the
transcorrelated Hamiltonian approach originally introduced by Boys and
Handy[30] with later improvements by Ten-no[31] and Luo[32], singularity-free
Hamiltonians are generated through a similarity transformation of the
Hamiltonian with a geminal correlation operator $\hat{A}$,
$\hat{H}\to\hat{\bar{H}}=e^{-\hat{A}}\hat{H}e^{\hat{A}}\quad.$ (1)
Yanai and Shiozaki[33] utilized ideas from both the canonical transformation
theory[34, 35] and the transcorrelated approach to construct canonical
transcorrelated (CT-F12) Hamiltonians, where unlike the previous works on the
transcorrelation theory, they used a unitary geminal operator which ensures
that the transformed Hamiltonian is Hermitian, making the formalism quite
robust and easy to use. In an earlier work by some of us[36], the CT-F12
Hamiltonian, in conjunction with the UCCSD ansatz and VQE algorithm was able
to produce near cc-pVTZ quality ground state correlation energies of several
small molecular species, with the much smaller 6-31g basis. McArdle and
Tew[37] and more recently, Sokolov and co-workers[38], also employed the
transcorrelated approach to improve the accuracy of quantum simulations, where
they had to make use of the imaginary-time evolution algorithms due to the
non-Hermiticity of their transformed Hamiltonian. In another application of
explicit correlation strategies in quantum computing, Schleich and co-
workers[39] recently employed the $[2]_{R12}$ formalism developed by Valeev
and Torheyden[40], where one- and two-body reduced density matrices obtained
from the quantum simulation was used to formulate a correction to the energy.
However, these a posteriori corrections have only been developed for ground-
state correlation energies, while a priori strategies like CT-F12 can be
potentially combined with any many-body quantum theory to calculate different
molecular properties corresponding to both ground and excited states.
In this work, we look to extend the CT-F12 formalism to the simulation of
molecular excited states as many interesting chemical phenomena in nature
involve excited states in one way or the other. Preliminary investigations
with the CT-F12 Hamiltonian revealed a very strong bias towards the ground-
state and the basis-set convergence of excited-state properties like
excitation energies was noticeably slower than the regular Hamiltonian itself.
Similar observations were also noticed in the framework of explicitly
correlated coupled-cluster response theory [41]. One reason for this
unbalanced description of the ground and excited states can be attributed to
the absence of virtual orbitals in the definition of the geminal operator.
This is quite obvious in the case of valence excited states where an accurate
simulation would require the inclusion of dynamic correlation effects between
an electron in occupied and virtual orbital in the effective Hamiltonian. For
example, the ${}^{1}P(2p\leftarrow 2s)$ state of the Be atom would be very
poorly described with the above formalism as both the occupied orbitals (1s
and 2s) are of S symmetry and hence the resulting pair won’t contribute at all
to the given excited state[42]. Furthermore, the basis-set convergence of the
energies of Rydberg-like excited states are dominated by non-dynamical
electron correlation effects, and one needs to incorporate orbital relaxations
in the effective Hamiltonian for their accurate description.
In this work, we have added pairs involving virtual orbitals in the definition
of the geminal operator along with the introduction of a singles operator in
the similarity transformation procedure. It should be noted that Watson et
al.[43] had implemented a similar approach for accurate molecular computations
on a classical computer with a minimal basis. In this paper, we develop an
improved recipe for obtaining the singles operator (see sec. II for details)
and apply the method to achieve a massive reduction in quantum resources for
the molecular ground and excited state calculations on a quantum computer.
Within the framework of VQE algorithms, different strategies have been
developed to compute the energies of the low-lying excited states of molecular
species[7, 44]. State-specific methods, that compute one excited state at a
time, include folded spectrum method [7] and orthogonality constrained VQE
method[44]. A more general approach, Quantum Krylov subspace expansion
methods[45, 46, 47, 48] diagonalize the Hamiltonian in a small subspace and
can provide a number of low-lying excited states together. In addition,
subspace expansion based on excited determinants has been proposed[49, 50],
including equation of motion (EOM) operator based qEOM method[51] that
provides a size-intensive approach for excitation energy calculations. In
order to demonstrate the advantage of transcorrelated methods in molecular
simulations on a quantum computer, we use a classical simulator to compute the
ground state energies through fermionic-ADAPT-VQE[9] and excitation energies
using the formalism of qEOM implemented on top of the ground state
calculation.
This manuscript is organized as follows: Section II introduces the formalism
of the canonical transcorrelation procedure. Computational details are
provided in Sec. III. In Sec. IV, we assess the accuracy of the
transcorrelated Hamiltonian in calculations of ground and excited states
energies of a number of small molecular species and estimate the potential
reduction in quantum resources with this approach. We give a summary of our
findings in Sec. V.
## II Theory
The molecular Hamiltonian in the second-quantized formalism can be written as,
$\hat{H}=h_{\nu}^{\mu}\hat{E}_{\mu}^{\nu}+\frac{1}{2}g_{\nu\kappa}^{\mu\lambda}\hat{E}_{\mu\lambda}^{\nu\kappa}$
(2)
where $h_{\nu}^{\mu}$ and $g_{\nu\kappa}^{\mu\lambda}$ refer to the one and
two-electron elements of the Hamiltonian and $\mu$, $\nu$, $\kappa$, $\lambda$
indices refer to the orbitals in the infinite orbital basis. Please refer to
Fig. 1 for a detailed description of the orbital spaces along with their
labels, used in this work. Here, $\hat{E}_{\mu}^{\nu}$ is the spin-free or
spin-summed excitation operator,
$E_{\mu}^{\nu}=a_{\mu\sigma}^{\dagger}a_{\nu\sigma}$, where
$a_{\mu\sigma}^{\dagger}$ and $a_{\mu\sigma}$ ($\sigma\in\\{\alpha\beta\\}$)
are the usual creation and annihilation operators respectively, with $\sigma$
referring to the the spin label. We have followed the Einstein summation
convention throughout this work.
Figure 1: Schematic notation used for different orbital spaces. OBS here
refers to the finite orbital basis while CABS is the complementary auxiliary
basis set, the orthogonal complement to the OBS space.
The canonical transcorrelated (CT) theory[33] aims to incorporate the missing
dynamic electron correlation effects into an effective Hamiltonian through a
similarity transformation,
$\hat{\bar{H}}=e^{\hat{A}^{\dagger}}\hat{H}e^{\hat{A}}$, where $\hat{A}$ is
usually an anti-Hermitian many-body operator which makes
$e^{\hat{A}^{\dagger}}$ and $e^{\hat{A}}$ operations unitary, thus maintaining
the Hermiticity of the Hamiltonian. Utilizing the Baker–Campbell–Hausdorff
(BCH) expansion, the effective Hamiltonian can be expressed in terms of
(nested) commutators,
$\hat{\bar{H}}=\hat{H}+[\hat{H},\hat{A}]+\frac{1}{2!}[[\hat{H},\hat{A}],\hat{A}]+\ldots.$
(3)
The CT theory introduces additional approximations of a) restricting the above
expansion to only double commutators and b) approximating the full Hamiltonian
($\hat{H}$) by its mean-field constituent, the Fock operator($\hat{F}$) in the
double commutator term,
$\hat{\bar{H}}\approx\hat{H}+{[\hat{H},\hat{A}]}_{1,2}+\frac{1}{2}{{[[\hat{F},\hat{A}]}_{1,2},\hat{A}]}_{1,2}\quad.$
(4)
The resulting CT-F12 Hamiltonian is correct at least through second-order in
perturbation[52]. It should be noted that similar approximations have also
been used in CC-F12 theories as well[53]. The notation $[..]_{1,2}$ means only
one- and two-body operators generated from the given commutator are retained
directly while the three-body operators are included through an approximate
decomposition into one- and two-body operators using the extended normal
ordering approach of Mukherjee and Kutzelnigg[54, 55, 56]. In spin orbitals
representation, the full decomposition can be written as[57],
$\displaystyle a_{stu}^{pqr}\hskip 0.72229pt=$
$\displaystyle\tilde{a}_{stu}^{pqr}+9\left(D_{s}^{p}\wedge
a_{tu}^{qr}\right)-36\left(D_{s}^{p}\wedge D_{t}^{q}\wedge D_{u}^{r}\right)$
(5) $\displaystyle+9\left(D_{st}^{pq}\wedge
a_{u}^{r}\right)+24\left(D_{s}^{p}\wedge
D_{t}^{q}D_{u}^{r}\right)-9\left(D_{st}^{pq}\wedge D_{u}^{r}\right)$
$\displaystyle+\lambda^{pqr}_{stu},$
where $\wedge$ denotes antisymmetrization over all upper and lower indices[57]
with the corresponding prefactor of $(\frac{1}{n})^{2}$, where $n$ is the
particle rank of the original undecomposed operator (n = 6 for three-body
decompositions). In this work, the first term $\tilde{a}_{stu}^{pqr}$, which
is the three-body fluctuation operator in normal ordered form with respect to
a reference and the last term $\lambda^{pqr}_{stu}$ which refers to the three-
body density cumulants, have been dropped from the above decomposition. The
final spin-free equations for this approximate decomposition can be found in
Ref.[56]. The mean-field one-body Fock operator in equation 3. is defined as,
$\begin{split}&\hat{F}=f_{\nu}^{\mu}\hat{E}_{\mu}^{\nu},\\\
&f_{\nu}^{\mu}=h_{\nu}^{\mu}+D_{\kappa}^{\lambda}\left(g_{\nu\kappa}^{\mu\lambda}-\frac{1}{2}g_{\kappa\nu}^{\mu\lambda}\right)\end{split}$
(6)
where,
$D_{\nu}^{\mu}=\left\langle\Psi_{0}\left|\hat{E}_{\nu}^{\mu}\right|\Psi_{0}\right\rangle$
is the one-body reduced density matrix associated with the reference
wavefunction $\Psi_{0}$. We have used the following form of the transformation
operator $\hat{A}$ in this work,
$\begin{split}&\hat{A}=\hat{A}^{\mathrm{F}12}+\hat{S^{\prime}},\\\
&\hat{A}^{\mathrm{F}12}=\frac{1}{2}G_{pq}^{\alpha\beta}\left(\hat{E}_{pq}^{\alpha\beta}-\hat{E}_{\alpha\beta}^{pq}\right),\\\
&\hat{S^{\prime}}=G_{p}^{\alpha}\left(\hat{E}_{p}^{\alpha}-\hat{E}_{\alpha}^{p}\right).\end{split}$
(7)
The amplitudes corresponding to the geminal operator $\hat{A}^{\mathrm{F}12}$
is defined as,
$G_{pq}^{\alpha\beta}=\frac{3}{8}\left\langle\alpha\beta\left|\hat{Q}_{12}\hat{F}_{12}\right|pq\right\rangle+\frac{1}{8}\left\langle\alpha\beta\left|\hat{Q}_{12}\hat{F}_{12}\right|qp\right\rangle.$
(8)
Here, we have made use of the SP ansatz[58, 59] of Ten-no where the geminal
amplitudes (1/8, 3/8) are fixed and are obtained by satisfying the first-order
cusp conditions for the singlet and triplet electron pairs respectively. We
also chose a slater-type geminal (STG) as the two-body correlation factor,
$\hat{F}_{12}(r_{12})=-\gamma^{-1}\exp\left(-\gamma r_{12}\right),$ (9)
where $\gamma$ is a scale-length parameter whose values are in practice tuned
to a given orbital basis set[60]. The strong orthogonality projector
$\hat{Q}_{12}$,
$\hat{Q}_{12}=1-\hat{V}_{1}\hat{V}_{2},$ (10)
where $\hat{V}_{i}$ projects the one-electron states into virtual orbitals
(a,b) of the orbital basis set (OBS), ensures that geminal matrix elements
involving products of virtual orbitals like $\langle
ab|\hat{Q}_{12}\hat{F}_{12}|pq\rangle=0$. Thus, all the geminal matrix
elements considered here contain at least one external (CABS) index.
In an earlier work by some of us[36], only occupied-occupied pairs were
included in the definition of the geminal operator. This made the
transcorrelated Hamiltonian very biased towards the ground state wavefunction.
The orbital pairs involving the virtual orbitals, especially the chemically
important ones (usually defined in active spaces) can contribute very
significantly to the excited state wavefunction and thus one needs to satisfy
the cusp conditions for such pairs as well. Thus, we included the occupied-
virtual and virtual-virtual geminal pairs as well in this work, so that our
Hamiltonian can treat multiple states at an equal footing. This approach is
very common in quasi-degenerate perturbation theory[61] where a multi-
configuration reference wavefunction is deployed instead of the regular
Hartree-Fock wavefunction to remove any kind of biasedness from the
Hamiltonian. However, not all virtual orbitals are equally important and one
can always generate an active space instead. One way to do this is to look at
the eigenvalues (also called occupation number) of the virtual-virtual block
of the MP2 one-body reduced density matrix[41]. and choose only those natural
virtual orbitals with occupation numbers greater than a given threshold.
Furthermore, we have also added a singles operator in the similarity
transformation procedure to incorporate orbital relaxation effects in the
Hamiltonian. In the equation of motion based formalisms, the excited state
wavefunction is often characterized by dominant contributions from the singles
excitation operator (for example $R^{a}_{i}$ amplitudes in EOM-CCSD) and thus
addition of quality singles amplitudes is essential. In order to define our
singles operator, we look towards the “CABS singles” approach usually employed
in the explicit correlation theory to accelerate the basis set convergence of
the energy of the reference wavefunction by allowing for orbital rotations
between the occupied and the missing virtual space i.e. the CABS space.
Following the works of Valeev and Kong[26], these amplitudes have been
determined from the Rayleigh-Schrödinger perturbation theory, with the
following partitioning of the Hamiltonian,
$\hat{H}^{(0)}=\left(\begin{array}[]{ccc}F_{j}^{i}&0&0\\\
0&F_{b}^{a}&F_{b}^{x}\\\ 0&F_{y}^{a}&F_{y}^{x}\end{array}\right),\hskip
3.61371pt\hat{H}^{(1)}=\left(\begin{array}[]{ccc}0&F_{i}^{a}&F_{i}^{x}\\\
F_{a}^{j}&0&0\\\ F_{y}^{j}&0&0\end{array}\right)$ (11)
where $F^{i}_{j}$, $F^{a}_{b}$, $F^{x}_{i}$ and $F^{y}_{x}$ refer to the
occupied-occupied, virtual-virtual, occupied-CABS and CABS-CABS block of the
Fock matrix respectively. In this approach, the occupied-CABS ($G^{x}_{i}$)
block of the singles amplitudes can be obtained by solving the following
equation,
$F^{j}_{i}G^{x}_{j}-F^{x}_{y}G^{y}_{i}=F^{x}_{i}.$ (12)
We chose the perturbative formulation of the “CABS singles” approach in this
work as it has been shown to work quite well for small basis sets[26]. One can
easily extend the above equation to solve for the virtual-CABS ($G^{x}_{a}$)
component of the singles amplitudes as well but this naive approach often
leads to very poor results as the resulting equations are generally very
poorly conditioned due to the small differences in orbital energies
($F^{\mu}_{\mu}$) between virtual and CABS orbitals. In order to overcome
this, we replace the virtual-virtual block of the Fock matrix by a constant
parameter $\epsilon$ which we denote as “shift” in this work,
$\epsilon*G^{x}_{a}-F^{x}_{y}G^{y}_{a}=F^{x}_{a}.$ (13)
This shift parameter is in practice very close to the HOMO energy but can be
adjusted for a given molecule and basis set to add optimal orbital relaxation
effects in the transcorrelated Hamiltonian. This concept is very similar to
the regularization strategies employed in the single reference-based
perturbation theories to treat multi-reference problems[62]. In other
approximations, following Shiozaki and Yanai,[33] $G^{cx}_{ab}$ types of
geminal amplitudes have been ignored as incorporating them would require
addition of full three body operators and not the approximated ones used in
this work.
Finally, the transcorrelated Hamiltonian has the following form,
$\hat{\bar{H}}=\bar{h}_{q}^{p}\hat{E}_{p}^{q}+\frac{1}{2}\bar{g}_{rs}^{pq}\hat{E}_{pq}^{rs},$
(14)
where $\bar{h}_{q}^{p}$ and $\bar{g}_{rs}^{pq}$ are the “perturbed” one- and
two-body interaction terms. The transcorrelated Hamiltonian ($\hat{\bar{H}}$)
is Hermitian but its two-body interaction term has a lower symmetry compared
to the original Hamiltonian:
$\overline{g}^{pq}_{rs}\neq\overline{g}^{ps}_{rq}$ while
$g^{pq}_{rs}=g^{ps}_{rq}$. The construction of the transcorrelated Hamiltonian
(with fully factorized equations) scale as $\mathcal{O}$($N^{6}$), where $N$
is the number of orbitals, with a quadratic dependence on the size of the CABS
basis when approach C[63] is used to evaluate the intermediate B.
## III Computational details
The second quantized expressions required to construct the transcorrelated
Hamiltonian were derived using a python module[64] that automates the
evaluation of single and double commutators of the Hamiltonian with excitation
and de-excitation operators of different orders using Wick’s theorem. The
resulting expressions were automatically converted to the einsum tensor
contraction routines from the numpy[65] library to generate the final
transcorrelated Hamiltonian. All the integrals and intermediates (V, X and
B)[26] were calculated using the MPQC[66] software. We used the approach C to
evaluate the matrix elements of the B intermediate.
In this study, we consider four small hydrogen-containing molecules: H2
(r(H-H) = 0.7 Å), LiH (r(Li-H) = 1.5957 Å), H2O (r(O-H) = 0.957 Å, $\theta$
(H-O-H) = 104.5°) and NH3 (r(N-H) = 1.09 Å, $\theta$ (H-N-H) = 109.427°).
6-31g basis set was used as the OBS in the calculations involving the H2
molecule while the ANO-RCC-MIN[67] basis set was employed for all others. In
the canonical transcorrelation procedure, aug-cc-pVTZ-OptRI[68] basis set was
used as the CABS basis for all the molecules studied in this work except the
LiH molecule where cc-pVDZ-F12-OptRI[69] basis set was used instead due to the
non-availability of the former. All the transcorrelated calculations utilized
the CABS+ approach[70]. Since, optimized $\gamma$ values are only available
for larger basis sets, we chose those $\gamma$ values for a given molecule and
basis set which produced ground state energies in the close vicinity of the
cc-pVTZ value. Similarly, those values of the shift parameter were chosen
which minimized the maximum deviation of the excitation energies from the
reference values. For a detailed analysis, we compare the performance of six
types of Hamiltonians in this work. $\hat{H}$ refers to the regular
untransformed Hamiltonian while $\hat{H}_{\text{F12}}^{(ij)}$ and
$\hat{H}_{\text{F12}}^{(pq)}$ refer to the transcorrelated Hamiltonians
generated by the doubles geminal operator defined by occupied-occupied and
all-all orbital pairs respectively. Consequently, addition of the singles
operator in the similarity transformation procedure leads to S’ +
$\hat{H}_{\text{F12}}^{(ij)}$ and S’ + $\hat{H}_{\text{F12}}^{(pq)}$
transcorelated Hamiltonians. Finally, S’ + $\hat{H}$ is obtained from the
similarity transformation of the regular Hamiltonian with the singles operator
only. The ground state energies associated with all these Hamiltonians were
calculated using the fermionic-ADAPT-VQE method (implemented clasically[71])
with Jordan-Wigner mapping in conjunction with the UCCSD ansatz. Excitation
energies were calculated using the qEOM formalism[51] on top of the ground
state calculations. The reference cc-pVTZ ground and excited state energies
were calculated classically using CCSD and EOM-CCSD methods respectively using
the PySCF software[72]
## IV Results and Discussions
We test the performance of the transcorrelated Hamiltonians by doing quantum
simulations of both ground and excited states on a number of small hydrogen
containing molecules: H2, LiH, H2O and NH3 using 6-31g and the minimal basis
set ANO-RCC-MB. We compare the ground state energies and excitation energies
of the few lowest-lying excited states of these molecules with the
corresponding values obtained with a much larger and more accurate basis set
cc-pVTZ.
Table 1: Deviations obtained in the ground state energy (mEh) and the excitation energies (eV) of the four lowest-lying excited states of the H2 molecule using the 6-31g basis set from the reference cc-pVTZ values (second column) for the six different Hamiltonians (see text for details). Parameters used: ($\gamma$, shift) = (0.7, -0.4) States | Reference | $\hat{H}$ | $\hat{H}_{\text{F12}}^{(ij)}$ | $\hat{H}_{\text{F12}}^{(pq)}$ | S’ + $\hat{H}_{\text{F12}}^{(ij)}$ | S’ + $\hat{H}_{\text{F12}}^{(pq)}$ | S’ + $\hat{H}$
---|---|---|---|---|---|---|---
| cc-pVTZ | | | 6-31G | |
S0 | -1.17101 | 20.8 | 6.8 | 3.3 | 2.5 | -1.6 | 16.0
T1 | 11.30 | 0.08 | 0.46 | 0.50 | 0.24 | 0.25 | -0.13
S1 | 13.89 | 1.76 | 2.14 | 1.45 | 0.26 | -0.15 | -0.11
T2 | 15.25 | 7.84 | 8.23 | 8.15 | -0.23 | -0.31 | -0.60
S2 | 17.60 | 10.75 | 11.14 | 10.43 | 1.01 | 0.17 | 0.62
### IV.1 H2
Table 1 lists the deviations obtained in the ground state energy (mEh) and the
excitation energies (eV) of the four lowest-lying excited states (arranged in
the order of increasing energies) of the H2 molecule using 6-31g basis set for
all the six types of Hamiltonians considered in this work, from the reference
cc-pVTZ values (second column). Si and Ti symbols in the first column refer to
the $i^{th}$ singlet and triplet states respectively. The values of $\gamma$
and shift parameters used in these calculations were 0.7 and -0.4
respectively. The ground state energy with the regular Hamiltonian is around
21 mEh away from the reference value. While all the transcorrelated
Hamiltonians bring this difference down, the ones with the singles operator in
the similarity transformation procedure, S’ + $\hat{H}_{\text{F12}}^{(ij)}$
and S’ + $\hat{H}_{\text{F12}}^{(pq)}$ perform the best with deviations of
only 2.5 mEh and -1.6 mEh respectively. Thus, the S’ +
$\hat{H}_{\text{F12}}^{(pq)}$ Hamiltonian produces even better energies than
the reference cc-pVTZ values and is in fact identical to the total energy
obtained using the cc-pVQZ basis, which is quite close to the complete basis
limit (CBS). Looking at the excitation energies, the
$\hat{H}_{\text{F12}}^{(ij)}$ Hamiltonian gives larger deviations than the
regular Hamiltonian itself. The deviations grow sharply as we move towards the
higher energy excited states, with a max deviation of 11.14 eV observed for
the S2 excited state. This clearly shows that an unbalanced description of the
ground and excited states is obtained when geminal operators are defined with
only occupied-occupied orbital pairs. On adding all the orbital pairs in the
geminal operator ($\hat{H}_{\text{F12}}^{(pq)}$), the max deviation is lowered
but only slightly to 10.43 eV. Adding the singles operator in the similarity
transformation on the other hand has quite a dramatic effect with a max
deviation of 1.01 eV and 0.31 eV for the S’ + $\hat{H}_{\text{F12}}^{(ij)}$
and S’ + $\hat{H}_{\text{F12}}^{(pq)}$ Hamiltonians respectively. For the
triplet excited states (T1, T2), the S’ + $\hat{H}_{\text{F12}}^{(ij)}$
Hamiltonian performs slightly better over its $pq$ counterpart but the
magnitude of improvement in excitation energies for these states are only 0.01
eV and 0.08 eV respectively. It should be noted that the correction due to the
S’ + $\hat{H}_{\text{F12}}^{(ij)}$ Hamiltonian can be seen to have a state-
specific nature. It will always be biased towards those excited states with
dominant contributions from configurations with the “regular” occupied pairs.
On the other hand, the S’ + $\hat{H}_{\text{F12}}^{(pq)}$ Hamiltonian is more
reliable as it also takes care of the missing dynamic correlation effects due
to electrons in occupied-virtual and virtual-virtual orbital pairs. Thus, it
was able to reduce the max deviation in excitation energies from 1.01 eV (S’ +
$\hat{H}_{\text{F12}}^{(ij)}$) to 0.31 eV as seen in table 1. From these
results, it’s quite clear that the S’ + $\hat{H}_{\text{F12}}^{(pq)}$
Hamiltonian should be preferred for accurate simulation of both ground and
excited states.
Furthermore, one can easily see that the basis set convergence of the energies
of these excited states are not dominated by dynamical electron correlation
effects and hence can’t be captured by the geminal operators alone. The last
column of the table illustrates this even more clearly where addition of just
the singles operator to the regular Hamiltonian (S’ + $\hat{H}$) lowers the
max deviation from 17.6 eV (regular Hamiltonian) to 0.62 eV. However, due to
the lack of transcorrelation procedure, the deviations in the ground state
energy for this Hamiltonian still remains quite high at around 16 mEh compared
to the 21 mEh obtained using the regular Hamiltonian. Thus, the quality of the
ground state wavefunction remains to be poor for the S’ + $\hat{H}$
Hamiltonian even though the differences in the ground and excited state
energies are quite accurate due to error cancellations.
Table 2: Deviations obtained in the ground state energy (mEh) and the excitation energies (eV) of the six lowest-lying excited states of the LiH molecule using the ANO-RCC-MIN basis set from the reference cc-pVTZ values (second column) for the six different Hamiltonians (see text for details). The superscript in the first column indicates the degeneracy of the given excited state. Parameters used: ($\gamma$, shift) = (0.7, -0.4) States | Reference | $\hat{H}$ | $\hat{H}_{\text{F12}}^{(ij)}$ | $\hat{H}_{\text{F12}}^{(pq)}$ | S’ + $\hat{H}_{\text{F12}}^{(ij)}$ | S’ + $\hat{H}_{\text{F12}}^{(pq)}$ | S’ + $\hat{H}$
---|---|---|---|---|---|---|---
| cc-pVTZ | | | ANO-RCC-MB | |
S0 | -8.02230 | 29.3 | 15.1 | 15.0 | 8.9 | 8.4 | 27.1
T1 | 3.26 | -0.52 | -0.14 | -0.12 | -0.18 | -0.16 | -0.64
S1 | 3.62 | -0.46 | -0.08 | -0.10 | -0.09 | -0.11 | -0.54
T2(2) | 4.24 | -0.43 | -0.04 | -0.08 | 0.00 | -0.04 | -0.48
S2(2) | 4.61 | -0.47 | -0.07 | -0.14 | -0.09 | -0.15 | -0.57
T3 | 5.77 | 2.02 | 2.41 | 2.33 | 0.24 | 0.14 | 0.20
S3 | 6.37 | 7.06 | 7.46 | 6.44 | 1.65 | 0.55 | 1.17
### IV.2 LiH
Table 2 tabulates the deviation in ground state and excitation energies of six
lowest-lying excited states of the LiH molecule in an identical layout as the
H2 molecule using the same values of $\gamma$ and shift parameters. Here, we
have used a minimal basis set ANO-RCC-MB and froze the core electrons in both
ground and excited state simulations in order to lower the number of required
qubits. The parenthesis in the first column indicates the degeneracy of the
excited state. For example, the second singlet and triplet excited states (S2,
T2) are doubly degenerate. For the ground state energy, just like before, the
transcorrelated Hamiltonians with the singles operator yield the lowest
deviations of 8.9 mEh (S’ + $\hat{H}_{\text{F12}}^{(ij)}$) and 8.4 mEh (S’ +
$\hat{H}_{\text{F12}}^{(pq)}$) respectively from the reference value. Adding
all the orbital pairs to the geminal operator and addition of the singles
operator improves the ground state energy the most. However, unlike the H2
molecule, the cc-pVDZ-F12-OptRI basis set was used as the CABS basis in the
generation of the transcorrelated Hamiltonian due to the non-availability of
the aug-cc-pVTZ-OptRI basis set for Li. We observed that the performance of
the cc-pVDZ-F12-OptRI basis is not as optimal compared to the aug-cc-pVTZ-
OptRI basis set when ANO-RCC-MB basis set is used as the OBS. Thus, these
deviations can be reduced even further with the help of an optimized CABS
basis for minimal basis sets. In the case of the excitation energies, the four
lowest-lying excited states of the LiH molecule have a weaker dependence on
the size of the basis set compared to the $H_{2}$ molecule, with a max
deviation of around 0.5 eV for the regular Hamiltonian. Contrary to the
$H_{2}$ molecule, the regular transcorrelated Hamiltonians (without singles
operators) are able to reduce the max deviation to 0.14 eV. The energies of
these excited states are thus dominated by the dynamical electron correlation
effects. Thus, no major improvements are observed by adding the singles
operators for these four states. Also, the performance of both S’ +
$\hat{H}_{\text{F12}}^{(ij)}$ and S’ + $\hat{H}_{\text{F12}}^{(pq)}$
Hamiltonians are very similar for these states with a maximum difference of
0.06 eV between the two. However, the deviation for the fifth (T3) and sixth
(S3) excited states rises sharply to 2.02 eV and 7.06 eV respectively for the
regular Hamiltonian which gets further increased to 2.41 eV and 7.46 eV for
the $\hat{H}_{\text{F12}}^{(ij)}$ Hamiltonian. The
$\hat{H}_{\text{F12}}^{(pq)}$ Hamiltonian decreases these deviations only
slightly to 2.33 eV and 6.44 eV respectively. The orbital relaxation effects
seem to be very important for these excited states, as can be seen from the
last column of table 2. Indeed, the S’ + $\hat{H}_{\text{F12}}^{(ij)}$ and S’
+ $\hat{H}_{\text{F12}}^{(pq)}$ Hamiltonians are able to bring the max
deviation down to 1.65 eV and 0.55 eV respectively. Thus, just like the H2
molecule, the results obtained using the S’ + $\hat{H}_{\text{F12}}^{(pq)}$
are the most accurate for both ground and excited states.
Table 3: Deviations obtained in the ground state energy (mEh) and the excitation energies (eV) of the seven lowest-lying excited states of the H2O molecule using the ANO-RCC-MIN basis set from the reference cc-pVTZ values (second column) for the six different Hamiltonians (see text for details). Parameters used: ($\gamma$, shift) = (1.4, -0.15) States | Reference | $\hat{H}$ | $\hat{H}_{\text{F12}}^{(ij)}$ | $\hat{H}_{\text{F12}}^{(pq)}$ | S’ + $\hat{H}_{\text{F12}}^{(ij)}$ | S’ + $\hat{H}_{\text{F12}}^{(pq)}$ | S’ + $\hat{H}$
---|---|---|---|---|---|---|---
| cc-pVTZ | | | ANO-RCC-MB | |
S0 | -76.32455 | 342.0 | 151.0 | 121.9 | 33.1 | -1.7 | 233.5
T1 | 7.56 | 3.68 | 5.16 | 3.88 | 1.50 | -0.14 | -0.05
S1 | 8.12 | 4.56 | 6.07 | 4.65 | 1.94 | 0.18 | 0.37
T2 | 9.82 | 3.46 | 4.82 | 3.48 | 1.57 | -0.06 | 0.12
T3 | 9.87 | 4.84 | 6.38 | 4.71 | 1.98 | 0.01 | 0.39
S2 | 10.14 | 5.16 | 6.53 | 4.99 | 2.33 | 0.25 | 0.66
S3 | 10.61 | 4.99 | 6.55 | 4.79 | 2.35 | 0.45 | 0.91
T4 | 11.92 | 3.89 | 5.29 | 3.78 | 1.56 | -0.17 | 0.15
### IV.3 H2O
The calculations on the H2O molecule also employed ANO-RCC-MB basis set and
frozen core settings with the values of $\gamma$ and shift parameter set to
1.4 and -0.15 respectively. From table 3, one can see the regular Hamiltonian
is 342 mEh away from the reference value. The $\hat{H}_{\text{F12}}^{(ij)}$
and $\hat{H}_{\text{F12}}^{(pq)}$ Hamiltonians are able to reduce it to 151
mEh and $\sim$ 122 mEh respectively. Thus, adding all the orbital pairs in the
geminal operator improves the ground state energy by around 29 mEh which is
quite significant. After the addition of the singles operator, we obtained
deviations of 33.1 mEh and -1.7 mEh for the S’ + $\hat{H}_{\text{F12}}^{(ij)}$
and S’ + $\hat{H}_{\text{F12}}^{(pq)}$ Hamiltonians respectively. Thus, just
like in the case of the H2 molecule, we were able to obtain better quality
ground state energies than the reference cc-pVTZ values. The excitation
energies of all the seven lowest-lying excited states of the $H_{2}$O
molecule, unlike the LiH molecule, have a stronger dependence on the size of
the basis set with a max deviation of $\sim$ 5.2 eV for the fifth excited
state (S2) using the regular Hamiltonian. The deviations increase even further
with the $\hat{H}_{\text{F12}}^{(ij)}$ Hamiltonian with a max deviation of
$\sim$ 6.6 eV, again illustrating the unbalanced treatment of the ground and
excited states by this Hamiltonian. Even the $\hat{H}_{\text{F12}}^{(pq)}$
Hamiltonian doesn’t offer any improvement over the regular Hamiltonian and
makes the deviations slightly worse for the three lowest-lying excited states
(T1, S1, T2) while only minor improvements were noted for the next four (T3,
S2, S3, T4). The low-lying excited states of the water molecule is
characterized by a mixture of Rydberg and valence correlation effects. In
essence, the $\hat{H}_{\text{F12}}^{(ij)}$ Hamiltonian has the same effect as
adding polarization functions for a more accurate description of the valence
correlation effects. However, one would require diffuse functions as well in
order to accurately describe the Rydberg character of these states. The effect
of the singles operator can be seen as an injection of diffusivity into the
Hamiltonian. Indeed, the S’ + $\hat{H}_{\text{F12}}^{(pq)}$ Hamiltonian is
able to bring down the max deviation from $\sim$ 5.2 eV (regular H) to $\sim$
0.4 eV. The S’ + $\hat{H}_{\text{F12}}^{(ij)}$ Hamiltonian on the other hand
still produces a max deviation of $\sim$ 2.3 eV. Thus, addition of all orbital
pairs in the geminal operator is very important for an accurate description of
both ground and excited states.
Table 4: Deviations obtained in the ground state energy (mEh) and the excitation energies (eV) of the eight lowest-lying excited states of the NH3 molecule using the ANO-RCC-MIN basis set from the reference cc-pVTZ values (second column) for the six different Hamiltonians (see text for details). Parameters used: ($\gamma$, shift) = (1.1, -0.30) States | Reference | $\hat{H}$ | $\hat{H}_{\text{F12}}^{(ij)}$ | $\hat{H}_{\text{F12}}^{(pq)}$ | S’ + $\hat{H}_{\text{F12}}^{(ij)}$ | S’ + $\hat{H}_{\text{F12}}^{(pq)}$ | S’ + $\hat{H}$
---|---|---|---|---|---|---|---
| cc-pVTZ | | | ANO-RCC-MB | |
S0 | -56.45043 | 270.6 | 105.3 | 72.6 | 24.3 | -13.4 | 197.7
T1 | 6.06 | 3.27 | 4.66 | 3.24 | 1.85 | 0.13 | 0.42
S1 | 6.62 | 4.06 | 5.49 | 3.86 | 2.44 | 0.53 | 1.00
T2 | 8.21 | 3.69 | 5.14 | 3.47 | 2.12 | 0.16 | 0.67
T3 | 8.21 | 3.70 | 5.15 | 3.48 | 2.13 | 0.16 | 0.68
S2 | 8.78 | 4.70 | 6.20 | 4.20 | 2.80 | 0.55 | 1.31
S3 | 8.78 | 4.71 | 6.21 | 4.21 | 2.81 | 0.55 | 1.32
T4 | 10.92 | 3.38 | 4.46 | 3.20 | 1.18 | -0.38 | 0.08
T5 | 10.93 | 3.38 | 4.46 | 3.19 | 1.17 | -0.38 | 0.08
### IV.4 NH3
For the NH3 molecule, the $\gamma$ and shift parameters were chosen to be 1.1
and -0.3 respectively, along with frozen-core settings and ANO-RCC-MB basis
set. The trends for the NH3 molecule as seen from the table 4 are quite
consistent with the previous calculations. The S’ +
$\hat{H}_{\text{F12}}^{(pq)}$ Hamiltonian again provides ground state energies
between cc-pVTZ and cc-pVQZ quality while the regular Hamiltonian yields a
deviation of $\sim$ 271 mEh from the cc-pVTZ value. Looking at the table, one
can see that three pairs ((T2,T3), (S2,S3), ((T4,T5))) of excited states are
nearly degenerate resulting in the appearance of identical deviations with all
the Hamiltonians for these states. For the excitation energies, the max
deviation produced by the S’ + $\hat{H}_{\text{F12}}^{(pq)}$ Hamiltonian is
around 0.5 eV compared to the 4.7 eV and 2.8 eV for the regular and S’ +
$\hat{H}_{\text{F12}}^{(ij)}$ Hamiltonians respectively.
Table 5: Estimation of quantum resources required to simulate the ground state of different molecules and basis sets studied in this work using the UCCSD ansatz assuming Jordan-Wigner mapping and frozen-core settings. All excitation operators are considered along with no circuit transpilation or truncation of the Hamiltonian elements Molecules | Basis | Orbitals | Qubits | Parameters | CNOT Gates
---|---|---|---|---|---
H2 | 6-31G | 4 | 8 | 15 | 768
| cc-pVTZ | 28 | 56 | 783 | 341280
LiH | ANO-RCC-MB | 5 | 10 | 24 | 1616
| cc-pVTZ | 43 | 86 | 1848 | 1249080
H2O | ANO-RCC-MB | 6 | 12 | 92 | 8064
| cc-pVTZ | 57 | 114 | 61904 | 47224272
NH3 | ANO-RCC-MB | 7 | 14 | 204 | 21072
| cc-pVTZ | 71 | 142 | 98892 | 93571664
### IV.5 Quantum resource reduction
Table 5 shows an estimate of the quantum resources required to simulate the
ground state of the molecules studied in this work along with the
corresponding basis sets assuming Jordan-Wigner mapping and frozen-core
settings. All our quantum calculations utilized the UCCSD ansatz and the
number of parameters in 5 refer to the total singles and doubles excitation
operators for the given molecule and basis set. Here, we use the number of
CNOT gates as a measure of the quantum circuit complexity. In order to
estimate the number of CNOT gates, we used the second-quantized particle-hole
formalism for describing the excitation operators and utilized the circuit
designs from Ref. [73]. The number of CNOT gates required for the
exponentiation of a given singles and doubles excitation operator was
calculated and summed to obtain the final numbers. These estimates don’t take
into account any circuit optimization or transpilation and doesn’t use any
kind of truncation schemes for the Hamiltonian matrix elements. However, they
are very useful to describe in a qualitative sense, the massive increase in
quantum resource requirements as the number of qubits increases. For example,
in the case of H2O and NH3 molecules, going from the minimal basis set of ANO-
RCC-MB to the cc-pVTZ basis set results in an increase in the number of CNOT
gates by more than 3 orders of magnitude. A smaller number of CNOT gates
corresponds to a shallow circuit with lower gate errors making the
transcorrelated formalism more suitable for quantum simulations on NISQ
devices. Furthermore, we can also reduce the number of measurements in the
qEOM procedure quite significantly. The measurement of each single matrix
element in the qEOM generalized eigenvalue problem scales as
$\mathcal{O}$($N^{4}$), where N is the number of qubits[51]. For the the NH3
molecule, this would translate into a reduction in the number of measurements
for a given matrix element by a factor of $10^{4}$ $((142/14)^{4})$.
## V Conclusions
We used the canonical transcorrelated theory to construct compact ab initio
Hamiltonians that can drastically reduce the quantum resources required for
accurate simulations of both ground and excited states of molecular systems.
In a work by some of the present authors[36], the transcorrelated Hamiltonians
that were obtained through a similarity transformation of the Hamiltonian with
an explicitly correlated two-body unitary operator greatly accelerated the
recovery of the ground state correlation energies with respect to the size of
the basis set. However, the convergence of the excited state properties like
excitation energies with these Hamiltonians show a completely different trend
and was found to be even slower than the regular Hamiltonian itself. This is
not surprising since excited states can have a very different character than
the ground state. For example, the low-lying excited states of the water
molecule are characterized by a mixture of Rydberg and valence correlation
effects but the traditional explicitly correlated methods can only capture the
missing dynamical electron correlation. Also, the previous formalism was not
able to recover the missing dynamical correlation effects between electrons in
occupied and virtual orbitals due to the absence of orbital pairs involving
virtual orbitals in the definition of the two-body geminal operator. In this
work, we have addressed all these points by re-defining the two-body geminal
operator to include all orbital pairs and added a singles operator in the
similarity transformation procedure to account for the orbital relaxation
effects, resulting in a balanced treatment of both ground and excited states.
The new transcorrelated Hamiltonians can produce ground state energies
comparable to the cc-pVTZ basis, even with a minimal basis set. Furthermore,
it can reduce the errors in the excitation energies by more than an order of
magnitude. This can potentially lead to more than one and three orders of
magnitude reduction in the number of qubits and CNOT gates respectively, in
the VQE procedure. Consequently, the quantum simulations with the
transcorrelated Hamiltonian are expected to be more noise resilient on NISQ
devices.
###### Acknowledgements.
Research presented in this article was supported by the Laboratory Directed
Research and Development (LDRD) program of Los Alamos National Laboratory
(LANL) under project number 20200056DR. LANL is operated by Triad National
Security, LLC, for the National Nuclear Security Administration of U.S.
Department of Energy (contract no. 89233218CNA000001). We thank LANL
Institutional Computing (IC) program for access to HPC resources. AK
acknowledges the help of Tanvi Gujarati (IBM, USA) for help in quantum
resource estimations.
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|
# Search for secluded dark matter towards the Galactic Centre with the ANTARES
neutrino telescope
A. Albert Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg,
France Université de Haute Alsace, F-68100 Mulhouse, France S. Alves IFIC -
Instituto de Física Corpuscular (CSIC - Universitat de València) c/
Catedrático José Beltrán, 2 E-46980 Paterna, Valencia, Spain M. André
Technical University of Catalonia, Laboratory of Applied Bioacoustics, Rambla
Exposició, 08800 Vilanova i la Geltrú, Barcelona, Spain M. Anghinolfi INFN -
Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy G. Anton
Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen Centre for
Astroparticle Physics, Erwin-Rommel-Str. 1, 91058 Erlangen, Germany M. Ardid
Institut d’Investigació per a la Gestió Integrada de les Zones Costaneres
(IGIC) - Universitat Politècnica de València. C/ Paranimf 1, 46730 Gandia,
Spain S. Ardid Institut d’Investigació per a la Gestió Integrada de les
Zones Costaneres (IGIC) - Universitat Politècnica de València. C/ Paranimf 1,
46730 Gandia, Spain J.-J. Aubert Aix Marseille Univ, CNRS/IN2P3, CPPM,
Marseille, France J. Aublin Université de Paris, CNRS, Astroparticule et
Cosmologie, F-75013 Paris, France B. Baret Université de Paris, CNRS,
Astroparticule et Cosmologie, F-75013 Paris, France S. Basa Aix Marseille
Univ, CNRS, CNES, LAM, Marseille, France B. Belhorma National Center for
Energy Sciences and Nuclear Techniques, B.P.1382, R. P.10001 12, Morocco M.
Bendahman Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013
Paris, France University Mohammed V in Rabat, Faculty of Sciences, 4 av. Ibn
Battouta, B.P. 1014, R.P. 10000 Rabat, Morocco F. Benfenati INFN - Sezione
di Bologna, Viale Berti-Pichat 6/2, 40127 Bologna, Italy Dipartimento di
Fisica e Astronomia dell’Università, Viale Berti Pichat 6/2, 40127 Bologna,
Italy V. Bertin Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France S.
Biagi INFN - Laboratori Nazionali del Sud (LNS), Via S. Sofia 62, 95123
Catania, Italy M. Bissinger Friedrich-Alexander-Universität Erlangen-
Nürnberg, Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1,
91058 Erlangen, Germany J. Boumaaza University Mohammed V in Rabat, Faculty
of Sciences, 4 av. Ibn Battouta, B.P. 1014, R.P. 10000 Rabat, Morocco M.
Bouta University Mohammed I, Laboratory of Physics of Matter and Radiations,
B.P.717, Oujda 6000, Morocco M.C. Bouwhuis Nikhef, Science Park, Amsterdam,
The Netherlands H. Brânzaş Institute of Space Science, RO-077125 Bucharest,
Măgurele, Romania R. Bruijn Nikhef, Science Park, Amsterdam, The Netherlands
Universiteit van Amsterdam, Instituut voor Hoge-Energie Fysica, Science Park
105, 1098 XG Amsterdam, The Netherlands J. Brunner Aix Marseille Univ,
CNRS/IN2P3, CPPM, Marseille, France J. Busto Aix Marseille Univ, CNRS/IN2P3,
CPPM, Marseille, France B. Caiffi INFN - Sezione di Genova, Via Dodecaneso
33, 16146 Genova, Italy D. Calvo IFIC - Instituto de Física Corpuscular
(CSIC - Universitat de València) c/ Catedrático José Beltrán, 2 E-46980
Paterna, Valencia, Spain A. Capone INFN - Sezione di Roma, P.le Aldo Moro 2,
00185 Roma, Italy Dipartimento di Fisica dell’Università La Sapienza, P.le
Aldo Moro 2, 00185 Roma, Italy L. Caramete Institute of Space Science,
RO-077125 Bucharest, Măgurele, Romania J. Carr Aix Marseille Univ,
CNRS/IN2P3, CPPM, Marseille, France V. Carretero IFIC - Instituto de Física
Corpuscular (CSIC - Universitat de València) c/ Catedrático José Beltrán, 2
E-46980 Paterna, Valencia, Spain S. Celli INFN - Sezione di Roma, P.le Aldo
Moro 2, 00185 Roma, Italy Dipartimento di Fisica dell’Università La Sapienza,
P.le Aldo Moro 2, 00185 Roma, Italy M. Chabab LPHEA, Faculty of Science -
Semlali, Cadi Ayyad University, P.O.B. 2390, Marrakech, Morocco. T. N. Chau
Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France
R. Cherkaoui El Moursli University Mohammed V in Rabat, Faculty of Sciences,
4 av. Ibn Battouta, B.P. 1014, R.P. 10000 Rabat, Morocco T. Chiarusi INFN -
Sezione di Bologna, Viale Berti-Pichat 6/2, 40127 Bologna, Italy M. Circella
INFN - Sezione di Bari, Via E. Orabona 4, 70126 Bari, Italy A. Coleiro
Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France
R. Coniglione INFN - Laboratori Nazionali del Sud (LNS), Via S. Sofia 62,
95123 Catania, Italy P. Coyle Aix Marseille Univ, CNRS/IN2P3, CPPM,
Marseille, France A. Creusot Université de Paris, CNRS, Astroparticule et
Cosmologie, F-75013 Paris, France A. F. Díaz Department of Computer
Architecture and Technology/CITIC, University of Granada, 18071 Granada, Spain
G. de Wasseige Université de Paris, CNRS, Astroparticule et Cosmologie,
F-75013 Paris, France C. Distefano INFN - Laboratori Nazionali del Sud
(LNS), Via S. Sofia 62, 95123 Catania, Italy I. Di Palma INFN - Sezione di
Roma, P.le Aldo Moro 2, 00185 Roma, Italy Dipartimento di Fisica
dell’Università La Sapienza, P.le Aldo Moro 2, 00185 Roma, Italy A. Domi
Nikhef, Science Park, Amsterdam, The Netherlands Universiteit van Amsterdam,
Instituut voor Hoge-Energie Fysica, Science Park 105, 1098 XG Amsterdam, The
Netherlands C. Donzaud Université de Paris, CNRS, Astroparticule et
Cosmologie, F-75013 Paris, France Université Paris-Sud, 91405 Orsay Cedex,
France D. Dornic Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France D.
Drouhin Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg,
France Université de Haute Alsace, F-68100 Mulhouse, France T. Eberl
Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen Centre for
Astroparticle Physics, Erwin-Rommel-Str. 1, 91058 Erlangen, Germany T. van
Eeden Nikhef, Science Park, Amsterdam, The Netherlands D. van Eijk Nikhef,
Science Park, Amsterdam, The Netherlands N. El Khayati University Mohammed V
in Rabat, Faculty of Sciences, 4 av. Ibn Battouta, B.P. 1014, R.P. 10000
Rabat, Morocco A. Enzenhöfer Aix Marseille Univ, CNRS/IN2P3, CPPM,
Marseille, France P. Fermani INFN - Sezione di Roma, P.le Aldo Moro 2, 00185
Roma, Italy Dipartimento di Fisica dell’Università La Sapienza, P.le Aldo
Moro 2, 00185 Roma, Italy G. Ferrara INFN - Laboratori Nazionali del Sud
(LNS), Via S. Sofia 62, 95123 Catania, Italy F. Filippini INFN - Sezione di
Bologna, Viale Berti-Pichat 6/2, 40127 Bologna, Italy Dipartimento di Fisica
e Astronomia dell’Università, Viale Berti Pichat 6/2, 40127 Bologna, Italy L.
Fusco Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France Y. Gatelet
Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France
P. Gay Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013
Paris, France Laboratoire de Physique Corpusculaire, Clermont Université,
Université Blaise Pascal, CNRS/IN2P3, BP 10448, F-63000 Clermont-Ferrand,
France H. Glotin LIS, UMR Université de Toulon, Aix Marseille Université,
CNRS, 83041 Toulon, France R. Gozzini IFIC - Instituto de Física Corpuscular
(CSIC - Universitat de València) c/ Catedrático José Beltrán, 2 E-46980
Paterna, Valencia, Spain R. Gracia Ruiz Nikhef, Science Park, Amsterdam, The
Netherlands K. Graf Friedrich-Alexander-Universität Erlangen-Nürnberg,
Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1, 91058
Erlangen, Germany C. Guidi INFN - Sezione di Genova, Via Dodecaneso 33,
16146 Genova, Italy Dipartimento di Fisica dell’Università, Via Dodecaneso
33, 16146 Genova, Italy S. Hallmann Friedrich-Alexander-Universität
Erlangen-Nürnberg, Erlangen Centre for Astroparticle Physics, Erwin-Rommel-
Str. 1, 91058 Erlangen, Germany H. van Haren Royal Netherlands Institute for
Sea Research (NIOZ), Landsdiep 4, 1797 SZ ’t Horntje (Texel), the Netherlands
A.J. Heijboer Nikhef, Science Park, Amsterdam, The Netherlands Y. Hello
Géoazur, UCA, CNRS, IRD, Observatoire de la Côte d’Azur, Sophia Antipolis,
France J.J. Hernández-Rey IFIC - Instituto de Física Corpuscular (CSIC -
Universitat de València) c/ Catedrático José Beltrán, 2 E-46980 Paterna,
Valencia, Spain J. Hößl Friedrich-Alexander-Universität Erlangen-Nürnberg,
Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1, 91058
Erlangen, Germany J. Hofestädt Friedrich-Alexander-Universität Erlangen-
Nürnberg, Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1,
91058 Erlangen, Germany F. Huang Aix Marseille Univ, CNRS/IN2P3, CPPM,
Marseille, France G. Illuminati Université de Paris, CNRS, Astroparticule et
Cosmologie, F-75013 Paris, France INFN - Sezione di Bologna, Viale Berti-
Pichat 6/2, 40127 Bologna, Italy Dipartimento di Fisica e Astronomia
dell’Università, Viale Berti Pichat 6/2, 40127 Bologna, Italy C. W. James
International Centre for Radio Astronomy Research - Curtin University,
Bentley, WA 6102, Australia B. Jisse-Jung Nikhef, Science Park, Amsterdam,
The Netherlands M. de Jong Nikhef, Science Park, Amsterdam, The Netherlands
Huygens-Kamerlingh Onnes Laboratorium, Universiteit Leiden, The Netherlands
P. de Jong Nikhef, Science Park, Amsterdam, The Netherlands Universiteit van
Amsterdam, Instituut voor Hoge-Energie Fysica, Science Park 105, 1098 XG
Amsterdam, The Netherlands M. Kadler Institut für Theoretische Physik und
Astrophysik, Universität Würzburg, Emil-Fischer Str. 31, 97074 Würzburg,
Germany O. Kalekin Friedrich-Alexander-Universität Erlangen-Nürnberg,
Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1, 91058
Erlangen, Germany U. Katz Friedrich-Alexander-Universität Erlangen-Nürnberg,
Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1, 91058
Erlangen, Germany N.R. Khan-Chowdhury IFIC - Instituto de Física Corpuscular
(CSIC - Universitat de València) c/ Catedrático José Beltrán, 2 E-46980
Paterna, Valencia, Spain A. Kouchner Université de Paris, CNRS,
Astroparticule et Cosmologie, F-75013 Paris, France I. Kreykenbohm Dr.
Remeis-Sternwarte and ECAP, Friedrich-Alexander-Universität Erlangen-Nürnberg,
Sternwartstr. 7, 96049 Bamberg, Germany V. Kulikovskiy INFN - Sezione di
Genova, Via Dodecaneso 33, 16146 Genova, Italy C. Lagunas Gualda Deutsches
Elektronen Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany Institut
für Physik, Humboldt-Universität zu Berlin, D-12489 Berlin, Germany R.
Lahmann Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen Centre
for Astroparticle Physics, Erwin-Rommel-Str. 1, 91058 Erlangen, Germany R. Le
Breton Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013
Paris, France S. LeStum Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille,
France D. Lefèvre Mediterranean Institute of Oceanography (MIO), Aix-
Marseille University, 13288, Marseille, Cedex 9, France; Université du Sud
Toulon-Var, CNRS-INSU/IRD UM 110, 83957, La Garde Cedex, France E. Leonora
INFN - Sezione di Catania, Via S. Sofia 64, 95123 Catania, Italy G. Levi
INFN - Sezione di Bologna, Viale Berti-Pichat 6/2, 40127 Bologna, Italy
Dipartimento di Fisica e Astronomia dell’Università, Viale Berti Pichat 6/2,
40127 Bologna, Italy M. Lincetto Aix Marseille Univ, CNRS/IN2P3, CPPM,
Marseille, France D. Lopez-Coto Dpto. de Física Teórica y del Cosmos &
C.A.F.P.E., University of Granada, 18071 Granada, Spain S. Loucatos
Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France
IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France L. Maderer
Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France
J. Manczak IFIC - Instituto de Física Corpuscular (CSIC - Universitat de
València) c/ Catedrático José Beltrán, 2 E-46980 Paterna, Valencia, Spain M.
Marcelin Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France A. Margiotta
INFN - Sezione di Bologna, Viale Berti-Pichat 6/2, 40127 Bologna, Italy
Dipartimento di Fisica e Astronomia dell’Università, Viale Berti Pichat 6/2,
40127 Bologna, Italy A. Marinelli INFN - Sezione di Napoli, Via Cintia 80126
Napoli, Italy J.A. Martínez-Mora Institut d’Investigació per a la Gestió
Integrada de les Zones Costaneres (IGIC) - Universitat Politècnica de
València. C/ Paranimf 1, 46730 Gandia, Spain B. Martino Aix Marseille Univ,
CNRS/IN2P3, CPPM, Marseille, France K. Melis Nikhef, Science Park,
Amsterdam, The Netherlands Universiteit van Amsterdam, Instituut voor Hoge-
Energie Fysica, Science Park 105, 1098 XG Amsterdam, The Netherlands P.
Migliozzi INFN - Sezione di Napoli, Via Cintia 80126 Napoli, Italy A. Moussa
University Mohammed I, Laboratory of Physics of Matter and Radiations,
B.P.717, Oujda 6000, Morocco R. Muller Nikhef, Science Park, Amsterdam, The
Netherlands L. Nauta Nikhef, Science Park, Amsterdam, The Netherlands S.
Navas Dpto. de Física Teórica y del Cosmos & C.A.F.P.E., University of
Granada, 18071 Granada, Spain E. Nezri Aix Marseille Univ, CNRS, CNES, LAM,
Marseille, France B. Ó Fearraigh Nikhef, Science Park, Amsterdam, The
Netherlands A. Păun Institute of Space Science, RO-077125 Bucharest,
Măgurele, Romania G.E. Păvălaş Institute of Space Science, RO-077125
Bucharest, Măgurele, Romania C. Pellegrino INFN - Sezione di Bologna, Viale
Berti-Pichat 6/2, 40127 Bologna, Italy Museo Storico della Fisica e Centro
Studi e Ricerche Enrico Fermi, Piazza del Viminale 1, 00184, Roma INFN -
CNAF, Viale C. Berti Pichat 6/2, 40127, Bologna M. Perrin-Terrin Aix
Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France V. Pestel Nikhef,
Science Park, Amsterdam, The Netherlands P. Piattelli INFN - Laboratori
Nazionali del Sud (LNS), Via S. Sofia 62, 95123 Catania, Italy C. Pieterse
IFIC - Instituto de Física Corpuscular (CSIC - Universitat de València) c/
Catedrático José Beltrán, 2 E-46980 Paterna, Valencia, Spain C. Poirè
Institut d’Investigació per a la Gestió Integrada de les Zones Costaneres
(IGIC) - Universitat Politècnica de València. C/ Paranimf 1, 46730 Gandia,
Spain V. Popa Institute of Space Science, RO-077125 Bucharest, Măgurele,
Romania T. Pradier Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000
Strasbourg, France N. Randazzo INFN - Sezione di Catania, Via S. Sofia 64,
95123 Catania, Italy D. Real IFIC - Instituto de Física Corpuscular (CSIC -
Universitat de València) c/ Catedrático José Beltrán, 2 E-46980 Paterna,
Valencia, Spain S. Reck Friedrich-Alexander-Universität Erlangen-Nürnberg,
Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1, 91058
Erlangen, Germany G. Riccobene INFN - Laboratori Nazionali del Sud (LNS),
Via S. Sofia 62, 95123 Catania, Italy A. Romanov INFN - Sezione di Genova,
Via Dodecaneso 33, 16146 Genova, Italy Dipartimento di Fisica
dell’Università, Via Dodecaneso 33, 16146 Genova, Italy F. Sala Laboratoire
de Physique Théorique et Hautes Énergies, CNRS, Sorbonne Université, Paris,
France A. Sánchez-Losa IFIC - Instituto de Física Corpuscular (CSIC -
Universitat de València) c/ Catedrático José Beltrán, 2 E-46980 Paterna,
Valencia, Spain INFN - Sezione di Bari, Via E. Orabona 4, 70126 Bari, Italy
F. Salesa Greus IFIC - Instituto de Física Corpuscular (CSIC - Universitat de
València) c/ Catedrático José Beltrán, 2 E-46980 Paterna, Valencia, Spain D.
F. E. Samtleben Nikhef, Science Park, Amsterdam, The Netherlands Huygens-
Kamerlingh Onnes Laboratorium, Universiteit Leiden, The Netherlands M.
Sanguineti INFN - Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy
Dipartimento di Fisica dell’Università, Via Dodecaneso 33, 16146 Genova, Italy
P. Sapienza INFN - Laboratori Nazionali del Sud (LNS), Via S. Sofia 62, 95123
Catania, Italy J. Schnabel Friedrich-Alexander-Universität Erlangen-
Nürnberg, Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1,
91058 Erlangen, Germany J. Schumann Friedrich-Alexander-Universität
Erlangen-Nürnberg, Erlangen Centre for Astroparticle Physics, Erwin-Rommel-
Str. 1, 91058 Erlangen, Germany F. Schüssler IRFU, CEA, Université Paris-
Saclay, F-91191 Gif-sur-Yvette, France J. Seneca Nikhef, Science Park,
Amsterdam, The Netherlands M. Spurio INFN - Sezione di Bologna, Viale Berti-
Pichat 6/2, 40127 Bologna, Italy Dipartimento di Fisica e Astronomia
dell’Università, Viale Berti Pichat 6/2, 40127 Bologna, Italy Th. Stolarczyk
IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France M. Taiuti
INFN - Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy Dipartimento
di Fisica dell’Università, Via Dodecaneso 33, 16146 Genova, Italy Y. Tayalati
University Mohammed V in Rabat, Faculty of Sciences, 4 av. Ibn Battouta, B.P.
1014, R.P. 10000 Rabat, Morocco S.J. Tingay International Centre for Radio
Astronomy Research - Curtin University, Bentley, WA 6102, Australia B.
Vallage Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013
Paris, France IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette,
France V. Van Elewyck Université de Paris, CNRS, Astroparticule et
Cosmologie, F-75013 Paris, France Institut Universitaire de France, 75005
Paris, France F. Versari Université de Paris, CNRS, Astroparticule et
Cosmologie, F-75013 Paris, France INFN - Sezione di Bologna, Viale Berti-
Pichat 6/2, 40127 Bologna, Italy Dipartimento di Fisica e Astronomia
dell’Università, Viale Berti Pichat 6/2, 40127 Bologna, Italy S. Viola INFN
- Laboratori Nazionali del Sud (LNS), Via S. Sofia 62, 95123 Catania, Italy
D. Vivolo INFN - Sezione di Napoli, Via Cintia 80126 Napoli, Italy
Dipartimento di Fisica dell’Università Federico II di Napoli, Via Cintia
80126, Napoli, Italy J. Wilms Dr. Remeis-Sternwarte and ECAP, Friedrich-
Alexander-Universität Erlangen-Nürnberg, Sternwartstr. 7, 96049 Bamberg,
Germany S. Zavatarelli INFN - Sezione di Genova, Via Dodecaneso 33, 16146
Genova, Italy A. Zegarelli INFN - Sezione di Roma, P.le Aldo Moro 2, 00185
Roma, Italy Dipartimento di Fisica dell’Università La Sapienza, P.le Aldo
Moro 2, 00185 Roma, Italy J.D. Zornoza IFIC - Instituto de Física
Corpuscular (CSIC - Universitat de València) c/ Catedrático José Beltrán, 2
E-46980 Paterna, Valencia, Spain J. Zúñiga IFIC - Instituto de Física
Corpuscular (CSIC - Universitat de València) c/ Catedrático José Beltrán, 2
E-46980 Paterna, Valencia, Spain (The ANTARES Collaboration)
###### Abstract
Searches for dark matter (DM) have not provided any solid evidence for the
existence of weakly interacting massive particles in the GeV-TeV mass range.
Coincidentally, the scale of new physics is being pushed by collider searches
well beyond the TeV domain. This situation strongly motivates the exploration
of DM masses much larger than a TeV. Secluded scenarios contain a natural way
around the unitarity bound on the DM mass, via the early matter domination
induced by the mediator of its interactions with the Standard Model. High-
energy neutrinos constitute one of the very few direct accesses to energy
scales above a few TeV. An indirect search for secluded DM signals has been
performed with the ANTARES neutrino telescope using data from 2007 to 2015.
Upper limits on the DM annihilation cross section for DM masses up to 6 PeV
are presented and discussed.
###### Contents
1. 1 Introduction
2. 2 Neutrinos from Dark Matter annihilations
3. 3 Detector and data set
4. 4 Analysis method
1. 4.1 Data Selection
2. 4.2 Signal Identification
5. 5 Results and discussion
6. 6 Conclusions
## 1 Introduction
Astrophysical and cosmological observations point to the existence of non-
luminous matter beyond that contained in the Standard Model (SM) of particle
physics. Among the many proposed candidates for such dark matter (DM), weakly
interacting massive particles, with a mass at the electroweak scale, have been
long looked for. They annihilate to ordinary particles detectable far from
their source, are scattered by ordinary matter, and can be produced at
colliders. No clear evidence for their existence has emerged so far from data.
This situation is encouraging the exploration of new regions of the DM
parameter space, and indeed recent years have seen a growing theoretical
interest in DM candidates heavier than about 10 TeV. This mass range is of
even more interest in light of the empty-handed searches for physics beyond
the Standard Model at the LHC, which push new physics models at scales larger
than a few TeV, see e.g. [1, 2]. In turn, these models may naturally host dark
matter candidates with a mass in a similar range, as known since a long time,
for example in supersymmetric theories [3].
Considerations of unitarity of DM annihilation processes imply the existence
of a well-known upper limit, of about 100 TeV, on the DM mass [4], see e.g.
[5, 6] for recent appraisals. This limit holds if some conditions about the
cosmological history of the universe and of DM are respected, and can for
example be easily evaded if the universe was matter dominated between the
freeze-out of dark matter interactions and Big Bang nucleosynthesis, see e.g.
[7]. Secluded DM models [8] naturally constitute a very economical framework
that realises the needed early-matter domination [9, 10, 11, 12]. Here the
dark matter particle interacts sizeably with a mediator, in turn feebly
interacting with SM particles. In these scenarios, the unitarity bound on the
mass of thermal dark matter is avoided thanks to the late time entropy
injection from decays of the mediators, which are responsible for the early-
matter domination. Dark matter masses of 100 TeV and above are therefore
allowed. Such models provide large signals in the so-called indirect detection
searches (because controlled by the dark matter-mediator interaction) with
almost no signal in direct detection and collider experiments (because
controlled by the small mediator-SM coupling). From a technical point of view,
a reliable phenomenological computation of the spectra of SM particles arising
from DM annihilations in secluded model is possible, providing an excellent
motivationfor indirect searches [12]. Indeed, the relevant energy scale is not
the heavy dark matter mass (that would demand a resummation of electroweak
radiation for111 Through all the text, units are chosen such that $c=\hbar=1$.
$m_{\text{DM}}>O(10)$ TeV, see [13] for a recent study that addressed this
challenge), but rather the sub-TeV mediator mass, where the first order
treatment of electroweak corrections [14] implemented in the tool PPPC4DMID
[15] is well under control. Therefore, despite the absence of prior bounds on
the mass of the mediator $m_{V}$ from theory, a reliable computation of the
indirect detection signals with [15] is only possible when $m_{V}<O(10)$ TeV,
otherwise electroweak radiation should be resummed also to compute decays of
the mediator. This condition on the masses also implies that the interaction
between DM particles is long-range, giving rise to phenomena like Sommerfeld
enhancement [16, 17] and bound state formation [18, 19, 20], which
significantly enhance the DM signal at present times with respect to the
‘standard’ case of short-range interactions.
Neutrino telescopes have been used for indirect searches of DM (see for
instance [21] for a recent review). The ANTARES detector has been used before
to search for DM accumulated in the Earth [22], the Sun [23] and the Galactic
Centre [24]. Moreover, there has been a specific search for secluded DM with
ANTARES looking at the Sun [25], and in the public data from IceCube [26, 27].
However, the Sun is not the best source to explore heavy DM due to absorption
of resulting particles in this dense medium, even high-energy neutrinos. Thus,
it seems for this case more appropriate to look at the Galactic Centre and
high-energy neutrinos constitute one of the very few direct accesses to energy
scales above a few TeV. This places the ANTARES telescope in a privileged
position to test this relatively unexplored mass range for dark matter, via
the search for neutrinos possibly coming from dark matter annihilations or
decays. This position is reinforced by the favourable geographical location of
the telescope with respect to the position of the Galactic Centre, where most
of the indirect signal from dark matter is expected to originate. It appears
therefore very well motivated to exploit ANTARES data to test models of dark
matter heavier than a few TeV.
This paper is organised as follows. The production spectra for heavy secluded
dark matter are detailed in Section 2. A description of the experimental setup
is presented in Section 3 (as for the detector and data set used) and 4 (as
for the analysis method). The results of this work are exposed and discussed
in Section 5, summarised and placed in further context in Section 6.
## 2 Neutrinos from Dark Matter annihilations
The neutrino signal at the ANTARES site arises from the annihilation of a pair
of dark matter particles into two mediators. They then decay into neutrinos
and/or other SM particles, which in turn will produce neutrinos via showering
and decays. The mediator lifetime is required to be shorter than about 0.1
seconds to respect limits from Big Bang nucleosynthesis [28]. With this
constraint, the mediator decay process is instantaneous from the astrophysical
point of view, and takes place entirely in the source of interest. The
baryonic matter density in the Galactic Centre is not enough to cause
distortions or absorption effects in outcoming neutrino spectra. The formation
of positronium-like bound states of DM can sizeably contribute to the signal
of interest for ANTARES, via the decay of the bound state into two or more
mediators [18]. The dark matter annihilation cross section, for which limits
will be presented here, is then to be intended as an effective cross section
taking into account also the bound state contribution (see e.g. [29, 11] for
more details).
The energy spectra of the neutrinos per single dark matter annihilation are
computed in two steps. First, the energy spectra of neutrinos from the decay
of a mediator at rest are obtained with the PPPC4DMID tool [15]. Second,
spectra are boosted to the centre of mass frame of the dark matter pair that
annihilates (see [30] for more details on this procedure). Flavour
oscillations then occur between the source and the detector site. In this
analysis, the production of three neutrino flavours was considered in the
Galactic Centre, and oscillated in the long-baseline approximation to obtain
spectra at the Earth surface. Figure 1 shows these spectra for two benchmark
values of the mediator mass $m_{V}$, and for a DM mass of 50 TeV. When
$m_{V}=50$ GeV, electroweak corrections to the spectra are not important.
Considering as an example the $V\to\nu_{\mu}\bar{\nu}_{\mu}$ channel, one can
then understand the shape of its spectrum as follows: in the mediator frame,
the spectrum consists of a delta-function, the energy of each neutrino is half
the mass of the mediator. When this delta is boosted to the frame of the DM
pair, it gives rise to neutrinos spread over all energies and up to the DM
mass, as visible in the left-hand panel of Figure 1. Instead, when
$m_{V}=1000$ GeV electroweak corrections are important, and they are for
example responsible for the “bump” visible at low energies in the
$V\to\nu_{\mu}\bar{\nu}_{\mu}$ channel: the neutrinos from the decay of the
mediator can radiate a $W$ or a $Z$ boson, which in turn will give rise to
more neutrinos at smaller energies. Analogous considerations apply to the
other $V$ decay channels. Apart from oscillation effects, the primary energy
spectra above coincide with the spectra at the ANTARES location, as neutrinos
of these energies propagate undisturbed in the Galaxy. In this analysis, the
following decay channels of mediators $V$ into Standard Model particles have
been considered:
$V\rightarrow\mu^{+}\mu^{-},\;\tau^{+}\tau^{-},\;b\bar{b},\;\nu_{\mu}\bar{\nu}_{\mu}\,.$
(1)
Each of these channels is treated independently with a branching ratio of
100%.
Figure 1: Energy distribution of the muon neutrinos plus antineutrinos at
Earth location, per single annihilation into two mediators $V$ of a pair of DM
particles each with mass of 50 TeV. The mediator decays to the SM pair
indicated in the legend, then all (anti)neutrino flavours coming from that
specific pair are included and contribute via long-distance oscillations to
the muon (anti)neutrinos at Earth location. The mediator mass is 50 GeV in the
left-hand plot and 1 TeV in the right-hand one.
## 3 Detector and data set
The ANTARES neutrino detector is situated underwater in the Mediterranean Sea
40 km offshore from Toulon (France). It is composed of 12 lines instrumented
with photomultiplier tubes for the detection of Cherenkov light [31]. ANTARES
records Cherenkov light induced by charged particles originated in the
interaction of a neutrino inside the detector or in the volume around it.
Based on these recorded signals, the neutrino energy and arrival direction are
reconstructed and constitute the main information of processed data. In the
text that follows the term neutrinos stands for $\nu$ and ${\bar{\nu}}$, as
the events generated by their interactions are seen indistinguishably in
neutrino telescopes. Muons produced in cosmic ray interactions in the
atmosphere form a very large background which is suppressed in analyses by
considering only events with arrival directions crossing the Earth. The
Galactic Centre, located at a declination of $-29.01^{\circ}$S, is visible
from the detector latitude about 70$\%$ of the time [32]. In this analysis, 9
years of muon tracks, mostly induced by upward-going $\nu_{\mu}$ charged
current (CC) interactions and collected between May 2007 and December 2015,
were searched. This sample is composed of 7637 reconstructed tracks recorded
over 2101.6 days of effective livetime. Tracks are reconstructed with a good
angular resolution of the order of 1∘ at the energies relevant for this search
[33]; this data set coincides with the one analysed in previous works [34].
Tracks are reconstructed from the calibrated positions [35] and calibrated hit
times [36] of photomultiplier hits recorded in coincidence with the event. A
quality parameter $\Lambda$ is associated to each reconstructed track, based
on a maximum likelihood obtained for the reconstruction fit [37]. In its
geometrical layout, the ANTARES detector is designed for the detection of
astrophysical neutrino fluxes, which ensures a good coverage of the energy
range necessary for neutrinos from heavy dark matter annihilation. The amount
of Cherenkov photons induced in the paths of the propagating charged particles
is proportional to the amount of deposited energy and, consequently, the
number of hit optical modules, $N_{\mbox{\tiny{HIT}}}$, is a proxy of the
neutrino energy $E_{\nu}$. A set of simulated data has been produced in
correspondence with the environmental and trigger conditions of each ANTARES
data run [38]. To reproduce the expected signal from secluded dark matter, the
simulated event energy is weighted with a factor obtained according to the
energy distributions of each annihilation channel computed following [12] and
shown in Figure 1.
## 4 Analysis method
The signature of secluded dark matter annihilation would be, as other dark
matter signals, very difficult to distinctively identify. In this analysis, a
stricter event selection has been applied with respect to previous searches
for weakly interacting massive particles [24], to setup a more assertive test
of the non-standard scenario, as detailed in Section 4.1. With the preliminary
event selection described in this section, the sample is cleaned off the
majority of atmospheric muons mis-reconstructed as upgoing that failed to be
removed by standard analysis cuts. The remaining atmospheric neutrinos plus a
possible component from dark matter annihilations compose our ‘pre-selected’
data sample. An unbinned maximum likelihood method is applied to this pre-
selected sample to search for signals of secluded dark matter over the
underlying background of atmospheric neutrinos. The discrimination between
atmospheric neutrinos and neutrinos from dark matter annihilation is based on
a space and morphology information on the location of the source, and on a
spectral information based on the knowledge of the energy distribution of each
DM annihilation channel. This method has been used in previous analyses such
as [34, 24].
### 4.1 Data Selection
A set of relaxed starting cuts is initially applied to the data sample in
order to reduce a large fraction of background from atmospheric muons, and
perform consistency checks between data and Monte Carlo simulation. Similarly
to other DM analyses by the ANTARES Collaboration [24, 37, 34], these cuts
regard quality indicators of the reconstructed events: the likelihood
$\Lambda$ for the linear fit interpolating the hit pattern, and the angular
uncertainty $\beta$ estimating the angular error on the track arrival
direction. The condition for the reconstructed event to be coming from across
the Earth is required with a cut on $\theta$, zenith angle of the
reconstructed track (with respect to an axis pointing up to the vertical).
Initially, events fulfilling $\Lambda>-5.6$, $\beta<1^{\circ}$ and
$\theta<90^{\circ}$ are selected. With these starting cuts, the suppression of
atmospheric muons, electronic noise and poorly reconstructed tracks is
ensured. A stricter set of variable cuts is then applied as summarised in what
follows. The data selection was optimised by maximizing the sensitivity on the
signal by varying $\Lambda$ (from $-5.4$ to $-4.8$ in steps of 0.2). A cut on
the neutrino energy is introduced, motivated by the shape of the energy
distribution of annihilating secluded dark matter (see Figure 1), which
favours events in the high-energy end of the spectrum. On the contrary, a
power law describes the atmospheric neutrino spectrum. In this analysis the
number of recorded light hits $N_{\mathrm{HIT}}$ is used as a proxy for the
reconstructed neutrino energy. In addition to the cuts on zenith angle,
$\Lambda$ and $\beta$, $N_{\mathrm{HIT}}$ is varied up to 200 in steps of 1.
The cut value leading to the strongest sensitivity in velocity averaged cross
section for DM annihilation $\langle\sigma v\rangle$ is chosen and applied to
the unblinded data set. While the best value for $\Lambda$ remains fixed at
$-5.2$, consistently with previous similar analyses [34], the cut value on
$N_{\mathrm{HIT}}$ varies according to the dark matter mass $m_{\mathrm{DM}}$,
annihilation channel, and mediator mass $m_{V}$; the corresponding values are
reported in Table 1. The flux of signal events is obtained dividing the number
of signal events (or limit) by the integrated acceptance, defined as the
integral of the effective area $A_{\mathrm{EFF}}$ weighted by the dark matter
annihilation spectrum
$\mathcal{A}(m_{\mathrm{DM}})=\int_{0}^{m_{\mathrm{DM}}}A_{\mathrm{EFF}}^{\nu}(E_{\nu})\frac{dN_{\nu}(E_{\nu})}{dE_{\nu}}dE_{\nu}+A_{\mathrm{EFF}}^{\bar{\nu}}(E_{\nu})\frac{dN_{\bar{\nu}}(E_{\bar{\nu}})}{dE_{\bar{\nu}}}dE_{\bar{\nu}}.$
(2)
Tightening any cut improves the purity of the signal sample but reduces the
acceptance. Initially, values for the $N_{\mathrm{HIT}}$ cuts have been chosen
such to reduce the acceptance to 90%, 75%, 50%, 25%, 10% with respect to the
uncut value (where uncut means including all other cuts). However, the values
25% and 10% result in a suppression of too many events for performing the
likelihood analysis. In these cases, it was impossible to successfully scan
the skymap looking for a signal cluster, and therefore these cut values were
not further considered in the analysis. The corresponding acceptances are
shown in Figure 2 for the four annihilation modes $V\rightarrow b\bar{b}$,
$V\rightarrow\mu^{+}\mu^{-}$, $V\rightarrow\nu_{\mu}\bar{\nu}_{\mu}$ and
$V\rightarrow\tau^{+}\tau^{-}$.
${\begin{array}[]{cc|ccccccccc|cccccc|}&&&&&&&&m_{\mathrm{DM}}&&&&&&\\\
&&&&&&\mathrm{TeV}&&&&&&&\mathrm{PeV}&\\\
&{\mathrm{channel}}&3&15&30&50&100&150&200&400&600&1&1.5&2.5&4&6\\\
\hline\cr\hline\cr&\mu&31&33&35&36&38&39&40&77&82&87&92&97&102&106\\\
m_{V}=&\tau&31&33&34&35&37&38&39&74&77&82&86&91&96&99\\\
\mathrm{50\,GeV}&b&29&31&32&32&33&34&35&36&37&38&39&40&75&78\\\
&\nu_{\mu}&31&34&36&38&40&75&78&86&91&97&102&107&111&113\\\
\hline\cr&\mu&31&33&35&36&38&39&40&77&82&87&92&97&102&106\\\
m_{V}=&\tau&31&33&34&35&37&38&39&74&77&82&86&91&96&99\\\
\mathrm{250\,GeV}&b&29&31&32&32&33&34&35&36&37&38&39&71&75&78\\\
&\nu_{\mu}&31&34&36&38&70&75&78&86&91&97&102&107&111&113\\\
\hline\cr&\mu&31&33&35&36&38&39&52&77&82&87&92&97&102&106\\\
m_{V}=&\tau&31&33&34&35&37&38&39&74&77&82&86&91&96&99\\\
\mathrm{1\,TeV}&b&29&31&32&32&33&34&35&36&37&38&39&71&75&78\\\
&\nu_{\mu}&31&34&36&38&40&75&78&86&91&97&102&107&111&113\\\ \hline\cr\\\
\end{array}}$
Table 1: Cut values on the number of hits $N_{\mathrm{HIT}}$, optimised for
best sensitivity for each dark matter mass $m_{\mathrm{DM}}$, annihilation
channel, mediator mass $m_{V}$.
Integrated acceptance [m2]
$b$channel$\mu$channel
Integrated acceptance [m2]
$\nu_{\mu}$channel$m_{\textrm{DM}}$[GeV]$m_{\textrm{DM}}$[GeV]$m_{\textrm{DM}}$[GeV]$m_{\textrm{DM}}$[GeV]$\tau$channel
Integrated acceptance [m2]
Integrated acceptance [m2]
Figure 2: Integrated acceptances, as defined in Equation 2, computed for four
annihilation spectra (indicated over each panel). Acceptances computed without
any $N_{\mathrm{HIT}}$ cut are shown with a red line; different cut values in
$N_{\mathrm{HIT}}$ as reported in Table 1, leading to a reduction of
acceptance from 90% to 10%
, are indicated with blue shades from brighter to darker respectively.
The search for a signal of secluded dark matter is optimised in a $blind$ way,
according to which the events are shuffled in right ascension to ensure the
unbiased optimisation of selection cuts. When blinding, a random number
$\alpha_{\mathrm{blind}}\in[-180^{\circ},180^{\circ}]$ is assigned as right
ascension of the reconstructed arrival direction. After establishing the best
sensitivities, the original right ascension coordinate is set back. The
blinding procedure does not alter the expected sky distribution of atmospheric
neutrinos, in which the declination coordinate is maintained.
### 4.2 Signal Identification
Based on the information about the expected signal, an unbinned maximum
likelihood algorithm has been used as a search method. Unbinned likelihood is
a fitting method based on the prior knowledge of probability distribution
functions (PDFs) of signal and background discriminating variables. The method
used here does not differ from the one already applied in other ANTARES
analyses such as [34, 24], and comprehends the following steps:
1. 1.
Computation of PDFs for the signal, based on the spectra described in [12] and
DM halo morphology described with Navarro–Frenk–White (NFW) profile of spatial
mass distribution of DM profile [39]
$\rho_{DM}(r)=\rho_{s}\;r_{s}/(r(1+r^{2}/r_{s}^{2}))$, with
$\rho_{s}=1.40\cdot 10^{7}M_{\odot}$/kpc3 and $r_{s}=16.1$ kpc [24].
Computation of PDF for the distribution of atmospheric background events is
obtained from blind data.
2. 2.
Generation of $10^{4}$ pseudo-skymaps for each DM parameter choice
($m_{\mathrm{DM}}$, $m_{V}$, annihilation channel). A number of signal events
scanned in steps of 1 between 0 and 50 is injected in addition to the total
number of events taken from the background sample. Each choice of a number of
signal events makes a population of pseudo-skymaps.
3. 3.
Maximisation of likelihood yielding a test statistic (TS) distribution for
each population of pseudo-skymaps. A convolution with a Poisson function is
performed to include fluctuations expected in the distribution of signal
events.
4. 4.
Computation of 90% confidence level (CL) median upper limit in number of
detectable events that will be referred to as sensitivity according to the
Neyman method [40]. The flux sensitivity is obtained dividing the sensitivity
on number of events by the integrated acceptance.
5. 5.
Unblinding: determine the likelihood of the real data distribution and, if no
evidence of excess, computation of limits at 90% CL on flux and velocity
averaged annihilation cross section $\langle\sigma v\rangle$. These results
are presented in Section 5.
In order to quantify the signal component, a TS is defined as the ratio
between the maximum likelihood and the likelihood of the pure background
sample. Sensitivities at 90% CL are obtained comparing the TS distribution for
different numbers of injected signal events with the median of pure background
distribution, and selecting the population which is confused with background
less than 10% of the times. The number of events $n_{s}^{*}$ reconstructed
with maximum likelihood in each set of pseudo-skymaps is subject to
fluctuations following a Poisson distribution. To include fluctuations, a
transformation through a Poisson function, $\mathcal{P}$, is performed,
returning the TS distribution $P(\mathrm{TS})$ as a function of the Poissonian
mean $\mu$:
$P\left(\mathrm{TS}(\mu)\right)=\sum_{n^{*}_{s}=1}^{N}P\left(\mathrm{TS}(n_{s}^{*})\right)\,\mathcal{P}(n_{s}^{*},\mu).$
(3)
As in similar analyses [34], to take into account systematics on the expected
number of $\nu_{\mu}$ CC reconstructed events, a smearing of the test
statistic with a 15% width Gaussian is performed [33].
## 5 Results and discussion
This search for heavy secluded dark matter is performed as a function of three
free parameters: the dark matter candidate mass $m_{\mathrm{DM}}$, the
mediator mass $m_{V}$ (with general condition $m_{V}\ll m_{\mathrm{DM}}$) and
the annihilation channel. As mentioned before, the galactic DM halo profile
has been fixed to the NFW parameterisation [39]. As explained in the previous
section, a set of optimal cuts, identified independently for each parameter
choice (14 dark matter masses, 3 mediator masses and four annihilation
channels) is applied to the data in $14\times 3\times 4$ unblindings. Data is
found to be consistent with the background-only hypothesis. Upper limits at
90% CL on the thermally averaged cross section for self-conjugate DM pair
annihilation are computed for a light ($m_{V}=50$ GeV), a medium ($m_{V}=250$
GeV) and a heavy ($m_{V}=1$ TeV) mediator. The most stringent limits are
obtained in the direct channel $\mathrm{DM}\,\mathrm{DM}\rightarrow
2\,V\rightarrow 4\nu$, which is due to the spectral shape of the 4$\nu$
annihilation mode, which among those considered yields the largest fraction of
neutrinos at high energies. Each annihilation mode is independently considered
with a branching ratio of 100%. Figures 3 and 4 display the results of the
upper limits for each channel separately, alongside with the sensitivities and
corresponding $1\sigma$ and $2\sigma$ containment bands shaded in green and
yellow respectively. 1000 sets of simulated data are generated and used to
determine the sensitivity as the mean expected exclusion, and the bands as the
680 and 950 closest lines to the sensitivity. This procedure allows one to
visualise possible statistical fluctuations of the background: the fact that
our limits stay within the bands means that data are compatible with the
background hypothesis at better than $2\sigma$. Limits have been raised to be
equal to sensitivities in case of underfluctuations, analogously to similar
ANTARES analyses [34].
Figure 3: Upper limits at 90% CL on the thermally averaged DM pair
annihilation cross section $\langle\sigma v\rangle$ for a mediator mass
$m_{V}$= 1 TeV, with 1$\sigma$ and 2$\sigma$ containment bands, for $4\mu$
(top panel, limits as blue boxes) and $4\tau$ (bottom panel, limits as orange
triangles) final states.
Figure 4: Upper limits at 90% CL on the thermally averaged DM pair
annihilation cross section $\langle\sigma v\rangle$ for a mediator mass
$m_{V}$= 1 TeV, with 1$\sigma$ and 2$\sigma$ containment bands, for $4b$ (top
panel, limits as cyan boxes) and $4\nu$ (bottom panel, limits as brown
triangles) final states.
To understand which DM models are tested, Figures 5 and 6 display also the
upper limits together with the two lines indicating the unitarity limit on the
annihilation cross section. They assume, respectively, that annihilation is
dominated by s-wave processes [4, 6]
$\sigma v<\frac{4\pi}{v}\frac{1}{m_{\text{DM}}^{2}},$ (4)
or that DM is a composite state with size $R\simeq(10~{}\text{GeV})^{-1}$
$\sigma
v<\frac{4\pi}{v}\frac{1}{m_{\text{DM}}^{2}}\Big{(}1+m_{\text{DM}}vR\Big{)}^{2}\,.$
(5)
The extra term in Eq. (5) with respect to Eq. (4) is the result of the sum
over all partial waves $\sum_{j=0}^{j_{\text{max}}}(2j+1)$, where
$j_{\text{max}}=m_{\text{DM}}vR$, see e.g. [4, 6] for more details. These two
lines should be regarded as rough indications of which models are tested by
the searches presented in this paper. One for example learns that the DM
models, for which DM masses heavier than 100 TeV can be tested, are those
where more than a single partial wave contributes significantly to the
annihilation cross section. Composite DM is a limiting case where a large
number of partial waves contributes, see e.g. [41, 42, 43]. Other DM models
that evade the unitarity limit feature an indirect detection phenomenology
analogous to the one of secluded models (e.g., supercooled composite DM [44,
45]), so they are also constrained by the searches presented here. The
interest of the limits presented in this paper goes therefore beyond the
cosmological histories with early matter domination, sketched in the
introduction.
## 6 Conclusions
A search in ANTARES data from 2007 to 2015 for a signal, coming from the
Galactic Centre, due to the annihilation of secluded dark matter particles was
presented. Data were found to be consistent with the background-only
hypothesis, so that limits on the velocity averaged cross sections for
annihilation were placed for DM candidate masses between 3 TeV and 6 PeV.
These limits have been compared with theoretical expectations for the maximal
possible annihilation signals in different models.
Previous DM searches with ANTARES have used the information on the energy of
each event (i.e. the number of hits) as an input variable for the likelihood,
and then computed limits integrating between the minimal ANTARES sensitivity
and the largest energy allowed by the signal model [24]. This search instead
constitutes the first case where the information on the energy is used to
preselect events, namely the lowest energy has been varied with the parameters
of the signal model tested, to optimise the sensitivity of ANTARES in testing
it. To the best of our knowledge, this also constitutes the first time that
any telescope tested annihilation signals from DM with masses up to the PeV
range.
## Acknowledgements
The authors acknowledge the financial support of the funding agencies: Centre
National de la Recherche Scientifique (CNRS), Commissariat à l’énergie
atomique et aux énergies alternatives (CEA), Commission Européenne (FEDER fund
and Marie Curie Program), Institut Universitaire de France (IUF), LabEx
UnivEarthS (ANR-10-LABX-0023 and ANR-18-IDEX-0001), Région Île-de-France (DIM-
ACAV), Région Alsace (contrat CPER), Région Provence-Alpes-Côte d’Azur,
Département du Var and Ville de La Seyne-sur-Mer, France; Bundesministerium
für Bildung und Forschung (BMBF), Germany; Istituto Nazionale di Fisica
Nucleare (INFN), Italy; Nederlandse organisatie voor Wetenschappelijk
Onderzoek (NWO), the Netherlands; Executive Unit for Financing Higher
Education, Research, Development and Innovation (UEFISCDI), Romania;
Ministerio de Ciencia, Innovación, Investigación y Universidades (MCIU):
Programa Estatal de Generación de Conocimiento (refs. PGC2018-096663-B-C41,
-A-C42, -B-C43, -B-C44) (MCIU/FEDER), Generalitat Valenciana: Prometeo
(PROMETEO/2020/019), Grisolía (refs. GRISOLIA/2018/119, /2021/192) and GenT
(refs. CIDEGENT/2018/034, /2019/043, /2020/049, /2021/023) programs, Junta de
Andalucía (ref. A-FQM-053-UGR18), La Caixa Foundation (ref. LCF/BQ/IN17/
11620019), EU: MSC program (ref. 101025085), Spain; Ministry of Higher
Education, Scientific Research and Innovation, Morocco, and the Arab Fund for
Economic and Social Development, Kuwait. We also acknowledge the technical
support of Ifremer, AIM and Foselev Marine for the sea operation and the CC-
IN2P3 for the computing facilities. F. Sala acknowledges funding support from
the Initiative Physique des Infinis (IPI), a research training program of the
Idex SUPER at Sorbonne Université.
Figure 5: Upper limits at 90% CL on the thermally averaged cross section for
DM pair annihilation $\langle\sigma v\rangle$, from the analysis of 9 years of
ANTARES data, for mediator masses $m_{V}$= 50 GeV, and for the mediator decay
channels 4$\mu$, 4$\tau$, 4$b$, 4$\nu_{\mu}$. The dashed lines denote the
unitarity limit on the DM annihilation cross section in two limiting cases,
one where only the $s$-wave dominates the scattering, and one where DM is a
composite object with size $R\simeq(10~{}\text{GeV})^{-1}$. In each of these
cases, the parameter space above the related line is theoretically
inaccessible, see text for more details.
Figure 6: Same as Figure 5 for mediator masses $m_{V}$= 250 GeV (top panel)
and 1 TeV (bottom panel).
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|
# Stability bounds of a delay visco-elastic rheological model with substrate
friction
Malik A. Dawi 1 Jose J. Muñoz1,2∗
1Laboratori de Càlcul Numèric (LaCàN)
Universitat Politècnica de Catalunya, Barcelona, Spain
2Dept. of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain
Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Barcelona,
Spain.
<EMAIL_ADDRESS>
###### Abstract
Cells and tissues exhibit oscillatory deformations during remodelling,
migration or embryogenesis. Although it has been shown that these oscillations
correlate with cell biochemical signalling, it is yet unclear the role of
these oscillations in triggering drastic cell reorganisation events or
instabilities, and the coupling of this oscillatory response with tested
visco-elastic properties.
We here present a rheological model that incorporates elastic, viscous and
frictional components, and that is able to generate oscillatory response
through a delay adaptive process of the rest-length. We analyse its stability
properties as a function of the model parameters and deduce analytical bounds
of the stable domain. While increasing values of the delay and remodelling
rate render the model unstable, we also show that increasing friction with the
substrate destabilise the oscillatory response. Furthermore, we numerically
verify that the extension of the model to non-linear strain measures is able
to generate sustained oscillations that alternate between stable and unstable
regions.
keywords:Oscillations, Delay differential equations, Visco-elasticity,
friction , stability, rheology, cells.
## 1 Introduction
Oscillatory cell deformations are ubiquitous and have been quantified _in
vitro_ [18, 20] and _in vivo_ , for instance in the segmented clock of mice
[27] or during _Drosophila_ fly dorsal closure [23]. These oscillations have
been associated to biochemical dynamics [13], signalling delays [19] or myosin
concentration fluctuations [7]. We here present a rheological model that
explicitly incorporates the delay between the cell length adaptation and the
current stretch.
Time delay has been included in numerous models in biology, with applications
in biochemical negative feedback [15], cell growth and division [1, 11], or
cell maturation [10], but are less common in biomechanics. In our case we
introduce this delay in an evolution law of the cell or tissue rest-length.
Such models with varying rest-length have been applied to stress relaxation
[14], morphogenesis [5], cortical mechanics [8], or endocytosis [4]. They have
the advantage of including a measurable quantity, the rest-length [26], and
also furnishing the observed visco-elastic response. We will here adapt these
models and include the delay response in conjunction with frictional or
adhesive forces from the environment or substrate.
Our visco-elastic model mimics the standard linear solid, but expressed in
terms of delay rest-length changes, which provides the oscillatory character
of the deformation. The stability of such system has been described in [17] or
in [3] for planar frictionless dynamics of monolayers. We here extend such
analysis to a frictional substrate, and deduce the stability conditions as a
function of viscous, stiffness and friction parameters.
The stability analysis is usually carried out through the inspection of the
characteristic equation [2, 22], or semi-discretisation methods [12, 25]. We
resort to the former method, and by analysing the associated Lambert function
[21, 6], we deduce strict and simple bounds of the stability region. We
compare our analysis with some numerical solutions of the Delay Differential
Equations (DDEs).
The article is organised as follows. We describe the visco-elastic model in
Section 2 together with the delay evolution law of the rest-length. In Section
3 the stability of a linear model is analysed, and some bounds as a function
of the model parameters are given. A non-linear extension is presented in
Section 4, which is solved numerically and is analysed with the help of the
results obtained in the linearised model. Our findings are finally discussed
in the Conclusions section.
## 2 Visco-elastic model with delay
We consider a material rheology that mimics the solid standard mode: a purely
elastic stress $\sigma^{e}$ in parallel with a visco-elastic stress
$\sigma^{v}$. Figure 1 shows schematically the two branches. We assume a one-
dimensional domain $D=\left[0,l(t)\right]$, with $l(t)$ a time dependent
apparent (measurable) length of the domain.
The total stress $\sigma$ in $D$ is given by the sum of elastic and
viscoelastic contributions,
$\displaystyle\sigma=\sigma^{e}+\sigma^{v},$
where each stress component is given by
$\sigma^{e}=k_{1}\varepsilon(l(t),l_{0})$ and
$\sigma^{v}=k_{2}\varepsilon^{e}(l(t),L(t))$, with $k_{1}$ and $k_{2}$ the
associated stiffness parameters. The strain measures $\varepsilon(l(t),l_{0})$
and $\varepsilon^{e}(l(t),L(t))$ will be detailed in the next sections for the
linear and non-linear models. As yet we mention that they depend, in addition
to $l(t)$, on the initial length $l_{0}=l(0)$ and the rest-length $L(t)$ of
the visco-elastic branch. This rest-length can be interpreted as an internal
variable, whose evolution mimics the viscous response of Maxwell models [16].
Figure 1: Schematic view of 1-dimensional model, illustrating both elastic and
visco-elastic branches with dissipative friction.
More specifically, $L(t)$ changes according to the following evolution law
$\dot{L}(t)=\gamma(l(t-\tau)-L(t-\tau)),t>0.$ (1)
Henceforth we denote by a superimposed dot the time derivatives, i.e.
$\dot{(\bullet)}=d(\bullet)/dt$. Parameter $\gamma>0$ is the _remodelling
rate_ , which measures the rate at which the cell adapts its length to the
difference $l(t-\tau)-L(t-\tau)$. We have introduced the delay parameter
$\tau\geq 0$ which aims at mimicking the measured time-lag between the
chemical signalling and the internal mechanical remodelling in the cell, as
measured in different systems such as Drosophila dorsal closure [7] or in
wound healing [28], and which in these cases is in the order of a few minutes.
We also include in our model a viscous friction $\sigma_{\eta}$ with the
external substrate or environment, and given by an external force
$\sigma_{\eta}(t)=-\eta\dot{l}(t)$, with $\eta\geq 0$ a viscous coefficient
(see Figure 1). In total, the balance law,
$\sigma_{\eta}=\sigma^{e}+\sigma^{v}$ reads in our case
$-\eta\dot{l}(t)=k_{1}\varepsilon^{e}(l(t),l_{0})+k_{2}\varepsilon^{e}(l(t),L(t)),\
t>0,$ (2)
which should be solved together with the evolution law in (1). Due to the
presence of the delay $\tau$, initial conditions must be specified for
$t\in[-\tau,0]$. For simplicity, we assume constant values
$\displaystyle l(t)=l_{0},\ t\in[-\tau,0],$ (3) $\displaystyle L(t)=L_{0},\
t\in[-\tau,0],$ (4)
with $l_{0}$ and $L_{0}$ given constants. In the next sections we will analyse
the stability and oscillatory regime of the system of Delay Differential
Equations (DDE) for linear and non-linear definitions of the strain measures
$\varepsilon$ and $\varepsilon^{e}$.
## 3 Stability analysis of linear model
### 3.1 Characteristic equations and analytical bounds
In order to ease the stability analysis, we assume here linear definitions of
the strain measures:
$\displaystyle\begin{aligned} \varepsilon(l(t),l_{0})&=l(t)-l_{0},\\\
\varepsilon^{e}(l(t),L(t))&=l(t)-L(t).\end{aligned}$
Inserting these expression into the balance equation (2), the set of DDE turn
into the following form:
$\displaystyle-\eta\dot{l}(t)$
$\displaystyle=k_{1}\left(l(t)-l_{0}\right)+k_{2}\left(l(t)-L(t)\right),$
$\displaystyle t>0$ (5) $\displaystyle\dot{L}(t)$
$\displaystyle=\gamma(l(t-\tau)-L(t-\tau)),$ $\displaystyle t>0$ (6)
with the initial conditions in (3). The coupled system of DDE can be written
in a compact form as
$\dot{\mathcal{L}}(t)+\mathbf{A}\mathcal{L}(t)+\mathbf{B}\mathcal{L}(t-\tau)+\mathbf{c}=\mathbf{0},t>0,$
(7)
with
$\mathcal{L}(t)=\left\\{\begin{array}[]{c}l(t)\\\ L(t)\end{array}\right\\}\ ;\
\mathbf{A}=\left[\begin{array}[]{cc}\frac{k_{1}+k_{2}}{\eta}&-\frac{k_{2}}{\eta}\\\
0&0\end{array}\right]\ ;\ \mathbf{B}=\left[\begin{array}[]{cc}0&0\\\
-\gamma&\gamma\end{array}\right]\ ;\
\mathbf{c}=\left\\{\begin{array}[]{c}\frac{k_{1}l_{0}}{\eta}\\\
0\end{array}\right\\}.$
Generally, the solution of the coupled system of DDE in (7) is characterized
qualitatively (e.g. asymptotic, synchronous, oscillatory) by the exponents or
the roots of the characteristic function [9, 22]. In order to obtain this
characteristic function, one might search for a solution in the form,
$\mathcal{L}(t)=\sum_{i}e^{m_{i}t}\mathcal{L}_{i}+\mathcal{L}_{0},$ (8)
where $\mathcal{L}_{0}$ and $\mathcal{L}_{i}$ are constant vectors that depend
on the chosen initial values, and $m_{i}\in\mathbb{C}$ are the characteristic
exponents. Clearly if all the exponent have negative real parts, i.e.
$Re(m_{i})<0$, the solution is asymptotically stable with time. Substituting
Eq. (8) into Eq. (7) gives for each term in the summation
$\left(m_{i}\mathbf{I}+\mathbf{A}+\mathbf{B}e^{-m_{i}\tau}\right)\mathcal{L}_{i}=\mathbf{0}.$
We remark that the above linear transformation must hold regardless of the
initial conditions, that is to say, the determinant must always vanish. This
allows us to express the characteristic function of the system as the
determinant of the above matrix, which gives
$f(m):=m^{2}+\gamma me^{-m\tau}+\frac{k_{1}+k_{2}}{\eta}m+\frac{\gamma
k_{1}}{\eta}e^{-m\tau}=0.$ (9)
We decompose the characteristic function to real and imaginary parts by
substituting $m=\alpha+i\beta$ and then separating each part, leading to the
following non-linear system of equations,
$\displaystyle\text{Re}\;f(m)$
$\displaystyle=\alpha^{2}-\beta^{2}+\frac{k_{1}+k_{2}}{\eta}\alpha+\gamma
e^{-\alpha\tau}\left(\left(\alpha+\frac{k_{1}}{\eta}\right)\cos(\beta\tau)+\beta\sin(\beta\tau)\right),$
(10) $\displaystyle\text{Im}\;f(m)$
$\displaystyle=2\alpha\beta+\frac{k_{1}+k_{2}}{\eta}\beta+\gamma
e^{-\alpha\tau}\left(\beta\cos(\beta\tau)-\left(\alpha+\frac{k_{1}}{\eta}\right)\sin(\beta\tau)\right).$
The stability regions in the parameters space are defined by the borders where
the number of unstable exponents changes, which means, at least one
characteristic exponents crosses the imaginary axes from left to right. In
such case Eq. (10) will have at least one solution with positive $\alpha$.
Here, we have constructed the phase diagram by solving the system in Eq. (10)
numerically while monitoring the values of $\alpha$ (see Fig. 2). If there is
at least one root with a positive $\alpha$ the solution was considered
unstable.
Figure 2: Phase diagrams for different pairs of material parameters. (a) Plane
$(k_{1},k_{2})$, (b) plane $(k_{1},\eta)$, (c) plane $(k_{2},\eta)$ and (d)
plane $(\tau,\gamma)$. The curves show stability borders for different values
of the off-plane parameters. Continuous lines are obtained with the numerical
solution of Eq. (10). Dashed lines represent the sufficient stability
condition in Eq. (11). The regions which are labeled as stable are those with
negative values for $\alpha$ and those label as unstable indicate the regions
with at least a single positive $\alpha$.
With the aim of furnishing a practical bound for detecting stable solutions,
we also give the following result:
###### Proposition 1.
The solution of the system of delay differential equations in Eq. (7) with
initial conditions in Eq. (3) is stable as long as,
$k_{1}+k_{2}-\gamma\eta-k_{1}\gamma\tau>0.$ (11)
Proof. Condition (11) is derived resorting to the results in [24], and
analysing the so-called D-curves defined as,
$\displaystyle R(\omega):=\text{Re}\;f(i\omega)$
$\displaystyle=-\omega^{2}+\gamma\left(\frac{k_{1}}{\eta}\cos(\omega\tau)+\omega\sin(\omega\tau)\right)$
(12) $\displaystyle S(\omega):=\text{Im}\;f(i\omega)$
$\displaystyle=\frac{k_{1}+k_{2}}{\eta}\omega+\gamma\left(\omega\cos(\omega\tau)-\frac{k_{1}}{\eta}\sin(\omega\tau)\right)$
(13)
with $\omega\in[0,+\infty)$. The functions $R(\omega)$ and $S(\omega)$ provide
infinite parametric curves that mark the region with constant number of
unstable characteristic exponents. In particular, we resort to Theorem 2.19 in
[24], which indicates that the zeros of Eq. (9) have no real positive parts if
and only if,
$S(\rho_{k})\neq 0\quad k=1,..,r,$ (14)
and
$\sum_{k=1}^{r}(-1)^{k}S(\rho_{k})=-1,$ (15)
where $\rho_{1}\geq...\geq\rho_{r}\geq 0$ are the non-negative roots of
$R(\omega)$, with $r$ being an odd number. Moreover, we introduce a polynomial
$S^{-}(\omega)$ which defines a lower bound for the function $S(\omega)$ such
that,
$0<S^{-}(\omega)\leq S(\omega)\quad\quad\text{for}\quad\omega\in(0,+\infty).$
(16)
In case that $S(\omega)$ satisfies the stability conditions in Eq. (14) and
(15), $S^{-}(\omega)$ will also satisfy them by construction. An adequate
choice for the polynomial $S^{-}(\omega)$ can be obtained by exploiting the
following inequalities,
$\cos(\omega\tau)\geq-1,\quad-\sin(\omega\tau)\geq-\omega\tau\quad\quad\text{for}\quad\omega\in(0,+\infty)$
which lead to,
$S^{-}(\omega)=\Big{(}\frac{k_{1}+k_{2}}{\eta}-\gamma-\frac{k_{1}}{\eta}\gamma\tau\Big{)}\omega.$
Since $\omega>0$, the condition in Eq. (16) is satisfied as long as
$k_{1}+k_{2}-\gamma\eta-k_{1}\gamma\tau>0.\quad\quad\qed$
We point out that the main benefit of Proposition 1 is that it counts in the
whole space of system parameters, giving the opportunity to cross check the
stability taking into account the relative variations of system parameters. In
the phase diagrams in the parametric space, condition (11) is indicated by the
dashed lines in Fig. 2. As it can be observed, it indicates stability regions
that are smaller then those obtained by solving numerically Eq. (10). These
plots emphasise the fact that although the bound in Eq. (16) does not provide
a necessary condition, it provides a useful sufficient stability condition.
We remark also two salient conclusion from the expression in the bound, which
are also confirmed in the phase diagrams: increasing values of $\gamma\tau$
have an unstable effect in the lengths $l(t)$ and $L(t)$, as previously
encountered in other models [17], while decreasing values of $\eta$ may render
the oscillations stable. This is an unexpected result, since increasing
viscosity has in general a stabilising or damping effects in mechanics. This
can be explained by highlighting the retardation or delay that viscosity
entails in the stress response, similar to an increase of $\tau$.
### 3.2 Numerical simulations
In order to verify the obtained stability limits, we have preformed some
numerical tests considering the one-dimensional model presented in Fig. 1. The
test mimics a previous compression state that is given by the following
initial conditions,
$\displaystyle l(t)=L(t)=1,$ $\displaystyle\tau<t\leq 0$ (17) $\displaystyle
l(-\tau)=0.9,L(-\tau)=1.$ (18)
In order to compare our results with previous values in the literature and
with more general boundary conditions, we will also test different prescribed
values of $l(t)$ and additional external forces. Indeed, in the presence of a
constant external force $f$, the equilibrium equation in (2) reads,
$\displaystyle-\eta\dot{l}(t)+f$
$\displaystyle=k_{1}\left(l(t)-l_{0}\right)+k_{2}\left(l(t)-L(t)\right)$ (19)
$\displaystyle\dot{L}(t)$
$\displaystyle=\gamma\left(l(t-\tau)-L(t-\tau)\right)$ (20)
#### 3.2.1 Unloaded free conditions
A backward Euler implicit time discretisation of equations in (19) yields the
following set of equations, which are computed sequentially,
$\begin{split}L_{n+1}&=\Delta t\gamma(l_{n-\tau}-L_{n-\tau})\\\
l_{n+1}&=\frac{1}{(\eta/\Delta t+k_{1}+k_{2})}\left(\frac{\eta}{\Delta
t}l_{n}+f_{n+1}+k_{1}L_{0}+k_{2}L_{n+1}\right)\end{split}$ (21)
We here consider the case $f_{n}=0,n=0,1,2\ldots,200/\Delta t$ and $\Delta
t=0.01$, which is found sufficiently accurate when being compared with smaller
values. The resulting evolution of $l_{n}$ and $L_{n}$ is consistent with the
stability analysis of the previous section. The presence of the delay $\tau>0$
produces oscillatory solutions for $l$ and $L$, as it can be seen in Fig. 3.
The stability of these oscillations depends on the model parameters as
indicated in the stability diagrams in Fig. 2. The first case in Fig. 3a
corresponds to stable oscillations, with parameters inside the stability
domain, while the second case in Fig, 3b yields unstable oscillations, with
parameters that exceed the stability limits.
(a) Model parameters: $k_{1}=2$, $k_{2}=3$, $\eta=8$, $\gamma=0.5$, $\tau=6$
(b) Model parameters: $k_{1}=3$, $k_{2}=2$, $\eta=8$, $\gamma=0.5$, $\tau=6$
Figure 3: Time evolution of current length and rest-length for free unloaded
conditions. (a) Parameters belonging to the stable domain. (b) Choice of
parameters that lie outside of the stable domain.
#### 3.2.2 Prescribed deformation
We here choose a constant value of the apparent length $l(t)$, with an initial
discontinuity:
$\displaystyle L(t)=L_{0}=1,$ $\displaystyle\ -\tau\leq t\leq 0,$
$\displaystyle l(-\tau)=0.9,l=l_{0}=1,$ $\displaystyle\ -\tau<t.$
In this case, $\dot{l}(t)=0,t>0$, so the the first differential gives us a
reaction force term equal to $k_{2}(l_{0}-L(t))$, while the DDE reads
$\displaystyle\dot{L}$ $\displaystyle=\gamma(l_{0}-L(t-\tau)).$
This DDE (or equivalent forms) has been extensively studied [22, 17], and is
known to yield oscillatory values of rest-length $L(t)$ whenever
$\gamma\tau>\frac{1}{e}$, and unstable oscillations whenever
$\gamma\tau>\frac{\pi}{2}$. This has been confirmed by the numerical
simulations in Fig. 4.
(a) Model parameters: $\gamma=0.35$, $\tau=4$
(b) Model parameters: $\gamma=0.35$, $\tau=5$
Figure 4: The evolution of the rest-length with fixed values for the apparent
length $l(t)$. The stability is in this case identical to the friction-less
models [17]: (a) Oscillatory solution when $\tau\gamma>\frac{1}{e}$, (b)
unstable solution arise whenever $\tau\gamma>\frac{\pi}{2}$.
#### 3.2.3 Prescribed forces
We now impose and external force $f=0.2$. Since this value only affects the
value of the vector $\mathbf{c}$ in Eq. (5), the stability is consequently
unaffected by the value of $f$. The plots in Fig. 5 confirm this fact. These
plots show the apparent length as a function of time, while the rest-length is
shown as the contourplot on the varying domain $x\in[0,l(t)]$.
(a) Model parameters: $k_{1}=1$, $k_{2}=1$, $\eta=1$, $\gamma=0.5$, $\tau=6$
(b) Model parameters: $k_{1}=1$, $k_{2}=1$, $\eta=3$, $\gamma=0.6$, $\tau=6$
Figure 5: The evolution of the current length and the rest-length (color map)
with prescribed compression forces $f$ ($f(x=0)=0.2,\quad f(x=1)=-0.2$). (a)
the solution inside the stability domain. (b) the time evolution as the
stability limit is exceeded.
## 4 Extension to non-linear: strain–based model
We now use a non-dimensional definition of the strains
$\displaystyle\varepsilon(l(t),l_{0})$
$\displaystyle=\frac{l(t)-l_{0}}{l_{0}},$
$\displaystyle\varepsilon^{e}(l(t),L(t))$
$\displaystyle=\frac{l(t)-L(t)}{L(t)}.$
While this is a more common strain measure, with non-dimensional values, these
expressions, when inserted into the equilibrium equations in (2) yield a set
of non-linear DDE:
$\displaystyle-\eta\dot{l}(t)$
$\displaystyle=k_{1}\left(\frac{l(t)-l_{0}}{l_{0}}\right)+k_{2}\left(\frac{l(t)-L(t)}{L(t)}\right),$
(22) $\displaystyle\dot{L}(t)$
$\displaystyle=\gamma\left(l(t-\tau)-L(t-\tau)\right).$ (23)
We aim at studying the oscillatory character and stability of these equations.
However, due to their non-linearity we cannot directly apply the methodology
previously presented. We aim instead at analysing the linearised form of
equation (22) at time $t_{0}$. By setting $\delta l(t)=l(t)-l(t_{0})$ and
$\delta L(t)=L(t)-L(t_{0})$, the linear terms read,
$\displaystyle-\eta\delta\dot{l}(t)=$ $\displaystyle\frac{k_{1}}{l_{0}}\delta
l+\frac{k_{2}}{L(t_{0})}\delta l(t)-\frac{k_{2}l(t_{0})}{L(t_{0})^{2}}\delta
L(t).$ (24)
It then follows that by defining the modified stiffness parameters,
$\displaystyle\hat{k}_{1}$
$\displaystyle=\frac{k_{1}}{l(t_{0})}+\frac{k_{2}}{L(t_{0})}\left(1-\frac{l(t_{0})}{L(t_{0})}\right),$
(25) $\displaystyle\hat{k}_{2}$
$\displaystyle=\frac{k_{2}l(t_{0})}{L(t_{0})^{2}},$ (26)
equation (24) is equivalent to the linear terms in the equilibrium equation in
(5), but replacing $(k_{1},k_{2})$ by $(\hat{k}_{1},\hat{k}_{2})$ and in terms
of $\delta l(t)$ and $\delta L(t)$ instead of $l(t)$ and $L(t)$. This allows
us to understand some of the numerical solutions obtained for the non-linear
case.
Figure 6a shows the time evolution of $l(t)$ and $L(t)$, which are sustained,
that is, their asymptotic behaviour does not increase nor decrease. We plot in
the parametric space of $k_{1}$ and $k_{2}$ the modified parameters
$\hat{k}_{1}$ and $\hat{k}_{2}$ for each time $t_{0}$, as shown in Fig. 6b. It
can be observed that although the initial values are located in the unstable
region, they in turn oscillate between the unstable and stable region,
reaching a limit cycle that alternates between the two domains.
(a) Time evolution of current length and rest-length
(b) The evolution of $\tilde{k_{1}}$ and $\tilde{k_{2}}$.
Figure 6: Numerical solution with sustained oscillations of the non-linear
model. Parameters: $k_{1}=1$, $k_{2}=1$, $\eta=3$, $\gamma=0.6$, $\tau=5$
We have also tested other parameter settings, with an initial location of
($\hat{k}_{1},\hat{k}_{2}$) in the parametric space farther from the stability
boundary (see Fig. 7). In this case, the system exhibits oscillations that
reach the singular value $L(t)=0$ for some $t>0$, which renders the DDEs in
(22) ill-posed. Instead, when using values that are farther inside the
stability region, as it is the case in Fig. 8, the oscillations stabilise
before reaching this singular value. Although we are not able to furnish
bounds for non-linear stability, we can explain the presence of stable,
sustained, or unstable (or singular) oscillations according to the distance of
the initial value of $(\hat{k}_{1},\hat{k}_{2})$ to the stability boundary of
the linear case.
(a) Time evolution of current length and rest-length
(b) The evolution of $\tilde{k_{1}}$ and $\tilde{k_{2}}$
Figure 7: Numerical solution with unstable oscillations on the non-linear
model. Parameters: $k_{1}=2$, $k_{2}=1$, $\eta=3$, $\gamma=0.6$, $\tau=5$
(a) Time evolution of current length and rest-length
(b) The evolution of $\tilde{k_{1}}$ and $\tilde{k_{2}}$
Figure 8: Numerical solution with stable oscillations of the non-linear model.
Parameter $k_{1}=1$, $k_{2}=2$, $\eta=3$, $\gamma=0.6$, $\tau=5$
## 5 Conclusions
Motivated by the presence of delays and visco-elastic response of tissues, we
have presented a rheological model that includes elastic and viscous
contributions, and also exhibits oscillatory behaviour.
We have analysed the stability of he model when using a linear strain measure
and as a function of the model parameters. We have recovered previous results,
which show that increasing values of the delay $\tau$ and the remodelling rate
$\gamma$ (a quantity that is inversely proportional to tissue viscosity),
render the oscillations unstable. Remarkably, increasing values of the viscous
friction of the domain with respect to external boundary also destabilise the
system.
By studying the characteristic function of the DDE we have provided sufficient
conditions of stability and bounds to the stability region. This analysis have
also allowed us to explain the presence of sustained oscillations in a non-
linear version of the model. This persistent oscillations in the tissue
deformations are frequently observed [18, 20], and in our model are due to the
transition between stable and unstable domains.
We note that despite visco-elastic models based on rest-length changes are
increasingly common [4, 5, 14], their stability in the presence of delayed
response has not been studied. We here provide such an analysis which may also
help to explain the observed sudden deformations in embryo development and
morphogenesis.
## acknowledgements
JJM and MD have been financially supported by the Spanish Ministry of Science,
Innovation and Universities (MICINN) with grant DPI2016-74929-R and by the
local government _Generalitat de Catalunya_ with grant 2017 SGR 1278.
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11footnotetext: ∗ Indicates equal contribution.22footnotetext: ${\dagger}$
Indicates corresponding author.
# Seq1F1B: Efficient Sequence-Level Pipeline Parallelism for Large Language
Model Training
Ao Sun1,∗ Weilin Zhao2,∗ Xu Han2,† Cheng Yang1,† Zhiyuan Liu2 Chuan Shi1
Maosong Sun2
1 Beijing University of Posts and Telecommunications, Beijing, China.
2 NLP Group, DCST, IAI, BNRIST, Tsinghua University, Beijing, China.
{maydomine<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract
The emergence of large language models (LLMs) relies heavily on distributed
training strategies, among which pipeline parallelism plays a crucial role. As
LLMs’ training sequence length extends to 32k or even 128k, the current
pipeline parallel methods face severe bottlenecks, including high memory
footprints and substantial pipeline bubbles, greatly hindering model
scalability and training throughput. To enhance memory efficiency and training
throughput, in this work, we introduce an efficient sequence-level one-
forward-one-backward (1F1B) pipeline scheduling method tailored for training
LLMs on long sequences named Seq1F1B. Seq1F1B decomposes batch-level
schedulable units into finer sequence-level units, reducing bubble size and
memory footprint. Considering that Seq1F1B may produce slight extra bubbles if
sequences are split evenly, we design a computation-wise strategy to partition
input sequences and mitigate this side effect. Compared to competitive
pipeline baseline methods such as Megatron 1F1B pipeline parallelism, our
method achieves higher training throughput with less memory footprint.
Notably, Seq1F1B efficiently trains a LLM with 30B parameters on sequences up
to 64k using 64 NVIDIA A100 GPUs without recomputation strategies, a feat
unachievable with existing methods. Our source code is based on Megatron-LM,
and now is avaiable at: https://github.com/MayDomine/Seq1F1B.git.
## 1 Introduction
In recent years, there has been a growing interest in large language models
(LLMs), which have revolutionized various tasks in natural language processing
Touvron et al. [2023], Reid et al. [2024], Jiang et al. [2024], Anil et al.
[2023]. Efficient distributed training strategies Korthikanti et al. [2023],
Rasley et al. [2020], Rajbhandari et al. [2020], Shoeybi et al. [2019],
Narayanan et al. [2021a] play a crucial role in training these large models.
Among these strategies, pipeline parallelism Shoeybi et al. [2019], Huang et
al. [2019], Yang et al. [2021], Qi et al. [2024], Li et al. [2021] stands out
due to its low communication bandwidth requirements and high scalability when
integrated with other distributed training strategies.
Fundamentally, pipeline parallelism involves partitioning the model into
multiple stages, with each computing device responsible for processing a stage
consisting of consecutive layers. This setup inherently leads to “bubbles”—the
idle time caused by the dependencies between the computation of sharded
layers. Several scheduling strategies, such as GPipe Huang et al. [2019], are
proposed to address this problem, significantly reducing pipeline bubbles by
splitting each mini-batch of training examples into several micro-batches.
These strategies come at the expense of increased memory usage, as each stage
must store all hidden states of the micro-batches generated during the forward
passes until the backward passes are completed. Based on GPipe, TeraPipe Li et
al. [2021] further splits each micro-batch along the sequence dimension into
micro-sequences to further reduce pipeline bubbles, but still suffer from the
high memory demand for storing the hidden states of micro-batches.
To address the issue of high memory demand, one-forward-one-backward (1F1B)
scheduling strategies are proposed Harlap et al. [2018], Fan et al. [2021],
Narayanan et al. [2021a] to rewrite GPipe to schedule backward passes in
advance and make backward passes have higher execution priority than forward
passes, without affecting final results. By adopting 1F1B parallel strategies,
the memory demand for storing hidden states can be significantly reduced
without adding extra pipeline bubbles. Other methods such as zero-bubble-
pipeline Qi et al. [2024] and 1F1B-I (1F1B with interleaved stages) Narayanan
et al. [2021a] seek to further reduce the bubbles of 1F1B strategies but at
the cost of more memory overhead and communication cost. Generally, optimizing
pipeline parallelism continues to handle trade-offs between bubble ratio and
memory overhead.
On the other hand, some recent efforts Buckman and Gelada , Reid et al. [2024]
have noticed that long-sequence training benefits LLMs in many aspects,
leading to increasingly longer training contexts for LLMs. However, LLMs can
not simply support training longer sequences due to the quadratic time and
memory complexities of Transformer attention modules in terms of sequence
length Vaswani et al. [2017]. Several efforts Dao et al. [2022], Ding et al.
[2023], Jacobs et al. [2023] to build efficient attention modules have been
proposed to address this issue. Even so, the challenge caused by long
sequences extends beyond attention modules. In distributed training scenarios,
long sequences may cause various parallel methods to fail. For pipeline
parallelism, such as GPipe Huang et al. [2019] and 1F1BHarlap et al. [2018],
Fan et al. [2021], Narayanan et al. [2021a], Qi et al. [2024], these methods
can only use micro-batches as the minimal scheduling units. In extreme cases,
a single micro-batch consisting of long sequences can lead to memory overflow.
Long sequences make the issue of high memory demand in pipeline parallelism
more serious.
A straightforward approach to solving such a problem is to split the micro-
batches along the sequence dimension. However, such simple modification is
challenging for existing 1F1B scheduling strategies such as Qi et al. [2024]
and 1F1B-I Narayanan et al. [2021a] because there are computation dependencies
between forward and backward passes across micro-sequences, making a direct
split along the sequence dimension unfeasible.
To solve such a challenge, we introduce the Seq1F1B, an efficient sequence-
level 1F1B schedule, which has higher efficiency and lower memory demands than
the traditional 1F1B methods. In detail, we introduce a partially ordered
scheduling queue in Seq1F1B to replace the first-in-first-out (FIFO)
scheduling queue in 1F1B, rewriting the scheduling strategy to preserve the
exact forward and backward semantics while providing synchronous pipeline
parallelism. To further improve Seq1F1B, we propose a strategy for balancing
the workload across sub-sequence computations. More specifically, we balance
the sub-sequence computation by designing a solution based on floating-point
operations on each sub-sequence. In this design, we addressed the imbalance of
computational workload caused by the attention mechanism by splitting
sequences based on computational workloads rather than simply dividing them
evenly along the sequence dimension.
Sufficient experiments demonstrate that Seq1F1B significantly outperforms both
the 1F1B and 1F1B-I scheduling strategies in terms of memory efficiency and
training throughput for training LLMs, with the sequence length ranging from
16k to 128k and the model size ranging from 2.7B to 32B. From the experimental
results, the efficiency of Seq1F1B becomes more pronounced as the sequence
length increases and Seq1F1B supports efficiently training a GPT with 30B
parameters on sequences up to 64k tokens using 64 NVIDIA A100 GPUs without any
recomputation strategies, which is unachievable with existing pipeline
parallel methods.
## 2 Related Work
Training large language models requires using a mixture of parallel
strategies, the most important of which are data parallelism, tensor
parallelism, and pipeline parallelism. Data parallelism scales training models
by distributing data across multiple devicesGoyal et al. [2017], Zinkevich et
al. [2010], Li et al. [2020], Xing et al. [2015], each device hosting a model
replica and synchronizing gradients. Zero redundancy optimizer (ZeRO)
Rajbhandari et al. [2020], Rasley et al. [2020], Ren et al. [2021] enhances
data parallelism’s memory efficiency by partitioning model parameters across
devices at the cost of significant communication. Tensor parallelismShoeybi et
al. [2019], Korthikanti et al. [2023] parallelize computation by partitioning
matrix multiplication. In such way, Tensor parallelism effectively enhances
computation efficiency but introduces high communication costs of aggregating
the results of matrix multiplication, making it commonly used within the
multiple workers of a single node. Since this paper focuses on improving
pipeline parallelism, we will show more details for pipeline parallelism next.
For pipeline parallelism, schedules can be broadly categorized into two main
types: synchronous and asynchronous. Asynchronous schedules such as
asynchronous PipeDream Harlap et al. [2018] and PipeMare Yang et al. [2021]
can achieve bubble-free but suffer from the performance degradation of final
trained models because they use outdated parameters to compute gradient
updates. In view of this, our work focuses on synchronous pipeline schedules,
as they ensure consistent semantics across different model parallel
strategies. GPipeHuang et al. [2019], Li et al. [2021] and 1F1BFan et al.
[2021], Narayanan et al. [2021a, b] are the most commonly used pipeline
schedules following synchronous settings. Many other works are built upon
these two foundation schedules.
The original GPipeHuang et al. [2019] simply divides a mini-batch into several
micro-batches. The scheduling process of GPipe has only two phases: the
forward and the backward phases. Only after all forward passes for the micro-
batches within a batch are complete will the backward passes be executed.
During the forward phase, the hidden states of each micro-batch are enqueued
into a first-in-first-out (FIFO) queue $Q$. During the backward phase, these
hidden states are dequeued for their corresponding backward passes. Since the
backward phase happens after all hidden states are queued, GPipe exhibits an
$O(M)$ memory consumption, where $M$ represents the number of micro-batches.
Based on GPipe, other methods such as TeraPipe Li et al. [2021] and Chimera Li
and Hoefler [2021] further optimize the bubble ratio of GPipe through
different techniques. TeraPipe relies on the observation of causal language
modeling — the computation of a given input token only depends on its previous
tokens. Specifically, TeraPipe divides GPipe’s micro-batch into multiple token
spans and replaces the FIFO queue with a last-in-first-out (LIFO) queue to
ensure the correct computation of gradients during attention backward passes.
By leveraging finer scheduling units, TeraPipe effectively reduces the bubble
ratio while being more memory-efficient than GPipe. Chimera adopts
bidirectional pipeline parallelism, where each computing device is responsible
for the workload of multiple stages. While this approach reduces the bubble
ratio, each device has to store redundant parameters (as stages are not evenly
distributed across devices), leading to increased memory usage.
Different from GPipe, which performs backward passes after completing all
forward passes, 1F1B Narayanan et al. [2021a], Fan et al. [2021] alternates
between forward and backward passes (adopting a one-forward-one-backward
pattern) to keep the number of hidden states in the FIFO queue $Q$ constant.
Regardless of the number of micro-batches, 1F1B mitigates excessive memory
usage. Based on 1F1B, 1F1B-INarayanan et al. [2021a] schedule enlarges the
number of pipeline stages, and each device is assigned multiple stages. By
interleaving stages among devices, 1F1B-I reduces the bubble ratio at the cost
of adding more communication operators and slightly increasing memory
consumption. Zero-bubble-pipelineQi et al. [2024] divides the backward passes
into obtaining weight and input gradients separately. Zero-bubble-pipelineQi
et al. [2024] achieves higher pipeline efficiency by delaying weight gradient
computation and using dynamic programming to optimize the schedule. Zero-
bubble-pipeline approach nearly achieves zero-bubble pipeline efficiency but
brings more memory footprint caused by such delayment.
## 3 Methodology
In this section, we detail how our method works, beginning with a preliminary
overview to introduce characteristics of the 1F1B schedule and language
modeling. Then, we can prove why it’s feasible to schedule at the sequence
dimension for micro-batches in 1F1B. Following that, We will then explain how
Seq1F1B works in detail and how to meet the exact semantics of original
language modeling.
Building on this, we will discuss how different sequence-splitting strategies
impact the scheduling order in pipeline parallelism, and we will construct an
optimal solution based on the theoretical, computational load to address the
load-balancing issues associated with sequence-splitting strategies, thereby
enhancing the efficiency of our method.
### 3.1 Preliminary
1F1B includes three phases during one iteration: warm-up, steady, and cooling-
down phase. Assume a 1F1B scheduling scenario where we have $P$ workers, each
responsible for one pipeline stage, such that the size of pipeline parallelism
is $P$ and the number of micro-batches is $M$. Each worker denoted as $i$,
executes a forward pass during the warm-up phase. The number of warm-up micro-
batches for each worker is determined by the Eq. 1.
$\centering\text{w}_{i}=\begin{cases}P-i-1&\text{if }M>P\\\ M&\text{if }M\leq
P\end{cases}\@add@centering$ (1)
When $w_{i}$ equals $M$, 1F1B degrades to the behavior of GPipe. Otherwise,
during the warm-up phase, a worker responsible for an earlier stage performs
one more forward pass than a worker for a subsequent stage. Each forward pass
results in a hidden state that is enqueued in a FIFO queue $Q$ to be used
later for gradient computation during the backward pass. In the steady phase,
each worker performs one forward pass and enqueues the resulting hidden state
into $Q$. Following each forward pass, a hidden state is dequeued from $Q$ and
immediately to perform a backward pass for gradient computation, which is
where the "one-forward-one-backward" (1F1B) name comes from. It is noted that
the bubble ratio is minimal during the steady phase, and the number of one-
forward-one-backward passes in this phase is given by: $M-\text{w}_{i}$. Thus,
as $M$ increases, the proportion of the steady phase increases, which reduces
the bubble ratio. After the steady phase, the 1F1B scheduling enters the
cooling-down phase, which is symmetric to the warm-up phase and involves
performing the same number of backward passes as in the warm-up.
The primary optimization of 1F1B is to ensure that the memory consumption of
the hidden states is independent of $M$. The peak memory consumption for the
hidden states is determined by the number of items in the queue $Q$ at the end
of the warm-up phase, where each worker holds $w_{i}$ hidden states. Assuming
the total memory consumption of all hidden states is $A$, the peak memory
consumption of worker $i$ is $w_{i}\frac{A}{P}$. During the steady and
cooling-down phases, this consumption does not increase.
Figure 1: Execution timeline for the 1F1B and Seq1F1B schedules. Blank spaces
represent idle time, also known as bubbles. The top figure illustrates the
original 1F1B schedule, where each micro-batch is labeled with an ID. The
bottom figure illustrates our Seq1F1B schedule, where the input is split into
two sequences for better illustration. In Seq1F1B’s illustration, light-
colored areas represent the first sequence, while dark-colored areas represent
the second sequence. Notice that the forward pass for the dark-colored
sequence follows the light-colored sequence, whereas, for the backward pass,
the dark-colored sequence precedes the light-colored sequence. Figure 2:
Execution timeline for the 1F1B-I and Seq1F1B-I schedules. The top figure
illustrates the 1F1B-I schedule, where each micro-batch is labeled with an ID,
and different colors distinguish the forward/backward passes of different
stages. The lower part of the figure shows the Seq1F1B-I schedule, where the
input is split into two segments. In Seq1F1B-I, the light-colored areas
represent the first sequence and the dark-colored areas represent the second
sequence.
Language modeling is the most common unsupervised objective in training
language models. In Language modeling’s objective, each token is predicted
sequentially while conditioned on the preceding tokens, embodying the
principles of sequential generation, as formulated in Eq. 2.
$P(\mathbf{x})=\prod_{t=1}^{T}P(x_{t}\mid x_{1},x_{2},\ldots,x_{t-1})$ (2)
In the context of language modeling using Transformers, the unidirectional
attention mechanism ensures that each token in a sequence can only see its
predecessors, including itself.
Given a sequence of tokens $x_{0},x_{1},\ldots,x_{n}$, the output of the
attention mechanism for each token can be computed as follows. Each token
$t_{i}$ is associated with a query vector $q_{i}$, a key vector $k_{i}$, and a
value vector $v_{i}$, which will be used for attention computation. The output
for each token $t_{i}$, denoted as $O_{i}$, is computed by attending over all
previous tokens up to $t_{i}$, as formulated in Eq. 3.
$O_{i}=\text{softmax}\left(\frac{q_{i}\cdot[k_{0},\ldots,k_{i}]^{T}}{\sqrt{d_{k}}}\right)[v_{0},\ldots,v_{i}]$
(3)
Based on these characteristics, it becomes clear that to partition Transformer
computation across the sequence dimension, the attention mechanism must retain
the key and value vectors of all preceding tokens. The forward and backward
passes also need to maintain a specific order. The forward computation of each
token must follow the completion of its predecessor’s computation, while the
backward pass requires the subsequent token’s gradients to complete its
computation.
### 3.2 Seq1F1B
From the illustration 1, we observe that the original 1F1B schedule cannot
accommodate the splitting of micro-batches along the sequence dimension
because the last stage needs to immediately execute a backward pass after
forwarding a micro-batch. A straightforward adaptation method is to divide
each original 1F1B micro-batch into $k$ segments and then execute a $k$FkB
pipeline Li et al. [2021]. Although this schedule can reduce some bubbles in
1F1B, it does not save memory usage.
To achieve a more efficient sequence-level 1F1B pipeline schedule, we propose
Seq1F1B, which is a handcrafted 1F1B schedule for sequence-level input.
Specifically, Seq1F1B partitions the model into consecutive sets of layers and
assigns each worker with the corresponding set (a.k.a pipeline stages). Then,
Seq1F1B initializes the schedule part. Similar to 1F1B, the schedule is
divided into three phases: warm-up, steady, and cooling-down.
$\centering\text{w}_{i}=\begin{cases}P-i-2+k&\text{if }M>P\\\ M&\text{if
}M\leq P\end{cases}\@add@centering$ (4)
During the warm-up phase, the number of warm-up micro-batches of each worker
$i$ is calculated according to Eq. 4, in which $k$ represents the number of
splits in the sequence. This equation ensures that the last stage can perform
a backward pass on the last sequence segment of the first batch when entering
the steady phase, and the worker responsible for each stage performs one more
forward pass than the worker responsible for the subsequent stage. Here, we
construct a partially ordered queue $Q_{s}$, where each pop returns the tail
sequence from the earliest batch that has enqueued. This satisfies the first-
in-first-out principle in the batch dimension and the first-in-last-out
principle in the sequence dimension. In each iteration of the warm-up phase,
workers execute one forward pass and enqueue the corresponding hidden states
onto $Q_{s}$.
In the steady phase, after each worker completes a forward pass, it dequeues
from $Q_{s}$ and performs a backward pass on the dequeued hidden states,
following the standard 1F1B process, except that the units for forward and
backward passes become a sequence segment.
In the cooling-down phase, workers dequeue the remaining warm-up hidden states
from $Q_{s}$ and perform backward passes sequentially.
From the timeline shown in Figure 1, it is evident that the Seq1F1B schedule
offers shorter execution time and significantly fewer bubbles compared to the
original 1F1B schedule. Meanwhile, it can be clearly seen that each worker now
has less memory consumption since the micro-sequence is smaller than the
micro-batch. Another observation is that optimizations similar to the zero-
bubble pipeline can also be applied to Seq1F1B by delaying the gradient
computation associated with weights in the backward pass. We will discuss this
in the following section.
### 3.3 Seq1F1B-I
Figure 3: Execution timeline for the zero-bubble-pipeline’s ZBH1 and Seq1F1B
schedule intergrated with zero-bubble-pipeline’ ZBH1. Where each micro-batch
is labeled with an ID and different colors distinguish the
forward/backward/weight computation of different stages. OPT stands for
optimizer step.
1F1B-I Narayanan et al. [2021a] achieves better efficiency by modifying the
1F1B schedule to support interleaved stages among workers. In 1F1B-I, each
worker is assigned multiple stages. Suppose we have $P$ workers and $V$ stages
$\\{s_{1},s_{2},\ldots,s_{V}\\}$ in our pipeline, where $V$ is a multiple of
$P$. Each worker $i$ will handle $n$ stages
$\\{s_{i},s_{i+P},s_{i+2P},\ldots,s_{i+(n-1)P}\\}$, where $n=\frac{V}{P}$. The
number of warm-up micro-batches of each worker $i$ in 1F1B-I is given in Eq.
5.
$w_{i}=(P-i-1)\times 2+(n-1)\times P$ (5)
After completing $P$ iterations of forward/backward passes, each worker
switches its context to the next stage it is responsible for. From the Figure
2, the above part shows a 1F1B-I pipeline with $P$ as 4 and $V$ as 8, in which
each worker handles 2 stages. 1F1B-I’s schedule reduces the bubble ratio by
interleaving stages among workers. However, this interleaving slightly
increases memory consumption, as the number of warm-up micro-batches $w_{i}$
is greater compared to 1F1B.
Similar to 1F1B-I, Seq1F1B-I further modifies 1F1B-I to achieve sequence-level
scheduling, as shown in bottom part of Figure 2. From the Figure 2, Seq1F1B-I
effectively reduces pipeline bubbles and maintains less memory footprint of
hidden state compared with 1F1B-I. Seq1F1B-I defines the number of warm-up
micro-batches as in Eq. 6.
$w_{i}=(P-i-1)\times 2+(n-1)\times P+k-1$ (6)
in which $k$ represents the number of splits in the sequence. By using the
partially ordered queue, Seq1F1B-I maintains strict order of forward/backward
pass and ensures the consistent semantics of gradient updates. From the
perspective of pipeline bubbles, Seq1F1B-I outperforms both Seq1F1B and
1F1B-I. In terms of memory demands, Seq1F1B-I requires slightly more memory
than Seq1F1B but significantly less than 1F1B-I.
### 3.4 Integration with Zero-bubble-pipeline
From the illustration3, we can see Seq1F1B can integrate with ZB1P method and
further reduce bubbles while reducing memory demands by splitting sequence.
Such integration outperforms simple ZB1P in both memory demands and pipeline
bubbles since sequence-level pipelines naturally have fewer bubbles.
Furthermore, Seq1F1B can integrate with ZB2P and ZBV methods too.
Theoretically, introducing a zero-bubble-pipeline to Seq1F1B should be more
efficient. Even though, such a fine-grained handcraft schedule may have
performance degradation under some settings. We hope our work inspires future
work to solve this problem.
### 3.5 Workload Balance
In this section, we detail the strategy of sequence partition and workload
balance consideration. Previous works, such as Li et al. [2021], have
discussed strategies for sequence partitioning. To achieve efficient pipeline
scheduling, it is crucial that the processing times for each subsequence are
approximately equal to avoid pipeline bubbles. Based on this premise, we
design a computation-wise partition strategy by estimating the FLOPs of
sequences and constructing a theoretical solution aiming to make the FLOPs of
all subsequences as closely as possible.
For a input sequence $S=(x_{1},x_{2},\cdots,x_{n})$, we devide it into $k$
segments $S=[S_{1},\cdots,S_{k}]$. Each segment having a length of $n_{i}$,
where $\sum_{i=1}^{k}n_{i}=n$. We expect the computational amount of each
segment to be roughly the same, that is
$\text{FLOPs}(S_{1})=\text{FLOPs}(S_{2})=\cdots=\text{FLOPs}(S_{k})={\frac{\text{FLOPs}(S)}{k}}.$
(7)
Specifically, we use the method proposed in Hoffmann et al. [2022] to estimate
the FLOPs for each subsequence, as formulated in Eq. 8,
$\begin{aligned}
\text{FLOPs}(S_{i})=2~{}n_{i}~{}P+2~{}L~{}n_{i}\left(\sum_{j=0}^{i}n_{j}\right)d,\forall
i=1\dots k;\ \ \text{FLOPs}(S)=2~{}n~{}P+2~{}L~{}n^{2}d\end{aligned},$ (8)
in which, $L$ is a number of layers, $d$ is dimension of the model, and $P$ is
the total number of parameters in the model. We have $k$ variables in Eq. 8
and $k$ equations in Eq. 7. Therefore, we can set up the equation to get the
optimial segmentation.
## 4 Experiments
### 4.1 Experimental Settings
Table 1: Settings used in experiments for training LLMs. Model | Number of | Attention | Hidden | Sequence | PP | TP | Number of
---|---|---|---|---|---|---|---
Size | Layers | Heads | Size | Length | Size | Size | Micro-batches
2.7B | 32 | 32 | 2560 | 16k / 24k / 32k | 8 | 1 | 32 / 64
7B | 32 | 32 | 4096 | 32k / 64k / 128k | 4 | 8 | 16 / 32
13B | 40 | 40 | 5120 | 32k / 64k / 128k | 4 | 8 | 16 / 32
30B | 64 | 64 | 6144 | 32k / 48k / 64k | 8 | 8 | 32 / 64
Figure 4: Peak Memory consumption of tranining a series of models under
varying sequence lengths and fixed batch settings. “X” means experiments ran
out of memory. We take the maximum memory consumption between all workers for
better clarification.
In experiments, we measure our methods and 1F1B and 1F1B-I under variable
sequence lengths, different numbers of micro-batches, different numbers of
GPUs, different pipeline parallel sizes and tensor parallel sizes. Compared
methods are as follows:
* •
Seq1F1B: Seq1F1B with computation-wise sequence partition strategy.
* •
Seq1F1B-I: Seq1F1B with interleaved stages and computation-wise sequence
partition strategy.
* •
1F1B/1F1B-I: 1F1B and 1F1B with interleaved stages in Megatron implementation.
* •
Seq1F1B w/o cwp: Seq1F1B without computation-wise sequence partition strategy.
* •
Seq1F1B-I w/o cwp: Seq1F1B-I without computation-wise sequence partition
strategy.
All assessments are based on GPT model and model configuration are listed in
Table 1. All experiments focus on long-sequence training since a lot of work
has metioned the importance. For hyperparameter configurations, we set the
number of sequence splits to four and each worker managing two stages in
interleaved settings. Our implementation is based on the open-source Megatron-
LM project Narayanan et al. [2021a] and ensures reproducibility. We adopts
Megatron-V3Korthikanti et al. [2023]’s tensor parallelism in all experiments
since it is necessary for long sequence training.
Our experiments include three cluster settings: 1) 1 node with 8 NVIDIA A100
SXM 80G GPUs interconnected by NvLink. 2) 4 nodes interconnected by a RoCE
RDMA network and each node has 8 NVIDIA A100 SXM 80G GPUs interconnected by
NvLink. 3) 8 nodes interconnected by a RoCE RDMA network and each node has 8
NVIDIA A100 SXM 80G GPUs interconnected by NvLink. Each measurement in the
experiment is repeated 100 times, and the standard deviation is recorded.
### 4.2 Main Results
Table 2: 2.7B GPT training experiments with pipeline parallel size of 8 under
8xA100 setting.
Model Size | 2.7b
---|---
Sequence Length | 16384 | 24576 | 32768
Micro-batch | 16 | 32 | 16 | 32 | 16 | 32
Throughput | 1F1B | 32.0±0.0 | 37.1±0.0 | 27.0±0.0 | 31.4±0.0 | OOM | OOM
(Thousands | 1F1B-I | 36.4±0.0 | 39.7±0.0 | OOM | OOM | OOM | OOM
Tokens/s) | Seq1F1B | 37.3±0.0 | 38.9±0.3 | 32.6±0.0 | 34.2±0.0 | 28.8±0.0 | 30.1±0.2
Seq1F1B-I | 38.0±0.0 | 38.9±0.0 | 33.3±0.0 | 34.3±0.0 | 29.5±0.0 | 30.3±0.0
TFLOPS | 1F1B | 96.9±0.0 | 112.3±0.0 | 95.5±0.1 | 111.1±0.1 | OOM | OOM
per device | 1F1B-I | 110.3±0.1 | 120.2±0.1 | OOM | OOM | OOM | OOM
Seq1F1B | 113.1±0.0 | 117.8±0.8 | 115.2±0.1 | 120.9±0.1 | 116.5±0.1 | 122.0±1.0
| Seq1F1B-I | 115.2±0.0 | 118.0±0.0 | 118.0±0.1 | 121.3±0.1 | 119.4±0.0 | 122.7±0.0
Table 3: 7B GPT training experiments with pipeline parallel size of 4 and
tensor parallel size of 8 under 32xA100 setting.
Model Size | 7b
---|---
Sequence Length | 32768 | 65536 | 131072
Micro-batch | 8 | 16 | 8 | 16 | 8 | 16
Throughput | 1F1B | 48.2±0.1 | 55.3±0.2 | 37.3±0.0 | 43.1±0.0 | OOM | OOM
(Thousands | 1F1B-I | 53.0±0.3 | 56.3±0.4 | 41.7±0.1 | 44.7±0.0 | OOM | OOM
Tokens/s) | Seq1F1B | 53.5±0.3 | 55.8±0.1 | 43.3±0.0 | 45.0±0.1 | 30.4±0.0 | 31.6±0.0
Seq1F1B-I | 47.2±0.9 | 46.2±0.8 | 40.9±0.4 | 41.0±0.3 | 30.0±0.0 | 30.4±0.0
TFLOPS | 1F1B | 99.7±0.2 | 114.5±0.4 | 107.5±0.0 | 124.0±0.1 | OOM | OOM
per device | 1F1B-I | 109.5±0.7 | 116.5±0.8 | 120.0±0.2 | 128.7±0.1 | OOM | OOM
Seq1F1B | 110.6±0.5 | 115.3±0.2 | 124.6±0.1 | 129.7±0.5 | 136.7±0.1 | 142.1±0.0
| Seq1F1B-I | 97.7±1.8 | 95.5±1.6 | 117.8±1.3 | 118.0±0.8 | 135.1±0.2 | 136.6±0.2
Table 4: 13B GPT training experiments with pipeline parallel size of 4 and
tensor parallel size of 8 under 32xA100 setting.
Model Size | 13b
---|---
Sequence Length | 32768 | 49152 | 65536
Micro-batch | 8 | 16 | 8 | 16 | 8 | 16
Throughput | 1F1B | 28.9±0.1 | 33.4±0.1 | 25.3±0.1 | 29.3±0.1 | 22.6±0.1 | 30.0±0.0
(Thousands | 1F1B-I | 32.2±0.2 | 34.4±0.1 | 28.2±0.2 | 30.6±0.1 | OOM | OOM
Tokens/s) | Seq1F1B | 32.9±0.1 | 34.3±0.1 | 29.5±0.1 | 30.8±0.0 | 26.7±0.0 | 27.8±0.0
Seq1F1B-I | 29.7±0.4 | 29.8±0.3 | 28.0±0.2 | 28.3±0.1 | 26.4±0.1 | 26.8±0.1
TFLOPS | 1F1B | 106.7±0.2 | 123.0±0.5 | 109.5±0.5 | 126.2±0.6 | 111.9±0.5 | 135.1±0.2
per device | 1F1B-I | 118.6±0.6 | 126.9±0.4 | 121.9±0.7 | 132.2±0.4 | OOM | OOM
Seq1F1B | 121.2±0.2 | 126.6±0.3 | 127.3±0.4 | 133.1±0.2 | 132.5±0.0 | 137.9±0.0
| Seq1F1B-I | 109.7±1.4 | 110.0±1.1 | 121.0±1.1 | 122.1±0.4 | 130.6±0.3 | 132.8±0.3
Table 5: 30B GPT training experiments with pipeline parallel size of 8 and
tensor parallel size of 8 under 64xA100 setting.
Model Size | 30b
---|---
Sequence Length | 32768 | 49152 | 65536
Micro-batch | 8 | 16 | 8 | 16 | 8 | 16
Throughput | 1F1B | 26.4±0.1 | 31.2±0.2 | OOM | OOM | OOM | OOM
(Thousands | 1F1B-I | OOM | OOM | OOM | OOM | OOM | OOM
Tokens/s) | Seq1F1B | 31.3±0.1 | 33.1±0.2 | 28.2±0.1 | 29.6±0.1 | 25.5±0.0 | 26.8±0.0
Seq1F1B-I | 28.0±0.4 | 28.4±0.2 | 26.5±0.2 | 27.1±0.2 | 24.8±0.1 | 25.2±0.1
TFLOPS | 1F1B | 104.8±0.3 | 123.9±0.7 | OOM | OOM | OOM | OOM
per device | 1F1B-I | OOM | OOM | OOM | OOM | OOM | OOM
Seq1F1B | 124.5±0.2 | 131.5±0.6 | 129.4±0.3 | 135.6±0.3 | 132.6±0.0 | 139.2±0.0
| Seq1F1B-I | 111.1±1.6 | 113.0±1.0 | 121.5±1.1 | 124.2±0.8 | 128.6±0.3 | 130.9±0.6
In Figure 4, we compared the memory consumption of our method with that of
1F1B and 1F1B-I. As can be seen, our method consistently requires less memory
across all settings, and notably, it can support training a 30B model on a
64xA100 cluster, which is impossible for the traditional combination of
pipeline and tensor parallelism. Additionally, we recorded TFLOPS(teraFLOPS)
per GPU in our experiments to measure the hardware utilization of different
methods. From the Table 2, 3, 4 and 5, our method Seq1F1B outperforms 1F1B and
1F1B-I under almost all settings in both training throughput and teraFLOPS.
However, as observed in Table 3,4,5, the Seq1F1B-I may have a performance
degradation under multi-node settings. This could be due to the overly fine-
grained interleaving of stage partitioning and input sequence partitioning,
which also implies more communication calls in Tensor Parallelism (although
the total communication volume remains unchanged), potentially leading to a
decrease in performance. Another observation is that the efficiency of Seq1F1B
becomes more pronounced as the sequence length increases. This is because the
computation time for each micro-sequence extends with longer sequences,
thereby enhancing the benefits derived from sequence partitioning.
### 4.3 Ablation Results
We also conducted all experiments using Seq1F1B without computation-wise
partitioning (Seq1F1B w/o cwp) and Seq1F1B-I without computation-wise
partitioning (Seq1F1B-I w/o cwp) to evaluate the effectiveness of our
computation-wise partition strategy. Under identical settings, employing the
computation-wise partition strategy leads to performance enhancements ranging
from approximately 10-30% for Seq1F1B compared to simply splitting the
sequence.
Across all experimental scales, Seq1F1B consistently surpassed Seq1F1B w/o cwp
in performance. Table 6 highlights the ablation performance for a 2.7B model
with a sequence length of 32k, demonstrating a performance boost of
approximately 28% due to the computation-wise partitioning.
## 5 Conclusion
Table 6: The Ablation experiments based on 2.7B GPT of sequence partitioning
strategies, where “w/o cwp” indicates the absence of a computation-wise
partitioning strategy.
Method | TFLOPS/device | SpeedUp
---|---|---
Seq1F1B w/o cwp | 94.8±0.1 | -
Seq1F1B | 122.0±1.0 | 1.28x
Seq1F1B-I w/o cwp | 103.5±0.1 | -
Seq1F1B-I | 122.7±0.0 | 1.18x
In this paper, we present Seq1F1B, an efficient 1F1B pipeline parallel
scheduling method orienting to training Transformer-based LLMs on long
sequences by decomposing the batch-level schedulable units used by typical
1F1B methods into more fine-grained sequence-level units. To achieve a better
workload balance of the sequence-level pipeline, we design a computation-wise
sequence partition strategy to partition the sequences well. Meanwhile Seq1F1B
can integrate with other pipeline parallel methods such as 1F1B with
interleaved stage or zero-bubble-pipeline. Our evaluations demonstrate that
Seq1F1B outperforms the 1F1B and 1F1B-I scheduling strategies regarding memory
efficiency and training throughput under variable sequence lengths and model
sizes. Moreover, Seq1F1B can support the efficient training of a 30B GPT model
on sequences up to 64k in length using 64xA100 GPUs, without using
recomputation strategies, which is unachievable with existing pipeline
parallel methods. In the future, we will thoroughly combine our method with
other distributed methods to achieve better LLM training acceleration. In
addition, we will systematically release our code to support the community in
training LLMs to process longer sequences more efficiently.
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§ ABSTRACT
Magnetic materials with noncollinear spin textures are promising for spintronic applications. To realize practical devices, control over the length and energy scales of such spin textures is imperative. The chiral helimagnets and exhibit analogous magnetic phase diagrams with different real-space periodicities and field dependence, positioning them as model systems for studying the relative strengths of the microscopic mechanisms giving rise to exotic spin textures. Here, we carry out a comparative study of the electronic structures of and using angle-resolved photoemission spectroscopy and density functional theory. We show that bands in are more dispersive than their counterparts in and connect this result to bonding and orbital overlap in these materials. We also unambiguously distinguish exchange splitting from surface termination effects by studying the dependence of their photoemission spectra on polarization, temperature, and beam size. We find strong evidence that hybridization between intercalant and host lattice electronic states mediates the magnetic exchange interactions in these materials, suggesting that band engineering is a route toward tuning their spin textures. Overall, these results underscore how the modular nature of intercalated transition metal dichalcogenides translates variation in composition and electronic structure to complex magnetism.
§ INTRODUCTION
Next-generation spintronic devices utilize the spin degree of freedom to store information. fert2013,parkin2015 Magnetic materials in which spins order in topologically protected quasiparticles, such as skyrmions or magnetic solitons, are promising platforms for realizing such devices.togawa2016,tokura2021 These chiral spin textures can be manipulated with currents and magnetic fields, which is appealing for various applications in memory, logic, and unconventional computing.tey2022 For practical spintronic devices, optimizing the energy and length scales of the spin textures is important: stability at operationally accessible temperatures and fields as well as high density in thin-film architectures are broadly desirable. Strategies to control the microscopic mechanisms that give rise to complex magnetism are thus needed. In terms of materials design, this can be broadly achieved by tailoring the interactions between spin centers as directed by their spatial arrangements and coordination environments.
The chiral helimagnets and are especially well-suited for device schemes implementing noncollinear spin textures because of their anisotropic layered structures, which are compatible with thin-film architectures.togawa2015,yamasaki2017,wang2017,zhang2022,osorio2022 In these materials, the $S=3/2$ Cr$^{3+}$ centers occupy pseudo-octahedral sites between layers of $2H$-NbS2 or $2H$-TaS2,parkin1980a,parkin1980 forming a $\sqrt{3} \times \sqrt{3}$ superlattice.rouxel1971 They exhibit easy-plane ferromagnetic (FM) behavior with chiral magnetic ordering out-of-plane: the Cr superlattice breaks the inversion symmetry of the transition metal dichalcogenide (TMD) host lattice along the crystallographic $c$-axis, giving rise to a Dzyaloshinskii–Moriya (DM) interaction, also known as antisymmetric exchange.dzyaloshinsky1958,moriya1960 The DM interaction favors out-of-plane spin canting, which competes with FM exchange to produce one-dimensional helical spin textures that propagate along [001]. Importantly, the application of an in-plane magnetic field creates a chiral soliton lattice (CSL) phase with tunable periodicities up to a critical field, $H_\mathrm{c}$, above which a forced ferromagnetic (FFM) state is observed.miyadai_magnetic_1983,togawa_chiral_2012-1,ghimire2013,zhang2021,obeysekera2021,du2021 Both and have Curie temperatures, $T_{\mathrm{C}}$, well above 100 K, and nanoscale soliton wavelengths tunable with fields of 1.5 T or less, thus providing a richly accessible phase space for manipulating chiral spin textures.
Although the magnetic phase diagrams for and are qualitatively analogous, the periodicities and stabilities of their magnetic solitons differ somewhat. Literature reports have established that consistently exhibits a higher $T_{\mathrm{C}}$,togawa_chiral_2012-1,ghimire2013,du2021,kousaka2022,meng2023, higher $H_\mathrm{c}$,ghimire2013,zhang2021,obeysekera2021,meng2023 and a shorter soliton wavelength than the Nb analogue.togawa_chiral_2012-1,kousaka2016,zhang2021,du2021 These observations imply that changing the host lattice from NbS2 to TaS2 alters the relative strengths of magnetic coupling among Cr centers, manifesting in quantitative changes to their magnetic phase diagrams. However, the origin of magnetic exchange interactions in these materials is still a matter of debate,sirica2016,sirica2021,qin2022,hicken2022 and comparative studies have been scant.hicken2022 A detailed investigation of the electronic structures of both and is thus motivated by the fact that these materials are natural platforms for studying how the length and energy scales of chiral spin textures can be tuned through materials chemistry.
Herein, we present a comprehensive investigation of the electronic structures of and using angle-resolved photoemission spectroscopy (ARPES) and density functional theory (DFT) calculations. We show that the Ta analogue has more dispersive bands, consistent with greater orbital overlap in the case of Ta, and discuss implications for their magnetic properties and the rational design of chiral helimagnets. Using polarization-dependent ARPES and orbital-projected DFT calculations, we assign the parity and orbital character of bands, finding excellent agreement between theory and experiment with the exception of additional band splitting near the Fermi level observed in ARPES but not predicted in DFT. ARPES data collected with smaller beam sizes reveal spatial variation in this band splitting, consistent with different surface terminations on the as-cleaved samples. Thus, we distinguish exchange splitting from surface vs. bulk splitting for the first time in these materials. Our findings establish a high degree of similarity in the electronic structures of and and highlight the relevance of their polar layered nature in interpreting surface-sensitive spectroscopic investigations. More generally, these results suggest that careful band structure engineering and Fermi level tuning may prove to be fruitful avenues toward optimizing the spin textures in intercalated TMDs.
First, we briefly outline the electronic and magnetic properties of ($M$ = Nb or Ta) as established in the existing literature. According to a simple electron counting scheme, these compounds can be considered as alternating layers of [Cr_1/3]+ and [$M$S2]-. The intercalant layers consist of Cr^3+ centers occupying 1/3 of the trigonally distorted pseudo-octahedral interstitial sites between layers of $2H$-$M$S2 ($M$ = Nb or Ta). These intercalant layers donate one electron per formula unit to the $M$S2 host lattice layers (Figure <ref>a). The qualitative local $d$-orbital splitting diagrams for the Cr^3+ ($D_{3d}$) and $M$^3+ ($D_{3h}$) centers are shown in Figure <ref>b–c.xie_structure_2022 The electronic structure of the periodic solids are more complex, and in reality, the half-filled TMD bands have both $d_{z^2}$ and $d_{xy}$/$d_{x^2-y^2}$ character.mattheiss1973,yee1991,whangbo1992 Nevertheless, this simplified picture captures (1) charge transfer from the intercalant species to the highest-lying $M$ $d$ bands of the host lattice, and (2) the inherently polar nature of these layered intercalation compounds.
The qualitative magnetic properties of and below $T_{\mathrm{C}}$ are summarized in Figure <ref>d–e. Within each [Cr_1/3]+ layer, the Cr spins exhibit FM coupling through an in-plane exchange constant, $J_{\parallel}$. Between adjacent [Cr_1/3]+ layers, FM coupling through an out-of-plane exchange constant, $J_{\perp}$, competes energetically with spin canting through a DM interaction term, $D$. At zero field, this results in a continuous helical arrangement of spins, or a CHM ground state, with the magnetic soliton wavelength determined by the ratio of $J_{\perp}$ and $D$.chapman2014spin,aczel2018 With increasing $H \perp c$, FM regions aligned with the field grow, effectively unwinding the CHM state to create the CSL phase, in which the distance separating adjacent solitons is a function of the magnitude of $H$. Finally, with fields larger than $H_\mathrm{c}$, an FFM state with saturated magnetization is obtained.miyadai_magnetic_1983,togawa_chiral_2012-1,ghimire2013,han2017,zhang2021,obeysekera2021,meng2023
(a) Crystal structure of , showing formal charges for the $M$S2 and Cr layers from a simple electron-counting picture. (b) and (c) Qualitative $d$-orbital splitting diagrams for isolated Cr and $M$ centers from the local ligand field in . (d) Schematic illustration of the magnetic structure of in the chiral helimagnetic (CHM) state. (e) Schematic representations of spin textures evolving CHM to chiral soliton lattice (CSL) to forced ferromagnetic (FFM) states with increasing applied magnetic field $H \perp c$. (f) and (g) $M(H)$ data for and , respectively, showing transitions between CHM, CSL, and FFM states.
In this study, we investigate the electronic structure of and in a comparative context to tease out differences between the two compounds and connect these to their magnetic phase diagrams. To do so, we grew and characterized single crystals, verified their chiral spin textures with magnetometry, carried out a comprehensive suite of ARPES measurements, and conducted DFT band structure calculations, as detailed below.
§ RESULTS
§.§ Synthesis, Structure, and Magnetism
Single crystals of and were grown via chemical vapor transport using iodine as a transport agent. X-ray diffraction confirmed that both materials crystallize in the noncentrosymmetric space group $P6_322$, with the Cr centers forming a $\sqrt{3} \times \sqrt{3}$ superlattice (Figure <ref> and Tables <ref>–<ref>). exhibits a slightly larger in-plane lattice parameter and smaller out-of-plane lattice parameter ($a$ = 5.7400(7) Å and $c$ = 12.1082(14) Å) compared to ($a$ = 5.7155(5) Å and $c$ = 12.1751(12) Å). Raman spectroscopy revealed sharp vibrational modes associated with the $\sqrt{3} \times \sqrt{3}$ superlatticesfan2021 (Figure <ref>), and energy dispersive X-ray spectroscopy indicated Cr:Nb and Cr:Ta ratios of 0.33(1):1 (Figures <ref> and <ref>).
The metamagnetic transitions across these states with applied magnetic field are observed in the $M(H)$ data for single crystals of and shown in Figure <ref>f–g, confirming the characteristic spin textures in our samples.miyadai_magnetic_1983,zhang2021,obeysekera2021 Both compounds exhibit similar saturation moments (2.7 $\mu_{\mathrm{B}}$/Cr for and 2.8 $\mu_{\mathrm{B}}$/Cr for ), close to the expected spin-only value of 3 $\mu_{\mathrm{B}}$/Cr. The analogous transitions are observed at fields more than an order of magnitude larger for than , with $H_\mathrm{c}$ values of about 0.45 mT for and 16 mT for . This is consistent with shorter soliton wavelengths in the Ta analogue.zhang2021,obeysekera2021,du2021 The $M(T)$ data show pronounced peaks at 110 and 133 K for and , respectively, corresponding to the onset of chiral helimagnetism below these temperatures (Figures <ref>–<ref>).
§.§ Superlattice Effects on Electronic Structure
After obtaining structural and magnetic evidence of highly ordered $\sqrt{3} \times \sqrt{3}$ Cr superlattices in and , we sought to investigate their influence on the electronic structure of these materials. Figure <ref>a illustrates the real-space $1 \times 1$ primitive unit cell for the host lattice TMD and the $\sqrt{3} \times \sqrt{3}$ superlattice unit cell for the intercalated compounds along [001]. The $\sqrt{3} \times \sqrt{3}$ unit cell is rotated by 30° compared to the $1 \times 1$ unit cell. In reciprocal space, the $\sqrt{3} \times \sqrt{3}$ superlattice defines a smaller Brillouin zone that is likewise rotated by 30° relative to the primitive Brillouin zone (Figure <ref>b). To probe the electronic effects of Cr intercalation, we first examined the symmetries of the experimental Fermi surfaces and band dispersions of and using ARPES.
(a) Real-space crystal structure of ($M$ = Nb or Ta) viewed along the crystallographic $c$-axis, with overlaid unit cells for the $1 \times 1$ primitive $M$S2 lattice (dashed green) and the $\sqrt{3} \times \sqrt{3}$ Cr superlattice (solid gray). (b) Surface Brillouin zones for the $1 \times 1$ primitive lattice and $\sqrt{3} \times \sqrt{3}$ superlattice. (c) and (d) ARPES Fermi surfaces of and with the dashed and solid overlaid lines corresponding to the primitive and $\sqrt{3} \times \sqrt{3}$ superlattice Brillouin zones, respectively. (e) and (f) ARPES band dispersions for and along the $\Gamma$–$\mathrm{K}_0$ direction, showing folding of features from $\Gamma$ to $\mathrm{K}_0$ and vice versa (18 K, $h\nu$ = 79 eV).
As shown in Figure <ref>c–d, the Fermi surfaces of both and below $T_{\mathrm{C}}$ ($h\nu$ = 79 eV) display multiple nested barrels around $\Gamma$ and $\mathrm{K}$ of the primitive Brillouin zone, which is indicated by the dashed green hexagons. Notably, six-fold symmetry is clearly observed around the primitive $\mathrm{K}$ (denoted as $\mathrm{K_0}$), in contrast with three-fold symmetry around $\mathrm{K}$ of the host lattice materials $2H$-NbS2 and $2H$-TaS2.elyoubi2021,zhao2017 Additionally, in the intercalated materials, three-fold symmetry is introduced at $\mathrm{K}$ of the $\sqrt{3} \times \sqrt{3}$ superlattice Brillouin zone (denoted as $\mathrm{K_{SL}}$), which is indicated by the solid gray hexagons in Figure <ref>c–d. Hence, the Fermi surfaces of and display the expected symmetries associated with reconstruction and band folding from the $\sqrt{3} \times \sqrt{3}$ Cr superlattice.
The ARPES dispersions show clear evidence of band folding as well (Figure <ref>e–f). Cuts along the $\Gamma$–$\mathrm{K_0}$ direction show the same features at both $\Gamma$ and $\mathrm{K_0}$: both materials display several nested hole pockets and parabolic bands below $E_{\mathrm{F}}$. In contrast, for the host TMDs $2H$-NbS2 and $2H$-TaS2, the bands crossing $E_{\mathrm{F}}$ have different dispersions and energies at $\Gamma$ and $\mathrm{K}$. In the Cr-intercalated materials, the $\sqrt{3} \times \sqrt{3}$ superlattice folds the primitive lattice $\Gamma$ to $\mathrm{K}$ and vice versa, as they both become $\Gamma$ of the superlattice Brillouin zone (denoted as $\mathrm{\Gamma_{SL}}$ in Figure <ref>b). Thus, the presence of the same features at both $\Gamma$ and $\mathrm{K_0}$ in the ARPES of the Cr-intercalated materials is consistent with $\sqrt{3} \times \sqrt{3}$ superlattice band folding.
The band folding in the ARPES data reveals that the $\sqrt{3} \times \sqrt{3}$ Cr superlattice potential is strong in both and . Broadly, this electronic reconstruction is in line with previous literature reports on ,sirica2016,qin2022 as well as other intercalated TMDs with $\sqrt{3} \times \sqrt{3}$ transition metal superlattices.tanaka2022,yang2022,popcevic2022,edwards2023 The features observed in both materials are qualitatively similar; however, at a glance, the hole pockets in appear to be larger than those found in . To contextualize differences in the experimental electronic structures of and , we turned to DFT calculations and quantitative analysis of their band dispersions.
§.§ Relative Band Dispersions
To understand the relative differences between the band structures of and , we started by comparing the host lattice materials, $2H$-NbS2 and $2H$-TaS2. DFT band structure calculations of $2H$-NbS2 and $2H$-TaS2 show that the bands crossing $E_{\mathrm{F}}$ in $2H$-TaS2 are more dispersive compared to the analogous bands in $2H$-NbS2. This can be clearly visualized by comparing the relative spread of the maxima and minima of these respective bands, as illustrated in Figure <ref>a–b: the more dispersive bands in $2H$-TaS2 have a higher-energy maximum and lower-energy minimum compared to $2H$-NbS2. These bands have predominantly Nb or Ta $d_{z^2}$ and $d_{xy}$/$d_{x^2-y^2}$ character, with additional contribution from S $p$ states.zhao2017,elyoubi2021
(a) and (b) DFT band structures of $2H$-NbS2 and $2H$-TaS2, with maxima and minima of the bands crossing $E_{\mathrm{F}}$ indicated by solid navy and mint lines, respectively. (c) and (d) Spin-polarized band structures for and in the FM state, with spin up and spin down bands indicated in red and blue, respectively. (e) and (f) ARPES dispersions of and (18 K, $h\nu$ = 46 eV), with blue circles indicating the peak center positions of the most intense feature from MDC analysis. The Fermi velocities, $v_{\mathrm{F}}$, are obtained from linear fits to the centers between 0 and $-50$ meV.
Next, we calculated the band structures of the Cr-intercalated materials and compared the results to our ARPES data. Due to the surface-sensitive nature of ARPES, we do not expect to experimentally resolve signatures of the CHM state, i.e. out-of-plane spin textures with length scales on the order of tens of nm. Hence, we use spin-polarized band structure calculations of and in their FM states, with the magnetization vector along [100], as proxies for the electronic structure near the surface (Figure <ref>c–d and Figures <ref>–<ref>). Three distinct changes are evident in the DFT band structures of and compared to the host lattices: (1) folding due to the $\sqrt{3} \times \sqrt{3}$ superlattice potential, (2) raising of $E_{\mathrm{F}}$ due to electron transfer from Cr to the host lattice, and (3) introduction of new bands crossing $E_{\mathrm{F}}$ due to Cr–Nb or Cr–Ta hybridization and FM exchange splitting.
Although the Cr-intercalated materials have more complex electronic structures than the host lattices, DFT calculations show that the Ta analogue again has more dispersive bands than the Nb analogue. The amount of charge transfer from Cr to the host lattice appears to be very similar for both materials, as shown by the calculated and experimental magnetic moments (Tables <ref> and <ref>). Thus, the shift of $E_{\mathrm{F}}$ upon intercalation is almost identical. This results in larger hole pockets around $\Gamma$ and $\mathrm{K_0}$ in than and an extra spin-up band crossing $E_{\mathrm{F}}$ at $\Gamma$ and $\mathrm{K}_0$ in . Notably, the ARPES dispersions of and at 18 K ($h\nu$ = 46 eV) show clearly that the most intense hole pocket around $\Gamma$ in the $\Gamma$–$\mathrm{M_{SL}}$ direction is considerably larger at $E = E_{\mathrm{F}}$ in compared to (Figure <ref>e–f), with Fermi wavevectors, $k_{\mathrm{F}}$, of 0.10 Å for and 0.19 Å for . By fitting the momentum distribution curves (MDCs) to Lorentzians between 0 and $-50$ meV, we extracted Fermi velocities, $v_{\mathrm{F}}$, of $1.47(5) \times 10^5$ m/s for and $1.8(2) \times 10^5$ m/s for —thus experimentally quantifying the relative band dispersions between the two systems. The larger experimental $v_{\mathrm{F}}$ for the Ta analogue mirrors the relative trends from the DFT band structures.
§.§ Orbital Character Assignments
To gain insight into the orbital character of the bands, we studied their polarization dependence in ARPES. For the photoemission process, the matrix element term can be described by
\[ | M^{\mathbf{k}}_{f,i}|^2 \propto | \langle \phi^{\mathbf{k}}_f | \hat{\epsilon} \cdot \mathbf{r} | \phi^{\mathbf{k}}_i \rangle |^2 \]
where $\hat{\epsilon}$ is the unit vector along the polarization direction of the light.damascelli2003 The final state wavefunction of the photoelectron, $\phi^{\mathbf{k}}_f$, can be described by a plane wave state, $e^{i\mathbf{kr}}$, with even parity with respect to the mirror plane defined by the analyzer slit and the normal to the sample surface (Figure <ref>a). To obtain a nonvanishing matrix element, $\hat{\epsilon}$ must be even (odd) for an even (odd) initial state wavefunction, $\phi^{\mathbf{k}}_i$. Based on the symmetry operations of space group $P6_322$ (point group $D_6$), and taking $z$ to be parallel to the crystallographic $c$-axis, we expect the even $a_1$ ($d_{z^2}$) states of both Cr and Nb or Ta to be visible with linear horizontal (LH) polarized light (even $\hat{\epsilon}$) but not linear vertical (LV) polarized light (odd $\hat{\epsilon}$).
The $e_1$ ($d_{xz}$, $d_{yz}$) and $e_2$ ($d_{xy}$, $d_{x^2-y^2}$) sets are not symmetric overall with respect to any scattering plane containing the sample surface normal. We illustrate this by considering a horizontal analyzer slit aligned to the $\Gamma$–$\mathrm{K}_0$ direction, and defining $x$ as parallel to the crystallographic $a$-axis of the superlattice unit cell. The resulting scattering plane is the $xz$ plane (Figure <ref>a) and contains $M$–S and Cr–S bonds (Figure <ref>b). As shown in Figure <ref>c, the $d_{xz}$ and $d_{x^2-y^2}$ orbitals are even with respect to the $xz$ plane, but the $d_{yz}$ and $d_{xy}$ orbitals (the other components of the $e_1$ and $e_2$ sets) are odd. Thus, the $e_1$ and $e_2$ sets are not symmetric collectively, and may be visible with both LH and LV polarization.
(a) Geometry of ARPES data collection for a horizontal analyzer slit aligned with the $xz$ scattering plane of the sample. (b) Surface Brillouin zones for the primitive lattice (dotted green) and $\sqrt{3} \times \sqrt{3}$ superlattice (solid gray), along with real space projections of local coordination environments for $M$ = Nb or Ta and Cr. (c) Symmetries of $d$ orbitals for $M$ = Nb or Ta and Cr for the scattering plane defined as the $xz$ plane aligned with the $\Gamma$–$\mathrm{K}_0$ direction.
ARPES data of measured with LV polarization (Figure <ref>a) show stronger intensity from the innermost parabolic bands centered at $\Gamma$ and especially $\mathrm{K_0}$. In contrast, with LH polarization (Figure <ref>b), the sharp outermost dispersive bands around $\Gamma$ and $\mathrm{K_0}$ are more prominent, as well as two sets of more diffuse electron pockets with minima at $\mathrm{M_{SL}}$ and $\mathrm{K_{SL}}$. To compare with the polarization-dependent ARPES data, we plotted the orbital-projected DFT band structure as a function of even ($d_{z^2}$) vs. not symmetric ($d_{xy}$/$d_{x^2-y^2}$ and $d_{xz}$/$d_{yz}$) states in Figure <ref>c. At $\Gamma$/$\mathrm{K_0}$, the innermost parabolic bands have predominantly $d_{xy}$/$d_{x^2-y^2}$ and $d_{xz}$/$d_{yz}$ character, whereas the outermost dispersive bands and electron pockets have more $d_{z^2}$ character. These parities are qualitatively consistent with the experimentally observed polarization dependence. The DFT band structure as a function of Cr vs. Nb character (<ref>d) indicates that all of the parabolic bands at $\Gamma$/$\mathrm{K_0}$ are predominantly Nb-derived, while the electron pockets with minima at $\mathrm{M_{SL}}$ and $\mathrm{K_{SL}}$ are composed of mixed Cr and Nb states. The polarization dependence of the host lattice bands more visible in LV polarization is consistent with $d_{xy}$/$d_{x^2-y^2}$ states folded to $\Gamma$ from $\mathrm{K_0}$ by the $\sqrt{3} \times \sqrt{3}$ superlattice potential.elyoubi2021
(a) and (b) ARPES band dispersions for with linear vertical (LV) and linear horizontal (LH) polarized photons, respectively (18 K, $h\nu$ = 79 eV). (c) and (d) DFT orbital-projected band structures of in the FM state, showing in-plane vs. out-of-plane character, and Cr vs. Nb character, respectively. (e–h) The same as (a–d) for , and Cr vs. Ta character in (h).
The polarization-dependent ARPES data for are similar to those for . The innermost bands at $\Gamma$ and $\mathrm{K_0}$ are more prominent in LV polarization (Figure <ref>e), whereas the outer bands around $\Gamma$ and $\mathrm{K_0}$ and more diffuse electron pockets with minima at $\mathrm{M_{SL}}$ and $\mathrm{K_{SL}}$ are more intense in LH polarization (Figure <ref>f). The orbital-projected DFT band structure reveals analogous parities to the Nb analogue (Figure <ref>g) and similar atomic parentage (Figure <ref>h), albeit with more Cr character in the vicinity of $E_\mathrm{F}$. Less Cr–Ta hybridization may be an effect of the slightly longer $c$ lattice parameter in .
For a more quantitative enumeration of the bands near $E_\mathrm{F}$ observed in ARPES, we fitted the momentum distribution curves (MDCs) of the data collected with LH polarization using multiple Lorentzian peaks along the cuts shown in Figure <ref>a–c. We refer to the dispersive features around $\Gamma$ near $E_{\mathrm{F}}$ as the $\alpha$, $\beta$, and $\gamma$ bands, respectively, and two parabolic bands below $E_{\mathrm{F}}$ as $\delta_1$ and $\delta_2$. Comparison of the full width at half maximum (FWHM) values from fits to the $\Gamma$–$\mathrm{M_{SL}}$ MDCs within 200 meV of $E_{\mathrm{F}}$ indicates that the two middle bands have similar FWHMs, while the outermost band (corresponding to the shallow electron pocket) is considerably broader (Figure <ref>d). We therefore assign the middle two features as split $\beta_1$ and $\beta_2$ bands in the vicinity of $E_{\mathrm{F}}$. At higher binding energies, the MDCs can be fitted well with two copies of the electron pocket bands split by about 250 meV, which we refer to as $\gamma_1$ and $\gamma_2$. Comparing the peak center positions from MDC fitting (Figure <ref>e) with the DFT band structure (Figure <ref>f) shows good qualitative agreement, other than the apparent doubling of the $\beta$ and $\gamma$ bands observed in the ARPES data.
(a) ARPES Fermi surface of . Dashed and solid overlaid lines indicate the primitive and $\sqrt{3} \times \sqrt{3}$ superlattice Brillouin zones, respectively. (b) and (c) MDCs along the $\Gamma$–$\mathrm{M_{SL}}$ and $\mathrm{M_{SL}}$–$\mathrm{K_{SL}}$ directions (cuts indicated by bold blue and teal lines in (a)). Gray lines are multi-Lorentzian fits, and colored circles indicate peak center positions, with band assignments labeled. (d) MDCs along the $\Gamma$–$\mathrm{M_{SL}}$ direction (18 K, $h\nu$ = 46 eV). Shaded regions correspond to the full width at half maximum values determined by multi-Lorentzian fits. (e) Sketch of the proposed band structure of derived from the MDC fits (peak center positions shown by gray circles). (f) DFT band structure of , with corresponding band assignments from MDC analysis indicated by colored overlays.
§.§ Temperature Evolution of Band Structure
Due to the aforementioned band splitting, we sought to probe the effect of magnetic ordering on the electronic structures of and by comparing ARPES data collected below and above $T_{\mathrm{C}}$. The dispersion of at 18 K (Figure <ref>a) vs. 145 K (Figure <ref>b) shows that the hole pockets around $\Gamma$ appear smaller at 18 K compared to 145 K. Nevertheless, multi-Lorentzian fits to the MDCs at $E - E_{\mathrm{F}} = -15$ meV show that the outer dispersive bands around $\Gamma$ crossing $E_{\mathrm{F}}$ display the same splitting at 18 and 145 K, as indicated by the labeled $\beta_1$, $\beta_2$, and $\gamma_2$ peaks in Figure <ref>c–d. A similar change in hole pocket sizes is evident in the dispersions of at 18 K (Figure <ref>e) and 170 K (Figure <ref>f). As with , fitting the MDCs at $E - E_{\mathrm{F}} = -15$ meV indicates that the splitting of the outer bands is observed at both 18 and 170 K (Figure <ref>g–h). For all the MDCs, we modeled the inner features around $\Gamma$ with Lorentzian peaks as well, but we note that ascertaining the effects of temperature on these bands is more challenging due to the lower intensities and a non-negligible background component from inelastic scattering. Nonetheless, the persistence of the $\beta_1$, $\beta_2$, and $\gamma_2$ splitting above $T_{\mathrm{C}}$ and the consistency in its magnitude for both materials prompted us to consider non-magnetic origins.
(a) and (b) ARPES dispersions for taken at 18 and 145 K, respectively. (c) and (d) Momentum distribution curves for for $E - E_{\mathrm{F}} = -15$ meV, and fits to multiple Lorentzian peaks (dotted lines), taken at 18 and 145 K, respectively. (e) and (f) ARPES dispersions for taken at 18 and 170 K, respectively. (g) and (h) Momentum distribution curves for for $E - E_{\mathrm{F}} = -15$ meV, and fits to multiple Lorentzian peaks (dotted lines), taken at 18 and 170 K, respectively. All data were measured with $h\nu = 79$ eV and LH polarization.
§.§ ARPES Measurements with Micron-scale Probes
Motivated by the polar nature of these materials and the observation of unexplained band splitting, we carried out ARPES experiments on with a smaller beam size (2–15 µm) to investigate the possible impact of nonuniform sample surfaces. We identified three types of distinct areas based on their Fermi surfaces (Figure <ref>a–c), core level spectra (Figure <ref>d–e), and band dispersions (Figure <ref>f–h). Spots with the simplest Fermi surfaces and the largest hole pockets around $\Gamma$ and $\mathrm{K_0}$ (Figure <ref>a) have the weakest Cr $2p$ core level spectra (Figure <ref>d). Spots with Fermi surfaces representative of the majority of the samples, with the aforementioned $\beta$ and $\gamma$ band doubling (Figure <ref>b), exhibit Cr $2p$ core level signals of intermediate intensity. Finally, spots with Fermi surfaces missing the broadest outermost pockets around $\Gamma$ and $\mathrm{K_0}$ (Figure <ref>c) show the strongest Cr $2p$ core level spectra. Based on the Cr core level intensities, these areas appear to correspond to low, intermediate, and high relative Cr surface concentrations, respectively. The trend in the S $2p$ core levels from the same spots corroborate this assignment: with decreasing Cr surface coverage, an S peak at lower binding energies grows in (indicated by the green arrow in Figure <ref>e), consistent with more reduced S sites on the surface that are not sharing electron density with Cr.
(a–c) ARPES Fermi surfaces of taken in regions of low, intermediate, and high surface Cr coverage, respectively (20 K, $h\nu$ = 120 eV). (d) and (e) Cr $2p$ and S $2p$ core level spectra, respectively, with the green arrow in (e) indicating the S $2p$ feature at low binding energy. (f–h) ARPES band dispersions for the same regions with low, intermediate, and high surface Cr coverage shown in (a–c). Blue arrows in (g) and (h) indicate non-dispersive features. (i) DFT orbital-projected band structure for showing atomic origin of bands.
The ARPES dispersions from these spots also exhibit notable differences. The “low Cr” spot (Figure <ref>f) exhibits less $\sqrt{3} \times \sqrt{3}$ superlattice reconstruction than the other spots (as seen from the apparent three-fold symmetry around $\mathrm{K_0}$ and different sized hole pockets at $\Gamma$ and $\mathrm{K_0}$) and resembles $2H$-NbS2 with $E_{\mathrm{F}}$ shifted up by approximately 250 meV.elyoubi2021 The intense “X”-shaped feature at $\Gamma$ located at about $-1.6$ eV in “low Cr” is shifted down to about $-2.0$ eV in both “intermediate Cr” (Figure <ref>g) and “high Cr” (Figure <ref>h) consistent with the latter two sampling more electron-doped states on average. In the “intermediate Cr” spot, the $\gamma$ band electron pockets near $E_{\mathrm{F}}$ are split by about 250 meV, as they are in other spectra measured with larger beam sizes. In the “high Cr” spot, the electron pockets near $E_{\mathrm{F}}$ are not noticeably split; instead, only a single set of features resembling the lower $\gamma_1$ band in “intermediate Cr” is observed. Additionally, the presence of flat bands in the “intermediate Cr” and “high Cr” spots (where they are especially prominent), as indicated by the blue arrows in Figure <ref>g–h, coincides with Cr states in the orbital-projected DFT band structures (Figure <ref>i), lending further support to the surface coverage assignments.
§ DISCUSSION
§.§ Band Structure and Magnetic Exchange Interactions
Taking the results from both ARPES and DFT into account, the most pronounced difference in the band structures of and is the more dispersive bands in the Ta analogue. The origin appears to be steeper dispersions in $2H$-TaS2 compared to $2H$-NbS2, i.e. the relative band dispersions of the host lattice materials are retained after Cr intercalation. This trend can be attributed to better overlap facilitated by more extended Ta $5d$ orbitals compared to the Nb $4d$ orbitals. For a more detailed explanation, we briefly discuss the salient bonding interactions in both host lattice materials.
(a) Orbital-projected band structure calculation for $2H$-NbS2, showing Nb $4d$ and S $3p$ character. (b) Orbital-projected band structure calculation for $2H$-TaS2, showing Ta $5d$ and S $3p$ character. (c) Projected density of states (DOS) calculation for $2H$-NbS2. (d) Projected DOS calculation for $2H$-TaS2. The energy spread of the Nb $4d$ and Ta $5d$ bands is indicated on the right.
The bands within about 6 eV of the Fermi level in $2H$-$M$S2 ($M$ = Nb or Ta) are comprised of $M$ $d$ states and S $3p$ states, indicating that $M$–S $d$–$p$ and $M$–$M$ $d$–$d$ interactions are those relevant to determining the strength of the bonding and the resulting dispersivity of the bands. Mixing among the $M$ $d$ orbitals results in the formation of a hybridization gap within the $d$ manifold: the bands crossing $E_\mathrm{F}$ are composed of $d_{z^2}$, $d_{xy}$, and $d_{x^2-y^2}$ orbitals, while the higher-lying $d$ bands have more $d_{xz}$ and $d_{yz}$ character.mattheiss1973 TaS2 has a slightly smaller in-plane lattice constant,fisher1980,meetsma1990 and the $5d$ orbitals are more spatially extended than the $4d$ orbitals in NbS2. This leads to better relative overlap in the Ta analogue, both in terms of $M$–S $d$–$p$ bonds and next-nearest-neighbor $M$–$M$ $d$–$d$ interactions. Hence, overall, the $d$ manifold of TaS2 is more dispersive than that of NbS2, as shown in the band structure and density of states (DOS) calculations in Figure <ref>b and c. In turn, the bands crossing $E_\mathrm{F}$ are also more dispersive in the Ta analogue.
These arguments can also be used to explain why the host lattice-derived bands are more dispersive in than , which we have observed in ARPES and DFT. From the crystal structures, the Ta–S bonds in are slightly shorter (2.488(3) Å on average) than the Nb–S bonds in (2.4931(11) Å on average), and the in-plane lattice constant in is slightly smaller (5.7155(5) Å vs. 5.7400(7) Å), suggestive of stronger Ta–S overlap compared to Nb–S overlap. In addition, the in-plane electrical conductivity of is more than an order of magnitude higher than that of , which is consistent with the more dispersive bands and larger hole pockets in the Ta analogue expected from this analysis.ghimire2013,obeysekera2021
The more dispersive bands in may have implications for the relative strengths of magnetic exchange interactions in these materials. As previously established in the literature, has a higher $T_\mathrm{C}$ than (133 K vs. 110 K for our samples). Given the extensive mixing of intercalant- and host lattice-derived states near $E_{\mathrm{F}}$ shown in DFT and supported by ARPES, our findings suggest that the magnetic exchange interactions involve some degree of itinerancy for Cr-based spins. Nevertheless, due to the considerable distance ($> 5.7$ Å) between Cr sites, magnetic exchange is likely still mediated through TMD-derived states. The modestly higher $T_\mathrm{C}$ of may reflect stronger FM coupling (i.e. larger $J_{\parallel}$ and $J_{\perp}$) in this material, which could arise from itinerant carriers with higher $v_{\mathrm{F}}$. Nevertheless, the shorter soliton wavelength and larger $H_\mathrm{c}$ imply that the ratio $D/J_{\perp}$ is considerably larger than in the Nb analogue, i.e. $D$ (the DM interaction term) increases more than $J_{\perp}$ in going from Nb to Ta. Higher $D$ in the Ta analogue can be ascribed to larger spin–orbit coupling (SOC),zhang2021,obeysekera2021,du2021 and appears to affect the length and energy scales of the chiral spin textures more than changes in $J$.
The fact that and have qualitatively analogous magnetic phase diagrams despite different Fermi wavevectors, $k_\mathrm{F}$, supports the notion that the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction does not adequately describe exchange coupling in these materials.ghimire2013,sirica2021 According to the RKKY formalism, the sign and magnitude of $J$ would depend closely on the magnitude of $k_\mathrm{F}$, which does not appear to be the case for and . However, the exchange interactions in other magnetic intercalated TMDs, such as Fe_$x$NbS2 and Fe_$x$TaS2, have been proposed to be more RKKY-like.ko2011,zheng2021,wu2022 Based on our results, we posit that TaS2-based materials may have more dispersive bands than NbS2-based materials for intercalants other than Cr. This difference (along with SOC) may be pertinent to the disparate magnetic properties of Nb and Ta analogues in other families of intercalated TMDs (e.g. Fe_1/3NbS2 is an antiferromagnet, whereas Fe_1/3TaS2 is a ferromagnet).
The most obvious route toward band engineering in and is through modulation of the Cr concentration. However, a delicate balance exists between disorder and vacancies in the Cr superlattice and the integrity of the desired spin textures.dyadkin2015,kousaka2022,goodge2023 Co-intercalation of another species into the interstitial spacepan2023 or substitutional doping on the TMD sublattice could constitute other pathways toward tuning the filling level while maintaining a well-ordered Cr superlattice (and hence a globally defined $D$). Future studies in these directions could more definitively probe the effects of band dispersion on the resultant spin textures. We note additionally that the sensitivity of the observed surface states on Cr concentration—as discussed in more detail in the next section—suggests that the surface electronic structure is amenable to tuning through further functionalization.
§.§ Exchange Splitting vs. Surface Termination Effects
Previous ARPES studies on have also reported band splitting near $E_{\mathrm{F}}$ that appears similar to our assignment of $\beta_1$, $\beta_2$, and $\gamma_2$ bands. These works interpreted this phenomenon as exchange splitting.sirica2016,sirica2021,qin2022 However, we observed good agreement between the exchange splitting predicted by our FM spin-polarized DFT band structures and observed in our ARPES results. Specifically, according to the spin-polarized DFT band structure calculations shown in Figure <ref>c–d, the $\alpha$ and $\beta$ bands (as labeled in Figure <ref>f) are an exchange-split pair. Because of different mixing in the spin-majority and spin-minority channels, $\alpha$ has more $d_{xy}$/$d_{x^2-y^2}$ character, while $\beta$ has more $d_{z^2}$ character (Figure <ref>c and g). The corresponding bands observed in ARPES show the expected polarization dependence: the $\alpha$ band, which just touches $E_\mathrm{F}$, is much more visible in LV polarization (Figure <ref>a and e), whereas the $\beta$ band is more prominent in LH polarization (Figure <ref>b and f). This prompted us to consider alternative explanations for the observation of more bands in ARPES than predicted by DFT.
Instead, taking the polar nature of intercalated TMDs into account, we surmised that the observed doubling of $\beta_1$/$\beta_2$ and $\gamma_1$/$\gamma_2$ bands could be attributed to surface termination effects. Previous work suggests that spatially distinct areas of Cr- and $M$S2-termination exist on cleaved surfaces: STM studies have observed islands of intercalants on cleaved crystals of intercalated TMDs,sirica2016,lim_tunable_2022 and recent ARPES studies of V_1/3NbS2edwards2023 and Co_1/3NbS2zhang2023 reported termination-dependent surface states. Thus, to understand the expected effects of surface termination on the ARPES data, we consider the charge distributions for Cr-terminated and $M$S2-terminated surfaces. As summarized in Figure <ref>a, the Cr^3+ centers formally donate one electron per $M$S2 formula unit, leading to alternating layers with $+1$ and $-1$ formal charges. The phenomenon of charge redistribution at polar-to-nonpolar interfaces to prevent a polar catastrophe (i.e. diverging electrostatic potential) is well-documented,nakagawa2006 and we expect analogous redistribution to occur at both the [$M$S2]–vacuum interface and the [Cr_1/3]–vacuum interface. Assuming fully occupied (unoccupied) Cr sites for the Cr- ($M$S2-) terminated regions, the surface formal charges expected from simple electron counting are shown in Figure <ref>. In short, $M$S2-terminated regions should exhibit $M$S2-derived surface states that are hole-doped relative to the bulk, originating from partial electron transfer from the surface TMD layer to compensate for its polarity.
Schematic illustration of local Cr clustering on cleaved surfaces of samples, with two distinct regions corresponding to predominantly $M$S2 and Cr terminations. Formal charges given for the two regions are based on a simple electron-counting picture, assuming completely full (absent) Cr coverage in the Cr- ($M$S2-) terminated regions. The “low Cr,” “intermediate Cr,” and “high Cr” sampling areas (colored boxes) thus have average surface stoichiometries of 0, 1/6, and 1/3.
Building upon this picture, we propose that the three distinct spectral signatures observed with mesoscopic probes (Figure <ref>) can be explained by sampling two different surface terminations, as illustrated in Figure <ref>. The “low Cr” and “high Cr” spots correspond to areas with almost exclusively $M$S2 termination and Cr termination, respectively. The “intermediate Cr” spots contain both terminations, resulting in doubled $\beta$ and $\gamma$ bands that are also observed in data collected with larger probes. The $\beta$ and $\gamma$ bands at lower binding energies are associated with $M$S2-terminated regions. Such effects are broadly consistent with those observed on surfaces of other polar layered materials.hossain2008 Hence, we propose that these additional band doublings do not reflect magnetic ordering, but rather originate from charge redistribution. More generally, the possible contribution of surface states should be considered in other cases where unexpected bands are observed in ARPES studies on magnetic intercalated TMDs.
§ CONCLUSIONS
The electronic structures of the chiral helimagnets and have been investigated using ARPES and DFT. Compared to the host lattice materials $2H$-NbS2 and $2H$-TaS2, the Cr-intercalated materials exhibit band folding from the $\sqrt{3} \times \sqrt{3}$ Cr superlattice, higher $E_{\mathrm{F}}$ from electron transfer from Cr to the host TMD, exchange splitting from the in-plane FM ordering of Cr moments, and new bands from hybridization between Cr and TMD states. The chief difference between the band structures of $2H$-NbS2 and $2H$-TaS2—higher band dispersions in the Ta analogue—are retained after Cr intercalation, resulting in a larger $v_{\mathrm{F}}$ in . This finding may have implications for the higher $T_{\mathrm{C}}$ in via stronger FM coupling mediated by itinerant carriers.
By studying the polarization dependence in ARPES and fitting the MDCs, we find that the experimentally observed band structures agree well with the orbital-projected DFT band structures. The primary features at $E_{\mathrm{F}}$ in both materials consist of dispersive hole pockets at $\Gamma$ (and $\mathrm{K}_0$) and shallow electron pockets centered around $\mathrm{K_{SL}}$. Notably, many bands near $E_{\mathrm{F}}$ have significant Cr character in both materials, indicating that a rigid band model is insufficient for modeling the effects of Cr intercalation. Additional copies of bands crossing $E_{\mathrm{F}}$ that are not predicted by DFT are assigned to surface states originating from TMD-terminated regions. The observation of three distinct regions in ARPES experiments with smaller spot sizes is consistent with Cr, TMD, and mixed surface terminations. These results indicate that the polar nature of the surfaces of intercalated TMDs affects the band splitting observed in ARPES data.
It has been well-established that and have analogous magnetic phase diagrams with different energy scales and different wavelengths of magnetic solitons. Our results show that the electronic structures of these two isostructural materials are broadly analogous, with more dispersive bands in the Ta analogue. This finding suggests that band structure engineering and Fermi level tuning may allow for further modulation of the magnitude of $J$, and hence the length and energy scales of magnetic solitons in these materials. To maintain the $\sqrt{3} \times \sqrt{3}$ Cr superlattice, substitutional doping on the TMD sites or co-intercalation of other charged species may be viable routes toward controlling the properties of these chiral helimagnets, which are promising platforms for studying the interplay between electronic structure and the microscopic mechanisms underlying noncollinear magnetism.
§ METHODS
§.§ Crystal Growth
Single crystals of and were grown using chemical vapor transport using iodine as a transport agent. For , high-purity powders of elemental Cr, Nb, and S in a 0.6:1:2 ratio and 5 mg/cm$^3$ of I_2 were sealed under vacuum in a fused quartz ampoule approximately 48 cm long. The ampoule was placed in a three-zone tube furnace with the hot end zone and middle zone maintained at 1050
°C and the cold (growth) zone maintained at 850 °C for 14 days before cooling to room temperature. For , high-purity powders of elemental Cr, Ta, and S in a 0.47:1:2.1 ratio and 2 mg/cm$^3$ of I_2 were sealed under vacuum in a fused quartz ampoule approximately 25 cm long. The ampoule was placed in a two-zone tube furnace with the hot zone maintained at 1100 °C and the cold (growth) zone maintained at 1000 °C for 14 days before cooling to room temperature. Shiny plate-shaped crystals with a silvery metallic luster and hexagonal habit up to $4 \times 4 \times 0.5$ mm in size were obtained.
§.§ Structural and Compositional Characterization
Single crystal X-ray diffraction was collected on a Rigaku XtaLAB P200 with Mo K$\alpha$ radiation at 295 K. Data reduction and scaling and empirical absorption correction were performed in CrysAlis Pro. Structures were solved by direct methods using SHELXTsheldrick2015 and refined against $F^2$ on all data by full-matrix least squares with SHELXL.sheldrick2015a Raman spectroscopy was collected on a Horiba LabRAM HR Evolution with an ultra-low frequency filter using 633 nm laser excitation and powers between 1 and 8 mW. Energy dispersive X-ray spectroscopy was acquired on a FEI Quanta 3D FEG or a Scios 2 DualBeam scanning electron microscope with an accelerating voltage of 20 kV.
§.§ Magnetometry
DC magnetization measurements were carried out on a Quantum Design Physical Property Measurement System Dynacool equipped with a 12 T magnet using either the Vibrating Sample Magnetometer option or the AC Measurement System II option. Single crystals were affixed to quartz sample holders with GE Varnish such that the magnetic field was applied perpendicular to the crystallographic $c$ axis.
§.§ ARPES Measurements
ARPES data were collected at the Quantum Materials Spectroscopy Centre (QMSC) of the Canadian Light Source (CLS) and at Beamline 7.0.2 of the Advanced Light Source (ALS) on both the microARPES and nanoARPES endstations using Scienta Omicron R4000 hemispherical electron analyzers. The beam spot sizes were approximately $20 \times 100$ µm at QMSC, $15 \times 15$ µm on microARPES, and $2 \times 2$ µm on nanoARPES. Results were reproduced on multiple samples at both beamlines with the exception of spatial variation observed with smaller spot sizes. Samples were cooled down to the base temperature of 20 K or below and cleaved in situ by carefully knocking off alumina posts affixed to the top surface of the sample with silver epoxy. All measurements were conducted at pressures lower than $5 \times 10^{-11}$ Torr. The primary datasets were collected at photon energies of 46, 79, and 120 eV with linear horizontal and linear vertical polarizations. Data analysis was performed using the PyARPES software package.stansbury2020
§.§ DFT Calculations
First-principles calculations were performed by using the open source plane-wave code Quantum Espresso (QE).QE The optimized norm-conserving Vanderbilt (ONCV) pseudopotentials from the PseudoDojo project ONCV1,van2018pseudodojo were applied. The kinetic energy cut-off for wavefunctions were set to 86 Ry for all the self-consistent calculations; for these calculations, the experimental lattice constants obtained from X-ray diffraction were used.fisher1980,meetsma1990 A $\Gamma$-centered $4\times4\times2$ $\it{k}$-mesh was sampled in the Brillouin zone for both and , and a $8\times8\times2$ $\it{k}$-mesh for both $2H$-NbS2 and $2H$-TaS2. The Perdew–Burke–Ernzerhof (PBE) functional PBE1997 of the spin-polarized generalized gradient approximation (GGA) was used to describe the exchange correlation of electrons.
§ ACKNOWLEDGMENTS
We thank Sinéad Griffin and Ryan Day for helpful discussions, and thank Nicholas Settineri for assistance in obtaining the single crystal XRD data. The experimental work is based upon work supported by the Air Force Office of Scientific Research under AFOSR Award No. FA9550-20-1-0007. L.S.X. acknowledges support from the Arnold and Mabel Beckman Foundation (Award No. 51532) and L'Oréal USA (Award No. 52025) for postdoctoral fellowships. O.G. acknowledges support from an NSF Graduate Research Fellowship grant DGE 1752814, and National GEM Consortium Fellowship. S.H.R. was supported by the QSA, supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers. S.H. acknowledges support from the Blavatnik Innovation Fellowship. Y.P. acknowledges the financial support from the Air Force Office of Scientific Research under AFOSR Award No. FA9550-21-1-0087. Confocal Raman spectroscopy was supported by a Defense University Research Instrumentation Program grant through the Office of Naval Research under Award No. N00014-20-1-2599 (D.K.B.). Part of the research described in this paper was performed at the Canadian Light Source, a national research facility of the University of Saskatchewan, which is supported by the Canada Foundation for Innovation (CFI), the Natural Sciences and Engineering Research Council (NSERC), the National Research Council (NRC), the Canadian Institutes of Health Research (CIHR), the Government of Saskatchewan, and the University of Saskatchewan. This research used resources of the Advanced Light Source, which is a DOE Office of Science User Facility under contract no. DE-AC02-05CH11231. Other instrumentation used in this work was supported by grants from the Canadian Institute for Advanced Research (CIFAR–Azrieli Global Scholar, Award No. GS21-011), the Gordon and Betty Moore Foundation EPiQS Initiative (Award no. 10637), the W.M. Keck Foundation (Award No. 993922), and the 3M Foundation through the 3M Non-Tenured Faculty Award (No. 67507585). The computational part used resources of the Center for Functional Nanomaterials, which is a U.S. DOE Office of Science Facility, and the Scientific Data and Computing Center, a component of the Computational Science Initiative, at Brookhaven National Laboratory under Contract No. DE-SC0012704, and the lux supercomputer at the University of California, Santa Cruz, funded by NSF MRI, Grant No. AST 1828315, and used Stampede supercomputer at the University of Texas at Austin's Texas Advanced Computing Center (TACC) through allocation DMR160106 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296. This research was undertaken thanks in part to funding from the Max Planck–UBC–UTokyo Centre for Quantum Materials and the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program. This project is also funded by the Mitacs Accelerate Program; the QuantEmX Program of the Institute for Complex Adaptive Matter (ICAM); the Moore EPiQS Program (A.D.); and the CIFAR Quantum Materials Program (A.D.).
§ COMPETING INTERESTS
The authors declare no competing interests.
Missing 'biblatex' package
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titleSkyrmions on the track
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titleMemory on the racetrack
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American Physical Society
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American Physical Society
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titleResponse of the chiral soliton lattice to spin-polarized currents
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journaltitlePhilos. Mag. B
title3d Transition-Metal Intercalates of the Niobium and Tantalum Dichalcogenides. I. Magnetic Properties
issn1364-2812, 1463-6417
journaltitlePhilos. Mag. B
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journaltitleBull. Soc. Chim. Fr.
titleEtude générale de systèmes $M_x$NbS2 (M élément de transition de la première période)
abstractA thermodynamic theory of “weak” ferromagnetism of α-Fe2O3, MnCO3 and CoCO3 is developed on the basis of landau's theory of phase transitions of the second kind. It is shown that the “weak” ferromagnetism is due to the relativistic spin-lattice and the magnetic dipole interactions. A strong dependence of the properties of “weak” ferromagnetics on the magnetic crystalline symmetry is noted and the behaviour of these ferromagnetics in a magnetic field is studied.
journaltitleJ. Phys. Chem. Solids
titleA thermodynamic theory of “weak” ferromagnetism of antiferromagnetics
abstractA theory of anisotropic superexchange interaction is developed by extending the Anderson theory of superexchange to include spin-orbit coupling. The antisymmetric spin coupling suggested by Dzialoshinski from purely symmetry grounds and the symmetric pseudodipolar interaction are derived. Their orders of magnitudes are estimated to be (Δgg) and (Δgg)2 times the isotropic superexchange energy, respectively. Higher order spin couplings are also discussed. As an example of antisymmetric spin coupling the case of CuCl2·2H2O is illustrated. In CuCl2·2H2O, a spin arrangement which is different from one accepted so far is proposed. This antisymmetric interaction is shown to be responsible for weak ferromagnetism in α-Fe2O3, MnCO3, and CrF3. The paramagnetic susceptibility perpendicular to the trigonal axis is expected to increase very sharply near the Néel temperature as the temperature is lowered, as was actually observed in CrF3.
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titleAnisotropic Superexchange Interaction and Weak Ferromagnetism
issn0031-9015, 1347-4073
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titleMagnetic Properties of Cr_1/3NbS2
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titleChiral Magnetic Soliton Lattice on a Chiral Helimagnet
issn1098-0121, 1550-235X
journaltitlePhys. Rev. B
titleMagnetic Phase Transition in Single Crystals of the Chiral Helimagnet Cr_1/3NbS2
liliaxie/Zotero/storage/QP5VGB9F/Ghimire et al. - 2013 - Magnetic phase transition in single crystals of th.pdf
issn0935-9648, 1521-4095
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titleChiral Helimagnetism and One-Dimensional Magnetic Solitons in a Cr-Intercalated Transition Metal Dichalcogenide
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titleThe Magneto-Transport Properties of Cr_1/3TaS2 with Chiral Magnetic Solitons
issn0027-8424, 1091-6490
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titleAn Emergence of Chiral Helimagnetism or Ferromagnetism Governed by Cr Intercalation in a Dichalcogenide CrNb3S6
abstractThe chiral magnetic soliton, a topological kinklike spin texture, has significant applications in spintronic components. In this work, a crossover of critical behavior is found in Cr1/3TaS2, a chiral magnetic soliton host with the highest TC to date. Angular-dependent magnetization reveals that Cr1/3TaS2 exhibits an easy orientation within the isotropic ab plane, but displays anisotropy with the c axis. By using a modified iterative method, two distinct sets of critical exponents, including β−=0.3190(1) and γ−=1.263(8) for T≤TC, and β+=0.3475(2) and γ+=1.385(5) for T≥TC, are acquired on both sides of the transition. Analysis of the exponents indicates a crossover of the magnetic interaction from a three-dimensional Ising type below TC to a three-dimensional Heisenberg type above TC, implying nontrivial magnetism in this system. Based on universality scaling, a detailed H−T phase diagram around TC is constructed for H⊥c. The crossover of the critical behavior in Cr1/3TaS2 is peculiar to chiral magnetic soliton hosts, suggesting that the three-dimensional magnetic coupling is replaced by a one-dimensional one in the chiral magnetic soliton phase via a phase transition.
journaltitlePhys. Rev. B
titleCrossover of critical behavior and nontrivial magnetism in the chiral soliton lattice host Cr_1/3TaS2
abstractWe report long periodic chiral helimagnetic orderings in ferromagnetic inorganic compounds CrM3S6 (M = Nb and Ta) with a chiral space group of P6322. Magnetization in polycrystalline samples and high resolution powder neutron diffraction were measured. Our powder neutron diffraction measurements in CrM3S6 successfully separated nuclear and magnetic satellite peaks, having the period of hundreds of angstroms along the c— axis. Therefore, we propose that the magnetic ordering in ferromagnetic CrM3S6 is not ferromagnetic, but long periodic chiral helimagnetic ordering.
journaltitleJ. Phys. Conf. Ser.
titleLong Periodic Helimagnetic Ordering in Cr$M$3S6 ($M$ = Nb and Ta)
American Physical Society
journaltitlePhys. Rev. B
titleElectronic Structure of the Chiral Helimagnet and $3d$-Intercalated Transition Metal Dichalcogenide Cr_1/3NbS2
American Physical Society
journaltitlePhys. Rev. B
titleDisentangling Electronic, Lattice, and Spin Dynamics in the Chiral Helimagnet Cr_1/3NbS2
American Physical Society
journaltitlePhys. Rev. B
titlePersistent Exchange Splitting in the Chiral Helimagnet Cr_1/3NbS2
journaltitlePhys. Rev. B
titleEnergy-Gap Driven Low-Temperature Magnetic and Transport Properties in Cr_1/3$M$S2 ($M$ = Nb, Ta)
issn0002-7863, 1520-5126
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titleStructure and Magnetism of Iron- and Chromium-Intercalated Niobium and Tantalum Disulfides
abstractThe nonrelativistic augmented-plane-wave (APW) method is applied to calculate the electronic band structures of several transition-metal-dichalcogenide (TX2) layer compounds, including materials with the C6(1T−HfS2,1T−TaS2), C27(2H−TaS2,2H−NbSe2), and C7(2H−MoS2) structure types. These calculations involve crystal potentials that are derived from neutral-atom charge densities. The results of these calculations confirm that the group-IVB (1T−HfS2) and group-VIB (2H−MoS2) compounds are semiconductors; the calculated indirect band gaps of 2.7 and 1.2 eV are in reasonable agreement with the observed values of 2.0 and 1.4 eV, respectively. Metallic behavior is predicted for the intermediate group-VB compounds 1T−TaS2, 2H−TaS2, and 2H−NbSe2. A novel feature of the metal d bands in the 2H−TX2 compounds is the occurence of a 1-eV hybridization gap within the dz2 and dxy, dx2−y2 manifolds. This splits off a pair of hybridized d bands which are half-filled in 2H−TaS2 and 2H−NbSe2 and completely filled in 2H−MoS2. As a result of this hybridization gap, the valence or conduction bandwidths in each of these 2H−TX2 compounds are reduced to about 1 eV.
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titleBand Structures of Transition-Metal-Dichalcogenide Layer Compounds
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shorttitleUtility of semilocalized bonding schemes in extended systems
titleUtility of semilocalized bonding schemes in extended systems: three-center metal-metal bonding in molybdenum sulfide (MoS2), niobium tantalum sulfide bronze (H_$x$(Nb,Ta)S2), and zirconium sulfide (ZrS)
ZE884CV/ic00010a019.html:text/html;Full Text PDF:/Users/liliaxie/Zotero/storage/RARPPQ32/Yee and Hughbanks - 1991 - Utility of semilocalized bonding schemes in extend.pdf:application/pdf
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titleAnalogies between the concepts of molecular chemistry and solid-state physics concerning structural instabilities. Electronic origin of the structural modulations in layered transition metal dichalcogenides
nalogies between the concepts of molecular chemis.pdf:application/pdf
AIP Publishing LLC
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titleSpin structure of the anisotropic helimagnet Cr_1/3NbS2 in a magnetic field
abstractThe topologically protected, chiral soliton lattice is a unique state of matter offering intriguing functionality, and it may serve as a robust platform for storing and transporting information in future spintronic devices. While the monoaxial chiral magnet Cr1∕3NbS2 is known to host this exotic state in an applied magnetic field, its detailed microscopic origin has remained a matter of debate. Here, we work towards addressing this open question by measuring the spin wave spectrum of Cr1∕3NbS2 over the entire Brillouin zone with inelastic neutron scattering. The well-defined spin wave modes allow us to determine the values of several microscopic interactions for this system. The experimental data are well-explained by a Heisenberg Hamiltonian with exchange constants up to the third nearest neighbor and an easy plane magnetocrystalline anisotropy term. Our work shows that both the second and third nearest neighbor exchange interactions contribute to the formation of the helimagnetic and chiral soliton lattice states in this robust three-dimensional magnet.
journaltitleAppl. Phys. Lett.
titleExtended exchange interactions stabilize long-period magnetic structures in Cr_1/3NbS2
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titleTricritical point and phase diagram based on critical scaling in the monoaxial chiral helimagnet Cr_1/3NbS2
journaltitleNano Lett.
titleExcitations of Intercalated Metal Monolayers in Transition Metal Dichalcogenides
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titleFermiology and Electron-Phonon Coupling in the $2H$ and $3R$ Polytypes of NbS2
family=van Wezel,
American Physical Society
journaltitlePhys. Rev. B
titleOrbital Selectivity Causing Anisotropy and Particle-Hole Asymmetry in the Charge Density Wave Gap of $2H$-TaS2
abstractWe study the mechanism of the exceptionally large anomalous Hall effect (AHE) in the noncentrosymmetric antiferromagnet CoNb3S6 by angle-resolved photoemission spectroscopy (ARPES) and magnetotransport measurements. From ARPES measurements of CoNb3S6 and its family compounds (FeNb3S6 and NiNb3S6), we find a band dispersion unique to the Co intercalation existing near the Fermi level. We further demonstrate that a slight deficiency of sulfur in CoNb3S6 eliminates the ferromagnetism and the AHE simultaneously while hardly changing the band structure, indicating that the weak ferromagnetism is responsible for the emergence of the large AHE. Based on our results, we propose Weyl points near the Fermi level to cause the large AHE.
journaltitlePhys. Rev. B
titleLarge anomalous Hall effect induced by weak ferromagnetism in the noncentrosymmetric antiferromagnet CoNb3S6
American Physical Society
journaltitlePhys. Rev. B
titleVisualizing the Out-of-Plane Electronic Dispersions in an Intercalated Transition Metal Dichalcogenide
journaltitlePhys. Rev. B
titleRole of Intercalated Cobalt in the Electronic Structure of Co_1/3NbS2
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titleGiant valley-Zeeman coupling in the surface layer of an intercalated transition metal dichalcogenide
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titleAngle-Resolved Photoemission Studies of the Cuprate Superconductors
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titleStoichiometry, structure, and physical properties of niobium disulfide
journaltitleActa Cryst. C
titleStructure of $2H$-TaS2
issn0031-9007, 1079-7114
journaltitlePhys. Rev. Lett.
titleRKKY Ferromagnetism with Ising-Like Spin States in Intercalated Fe_1/4TaS2
liliaxie/Zotero/storage/XBKIDHGE/Ko et al. - 2011 - RKKY Ferromagnetism with Ising-Like Spin States in.pdf
abstractDzyaloshinskii–Moriya interaction (DMI) is vital to form various chiral spin textures, novel behaviors of magnons and permits their potential applications in energy-efficient spintronic devices. Here, we realize a sizable bulk DMI in a transition metal dichalcogenide (TMD) 2H-TaS2 by intercalating Fe atoms, which form the chiral supercells with broken spatial inversion symmetry and also act as the source of magnetic orderings. Using a newly developed protonic gate technology, gate-controlled protons intercalation could further change the carrier density and intensely tune DMI via the Ruderman–Kittel–Kasuya–Yosida mechanism. The resultant giant topological Hall resistivity $${\rho }_{{xy}}ˆ{T}$$of $$1.41{\mathrm{\mu}} \Omega \cdot {{\mathrm{cm}}}$$at $${V}_{g}=-5.2{\mathrm{V}}$$(about $$424 \%$$larger than the zero-bias value) is larger than most known chiral magnets. Theoretical analysis indicates that such a large topological Hall effect originates from the two-dimensional Bloch-type chiral spin textures stabilized by DMI, while the large anomalous Hall effect comes from the gapped Dirac nodal lines by spin–orbit interaction. Dual-intercalation in 2H-TaS2 provides a model system to reveal the nature of DMI in the large family of TMDs and a promising way of gate tuning of DMI, which further enables an electrical control of the chiral spin textures and related electromagnetic phenomena.
journaltitleNat. Commun.
titleTailoring Dzyaloshinskii–Moriya interaction in a transition metal dichalcogenide by dual-intercalation
American Physical Society
journaltitlePhys. Rev. X
titleHighly Tunable Magnetic Phases in Transition-Metal Dichalcogenide Fe_$1/3+\delta$NbS2
journaltitlePhys. Rev. B
titleStructural Disorder versus Chiral Magnetism in Cr_1/3NbS2
abstractTransition metal-intercalated transition metal dichalcogenides (TMDs) are promising platforms for next-generation spintronic devices based on their wide range of electronic and magnetic phases, which can be tuned by varying the host lattice or the identity of the intercalant, along with its stoichiometry and spatial order. Some of these compounds host a chiral magnetic phase in which the helical winding of magnetic moments propagates along a high-symmetry crystalline axis. Previous studies have demonstrated that variation in intercalant concentrations can have a dramatic impact on the formation of chiral domains and ensemble magnetic properties. However, a systematic and comprehensive study of how atomic-scale order and disorder impacts collective magnetic behavior are so far lacking. Here, we leverage a combination of imaging modes in the (scanning) transmission electron microscope (S/TEM) to directly probe (dis)order across multiple length scales and show how subtle changes in the atomic lattice can be leveraged to tune the mesoscale spin textures and bulk magnetic response, with direct implications for the fundamental understanding and technological implementation of such compounds.
notearXiv:2305.06656 [cond-mat]
titleConsequences and control of multi-scale (dis)order in chiral magnetic textures
journaltitlePhys. Rev. Mater.
titleFe and Cr co-intercalation in $2H$-NbS2 single crystals for realization of perpendicular magnetic anisotropy and large anomalous Hall effect
issn1530-6984, 1530-6992
journaltitleNano Lett.
titleTunable Single-Atomic Charges on a Cleaved Intercalated Transition Metal Dichalcogenide
abstractThe Dirac equation combines the two cornerstones of modern physics-quantum mechanics and relativity. There are several manifestations of the Dirac equation in condensed matter systems, such as the quasiparticle dispersion in graphene, topological insulators, Dirac semimetals (DSMs), Weyl semimetals, and d-wave high-temperature superconductors. In a DSM, the massless Dirac fermion has zero chirality, leading to surface states connected adiabatically to a topologically trivial surface state as well as vanishing anomalous Hall effect (AHE). Recently, it is predicted that in the nonrelativistic limit of certain antiferromagnets, there exists a type of chiral 'Dirac-like' fermion, whose dispersion manifests four-fold degenerate crossing points formed by doubly degenerate linear bands, with topologically protected Fermi arcs. Such unconventional chiral fermion, protected by a hidden SU(2) symmetry in the hierarchy of an enhanced crystallographic group, namely spin space group, is not experimentally verified yet. Here, by combining neutron diffraction, angle-resolved photoemission spectroscopy and first-principles calculations, we reveal the existence of the Fermi-arc surface states induced by chiral Dirac-like fermions in collinear antiferromagnet CoNb3S6, which caught great interest due to its surprisingly large AHE. Our transport measurements and theoretical calculations provide a scenario that large Berry curvature embedded in the chiral fermions and weak symmetry breaking are responsible for the emergent AHE. Our work evidences the existence of chiral Dirac-like fermion in CoNb3S6, paving an avenue for exploring new emergent phenomena in quantum materials with unconventional quasiparticle excitations.
notearXiv:2301.12201 [cond-mat]
titleChiral Dirac fermion in a collinear antiferromagnet
journaltitleNat. Mater.
titleWhy some interfaces cannot be sharp
journaltitleNat. Phys.
titleIn situ doping control of the surface of high-temperature superconductors
International Union of Crystallography
abstractThe new computer program SHELXT employs a novel dual-space algorithm to solve the phase problem for single-crystal reflection data expanded to the space group P1. Missing data are taken into account and the resolution extended if necessary. All space groups in the specified Laue group are tested to find which are consistent with the P1 phases. After applying the resulting origin shifts and space-group symmetry, the solutions are subject to further dual-space recycling followed by a peak search and summation of the electron density around each peak. Elements are assigned to give the best fit to the integrated peak densities and if necessary additional elements are considered. An isotropic refinement is followed for non-centrosymmetric space groups by the calculation of a Flack parameter and, if appropriate, inversion of the structure. The structure is assembled to maximize its connectivity and centred optimally in the unit cell. SHELXT has already solved many thousand structures with a high success rate, and is optimized for multiprocessor computers. It is, however, unsuitable for severely disordered and twinned structures because it is based on the assumption that the structure consists of atoms.
journaltitleActa Cryst. A
titleSHELXT – Integrated Space-Group and Crystal-Structure Determination
International Union of Crystallography
abstractThe improvements in the crystal structure refinement program SHELXL have been closely coupled with the development and increasing importance of the CIF (Crystallographic Information Framework) format for validating and archiving crystal structures. An important simplification is that now only one file in CIF format (for convenience, referred to simply as `a CIF') containing embedded reflection data and SHELXL instructions is needed for a complete structure archive; the program SHREDCIF can be used to extract the .hkl and .ins files required for further refinement with SHELXL. Recent developments in SHELXL facilitate refinement against neutron diffraction data, the treatment of H atoms, the determination of absolute structure, the input of partial structure factors and the refinement of twinned and disordered structures. SHELXL is available free to academics for the Windows, Linux and Mac OS X operating systems, and is particularly suitable for multiple-core processors.
journaltitleActa Cryst. C
titleCrystal Structure Refinement with SHELXL
titlePyARPES: An Analysis Framework for Multimodal Angle-Resolved Photoemission Spectroscopies
family=Dal Corso,
abstractQUANTUM ESPRESSO is an integrated suite of computer codes for electronic-structure calculations and materials modeling, based on density-functional theory, plane waves, and pseudopotentials (norm-conserving, ultrasoft, and projector-augmented wave). The acronym ESPRESSO stands for opEn Source Package for Research in Electronic Structure, Simulation, and Optimization. It is freely available to researchers around the world under the terms of the GNU General Public License. QUANTUM ESPRESSO builds upon newly-restructured electronic-structure codes that have been developed and tested by some of the original authors of novel electronic-structure algorithms and applied in the last twenty years by some of the leading materials modeling groups worldwide. Innovation and efficiency are still its main focus, with special attention paid to massively parallel architectures, and a great effort being devoted to user friendliness. QUANTUM ESPRESSO is evolving towards a distribution of independent and interoperable codes in the spirit of an open-source project, where researchers active in the field of electronic-structure calculations are encouraged to participate in the project by contributing their own codes or by implementing their own ideas into existing codes.
journaltitleJ. Phys.: Condens. Matter
titleQUANTUM ESPRESSO: A Modular and Open-Source Software Project for Quantum Simulations of Materials.
American Physical Society
abstractFully nonlocal two-projector norm-conserving pseudopotentials are shown to be compatible with a systematic approach to the optimization of convergence with the size of the plane-wave basis. A reformulation of the optimization is developed, including the ability to apply it to positive-energy atomic scattering states and to enforce greater continuity in the pseudopotential. The generalization of norm conservation to multiple projectors is reviewed and recast for the present purposes. Comparisons among the results of all-electron and one- and two-projector norm-conserving pseudopotential calculations of lattice constants and bulk moduli are made for a group of solids chosen to represent a variety of types of bonding and a sampling of the periodic table.
journaltitlePhys. Rev. B
titleOptimized Norm-Conserving Vanderbilt Pseudopotentials
journaltitleComput. Phys. Commun.
titleThe PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table
journaltitlePhys. Rev. Lett.
titleGeneralized Gradient Approximation Made Simple
journaltitleNano Lett.
titleExcitations of Intercalated Metal Monolayers in Transition Metal Dichalcogenides
family=Dal Corso,
abstractQUANTUM ESPRESSO is an integrated suite of computer codes for electronic-structure calculations and materials modeling, based on density-functional theory, plane waves, and pseudopotentials (norm-conserving, ultrasoft, and projector-augmented wave). The acronym ESPRESSO stands for opEn Source Package for Research in Electronic Structure, Simulation, and Optimization. It is freely available to researchers around the world under the terms of the GNU General Public License. QUANTUM ESPRESSO builds upon newly-restructured electronic-structure codes that have been developed and tested by some of the original authors of novel electronic-structure algorithms and applied in the last twenty years by some of the leading materials modeling groups worldwide. Innovation and efficiency are still its main focus, with special attention paid to massively parallel architectures, and a great effort being devoted to user friendliness. QUANTUM ESPRESSO is evolving towards a distribution of independent and interoperable codes in the spirit of an open-source project, where researchers active in the field of electronic-structure calculations are encouraged to participate in the project by contributing their own codes or by implementing their own ideas into existing codes.
journaltitleJ. Phys.: Condens. Matter
titleQUANTUM ESPRESSO: A Modular and Open-Source Software Project for Quantum Simulations of Materials.
American Physical Society
abstractFully nonlocal two-projector norm-conserving pseudopotentials are shown to be compatible with a systematic approach to the optimization of convergence with the size of the plane-wave basis. A reformulation of the optimization is developed, including the ability to apply it to positive-energy atomic scattering states and to enforce greater continuity in the pseudopotential. The generalization of norm conservation to multiple projectors is reviewed and recast for the present purposes. Comparisons among the results of all-electron and one- and two-projector norm-conserving pseudopotential calculations of lattice constants and bulk moduli are made for a group of solids chosen to represent a variety of types of bonding and a sampling of the periodic table.
journaltitlePhys. Rev. B
titleOptimized Norm-Conserving Vanderbilt Pseudopotentials
journaltitleComput. Phys. Commun.
titleThe PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table
journaltitlePhys. Rev. Lett.
titleGeneralized Gradient Approximation Made Simple
issn1098-0121, 1550-235X
journaltitlePhys. Rev. B
titleMagnetic Phase Transition in Single Crystals of the Chiral Helimagnet Cr_1/3NbS2
liliaxie/Zotero/storage/QP5VGB9F/Ghimire et al. - 2013 - Magnetic phase transition in single crystals of th.pdf
journaltitlePhys. Rev. B
titleOut-of-Plane Spin-Orientation Dependent Magnetotransport Properties in the Anisotropic Helimagnet Cr_1/3NbS2
American Physical Society
journaltitlePhys. Rev. B
titleDisentangling Electronic, Lattice, and Spin Dynamics in the Chiral Helimagnet Cr_1/3NbS2
American Physical Society
journaltitlePhys. Rev. B
titlePersistent Exchange Splitting in the Chiral Helimagnet Cr_1/3NbS2
journaltitlePhys. Rev. B
titleMagnetism of chromia from first-principles calculations
journaltitlePhys. Rev. B
titleEffective Coulomb interaction in transition metals from constrained random-phase approximation
journaltitlePhys. Rev. B
titleCharge density wave phase, Mottness, and ferromagnetism in monolayer $1T$-NbSe2
journaltitlePhys. Rev. Lett.
titlePhonon-Assisted Magnetic Mott-Insulating State in the Charge Density Wave Phase of Single-Layer $1T$-NbSe2
issn0031-9015, 1347-4073
journaltitleJ. Phys. Soc. Jpn.
titleMagnetic Properties of Cr_1/3NbS2
journaltitleJ. Solid State Chem.
titleOn the magnetic behavior of new $2H$-NbS2-type derivatives
issn0935-9648, 1521-4095
journaltitleAdv. Mater.
titleChiral Helimagnetism and One-Dimensional Magnetic Solitons in a Cr-Intercalated Transition Metal Dichalcogenide
issn2199-160X, 2199-160X
journaltitleAdv. Electron. Mater.
titleThe Magneto-Transport Properties of Cr_1/3TaS2 with Chiral Magnetic Solitons
issn0027-8424, 1091-6490
journaltitleProc. Natl. Acad. Sci.
titleTopological Spin/Structure Couplings in Layered Chiral Magnet Cr_1/3TaS2: The Discovery of Spiral Magnetic Superstructure
Supporting Information:
Comparative Electronic Structures of the Chiral Helimagnets and
Lilia S. Xie, Oscar Gonzalez, Kejun Li, Matteo Michiardi, Sergey Gorovikov, Sae Hee Ryu, Shannon S. Fender, Marta Zonno, Na Hyun Jo, Sergey Zhdanovich, Chris Jozwiak, Aaron Bostwick, Samra Husremović, Matthew P. Erodici, Cameron Mollazadeh, Andrea Damascelli, Eli Rotenberg, Yuan Ping, D. Kwabena Bediako
§ SINGLE CRYSTAL X-RAY DIFFRACTION
Crystal structures of and from single crystal X-ray diffraction. Representations are shown for and , respectively, along the $a$ crystallographic axis in (a) and (b), and $c$ crystallographic axis in (c) and (d).
Crystal data and structure refinement for and .
Empirical formula CrNb3S6 CrTa3S6
Formula weight (g/mol) 523.09 787.21
Temperature (K) 293(2) 293(2)
Wavelength (Å) 0.71073 0.71073
Crystal system Hexagonal Hexagonal
Space group $P6_322$ $P6_322$
$a$ (Å) 5.7400(7) 5.7155(5)
$c$ (Å) 12.1082(14) 12.1751(12)
Volume (Å$^{-3}$) 345.49(9) 344.44(7)
$Z$ 2 2
Density (calculated) (g/cm$^3$) 3.402 5.680
Absorption coefficient (mm$^{-1}$) 8.082 50.733
$F(000)$ 486 678
Crystal size (mm$^3$) $0.033 \times 0.017 \times 0.013$ $0.119 \times 0.067 \times 0.025$
$\theta$ ($\degree$) 3.365 to 29.272 4.117 to 29.531
Index ranges $-6 \leq h \leq 7$ $-7 \leq h \leq 7$
$-7 \leq k \leq 6$ $-7 \leq k \leq 7$
$-14 \leq l \leq 16$ $-16 \leq l \leq 16$
Reflections collected 2684 2641
Independent reflections 297 302
Completeness to $\theta_\mathrm{full}$ 1.000 0.994
Absorption correction Semi-empirical from equivalents Semi-empirical from equivalents
Refinement method Full-matrix least-squares on $F^2$ Full-matrix least-squares on $F^2$
Data / restraints / parameters 297 / 0 / 17 302 / 0 / 17
Goodness-of-fit on $F^2$ 1.312 1.156
Final $R$ indices [$I > 2\sigma(I)$] $R_1$ = 0.0331, $wR_2$ = 0.0848 $R_1$ = 0.0381, $wR_2$ = 0.1166
$R$ indices (all data) $R_1$ = 0.0435, $wR_2$ = 0.0880 $R_1$ = 0.0437, $wR_2$ = 0.1213
Largest diff. peak and hole ($e$ Å$^{-3}$) 1.81 and $-0.65$ 4.83 and $-1.89$
Atomic coordinates, Wyckoff positions, and equivalent isotropic displacement parameters for .
Atom Labels $x$ $y$ $z$ Site $U_{\mathrm{iso}}$
Cr01 2/3 1/3 3/4 $2c$ 0.0067(6)
Nb02 0 0 1/2 $2a$ 0.0037(4)
Nb03 1/3 2/3 0.50283(6) $4f$ 0.00236(3)
S04 0.6680(3) 0.6675(3) 0.63086(11) $12i$ 0.0048(4)
Atomic coordinates, Wyckoff positions, and equivalent isotropic displacement parameters for .
Atom Labels $x$ $y$ $z$ Site $U_{\mathrm{iso}}$
Cr01 2/3 1/3 3/4 $2c$ 0.0098(7)
Ta02 0 0 1/2 $2a$ 0.0047(5)
Ta03 1/3 2/3 0.50232(4) $4f$ 0.0048(5)
S04 0.6679(3) 0.6685(3) 0.6304(2) $12i$ 0.0058(7)
§ RAMAN SPECTROSCOPY
Raman spectra of and , with modes associated with the $\sqrt{3} \times \sqrt{3}$ superlattice labeled as “SL” and modes associated with the host lattice materials labeled according to symmetry.fan2021
§ ENERGY DISPERSIVE X-RAY SPECTROSCOPY
Representative dispersive X-ray spectroscopy data for a single crystal of with peaks corresponding to Cr, Nb, and S labeled. The atomic ratio determined by fitting the Cr K$\alpha_1$, Nb L$\alpha_1$, and S K$\alpha_1$ peaks was 1.00:3.00:6.30., corresponding to a formula of Cr_0.33NbS_2.10.
Representative dispersive X-ray spectroscopy data for a single crystal of with peaks corresponding to Cr, Ta, and S labeled. The atomic ratio determined by fitting the Cr K$\alpha_1$, Ta M$\alpha_1$, and S K$\alpha_1$ peaks was 1.00:3.07:5.70, corresponding to a formula of Cr_0.33TaS_1.86.
§ MAGNETOMETRY
Temperature-dependent magnetic susceptibility for , measured with $H \perp c$ = 50 Oe.
Temperature-dependent magnetic susceptibility for , measured with $H \perp c$ = 200 Oe.
§ DFT CALCULATIONS
To understand the band structures obtained from ARPES measurements, first-principles calculations for pristine $2H$-NbS2, $2H$-TaS2, and the Cr-intercalated analogs of these host lattices were performed by using the open source plane-wave code Quantum Espresso (QE).QE The optimized norm-conserving Vanderbilt (ONCV) pseudopotentials from the PseudoDojo project ONCV1,van2018pseudodojo were applied. The kinetic energy cut-off for wavefunctions were set to 86 Ry for all the self-consistent calculations; for these calculations, the experimental lattice constants obtained from X-ray diffraction were used. A $\Gamma$-centered $4\times4\times2$ $\it{k}$-mesh was sampled in the Brillouin zone for both and , and a $8\times8\times2$ $\it{k}$-mesh for both $2H$-NbS2 and $2H$-TaS2. The Perdew–Burke–Ernzerhof (PBE) functional PBE1997 of the spin-polarized generalized gradient approximation (GGA) was used to describe the exchange-correlation of electrons. Previous studies incorporating first-principles calculations at the PBE levelghimire2013,bornstein2015,sirica2021 and at the GGA+$U$ level qin2022 used an on-site Coulomb interaction, $U$, of 4 eV for Cr. In this work, different $U$ parameters for Cr, Nb, and Ta were explored. The results obtained from PBE are shown and compared to the experiments in the main text. Calculations obtained for the spin-polarized, orbital-projected, $k_z$-dependent band structures, and the Fermi surfaces are shown in comparison with the results.
§.§ U Parameter for Band Structure from GGA+U Calculations
Cr, Nb, and Ta are transition metals for which the $U$ parameter can be used to describe the on-site Coulomb interaction between localized $d$ electrons.shi2009magnetism,csacsiouglu2011effective However, whether or not the $U$ parameters for the aforementioned transition metals are important for the band structure calculations of and is not clear. It needs more investigation beyond the previous studies using PBEghimire2013,bornstein2015,sirica2021 and the study using GGA+$U$ with a $U$ value of 4 eV for Cr.qin2022 Here, the on-site Coulomb interaction $U$ parameters for Cr and Nb are tested using the values close to those from Ref. <cit.>. The $U$ parameters are discussed in order to clarify the effect of $U$ parameters on the band structure and the necessity of adopting $U$ parameters.
Looking at the band structures with varying $U$(Cr) in Figure <ref>a, overall the band structures of interest within $-$1 eV to 0 eV do not change much with $U$(Cr). Moreover, the band structures near the Fermi level do not show qualitative changes, except for the minor upshift of the bands at $\Gamma$ with increasing $U$(Cr).
The small change of this part of band structures may be explained by the fact that $d$ orbitals of Cr are minor in the composition as shown in Figure <ref>c of the partial density of states (PDOS), and that the $d$ electrons near the Fermi level are nearly delocalized and thus not affected by the $U$ parameter. Likewise, $U$(Nb) can be found to be not important to the band structures of interest from Figure <ref>b. Thus, the band structures from PBE are shown in the main text and compared with experiments.
(a) and (b) Band structures of with the on-site Coulomb interaction for Cr and Nb represented as $U$(Cr) and $U$(Nb), respectively. (c) density of states of calculated with PBE. (a) $U$(Nb) is set to 0 eV and the effect of $U$(Cr) on the band structure is shown when varied from 0 to 4 eV. (b) $U$(Cr) is set to 0 eV and the effect of $U$(Nb) on the band structure is shown when varied from 0 to 3 eV.
(a) and (b) $k_z$-projected band structures of and , respectively, calculated using the PBE functional.
Magnetic moment of .
Stoichiometry Saturation magnetization ($\mu_\text{B}$) Reference Notes
1/3 2.9 <cit.> $M(H)$ (4.2 K)
1/3 3.89 <cit.> $M(H)$ (2 K)
1/3 3.2 <cit.> $M(H)$ (2 K)
0.33(1) 2.68 This work $M(H)$ (2 K)
1/3 2.66 This work DFT (PBE)
Magnetic moment of .
Stoichiometry Saturation magnetization ($\mu_\text{B}$) Reference Notes
1/3 2.97 <cit.> $M(H)$ (2 K)
1/3 2.73 <cit.> $M(H)$ (2 K)
1/3 2.73 <cit.> $M(H)$ (2 K)
0.33(1) 2.82 This work $M(H)$ (2 K)
1/3 2.71 This work DFT (PBE)
Missing 'biblatex' package
The bibliography requires the 'biblatex' package.
abstractMagnetic skyrmions are nanoscale spin configurations that hold promise as information carriers in ultradense memory and logic devices owing to the extremely low spin-polarized currents needed to move them.
journaltitleNat. Nanotechnol.
titleSkyrmions on the track
abstractRacetrack memory stores digital data in the magnetic domain walls of nanowires. This technology promises to yield information storage devices with high reliability, performance and capacity.
journaltitleNat. Nanotechnol.
titleMemory on the racetrack
issn0031-9015, 1347-4073
journaltitleJ. Phys. Soc. Jpn.
titleSymmetry, Structure, and Dynamics of Monoaxial Chiral Magnets
liliaxie/Zotero/storage/BTC5V7DL/Togawa et al. - 2016 - Symmetry, Structure, and Dynamics of Monoaxial Chi.pdf
journaltitleChem. Rev.
titleMagnetic Skyrmion Materials
abstractThe realization of chiral spin textures, comprising myriad distinct, nanoscale arrangements of spins with topological properties, has established pathways for engineering robust, energy-efficient, and scalable elements for non-volatile nanoelectronics. Particularly, current-induced manipulation of spin textures in nanowire racetracks and tunnel junction based devices are actively investigated for applications in memory, logic, and unconventional computing. In this Article, we paint a background on the progress of spin textures, as well as the relevant state-of-the-art techniques used for their development. In particular, we clarify the competing energy landscape of chiral spin textures─skyrmions and chiral domain walls, to tune their size, density, and zero-field stability. Next, we discuss the spin texture phenomenology and their response to extrinsic factors arising from geometric constraints, interwire interactions, and thermal-electrical effects. Finally, we reveal promising chiral spintronic memory and neuromorphic devices and discuss emerging material and device engineering opportunities.
journaltitleACS Appl. Electron. Mater.
titleChiral Spin Textures for Next-Generation Memory and Unconventional Computing
journaltitlePhys. Rev. B
titleMagnetic soliton confinement and discretization effects arising from macroscopic coherence in a chiral spin soliton lattice
journaltitle2D Mater.
titleExfoliation and van Der Waals Heterostructure Assembly of Intercalated Ferromagnet Cr_1/3TaS2
American Physical Society
journaltitlePhys. Rev. Lett.
titleControlling the Topological Sector of Magnetic Solitons in Exfoliated Cr_1/3NbS2
journaltitleRare Met.
titleMagnetic soliton confinement and discretization effects in Cr_1/3TaS2 nanoflakes
American Physical Society
journaltitlePhys. Rev. B
titleResponse of the chiral soliton lattice to spin-polarized currents
issn1364-2812, 1463-6417
journaltitlePhilos. Mag. B
title3d Transition-Metal Intercalates of the Niobium and Tantalum Dichalcogenides. I. Magnetic Properties
issn1364-2812, 1463-6417
journaltitlePhilos. Mag. B
title3d Transition-Metal Intercalates of the Niobium and Tantalum Dichalcogenides. II. Transport Properties
journaltitleBull. Soc. Chim. Fr.
titleEtude générale de systèmes $M_x$NbS2 (M élément de transition de la première période)
abstractA thermodynamic theory of “weak” ferromagnetism of α-Fe2O3, MnCO3 and CoCO3 is developed on the basis of landau's theory of phase transitions of the second kind. It is shown that the “weak” ferromagnetism is due to the relativistic spin-lattice and the magnetic dipole interactions. A strong dependence of the properties of “weak” ferromagnetics on the magnetic crystalline symmetry is noted and the behaviour of these ferromagnetics in a magnetic field is studied.
journaltitleJ. Phys. Chem. Solids
titleA thermodynamic theory of “weak” ferromagnetism of antiferromagnetics
abstractA theory of anisotropic superexchange interaction is developed by extending the Anderson theory of superexchange to include spin-orbit coupling. The antisymmetric spin coupling suggested by Dzialoshinski from purely symmetry grounds and the symmetric pseudodipolar interaction are derived. Their orders of magnitudes are estimated to be (Δgg) and (Δgg)2 times the isotropic superexchange energy, respectively. Higher order spin couplings are also discussed. As an example of antisymmetric spin coupling the case of CuCl2·2H2O is illustrated. In CuCl2·2H2O, a spin arrangement which is different from one accepted so far is proposed. This antisymmetric interaction is shown to be responsible for weak ferromagnetism in α-Fe2O3, MnCO3, and CrF3. The paramagnetic susceptibility perpendicular to the trigonal axis is expected to increase very sharply near the Néel temperature as the temperature is lowered, as was actually observed in CrF3.
journaltitlePhys. Rev.
titleAnisotropic Superexchange Interaction and Weak Ferromagnetism
issn0031-9015, 1347-4073
journaltitleJ. Phys. Soc. Jpn.
titleMagnetic Properties of Cr_1/3NbS2
issn0031-9007, 1079-7114
journaltitlePhys. Rev. Lett.
titleChiral Magnetic Soliton Lattice on a Chiral Helimagnet
issn1098-0121, 1550-235X
journaltitlePhys. Rev. B
titleMagnetic Phase Transition in Single Crystals of the Chiral Helimagnet Cr_1/3NbS2
liliaxie/Zotero/storage/QP5VGB9F/Ghimire et al. - 2013 - Magnetic phase transition in single crystals of th.pdf
issn0935-9648, 1521-4095
journaltitleAdv. Mater.
titleChiral Helimagnetism and One-Dimensional Magnetic Solitons in a Cr-Intercalated Transition Metal Dichalcogenide
issn2199-160X, 2199-160X
journaltitleAdv. Electron. Mater.
titleThe Magneto-Transport Properties of Cr_1/3TaS2 with Chiral Magnetic Solitons
issn0027-8424, 1091-6490
journaltitleProc. Natl. Acad. Sci.
titleTopological Spin/Structure Couplings in Layered Chiral Magnet Cr_1/3TaS2: The Discovery of Spiral Magnetic Superstructure
journaltitleAPL Mater.
titleAn Emergence of Chiral Helimagnetism or Ferromagnetism Governed by Cr Intercalation in a Dichalcogenide CrNb3S6
abstractThe chiral magnetic soliton, a topological kinklike spin texture, has significant applications in spintronic components. In this work, a crossover of critical behavior is found in Cr1/3TaS2, a chiral magnetic soliton host with the highest TC to date. Angular-dependent magnetization reveals that Cr1/3TaS2 exhibits an easy orientation within the isotropic ab plane, but displays anisotropy with the c axis. By using a modified iterative method, two distinct sets of critical exponents, including β−=0.3190(1) and γ−=1.263(8) for T≤TC, and β+=0.3475(2) and γ+=1.385(5) for T≥TC, are acquired on both sides of the transition. Analysis of the exponents indicates a crossover of the magnetic interaction from a three-dimensional Ising type below TC to a three-dimensional Heisenberg type above TC, implying nontrivial magnetism in this system. Based on universality scaling, a detailed H−T phase diagram around TC is constructed for H⊥c. The crossover of the critical behavior in Cr1/3TaS2 is peculiar to chiral magnetic soliton hosts, suggesting that the three-dimensional magnetic coupling is replaced by a one-dimensional one in the chiral magnetic soliton phase via a phase transition.
journaltitlePhys. Rev. B
titleCrossover of critical behavior and nontrivial magnetism in the chiral soliton lattice host Cr_1/3TaS2
abstractWe report long periodic chiral helimagnetic orderings in ferromagnetic inorganic compounds CrM3S6 (M = Nb and Ta) with a chiral space group of P6322. Magnetization in polycrystalline samples and high resolution powder neutron diffraction were measured. Our powder neutron diffraction measurements in CrM3S6 successfully separated nuclear and magnetic satellite peaks, having the period of hundreds of angstroms along the c— axis. Therefore, we propose that the magnetic ordering in ferromagnetic CrM3S6 is not ferromagnetic, but long periodic chiral helimagnetic ordering.
journaltitleJ. Phys. Conf. Ser.
titleLong Periodic Helimagnetic Ordering in Cr$M$3S6 ($M$ = Nb and Ta)
American Physical Society
journaltitlePhys. Rev. B
titleElectronic Structure of the Chiral Helimagnet and $3d$-Intercalated Transition Metal Dichalcogenide Cr_1/3NbS2
American Physical Society
journaltitlePhys. Rev. B
titleDisentangling Electronic, Lattice, and Spin Dynamics in the Chiral Helimagnet Cr_1/3NbS2
American Physical Society
journaltitlePhys. Rev. B
titlePersistent Exchange Splitting in the Chiral Helimagnet Cr_1/3NbS2
journaltitlePhys. Rev. B
titleEnergy-Gap Driven Low-Temperature Magnetic and Transport Properties in Cr_1/3$M$S2 ($M$ = Nb, Ta)
issn0002-7863, 1520-5126
journaltitleJ. Am. Chem. Soc.
titleStructure and Magnetism of Iron- and Chromium-Intercalated Niobium and Tantalum Disulfides
abstractThe nonrelativistic augmented-plane-wave (APW) method is applied to calculate the electronic band structures of several transition-metal-dichalcogenide (TX2) layer compounds, including materials with the C6(1T−HfS2,1T−TaS2), C27(2H−TaS2,2H−NbSe2), and C7(2H−MoS2) structure types. These calculations involve crystal potentials that are derived from neutral-atom charge densities. The results of these calculations confirm that the group-IVB (1T−HfS2) and group-VIB (2H−MoS2) compounds are semiconductors; the calculated indirect band gaps of 2.7 and 1.2 eV are in reasonable agreement with the observed values of 2.0 and 1.4 eV, respectively. Metallic behavior is predicted for the intermediate group-VB compounds 1T−TaS2, 2H−TaS2, and 2H−NbSe2. A novel feature of the metal d bands in the 2H−TX2 compounds is the occurence of a 1-eV hybridization gap within the dz2 and dxy, dx2−y2 manifolds. This splits off a pair of hybridized d bands which are half-filled in 2H−TaS2 and 2H−NbSe2 and completely filled in 2H−MoS2. As a result of this hybridization gap, the valence or conduction bandwidths in each of these 2H−TX2 compounds are reduced to about 1 eV.
journaltitlePhys. Rev. B
titleBand Structures of Transition-Metal-Dichalcogenide Layer Compounds
journaltitleInorg. Chem.
shorttitleUtility of semilocalized bonding schemes in extended systems
titleUtility of semilocalized bonding schemes in extended systems: three-center metal-metal bonding in molybdenum sulfide (MoS2), niobium tantalum sulfide bronze (H_$x$(Nb,Ta)S2), and zirconium sulfide (ZrS)
ZE884CV/ic00010a019.html:text/html;Full Text PDF:/Users/liliaxie/Zotero/storage/RARPPQ32/Yee and Hughbanks - 1991 - Utility of semilocalized bonding schemes in extend.pdf:application/pdf
journaltitleJ.. Am. Chem. Soc.
titleAnalogies between the concepts of molecular chemistry and solid-state physics concerning structural instabilities. Electronic origin of the structural modulations in layered transition metal dichalcogenides
nalogies between the concepts of molecular chemis.pdf:application/pdf
AIP Publishing LLC
journaltitleAppl. Phys. Lett.
titleSpin structure of the anisotropic helimagnet Cr_1/3NbS2 in a magnetic field
abstractThe topologically protected, chiral soliton lattice is a unique state of matter offering intriguing functionality, and it may serve as a robust platform for storing and transporting information in future spintronic devices. While the monoaxial chiral magnet Cr1∕3NbS2 is known to host this exotic state in an applied magnetic field, its detailed microscopic origin has remained a matter of debate. Here, we work towards addressing this open question by measuring the spin wave spectrum of Cr1∕3NbS2 over the entire Brillouin zone with inelastic neutron scattering. The well-defined spin wave modes allow us to determine the values of several microscopic interactions for this system. The experimental data are well-explained by a Heisenberg Hamiltonian with exchange constants up to the third nearest neighbor and an easy plane magnetocrystalline anisotropy term. Our work shows that both the second and third nearest neighbor exchange interactions contribute to the formation of the helimagnetic and chiral soliton lattice states in this robust three-dimensional magnet.
journaltitleAppl. Phys. Lett.
titleExtended exchange interactions stabilize long-period magnetic structures in Cr_1/3NbS2
:/Users/liliaxie/Zotero/storage/QBJM8LEA/Aczel et al. - 2018 - Extended exchange interactions stabilize long-peri.pdf:application/pdf;Snapshot:/Users/liliaxie/Zotero/storage/NP9QSWDF/Extended-exchange-interactions-stabilize-long.html:text/html
journaltitlePhys. Rev. B
titleTricritical point and phase diagram based on critical scaling in the monoaxial chiral helimagnet Cr_1/3NbS2
journaltitleNano Lett.
titleExcitations of Intercalated Metal Monolayers in Transition Metal Dichalcogenides
journaltitlePhys. Rev. B
titleFermiology and Electron-Phonon Coupling in the $2H$ and $3R$ Polytypes of NbS2
family=van Wezel,
American Physical Society
journaltitlePhys. Rev. B
titleOrbital Selectivity Causing Anisotropy and Particle-Hole Asymmetry in the Charge Density Wave Gap of $2H$-TaS2
abstractWe study the mechanism of the exceptionally large anomalous Hall effect (AHE) in the noncentrosymmetric antiferromagnet CoNb3S6 by angle-resolved photoemission spectroscopy (ARPES) and magnetotransport measurements. From ARPES measurements of CoNb3S6 and its family compounds (FeNb3S6 and NiNb3S6), we find a band dispersion unique to the Co intercalation existing near the Fermi level. We further demonstrate that a slight deficiency of sulfur in CoNb3S6 eliminates the ferromagnetism and the AHE simultaneously while hardly changing the band structure, indicating that the weak ferromagnetism is responsible for the emergence of the large AHE. Based on our results, we propose Weyl points near the Fermi level to cause the large AHE.
journaltitlePhys. Rev. B
titleLarge anomalous Hall effect induced by weak ferromagnetism in the noncentrosymmetric antiferromagnet CoNb3S6
American Physical Society
journaltitlePhys. Rev. B
titleVisualizing the Out-of-Plane Electronic Dispersions in an Intercalated Transition Metal Dichalcogenide
journaltitlePhys. Rev. B
titleRole of Intercalated Cobalt in the Electronic Structure of Co_1/3NbS2
journaltitleNat. Mater.
titleGiant valley-Zeeman coupling in the surface layer of an intercalated transition metal dichalcogenide
journaltitleRev. Mod. Phys.
titleAngle-Resolved Photoemission Studies of the Cuprate Superconductors
journaltitleInorg. Chem.
titleStoichiometry, structure, and physical properties of niobium disulfide
journaltitleActa Cryst. C
titleStructure of $2H$-TaS2
issn0031-9007, 1079-7114
journaltitlePhys. Rev. Lett.
titleRKKY Ferromagnetism with Ising-Like Spin States in Intercalated Fe_1/4TaS2
liliaxie/Zotero/storage/XBKIDHGE/Ko et al. - 2011 - RKKY Ferromagnetism with Ising-Like Spin States in.pdf
abstractDzyaloshinskii–Moriya interaction (DMI) is vital to form various chiral spin textures, novel behaviors of magnons and permits their potential applications in energy-efficient spintronic devices. Here, we realize a sizable bulk DMI in a transition metal dichalcogenide (TMD) 2H-TaS2 by intercalating Fe atoms, which form the chiral supercells with broken spatial inversion symmetry and also act as the source of magnetic orderings. Using a newly developed protonic gate technology, gate-controlled protons intercalation could further change the carrier density and intensely tune DMI via the Ruderman–Kittel–Kasuya–Yosida mechanism. The resultant giant topological Hall resistivity $${\rho }_{{xy}}ˆ{T}$$of $$1.41{\mathrm{\mu}} \Omega \cdot {{\mathrm{cm}}}$$at $${V}_{g}=-5.2{\mathrm{V}}$$(about $$424 \%$$larger than the zero-bias value) is larger than most known chiral magnets. Theoretical analysis indicates that such a large topological Hall effect originates from the two-dimensional Bloch-type chiral spin textures stabilized by DMI, while the large anomalous Hall effect comes from the gapped Dirac nodal lines by spin–orbit interaction. Dual-intercalation in 2H-TaS2 provides a model system to reveal the nature of DMI in the large family of TMDs and a promising way of gate tuning of DMI, which further enables an electrical control of the chiral spin textures and related electromagnetic phenomena.
journaltitleNat. Commun.
titleTailoring Dzyaloshinskii–Moriya interaction in a transition metal dichalcogenide by dual-intercalation
American Physical Society
journaltitlePhys. Rev. X
titleHighly Tunable Magnetic Phases in Transition-Metal Dichalcogenide Fe_$1/3+\delta$NbS2
journaltitlePhys. Rev. B
titleStructural Disorder versus Chiral Magnetism in Cr_1/3NbS2
abstractTransition metal-intercalated transition metal dichalcogenides (TMDs) are promising platforms for next-generation spintronic devices based on their wide range of electronic and magnetic phases, which can be tuned by varying the host lattice or the identity of the intercalant, along with its stoichiometry and spatial order. Some of these compounds host a chiral magnetic phase in which the helical winding of magnetic moments propagates along a high-symmetry crystalline axis. Previous studies have demonstrated that variation in intercalant concentrations can have a dramatic impact on the formation of chiral domains and ensemble magnetic properties. However, a systematic and comprehensive study of how atomic-scale order and disorder impacts collective magnetic behavior are so far lacking. Here, we leverage a combination of imaging modes in the (scanning) transmission electron microscope (S/TEM) to directly probe (dis)order across multiple length scales and show how subtle changes in the atomic lattice can be leveraged to tune the mesoscale spin textures and bulk magnetic response, with direct implications for the fundamental understanding and technological implementation of such compounds.
notearXiv:2305.06656 [cond-mat]
titleConsequences and control of multi-scale (dis)order in chiral magnetic textures
journaltitlePhys. Rev. Mater.
titleFe and Cr co-intercalation in $2H$-NbS2 single crystals for realization of perpendicular magnetic anisotropy and large anomalous Hall effect
issn1530-6984, 1530-6992
journaltitleNano Lett.
titleTunable Single-Atomic Charges on a Cleaved Intercalated Transition Metal Dichalcogenide
abstractThe Dirac equation combines the two cornerstones of modern physics-quantum mechanics and relativity. There are several manifestations of the Dirac equation in condensed matter systems, such as the quasiparticle dispersion in graphene, topological insulators, Dirac semimetals (DSMs), Weyl semimetals, and d-wave high-temperature superconductors. In a DSM, the massless Dirac fermion has zero chirality, leading to surface states connected adiabatically to a topologically trivial surface state as well as vanishing anomalous Hall effect (AHE). Recently, it is predicted that in the nonrelativistic limit of certain antiferromagnets, there exists a type of chiral 'Dirac-like' fermion, whose dispersion manifests four-fold degenerate crossing points formed by doubly degenerate linear bands, with topologically protected Fermi arcs. Such unconventional chiral fermion, protected by a hidden SU(2) symmetry in the hierarchy of an enhanced crystallographic group, namely spin space group, is not experimentally verified yet. Here, by combining neutron diffraction, angle-resolved photoemission spectroscopy and first-principles calculations, we reveal the existence of the Fermi-arc surface states induced by chiral Dirac-like fermions in collinear antiferromagnet CoNb3S6, which caught great interest due to its surprisingly large AHE. Our transport measurements and theoretical calculations provide a scenario that large Berry curvature embedded in the chiral fermions and weak symmetry breaking are responsible for the emergent AHE. Our work evidences the existence of chiral Dirac-like fermion in CoNb3S6, paving an avenue for exploring new emergent phenomena in quantum materials with unconventional quasiparticle excitations.
notearXiv:2301.12201 [cond-mat]
titleChiral Dirac fermion in a collinear antiferromagnet
journaltitleNat. Mater.
titleWhy some interfaces cannot be sharp
journaltitleNat. Phys.
titleIn situ doping control of the surface of high-temperature superconductors
International Union of Crystallography
abstractThe new computer program SHELXT employs a novel dual-space algorithm to solve the phase problem for single-crystal reflection data expanded to the space group P1. Missing data are taken into account and the resolution extended if necessary. All space groups in the specified Laue group are tested to find which are consistent with the P1 phases. After applying the resulting origin shifts and space-group symmetry, the solutions are subject to further dual-space recycling followed by a peak search and summation of the electron density around each peak. Elements are assigned to give the best fit to the integrated peak densities and if necessary additional elements are considered. An isotropic refinement is followed for non-centrosymmetric space groups by the calculation of a Flack parameter and, if appropriate, inversion of the structure. The structure is assembled to maximize its connectivity and centred optimally in the unit cell. SHELXT has already solved many thousand structures with a high success rate, and is optimized for multiprocessor computers. It is, however, unsuitable for severely disordered and twinned structures because it is based on the assumption that the structure consists of atoms.
journaltitleActa Cryst. A
titleSHELXT – Integrated Space-Group and Crystal-Structure Determination
International Union of Crystallography
abstractThe improvements in the crystal structure refinement program SHELXL have been closely coupled with the development and increasing importance of the CIF (Crystallographic Information Framework) format for validating and archiving crystal structures. An important simplification is that now only one file in CIF format (for convenience, referred to simply as `a CIF') containing embedded reflection data and SHELXL instructions is needed for a complete structure archive; the program SHREDCIF can be used to extract the .hkl and .ins files required for further refinement with SHELXL. Recent developments in SHELXL facilitate refinement against neutron diffraction data, the treatment of H atoms, the determination of absolute structure, the input of partial structure factors and the refinement of twinned and disordered structures. SHELXL is available free to academics for the Windows, Linux and Mac OS X operating systems, and is particularly suitable for multiple-core processors.
journaltitleActa Cryst. C
titleCrystal Structure Refinement with SHELXL
titlePyARPES: An Analysis Framework for Multimodal Angle-Resolved Photoemission Spectroscopies
family=Dal Corso,
abstractQUANTUM ESPRESSO is an integrated suite of computer codes for electronic-structure calculations and materials modeling, based on density-functional theory, plane waves, and pseudopotentials (norm-conserving, ultrasoft, and projector-augmented wave). The acronym ESPRESSO stands for opEn Source Package for Research in Electronic Structure, Simulation, and Optimization. It is freely available to researchers around the world under the terms of the GNU General Public License. QUANTUM ESPRESSO builds upon newly-restructured electronic-structure codes that have been developed and tested by some of the original authors of novel electronic-structure algorithms and applied in the last twenty years by some of the leading materials modeling groups worldwide. Innovation and efficiency are still its main focus, with special attention paid to massively parallel architectures, and a great effort being devoted to user friendliness. QUANTUM ESPRESSO is evolving towards a distribution of independent and interoperable codes in the spirit of an open-source project, where researchers active in the field of electronic-structure calculations are encouraged to participate in the project by contributing their own codes or by implementing their own ideas into existing codes.
journaltitleJ. Phys.: Condens. Matter
titleQUANTUM ESPRESSO: A Modular and Open-Source Software Project for Quantum Simulations of Materials.
American Physical Society
abstractFully nonlocal two-projector norm-conserving pseudopotentials are shown to be compatible with a systematic approach to the optimization of convergence with the size of the plane-wave basis. A reformulation of the optimization is developed, including the ability to apply it to positive-energy atomic scattering states and to enforce greater continuity in the pseudopotential. The generalization of norm conservation to multiple projectors is reviewed and recast for the present purposes. Comparisons among the results of all-electron and one- and two-projector norm-conserving pseudopotential calculations of lattice constants and bulk moduli are made for a group of solids chosen to represent a variety of types of bonding and a sampling of the periodic table.
journaltitlePhys. Rev. B
titleOptimized Norm-Conserving Vanderbilt Pseudopotentials
journaltitleComput. Phys. Commun.
titleThe PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table
journaltitlePhys. Rev. Lett.
titleGeneralized Gradient Approximation Made Simple
journaltitleNano Lett.
titleExcitations of Intercalated Metal Monolayers in Transition Metal Dichalcogenides
family=Dal Corso,
abstractQUANTUM ESPRESSO is an integrated suite of computer codes for electronic-structure calculations and materials modeling, based on density-functional theory, plane waves, and pseudopotentials (norm-conserving, ultrasoft, and projector-augmented wave). The acronym ESPRESSO stands for opEn Source Package for Research in Electronic Structure, Simulation, and Optimization. It is freely available to researchers around the world under the terms of the GNU General Public License. QUANTUM ESPRESSO builds upon newly-restructured electronic-structure codes that have been developed and tested by some of the original authors of novel electronic-structure algorithms and applied in the last twenty years by some of the leading materials modeling groups worldwide. Innovation and efficiency are still its main focus, with special attention paid to massively parallel architectures, and a great effort being devoted to user friendliness. QUANTUM ESPRESSO is evolving towards a distribution of independent and interoperable codes in the spirit of an open-source project, where researchers active in the field of electronic-structure calculations are encouraged to participate in the project by contributing their own codes or by implementing their own ideas into existing codes.
journaltitleJ. Phys.: Condens. Matter
titleQUANTUM ESPRESSO: A Modular and Open-Source Software Project for Quantum Simulations of Materials.
American Physical Society
abstractFully nonlocal two-projector norm-conserving pseudopotentials are shown to be compatible with a systematic approach to the optimization of convergence with the size of the plane-wave basis. A reformulation of the optimization is developed, including the ability to apply it to positive-energy atomic scattering states and to enforce greater continuity in the pseudopotential. The generalization of norm conservation to multiple projectors is reviewed and recast for the present purposes. Comparisons among the results of all-electron and one- and two-projector norm-conserving pseudopotential calculations of lattice constants and bulk moduli are made for a group of solids chosen to represent a variety of types of bonding and a sampling of the periodic table.
journaltitlePhys. Rev. B
titleOptimized Norm-Conserving Vanderbilt Pseudopotentials
journaltitleComput. Phys. Commun.
titleThe PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table
journaltitlePhys. Rev. Lett.
titleGeneralized Gradient Approximation Made Simple
issn1098-0121, 1550-235X
journaltitlePhys. Rev. B
titleMagnetic Phase Transition in Single Crystals of the Chiral Helimagnet Cr_1/3NbS2
liliaxie/Zotero/storage/QP5VGB9F/Ghimire et al. - 2013 - Magnetic phase transition in single crystals of th.pdf
journaltitlePhys. Rev. B
titleOut-of-Plane Spin-Orientation Dependent Magnetotransport Properties in the Anisotropic Helimagnet Cr_1/3NbS2
American Physical Society
journaltitlePhys. Rev. B
titleDisentangling Electronic, Lattice, and Spin Dynamics in the Chiral Helimagnet Cr_1/3NbS2
American Physical Society
journaltitlePhys. Rev. B
titlePersistent Exchange Splitting in the Chiral Helimagnet Cr_1/3NbS2
journaltitlePhys. Rev. B
titleMagnetism of chromia from first-principles calculations
journaltitlePhys. Rev. B
titleEffective Coulomb interaction in transition metals from constrained random-phase approximation
journaltitlePhys. Rev. B
titleCharge density wave phase, Mottness, and ferromagnetism in monolayer $1T$-NbSe2
journaltitlePhys. Rev. Lett.
titlePhonon-Assisted Magnetic Mott-Insulating State in the Charge Density Wave Phase of Single-Layer $1T$-NbSe2
issn0031-9015, 1347-4073
journaltitleJ. Phys. Soc. Jpn.
titleMagnetic Properties of Cr_1/3NbS2
journaltitleJ. Solid State Chem.
titleOn the magnetic behavior of new $2H$-NbS2-type derivatives
issn0935-9648, 1521-4095
journaltitleAdv. Mater.
titleChiral Helimagnetism and One-Dimensional Magnetic Solitons in a Cr-Intercalated Transition Metal Dichalcogenide
issn2199-160X, 2199-160X
journaltitleAdv. Electron. Mater.
titleThe Magneto-Transport Properties of Cr_1/3TaS2 with Chiral Magnetic Solitons
issn0027-8424, 1091-6490
journaltitleProc. Natl. Acad. Sci.
titleTopological Spin/Structure Couplings in Layered Chiral Magnet Cr_1/3TaS2: The Discovery of Spiral Magnetic Superstructure
|
201071-82Nancy, France 71
László Babai
Anandam Banerjee
Raghav Kulkarni
Vipul Naik
# Evasiveness and the Distribution of Prime Numbers
L. Babai University of Chicago, Chicago, IL, USA. , A. Banerjee
Northeastern University, Boston, MA, USA. , R. Kulkarni and V. Naik
###### Abstract.
A Boolean function on $N$ variables is called _evasive_ if its decision-tree
complexity is $N$. A sequence $B_{n}$ of Boolean functions is _eventually
evasive_ if $B_{n}$ is evasive for all sufficiently large $n$.
We confirm the eventual evasiveness of several classes of monotone graph
properties under widely accepted number theoretic hypotheses. In particular we
show that Chowla’s conjecture on Dirichlet primes implies that (a) for any
graph $H$, “forbidden subgraph $H$” is eventually evasive and (b) all
nontrivial monotone properties of graphs with $\leq n^{3/2-\epsilon}$ edges
are eventually evasive. ($n$ is the number of vertices.)
While Chowla’s conjecture is not known to follow from the Extended Riemann
Hypothesis (ERH, the Riemann Hypothesis for Dirichlet’s $L$ functions), we
show (b) with the bound $O(n^{5/4-\epsilon})$ under ERH.
We also prove unconditional results: (a′) for any graph $H$, the query
complexity of “forbidden subgraph $H$” is $\binom{n}{2}-O(1)$; (b′) for some
constant $c>0$, all nontrivial monotone properties of graphs with $\leq cn\log
n+O(1)$ edges are eventually evasive.
Even these weaker, unconditional results rely on deep results from number
theory such as Vinogradov’s theorem on the Goldbach conjecture.
Our technical contribution consists in connecting the topological framework of
Kahn, Saks, and Sturtevant (1984), as further developed by Chakrabarti, Khot,
and Shi (2002), with a deeper analysis of the orbital structure of permutation
groups and their connection to the distribution of prime numbers. Our
unconditional results include stronger versions and generalizations of some
result of Chakrabarti et al.
###### Key words and phrases:
Decision tree complexity, evasiveness, graph property, group action, Dirichlet
primes, Extended Riemann Hypothesis
###### 1991 Mathematics Subject Classification:
F.2.2, F.1.1, F.1.3
22footnotetext: Partially supported by NSF Grant CCF-0830370.
## 1\. Introduction
### 1.1. The framework
A graph property $P_{n}$ of $n$-vertex graphs is a collection of graphs on the
vertex set $[n]=\\{1,\dots,n\\}$ that is invariant under relabeling of the
vertices. A property $P_{n}$ is called monotone (decreasing) if it is
preserved under the deletion of edges. The trivial graph properties are the
empty set and the set of all graphs. A class of examples are the forbidden
subgraph properties: for a fixed graph $H$, let $Q_{n}^{H}$ denote the class
of $n$-vertex graphs that do not contain a (not necessarily induced) subgraph
isomorphic to $H$.
We view a set of labeled graphs on $n$ vertices as a Boolean function on the
$N=\binom{n}{2}$ variables describing adjacency. A Boolean function on $N$
variables is evasive if its deterministic query (decision-tree) complexity is
$N$.
The long-standing Aanderaa-Rosenberg-Karp conjecture asserts that every
nontrivial monotone graph property is evasive. The problem remains open even
for important special classes of monotone properties, such as the forbidden
subgraph properties.
### 1.2. History
In this note, $n$ always denotes the number of vertices of the graphs under
consideration.
Aanderaa and Rosenberg (1973) [17] conjectured a lower bound of
$\Omega(n^{2})$ on the query complexity of monotone graph properties. Rivest
and Vuillemin (1976) [19] verified this conjecture, proving an $n^{2}/16$
lower bound. Kleitman and Kwiatkowski (1980) [10] improved this to $n^{2}/9.$
Karp conjectured that nontrivial monotone graph properties were in fact
evasive. We refer to this statement as the Aanderaa-Rosenberg-Karp (ARK)
conjecture.
In their seminal paper, Kahn, Saks, and Sturtevant [11] observe that non-
evasiveness of monotone Boolean functions has strong topological consequences
(contracibility of the associated simplicial complex). They then use results
of R. Oliver about fixed points of group actions on such complexes to verify
the ARK conjecture when $n$ is a prime-power. As a by-product, they improve
the lower bound for general $n$ to $n^{2}/4.$
Since then, the topological approach of [11] has been influential in solving
various interesting special cases of the ARK conjecture. Yao (1988) [25]
proves that non-trivial monotone properties of bipartite graphs with a given
partition $(U,V)$ are evasive (require $|U||V|$ queries). Triesch (1996) [22]
shows (in the original model) that any monotone property of bipartite graphs
(all the graphs satisfying the property are bipartite) is evasive.
Chakrabarti, Khot, and Shi (2002) [3] introduce important new techniques which
we use; we improve over several of their results (see Section 1.4).
### 1.3. Prime numbers in arithmetic progressions
Dirichlet’s Theorem (1837) (cf. [5]) asserts that if $\gcd(a,m)=1$ then there
exist infinitely many primes $p\equiv a\pmod{m}$. Let $p(m,a)$ denote the
smallest such prime $p$. Let $p(m)=\max\\{p(m,a)\mid\gcd(a,m)=1\\}$. Linnik’s
celebrated theorem (1947) asserts that $p(m)=O(m^{L})$ for some absolute
constant $L$ (cf. [16, Chap. V.]). Heath-Brown [9] shows that $L\leq 5.5$.
Chowla [4] observes that under the Extended Riemann Hypothesis (ERH) we have
$L\leq 2+\epsilon$ for all $\epsilon>0$ and conjectures that $L\leq
1+\epsilon$ suffices:
###### Conjecture 1.1 (S. Chowla [4]).
For every $\epsilon>0$ and every $m$ we have $p(m)=O(m^{1+\epsilon})$.
This conjecture is widely believed; in fact, number theorists suggest as
plausible the stronger form $p(m)=O(m(\log m)^{2})$ [8]. Turán [23] proves the
tantalizing result that for almost all $a$ we have $p(m,a)=O(m\log m)$ .
Let us call a prime $p$ an $\epsilon$-near Fermat prime if there exists an
$s\geq 0$ such that $2^{s}\mid p-1$ and $\frac{p-1}{2^{s}}\leq p^{\epsilon}$.
We need the following weak form of Chowla’s conjecture:
###### Conjecture 1.2 (Weak Chowla Conjecture).
For every $\epsilon>0$ there exist infinitely many $\epsilon$-near Fermat
primes.
In other words, the weak conjecture says that for every $\epsilon$, for
infinitely many values of $s$ we have $p(2^{s},1)<(2^{s})^{1+\epsilon}$.
### 1.4. Main results
For a graph property $P$ we use $P_{n}$ to denote the set of graphs on vertex
set $[n]$ with property $P$. We say that $P$ is eventually evasive if $P_{n}$
is evasive for all sufficiently large $n$.
Our first set of results states that the “forbidden subgraph” property is
“almost evasive” under three different interpretations of this phrase.
###### Theorem 1.3 (Forbidden subgraphs).
For all graphs $H$, the forbidden subgraph property $Q_{n}^{H}$ (a) is
eventually evasive, assuming the Weak Chowla Conjecture; (b) is evasive for
almost all $n$ (unconditionally); and (c) has query complexity
$\binom{n}{2}-O(1)$ for all $n$ (unconditionally).
Part (b) says the asymptotic density of values of $n$ for which the problem is
not evasive is zero. Part (c) improves the bound $\binom{n}{2}-O(n)$ given in
[3]. Parts (a) and (c) will be proved in Section 3. We defer the proof of part
(b) to the journal version.
The term “monotone property of graphs with $\leq m$ edges” describes a
monotone property that fails for all graphs with more than $m$ edges.
###### Theorem 1.4 (Sparse graphs).
All nontrivial monotone properties of graphs with at most $f(n)$ edges are
eventually evasive, where (a) under Chowla’s Conjecture,
$f(n)=n^{3/2-\epsilon}$ for any $\epsilon>0$; (b) under ERH,
$f(n)=n^{5/4-\epsilon}$; and (c) unconditionally, $f(n)=cn\log n$ for some
constant $c>0$. (d) Unconditionally, all nontrivial monotone properties of
graphs with no cycle of length greater than $(n/4)(1-\epsilon)$ are eventually
evasive (for all $\epsilon>0$).
Part (c) of Theorem 1.4 will be proved in Section 4. Parts (a) and (b) follow
in Section 5. The proof of part (d) follows along the lines of part (c); we
defer the details to the journal version of this paper.
We note that the proofs of the unconditional results (c) and (d) in Theorem
1.4 rely on Haselgrove’s version [7] of Vinogradov’s Theorem on Goldbach’s
Conjecture (cf. Sec. 4.2).
Recall that a _topological subgraph_ of a graph $G$ is obtained by taking a
subgraph and replacing any induced path $u-\dots-v$ in the subgraph by an edge
$\\{u,v\\}$ (repeatedly) and deleting parallel edges. A minor of a graph is
obtained by taking a subgraph and contracting edges (repeatedly). If a class
of graphs is closed under taking minors then it is also closed under taking
topological subgraphs but not conversely; for instance, graphs with maximum
degree $\leq 3$ are closed under taking toopological subgraphs but every graph
is a minor of a regular graph of degree 3.
###### Corollary 1.5 (Excluded topological subgraphs).
Let $P$ be a nontrivial class of graphs closed under taking topological
subgraphs. Then $P$ is eventually evasive.
This unconditional result extends one of the results of Chakrabarti et al.
[3], namely, that nontrival classes of graphs closed under taking minors is
eventually evasive.
Corollary 1.5 follows from part (c) of Theorem 1.4 in the light of Mader’s
Theorem which states that if the average degree of a graph $G$ is greater than
$2^{\binom{k+1}{2}}$ then it contains a topological $K_{k}$ [13, 14].
Theorem 1.4 suggests a new stratification of the ARK Conjecture. For a
monotone (decreasing) graph property $P_{n}$, let
$\dim(P_{n}):=\max\\{|E(G)|-1\ |\ G\in P_{n}\\}.$
We can now restate the ARK Conjecture:
###### Conjecture 1.6.
If $P_{n}$ is a non-evasive, non-empty, monotone decreasing graph property
then $\dim(P_{n})=\binom{n}{2}-1.$
## 2\. Preliminaries
### 2.1. Group action
For the basics of group theory we refer to [18]. All groups in this paper are
finite. For groups $\Gamma_{1},\Gamma_{2}$ we use $\Gamma_{1}\leq\Gamma_{2}$
to denote that $\Gamma_{1}$ is a subgroup; and $\Gamma_{1}\lhd\Gamma_{2}$ to
denote that $\Gamma_{1}$ is a (not necessarily proper) normal subgroup. We say
that $\Gamma$ is a $p$-group if $|\Gamma|$ is a power of the prime $p$.
For a set $\Omega$ called the “permutation domain,” let
$\operatorname{Sym}(\Omega)$ denote the symmetric group on $\Omega$,
consisting of the $|\Omega|!$ permutations of $\Omega$. For
$\Omega=[n]=\\{1,\dots,n\\}$, we set $\Sigma_{n}=\operatorname{Sym}([n])$. For
a group $\Gamma$, a homomorphism
$\varphi\,:\,\Gamma\to\operatorname{Sym}(\Omega)$ is called a $\Gamma$-action
on $\Omega$. The action is faithful if $\ker(\varphi)=\\{1\\}$. For
$x\in\Omega$ and $\gamma\in\Gamma$ we denote by $x^{\gamma}$ the image of $x$
under $\varphi(\gamma)$. For $x\in\Omega$ we write
$x^{\Gamma}=\\{x^{\gamma}\,:\,\gamma\in\Gamma\\}$ and call it the orbit of $x$
under the $\Gamma$-action. The orbits partition $\Omega$.
Let $\binom{\Omega}{t}$ denote the set of $t$-subsets of $\Omega$. There is a
natural induced action
$\operatorname{Sym}(\Omega)\to\operatorname{Sym}(\binom{\Omega}{t})$ which
also defines a natural $\Gamma$-action on $\binom{\Omega}{t}$. We denote this
action by $\Gamma^{(t)}$. Similarly, there is a natural induced
$\Gamma$-action on $\Omega\times\Omega$. The orbits of this action are called
the orbitals of $\Gamma$. We shall need the undirected version of this
concept; we shall call the orbits of the $\Gamma$-action on
$\binom{\Omega}{2}$ the u-orbitals (undirected orbitals) of the
$\Gamma$-action.
By an action of the group $\Gamma$ on a structure $\mathfrak{X}$ such as a
group or a graph or a simplicial complex we mean a homomorphism
$\Gamma\to\operatorname{Aut}({\mathfrak{X}})$ where
$\operatorname{Aut}({\mathfrak{X}})$ denotes the automorphism group of
$\mathfrak{X}$.
Let $\Gamma$ and $\Delta$ be groups and let
$\psi\,:\,\Delta\to\operatorname{Aut}(\Gamma)$ be a $\Delta$-action on
$\Gamma$. These data uniquely define a group $\Theta=\Gamma\rtimes\Delta$, the
_semidirect product_ of $\Gamma$ and $\Delta$ with respect to $\psi$. This
group has order $|\Theta|=|\Gamma||\Delta|$ and has the following properites:
$\Theta$ has two subgroups $\Gamma^{*}\cong\Gamma$ and $\Delta^{*}\cong\Delta$
such that $\Gamma^{*}\lhd\Theta$; $\Gamma^{*}\cap\Delta^{*}=\\{1\\}$; and
$\Theta=\Gamma^{*}\Delta^{*}=\\{\gamma\delta\mid\gamma\in\Gamma^{*},\delta\in\Delta^{*}\\}$.
Moreover, identifying $\Gamma$ with $\Gamma^{*}$ and $\Delta$ with
$\Delta^{*}$, for all $\gamma\in\Gamma$ and $\delta\in\Delta$ we have
$\gamma^{\psi(\delta)}=\delta^{-1}\gamma\delta$.
$\Theta$ can be defined as the set $\Delta\times\Gamma$ under the group
operation
$(\delta_{1},\gamma_{1})(\delta_{2},\gamma_{2})=(\delta_{1}\delta_{2},\gamma_{1}^{\psi(\delta_{2})}\gamma_{2})\quad\quad(\delta_{i}\in\Delta,\gamma_{i}\in\Gamma).$
For more on semidirect products, which we use extensively, see [18, Chap. 7].
The group $\operatorname{AGL}(1,q)$ of affine transformations $x\mapsto ax+b$
of $\mathbb{F}_{q}$ ($a\in\mathbb{F}_{q}^{\times}$, $b\in\mathbb{F}_{q}$) acts
on $\mathbb{F}_{q}$. For each $d\mid q-1$, $\operatorname{AGL}(1,q)$ has a
unique subgroup of order $qd$; we call this subgroup $\Gamma(q,d)$. We note
that $\mathbb{F}_{q}^{+}\lhd\Gamma(q,d)$ and $\Gamma(q,d)/\mathbb{F}_{q}^{+}$
is cyclic of order $d$ and is isomorphic to a subgroup $\Delta$ of
$\operatorname{AGL}(1,q)$; $\Gamma(q,d)$ can be described as a semidirect
product $(\mathbb{F}_{q}^{+})\rtimes\Delta$.
### 2.2. Simplicial complexes and monotone graph properties
An abstract simplicial complex ${\mathcal{K}}$ on the set $\Omega$ is a subset
of the power-set of $\Omega$, closed under subsets: if $B\subset
A\in{{\mathcal{K}}}$ then $B\in{{\mathcal{K}}}$. The elements of
${\mathcal{K}}$ are called its faces. The dimension of $A\in{{\mathcal{K}}}$
is $\dim(A)=|A|-1$; the dimension of ${\mathcal{K}}$ is
$\dim({{\mathcal{K}}})=\max\\{\dim(A)\mid A\in{{\mathcal{K}}}\\}$. The Euler
characteristic of ${{\mathcal{K}}}$ is defined as
$\chi({{\mathcal{K}}}):=\sum_{A\in{{\mathcal{K}}},A\neq\emptyset}{(-1)^{\dim(A)}}.$
Let $[n]:=\\{1,2,\ldots,n\\}$ and $\Omega=\binom{[n]}{2}$. Let $P_{n}$ be a
subset of the power-set of $\Omega$, i. e., a set of graphs on the vertex set
$[n]$. We call $P_{n}$ a graph property if it is invariant under the induced
action $\Sigma_{n}^{(2)}$. We call this graph property monotone decreasing if
it is closed under subgraphs, i. e., it is a simplicial complex. We shall omit
the adjective “decreasing.”
### 2.3. Oliver’s Fixed Point Theorem
Let ${\mathcal{K}}\subseteq 2^{\Omega}$ be an abstract simplicial complex with
a $\Gamma$-action. The fixed point complex ${\mathcal{K}}_{\Gamma}$ action is
defined as follows. Let $\Omega_{1},\dots,\Omega_{k}$ be the $\Gamma$-orbits
on $\Omega$. Set
${\mathcal{K}}_{\Gamma}:=\\{S\subseteq[k]\mid\bigcup_{i\in
S}\Omega_{i}\in{\mathcal{K}}\\}.$
We say that a group $\Gamma$ satisfies Oliver’s condition if there exist (not
necessarily distinct) primes $p,q$ such that $\Gamma$ has a (not necessarily
proper) chain of subgroups $\Gamma_{2}\lhd\Gamma_{1}\lhd\Gamma$ such that
$\Gamma_{2}$ is a $p$-group, $\Gamma_{1}/\Gamma_{2}$ is cyclic, and
$\Gamma/\Gamma_{1}$ is a $q$-group.
###### Theorem 2.1 (Oliver [15]).
Assume the group $\Gamma$ satisfies Oliver’s condition. If $\Gamma$ acts on a
nonempty contractible simplicial complex ${\mathcal{K}}$ then
$\chi({\mathcal{K}}_{\Gamma})\equiv 1\pmod{q}.$ (2.1)
In particular, such an action must always have a nonempty invariant face.
### 2.4. The KSS approach and the general strategy
The topological approach to evasiveness, initiated by Kahn, Saks, and
Sturtevant, is based on the following key observation.
###### Lemma 2.2 (Kahn-Saks-Sturtevant [11]).
If $P_{n}$ is a non-evasive graph property then $P_{n}$ is contractible.
Kahn, Saks, and Sturtevant recognized that Lemma 2.2 brought Oliver’s Theorem
to bear on evasiveness. The combination of Lemma 2.2 and Theorem 2.1 suggests
the following general strategy, used by all authors in the area who have
employed the topological method, including this paper: We find primes $p,q$, a
group $\Gamma$ satisfying Oliver’s condition with these primes, and a
$\Gamma$-action on $P_{n}$, such that $\chi(P_{n})\equiv 0\pmod{q}$. By
Oliver’s Theorem and the KSS Lemma this implies that $P_{n}$ is evasive. The
novelty is in finding the right $\Gamma$.
KSS [11] made the assumption that $n$ is a prime power and used as
$\Gamma=\operatorname{AGL}(1,n)$, the group of affine transformations
$x\mapsto ax+b$ over the field of order $n$. While we use subgroups of such
groups as our building blocks, the attempt to combine these leads to hard
problems on the distribution of prime numbers.
Regarding the “forbidden subgraph” property, Chakrabarti, Khot, and Shi [3]
built considerable machinery which we use. Our conclusions are considerably
stronger than theirs; the additional techniques involved include a study of
the orbitals of certain metacyclic groups, a universality property of
cyclotomic graphs derivable using Weil’s character sum estimates, plus the
number theoretic reductions indicated.
For the “sparse graphs” result (Theorem 1.4) we need $\Gamma$ such that all
u-orbitals of $\Gamma$ are large and therefore
$(P_{n})_{\Gamma}=\\{\emptyset\\}$.
In both cases, we are forced to use rather large building blocks of size $q$,
say, where $q$ is a prime such that $q-1$ has a large divisor which is a prime
for Theorem 1.4 and a power of 2 for Theorem 1.3.
## 3\. Forbidden subgraphs
In this section we prove parts (a) and (c) of Theorem 1.3.
### 3.1. The CKS condition
A homomorphism of a graph $H$ to a graph $H^{\prime}$ is a map $f\,:\,V(H)\to
V(H^{\prime})$ such that $(\forall x,y\in V(H))(\\{x,y\\}\in
E(H)\Rightarrow\\{f(x),f(y)\\}\in E(H^{\prime}))$. (In particular,
$f^{-1}(x^{\prime})$ is an independent set in $H$ for all $x^{\prime}\in
V(H^{\prime})$.) Let $Q_{r}^{[[H]]}$ be the set of those $H^{\prime}$ with
$V(H^{\prime})=[r]$ that do not admit an $H\to H^{\prime}$ homomorphism. Let
further $T_{H}:=\min\\{2^{2^{t}}-1\ \mid\ 2^{2^{t}}\geq h\\}$ where $h$
denotes the number of vertices of $H$. The following is the main lemma of
Chakrabarti, Khot, and Shi [3].
###### Lemma 3.1 (Chakrabarti et al. [3]).
If $r\equiv 1\pmod{T_{H}}$ then $\chi(Q_{r}^{[[H]]})\equiv 0\pmod{2}$.
### 3.2. Cliques in generalized Paley graphs
Let $q$ be an odd prime power and $d$ an even divisor of $q-1.$ Consider the
graph $P(q,d)$ whose vertex set is $\mathbb{F}_{q}$ and the adjacency between
the vertices is defined as follows: $i\sim j\iff(i-j)^{d}=1.$ $P(q,d)$ is
called a generalized Paley graph.
###### Lemma 3.2.
If $(q-1)/d\leq q^{1/(2h)}$ then $P(q,d)$ contains a clique on $h$ vertices.
This follows from the following lemma which in turn can be proved by a routine
application of Weil’s character sum estimates (cf. [1]).
###### Lemma 3.3.
Let $a_{1},\ldots,a_{t}$ be distinct elements of the finite field
$\mathbb{F}_{q}.$ Assume $\ell\mid q-1$. Then the number of solutions
$x\in\mathbb{F}_{q}$ to the system of equations $(a_{i}+x)^{(q-1)/\ell}=1$ is
$\frac{q}{\ell^{t}}\pm t\sqrt{q}.$ ∎
Let $\Gamma(q,d)$ be the subgroup of order $qd$ of $\operatorname{AGL}(1,q)$
defined in Section 2.1. Each u-orbital of $\Gamma(q,d)$ is isomorphic to
$P(q,d)$. ∎
###### Corollary 3.4.
If ${\frac{q-1}{d}\leq q^{1/(2h)}}$ then each u-orbital of $\Gamma(q,d)$
contains a clique of size $h.$
### 3.3. $\epsilon$-near-Fermat primes
The numbers in the title were defined in Section 1.3. In this section we prove
Theorem 1.3, part (a).
###### Theorem 3.5.
Let $H$ be a graph on $h$ vertices. If there are infinitely many
$\frac{1}{2h}$-near-Fermat primes then $Q_{n}^{H}$ is eventually evasive.
Proof. Fix an odd prime $p\equiv 2\pmod{T_{H}}$ such that $p\geq|H|.$ If there
are infinitely many $\frac{1}{2h}$-near-Fermat primes then infinitely many of
them belong to the same residue class mod $p$, say $a+\mathbb{Z}p$. Let
$q_{i}$ be the $i$-th $\frac{1}{2h}$-near-Fermat prime such that $q_{i}\geq p$
and $q_{i}\equiv a\pmod{p}.$ Let $r^{\prime}=na^{-1}\pmod{p}$ and
$k^{\prime}=\sum_{i=1}^{r^{\prime}}q_{i}.$ Then $k^{\prime}\equiv n\pmod{p}$
and therefore $n=pk+k^{\prime}$ for some $k$.
Now in order to use Lemma 3.1, we need to write $n$ as a sum of $r$ terms
where $r\equiv 1\pmod{T_{H}}$. We already have $r^{\prime}$ of these terms; we
shall choose each of the remaining $r-r^{\prime}$ terms to be $p$ or $p^{2}$.
If there are $t$ terms equal to $p^{2}$ then this gives us a total of
$r=t+(k-tp)+r^{\prime}$ terms, so we need $t(p-1)\equiv
k+r^{\prime}\pmod{T_{H}}$. By assumption, $p-1\equiv 1\pmod{T_{H}}$; therefore
such a $t$ exists; for large enough $n$, it will also satisfy the constraints
$0\leq t\leq k/p$,
Let now
$\Lambda_{1}:=\left((\mathbb{F}^{+}_{p^{2}})^{t}\times(\mathbb{F}^{+}_{p})^{k-tp}\right)\rtimes\mathbb{F}_{p^{2}}^{\times}$
acting on $[pk]$ with $t$ orbits of size $p^{2}$ and $k-pt$ orbits of size $p$
as follows: on an orbit of size $p^{i}$ ($i=1,2$) the action is
$\operatorname{AGL}(1,p^{i})$. The additive groups act independently, with a
single multiplicative action on top. $\mathbb{F}_{p^{2}}^{\times}$ acts on
$\mathbb{F}_{p}^{+}$ through the group homomorphism
$\mathbb{F}_{p^{2}}^{\times}\to\mathbb{F}_{p}^{\times}$ defined by the map
$x\mapsto x^{p-1}$. Let $B_{j}$ denote an orbit of $\Lambda_{1}$ on $[kp]$.
Now the orbit of any pair $\\{u,v\\}\in{B_{j}\choose 2}$ is a clique of size
$|B_{j}|\geq p\geq h$, therefore a $\Lambda_{1}$-invariant graph cannot
contain an intra-cluster edge.
Let $d_{i}$ be the largest power of 2 that divides $q_{i}-1.$ Let $C_{i}$ be
the subgroup of $\mathbb{F}_{q_{i}}^{\times}$ of order $d_{i}.$ Let
$\displaystyle{\Lambda_{2}:=\prod_{i=1}^{r^{\prime}}\Gamma(q_{i},d_{i}),}$
acting on $[k^{\prime}]$ with $r^{\prime}$ orbits of sizes
$q_{1},\dots,q_{r^{\prime}}$ in the obvious manner.
From Lemma 3.2 we know that the orbit of any $\\{u,v\\}\in{[q_{i}]\choose 2}$
must contain a clique of size $h.$ Hence, an invariant graph cannot contain
any intra-cluster edge.
Overall, let $\Gamma:=\Lambda_{1}\times\Lambda_{2}$, acting on $[n].$ Since
$q_{i}\geq p,$ we have $\gcd(q_{i},p^{2}-1)=1.$ Thus, $\Gamma$ is a “$2$-group
extension of a cyclic extension of a $p$-group” and therefore satisfies
Oliver’s Condition (stated before Theorem 2.1). Hence, assuming $Q_{n}^{H}$ is
non-evasive, Lemma 2.2 and Theorem 2.1 imply
$\chi((Q_{n}^{H})_{\Gamma})\equiv 1\pmod{2}.$
On the other hand, we claim that the fixed-point complex
$(Q_{n}^{H})_{\Gamma}$ is isomorphic to $Q_{r}^{[[H]]}$. The (simple) proof
goes along the lines of Lemma 4.2 of [3]. Thus, by Lemma 3.1 we have
$\chi(Q_{r}^{[[H]]})\equiv 0\pmod{2},$ a contradiction. ∎
### 3.4. Unconditionally, $Q_{n}^{H}$ is only $O(1)$ away from being evasive
In this section, we prove part (c) of Theorem 1.3.
###### Theorem 3.6.
For every graph $H$ there exists a number $C_{H}$ such that the query
complexity of $Q_{n}^{H}$ is $\geq\binom{n}{2}-C_{H}.$
Proof. Let $h$ be the number of vertices of $H$. Let $p$ be the smallest prime
such that $p\geq h$ and $p\equiv 2\pmod{T_{H}}$. So $p<f(H)$ for some function
$f$ by Dirichlet’s Theorem (we don’t need any specific estimates here). Since
$p-1\equiv 1\pmod{T_{H}},$ we have $\gcd(p-1,T_{H})=1$ and therefore
$\gcd(p-1,pT_{H})=1$. Now, by the Chinese Remainder Theorem, select the
smallest positive integer $k^{\prime}$ satisfying $k^{\prime}\equiv
n\pmod{pT_{H}}$ and $k^{\prime}\equiv 1\pmod{p-1}$. Note that
$k^{\prime}<p^{2}T_{H}$. Let $k=(n-k^{\prime})/(pT_{H})$; so we have
$n=kpT_{H}+k^{\prime}$.
Let $N^{\prime}=\binom{n}{2}-\binom{k^{\prime}}{2}$. Consider the following
Boolean function $B_{n}^{H}$ on $N^{\prime}$ variables. Consider graphs $X$ on
the vertex set $[n]$ with the property that they have no edges among their
last $k^{\prime}$ vertices. These graphs can be viewed as Boolean functions of
the remaining $N^{\prime}$ variables. Now we say that such a graph has
property $B_{n}^{H}$ if it does not contain $H$ as a subgraph.
Claim. The function $B_{n}^{H}$ is evasive.
The Claim immediately implies that the query complexity of $Q_{n}^{H}$ is at
least $N^{\prime}$, proving the Theorem with
$C_{H}=\binom{k^{\prime}}{2}<p^{4}T_{H}^{2}<f(H)^{4}T_{H}^{2}$.
To prove the Claim, consider the groups
$\Lambda:=(\mathbb{F}_{p}^{+})^{kT_{H}}\rtimes\mathbb{F}_{p}^{\times}$ and
$\Gamma:=\Lambda\times\mathbb{Z}_{k^{\prime}}$. Here $\Lambda$ acts on
$[pkT_{H}]$ in the obvious way: we divide $[pkT_{H}]$ into $kT_{H}$ blocks of
size $p$; $\mathbb{F}_{p}^{+}$ acts on each block independently and
$\mathbb{F}_{p}^{\times}$ acts on the blocks simultaneously (diagonal action)
so on each block they combine to an $\operatorname{AGL}(1,p)$-action.
$\mathbb{Z}_{k^{\prime}}$ acts as a $k^{\prime}$-cycle on the remaining
$k^{\prime}$ vertices. So $\Gamma$ is a cyclic extension of a $p$-group
(because $\gcd(p-1,k^{\prime})=1$).
If $B_{n}^{H}$ is not evasive then from Theorem 2.1 and Lemma 2.2, we have
$\chi\left((B_{n}^{H})_{\Gamma}\right)=1$.
On the other hand we claim that, $(B_{n}^{H})_{\Gamma}\cong Q_{r}^{[[H]]},$
where $r=kT_{H}+1.$ The proof of this claim is exactly the same as the proof
of Lemma 4.2 of [3]. Thus, from Lemma 3.1, we conclude that
$\chi(Q_{r}^{[[H]]})$ is even. This contradicts the previous conclusion that
$\chi(Q_{r}^{[[H]]})=1.$ ∎
###### Remark 3.7.
Specific estimates on the smallest Dirichlet prime can be used to estimate
$C_{H}$. Linnik’s theorem implies $C_{H}<h^{O(1)}$, extending Theorem 3.6 to
strong lower bounds for variable $H$ up to $h=n^{c}$ for some positive
constant $c$.
## 4\. Sparse graphs: unconditional results
We prove part (c) of Theorem 1.4.
###### Theorem 4.1.
If the non-empty monotone graph property $P_{n}$ is not evasive then
$\dim(P_{n})=\Omega(n\log n).$
### 4.1. The basic group construction
Assume in this section that $n=p^{\alpha}k$ where $p$ is prime. Let
$\Delta_{k}\leq\Sigma_{k}$. We construct the group
$\Gamma_{0}(p^{\alpha},\Delta_{k})$ acting on $[n].$
Let $\Delta=(\mathbb{F}_{p^{\alpha}}^{\times}\times\Delta_{k})$. Let
$\Gamma_{0}(p^{\alpha},\Delta_{k})$ be the semidirect product
$(\mathbb{F}_{p^{\alpha}}^{+})^{k}\rtimes\Delta$ with respect to the
$\Delta$-action on $(\mathbb{F}_{p^{\alpha}}^{+})^{k}$ defined by
$(a,\sigma):(b_{1},\ldots,b_{k})\mapsto(ab_{\sigma^{-1}(1)},\ldots,ab_{\sigma^{-1}(k)}).$
We describe the action of $\Gamma_{0}(p^{\alpha},\Delta_{k})$ on $[n]$.
Partition $[n]$ into $k$ clusters of size $p^{\alpha}$ each. Identify each
cluster with the field of order $p^{\alpha},$ i.e., as a set,
$[n]=[k]\times\mathbb{F}_{p^{\alpha}}.$ The action of
$\gamma=(b_{1},\ldots,b_{k},a,\sigma)$ is described by
$\gamma:(x,y)\mapsto(\sigma(x),ay+b_{\sigma(x)}).$
An unordered pair $(i,j)\in[n]$ is termed an intra-cluster edge if both $i$
and $j$ are in the same cluster, otherwise it is termed an inter-cluster edge.
Note that every u-orbital under $\Gamma$ has only intra-cluster edges or only
inter-cluster edges. Denote by $m_{\operatorname{intra}}$ and
$m_{\operatorname{inter}}$ the minimum sizes of u-orbitals of intra-cluster
and inter-cluster edges respectively.
We denote by $m^{\prime}_{k}$ the minimum size of an orbit in $[k]$ under
$\Delta_{k}$ and by $m^{\prime\prime}_{k}$ the minimum size of a u-orbital in
$[k].$ We then have:
$m_{\operatorname{intra}}\geq\binom{p^{\alpha}}{2}\times m^{\prime}_{k},\qquad
m_{\operatorname{inter}}\geq(p^{\alpha})^{2}\times m^{\prime\prime}_{k}$
Let $m_{k}`:=\min\\{m^{\prime}_{k},m^{\prime\prime}_{k}\\}$ and define $m^{*}$
as the minimum size of a u-orbital in $[n].$ Then
$m^{*}=\min\\{m_{\operatorname{intra}},m_{\operatorname{inter}}\\}=\Omega(p^{2\alpha}m_{k})$
(4.1)
### 4.2. Vinogradov’s Theorem
The Goldbach Conjecture asserts that every even integer can be written as the
sum of two primes. Vinogradov’s Theorem [24] says that every sufficiently
large odd integer $k$ is the sum of three primes $k=p_{1}+p_{2}+p_{3}$. We use
here Haselgrove’s version [7] of Vinogradov’s theorem which states that we can
require the primes to be roughly equal: $p_{i}\sim k/3$. This can be combined
with the Prime Number Theorem to conclude that every sufficiently large even
integer $k$ is a sum of four roughly equal primes.
### 4.3. Construction of the group
Let $n=p^{\alpha}k$ where $p$ is prime. Assume $k$ is not bounded. Write $k$
as a sum of $t\leq 4$ roughly equal primes $p_{i}$. Let
$\Delta_{k}:=\prod_{i}\mathbb{Z}_{p_{i}}$ where $\mathbb{Z}_{p_{i}}$ denotes
the cyclic group of order $p_{i}$ and the direct product is taken over the
distinct $p_{i}$.
$\Delta_{k}$ acts on $[k]$ as follows: partition $k$ into parts of sizes
$p_{1},\dots,p_{t}$ and call these parts $[p_{i}].$ The group
$\mathbb{Z}_{p_{i}}$ acts as a cyclic group on the part $[p_{i}].$ In case of
repetitions, the same factor $\mathbb{Z}_{p_{i}}$ acts on all the parts of
size $p_{i}.$
We follow the notation of Section 4.1 and consider the group
$\Gamma_{0}(p^{\alpha},\Delta_{k})$ with our specific $\Delta_{k}$. We have
$m_{k}=\Omega(k)$ and hence we get, from equation (4.1):
###### Lemma 4.2.
Let $n=p^{\alpha}k$ where $p$ is a prime. For the group
$\Gamma_{0}(p^{\alpha},\Delta_{k})$, we have
$m^{*}=\Omega(p^{2\alpha}k)=\Omega(p^{\alpha}n),$ where $m^{*}$ denotes the
minimum size of a u-orbital.
### 4.4. Proof for the superlinear bound
Let $n=p^{\alpha}k$ where $p^{\alpha}$ is the largest prime power dividing
$n$; so $p^{\alpha}=\Omega(\log n)$; this will be a lower bound on the size of
u-orbitals. Our group $\Gamma$ will be of the general form discussed in
Section 4.1.
Case 1. $p^{\alpha}=\Omega(n^{2/3}).$
Let $\Gamma=\Gamma_{0}(p^{\alpha},\\{1\\})$. Following the notation of Section
4.1, we get $m_{k}^{\prime}=m_{k}^{\prime\prime}=1,$ and this yields that
$m^{*}=\Omega((p^{\alpha})^{2})=\Omega(n^{4/3})=\Omega(n\log n).$ Oliver’s
condition is easily verified for $\Gamma$.
Case 2. $k=\Omega(n^{1/3}).$
Consider the $\Gamma:=\Gamma_{0}(p^{\alpha},\Delta_{k})$ acting on $[n]$ where
$\Delta_{k}$ is as described in Section 4.3. The minimum possible size $m^{*}$
of a u-orbital is $\Omega(np^{\alpha})$ by Lemma 4.2. Finally, since
$p^{\alpha}=\Omega(\log n)$, we obtain $m^{*}=\Omega(n\log n).$
If all $p_{i}$ are co-prime to $p^{\alpha}-1$ then
$\mathbb{F}_{p^{\alpha}}^{\times}\times\Delta_{k}$ becomes a cyclic group and
$\Gamma$ becomes a cyclic extension of a $p$-group.
Since $p_{i}=\Omega(k)=\Omega(n^{1/3})$ for all $i$ and
$p^{\alpha}=O(n^{2/3})$, size considerations yield that at most one $p_{i}$
divides $p^{\alpha}-1$ and $p_{i}^{2}$ does not. Suppose, without loss of
generality, $p_{1}$ divides $p^{\alpha}-1.$ Let $p^{\alpha}-1=p_{1}d,$ then
$d$ must be co-prime to each $p_{i}.$ Thus,
$\Delta=(\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{d})\times(\mathbb{Z}_{p_{1}}\times\ldots\times\mathbb{Z}_{p_{t}})=(\mathbb{Z}_{d}\times\mathbb{Z}_{p_{2}}\times\ldots\times\mathbb{Z}_{p_{r}})\times(\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{1}}).$
Thus, $\Delta$ is a $p_{1}$-group extension of a cyclic group. Hence, $\Gamma$
satisfies Oliver’s Condition (cf. Theorem 2.1). ∎
###### Remark 4.3.
For almost all $n,$ our proof gives a better dimension lower bound of
$\Omega(n^{1+\frac{1+o(1)}{\ln\ln n}}).$
## 5\. Sparse graphs: conditional improvements
In this section we prove parts (a) and (b) of Theorem 1.4.
### 5.1. General Setup
Let $n=pk+r,$ where $p$ and $r$ are prime numbers. Let $q$ be a prime divisor
of $(r-1).$ We partition $[n]$ into two parts of size $pk$ and $r$, denoted by
$[pk]$ and $[r]$ respectively. We now construct a group $\Gamma(p,q,r)$ acting
on $[n]$ as a direct product of a group acting on $[pk]$ and a group acting on
$[r],$ as follows:
$\Gamma=\Gamma(p,q,r):=\Gamma_{0}(p,\Delta_{k})\times\Gamma(r,q)$
Here, $\Gamma_{0}(p,\Delta_{k})$ acts on $[pk]$ and is as defined in Section
4.3, and involves choosing a partition of $k$ into upto four primes that are
all $\Omega(k).$
$\Gamma(r,q)$ is defined as the semidirect product $\mathbb{F}_{r}^{+}\rtimes
C_{q},$ with $C_{q}$ viewed as a subgroup of the group
$\mathbb{F}_{r}^{\times}.$ It acts on $[r]$ as follows: We identify $[r]$ with
the field of size $r.$ Let $(b,a)$ be a typical element of $\Gamma_{r}$ where
$b\in\mathbb{F}_{r}$ and $a\in C_{q}.$ Then, $(b,a):x\mapsto ax+b.$
Thus, $\Gamma=\Gamma(p,q,r)$ acts on $[n].$ Let $m^{*}$ be the minimum size of
the orbit of any edge $(i,j)\in{[n]\choose 2}$ under the action of $\Gamma.$
One can show that
$m^{*}=\Omega(\min\\{p^{2}k,pkr,qr\\}).$ (5.1)
We shall choose $p,q,r$ carefully such that (a) the value of $m^{*}$ is large,
and (b) Oliver’s condition holds for $\Gamma(p,q,r)$.
### 5.2. ERH and Dirichlet primes
The Extended Riemann Hypothesis (ERH) implies the following strong version of
the Prime Number Theorem for arithmetic progressions. Let $\pi(n,D,a)$ denote
the numer of primes $p\leq n$, $p\equiv a\pmod{D}$. Then for $D<n$ we have
$\pi(n,D,a)=\frac{\operatorname{li}(n)}{\varphi(D)}+O(\sqrt{x}\ln x)$ (5.2)
where $\operatorname{li}(n)=\int_{2}^{n}dt/t$ and the constant implied by the
big-Oh notation is absolute (cf. [16, Ch. 7, eqn. (5.12)] or [2, Thm. 8.4.5]).
This result immediately implies “Bertrand’s Postulate for Dirichlet primes:”
###### Lemma 5.1 (Bertrand’s Postulate for Dirichlet primes).
Assume ERH. Suppose the sequence $D_{n}$ satisfies
$D_{n}=o(\sqrt{n}/\log^{2}n)$. Then for all sufficiently large $n$ and for any
$a_{n}$ relatively prime to $D_{n}$ there exists a prime $p\equiv
a_{n}\pmod{D_{n}}$ such that $\frac{n}{2}\leq p\leq n.$
### 5.3. With ERH but without Chowla
We want to write $n=pk+r,$ where $p$ and $r$ are primes, and with $q$ a prime
divisor of $r-1,$ as described in Section 5.1. Specifically, we try for:
$p=\Theta(n^{1/4}),\quad\frac{n}{4}\leq r\leq\frac{n}{2},\quad
q=\Theta(n^{1/4-\epsilon})$
We claim that under ERH, such a partition of $n$ is possible.
To see this, fix some $p=\Theta(n^{1/4})$ such that $\gcd(p,n)=1.$ Fix some
$q=\Theta(n^{1/4-\epsilon}).$ Now, $r\equiv 1\pmod{q}$ and $r\equiv n\pmod{p}$
solves to $r\equiv a\pmod{pq}$ for some $a$ such that $\gcd(a,pq)=1.$ Since
$pq=\Theta(n^{1/2-\epsilon}),$ we can conclude under ERH (using Lemma 5.1)
that there exists a prime $r\equiv a\pmod{pq}$ such that $\frac{n}{4}\leq
r\leq\frac{n}{2}.$ This gives us the desired partition. One can verify that
our $\Gamma$ satisfies Oliver’s Condition. Equation (5.1) gives
$m^{*}=\Omega(n^{5/4-\epsilon}).$ This completes the proof of part (b) of
Theorem 1.4. ∎
### 5.4. Stronger bound using Chowla’s conjecture
Let $a$ and $D$ be relatively prime. Let $p$ be the first prime such that
$p\equiv a\pmod{D}.$ Chowla’s conjecture tells us that $p=O(D^{1+\epsilon})$
for every $\epsilon>0.$ Using this, we show $m^{*}=\Omega(n^{3/2-\epsilon}).$
We can use Chowla’s conjecture, along with the general setup of Section 5.1,
to obtain a stronger lower bound on $m^{*}.$ The new bounds we hope to achieve
are:
$p=\Theta(\sqrt{n}),\quad n^{1-2.5\delta}\leq r\leq n^{1-0.5\delta},\quad
q=\Theta(n^{1/2-\delta})$
Such a partition is always possible assuming Chowla’s conjecture. To see this,
first fix $p=\Theta(n^{1/2}),$ then fix $q=\Theta(n^{1/2-2\delta})$ and find
the least solution for $r\equiv 1\pmod{q}$ and $r\equiv n\pmod{p},$ which is
equivalent to solving for $r\equiv a\pmod{pq}$ for some $a<pq.$ The least
solution will be greater than $pq$ unless $a$ happens to be a prime. In this
case, we add another constraint, say $r\equiv a+1\pmod{3}$ and resolve to get
the least solution greater than $pq.$ Note that $n^{1-2.5\delta}\leq r\leq
n^{1-0.5\delta}.$ Now, from Equation (5.1), we get the lower bound of
$m^{*}=\Omega(n^{3/2-4\delta}).$ This completes the proof of part (a) of
Theorem 1.4. ∎
### Acknowledgment.
Raghav Kulkarni expresses his gratitude to Sasha Razborov for bringing the
subject to his attention and for helpful initial discussions.
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The evolution of the Mdisc-Mstar and Mdot-Mstar correlations traces protoplanetary disc dispersal
Alice Somigliana
1,2email<EMAIL_ADDRESS>Leonardo Testi3,4,
Giovanni Rosotti5,
Claudia Toci1,
Giuseppe Lodato5,
Rossella Anania5,
Benoît Tabone6,
Marco Tazzari4,
Ralf Klessen7, 8,
Ugo Lebreuilly 9,
Patrick Hennebelle 9,
and Sergo Molinari 10
European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching bei München, Germany
Fakultat für Physik, Ludwig-Maximilians-Universität München, Scheinersts. 1, 81679 München, Germany
Dipartimento di Fisica e Astronomia, Universita' di Bologna, Via Gobetti 93/2, I-40122 Bologna, Italy
INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy
Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy
Université Paris-Saclay, CNRS, Institut d'Astrophysique Spatiale, Orsay, France
Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany
Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany
Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France
Istituto Nazionale di Astrofisica-IAPS, Via Fosso del Cavaliere 100, I-00133 Roma, Italy
Observational surveys of entire star-forming regions have provided evidence of power-law correlations between the disc integrated properties and the stellar mass, especially the disc mass ($M_{\mathrm{d}} \propto {M_{\star}}^{\lambda_{\mathrm{m}}}$) and the accretion rate ($\dot M \propto {M_{\star}}^{\lambda_{\mathrm{acc}}}$). Whether the secular disc evolution affects said correlations is still a matter of debate: while the purely viscous scenario has been investigated, other evolutionary mechanisms could have a different impact. In this paper, we study the time evolution of the slopes $\lambda_{\mathrm{m}}$ and $\lambda_{\mathrm{acc}}$ in the wind-driven and viscous-wind hybrid case and compare it to the purely viscous prediction. We use a combination of analytical calculations, where possible, and numerical simulations performed with the 1D population synthesis code , that we also present and release to the community. Assuming $M_{\mathrm{d}}(0) \propto {M_{\star}}^{\lambda_{\mathrm{m}, 0}}$ and $\dot M(0) \propto {M_{\star}}^{\lambda_{\mathrm{acc}, 0}}$ as initial conditions, we find that viscous and hybrid accretion preserve the power-law shape of the correlations, while evolving their slope; on the other hand, MHD winds change the shape of the correlations, bending them in the higher or lower end of the stellar mass spectrum depending on the scaling of the accretion timescale with the stellar mass. However, we show how a spread in the initial conditions conceals this behaviour, leading to power-law correlations with evolving slopes like in the viscous and hybrid case. We analyse the impact of disc dispersal, intrinsic in the wind model and due to internal photoevaporation in the viscous case: we find that the currently available sample sizes ($\sim 30$ discs at 5 Myr) introduce stochastic oscillations in the slopes evolution, which dominate over the physical signatures. We show that we could mitigate this issue by increasing the sample size: with $\sim 140$ discs at 5 Myr, corresponding to the complete Upper Sco sample, we would obtain small enough error bars to use the evolution of the slopes as a proxy for the driving mechanism of disc evolution. Finally, from our theoretical arguments we discuss how the observational claim of steepening slopes necessarily leads to an initially steeper $M_{\mathrm{d}} - M_{\star}$ correlation with respect to $\dot M - M_{\star}$.
The $M_{\mathrm{d}}-M_{\star}$ and $\dot M - M_{\star}$ correlations trace disc dispersal
Somigliana, A. et al.
§ INTRODUCTION
The secular evolution of protoplanetary discs is deeply intertwined with both the planet formation process [Morbidelli et al., 2012] and the accretion onto the central protostar [Hartmann et al., 1998]. Planetesimals, the building blocks of planets, form and evolve within the disc following the dynamics of either the gaseous or solid component, depending on their relative size and their coupling (or lack thereof) with the gas particles; on the other hand, the protostar is fed by the disc itself, through the accretion of material that loses angular momentum and drifts inwards. The ideal ground to explore the connection between protoplanetary discs and their host stars is provided by large surveys of entire star-forming regions, targeting the properties of both discs and protostars; the last decade has seen a significant observational effort in the direction of these population-level studies, also thanks to the advent of facilities like the Atacama Large Millimeter Array (ALMA) (see the PPVII reviews by Manara et al., 2023, Miotello et al., 2023).
Disc masses and accretion rates are arguably the most studied integrated disc properties. Accretion rates are inferred from the spectra of the central stars, which show an excess emission (especially prominent in the UV) when accretion is taking place; surveys performed across different star-forming regions [Muzerolle et al., 2003, Natta et al., 2004, Mohanty et al., 2005, Dullemond et al., 2006, Herczeg & Hillenbrand, 2008, Rigliaco et al., 2011, Manara et al., 2012, Alcalá et al., 2014, Manara et al., 2016, Alcalá et al., 2017, Manara et al., 2017, Venuti et al., 2019, Manara et al., 2020] agree on the presence of a power-law correlation between the accretion rate and the stellar mass, $\dot M \propto {M_{\star}}^{\lambda_{\mathrm{acc, obs}}}$ (following the notation of Somigliana et al., 2022). On the other hand, disc masses have traditionally been determined from observations of the sub-mm continuum emission of the solid component of discs; due to the large number of assumptions involved in converting sub-mm fluxes into total disc masses (see Miotello et al., 2023), one of the current main goals of the protoplanetary disc community is the accurate determination of total disc masses - both from dynamical constraints [Veronesi et al., 2021, Lodato et al., 2023] and direct measurements of the total gas content (e.g., Bergin et al., 2013, Anderson et al., 2022, Trapman et al., 2022). Despite the systematic uncertainties involved in their determination, dust-based disc masses also seem to show a power-law correlation with the stellar mass, $M_{\mathrm{d}} \propto {M_{\star}}^{\lambda_{\mathrm{m, obs}}}$, across different star-forming regions [Ansdell et al., 2016, Ansdell et al., 2017, Barenfeld et al., 2016, Pascucci et al., 2016, Testi et al., 2016, Testi et al., 2022, Sanchis et al., 2020].
The existence of the disc mass-stellar mass and accretion rate-stellar mass correlations is now generally accepted; however, there is no consensus on the physical reason behind their establishment and their evolution with time. While the $\dot M - M_{\star}$ correlation appears to have a roughly constant slope[Throughout this work, we use 'slope' as a synonym of power-law index, referring to the correlations in the logarithmic plane.] of $\lambda_{\mathrm{acc, obs}} \approx 1.8 \pm 0.2$ (as first suggested by Muzerolle et al., 2003 and supported by many of the following works mentioned above), the $M_{\mathrm{d}} - M_{\star}$ correlations is claimed to be steepening with time [Ansdell et al., 2017], from the lowest $\lambda_{\mathrm{m, obs}} ( t \sim 1$ Myr$ ) = 1.7 \pm 0.2$ (Taurus) to the highest $\lambda_{\mathrm{m, obs}} (t \sim 5$ Myr$) = 2.4 \pm 0.4$ (Upper Sco). Whether these correlations reflect the initial conditions of disc populations, or are rather a product of the secular evolution, is still under debate. Both possibilities have been discussed for the $\dot M - M_{\star}$ correlation: [Alexander & Armitage, 2006] have assumed it to hold as initial condition, favouring the correlation to be present in young populations, whereas [Dullemond et al., 2006] have derived it from a simple model of disc formation from a rotating collapsing core, which provided an explanation for evolved disc populations. At the same time, the claimed increase in the slope of $M_{\mathrm{d}} - M_{\star}$ does suggest an evolutionary trend; [Somigliana et al., 2022] have found that, assuming power-law correlations between both $M_{\mathrm{d}}$ and $\dot M$ and the stellar mass as initial conditions, secular evolution can indeed alter the slopes of the correlations themselves (see Section <ref> for details). However, their analysis was limited to the standard viscous evolution paradigm, whereas the driving mechanism of accretion is far from being constrained (see Manara et al., 2023 for a review).
The traditional viscous accretion model prescribes a macroscopic viscosity as the cause of redistribution of angular momentum within the disc [Lynden-Bell & Pringle, 1974, Pringle, 1981]. In this scenario, while part of the material loses angular momentum and moves radially closer to the star, some other material gains the same amount of angular momentum and moves further away, effectively increasing the disc size. The viscous paradigm can explain many key features of disc evolution, but it cannot account for disc dispersal - as determined from the observational evidence of exponentially decreasing fraction of both disc-bearing [Hernández et al., 2007] and accreting [Fedele et al., 2010] sources in star-forming regions with time; furthermore, the low levels of turbulence detected in discs [Pinte et al., 2016, Flaherty et al., 2018, Rosotti, 2023] appear incompatible with the observed evolution. While the discrepancy in the disc and accretion fraction can be mended considering mechanisms such as internal or external photoevaporation [Alexander et al., 2014, Winter et al., 2018], that effectively clear discs on timescales comparable with the observed decline, the tension between the expected and observed amount of turbulence does not appear to be solved yet. On the other hand, the MHD disc winds scenario offers a promising alternative. Pioneering work [Blandford & Payne, 1982, Ferreira, 1997] supported by recent numerical simulations (e.g., Béthune et al., 2017) demonstrated that MHD winds launched from the disc surface have the net effect of removing angular momentum as a consequence of the extraction of material; SMHD wind-driven accretion can even lead to disc dispersal [Armitage et al., 2013, Tabone et al., 2022]. Following disc evolution at population level in numerical simulations remains out of reach; however, three-dimensional core-collapse simulations have shown how non-ideal magnetohydrodynamics and ambipolar diffusion play a fundamental role in shaping the resulting population of early-type young stellar objects [Lebreuilly et al., 2021, Lebreuilly et al., 2024]. While some 3D studies of isolated disc formation have attempted to bridge the gap between Class 0/I and Class II stages [Machida & Hosokawa, 2013, Hennebelle et al., 2020, Xu & Kunz, 2021, Xu & Kunz, 2021, Machida & Basu, 2024, Mauxion et al., 2024], the high numerical cost of the simulations for 3D population synthesis does not allow to follow the evolution of the discs up to very evolved stages where they can be considered isolated from the surrounding environment. MHD wind-driven disc populations can however be modelled in 1D using simple prescriptions as proposed by [Suzuki et al., 2016] or [Tabone et al., 2022]. Detecting characteristic signatures of either of the two evolutionary prescriptions is a compelling issue [Long et al., 2022, Alexander et al., 2023, Somigliana et al., 2023, Trapman et al., 2023, Coleman et al., 2024].
In the context of the evolution of the correlations between the disc properties and the stellar mass, while the purely viscous scenario has been extensively studied by [Somigliana et al., 2022], the wind-driven paradigm remains unexplored; with this paper, we address this deficiency and investigate the impact of MHD wind-driven evolution on the $M_{\mathrm{d}} - M_{\star}$ and $\dot M - M_{\star}$ correlations, with a particular focus on their time evolution and the comparison with the purely viscous paradigm. We also extend the work of [Somigliana et al., 2022] by including internal photoevaporation to the viscous framework. We employ numerical simulations of populations of protoplanetary discs, performed with the population synthesis code , which we also introduce and release to the community.
The paper is structured as follows: in Section <ref>, we present and describe its main features, set up and solution algorithm; in Section <ref>, we discuss the time evolution of the $M_{\mathrm{d}} - M_{\star}$ and $\dot M - M_{\star}$ correlations in three evolutionary scenarios from the theoretical perspective; in Section <ref>, we show the impact of a spread in the initial conditions and dispersal mechanisms on the evolution of the slopes, and we present the numerical results obtained from realistic disc population synthesis; in Section <ref> we interpret the implications of our findings in the context of the observational determination of the slopes, and finally in Section <ref> we draw the conclusions of this work.
§ NUMERICAL METHODS:
In this Section we present the 1D population synthesis code [ and the output analysis library can be installed via the Python Package Index, and . The full documentation and tutorials are available at https://alicesomigliana.github.io/diskpop-docs/index.html. If you use in your work, please cite this paper (Somigliana et al. 2024).]. We describe the master equation for the secular evolution of discs (Section <ref>), the initial conditions to generate a synthetic population (Section <ref>), the solution algorithm (Section <ref>), and the user interface and output (Section <ref>). For a more detailed description, we refer to the code documentation; for a validation of the code, see Appendix <ref>.
§.§ Master equation
The master equation of protoplanetary disc evolution,
\begin{equation}
\label{eq:master_equation}
\begin{split}
\frac{\partial \Sigma}{\partial t} = \frac{3}{r} \frac{\partial}{\partial r} \left[ \frac{1}{\Omega r} \frac{\partial}{\partial r} \left( r^2 \alpha_{\mathrm{SS}} \Sigma {c_s}^2 \right) \right] + \frac{3}{2r} \frac{\partial}{\partial r} \left[ \frac{\alpha_{\mathrm{DW}} \Sigma {c_s}^2}{\Omega} \right] \\
- \frac{3 \alpha_{\mathrm{DW}} \Sigma {c_s}^2}{4 (\lambda-1)r^2 \Omega} - \dot{\Sigma}_{\mathrm{photo}},
\end{split}
\end{equation}
describes the time evolution of the gas surface density in the most general framework, where $\Sigma$ is the gas surface density, $\Omega$ the Keplerian orbital frequency, $\alpha_{\mathrm{SS}}$ the [Shakura & Sunyaev, 1973] $\alpha$ parameter, $\alpha_{\mathrm{DW}}$ the MHD equivalent of $\alpha_{\mathrm{SS}}$ [Tabone et al., 2022], $c_s$ the sound speed, and $\lambda$ the magnetic lever arm parameter, which quantifies the ratio of extracted to initial specific angular momentum. The four terms on the right hand side (RHS) refer to (i) the viscous torque, whose strength is parameterised by $\alpha_{\mathrm{SS}}$, (ii) the wind-driven accretion, which corresponds to an advection term, parameterised by $\alpha_{\mathrm{DW}}$, (iii) mass loss due to MHD disc winds, parameterised by $\lambda$ and (iv) mass loss due to other physical phenomena (in our case, we consider internal and external photoevaporation). Depending on the values of the specific parameters, Equation (<ref>) can describe a purely viscous ($\alpha_{\mathrm{DW}} = 0$), purely MHD wind-driven ($\alpha_{\mathrm{SS}} = 0$) or hybrid ($\alpha_{\mathrm{SS}}, \alpha_{\mathrm{DW}} \neq 0$) evolution, with ($\dot \Sigma_{\mathrm{photo}} \neq 0$) or without ($\dot \Sigma_{\mathrm{photo}} = 0$) the influence of photoevaporation. In the following, we briefly describe the various evolutionary scenarios and the available analytical solutions.
Viscously evolving discs. In the case of purely viscous evolution, the MHD winds parameter $\alpha_{\mathrm{DW}}$ is set to zero. If we also neglect the influence of photoevaporation, Equation (<ref>) reduces to the first term on the RHS and its solution depends on the functional form of the effective viscosity, parameterised as $\nu = \alpha_{\mathrm{SS}} c_s H$ (where $H$ is the vertical height of the disc). A popular analytical solution for viscous discs is the [Lynden-Bell & Pringle, 1974] self-similar solution, which assumes viscosity to scale as a power-law of the radius ($\nu \propto R^{\gamma}$).
MHD winds-driven evolution. There are two classes of analytical solutions to Equation (<ref>) in the MHD wind-driven scenario, associated with a specific prescription of $\alpha_{\mathrm{DW}}$ [Tabone et al., 2022]. We briefly describe their key features, and refer to the original paper for their derivation and an in-depth discussion.
* The simplest class of solutions (so-called hybrid solutions), which highlight the main features of wind-driven accretion in comparison to the viscous model, assume a constant $\alpha_{\mathrm{DW}}$ with time; these solutions depend on the value of $\psi \equiv \alpha_{\mathrm{DW}}/\alpha_{\mathrm{SS}}$, which quantifies the relative strength of the radial and vertical torque.
* Another class of solutions, which describe the unknown evolution of the magnetic field strength, assume a varying $\alpha_{\mathrm{DW}}$ with time. To obtain these, [Tabone et al., 2022] parameterised $\alpha_{\mathrm{DW}}(t) \propto \Sigma_{\mathrm{c}} (t)^{-\omega}$, with $\Sigma_{\mathrm{c}} = M_{\mathrm{d}}(t)/2 \pi {R_c}^2 (t)$ (where $R_c$ is a characteristic radius) and $\omega$ as a free parameter, and neglect the radial transport of angular momentum ($\alpha_{\mathrm{SS}} = 0$).
Photoevaporation. The generic $\dot \Sigma_{\mathrm{photo}}$ term in Equation (<ref>) allows to account for photoevaporative processes, both internal and external. The exact form of $\dot \Sigma_{\mathrm{photo}}$ depends on the specific model considered; therefore, the availability (or lack thereof) of analytical solutions needs to be considered case by case.
allows to evolve populations of discs analytically. In particular, as of this release, it includes implementations of the [Lynden-Bell & Pringle, 1974] self-similar solution and all the analytical solution proposed by [Tabone et al., 2022]. In the cases where Equation (<ref>) cannot be solved analytically, the code relies on the solution algorithm described in Section <ref>.
§.§ Initial conditions and parameters
Every simulation begins with the generation of a synthetic population of Young Stellar Objects (YSOs). Each YSO constitutes of a star and a disc, whose key initial parameters (stellar mass, disc mass, accretion rate, disc radius, evolutionary parameters $\alpha_{\mathrm{SS}}$, $\alpha_{\mathrm{DW}}$, $\lambda$, $\omega$...) can be set by the user. In the following, we describe the standard case where we consider the stellar masses to be distributed according to an Initial Mass Function (IMF) and correlating with the disc mass and radius, and briefly mention the other possible choices; for a deeper discussion, we refer to the documentation.
assembles YSOs by determining their parameters as follows:
* Stellar mass $M_{\star}$: determined following the [Kroupa, 2001] IMF. Other possible choices are a constant mass for all the stars in the population, or a set of custom stellar masses.
* Initial disc mass $M_{\mathrm{d}}$, accretion rate $\dot M$: determined from log-normal distributions of given width and mean value. In the standard case, considers an initial power-law correlation between the initial $M_{\mathrm{d}}$ and $\dot M$ and the stellar mass (see Section <ref> for a detailed discussion), where the normalisation at 1 M$_{\odot}$, the slope and the scatter around the power-laws are free parameters. If the correlations with the stellar mass are neglected, the user sets the mean value and spread of the distributions.
* Accretion parameters ($\alpha_{\mathrm{SS}}$, $\alpha_{\mathrm{DW}}$, $\lambda$, $\omega$): global properties of the whole population, given as input from the user. By setting the parameters controlling accretion, determines the disc radius $R_{\mathrm{d}}$ and accretion timescale $t_{\mathrm{acc}}$ - which are instead disc-specific and linked to the disc mass and accretion rate.
* Internal photoevaporation parameters ($\dot M_{\mathrm{wind}}$, $L_X$): the total photoevaporative mass-loss rate, $\dot M_{\mathrm{wind}}$, can either be set by the user or computed from the stellar X luminosity $L_X$ as [Owen et al., 2012]
\begin{equation*}
\dot M_{\mathrm{wind}} = 6.25 \times 10^{-9} \times \left( \frac{M_{\star}}{M_{\odot}} \right)^{-0.068} \left( \frac{L_X}{10^{30} \mathrm{erg}\mathrm{s}^{-1}} \right)^{1.14} M_{\odot} \mathrm{yr}^{-1}.
\end{equation*}
The surface mass-loss profile $\dot \Sigma_{\mathrm{photo}}$ (Equation B2 in Owen et al., 2012) is then scaled so that $\int 2 \pi R \dot \Sigma_{\mathrm{photo}} \rm{d}R$ is equal to $\dot M_{\mathrm{wind}}$. Like for $M_{\mathrm{d}}$ and $\dot M$, $\dot M_{\rm{wind}}$ (or equivalently $L_X$) is extracted from a log-normal distribution whose mean is determined assuming power-law correlations with the stellar mass, while the normalisation at 1 M$_{\odot}$, the slope and the width of the distribution are free parameters.
* External photoevaporation parameters ($FUV$): FUV flux experienced by each disc, in units of G$_{0}$[G$_{0}$ stands for the Habing unit [Habing], 1968], the flux integral over the range of wavelengths [912 - 2400] $\rm{\mathring{A}}$ weighted by the average value in the solar neighbourhood (1.6$\times 10^{-3}$ erg s$^{-1}$ cm$^{-2}$).. This parameter can be set to any value accessible in the FRIEDv2 grid of mass loss rate [Haworth et al., 2023], spanning from 1 to $10^{5}$ G$_{0}$.
§.§ Solution algorithm
After generating the initial population of YSOs as described above, proceeds to evolve it by integrating the master equation (<ref>). Our solution algorithm employs an operator splitting method: the original equation is separated into different parts over a time step, and the solution to each part is computed separately. Then, all the solutions are combined together to form a solution to the original equation. We split Equation (<ref>) into five different pieces, related to viscosity, wind-driven accretion onto the central star, wind-driven mass loss, internal and external photoevaporation respectively. Furthermore, includes the possibility to trace the dust evolution in the disc, which is split in radial drift and dust diffusion. In the following, we describe the solution algorithm for each process.
* Viscous accretion: the standard viscous solver is based on the freely available code by [Booth et al., 2017]. We assume a radial temperature profile $T \propto R^{-1/2}$, which results in $c_{s} \propto R^{-1/4}$ and $H/R \propto R^{1/4}$. Note that this implies $\nu \propto R$ (i.e., $\gamma = 1$), which will be the case from now on. We assume $H/R = 1/30$ at 1 AU and a mean molecular weight of $2.4$. We refer to the original paper for details on the algorithm.
* Wind-driven accretion: the second term in Equation (<ref>) is effectively an advection term. The general form of the advection equation for a quantity $q$ with velocity $v$ is $\partial_t q(x, t) + v \partial_x q(x, t) = 0$; in the case of wind-driven accretion, the advected quantity is $R \Sigma$, while the advection (inwards) velocity is given by $v_{\mathrm{DW}} = (3 \alpha_{\mathrm{DW}} H c_s)/2R$. We solve the advection equation with an explicit upwind algorithm (used also for dust radial drift).
* Wind-driven mass loss: the mass loss term (third in Equation <ref>) does not involve any partial derivative, and therefore is simply integrated in time multiplying by the time step.
* Internal photoevaporation: effectively, internal photoevaporation (implemented through the model of Owen et al., 2012) is another mass loss term - therefore, as above, its contribution is computed with a simple multiplication by the time step. Once the accretion rate of the disc drops below the photoevaporative mass loss rate, a gap opens in the disc at the radius of influence of photoevaporation: in the model of [Owen et al., 2012], the prescription changes depending on the radial location in the disc, with respect to the gap itself. Later, the gap continues to widen; when it eventually becomes larger than the disc, we stop the evolution and consider the disc as dispersed.
* External photoevaporation: for a given stellar mass and FUV flux experienced by the disc, the mass loss rate arising from external photoevaporation is obtained, at each radial position, from a bi-linear interpolation of the FRIEDv2 grid [Haworth et al., 2023] using the disc surface density at each radial cell. The outside-in depletion of material is implemented following the numerical approach of Sellek et al., 2020: we define the truncation radius, $R_{\mathrm{t}}$, as the position in the disc corresponding to the maximum photoevaporation rate (which is related to the optically thin/thick transition of the wind), and we remove material from each grid cell at $R>R_{\mathrm{t}}$ weighting on the total mass outside this radius. The mass loss attributed to the cell $i$ can be written as:
\begin{equation}
\dot{M}_{\mathrm{ext},i} = \dot{M}_{\mathrm{tot}} \frac{M_{i}}{M(R>R_{\mathrm{t}})},
\end{equation}
where $M_{i}$ is the mass contained in the cell $i$, and $\dot{M}_{\mathrm{tot}}$ is the total mass loss rate outside the truncation radius.
* Dust evolution[As the dust evolution module was forked from Richard Booth's repository, users of who wish to use dust in their work ought to cite [Booth] et al., 2017] together with this paper.: based on the two populations model by [Birnstiel et al., 2012] and the implementation of [Booth et al., 2017]. We consider the dust grain distribution to be described by two representative sizes, a constant monomer size and a time-dependent larger size, which can grow up to the limit imposed by the fragmentation and radial drift barriers. We evolve the dust fraction of both sizes following [Laibe & Price, 2014], and also include a diffusive term: the diffusion comes from the coupling with the turbulent gas, which has the effect of mixing the dust grains, counteracting gradients in concentration [Birnstiel et al., 2010]. The dust-gas relative velocities are computed following [Tanaka et al., 2005] and include feedback on the gas component. We refer to [Booth et al., 2017] for details on the numerical implementation. Dust evolution is included in the release of , however the scientific results presented in this work are based on gas simulations only.
The separate pieces of Equation (<ref>) must be solved over the same time step to be joined in a coherent solution. We calculate the time step for each process imposing the Courant-Friedrichs-Lewy (CFL) condition. The CFL condition reads $\Delta t = C \mathrm{ min}(\Delta x / v)$ and ensures that, within one time step $\Delta t$, the material moving at velocity $v$ does not flow further than one grid spacing $\Delta x$. The Courant number $C$ must be positive and smaller than 1, with $C = 1$ corresponding to the maximum allowed timestep to keep the algorithm stable. In our implementation, we pick $C = 0.5$. We use zero gradients boundary conditions, setting the value of the first and last cell in our grid to that of the second and second to last. We solve the equation on a radial grid of $10^3$ points with power-law spacing and exponent $1/2$, extending from $3 \times 10^{-3}$ au to $10^4$ au. From the physical point of view, this choice corresponds to assuming boundary layer accretion (see., e.g., Popham et al., 1993, Kley & Lin, 1996) - however the difference from magnetic truncation accretion is negligible beyond $\sim 10^{-3}$ au.
After each process has been solved separately, all the pieces are put back together to compute the new surface density, from which the integrated disc quantities are then calculated. As each disc evolves independently of the others in the population, the solver can easily be run in parallel.
§.§ User interface and output
The user interface of is a .json parameters file which includes all the user-dependent parameters. Aside from the number of objects in the population and the evolutionary mechanism, the user can set the chosen IMF (either Kroupa, 2001, single stellar mass, or custom input file), the distributions to draw the disc parameters from (single value, flat, normal, log-normal), as well as the normalisation, slope and spread of the correlations, the times at which snapshots are generated, and the initial dust-to-gas ratio. Furthermore, the user can determine a limit disc mass: this is to be intended as a threshold below which the disc would not be detectable anymore, and is therefore considered dispersed in the simulation as well. When a disc is dispersed, the corresponding YSO turns into a Class III object consisting of the central star only.
The output of is a .hdf5 file containing the properties of both the disc and the star at all chosen time steps for each YSO in the population: this includes the stellar mass, luminosity, temperature, disc mass, accretion rate, accretion timescale, gas and dust surface density, disc radius, dust grain sizes. The output can be easily read and analysed with the dedicated library , released with the code. For a more in-depth description of the parameters, the user interface and the output, we refer to the documentation.
§ THE TIME EVOLUTION OF THE CORRELATIONS BETWEEN DISC PROPERTIES AND STELLAR MASS UNDER DIFFERENT ACCRETION DRIVERS: ANALYTICAL CONSIDERATIONS
H]P1.5cm P5.5cm
0pt2.5ex Parameter Description
0pt2.5ex $\lambda$ Magnetic lever arm parameter
$\psi$ Wind-to-turbulent $\alpha$ ratio
$\omega$ Power-law index of $\alpha_{\mathrm{DW}}$ with $\Sigma_{\mathrm{c}}$
$\xi$ Mass ejection index
$f_{\mathrm{M}, 0}$ Initial mass ejection-to-accretion ratio
0pt2.5ex $\lambda_{\mathrm{m}}$ Power-law index of $M_{\mathrm{d}}$ with $M_{\star}$ (Eq. <ref>)
$\lambda_{\mathrm{acc}}$ Power-law index of $\dot M$ with $M_{\star}$ (Eq. <ref>)
$\zeta$ Power-law index of $R_{\mathrm{d}}$ with $M_{\star}$ (Eq. <ref>)
$\beta$ Power-law index of $H/R$ with $M_{\star}$
$\mu$ Power-law index of $t_{\mathrm{acc}}$ with $M_{\star}$
$\delta$ $\lambda_{\mathrm{m}} - \lambda_{\mathrm{acc}}$
Summary and description of the parameters used throughout the paper: the top block refers to the MHD parameters defined in [Tabone et al., 2022], while the bottom block shows the slopes of the correlations between the disc properties and the stellar mass.
The existence of power-law correlations between the main integrated disc properties - namely the disc mass and stellar accretion rate - and the stellar mass is supported by various surveys across a number of different star-forming regions (e.g., on $M_{\mathrm{d}}-M_{\star}$: Ansdell et al., 2016, Barenfeld et al., 2016, Pascucci et al., 2016, Testi et al., 2016; on $\dot M - M_{\star}$: Muzerolle et al., 2003, Natta et al., 2004, Mohanty et al., 2005, Alcalá et al., 2014, Manara et al., 2016, Alcalá et al., 2017, Manara et al., 2017, Venuti et al., 2019, Manara et al., 2020, Testi et al., 2022). However, whether the establishment and subsequent evolution of said correlations is a product of the secular evolution of discs, or rather an imprint of the initial conditions, remains unclear. [Somigliana et al., 2022] explored a combination of both possibilities, assuming the correlations to hold as initial conditions and investigating the impact of purely viscous evolution; we briefly recall their main theoretical results (Section <ref>) and extend their analysis to the hybrid (Section <ref>) and purely wind-driven (Section <ref>) models from the theoretical perspective. We note that the results presented in this work are based on gas simulations.
Following [Somigliana et al., 2022], we assume power-law correlations between the disc properties and the stellar mass to hold as initial conditions. We focus on the disc mass $M_{\mathrm{d}}$, the stellar accretion rate $\dot M$ and the disc radius $R_{\mathrm{d}}$, and label the slopes of their correlations with the stellar mass $\lambda_{\mathrm{m}}$, $\lambda_{\mathrm{acc}}$, and $\zeta$ respectively. The initial correlations are set as follows:
\begin{equation}
\begin{cases}
M_\mathrm{d} (0) \propto {M_{\star}}^{\lambda_{\mathrm{m}, 0}}, \\
\dot{M} (0) \propto {M_{\star}}^{\lambda_{\mathrm{acc}, 0}}, \\
R_{\mathrm{d}} (0) \propto {M_{\star}}^{\zeta_0}.
\end{cases}
\label{eq:assumptions}
\end{equation}
To analyse the impact of secular evolution on this set of initial conditions, we analytically determine the evolved expressions for $M_\mathrm{d} (t)$, $\dot{M} (t)$ and $R_{\mathrm{d}} (t)$ in the three different scenarios. Table <ref> summarises the parameters introduced in this Section.
§.§ Purely viscous model
The full calculations for the purely viscous case can be found in [Somigliana et al., 2022]. Here, we briefly remind the main assumptions and results, and we refer to the original paper for a detailed discussion.
As mentioned in Section <ref>, assuming a power-law scaling of viscosity with the disc radius ($\nu \propto R^{\gamma}$) allows to solve the viscous evolution equation analytically, recovering the so-called self-similar solution [Lynden-Bell & Pringle, 1974]. In this case, the disc mass and accretion rate read
\begin{equation}
M_\mathrm{d} (t) = M_{\mathrm{d}, 0} \left( 1 + \frac{t}{t_{\nu}} \right)^{1 - \eta},
\label{eq:discmass_ss}
\end{equation}
\begin{equation}
\dot M (t) = (\eta - 1) \frac{M_{\mathrm{d}, 0}}{t_{\nu}} \left( 1 + \frac{t}{t_{\nu}} \right)^{- \eta},
\label{eq:accrate_ss}
\end{equation}
where $\eta = (5/2 - \gamma)/(2 - \gamma)$ and the viscous timescale $t_{\nu} = {R_c}^2/[3(2-\gamma)^2 \nu(R = R_c)]$ at the characteristic radius $R_c$. Because $\dot M_0 \propto M_{\mathrm{d}, 0}/t_{\nu, 0}$, a power-law scaling of $M_{\mathrm{d}, 0}$ and $\dot M_0$ with the stellar mass implies the viscous timescale $t_{\nu, 0}$ to scale as a power-law with the stellar mass as well, which we define as $t_{\nu, 0} \propto {M_{\star}}^{\mu_0}$; furthermore, this scaling corresponds to the difference between the scaling of the disc mass with the stellar mass and of the accretion rate with the stellar mass. Defining $\delta_0 = \lambda_{\mathrm{m}, 0} - \lambda_{\mathrm{acc}, 0}$, in this case $\mu_0 = \delta_0$[The definition of $\delta_0$ might seem redundant at this stage, but it will become important in the following discussion.]; therefore, the scaling of $t_{\nu, 0}$ with the stellar mass is determined by the relative values of $\lambda_{\mathrm{m}, 0}$ and $\lambda_{\mathrm{acc}, 0}$. The main results of [Somigliana et al., 2022] are that (i) viscous evolution maintains the power-law shape of the correlations between the stellar mass and the disc parameters, however (ii) the slope of said correlations may evolve with time, depending on the initial conditions. This is because in a purely viscous framework, the $M_{\mathrm{d}}-\dot M$ correlation is bound to reach a linear correlation with slope unity [Lodato et al., 2017, Rosotti et al., 2017], which implies the two quantities to have the same dependence on the stellar mass, as $M_{\mathrm{d}} / \dot M \propto {M_{\star}}^{\lambda_{\mathrm{m}} - \lambda_{\mathrm{acc}}}$. Therefore, $\lambda_{\mathrm{m}}$ and $\lambda_{\mathrm{acc}}$ must eventually reach the same value, determined by the initial conditions as
\begin{equation}
\lambda_{\mathrm{m, evo}} = \lambda_{\mathrm{acc, evo}} = \frac{3 \lambda_{\mathrm{m}, 0} - \lambda_{\mathrm{acc}, 0}}{2}.
\label{eq:lambdam=lambdaac_evolved}
\end{equation}
Depending on the sign of $\delta_0 = \lambda_{\mathrm{m}, 0} - \lambda_{\mathrm{acc}, 0}$, the initial slopes can either
* steepen, i.e. $\lambda_{\mathrm{m, evo}} > \lambda_{\mathrm{m}, 0}$, if $\delta_0 > 0$ (implying also
$\lambda_{\mathrm{acc, evo}} > \lambda_{\mathrm{acc}, 0}$);
* flatten, i.e. $\lambda_{\mathrm{m, evo}} < \lambda_{\mathrm{m}, 0}$, if $\delta_0 < 0$ (implying also
$\lambda_{\mathrm{acc, evo}} < \lambda_{\mathrm{acc}, 0}$);
* remain constant, i.e. $\lambda_{\mathrm{m, evo}} = \lambda_{\mathrm{m}, 0}$, if $\delta_0 = 0$ (implying
also $\lambda_{\mathrm{acc, evo}} = \lambda_{\mathrm{acc}, 0}$).
Because in the viscous case $\delta_0 = \mu_0$, where we note again that $\mu_0$ is the slope of the correlation between the viscous timescale and the stellar mass ($t_{\nu, 0} \propto {M_{\star}}^{\mu_0}$), we can also interpret these scenarios from the viscous timescale perspective. If $\mu_0 > 0$, meaning that the viscous timescale increases with the stellar mass, discs around less massive stars will have shorter viscous timescales, which leads to a faster evolution, compared to discs around more massive stars, which will in turn have longer viscous timescales. This uneven evolution across the stellar mass spectrum leads to a steepening of the linear correlation, as is visualised by [Somigliana et al., 2022] in Figure 1. The same reasoning, but with opposite/constant trend, applies to the other two scenarios.
§.§ Hybrid model - omega = 0
In the hybrid viscous and MHD winds model, the general analytical solution by [Tabone et al., 2022] gives
\begin{equation}
M_\mathrm{d} (t) = M_0 \left( 1 + \frac{t}{(1 + \psi) t_{\mathrm{acc}, 0}} \right)^{-(\psi + 2 \xi + 1)/2},
\label{eq:mdisc_mhd_omegazero}
\end{equation}
\begin{equation}
\dot{M} (t) = \dot{M}_0 \left( 1 + \frac{t}{(1 + \psi) t_{\mathrm{acc}, 0}} \right)^{-(\psi + 4 \xi + 3)/2},
\label{eq:mdot_mhd_omegazero}
\end{equation}
where $\dot{M}_0$ is defined as
\begin{equation}
\dot{M}_0 = \frac{\psi + 1 + 2 \xi}{\psi + 1} \frac{M_0}{2 t_{\mathrm{acc}, 0}} \frac{1}{(1+f_{\mathrm{M}_0})};
\label{eq:mdot0_mhd_omegazero}
\end{equation}
Time evolution of the slopes of the $M_{\mathrm{d}}-M_{\star}$ and $\dot M-M_{\star}$ correlations, $\lambda_{\mathrm{m}}$ and $\lambda_{\mathrm{acc}}$ (blue and orange solid line respectively) in the hybrid scenario with $\alpha_{\mathrm{SS}} = \alpha_{\mathrm{DW}} = 10^{-3}$ ($\psi = 1$), $\lambda = 3$, $\beta = 0.5$, resulting in $\xi = 0.11$. The dashed lines represent the expected evolved value of both slopes, as in Equation (<ref>). For comparison, we include the viscous evolution as well, represented by grey solid (actual evolution) and dashed (expected evolved value) lines (see Somigliana et al., 2022 for a detailed discussion). The three panels show different values of $\mu_0$, slope of the $t_{\mathrm{acc}, 0} - M_{\star}$ correlation, which is directly linked to the difference between $\lambda_{\mathrm{m}, 0}$ and $\lambda_{\mathrm{acc}, 0}$ (see text for details): we expect the slopes to (i) decrease if $\mu_0 < 0$ (left panel), (ii) remain constant if $\mu_0 = 0$ (central panel), and (iii) increase if $\mu_0 > 0$ (right panel). Contrary to the viscous case, the slopes are not expected to reach the same value anymore, but rather settle to a constant difference of $- \xi/2$. This difference is always negative, meaning that the evolved $\dot M - M_{\star}$ correlation is always steeper than that of $M_{\mathrm{d}} - M_{\star}$ (explaining the lines crossing in the right panel).
in this notation, $\psi = \alpha_{\mathrm{DW}}/\alpha_{\mathrm{SS}}$ represents the relative strength of MHD winds and viscosity,
\begin{equation*}
\xi = \frac{1}{4} (\psi + 1) \left[ \sqrt{1 + \frac{4\psi}{(\lambda-1)(\psi + 1)^2}} - 1 \right]
\end{equation*}
is the mass ejection index quantifying the local mass loss rate to the local accretion rate, and $f_{\mathrm{M}, 0} = (R_{c, 0}/R_{in})^{\xi} - 1$ the dimensionless mass ejection-to-accretion ratio (with $R_{in}$ initial disc radius). If we neglect the MHD-driven mass loss ($\psi \ll 1$ and $\xi \ll 1$, which correspond to $f_{\mathrm{M}, 0} \ll 1$ as well), Equations (<ref>) and (<ref>) reduce to the viscous case; on the other hand, if mass loss is included, it depends on the radial extent of the disc through $f_{\mathrm{M}, 0} + 1$ - which has an impact on the initial accretion rate (Equation <ref>). Because the accretion timescale $t_{\mathrm{acc}}$ is a generalisation of $t_{\nu}$ in the MHD winds framework, the dependence of the two timescales on the stellar mass is exactly equivalent, and we will keep the same notation as above: $t_{\mathrm{acc}} \propto {M_{\star}}^{\mu}$. However, as mentioned above $\dot M_0$ depends on the stellar mass not only through $M_0$ and $t_{\mathrm{acc}, 0}$ as in the viscous case, but also through $f_{\mathrm{M},0} + 1$. As $f_{\mathrm{M},0} + 1 \propto {R_{c, 0}}^{\xi}$, and $R_{c, 0} \propto {M_{\star}}^{\zeta_0}$, the additional dependence will have a slope of $\zeta_0 \xi$. Therefore, in the MHD winds scenario we can link $\delta_0$ with $\mu_0$ as $\delta_0 = \mu_0 + \zeta_0 \xi$. The practical meaning of this difference is that, while in the viscous scenario the difference in slope between the two correlations depends only on the scaling of the viscous timescale with the stellar mass, in the hybrid scenario it depends also on the scaling between the disc radius and the stellar mass. It is important to note that $\xi$ is a small number, typically of the order of $\sim 0.1$, therefore the difference between the viscous and hybrid case is not particularly prominent. For evolved populations, the disc mass and accretion rate read [Tabone et al., 2022]
\begin{equation}
M_\mathrm{d} (t \gg t_{\mathrm{acc}}) \sim M_0 \left(\frac{t}{t_{\mathrm{acc}, 0}} \right)^{-(\psi + 2 \xi + 1)/2},
\label{eq:mdisc_mhd_omegazero_evo}
\end{equation}
\begin{equation}
\dot{M} (t \gg t_{\mathrm{acc}}) \sim \dot{M}_0 \left(\frac{t}{t_{\mathrm{acc}, 0}} \right)^{-(\psi + 4 \xi + 3)/2};
\label{eq:mdot_mhd_omegazero_evo}
\end{equation}
this brings the evolved slopes $\lambda_{\mathrm{m, evo}}$ and $\lambda_{\mathrm{acc, evo}}$ to
\begin{equation}
\begin{split}
\lambda_{\mathrm{m, evo}} = \lambda_{\mathrm{m}, 0} + \frac{1}{2} \mu_0 (\psi + 2\xi + 1), \\
\lambda_{\mathrm{acc, evo}} = \lambda_{\mathrm{acc}, 0} + \frac{1}{2} \mu_0 (\psi + 4\xi + 3),
\end{split}
\label{eq:lambda_evo_mhd}
\end{equation}
Time evolution of $M_{\mathrm{d}} - M_{\star}$ (left) and $\dot M - M_{\star}$ (right) in the pure wind model ($\omega \neq 0$). This plot is obtained with disc population synthesis modelling, without any spread in the initial conditions. Each dot represents a disc in the population at different ages as shown in the colour bar. The initial power-law correlation, shown in light blue, is lost as early as $\sim 1$ Myr (corresponding to $\sim 2 <t_{\mathrm{acc}, 0}>$ with these parameters) due to a downward bending corresponding to lower stellar masses. In this simulation, we have used $N = 100$ discs, $\alpha_{\mathrm{DW}} = 10^{-3}$, $\lambda = 3$, $\omega = 0.25$, $\lambda_{\mathrm{m}, 0} = 2.1$, $\lambda_{\mathrm{acc}, 0} = 1.5$. The set of MHD parameters is based on Tabone et al., 2022.
which reduces to Equation (<ref>) in the viscous case ($\psi \ll 1$, $\xi \ll 1$). Like viscosity, a hybrid secular evolution maintains the power-law shape of the correlation; moreover, Equation (<ref>) provides a theoretical prediction for the evolved slopes: they can steepen, flatten or remain the same as the initial conditions, depending on the involved parameters. As the terms in parentheses in Equation (<ref>) are sums of positive values, the sign of the evolved slopes depends on the sign of $\mu_0$ like in the viscous case. However, there is a difference from the viscous case: as in the hybrid scenario $\mu_0$ and $\delta_0$ do not coincide anymore, a constraint on the value of $\mu_0$ is translated into a constraint on $\delta_0 - \zeta_0 \xi$. In particular, the slopes will increase if $\delta_0 > \zeta_0 \xi$ (corresponding to $\mu_0 > 0$), whereas if $\delta_0 < \zeta_0 \xi$ (corresponding to $\mu_0 < 0$), the slopes will decrease; and finally, if $\delta_0 = \zeta_0 \xi$ (corresponding to $\mu_0 = 0$) the slopes will remain constant in time. Another difference from the viscous scenario is that $\lambda_{\mathrm{m, evo}}$ and $\lambda_{\mathrm{acc, evo}}$ are not expected to reach the same value anymore. The limit difference is given by $\delta_{\mathrm{evo}} = \lambda_{\mathrm{m, evo}} - \lambda_{\mathrm{acc, evo}}$: substituting the values from Equation (<ref>) one finds $\delta_{\mathrm{evo}} = \delta_0 - \mu_0 (\xi + 1)$, which can be further reduced to $\delta_{\mathrm{evo}} = \xi (2 \beta + \frac{1}{2})$ by using the definition of $\mu_0$ as the slope of the $t_{\mathrm{acc}, 0} - M_{\star}$ correlation. In this expression, $\beta$ is the slope of the correlation between the disc aspect ratio and the stellar mass, which comes from definition of the accretion timescale; with a standard $\beta = -1/2$ (a reasonable approximation of the value derived by radiative transfer simulation, see e.g., Sinclair et al., 2020), we obtain $\delta_{\mathrm{evo}} = -\xi/2$. As $\xi$ is positive by definition, in the hybrid scenario $\delta_{\mathrm{evo}}$ is always negative, meaning that in an evolved population, the correlation between the accretion rate and the stellar mass will necessarily be steeper than that between the disc mass and the stellar mass. However, we stress once more that $\xi$ is a small number and therefore the predicted difference is also small. Figure <ref> shows the evolution of the slopes from simulations (with no spread in the initial conditions) for the hybrid model (coloured) compared with the viscous case (grey), which matches the theoretical expectations discussed above. The list of parameters used in the simulations is available in Table <ref>.
Summarising, both the hybrid and viscous secular evolution preserve the power-law shape of the correlations between the disc properties and the stellar mass. The main difference is that the hybrid model does not predict the slopes of the $M_{\mathrm{d}} - M_{\star}$ and $\dot M - M_{\star}$ correlations to reach the same limit value (unlike the viscous case). The predicted difference in the evolved slopes is given by $\delta_{\mathrm{evo}} = -\xi/2$ (of the order of 0.1). The current observational uncertainties on $\delta_{\mathrm{evo}}$ range from 0.4 to 0.8 (Testi et al., 2022, see Table <ref>), 4 to 8 times larger than the predicted difference, making it not observable at this stage.
§.§ Pure wind - $\omega \neq 0$
The solution to the pure wind model (i.e., time-dependent $\alpha_{\mathrm{DW}}$ through $\alpha_{\mathrm{DW}}(t) \propto \Sigma_{\mathrm{c}}(t)^{- \omega}$) by [Tabone et al., 2022] gives
\begin{equation}
M_\mathrm{d} (t) = M_0 \left( 1 - \frac{\omega}{2 t_{\mathrm{acc}, 0}} t \right)^{1/\omega},
\label{eq:mdisc_mhd_omeganotzero}
\end{equation}
\begin{equation}
\dot{M} (t) = \frac{M_0}{2 t_{\mathrm{acc}, 0} (1+f_{\mathrm{M}, 0})} \left( 1 - \frac{\omega}{2 t_{\mathrm{acc}, 0}} t \right)^{-1+1/\omega};
\label{eq:mdot_mhd_omeganotzero}
\end{equation}
in this case, the functional form of Equation (<ref>) and (<ref>) does not allow us to derive a simple analytical expression for $M_\mathrm{d} (t \gg t_{\mathrm{acc}})$ and $\dot M (t \gg t_{\mathrm{acc}})$. Therefore, to explore the evolution of the correlations in the pure wind case, we fully rely on simulations. In order to account for the impact of secular evolution only, we input perfect correlations between the disc properties and the stellar mass - that is, we do not include any spread in the initial conditions. Figure <ref> shows the time evolution of $M_{\mathrm{d}}$ (left panel) and $\dot M$ (right panel) as a function of the stellar mass, from younger (darker) to older (lighter) populations. The input power-law correlation (light blue) corresponds to a line in the logarithmic plane; however, as early as $\sim 1$ Myr (corresponding to $\sim 2<t_{\mathrm{acc}, 0}>$ for this simulation), the input correlation starts to bend downwards at lower stellar masses. This behaviour reveals a significantly different trend from the viscous and hybrid model: in the pure wind scenario, the initial power-law shape of the correlations is not preserved by the secular evolution, but rather broken. In Figure <ref>, the faster evolution of discs around lower mass stars is the consequence of a positive $\mu_0$, implying a positive correlation between the stellar mass and the accretion timescale; a negative correlation between $M_{\star}$ and $t_{\mathrm{acc}, 0}$ (i.e., a negative $\mu_0$) would lead to faster evolution of discs around higher mass stars, causing the correlation to bend towards the other end of the stellar mass spectrum (see Figure <ref>). Another parameter that might impact the evolution of the correlation is the dispersal timescale, $t_{\mathrm{disp}}$; in the wind-driven case, $t_{\mathrm{disp}} \propto t_{\mathrm{acc}}$, therefore $t_{\mathrm{disp}}$ does not introduce any further dependence on the stellar mass.
§.§ Summary
In this Section we have discussed the time evolution of the correlations between the disc properties and the stellar mass from the theoretical point of view. We have performed analytical calculations for the viscous and hybrid scenario, finding that the power-law shape of the initial correlations is preserved during disc evolution. While the slopes are expected to evolve, increasing or decreasing depending on the initial conditions, the limit slopes for evolved populations differ in the two models. While viscosity requires the slope of the $M_{\mathrm{d}} - M_{\star}$ and $\dot M - M_{\star}$ correlations to reach the same value, in the hybrid scenario the two slopes always have different values; moreover, the evolved slope of $\dot M - M_{\star}$ is bound to be higher than that of $M_{\mathrm{d}} - M_{\star}$. Unfortunately however, the difference from the viscous model is a factor four too small to be observed within the current error bars. We have then confirmed our predictions running simulations without any spread in the initial conditions (Figure <ref>). In the pure wind case instead, the functional form of the solution to Equation (<ref>) does not allow for an analytical determination of the evolved slopes; therefore, we entirely relied on simulations. Our results show that, contrary to the viscous and hybrid case, wind-driven accretion does not maintain the power-law shape of the initial correlations: depending on the scaling of the accretion timescale with the stellar mass, the power-laws are broken because of a bending towards higher ($t_{\mathrm{acc}, 0}$ decreasing with $M_{\star}$) or lower ($t_{\mathrm{acc}, 0}$ increasing with $M_{\star}$) stellar masses.
§ POPULATION SYNTHESIS
In Section <ref> we have discussed analytical trends, and presented simulations with no spread to analyse the effect of secular disc evolution alone on the evolution of the slopes. In order to test our theoretical predictions against observational data, we need to account for both a spread in the initial conditions and disc dispersal mechanisms. In this Section, we discuss the impact of both factors on the $M_{\mathrm{d}} - M_{\star}$ and $\dot M - M_{\star}$ slopes and run realistic population synthesis simulations, to determine whether the model-dependent evolutionary features described in Section <ref> would be observable with the currently available data.
§.§ Effects of a spread in the initial conditions
Same as Figure <ref> with the addition of a spread in the initial correlations between the disc properties and the stellar mass ($\sigma_{M_{\mathrm{d}}}(0) = 0.65$ dex, $\sigma_{R_{\mathrm{d}}}(0) = 0.52$ dex). Despite the linear correlation being readily broken in theory (see Section <ref>), the scatter introduced by the spread in the initial conditions simulates the correlation also at more evolved ages. As an example, we show the fitted line at 5 Myr (the age of the oldest observed population) in both panels ($\log_{10}(M_{\mathrm{d}}/M_{\odot}) = 3.1 \log_{10}(M_{\star}/M_{\odot}) - 3.3$, $\log_{10}(\dot M/M_{\odot} \mathrm{yr}^{-1}) = 2.5 \log_{10}(M_{\star}/M_{\odot}) - 10.4$).
The introduction of an observationally-motivated spread in the initial conditions is crucial to produce realistic population synthesis model. In the purely viscous case, [Somigliana et al., 2022] have shown how the spread does not significantly impact the evolution of either the $M_{\mathrm{d}} - M_{\star}$ or $\dot M - M_{\star}$ correlations; the shape of the curves (grey in Figure <ref>) is unaffected, except for their starting point, and the statistical fluctuation - determined as the interval between the 25th and 75th percentile out of 100 realisations of numerical simulations with the same initial conditions - is of the order $\sim 0.1$ for both slopes. This is a factor two less than the smallest observational uncertainty, and therefore does not produce a detectable difference in the predicted results.
Following [Somigliana et al., 2022], we set $\sigma_{M_{\mathrm{d}}}(0) = 0.65$ dex and $\sigma_{R}(0) = 0.52$ dex (determined from Ansdell et al., 2017 and Testi et al., 2022) for the log-normal distributions of $M_{\mathrm{d}}(0)$ and $R_{\mathrm{d}}(0)$ in the hybrid scenario (with $\alpha_{\mathrm{SS}} = \alpha_{\mathrm{DW}} = 10^{-3}$ hence $\psi =1$, $\lambda = 3$, $\omega = 0$). We find that, just like in the purely viscous case, a spread in the initial conditions only shifts the starting point of the curves (coloured in Figure <ref>) and does not have any significant effect on the shape of the evolution of the slopes (see Figure 5-6-7 in Somigliana et al., 2022). The statistical fluctuation for both slopes is again of the order of 0.1, and therefore below the observational error and not impacting our predictions.
On the other hand, wind-driven models with increasing $\alpha_{\mathrm{DW}}$ in time and a spread in the initial conditions (Figure <ref>) behave quite differently from the theoretical expectation discussed in Section <ref>. As the spread introduces a stochastic component, the discs will have higher or lower masses and accretion rates with equal probability; the practical result for the initial correlations is that the bending towards lower stellar masses (approximately $\log_{10}(M_{\star}/M_{\odot}) < 0.5$ in Figure <ref>) is lost to the stochastic displacement of the discs in the $M_{\mathrm{d}} - M_{\star}$ and $\dot M - M_{\star}$ planes. Actually, the resulting distribution of discs both in the $M_{\mathrm{d}} - M_{\star}$ and $\dot M - M_{\star}$ log-log plane does simulate a linear correlation; this implies that, while the stellar mass and the disc properties should not exhibit a linear correlation already after a few $t_{\mathrm{acc}}$, the presence of a spread mimics such correlation, making the wind-driven scenario indistinguishable from the viscous and hybrid ones. The one feature that remains observable, despite the presence of a spread, is the removal of discs around more or less massive stars - depending on the value of $\mu_0$, as discussed in Section <ref>; in the simulation shown in Figure <ref> we have set $\mu_0 > 0$, implying that discs around less massive stars evolve more rapidly and are therefore more readily dispersed, as is visualised by the lack of sources around lower stellar masses at evolved ages.
Summarising, an initial power-law correlation between the disc properties and the stellar mass would keep its power-law shape under wind-driven evolution, like in the viscous or hybrid case; however, the interpretation of the observed correlations is different depending on the theoretical framework. While the viscous and hybrid models preserve an initially established correlation, making the characteristic evolution of the slopes $\lambda_{\mathrm{m}}$ and $\lambda_{\mathrm{acc}}$ a tracer of disc evolution itself, the apparent correlation observed in wind-driven populations is merely a signature of the initial spread, rather than the evolutionary mechanism at play.
Time evolution of the slope of the $M_{\mathrm{d}} - M_{\star}$ correlation for 15 statistical realisations in the purely viscous (left, $\alpha_{\mathrm{SS}} = 10^{-3}$), photoevaporative (centre, $\dot M_w = 4 \times 10^{-10} \mathrm{M}_{\odot}/\mathrm{yr}$, $\alpha_{\mathrm{SS}} = 10^{-3}$) and wind-driven (right, $\omega = 0.25$, $\lambda = 3$, $\alpha_{\mathrm{DW}} = 10^{-3}$) model. The dashed lines show the median evolution, while the shaded area represents the interval between the 25th and the 75th percentiles. The three rows show different sample sizes, increasing from top to bottom. The initial size of each population was chosen to obtain a certain number of discs at 5 Myr (the age of the oldest observed population, Upper Sco) with the different disc fractions. In the top row, we match the current size of the Upper Sco sample($\sim 30$ objects, Testi et al., 2022) with both accretion rate and disc mass measurements; the middle row shows double the current sample size ($\sim 60$ objects), while the bottom row assumes a complete sample ($\sim 140$ objects). While the viscous model produces a remarkably similar evolution for all simulations, the latter two show stochastic oscillations from one realisation to another, suggesting that disc dispersal impacts the observed slope more than the evolutionary model does - at least with the currently available sample sizes; increasing the number of sources significantly mitigates the oscillations. The slope of the $\dot M - M_{\star}$ correlation behaves the same way.
§.§ Accounting for disc dispersal: internal photoevaporation
Out of the three theoretical scenarios discussed so far, wind-driven evolution is the only one that manages to reproduce the disc and accretion fraction (as measured by Hernández et al., 2007 and Fedele et al., 2010 respectively) - and therefore, the only one whose predictions can reasonably be compared with observations. Traditionally, the problem of disc dispersal in viscous populations is addressed by including internal photoevaporation (see e.g., Hollenbach et al., 1994, Clarke et al., 2001, Alexander et al., 2006, Alexander et al., 2006): in this Section, we discuss the impact of internal photoevaporation on the previously described expectations for the evolution of the slopes in the purely viscous scenario. As the lack of analytical solutions to the general equation (<ref>) does not allow for analytical arguments, we base the following discussion on physical considerations.
Internal photoevaporation is a threshold process, that kicks in after the accretion rate drops below the photoevaporative mass-loss rate [Clarke et al., 2001]. The moment where the effect of photoevaporation becomes non-negligible depends therefore on the initial accretion rate: assuming for simplicity a fixed photoevaporation rate for the whole population, as the accretion rate scales positively with the stellar mass we can expect discs around lower mass stars to show the effects of photoevaporation earlier. Moreover, given that the disc mass also scales positively with the stellar mass, said sources correspond to the less massive ones in the population. From these considerations, we can expect discs with lower initial mass to be the first ones to be affected by photoevaporation, causing a steepening of the $M_{\mathrm{d}} - M_{\star}$ and $\dot M - M_{\star}$ correlations. However, photoevaporation also disperses discs: removing sources from the population may alter the expected behaviour, hence the need to perform numerical simulations to understand the evolution of the correlations for a population of discs undergoing internal photoevaporation. Our simulations remove discs either because the photoevaporative gap becomes too large, or because the disc mass or accretion rate fall below a certain detectability threshold. As we mentioned above, the dispersal timescale $t_{\mathrm{disp}}$ might also play a role, if it has a different scaling with the stellar mass with respect to the accretion timescale ($t_{\nu}$ in the viscous case). Our numerical implementation of internal photoevaporation follows [Owen et al., 2010], and we assume a mass-loss rate of $4 \times 10^{-10}$ M$_{\odot}$ yr$^{-1}$ - which allows us to reproduce the observed disc fraction for the set of parameters of our simulation - for all discs in the population; therefore, no further $M_{\star}$ dependence is introduced, and the $t_{\nu} - M_{\star}$ scaling is the only one that matters. We stress that a stellar mass-dependent photoevaporative rate is expected [Picogna et al., 2021]: we explore the influence of such dependence on disc observables in an upcoming work (Malanga et al. in prep.). We discuss the results of our simulations in the following Section.
§.§ What are the slopes tracing?
Comparison of the time evolution of $\lambda_{\mathrm{m}}$ (top) and $\lambda_{\mathrm{acc}}$ (bottom) between the viscous+photoevaporative and wind-driven case including measured slopes from four star-forming regions. To account for statistical fluctuations, each simulation combines 100 realisations of the same initial conditions: the lines show the median evolution, while the shaded area represent the interval between the 25th and 75th percentiles. The simulations in the three columns differ for the initial number of discs, determined to obtain a specific sample size at 5 Myr - currently available sample (left), double the currently available sample (centre) and the complete sample (right). Observed slopes from [Testi et al., 2022].
Figure <ref> shows the time evolution of the $M_{\mathrm{d}} - M_{\star}$ slope for 15 realisations with the same initial conditions for the viscous plus photoevaporative (central panel) and wind-driven (right panel) models, both reproducing the observed disc and accretion fractions, compared to the purely viscous scenario (left panel). The initial number of discs in the populations was determined to recover a certain number of objects at 5 Myr (increasing from top to bottom), and varies in the different simulations, as the decline (or lack thereof) of the disc fraction is model-dependent (see Figure 6 of Somigliana et al., 2023).
The number of objects in the simulation displayed in the first row was set to obtain $\sim$ 30 discs at 5 Myr, corresponding to the currently available sample size in Upper Sco, the oldest observed star-forming region. In the left panel, we see how the evolution of the slope in the purely viscous model is not significantly affected by a spread in the initial conditions: the single realisations resemble each other remarkably well, the only difference being the starting point of the curve (as found by Somigliana et al., 2022). On the other hand, the photoevaporative and wind-driven models have a dissimilar behaviour: each realisation can deviate substantially from the others, as we can particularly notice by the location, amplitude and direction of the bumps. The key difference between these models and the purely viscous case is disc dispersal: the stochastic nature of the slope evolution suggests that it does not trace the underlying secular disc evolution, like in the viscous scenario, but rather carry the signatures of disc dispersal itself - making it impossible to use the evolution of the slopes as a proxy for accretion mechanisms. There are two main factor that play a role in this context:
* Initial conditions and spread in the correlations. The exact evolution of the slopes will depend on the initial conditions, both of the disc mass-stellar mass correlations themselves and of the population-wide parameters. Furthermore, the removal of discs from the population would not impact the results of the fitting procedure only if there was perfect correlations between the disc parameters and the stellar mass; with a spread in the initial correlations, on the other hand, the results may differ depending on which discs in the population are dispersed;
* Small number statistics. Depending on the initial number of objects, disc removal can lead to small samples - so small that it might lead to low number statistics issues. This is the case for the top row of Figure <ref>, where the number of objects at 10 Myr is of the order of 10 or lower.
In this work, we have used one specific set of parameters (summarised in Table <ref>), determined following [Somigliana et al., 2022] (viscous model) and [Tabone et al., 2022] (hybrid and wind-driven), and we leave a deeper exploration of the parameters space to a future work. While the exact shape of the slope evolution, and therefore the accretion model signature, might depend on the initial conditions, the top panel of Figure <ref> shows that with the current sample sizes the noise dominates over the physical evolution. However, the currently available sample of the oldest star-forming region (Upper Sco) with both disc masses (derived from the millimetre flux, Barenfeld et al., 2016) and accretion rate [Manara et al., 2020] estimates is highly incomplete; it is therefore worth investigating whether a higher level of completeness would help reducing the entity of the oscillations, allowing to disentangle between the different evolutionary models.
The central and bottom rows of Figure <ref> show how a larger sample would impact the oscillations of the slope evolution. The simulations in the middle row are performed imposing a double sample size at 5 Myr with respect to the current one ($\sim 60$ discs), while in the bottom row we assume to have the complete Upper Sco sample, totalling $\sim 140$ discs (Carpenter et al. in prep.). We remind that we focus on Upper Sco as the oldest observed star-forming region, which makes it the most affected by disc dispersal.
As expected, statistical significance increases with a larger sample, leading to a decreased impact of the oscillations on the global slope evolution; with the complete sample, in particular, we can reduce the spread in the evolution by a factor of $\sim 2$ compared to the current available data. This argument confirms the importance of larger sample sizes in discriminating between the viscous and wind-driven models, as already suggested by [Alexander et al., 2023] in the context of the accretion rates distribution.
As we mentioned in Section <ref>, our simulations consider discs as dispersed if their masses or accretion rates fall below the imposed detectability threshold of $10^{-12}$ M$_{\odot}$ yr$^{-1}$. We have also included a threshold in accretion rates in post-processing to account for non-accreting objects. From the observational point of view, this latter selection depends on both the instrumental sensitivity and the definition of disc itself: how Class II objects are defined, and in turn how Class III sources are removed from the observed samples, impacts the resulting slope. Summarising, with the current sample sizes, the evolution of the slopes is significantly more affected by disc dispersal than it is by secular evolution; therefore, at the state of the art, it cannot be used as a proxy to disentangle between the different evolutionary models. Increasing the sample size would allow to reduce the effect of low number statistics, potentially allowing to observe the different evolution of the slopes under the two evolutionary mechanisms; we further discuss this possibility in the following Section.
H]|P1cm | P1cm P2cm | P1cm P1.9cm | P1.6cm P5.2cm | P0.2cm|
8c $\mu_0 = \delta_0 - \zeta_0 \xi$
1-7 red8-8black
$\xi$ $\mu_0$ $t_{\mathrm{acc}, 0} - M_{\star}$ $\zeta_0$ $R_{\mathrm{d}} - M_{\star}$ $\delta_0$ 1c!$M_{\mathrm{d}}-M{\star}$ (a) and $\dot M-M{\star}$ (b) 1c!$\Rightarrow$
3*0 bg_visc_light $ < 0 $ bg_visc_light $t_{\mathrm{acc}, 0} \downarrow M_{\star} \uparrow$ 2c|bg_visc_light any bg_visc_light $ < 0 $ 1c!(a) shallower than (b) bg_visc_light 1c!$=$ bg_visc_light
bg_visc $0$ bg_visc $t_{\mathrm{acc}, 0} \leftrightarrow M_{\star} \uparrow$ 2c|bg_visc any bg_visc 0 1c!(a) as steep as (b) bg_visc 1c!$=$ bg_visc
bg_visc_light $ > 0 $ bg_visc_light $t_{\mathrm{acc}, 0} \uparrow M_{\star} \uparrow$ 2c|bg_visc_light any bg_visc_light $ > 0 $ 1c!(a) steeper than (b)bg_visc_light 1c!$=$ bg_visc_light
bg_mhd_light bg_mhd_light bg_mhd_light $ \leq 0 $ bg_mhd_light $R_{\mathrm{d}} \downarrow \leftrightarrow M_{\star} \uparrow$ bg_mhd_light $ \leq 0 $ 1c! (a) shallower or as steep as (b) bg_mhd_light 1c! $=$ bg_mhd_light
bg_mhd_light -2*$ < 0 $ bg_mhd_light -2*$t_{\mathrm{acc}, 0} \downarrow M_{\star} \uparrow$ bg_mhd_light $ > 0 $ bg_mhd_light $R_{\mathrm{d}} \uparrow M_{\star} \uparrow$ bg_mhd_light $ (- \infty, \zeta_0 \xi) $ 1c!(a) shallower or steeper than (b)[ $\zeta_0 \xi $ is positive, therefore $\delta_0$ can either be negative (implying (a) shallower than (b)) or positive (implying (a) steeper than (b)).] bg_mhd_light 1c!$\neq$ bg_mhd_light
bg_mhd bg_mhd bg_mhd $ \leq 0 $ bg_mhd $R_{\mathrm{d}} \downarrow \leftrightarrow M_{\star} \uparrow$ bg_mhd $ \leq 0 $ 1c! (a) shallower or as steep as (b) bg_mhd 1c!$\circ$ bg_mhd
bg_mhd -2*$ 0 $ bg_mhd -2*$t_{\mathrm{acc}, 0} \leftrightarrow M_{\star} \uparrow$ bg_mhd $ > 0 $ bg_mhd $R_{\mathrm{d}} \uparrow M_{\star} \uparrow$ bg_mhd $ \geq 0 $ 1c!(a) steeper or as steep as (b) bg_mhd 1c!$=$ bg_mhd
bg_mhd_light bg_mhd_light bg_mhd_light $ \leq 0 $ bg_mhd_light $R_{\mathrm{d}} \downarrow \leftrightarrow M_{\star} \uparrow$ bg_mhd_light $\left(\zeta_0 \xi, +\infty \right)$ 1c!(a) shallower or steeper than (b)[ $\zeta_0 \xi $ is negative, therefore the same argument as in <ref> holds.] bg_mhd_light 1c! $\circ$ bg_mhd_light
-6*(0, 1) bg_mhd_light -2*$ > 0 $ bg_mhd_light -2*$t_{\mathrm{acc}, 0} \uparrow M_{\star} \uparrow$ bg_mhd_light $ > 0 $ bg_mhd_light $R_{\mathrm{d}} \uparrow M_{\star} \uparrow$ bg_mhd_light $\left(\zeta_0 \xi, +\infty \right)$ 1c!(a) steeper than (b) bg_mhd_light 1c!$=$ bg_mhd_light
1-7 red8-8black
H]|P1.6cm P5cm|
2c $\delta_{\mathrm{evo}} = \xi ( \zeta_0 - \mu_0), \quad \xi \neq 0$
$\delta_{\mathrm{evo}}$ $M_{\mathrm{d}}-M{\star}$ (a) and $\dot M-M{\star}$ (b)
< 0 (a) shallower than (b)
bg_2 0 bg_2 (a) as steep as (b)
> 0 (a) steeper than (b)
Summary of the different possible theoretical scenarios described in Section <ref> to visualise the relative signs of the parameters at play. From left to right in the top table, the columns show (i) $\xi$, a proxy for the evolutionary model (viscous if $\xi = 0$, hybrid or wind-driven otherwise); (ii) $\mu_0$, the slope of the $t_{\mathrm{acc}, 0} - M_{\star}$ correlation and its implication on the correlation itself; (iii) $\zeta_0$, the slope of the $R_{\mathrm{d}}-M_{\star}$ correlation, and its implication on the correlation itself; (iv) $\delta_0 = \lambda_{\mathrm{m}, 0} - \lambda_{\mathrm{acc}, 0}$, the difference between the initial slopes of the $M_{\mathrm{d}}-M_{\star}$ and $\dot M - M_{\star}$ correlations and its implication on their relative steepness. The final column summarises whether the signs of $\mu_0$ and $\delta_0$ are necessarily the same ($=$), necessarily opposite ($\neq$) or can be either ($\circ$). When discussing the implications on correlations, up(down)wards arrows represent an in(de)crease of the parameters, while horizontal arrows describes the lack of correlation. The different cell colours are purely meant to guide the eye. The top table links the initial conditions, while the bottom table summarises the implications of the evolved difference in the slopes.
§ THE OBSERVATIONAL RELEVANCE OF THE SLOPES
Observed star-forming regions have both a spread in the initial conditions in addition to some disc dispersal mechanism (be it photoevaporation or MHD winds); as we discussed in Section <ref>, with the current sample sizes, the statistical significance of the observationally-determined slopes is undermined and their evolution traces disc dispersal, rather than the accretion mechanism. In this Section, we perform a statistical analysis of our simulated slopes and compare them with the currently available measurements; furthermore, we show the relevance of measuring the slopes despite these limitations and discuss the conditions under which they allow us to put constraints on disc evolution.
§.§ Comparison of different evolutionary models
Figure <ref> shows the comparison of the evolution of the slopes between the viscous + photoevaporative (solid line) and wind-driven (dashed line) models for both the $M_{\mathrm{d}}-M_{\star}$ and $\dot M - M_{\star}$ correlations (top and bottom row, respectively), including the measured slopes in four star-forming regions from [Testi et al., 2022] as grey dots. As we mentioned above, both models lead to disc dispersal consistently with the observed disc and accretion fraction (shown in Figure 6 of Somigliana et al., 2023); the three columns show simulations performed with a different initial number of discs, increasing from left to right, to obtain a different sample size at 5 Myr according to the predicted decline of observed discs. Like in Figure <ref>, the number of objects at 5 Myr is $\sim 30$, $\sim 60$ and $\sim 140$ from left to right, increasing from the currently available measurements in Upper Sco to the virtually complete sample. To estimate the effect of statistical fluctuations, given by the spread in the initial conditions, we ran 100 simulations for each set up: the solid and dashed lines represent the median evolution, while the intervals between the 25th and 75th percentile are visualised by the shaded areas. The growing shaded area, particularly visible with smaller sample sizes, is representative of the decreasing amount of sources on which the fit is performed: with the current sample size (left column), which leads to $\sim 30$ discs at 5 Myr, we end up with a 1$\sigma$ deviation from the median value of $\sim 0.5 - 0.6$. Larger sample sizes significantly reduce the scatter, leading to $\sigma \sim 0.4$ with a double sample and $\sigma \sim 0.2$ for the complete sample, reducing the current one by a factor 3. As mentioned in Section <ref>, with the currently available number of objects the dominant role in the evolution of the slopes is played by disc dispersal, which makes it difficult to trace the imprint of the secular evolution. The expected Upper Sco complete sample (right column) allows for a better separation between the two models - particularly for the $M_{\mathrm{d}}-M_{\star}$ correlation: the expected slope in the two scenarios differs by $\sim 0.5$, while the typical uncertainty of the currently measured slopes is between 0.2 and 0.3. Larger sample sizes would further decrease this uncertainty, allowing us to discriminate between the two models based on the slope evolution.
The observed slopes (from Testi et al., 2022) are only included in the left column of Figure <ref> as they refer to the current sample size. The main source of uncertainty in the current measurements is given by Upper Sco, mainly due to the incomplete sample; moreover, it is worth pointing out that external photoevaporation is likely to play a significant role in this region (Anania et al. in prep.). This comparison is meant as a first glance of the parameters space of the observed slopes, and we anticipate a proper exploration of the initial conditions once the full sample will be available.
In the following Section, we discuss the other constraints that we can put on disc evolution, besides identifying the driving accretion mechanism.
§.§ What are the slopes telling us, then?
Despite not allowing to conclusively discriminate between different evolutionary scenarios with the current sample sizes, the slopes of the $M_{\mathrm{d}} - M_{\star}$ and $\dot M - M_{\star}$ correlations can still help with constraining other properties from the theoretical considerations presented in Section <ref>, which we summarise in Table <ref>. If we assume an evolutionary model to begin with, and we can estimate (directly or indirectly) either $\mu_0$ or $\delta_0$, we can constrain the other parameter. When discussing the observational determination of what we have so far referred to as initial conditions, it is important to clarify the meaning of "initial". deals with and evolves Class II, potentially Class III, objects; hence, the initial conditions we input refer to the beginning of the Class II phase, where the protostellar collapse is over and the disc is already formed. From the observational point of view, this means that we expect $\delta_0$ and $\mu_0$ to refer to young Class II objects - around, or younger than, approximately 1 Myr. Earlier phases like the Class 0 and I need a dedicated study, as the accretion of the protostellar envelope is expected to play a prominent role in those stages.
In the following, we discuss the constraints we can put in both directions and comment on their feasibility based on the currently available estimates of $\mu_0$ and $\delta_0$.
§.§.§ Constraining delta0 from mu0
[Ansdell et al., 2017] claimed $\lambda_{\mathrm{m}}$ to be increasing with time. As shown by [Somigliana et al., 2022] and discussed in Section <ref>, increasing slopes imply that discs around less massive stars evolve faster than those around more massive stars; this can be interpreted in terms of increasing accretion timescale with stellar mass, which corresponds to $\mu_0 > 0$ (with $t_{\mathrm{acc}, 0} \propto {M_{\star}}^{\mu_0}$). In this section we discuss the implications of the increasing slope scenario on the initial conditions, $\lambda_{\mathrm{m}, 0}$ and $\lambda_{\mathrm{acc}, 0}$.
The top panel of Table <ref> shows the relation between $\mu_0$ and $\delta_0$ in the different evolutionary models. As $\mu_0 = \delta_0 - \zeta_0 \xi$, in the viscous case (corresponding to $\xi = 0$) we have $\mu_0 = \delta_0$ as mentioned in Section <ref>. This means that, to recover the suggested increasing slopes scenario, $\delta_0$ necessarily needs to be positive - regardless of the value of any other parameters: this translates to the initial $M_{\mathrm{d}} - M_{\star}$ correlation being steeper than $\dot M - M_{\star}$. In the hybrid and wind-driven case, instead, the implication is less straightforward as a positive $\mu_0$ can lead to opposite signs of $\delta_0$: this is determined by the scaling of the disc radius with the stellar mass, which is suggested to be (weakly) positive from observational evidences (e.g., Andrews et al., 2018). In principle, as $\delta_0 \in (\zeta_0 \xi, + \infty)$, the sign of $\zeta_0$ determines whether negative values of $\delta_0$ are possible; however, as $\xi$ is a small number (0.1 in this work), only a limited area of the parameters space would lead to a negative $\delta_0$. Summarising, if we assume increasing slopes ($\mu_0 > 0$) we can constrain the sign of $\delta_0$ regardless of the evolutionary model assumed: in both cases $\delta_0$ needs to be positive, which leads to an initially steeper $M_{\mathrm{d}} - M_{\star}$ than $\dot M - M_{\star}$ correlation.
§.§.§ Constraining mu0 from deltaevo
Instead of assuming increasing slopes, we can start from the currently measured values of $\delta_{\mathrm{evo}}$ and estimate $\delta_0$ in the different evolutionary models. In the viscous case, because $\delta_{\mathrm{evo}} = 0$, we focus on the value of the single slopes instead: as $\lambda_{\mathrm{m, evo}} = \lambda_{\mathrm{acc, evo}} = \delta_0/2 + \lambda_{\mathrm{m}, 0}$, the measured final value of the slopes does not help in constraining $\delta_0$ as it also depends on $\lambda_{\mathrm{m}, 0}$. In the hybrid case, instead, we have $\delta_{\mathrm{evo}} = \xi(\zeta_0 - \mu_0)$, meaning that if we can determine the sign of $\delta_{\mathrm{evo}}$ we can constrain that of $\mu_0$ as well. While in principle the sign of $\zeta_0$ influences that of $\mu_0$, as we mentioned above $\zeta_0$ is likely a small number: therefore, effectively, $\delta_{\mathrm{evo}}$ and $\mu_0$ have opposite signs for the vast majority of the parameters space.
Assuming that the observed disc populations can be considered evolved enough for the above arguments to hold, we can estimate $\delta_{\mathrm{evo}}$ from the most recent and homogeneous measurements available of $\lambda_{\mathrm{m}}$ and $\lambda_{\mathrm{acc}}$ [Testi et al., 2022]. The resulting values of $\delta_{\mathrm{obs}}$ (which we label 'observed' as opposed to the theoretical expectation, 'evolved'), summarised in Table <ref>, are oscillating: out of the four regions L1668, Lupus, Chameleon I and Upper Sco, we find two positive and two negative median values. Moreover, in three cases out of four the uncertainties are so large that $\delta_{\mathrm{obs}}$ would be compatible with both a positive and a negative value. The difficulty in assessing the sign of $\delta_{\mathrm{obs}}$ from the current measurements of the slopes make constraining $\mu_0$ from $\delta_{\mathrm{evo}}$ not trivial. Larger sample sizes would give a better measurement of the slopes and reduce the uncertainty, leading to a more solid determination of the sign of $\delta_{\mathrm{obs}}$ - which would possibly allow to constrain $\mu_0$.
Summarising, the (admittedly not robust) observational evidence pointing towards increasing accretion timescale with stellar mass allows us to constrain the initial correlations between the stellar mass and disc parameters; regardless of the evolutionary model considered, the initial slope of the $M_{\mathrm{d}}-M_{\star}$ correlation needs to be larger than that of $\dot M - M_{\star}$. The other way around, constraining the slope of the accretion timescale - stellar mass correlation from the difference between $\lambda_{\mathrm{m}}$ and $\lambda_{\mathrm{acc}}$ at the present time, requires sample sizes larger by at least a factor two.
H]|P1.5cm | P3.5cm | P1.5cm |
Region Median age [Myr] $\delta_{\mathrm{obs}}$
L1668 1 $-0.3 \pm 0.5$
Lupus 2 $0.1 \pm 0.4$
Cha I 2.8 $-0.7 \pm 0.4$
Upper Sco 4.3 $0.7 \pm 0.8$
Values of $\delta$ derived from the currently available measurements of $\lambda_{\mathrm{m}}$ and $\lambda_{\mathrm{acc}}$ [Testi et al., 2022].
§ CONCLUSIONS
In this paper, we have investigated the impact of disc evolution models on the correlations between the stellar mass and the disc properties - especially the disc mass and the accretion rate. We have explored the purely viscous, wind-driven, viscous and wind hybrid, and photoevaporative models. Assuming power-law correlations to hold as initial conditions, $M_{\mathrm{d}}(0) \propto {M_{\star}}^{\lambda_{\mathrm{m}, 0}}$, $\dot M(0) \propto {M_{\star}}^{\lambda_{\mathrm{acc}, 0}}$, we performed analytical calculations (where possible) and population synthesis simulations for both evolutionary scenarios, and compared them with the purely viscous case discussed in [Somigliana et al., 2022]. Our main results are the following:
* The viscous and hybrid models change the slope of the initial correlations as function of the evolutionary time, but preserve their shape. In the wind-driven model, instead, the correlations deviate from the original power-law shape: this is visualised in the logarithmic plane as a bending of the linear correlation (see Figure <ref>). The bending direction is towards the less or more massive stars depending on the scaling of the accretion timescale with the stellar mass (positive and negative correlation respectively).
* The characteristic behaviour of the slopes in the wind-driven model is concealed by the presence of a spread in the initial conditions, which introduces a scatter in the correlations and makes it no longer possible to detect the bending (Figure <ref>). This leads to a considerably similar evolution of the correlations in the different accretion models.
* Performing our simulations with evolutionary models that match the disc dispersal timescales (intrinsic in the wind-driven model and including internal photoevaporation in the viscous case), we find that the evolution of the slopes is significantly impacted by the removal of discs from the population (Figure <ref>). Different realisations of the same simulation dramatically differ from one another, and show a stochastic behaviour with large variations (Figure <ref>). This has both a physical (presence of a spread in the initial conditions) and a statistical (low number of objects left after a few Myr of evolution) reason.
* While a proper exploration of the parameters space, outside of the scope of this work, would be needed to assess the impact of the initial conditions, with the currently available sample sizes the noise dominates over the physical evolution.
* Increasing the sample size can mitigate the effects of disc dispersal on the evolution of the slope by removing the stochastic effects. We find that, for our parameters choice, the complete sample of Upper Sco ($\sim 140$ sources) at 5 Myr would reduce the oscillations enough to make the slopes a proxy for the evolutionary model (Figure <ref>).
* While the currently available sample sizes do not yet allow to distinguish between the different evolutionary models, we can use them to put some constraints on the initial conditions. We find that in all evolutionary scenarios, the observational claim of increasing slopes leads to an initially steeper correlation between the disc mass and the stellar mass than between the accretion rate and the stellar mass. The other possible way, measuring the current slopes and inferring the correlation between the accretion timescale and the stellar mass from them, provides weaker constraints because of the high uncertainties in the current measurements.
* We have presented and released the 1D Python disc population synthesis code and its output analysis library .
In this work, we have shown how large enough samples of protoplanetary discs can provide a way of distinguishing between the evolutionary models (with a standard set of parameters) through the observation of the time evolution of the correlations between the disc properties and the stellar mass. We have shown how the stochastic fluctuations seen with the currently available observations could be significantly reduced if we had access to the complete Upper Sco sample, consisting of approximately 140 sources at 5 Myr. We strongly support the observational effort in the direction of obtaining larger amounts of data for evolved star-forming regions, and encourage the exploration of the parameters space beyond the standard case.
We thank an anonymous referee for their useful comments that helped us improve the clarity of the manuscript. This work was partly supported by the Italian Ministero dell’Istruzione, Università e Ricerca through the grant Progetti Premiali 2012-iALMA (CUP C52I13000140001), by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Ref no. 325594231 FOR 2634/2 TE 1024/2-1, by the DFG Cluster of Excellence Origins (www.origins-cluster.de). This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska- Curie grant agreement No 823823 (DUSTBUSTERS) and from the European Research Council (ERC) via the ERC Synergy Grant ECOGAL (grant 855130) and ERC Starting Grant DiscEvol (grant 101039651). GR acknowledges support from Fondazione Cariplo, grant No. 2022-1217. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
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§ PARAMETERS USED THROUGHOUT THE PAPER
Table <ref> shows the parameters used throughout the paper. The general simulation parameters refer to the initial correlations (distribution, slopes and spreads), while the following three tables are divided by evolutionary model and display the main parameters for each of them.
General simulation parameters
H]P3.5cm | P1.5cm
0pt1.5ex $\lambda_{\mathrm{m}, 0}$ 2.1
0pt1.5ex$\lambda_{\mathrm{acc}, 0}$ 1.5
0pt1.5ex$\sigma_{\mathrm{M_d}}$ $0.65$ dex
0pt1.5ex$\sigma_{\mathrm{R_d}}$ $0.52$ dex
0pt1.5ex$M_{\mathrm{d}}$, $R_{\mathrm{d}}$ distribution lognormal
Viscous model
H]P1.5cm | P1.5cm
0pt2.2ex $\alpha_{\mathrm{SS}}$ $10^{-3}$
Photoevaporative model
H]P1.5cm | P3cm
0pt1.5ex $\alpha_{\mathrm{SS}}$ $10^{-3}$
0pt1.5ex $\dot M_{w}$ $4 \times 10^{-10}$ M$_{\odot}$ yr$^{-1}$
Wind-driven model
H]P1.5cm | P1.5cm
0pt1.5ex $\alpha_{\mathrm{SS}}$ $10^{-3}$
0pt1.5ex $\alpha_{\mathrm{DW}}$ $10^{-3}$
0pt1.5ex $\lambda$ $3$
0pt1.5ex $\omega$ $0.25$
Values of the parameters used throughout the paper, unless explicitly stated otherwise.
§ MHD MODEL WITH MU<0
As mentioned in Section <ref>, the breaking of the linear correlation between the disc properties and the stellar mass happens towards higher or lower stellar masses depending on the value of $\mu_0$. Figure <ref> shows the a simulation $\mu_0 > 0$, while in Figure <ref> we show the opposite case. As $\mu_0$ is linked to $\delta_0$ through $\mu_0 = \delta_0 - \zeta_0 \xi$, if $\delta_0 < \zeta_0 \xi$ the resulting $\mu_0$ will be negative, leading to a specular bending of the correlation. Given that $\zeta_0 \xi$ is a small number ($\sim 0.1$ in our simulation), this generally requires a negative $\delta_0$. The simulation in Figure <ref> has $\lambda_{\mathrm{m}, 0} = 1.3$ and $\lambda_{\mathrm{acc}, 0} = 1.7$, resulting in $\delta_0 = -0.4$.
Same as Figure <ref>, but with a choice of initial slopes resulting in a negative $\mu_0$ ($\lambda_{\mathrm{m}, 0} = 1.3, \lambda_{\mathrm{acc}, 0} = 1.7$). The bending of the linear correlation happens towards larger disc masses, in agreement with the prediction.
§ VALIDATION OF
Figure <ref> shows the evolution of the gas surface density as a function of the disc radius in the cases of evolution driven by (i) viscosity, (ii) viscosity and internal photoevaporation, (iii) MHD winds, and (iv) viscosity and external photoevaporation for a single disc simulated with . The top left panel, corresponding to viscous evolution (i), shows the key feature of viscous spreading: while the global surface density decreases as a consequence of the accretion onto the central star, the radial extent of the disc increases. This is a consequence of the redistribution of angular momentum, that causes part of the disc material to move towards larger radii. The top right panel, where the disc evolves under the combined effect of viscosity and internal photoevaporation (ii), shows the characteristic two-timescales behaviour [Clarke et al., 2001]: the evolution is effectively viscous in the earliest stages, as long as the accretion rate is larger than the photoevaporative mass-loss rate; then, photoevaporation opens a cavity within the disc, which gets divided into an inner and an outer disc. The inner disc is less extended and therefore has a shorter viscous timescale, which means it evolves much faster and is quickly completely accreted onto the protostar; the outer disc instead keeps on evolving on timescales comparable to the original one, making photoevaporation a two-timescales process. The bottom left panel shows a disc evolving due to MHD winds (iii): the absolute value of the surface density drops faster than in the viscous model, because of the increase of $\alpha_{\mathrm{DW}}$ in time. Furthermore, as angular momentum is removed from the wind (together with material), the disc does not spread but rather shrinks in time, as expected from the theoretical prediction [Tabone et al., 2022]. Finally, the bottom right panel shows the evolution of a disc undergoing external photoevaporation combined with viscosity (iv): the latter dominates at the earliest stages, producing the characteristic features like viscous spreading, while the effect of external photoevaporation is visible at later ages as a truncation of the disc that also halts its spreading. In this case, the disc truncation and the outside-in depletion of disc material is the consequence of the photo-dissociation of gas molecules due to the FUV radiation emitted by massive stars and experienced by the disc. The efficiency of this process depends primarily on the stellar mass and the FUV flux experienced: given a fixed FUV flux, a disc around a lower mass star will lose its material to external winds more easily compared to a disc around a higher mass star, because of the higher gravitational bond of the system. For the same reason, more extended and less massive discs are more prone to external truncation.
Gas surface density as a function of radius for a disc generated with Diskpop at different times as the colour bar shows. The four models are purely viscous (top left, $\alpha_{\mathrm{SS}} = 10^{-3}$), viscous including internal photoevaporation (top right, $\alpha_{\mathrm{SS}} = 10^{-3}$, $\dot M_{\mathrm{w}} = 4 \times 10^{-10}$ M$_{\odot} \mathrm{yr^{-1}}$), wind-driven (bottom left, $\alpha_{\mathrm{DW}} = 10^{-3}$, $\lambda = 3$, $\omega = 0.25$) and viscous including external photoevaporation (bottom right, $\alpha_{\mathrm{SS}} = 10^{-3}$, FUV = 100 $G_0$) respectively.
Figure <ref> shows the isochrones in the $\dot M - M_{\mathrm{d}}$ plane at 0.1, 1 and 10 Myr for the three evolutionary models of viscosity, viscosity and photoevaporation, and MHD winds. Each dot represents a disc in the population, while the solid lines show the analytical prediction (when applicable): in the viscous case, the isochrones read [Lodato et al., 2017]
\begin{equation}
\dot M = \frac{M_{\rm{d}}}{2(2 - \gamma)t} \left[ 1 - \left( \frac{M_{\rm{d}}}{M_0} \right)^{2(2-\gamma)} \right],
\label{eq:theoretical_isochrone_viscous}
\end{equation}
while in the MHD wind-driven scenario [Tabone et al., 2022]
\begin{equation}
\dot M = \frac{1}{\omega (1 + f_{\rm{M}, 0})t} M_{\rm{d}} \left[ \left( \frac{M_{\rm{d}}}{M_0} \right)^{- \omega} - 1 \right].
\label{eq:theoretical_isochrone_MHD}
\end{equation}
The left panel shows the viscous model, where the discs tend more and more towards the theoretical isochrone during their evolution [Lodato et al., 2017]; the central panel includes internal photoevaporation, which has the effect of bending the isochrones once the accretion rate becomes comparable to the photoevaporative mass-loss rate [Somigliana et al., 2020]; finally, the right panel shows an MHD wind-driven population, where the scatter in the $\dot M - M_{\mathrm{d}}$ plane remains significant during the evolution - contrary to the viscous prediction [Somigliana et al., 2023].
Isochrones at 0.1, 1, and 10 Myr for disc populations undergoing viscous, viscous+internal photoevaporation and wind-driven evolution (left, centre, and right panel respectively), with the same parameters as Figure <ref>. Each dot represents a disc, while the solid lines show the theoretical isochrones at the corresponding age, where available.
|
# CALI: Coarse-to-Fine ALIgnments Based Unsupervised Domain Adaptation of
Traversability Prediction for Deployable Autonomous Navigation
Zheng Chen Luddy School of Informatics,
Computing, and Engineering
Indiana University - Bloomington
Indiana 47408
Email<EMAIL_ADDRESS>Durgakant Pushp Luddy School of Informatics,
Computing, and Engineering
Indiana University - Bloomington
Indiana 47408
Email<EMAIL_ADDRESS>Lantao Liu Luddy School of Informatics,
Computing, and Engineering
Indiana University - Bloomington
Indiana 47408
Email<EMAIL_ADDRESS>
###### Abstract
Traversability prediction is a fundamental perception capability for
autonomous navigation. The diversity of data in different domains imposes
significant gaps to the prediction performance of the perception model. In
this work, we make efforts to reduce the gaps by proposing a novel coarse-to-
fine unsupervised domain adaptation (UDA) model – CALI. Our aim is to transfer
the perception model with high data efficiency, eliminate the prohibitively
expensive data labeling, and improve the generalization capability during the
adaptation from easy-to-obtain source domains to various challenging target
domains. We prove that a combination of a coarse alignment and a fine
alignment can be beneficial to each other and further design a first-coarse-
then-fine alignment process. This proposed work bridges theoretical analyses
and algorithm designs, leading to an efficient UDA model with easy and stable
training. We show the advantages of our proposed model over multiple baselines
in several challenging domain adaptation setups. To further validate the
effectiveness of our model, we then combine our perception model with a visual
planner to build a navigation system and show the high reliability of our
model in complex natural environments where no labeled data is available.
111The paper is published in Robotics: Science and Systems (RSS) 2022.
The robot navigation demonstration can be seen in this video:
https://www.youtube.com/watch?v=Nqsegaq_x-o.
## I Introduction
We consider the deployment of autonomous robots in the real-world unstructured
field environments, where the environments can be extremely complex involving
random obstacles (e.g., big rocks, tree stumps, man-made objects), cross-
domain terrains (e.g., combinations of gravel, sand, wet, uneven surfaces), as
well as dense vegetation (tall and low grasses, shrubs, trees). Whenever a
robot is deployed in such an environment, it needs to understand which area of
the captured scene is navigable. A typical solution to this problem is the
visual traversability prediction that can be achieved by learning the scene
semantic segmentation.
Visual traversability prediction has been tackled by using deep neural
networks where the models are typically trained offline with well-labeled
datasets in limited scenarios. However, there is a gap between the data used
to train the model and the real world. It is usually challenging for existing
datasets to well approximate the true distributions of the unseen target
environments where the robot is deployed. Even incrementally collecting and
adding new training data on the fly cannot guarantee the target environments
to be well in-distribution included. In addition, annotating labels for dense
predictions, e.g., semantic segmentation, is prohibitively expensive.
Therefore, developing a generalization-aware deep model is crucial for robotic
systems considering the demands of the practical deployment of deep perception
models and the costs/limits of collecting new data in many robotic
applications, e.g., autonomous driving, search and rescue, and environmental
monitoring.
Figure 1: Transferring models from the available domain to the target domain.
The existing available data might be from either a simulator or collecting
data in certain environments, at a certain time, and with certain sensors. In
contrast, the target deployment might have significantly varying environments,
time, and sensors.
To tackle this challenge, a broadly studied framework is transfer learning
[24] which aims to transfer models between two domains – source domain and
target domain – that have related but different data distributions. The
prediction on target domain can be considered as a strong generalization since
testing data (in target domain) might fall out of the independently and
identically distributed (i.i.d.) assumption and follow a very different
distribution than the training data (in source domain). The “transfer” process
has significant meaning to our model development since we can view the
available public datasets [29, 8, 35, 17] as the source domain and treat the
data in the to-be-deployed environments as the target domain. In this case, we
have access to images and corresponding labels in source domain and images in
target domain, but no access to labels in target domain. Transferring models,
in this set-up, is called Unsupervised Domain Adaptation (UDA) [36, 40].
Domain Alignment (DA) [10, 12, 13, 32, 33] and Class Alignment (CA) [31] are
two conventional ways to tackle the UDA problem. DA treats the deep features
as a whole. It works well for image-level tasks such as image classification,
but has issues with pixel-level tasks such as semantic segmentation, as the
alignment of whole distributions ignores the class features and might misalign
class distributions, even the whole features from the source domain and target
domain are already well-aligned. CA is proposed to solve this issue for dense
predictions with multiple classes.
It is natural and necessary to use CA to tackle the UDA of semantic
segmentation as we need to consider aligning class features. However, CA can
be problematic and might fail to outperform the DA for segmentation, and in a
worse case, might have unacceptable negative transfer, which means the
performance with adaptation is even degraded than that without adaptation.
Intuitively, we need to consider more alignments in CA than in DA. Thus the
searching space might be more complicated, and training might be more unstable
and harder to converge to an expected minima, leading to larger prediction
errors.
To solve the issue of CA, we investigate the relationship of the upper bounds
of the prediction error on target domain between DA and CA and provide a
theoretical analysis of the upper bounds of target prediction error in UDA
setup, and bridge the theoretical analysis and algorithm design for UDA of
traversability prediction.
In summary, our contributions include
* •
We prove that with proper assumptions, the upper bound of CA is upper bounded
by the upper bound of DA. This indicates that constraining the training of CA
using DA can be beneficial. We then propose a novel concept of pseudo-
trilateral game structure (PTGS) for integrating DA and CA.
* •
We propose an efficient coarse-to-fine alignments based UDA model, named CALI,
for traversability prediction. The new proposal includes a trilateral network
structure, novel training losses, and an alternative training process. Our
model design is well supported by theoretical analysis. It is also easy and
stable to train and converge.
* •
We show significant advantages of our proposed model compared to several
baselines in multiple challenging public datasets and one self-collected
dataset. We combine the proposed segmentation model and a visual planner to
build a visual navigation system. The results show high safety and
effectiveness of our model.
## II Related Work
Semantic Segmentation: Semantic segmentation aims to predict a unique human-
defined semantic class for each pixel in the given images. With the prosperity
of deep neural networks, the performance of semantic segmentation has been
boosted significantly, especially by the advent of FCN [20] that first
proposes to use deep convolutional neural nets to predict segmentation.
Following works try to improve the FCN performance by multiple proposals,
e.g., using different sizes of kernels or dilation rates to aggregate multi-
scale features [6, 7, 38]; building image pyramids to create multi-resolution
inputs [41]; applying probabilistic graph to smooth the prediction [19];
compensating features in deeper level by an encoder-decoder structure [30],
and employing attention mechanism to capture the long-range dependencies among
pixels in a more compact and efficient way [28]. We can also see how excellent
the current semantic segmentation SOTA performance is from very recently
released work [42, 37]. However, all of those methods belong to fully-
supervised learning and the performance might catastrophically be degraded
when a domain shift exists between the training data and the data when
deploying. Considering the possible domain shift and developing adaptation-
aware models is extremely practical and urgent.
Unsupervised Domain Adaptation: The main approaches to tackle UDA include
adversarial training (a.k.a., distribution alignment) [10, 12, 13, 32, 31, 33,
21, 34] and self-training [43, 39, 23, 16]. Although self-training is becoming
another dominant method for segmentation UDA in terms of the empirical
results, it still lacks a sound theoretical foundation. In this paper, we only
focus on the alignment-based methods that not only keep close to the UDA
state-of-the-art (SOTA) performance but also are well supported by sound
theoretical analyses [1, 3, 2].
The alignment-based methods adapt models via aligning the distributions from
the source domain and target domain in an adversarial training process, i.e.,
making the deep features of source images and target images indistinguishable
to a discriminator net. Typical approaches to UDA include Domain Alignment
(DA) [10, 12, 13, 32, 33], which aligns the two domains using global features
(aligning the feature tensor from source or target as a whole) and Class
Alignment (CA) [31, 21, 34], which only considers aligning features of each
class from source and target, no matter whether the domain distributions are
aligned or not. In [31], the authors are inspired by the theoretical analysis
of [2] and propose a discrepancy-based model for aligning class features.
There is a clear relation between the theory guidance [2] and the design of
network, loss, and training methods. There are some recent works [21, 34]
similar to the proposed work in spirit and show improved results compared to
[31], but it is still unclear to relate the proposed algorithms with theory
and to understand why the structure/loss/training is designed as the presented
way.
## III Background and Preliminary Materials
We consider segmentation tasks where the input space is
$\mathcal{X}\subset\mathbb{R}^{H\times W\times 3}$, representing the input RGB
images, and the label space is
$\mathcal{Y}\subset\left\\{0,1\right\\}^{H\times W\times K}$, representing the
ground-truth $K$-class segmentation images, where the label for a single pixel
at $(h,w)$ is denoted by a one-hot vector $y^{(h,w)}\in\mathbb{R}^{K}$ whose
elements are by-default 0-valued except at location $(h,w)$ which is labeled
as 1. Domain adaptation has two domain distributions over
$\mathcal{X}\times\mathcal{Y}$, named source domain $\mathcal{D}_{S}$ and
target domain $\mathcal{D}_{T}$. In the setting of UDA for segmentation, we
have access to $m_{s}$ i.i.d. samples with labels
$\mathcal{U}_{S}=\left\\{\mathbf{x}_{si},\mathbf{y}_{si}\right\\}_{i=1}^{m_{s}}$
from $\mathcal{D}_{S}$ and $m_{t}$ i.i.d. samples without labels
$\mathcal{U}_{T}=\left\\{\mathbf{x}_{tj}\right\\}_{j=1}^{m_{t}}$ from
$\mathcal{D}_{T}$.
In the UDA problem, we need to reduce the prediction error on the target
domain. A hypothesis is a function $h:\mathcal{X}\rightarrow\mathcal{Y}$. We
denote the space of $h$ as $\mathcal{H}$. With the loss function
$l(\cdot,\cdot)$, the expected error of $h$ on $\mathcal{D}_{S}$ is defined as
$\epsilon_{S}(h):=\mathbb{E}_{(x,y)\sim\mathcal{D}_{S}}l(h(x),y).$ (1)
Similarly, we can define the expected error of $h$ on $\mathcal{D}_{T}$ as
$\epsilon_{T}(h):=\mathbb{E}_{(x,y)\sim\mathcal{D}_{T}}l(h(x),y).$ (2)
Two important upper bounds related to the source and target error are given in
[2]. Basically,
Theorem 1 For a hypothesis $h$,
$\epsilon_{T}(h)\leq\epsilon_{S}(h)+d_{1}(\mathcal{D}_{S},\mathcal{D}_{T})+\lambda,$
(3)
where $d_{1}(\cdot,\cdot)$ is the $L^{1}$ divergence for two distributions,
and the constant term $\lambda$ does not depend on any $h$. However, it is
claimed in [2] that the bound with $L^{1}$ divergence cannot be accurately
estimated from finite samples, and using $L^{1}$ divergence can unnecessarily
inflate the bound. Another divergence measure is thus introduced to replace
the $L^{1}$ divergence with a new bound derived.
Definition 1 Given two domain distributions $\mathcal{D}_{S}$ and
$\mathcal{D}_{T}$ over $\mathcal{X}$, and a hypothesis space $\mathcal{H}$
that has finite VC dimension, the $\mathcal{H}$-divergence between
$\mathcal{D}_{S}$ and $\mathcal{D}_{T}$ is defined as
$\displaystyle
d_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})=2\sup_{h\in\mathcal{H}}|$
$\displaystyle\text{P}_{x\sim\mathcal{D}_{S}}\left[h(x)=1\right]-$ (4)
$\displaystyle\text{P}_{x\sim\mathcal{D}_{T}}\left[h(x)=1\right]|,$
where $\text{P}_{x\sim\mathcal{D}_{S}}[h(x)=1]$ represents the probability of
$x$ belonging to $\mathcal{D}_{S}$. Same to
$\text{P}_{x\sim\mathcal{D}_{T}}\left[h(x)=1\right]$.
The $\mathcal{H}$-divergence resolves the issues in the $L^{1}$ divergence. If
we replace $d_{1}(\mathcal{D}_{S},\mathcal{D}_{T})$ in Eq. (3) with
$d_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})$, then a new upper bound for
$\epsilon_{T}(h)$, named as $\mathbb{UB}_{1}$, can be written as
$\displaystyle\epsilon_{T}(h)\leq\mathbb{UB}_{1},$ (5)
$\displaystyle\mathbb{UB}_{1}=\epsilon_{S}(h)+d_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+\lambda.$
An approach to compute the empirical $\mathcal{H}$-divergence is also proposed
in [2].
Lemma 1 For a symmetric hypothesis class $\mathcal{H}$ (one where for every
$h\in\mathcal{H}$, the inverse hypothesis $1-h$ is also in $\mathcal{H}$) and
two sample sets
$\mathcal{U}_{S}=\left\\{x_{i},i=1,\cdots,m_{s},x_{i}\sim\mathcal{D}_{S}\right\\}$
and
$\mathcal{U}_{T}=\left\\{x_{j},j=1,\cdots,m_{t},x_{j}\sim\mathcal{D}_{T}\right\\}$.
$\ \begin{aligned}
\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})=2\Bigg{(}\Bigg{.}1-\min_{\eta\in\mathcal{H}}\Bigg{[}\Bigg{.}&\frac{1}{m_{s}}\sum_{i=1}^{m_{s}}\mathbb{I}[\eta(x_{i})=0]+\\\
&\frac{1}{m_{t}}\sum_{j=1}^{m_{t}}\mathbb{I}[\eta(x_{j})=1]\Bigg{.}\Bigg{]}\Bigg{.}\Bigg{)},\end{aligned}$
(6)
where $\mathbb{I}[a]$ is an indicator function which is 1 if $a$ is true, and
$0$ otherwise.
The second upper bound is based on a new hypothesis called symmetric
difference hypothesis.
Definition 2 For a hypothesis space $\mathcal{H}$, the symmetric difference
hypothesis space $\mathcal{H}\Delta\mathcal{H}$ is the set of hypotheses
$g\in\mathcal{H}\Delta\mathcal{H}\Leftrightarrow g(x)=h(x)\oplus
h^{\prime}(x)~{}~{}~{}\text{for some }h,h^{\prime}\in\mathcal{H},$ (7)
where $\oplus$ denotes an XOR operation. Then we can define the
$\mathcal{H}\Delta\mathcal{H}$-distance as
$\displaystyle
d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})=2\sup_{h,h^{\prime}\in\mathcal{H}}|$
$\displaystyle\text{P}_{x\sim\mathcal{D}_{S}}\left[h(x)\neq
h^{\prime}(x)\right]-$ (8)
$\displaystyle\text{P}_{x\sim\mathcal{D}_{T}}\left[h(x)\neq
h^{\prime}(x)\right]|.$
Similar to Eq. (5), if we replace $d_{1}(\mathcal{D}_{S},\mathcal{D}_{T})$
with $d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})$, the
second upper bound for $\epsilon_{T}(h)$, named as $\mathbb{UB}_{2}$, can be
expressed as
$\displaystyle\epsilon_{T}(h)\leq\mathbb{UB}_{2},$ (9)
$\displaystyle\mathbb{UB}_{2}=\epsilon_{S}(h)+d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+\lambda,$
where $\lambda$ is the same term as in Eq. (3).
A standard way to achieve the alignment for deep models is to use the
adversarial training method, which is also used in Generative Adversarial
Networks (GANs). Therefore we explain the key concepts of adversarial training
using the example of GANs.
GAN is proposed to learn the distribution $p_{r}$ of a set of given data
$\left\\{\mathbf{x}\right\\}$ in an adversarial manner. The architecture
consists of two networks - a generator $G$, and a discriminator $D$. The $G$
is responsible for generating fake data (with distribution $p_{g}$) from
random noises $\mathbf{z}\sim p_{\mathbf{z}}$ to fool the discriminator $D$
that is instead to accurately distinguish between the fake data and the given
data. Optimization of a GAN involves a mini-maximization over a joint loss for
$G$ and $D$.
$\displaystyle\min_{G}\max_{D}V(G,D)$ (10) $\displaystyle
V(G,D)=\mathbb{E}_{\mathbf{x}\sim
p_{r}}\log\left[D(\mathbf{x})\right]+\mathbb{E}_{\mathbf{z}\sim
p_{\mathbf{z}}}\log\left[1-D(G(z))\right].$
where we use $1$ as the real label and $0$ as the fake label. Training with
Eq. (10) is a bilateral game where the distribution $p_{g}$ is aligned with
the distribution $p_{r}$.
The two bounds (Eq. (5) and Eq. (9)) for the target domain error are
separately given in [2]. It has been independently demonstrated that domain
alignment corresponds to optimizing over $\mathbb{UB}_{1}$ [10], where
optimization over the upper bound $\mathbb{UB}_{1}$ (Eq. (5) with the
divergence Eq. (6)) is proved as equivalent to Eq. (10) with a supervised
learning in the source domain, and that class alignment corresponds to
optimizing over $\mathbb{UB}_{2}$ [31], where the
$d_{\mathcal{H}\Delta\mathcal{H}}$ is approximated by the discrepancy between
two different classifiers.
Training DA is straightforward since we can easily define binary labels for
each domain, e.g., we can use 1 as the source domain label and 0 as the target
domain label. Adversarial training over the domain labels can achieve domain
alignment. For CA, however, it is difficult to implement as we do not have
target labels, hence the target class features are completely unknown to us,
thus leading naively using adversarial training over each class impossible.
The existing way well supported by theory to perform CA [31] is to indirectly
align class features by devising two different classifier hypotheses. The two
classifiers have to be well trained on the source domain and are able to
classify different classes in the source domain with different decision
boundaries. Then considering the shift between source and target domain, the
trained two classifiers might have disagreements on target domain classes.
Note since the two classifiers are already well trained on the source domain,
the agreements of the two classifiers represent those features in the target
domain that are close to the source domain, while in contrast, the features
where disagreements happen indicate that there is a large shift between source
and target. We use the disagreements to approximate the distance between
source and target. If we are able to minimize the disagreements of the two
classifiers, then features of each class between source and target will be
enforced to be well aligned.
Figure 2: An ideal iterative training process by integrating DA and CA.
## IV Methodology
In this work we further investigate the relation between the $\mathbb{UB}_{1}$
and $\mathbb{UB}_{2}$ and prove that $\mathbb{UB}_{1}$ turns out to be an
upper bound of $\mathbb{UB}_{2}$, meaning DA can be a necessary constraint to
CA. This is also consistent to our intuition: DA aligns features globally in a
coarse way while CA aligns features locally in a finer way. Constraining CA
with DA is actually a coarse-to-fine process, which makes the alignment
process efficient and stable. By carefully studying the internal structure of
existing DA and CA work, we propose a novel concept, pseudo-trilateral game
structure, for efficiently integrating DA and CA. We follow our theoretical
analysis and proposed PTGS to guide the development of CALI, including designs
of model structure, losses and training process.
Figure 3: Pseudo-trilateral game structure (PTGS). Three players are in the
game, a feature extractor $G$, a domain discriminator $D$ and a family of
classifiers $C_{s}$. The game between $G$ and $Cs$ is the CA while the game
between $G$ and $D$ is the DA. The DA and CA are connected by sharing the same
feature extractor $G$. Both $D$ and $C_{s}$ are trying to adjust the $G$ such
that the features between source and target generated from $G$ could be well
aligned globally and locally.
Notations used in this paper is explained as follows. we denote the
segmentation model $h$ as $h^{\theta,\phi}(x)=C^{\theta}(G^{\phi}(x))$ which
consists of a feature extractor $G^{\phi}$ parameterized by $\phi$ and a
classifier $C^{\theta}$ parameterized by $\theta$, and $x$ is a sample from
$\mathcal{U}_{S}$ or $\mathcal{U}_{T}$. If multiple classifiers are used, we
will denote the $j^{th}$ classifier as $C_{j}$. We denote the discriminator as
$D^{\psi}$ parameterized by $\psi$.
### IV-A Bounds Relation
We start by examining the relationship between the DA and the CA from the
perspective of target error bound. We propose to use this relation to improve
the segmentation performance of class alignment, which is desired for dense
prediction tasks. We provide the following theorem:
Theorem 2 If we assume there is a hypothesis space $\mathcal{H}$ for
segmentation model $h^{\theta,\phi}$ and a hypothesis space $\mathcal{H}_{D}$
for domain classifiers $D^{\psi}$, and
$\mathcal{H}\Delta\mathcal{H}\subset\mathcal{H}_{D}$, then we have
$\displaystyle\epsilon_{T}(h)$
$\displaystyle\leq\hat{\mathbb{UB}}_{2}\leq\hat{\mathbb{UB}}_{1},$ (11)
$\displaystyle\hat{\mathbb{UB}}_{1}$
$\displaystyle=\epsilon_{S}(h)+\frac{1}{2}d_{\mathcal{H}_{D}}(\mathcal{D}_{S},\mathcal{D}_{T})+\lambda,$
$\displaystyle\hat{\mathbb{UB}}_{2}$
$\displaystyle=\epsilon_{S}(h)+\frac{1}{2}d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+\lambda.$
The proof of this theorem is provided in Appendix. VI-A.
Essentially, we limit the hypothesis space $\mathcal{H}$ and $\mathcal{H}_{D}$
in Eq. (11) into the space of deep neural networks. Directly optimizing over
$\hat{\mathbb{UB}_{2}}$ might be hard to converge since
$\hat{\mathbb{UB}_{2}}$ is a tighter upper bound for the prediction error on
target domain. The bounds relation in Eq. (11) shows that the
$\hat{\mathbb{UB}}_{1}$ is an upper bound of $\hat{\mathbb{UB}}_{2}$. This
provides us a clue to improve the training process of class alignment, i.e.,
the domain alignment can be a global constraint and narrow down the searching
space for the class alignment. This also implies that integrating the domain
alignment and class alignment might boost the training efficiency as well as
the prediction performance of UDA. An ideal training process is illustrated in
Fig. 2 where the searching space of $\hat{\mathbb{UB}}_{2}$ (CA) is constantly
bounded by that of $\hat{\mathbb{UB}}_{1}$ (DA), ensuring the whole training
process converge stably. This inspires us to design a new model, and we are
explaining next in details about our model structures, losses and training
process.
### IV-B CALI Structure
The existing DA or CA works usually involve a bilateral game. In CA, the game
is between a feature extractor and a family of classifiers. The two players
are optimized over the discrepancy of the two classifiers (note here the two
players are the two classifiers vs. the feature extractor) in an opposite
manner. In DA, the game happens between a segmentation net and a domain
discriminator. The two players are optimized over the domain discrimination in
an opposite way. It has been empirically showed [33, 32] that DA performs well
if the domain alignment happens to the prediction probability (after
Softmax()). However, according to the identified relation in Eq. (11), the two
upper bounds $\hat{\mathbb{UB}}_{1}$ and $\hat{\mathbb{UB}}_{2}$ need to use
the same feature, hence we connect the domain alignment and class alignment
using a shared feature extractor and propose a novel concept called PTGS (see
Fig. 3) to illustrate an interesting structure to integrate DA and CA. Both
$C_{s}$ and $D$ have game with $G$, but there is no game between $C_{s}$ and
$D$, hence we call this game as pseudo-trilateral game. Furthermore, as
defined in Eq. (8), $h$ and $h^{{}^{\prime}}$ are two different hypotheses,
thus we have to ensure the classifiers in $C_{s}$ are different during the
training.
Following the concept of PTGS, we design the structure of our CALI model as
shown in Fig. 4. Four networks are involved, a shared feature extractor $G$, a
domain discriminator $D$ and two classifiers $C_{1}$ and $C_{2}$. $f$
represents the shared features; $P_{1}/O_{1}$ and $P_{2}/O_{2}$ are the
probability/class predictions for $C_{1}$ and $C_{2}$, respectively; $S/T$
represent the source domain label (1) and target domain label (0); and $L_{1}$
represents the $L_{1}$ distance measure between two probability distributions.
The one-way solid arrows indicate the forward propagation of the data flow
while the two-way dashed arrows indicate losses are generated. The red arrows
represent the source-related data while the blue ones represent the target-
related data. The orange two-way dashed line indicates the structural
regularization loss between the $C_{1}$ and $C_{2}$.
Figure 4: CALI network structure. See Section. IV-B for more details.
### IV-C CALI Losses
We denote raw images from source or target domain as $\mathbf{x}$, and the
label from source domain as $\mathbf{y}$. We use semantic labels in source
domain to train all of the nets, but the domain discriminator, in a supervised
way, see the solid red one-way arrow in Fig. 4. We need to minimize the
supervised segmentation loss since Eq. (11) and other related Eqs suggest that
the source prediction error is also part of the upper bound of target error.
The supervised segmentation loss for training CALI is defined as
$\displaystyle\mathcal{L}_{seg}(G,C_{1},C_{2})=\frac{1}{2}\Bigg{(}\Bigg{.}\mathbb{E}_{(\mathbf{x},\mathbf{y})\sim\mathcal{D}_{S}}\left[-\mathbf{y}\log(C_{1}(G(\mathbf{x})))\right]+$
(12)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathbb{E}_{(\mathbf{x},\mathbf{y})\sim\mathcal{D}_{S}}\left[-\mathbf{y}\log(C_{2}(G(\mathbf{x})))\right]\Bigg{.}\Bigg{)}$
$\displaystyle=-\frac{1}{2}\mathbb{E}_{(\mathbf{x},\mathbf{y})\sim\mathcal{D}_{S}}[\mathbf{y}_{S}\log\left((C_{1}(G(\mathbf{x}))\odot(C_{2}(G(\mathbf{x}))))\right)],$
where $\odot$ represents the element-wise multiplication between two tensors.
To perform domain alignment, we need to define the joint loss function for $G$
and $D$
$\mathcal{V}_{1}(G,D)=-\left(\mathcal{CE}_{S}(\mathbf{x})+\mathcal{CE}_{T}(\mathbf{x})\right),$
(13)
where no segmentation labels but domain labels are used, and we use the
standard cross-entropy to compute the domain classification loss for both
source ($\mathcal{CE}_{S}(\mathbf{x})$) and target data
($\mathcal{CE}_{T}(\mathbf{x})$). We have
$\displaystyle\mathcal{CE}_{S}(\mathbf{x})$
$\displaystyle=\mathbb{E}_{\mathbf{x}\sim\mathcal{D}_{S}}[\mathcal{CE}([1,0]^{T},[D(G(\mathbf{x})),1-D(G(\mathbf{x}))]^{T})]$
(14)
$\displaystyle=\mathbb{E}_{\mathbf{x}\sim\mathcal{D}_{S}}[-\log(D(G(\mathbf{x})))].$
and
$\displaystyle\mathcal{CE}_{T}(\mathbf{x})$
$\displaystyle=\mathbb{E}_{\mathbf{x}\sim\mathcal{D}_{T}}[\mathcal{CE}([0,1]^{T},[D(G(\mathbf{x})),1-D(G(\mathbf{x}))]^{T})]$
(15)
$\displaystyle=\mathbb{E}_{\mathbf{x}\sim\mathcal{D}_{T}}[-\log(1-D(G(\mathbf{x})))],$
Note we include $G$ in Eq. (14) since both the source data and target data are
passed through the feature extractor. This is different than standard GAN,
where the real data is directly fed to $D$, without passing through the
generator.
To perform class alignment, we need to define the joint loss function for $G$,
$C_{1}$, and $C_{2}$
$\displaystyle\mathcal{V}_{2}(G,C_{1},C_{2})=\mathbb{E}_{\mathbf{x}\sim\mathcal{D}_{T}}\left[d(C_{1}(G(\mathbf{x})),C_{2}(G(\mathbf{x})))\right],$
(16)
where $d(\cdot,\cdot)$ is the distance measure between two distributions from
the two classifiers. In this paper, we use the same $L_{1}$ distance in [31]
as the measure, thus $d(p,q)=\frac{1}{K}|p-q|_{1}$, where $p$ and $q$ are two
distributions and $K$ is the number of label classes.
To prevent $C_{1}$ and $C_{2}$ from converging to the same network throughout
the training, we use the cosine similarity as a weight regularization to
maximize the difference of the weights from $C_{1}$ and $C_{2}$, i.e.,
$\mathcal{WR}(C_{1},C_{2})=\frac{\mathbf{w}_{1}\cdot\mathbf{w}_{2}}{\left\|\mathbf{w}_{1}\right\|\left\|\mathbf{w}_{2}\right\|},$
(17)
where $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ are the weight vectors of $C_{1}$
and $C_{2}$, respectively.
### IV-D CALI Training
We integrate the training processes of domain alignment and class alignment to
systematically train our CALI model. To be consistent with Eq. (11), we adopt
an iterative mechanism that alternates between domain alignment and class
alignment. Our training process is pseudo-coded in Algorithm 1.
1 Input: Source dataset $\mathcal{U}_{s}$; Target dataset $\mathcal{U}_{t}$;
Initial model $G,C_{1},C_{2}$ and $D$; Maximum iterations $M$; Iteration
interval $I$.
2 Output: Updated model parameters $\phi_{G},\theta_{C_{1}},\theta_{C_{2}}$
and $\psi_{D}$
3 Initialization: is_domain=True; is_class=False;
4 for _m $\leftarrow$ 1 to M_ do
5 if _$m\%I==0$ and $m\neq 0$_ then
6 is_domain = not is_domain;
7 is_class = not is_class;
8
// Eq. (12)
9
$\min_{\phi_{G},\theta_{C_{1}},\theta_{C_{2}}}\mathcal{L}_{seg}(G,C_{1},C_{2});$
// Eq. (17)
10 $\min_{\theta_{C_{1}},\theta_{C_{2}}}\mathcal{WR}(C_{1},C_{2});$
11 if _is_domain_ then
// Eq. (13)
12 $\max_{\psi_{D}}\min_{\theta_{G}}\mathcal{V}_{1}(G,D);$
13
14 if _is_class_ then
// Eq. 16
15
$\max_{\theta_{C_{1}},\theta_{C_{2}}}\min_{\phi_{G}}\mathcal{V}_{2}(G,C_{1},C_{2});$
16
17
Return $\phi_{G}$, $\theta_{C_{1}}$, $\theta_{C_{2}}$ and $\psi_{D}$;
Algorithm 1 CALI Training Process
Note the adversarial training order of $\mathcal{V}_{1}$ in Algorithm 1 is
$\max_{\psi_{D}}\min_{\phi_{G}}$, instead of the
$\min_{\phi_{G}}\max_{\psi_{D}}$, meaning in each training iteration we first
train the feature extractor and then the discriminator. The reason for this
order is because we empirically find that the feature from $G$ is relatively
easy for $D$ to discriminate, hence if we train $D$ first, then the $D$ might
become an accurate discriminator in the early stage of training and there will
be no adversarial signals for training $G$, thus making the whole training
fail. The same order applies to training of the pair of $G$ and $Cs$ with
$\mathcal{V}_{2}$.
Figure 5: Qualitative results on adaptation GTA5$\rightarrow$Cityscapes.
Results of our proposed model is listed in the last second column. GT
represents the ground-truth labels.
### IV-E Visual Planner
We design a visual receding horizon planner to achieve feasible visual
navigation by combining the learned image segmentation. Specifically, first we
compute a library of motion primitives [14, 15]
$\mathcal{M}=\left\\{\mathbf{p}_{1},\mathbf{p}_{2},\cdots,\mathbf{p}_{n}\right\\}$
where each
$\mathbf{p}_{*}=\left\\{\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{m}\right\\}$
is a single primitive. We use
$\mathbf{x}_{*}=\begin{bmatrix}x&y&\psi\end{bmatrix}^{T}$ to denote a robot
pose. Then we project the motion primitives to the image plane and compute the
navigation cost function for each primitive based on the evaluation of
collision risk in image space and target progress. Finally, we select the
primitive with minimal cost to execute. The trajectory selection problem can
be defined as:
$\mathbf{p}_{optimal}=\underset{\mathbf{p}}{\text{argmin}}~{}w_{1}\cdot
C_{c}(\mathbf{p})+w_{2}\cdot C_{t}(\mathbf{p}),$ (18)
where $C_{c}(\mathbf{p})=\sum_{j}^{m}c_{c}^{j}$ and
$C_{t}(\mathbf{p})=\sum_{j}^{m}c_{t}^{j}$ are the collision cost and target
cost of one primitive $\mathbf{p}$, and $w_{1}$, $w_{2}$ are corresponding
weights, respectively.
To efficiently evaluate the collision risk in the learned segmentation images,
we first classify the classes in terms of their navigability, e.g., in off-
road environments, grass and mulch are classified as navigable while tree and
bush are classified as non-navigable. In this case, we are able to extract the
boundary between the navigable space and the non-navigable space. We treat the
boundary part close to the bottom line of the image as the obstacle boundary.
We further use the obstacle boundary to generate a Scaled Euclidean Distance
Field (SEDF), where the values fall in $[0,1]$, representing the risk level at
the pixel position. Examples of different SEDF with different scale factors
can be seen in Fig. 6.
Assume $\mathbf{x}^{j}$ is the $j^{th}$ pose in one primitive and its image
coordinates are $\left(u^{j},v^{j}\right)$, then the collision risk for
$\mathbf{x}^{j}$ is
$c_{c}^{j}=E[u^{j},v^{j}],$ (19)
where $E$ represents the SEDF image.
Figure 6: Different SEDFs with varying scale factors of (a) $\alpha=0.25$, (b)
$\alpha=0.55$ and (c) $\alpha=1.00$. Values range from 0 to 1 by the color
from blue to yellow. Figure 7: Qualitative results on adaptation
RUGD$\rightarrow$RELLIS. Results of our proposed model is listed in the last
second column. GT represents the ground-truth labels.
To evaluate target progress during the navigation progress, we propose to use
the distance on $SE(3)$ as the metric. We define three types of frames: world
frame $F_{w}$, primitive pose frame $F_{pj}$, and goal frame $F_{g}$. The
transformation of $F_{pj}$ in $F_{w}$ is denoted as $\mathbf{T}_{wpj}$ while
that of $F_{g}$ in $F_{w}$ is $\mathbf{T}_{wg}$. A typical approach to
represent the distance is to split a pose into a position and an orientation
and define two distances on $\mathbb{R}^{3}$ and $SO(3)$. Then the two
distances can be fused in a weighted manner with two strictly positive scaling
factors $a$ and $b$ and with an exponent parameter $p\in[1,\infty]$ [5]:
$\displaystyle d(\mathbf{T}_{wpj},\mathbf{T}_{wg})=\Bigg{[}\Bigg{.}$
$\displaystyle a\cdot d_{rot}(\mathbf{R}_{wpj},\mathbf{R}_{wg})^{p}+$ (20)
$\displaystyle b\cdot
d_{trans}(\mathbf{t}_{wpj},\mathbf{t}_{wg})^{p}\Bigg{.}\Bigg{]}^{1/p}.$
We use the Euclidean distance as
$d_{trans}(\mathbf{t}_{wpj},\mathbf{t}_{wg})$, the Riemannian distance over
$SO(3)$ as $d_{rot}(\mathbf{R}_{wpj},\mathbf{R}_{wg})$ and set $p$ as $2$.
Then the distance (target cost) between two transformation matrices can be
defined [25] as:
$\displaystyle c_{t}^{j}$ $\displaystyle=d(\mathbf{T}_{wpj},\mathbf{T}_{wg})$
(21)
$\displaystyle=\left[a\cdot\left\|\log(\mathbf{R}_{wpj}^{-1}\mathbf{R}_{wg})\right\|^{2}+b\cdot\left\|\mathbf{t}_{wpj}-\mathbf{t}_{wg}\right\|^{2}\right]^{1/2}.$
## V Experiments
Figure 8: Qualitative results on adaptation RUGD$\rightarrow$MESH. Results of
our proposed model is listed in the last column.
### V-A Datasets
We evaluate CALI together with several baseline methods on a few challenging
domain adaptation scenarios, where several public datasets, e.g., GTA5 [29],
Cityscapes [8], RUGD [35], RELLIS [17], as well as a small self-collected
dataset, named MESH (see the first column of Fig. 8), are investigated. The
GTA5 dataset contains $24966$ synthesized high-resolution images in the urban
environments from a video game and pixel-wise semantic annotations of 33
classes. The Cityscapes dataset consists of $5000$ finely annotated images
whose label is given for 19 commonly seen categories in urban environments,
e.g., road, sidewalk, tree, person, car, etc. The RUGD and RELLIS are two
recently released datasets that aim to evaluate segmentation performance in
off-road environments. The RUGD and the RELLIS contain 24 and 20 classes with
$8000$ and $6000$ images, respectively. RUGD and RELLIS cover various scenes
like trails, creeks, parks, villages, and puddle terrains. Our dataset, MESH,
includes features like grass, trees (particularly challenging in winter due to
foliage loss and monochromatic colors), mulch, etc. It helps us to further
validate the performance of our proposed model for traversability prediction
in challenging scenes, particularly the off-road environments.
TABLE I: Quantitative comparison of different methods in UDA of GTA5$\rightarrow$Cityscapes. mIoU* represents the average mIoU over all of classes. Class | SO | DA | CA | CALI
---|---|---|---|---
Road | 38.86 | 52.80 | 78.56 | 75.36
Sidewalk | 17.47 | 18.95 | 2.79 | 27.12
Building | 63.60 | 61.73 | 43.51 | 67.00
Sky | 58.08 | 54.35 | 46.59 | 60.49
Vegetation | 67.21 | 64.69 | 41.48 | 67.50
Terrain | 7.63 | 7.04 | 8.37 | 9.56
Person | 16.89 | 15.45 | 13.48 | 15.03
Car | 30.32 | 43.41 | 31.64 | 52.25
Pole | 11.61 | 12.38 | 9.68 | 11.91
mIoU* | 34.63 | 36.76 | 30.68 | 42.91
### V-B Implementation Details
To be consistent with our theoretical analysis, the implementation of CALI
only adopts the necessary indications by Eq. (11). First, Eq. (11) requires
that the input of the two upper bounds (one for DA and the other one for CA)
should be the same. Second, nothing else but only domain classification and
hypotheses discrepancy are involved in Eq. (11) and other related analyses
(Eq. (3) - Eq. (9)). Accordingly, we strictly follow the guidance of our
theoretical analyses. First, CALI performs DA in the intermediate-feature
level ($f$ in Fig. 4), instead of the output-feature level used in [33].
Second, we exclude the multiple additional tricks, e.g., entropy-based and
multi-level features based alignment, and class-ratio priors in [33] and
multi-steps training for feature extractor in [31]. We also implement baseline
methods without those techniques for a fair comparison. To avoid possible
degraded performance bought by a class imbalance in the used datasets, we
regroup those rare classes into classes with a higher pixel ratio. For
example, we treat the building, wall, and fence as the same class; the person
and rider as the same class in the adaptation of GTA5$\rightarrow$Cityscapes.
In the adaptation of RUGD$\rightarrow$RELLIS, we treat the tree, bush, and log
as the same class, and the rock and rockbed as the same class. Details about
remapping can be seen in Fig. 14 and Fig. 15 in Appendix. VI-B.
TABLE II: Quantitative comparison of different methods in UDA of RUGD$\rightarrow$RELLIS. mIoU* is the average mIoU over all of classes. Class | SO | DA | CA | CALI
---|---|---|---|---
Dirt | 0.00 | 0.53 | 3.23 | 0.01
Grass | 64.78 | 61.63 | 65.35 | 67.08
Tree | 40.79 | 45.93 | 41.51 | 55.80
Sky | 45.07 | 67.00 | 2.31 | 72.99
Building | 10.90 | 12.29 | 10.91 | 10.28
mIoU* | 32.31 | 37.48 | 24.66 | 41.23
Figure 9: Target discrepancy changes during training process of (a)
GTA5$\rightarrow$Cityscapes; (b) RUGD$\rightarrow$RELLIS; and (c)
RUGD$\rightarrow$MESH.
Figure 10: Using minmax can cause the collapse of training. Figure 11: An
example of collapsed trained model using minmax.
We use the PyTorch [26] framework for implementation. Training images from
source and target domains are cropped to be half of their original image
dimensions. The batch size is set to 1 and the weights of all batch
normalization layers are fixed. We use the ResNet-101 [11] pretrained on
ImageNet [9] as the model $G$ for extracting features. We use the ASPP module
in DeepLab-V2 [6] as the structure for $C_{1}$ and $C_{2}$. We use the similar
structure in [27] as the discriminator $D$, which consists of 5 convolution
layers with kernel $4\times 4$ and with channel size
$\left\\{64,128,256,512,1\right\\}$ and stride of 2. Each convolution layer is
followed by a Leaky-ReLU [22] parameterized by 0.2, but only the last
convolution layer is follwed by a Sigmoid function. During the training, we
use SGD [4] as the optimizer for $G,C_{1}$ and $C_{2}$ with a momentum of 0.9,
and use Adam [18] to optimize $D$ with $\beta_{1}=0.9,\beta_{2}=0.99$. We set
all SGD optimizers a weight decay of $5\text{e-}4$. The initial learning rates
of all SGDs for performing domain alignment are set to $2.5\text{e-}4$ and the
one of Adam is set as $1\text{e-}4$. For class alignment, the initial learning
rate of SGDs is set to $1\text{e-}3$. All of the learning rates are decayed by
a poly learning rate policy, where the initial learning rate is multiplied by
$(1-\frac{iter}{max\\_iters})^{power}$ with $power=0.9$. All experiments are
conducted on a single Nvidia Geforce RTX 2080 Super GPU.
### V-C Comparative Studies
We present comparative experimental results of our proposed model, CALI,
compared to different baseline methods – Source-Only (SO) method, Domain-
Alignment (DA) [33] method, and Class-Alignment [31] method. Specifically, we
first perform evaluations on a sim2real UDA in city-like environments, where
the source domain is represented by GTA5 while the target domain is the
Cityscapes. Then we consider a transfer of real2real in forest environments,
where the source domain and target domain are set as RUGD and RELLIS,
respectively. All models are trained with full access to the images and labels
in the source domain and with only access to the images in the target domain.
The labels in target datasets are only used for evaluation purposes. Finally,
we further validate our model performance for adapting from RUGD to our self-
collected dataset MESH.
To ensure a fair comparison, all the methods use the same feature extractor
$G$; both DA and CALI have the same domain discriminator $D$; both CA and CALI
have the same two classifiers $C_{1}$ and $C_{2}$. We also use the same
optimizers and optimization-related hyperparameters if any is used for models
under comparison.
We use the mean of Intersection over Union (mIoU) as the metric to evaluate
each class and overall segmentation performance on testing images. IoU is
computed as $\frac{n_{tp}}{n_{tp}+n_{fp}+n_{fn}}$, where
$n_{tp},n_{tn},n_{fp}$ and $n_{fn}$ are true positive, true negative, false
positive and false negative, respectively.
#### V-C1 GTA5$\rightarrow$Cityscapes
Quantitative comparison results of GTA5$\rightarrow$Cityscapes are shown in
Table. I, where segmentations are evaluated on 9 classes (as regrouped in Fig.
14). Our proposed method has significant advantages over multiple baseline
methods for most categories and overall performance (mIoU*).
Figure 12: Navigation behaviors in MESH$\\#1$ environment. The left-most
column: top-down view of the environment; Purple triangle: the starting point;
Blue star: the target point; We also show the segmentation (top row) and
planning results (bottom row) at four different moments during the navigation,
as shown from the second column to the last one. Figure 13: Navigation
behaviors in MESH$\\#2$ environment. Same legends with Fig. 12.
In our testing case, SO achieves the highest score for the class person even
without any domain adaptation. One possible reason for this is the deep
features of the source person and the target person from the model solely
trained on source domain, are already well-aligned. If we try to interfere
this well-aligned relation using unnecessary additional efforts, the target
prediction error might be increased (see the mIoU values of the person from
the other three methods). We call this phenomenon as negative transfer, which
also happens to other classes if we compare SO and DA/CA, e.g., sidewalk,
building, sky, vegetation, and so on. In contrast, CALI maintains an improved
performance compared to either SO or DA/CA. We validate our analytical method
for DA and CA (Section. IV-A) by a comparison between CALI and baselines. This
indicates either single DA or CA is problematic for semantic segmentation,
particularly when we strictly follow what the theory supports and do not
include any other training tricks (that might increase the training complexity
and make the training unstable). This implies that integration of DA and CA is
beneficial to each other with significant improvements, and more importantly,
CALI is well theoretically supported, and the training process is easy and
stable.
Fig. 5 shows the examples of qualitative comparison for UDA of
GTA5$\rightarrow$Cityscapes. We find that CALI prediction is less noisy
compared to the baselines methods as shown in the second and third columns
(sidewalk or car on-road), and shows better completeness (part of the car is
missing, see the fourth column).
#### V-C2 RUGD$\rightarrow$RELLIS
We show quantitative results of RUGD$\rightarrow$RELLIS in Table. II, where
only 5 classes222This is because other classes (in Fig. 15) frequently
appearing in source domain (RUGD) are extremely rare in target domain
(RELLIS), hence no prediction for those classes occurs especially considering
the domain shift. are evaluated. It shows the same trend as Table. I. Both
tables show that CA has the negative transfer (compared with SO) issue for
either sim2real or real2real UDA. However, if we constrain the training of CA
with DA, as in our proposed model, then the performance will be remarkably
improved. Some qualitative results are shown in Fig. 7.
#### V-C3 RUGD$\rightarrow$MESH
Our MESH dataset contains only unlabeled images that restrict us to show only
a qualitative comparison for the UDA of RUGD$\rightarrow$MESH, as shown in
Fig. 8. We have collected data in winter forest environments, which are
significantly different than the images in the source domain (RUGD) -
collected in a different season, e.g., summer or spring. These cross-season
scenarios make the prediction more challenging. However, it is more practical
to evaluate the UDA performance of cross-season scenarios, as we might have to
deploy our robot at any time, even with extreme weather conditions, but our
available datasets might be far from covering every season and every weather
condition. From Fig. 8, we can still see the obvious advantages of our
proposed CALI model over other baselines.
### V-D Discussions
In this section, we aim to discuss our model behaviors in more details.
Specifically, first we will explain the advantages of CALI over CA from the
perspective of training process. Second, we will show the vital influence of
mistakenly using wrong order of adversarial training.
The most important part in CA is the discrepancy between the two classifiers,
which is the only training force for the functionality of CA. It has been
empirically studied in [31] that the target prediction accuracy will increase
as the target discrepancy is decreasing, hence the discrepancy is also an
indicator showing if the training is on the right track. We compare the target
discrepancy changes of CALI and our baseline CA in Fig. 9, where the curves
for the three UDA scenarios are presented from (a) to (c) and we only show the
data before iteration 30k. It can be seen that before around iteration 2k, the
target discrepancy of both CALI and CA are drastically decreasing, but
thereafter, the discrepancy of CA starts to increase. On the other hand, if we
impose a DA constraint over the same CA (iteratively), leading to our proposed
CALI, then the target discrepancy will be decreasing as expected. This
validates that integrating DA and CA will make the training process of CA more
stable, thus improving the target prediction accuracy.
As mentioned in Algorithm 1, we have to use adversarial training order of
$\max_{\psi_{D}}\min_{\phi_{G}}$, instead of $\min_{\phi_{G}}\max_{\psi_{D}}$.
The reason for this is related to our designed net structure. Following the
guidance of Eq. (11), we use the same input to the two classifiers and the
domain discriminator, hence the discriminator has to receive the intermediate-
level feature as the input. If we use the order of
$\min_{\phi_{G}}\max_{\psi_{D}}$ in CALI, then the outputs of the
discriminator will be like Fig. 10, where the domain discriminator of CALI
will quickly converge to the optimal state and it can accurately discriminate
if the feature is from source or target domain. In this case, the adversarial
loss for updating the feature extractor will be near 0, hence the whole
training fails, which is validated by changes of the target discrepancy curve,
as shown in Fig. 10, where the discrepancy value is decreasing in a small
amount in the first few iterations and then quickly increase to a high level
that shows the training is divergent and the model is collapsed. This is also
verified by the prediction results at (and after) around iteration 1k, as
shown in Fig. 11, where the first row is the source images while the second
row is the target images.
### V-E Navigation Missions
To further show the effectiveness of our proposed model for real deployments,
we build a navigation system by combining the proposed CALI (trained with
RUGD$\rightarrow$MESH set-up) segmentation model with our visual planner. We
test behaviors of our navigation system in two different forest environments
(named MESH$\\#1$ in Fig. 12 and MESH$\\#2$ in Fig. 13), where our navigation
system shows high reliability. In navigation tasks, the image resolution is
$[400,300]$, and the inference time for pure segmentation inference is around
$33$ frame per second (FPS). However, since a complete perception system
requires several post-processing steps, such as navigability definition, noise
filtering, Scaled Euclidean Distance Field computation, motion primitive
evaluation and so on, the response time for the whole perception pipeline (in
python) is around $8$ FPS without any engineering optimization. The inference
of segmentation for navigation is performed on an Nvidia Tesla T4 GPU. We set
the linear velocity as $0.3m/s$ and control the angular velocity to track the
selected motion primitive. The path length is $32.26m$ in Fig. 12 and $28.63m$
in Fig. 13. Although the motion speed is slow in navigation tasks, as a proof
of concept and with a very basic motion planner, the system behavior is as
expected, and we have validated that the proposed CALI model is able to well
accomplish the navigation tasks in unstructured environments.
## VI Conclusion
We present CALI, a novel unsupervised domain adaptation model specifically
designed for semantic segmentation, which requires fine-grained alignments in
the level of class features. We carefully investigate the relationship between
a coarse alignment and a fine alignment in theory. The theoretical analysis
guides the design of the model structure, losses, and training process. We
have validated that the coarse alignment can serve as a constraint to the fine
alignment and integrating the two alignments can boost the UDA performance for
segmentation. The resultant model shows significant advantages over baselines
in various challenging UDA scenarios, e.g., sim2real and real2real. We also
demonstrate the proposed segmentation model can be well integrated with our
proposed visual planner to enable highly efficient navigation in off-road
environments.
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## Appendix
### VI-A Proof of Theorem 2
For a hypothesis $h$,
$\displaystyle\epsilon_{T}(h)$
$\displaystyle\leq\epsilon_{T}(h^{*})+\epsilon_{T}(h,h^{*})$ (22)
$\displaystyle=\epsilon_{T}(h^{*})+\epsilon_{S}(h,h^{*})-\epsilon_{S}(h,h^{*})+\epsilon_{T}(h,h^{*})$
$\displaystyle\leq\epsilon_{T}(h^{*})+\epsilon_{S}(h,h^{*})+|\epsilon_{T}(h,h^{*})-\epsilon_{S}(h,h^{*})|$
$\displaystyle\leq\epsilon_{T}(h^{*})+\epsilon_{S}(h,h^{*})+\frac{1}{2}d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})$
$\displaystyle\leq\epsilon_{T}(h^{*})+\epsilon_{S}(h)+\epsilon_{S}(h^{*})+\frac{1}{2}d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})$
$\displaystyle=\epsilon_{S}(h)+\frac{1}{2}d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+\epsilon_{S}(h^{*})+\epsilon_{T}(h^{*})$
$\displaystyle=\epsilon_{S}(h)+\frac{1}{2}d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+\lambda$
$\displaystyle=\epsilon_{S}(h)+\sup_{h,h^{{}^{\prime}}\in\mathcal{H}}|\text{P}_{\mathbf{x}\sim\mathcal{D}_{S}}\left[h(\mathbf{x})\neq
h^{{}^{\prime}}(\mathbf{x})\right]-$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{P}_{\mathbf{x}\sim\mathcal{D}_{T}}\left[h(\mathbf{x})\neq
h^{{}^{\prime}}(\mathbf{x})\right]|+\lambda$
$\displaystyle=\epsilon_{S}(h)+\sup_{g\in\mathcal{H}\Delta\mathcal{H}}|\text{P}_{\mathbf{x}\sim\mathcal{D}_{S}}\left[g(\mathbf{x})=1)\right]-$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{P}_{\mathbf{x}\sim\mathcal{D}_{T}}\left[g(\mathbf{x})=1\right]|+\lambda$
$\displaystyle=\epsilon_{S}(h)+\sup_{g\in\mathcal{H}\Delta\mathcal{H}}|\text{P}_{\mathbf{x}\sim\mathcal{D}_{S}}\left[g(\mathbf{x})=1)\right]+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{P}_{\mathbf{x}\sim\mathcal{D}_{T}}\left[g(\mathbf{x})=0\right]-1|+\lambda$
$\displaystyle\leq\epsilon_{S}(h)+\sup_{g\in\mathcal{H}\Delta\mathcal{H}}|\text{P}_{\mathbf{x}\sim\mathcal{D}_{S}}\left[g(\mathbf{x})=1)\right]+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{P}_{\mathbf{x}\sim\mathcal{D}_{T}}\left[g(\mathbf{x})=0\right]|-\inf_{g\in\mathcal{H}\Delta\mathcal{H}}1+\lambda$
$\displaystyle=\epsilon_{S}(h)+\sup_{g\in\mathcal{H}\Delta\mathcal{H}}|\text{P}_{\mathbf{x}\sim\mathcal{D}_{S}}\left[g(\mathbf{x})=1)\right]+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{P}_{\mathbf{x}\sim\mathcal{D}_{T}}\left[g(\mathbf{x})=0\right]|+\lambda-1$
$\displaystyle=\epsilon_{S}(h)+\frac{1}{2}d_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+1+\lambda-1$
$\displaystyle=\epsilon_{S}(h)+\frac{1}{2}d_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+\lambda,$
where $\lambda=\epsilon_{S}(h^{*})+\epsilon_{T}(h^{*})$ and $h^{*}$ is the
ideal joint hypothesis (see the Definition 2 in Section 4.2 of [2]).
We have the $4^{th}$, and the $8^{th}$ line because of the Lemma 3 [2]; the
$5^{th}$ line because of the Theorem 2 [2]; the last second line because of
the Lemma 2 [2]. We have the $11^{th}$ line because $\sup|f_{1}-f_{2}|=\sup
f_{1}-\inf f_{2}\leq\sup|f_{1}|-\inf
f_{2}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\blacksquare$
### VI-B Remapping of Label Space
We regroup the original label classes according to the semantic similarities
among classes. In GTA5 and Cityscapes, we cluster the building, wall and fence
as the same category; traffic light, traffic sign and pole as the same group;
car, train. bicycle, motorcycle, bus and truck as the same class; and treat
the person and rider as the same one. See Fig. 14. Similarly, we also have
regroupings for classes in RUGD and RELLIS, as can be seen in Fig. 15.
Figure 14: Lable remapping for GTA5$\rightarrow$Cityscapes. Name of each new
group is marked as bold. Figure 15: Lable remapping for
RUGD$\rightarrow$RELLIS and RUGD$\rightarrow$MESH. Name of each new group is
marked as bold.
|
HU-Mathematik-2021-06
HU-EP-21/51
SAGEX-21-37-E
Combinatorial Solution of the
Eclectic Spin Chain
Changrim Ahna, Luke Corcoranb, Matthias Staudachera,b
a Department of Physics, Ewha Womans University,
52 Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, S. Korea
b Institut für Physik, Humboldt-Universität zu Berlin,
Zum Großen Windkanal 2, 12489 Berlin, Germany
ahn<EMAIL_ADDRESS>
Abstract
The one-loop dilatation operator in the holomorphic 3-scalar sector of the
dynamical fishnet theory is studied. Due to the non-unitary nature of the
underlying field theory this operator, dubbed in [1] the eclectic spin chain
Hamiltonian, is non-diagonalisable. The corresponding spectrum of Jordan
blocks leads to logarithms in the two-point functions, which is characteristic
of logarithmic conformal field theories. It was conjectured in [2] that for
certain filling conditions and generic couplings the spectrum of the eclectic
model is equivalent to the spectrum of a simpler model, the hypereclectic spin
chain. We provide further evidence for this conjecture, and introduce a
generating function which fully characterises the Jordan block spectrum of the
simplified model. This function is found by purely combinatorial means and is
simply related to the $q$-binomial coefficient.
###### Contents
1. 1 Introduction and Overview
2. 2 The (Hyper)eclectic Spin Chain
1. 2.1 Hamiltonian
2. 2.2 Translation Operator and Cyclicity Classes
3. 2.3 Spectral Problem
3. 3 Hypereclectic with One Wall
1. 3.1 Warmup Examples
2. 3.2 Generating Function
4. 4 General Hypereclectic
1. 4.1 Warmup Examples
2. 4.2 General $L,M,K$
5. 5 Eclectic Spin Chain and Universality
1. 5.1 Eclectic Spin Chain and Level $S$
2. 5.2 Warmup Example
3. 5.3 General Argument for $K=1$
6. 6 Conclusions and Outlook
7. A Unexpected Shortening
8. B Universality Details for $K=1$
9. C Fine Tuning and Cyclicity Classes
## 1 Introduction and Overview
Integrability of gauge and string theories continues to generate exciting new
types of exactly solvable models, ranging from new spin chains to novel
quantum field theories and hitherto unstudied string theories. Frequently this
stems from deformations and/or subtle limit taking of previously investigated
systems. A case at hand is a certain double-scaling limit of the three-
parameter $\gamma$-deformation of $\mathcal{N}=4$ super Yang-Mills theory that
had originally been proposed in [3, 4]. It partially or fully breaks
R-symmetry and thus also supersymmetry, while apparently retaining
conformality and integrability in the planar limit. Its double-scaling limit
was then proposed in [5], combining a strong imaginary $\gamma$-twist with a
vanishing coupling constant. In this limit all gauge field interactions
decouple and one is left with an un-gauged quantum field theory of scalars
$\phi_{i}$ and fermions $\psi_{i}$, $i=1,2,3$. In light of [6] we refer to the
resulting theory as the dynamical fishnet theory, with interaction Lagrangian
$\displaystyle\mathcal{L}^{\text{int}}_{\text{DFN}}$
$\displaystyle=N_{c}\text{tr}\left(\xi_{1}^{2}\phi^{\dagger}_{2}\phi^{\dagger}_{3}\phi_{2}\phi_{3}+\xi_{2}^{2}\phi^{\dagger}_{3}\phi^{\dagger}_{1}\phi_{3}\phi_{1}+\xi_{3}^{2}\phi^{\dagger}_{1}\phi^{\dagger}_{2}\phi_{1}\phi_{2}\right)$
(1.1)
$\displaystyle+N_{c}\text{tr}\left(i\sqrt{\xi_{2}\xi_{3}}(\psi^{3}\phi_{1}\psi^{2}+\bar{\psi}_{3}\phi_{1}^{\dagger}\bar{\psi}_{2})+\text{cyclic}\right).$
This model can be further simplified by taking
$\xi_{1}=\xi_{2}=0,\xi_{3}\equiv\xi$. In this case we recover the bi-scalar
fishnet theory
$\mathcal{L}_{\text{FN}}^{\text{int}}=\xi^{2}N_{c}\text{tr}\left(\phi^{\dagger}_{1}\phi^{\dagger}_{2}\phi_{1}\phi_{2}\right).$
(1.2)
Notably, the theories (1.1) and (1.2) are non-unitary. However, the chiral
nature of these interactions leads to a vast simplification in the Feynman-
diagrammatic structure of many physical quantities. This raises the hope that
the integrability of these models might be more easily understood from first
principles. Recall that the origin of integrability of undeformed or
$\gamma$-deformed $\mathcal{N}=4$ SYM remains shrouded in mystery. On the
contrary, in the chiral models one often observes recursive structures in the
Feynman graphs, and the associated graph-building operator may sometimes be
shown to possess integrable properties [6, 7, 8]. In some cases, correlation
functions are represented by a single Feynman diagram. An example of this is
the fishnet Feynman integrals. These have been shown to enjoy a Yangian
symmetry [9], which in some cases has been sufficient to bootstrap the
integral [10, 11]. In a four-point limit these fishnet graphs reduce to the
celebrated Basso-Dixon correlators, for which integrability has been studied
from various perspectives [12, 13, 14]. The fishnet theory has also been
argued to possess at strong coupling a holographic dual [15, 16, 17].
As written, the theories (1.1) and (1.2) are not strictly conformal, even in
their planar limit [18, 19]. Double trace couplings are generated upon
renormalisation. However, it has been argued in [6, 20] that these coupling
may be fine-tuned as a function of $\xi^{2}$ such that the overall beta-
function becomes identically zero, while preserving integrability. As a
result, one expects to get an integrable logarithmic conformal field theory.
This is a consequence of the models’ non-unitarity: While the state space is
still reducible, it is no longer decomposable. The logarithmic nature of the
underlying CFT poses curious new challenges for the spectral problem of the
theory [1, 2]. In particular, in certain operator sectors the dilatation
operator is no longer diagonalisable. It is known that this leads to the
appearance of logarithms in the two-point functions [21]. For example, in the
simplest case where the dilatation operator acts on an operator pair
$\mathcal{O}_{1},\mathcal{O}_{2}$ as a $2\times 2$ Jordan cell
$\mathfrak{D}\begin{pmatrix}\mathcal{O}_{1}\\\
\mathcal{O}_{2}\end{pmatrix}=\begin{pmatrix}\Delta&1\\\
0&\Delta\end{pmatrix}\begin{pmatrix}\mathcal{O}_{1}\\\
\mathcal{O}_{2}\end{pmatrix},$ (1.3)
the two-point function can be brought into the form
$\langle\mathcal{O}_{i}(x)\mathcal{O}_{j}(0)\rangle=\frac{c}{|x|^{2\Delta}}\begin{pmatrix}\log
x^{2}&1\\\ 1&0\end{pmatrix}.$ (1.4)
An explicit example of this in the fishnet theory for length $5$ operators is
given in [7].
Logarithmic conformal field theories play an important role in two dimensions
[22]. There, due to their direct connection with two-dimensional statistical
mechanics models, they are of great physical interest. Important examples
include models of self-avoiding walks, polymers, and percolation; for recent
progress see [23] and references therein. Often the logarithmic scaling
violations occurring in these models are of both experimental and theoretical
interest. In fact, their mathematical analysis often shows intricate and novel
features as compared to the non-logarithmic case. In higher dimensions,
logarithmic CFTs have been much less studied. Still, given their success story
in two dimensions, it is natural to suspect that they will also be of
considerable value.
A systematic study of the dilatation operator in strongly twisted
$\mathcal{N}=4$ SYM was initiated in [1]. It was found that the mentioned non-
diagonalisability is ubiquitous in these models, leading to a rich structure
of Jordan cells. It was also pointed out that the standard methods of
integrability largely fail when applied to the model’s non-diagonalisable
sectors. This was then studied in more detail in a particularly simple
setting, namely at one-loop and with three scalars of equal chirality, in [2].
The resulting spin chain was dubbed the eclectic spin chain in [1, 2], and an
even simpler model, the hypereclectic spin chain was proposed, but not solved.
Interestingly, the latter appeared to possess an even richer spectrum of
Jordan decompositions as compared to the one in the generic eclectic model.
This phenomenon was called universality in [2].
The current work seamlessly continues [2], and proceeds to find the exact
solution of the hypereclectic model. Curiously, for the moment this does not
use at all the model’s integrability, but instead combines methods of linear
algebra and combinatorics. As a result we obtained an elegant generating
function for the spectrum of Jordan blocks. It is reminiscent of a partition
function, since it can be obtained by computing a trace over the state space
$\mathcal{Z}(q)=\text{tr}\hskip 1.42271ptq^{\hat{S}^{\prime}}\,,$ (1.5)
where $\hat{S}^{\prime}$ is a certain counting operator, which is diagonal in
the canonical basis of tensor product states of the spin chain, see end of
section 4.2. It uniquely encodes in full generality the sizes and
multiplicities of the Hamiltonian’s Jordan block decomposition:
$\displaystyle\mathcal{Z}(q)=\sum_{j}N_{j}[j]_{q}=N_{1}\,q^{0}+N_{2}\,\left(q^{-\frac{1}{2}}+q^{\frac{1}{2}}\right)+N_{3}\,\left(q^{-1}+q^{0}+q^{1}\right)+\ldots\,,$
(1.6)
where $N_{j}$ is the number of Jordan blocks of length $j$, and $[j]_{q}$ is a
$q$-analog of $j$, cf. (3.38). It is easy to see that the $\\{N_{j}\\}$ are
indeed uniquely fixed once one knows $\mathcal{Z}(q)$. We also derive formulas
expressing $\mathcal{Z}(q)$ more explicitly than (1.5) in terms of
$q$-binomial coefficients. For example, for the case corresponding to the
fishnet interaction Lagrangian (1.2), with $L-M$ fields $\phi_{1}$, $M-1$
fields $\phi_{2}$, and a single, non-interacting third field $\phi_{3}$, we
find for the one-loop spectrum of Jordan blocks in the cyclic sector the
(shifted) $q$-binomial coefficients
$Z_{L,M}(q)=\genfrac{[}{]}{0.0pt}{0}{L-1}{M-1}_{q}=\prod_{k=1}^{M-1}\frac{q^{\frac{L-k}{2}}-q^{-\frac{L-k}{2}}}{q^{\frac{k}{2}}-q^{-\frac{k}{2}}}\,.$
(1.7)
Interestingly, this result is also valid for the dynamical fishnet theory
interaction Lagrangian (1.1) (for generic couplings) due to the phenomenon of
universality already pointed out in [2].
The paper is organised as follows. Section 2 recalls the definitions of the
non-hermitian chiral spin chain models at hand. The hypereclectic spin chain
has a particularly simple Hamiltonian, essentially describing chiral right-
movers on a chain along with a number of impenetrable non-movers, which we
call walls. Section 3 derives the exact solution of this model in the case of
a single wall. The partition function method is introduced, and the solution
is expressed in terms of $q$-binomial coefficients. Section 4 generalises
these findings to an arbitrary number of walls. Section 5 analyses the generic
three-parameter eclectic model, and, for the case of a single wall, sketches a
proof of the universality hypothesis. Some remarks on the general case are
made. We end with the short, concluding section 6, where it is also pointed
out that the most important open issue seems to be our current inability to
use integrability to analyse these integrable models. A few appendices A,B,C
give further technical details.
## 2 The (Hyper)eclectic Spin Chain
In this section we collect basic facts about the models under consideration.
### 2.1 Hamiltonian
We consider local single-trace operators in the holomorphic 3-scalar sector of
the theory (1.1)
$\mathcal{O}_{j_{1},j_{2},\dots,j_{L}}(x)=\text{tr}\left(\phi_{j_{1}}\phi_{j_{2}}\cdots\phi_{j_{L}}(x)\right),\hskip
28.45274ptj_{i}\in\\{1,2,3\\}.$ (2.1)
In $\mathcal{N}=4$ SYM the one-loop dilatation operator in the analogous
sector can be written as a sum over permutation operators and enjoys an
$\mathfrak{su}(3)$ symmetry [24]. In the strongly twisted theory (1.1) this
symmetry is broken and the one-loop dilatation operator
$H_{\text{ec}}:\left(\mathbb{C}^{3}\right)^{\otimes
L}\rightarrow\left(\mathbb{C}^{3}\right)^{\otimes L}$ is a sum over chiral
permutation operators [2]
$H_{\text{ec}}=H_{1}+H_{2}+H_{3}=\sum_{i=1}^{L}\left(\xi_{1}\mathcal{H}_{1}^{i,i+1}+\xi_{2}\mathcal{H}_{2}^{i,i+1}+\xi_{3}\mathcal{H}_{3}^{i,i+1}\right).$
(2.2)
The chiral permutation operators
$\mathcal{H}_{i}:\mathbb{C}^{3}\otimes\mathbb{C}^{3}\rightarrow\mathbb{C}^{3}\otimes\mathbb{C}^{3}$
act as follows:
$\mathcal{H}_{1}\ket{32}=\ket{23},\qquad\mathcal{H}_{2}\ket{13}=\ket{31},\qquad\mathcal{H}_{3}\ket{21}=\ket{12},$
(2.3)
and annihilate all other states. Periodic boundary conditions are implemented
$\mathcal{H}_{i}^{L,L+1}\equiv\mathcal{H}_{i}^{L,1}$. We have simplified the
notation for the states of the spin chain by
$\ket{\phi_{j_{1}}\phi_{j_{2}}\cdots\phi_{j_{L}}}\rightarrow\ket{j_{1}j_{2}\cdots
j_{L}}.$ (2.4)
Therefore the Hamiltonian (2.2) scans a state for neighboring fields in chiral
order $\ket{32},\ket{13},$ or $\ket{21},$ and swaps them to anti-chiral order
$\ket{23},\ket{31},$ and $\ket{12}$ respectively. E.g. we have
$\displaystyle
H_{\text{ec}}\ket{321}=\xi_{1}\ket{231}+\xi_{3}\ket{312}+\xi_{2}\ket{123},$
(2.5) $\displaystyle H_{\text{ec}}\ket{123}=0.$
Setting $\xi_{1}=\xi_{2}=0,\xi_{3}\equiv\xi$ we recover the hypereclectic
model
$H_{3}=\xi\sum_{i=1}^{L}\mathcal{H}_{3}^{i,i+1}.$ (2.6)
The Hamiltonians (2.2) and (2.6) are block diagonal with respect to sectors of
fixed numbers $K$ of $\phi_{3}$ fields, $M-K$ of $\phi_{2}$ fields, and $L-M$
of $\phi_{1}$ fields. We define $V^{L,M,K}$ to be the vector subspace of
$\left(\mathbb{C}^{3}\right)^{\otimes L}$ corresponding to these numbers of
fields. Clearly we have
$\text{dim}\hskip 2.84544ptV^{L,M,K}=\frac{L!}{(L-M)!(M-K)!K!}.$ (2.7)
$H_{3}$ corresponds to the one-loop dilatation operator in the fishnet theory,
where we consider $K$ non-dynamical insertions $\phi_{3}$, which act as walls.
For $K=0$ this operator, although non-Hermitian, is diagonalisable via a
coordinate Bethe ansatz [1]. It corresponds essentially to a chiral version of
the XY-model [25].
### 2.2 Translation Operator and Cyclicity Classes
We can further reduce the state space by considering the translation
invariance of these Hamiltonians. Each $H_{i}$ commutes with the translation
operator $U$
$[H_{i},U]=0,\qquad i=1,2,3,$ (2.8)
where $U$ generates a shift along the chain
$U\ket{j_{1}j_{2}\cdots j_{L-1}j_{L}}=\ket{j_{L}j_{1}j_{2}\cdots j_{L-1}}.$
(2.9)
This further implies $[H_{\text{ec}},U]=0$. Therefore we can choose to work in
a basis where $U$ is diagonal. $U$ has $L$ distinct eigenvalues given by the
$L^{th}$ roots of unity
$\omega_{L}^{k}=e^{2\pi ik/L},\hskip 28.45274ptk=0,1,\dots,L-1.$ (2.10)
The $U$-eigenstates in $V^{L,M,K}$ with eigenvalue $\omega_{L}^{k}$ are said
to be in the $k^{th}$ cyclicity class $V^{L,M,K}_{k}$. The $k=0$ cyclicity
class $V^{L,M,K}_{k=0}$ is known as the cyclic sector. The states in the
$k^{th}$ cyclicity class are easily generated by acting repeatedly on a
reference elementary state111We call single ket states $\ket{j_{1}j_{2}\dots
j_{L}}$ elementary. In general states are linear combinations of these. with
$\omega_{L}^{-k}U$. For example, given $\ket{123}\in V^{3,2,1}$ we can form
the cyclic state
$\ket{123}+U\ket{123}+U^{2}\ket{123}=\ket{123}+\ket{312}+\ket{231},$ (2.11)
and states with $k=1$ or $k=2$
$\ket{123}+\omega_{3}^{-1}U\ket{123}+\omega_{3}^{-2}U^{2}\ket{123}=\ket{123}+e^{-2\pi
i/3}\ket{312}+e^{-4\pi i/3}\ket{231},$ (2.12)
$\ket{123}+\omega_{3}^{-2}U\ket{123}+\omega_{3}^{-4}U^{2}\ket{123}=\ket{123}+e^{-4\pi
i/3}\ket{312}+e^{-8\pi i/3}\ket{231}.$ (2.13)
For a given $L,M,K$ counting the number of states in $V^{L,M,K}$ with a given
cyclicity $k$ requires Pólya counting, see for example [26]. We denote the
states in the $k^{th}$ cyclicity class by
$\ket{j_{1}j_{2}\cdots
j_{L}}_{k}\equiv\sum_{l=0}^{L-1}(w^{-k}U)^{l}\ket{j_{1}j_{2}\cdots
j_{L}}\equiv\mathcal{C}_{k}\ket{j_{1}j_{2}\cdots j_{L}},$ (2.14)
where $\mathcal{C}_{k}$ is an (unnormalised) projector222Note that this
projection may also result in a zero vector.
$\mathcal{C}_{k}^{2}\propto\mathcal{C}_{k}$ onto the $k^{th}$ cyclicity class
$V^{L,M,K}_{k}$. For the hypereclectic spin chain we find it more natural to
consider a so-called static basis, which we describe at the beginning of
section 3.
### 2.3 Spectral Problem
Given a Hermitian Hamiltonian $H$ on an $n$-dimensional Hilbert space, it is
well-known that one can construct an orthonormal $H$-eigenbasis $\psi_{j}$,
$j=1,2,\dots,n$, such that
$H\psi_{j}=E_{j}\psi_{j}\qquad j=1,2,\dots,n,$ (2.15)
where $E_{j}\in\mathbb{C}$ are the (possibly degenerate) eigenvalues of $H$.
For non-Hermitian Hamiltonians diagonalisability is not guaranteed, and indeed
the (hyper)eclectic Hamiltonian is nilpotent and therefore non-diagonalisable
in sectors with $K>0$. In this case there is still an essentially unique333Up
to the ordering of the Jordan blocks. form to which the matrix can be brought,
namely its Jordan normal form. Furthermore, it is exactly the structure of the
Jordan normal form which determines how the logarithms appear in the two-point
functions [21].
Let $H^{L,M,K}_{\text{ec}}$ be the eclectic Hamiltonian (2.2) restricted to
$V^{L,M,K}$. Then there exists a set of generalised eigenstates
$\psi^{m_{j}}_{j}$, $j=1,\dots,N$, $m_{j}=1,\dots,l_{j}$, which satisfy
$H^{L,M,K}_{\text{ec}}\psi_{j}^{k}=\psi_{j}^{k-1},\qquad
H^{L,M,K}_{\text{ec}}\psi_{j}^{1}=0.$ (2.16)
We then say there are $N$ Jordan blocks labelled by $j$, each of length
$l_{j}$. We call $\psi_{j}^{l_{j}}$ the top state of the $j^{\text{th}}$
block. Each block has a true eigenstate $\psi^{1}_{j}$ of $H$ with eigenvalue
$0$. In more general situations each block has a generalised eigenvalue
$\mathcal{E}_{j}$ associated to it. However, in our case we have
$\mathcal{E}_{j}=0$ for each $j$ since $H^{L,M,K}_{\text{ec}}$ is nilpotent.
On a Jordan block of length $l$, $H^{L,M,K}_{\text{ec}}$ acts as the $l\times
l$ matrix
$J_{l}=\left(\begin{array}[]{ccccc}0&1&&&0\\\ &0&1&&\\\ &&0&\ddots&\\\
&&&\ddots&1\\\ 0&&&&0\end{array}\right).$ (2.17)
For the rest of this paper our goal will be to determine the Jordan block
spectrum of $H_{\text{ec}}$ as a function of the sector labels $L,M,K$. This
means finding the length and multiplicities of each of the blocks. For
example, consider the sector $L=5,M=3,K=1$, which contains $30$ total states.
For generic values444Interestingly, the couplings can be tuned to give a finer
Jordan block decomposition, see appendix C. of the couplings $\xi_{i}$ there
are $5$ Jordan blocks of length 5, and $5$ Jordan blocks of length 1. We
denote this as
$\text{JNF}_{531}=(5^{5},1^{5}).$ (2.18)
## 3 Hypereclectic with One Wall
In this section we describe a method to determine the full Jordan block
spectrum for the hypereclectic spin chain in sectors where $K=1$, i.e. there
is a single, non-moving $\phi_{3}$ field, which acts as a fixed wall. In these
sectors the model is equivalent to a chiral XY spin chain with open boundary
conditions. The sizes and multiplicities of the Jordan blocks can be read off
very simply from a generating function $Z_{L,M}(q)$, and the states of the
Jordan blocks are determined by algorithmic methods. Throughout this section
we denote the hypereclectic Hamiltonian $H_{3}\equiv H$ and set $\xi=1$.
Since the $\phi_{3}$ field does not move under the action of $H$, we can
further restrict to sectors with a fixed position of $\phi_{3}$. We will
restrict to static states of the form $\ket{j_{1}j_{2}\cdots j_{L-1}3}$, where
$j_{1},j_{2},\dots,j_{L-1}\in\\{1,2\\}$. We will refer to the subspace of
$V^{L,M,1}$ spanned by states of this form as $W^{L,M}$. We can access states
where $\phi_{3}$ is in a different position by acting with the translation
operator $U$, so that the Hilbert space decomposes
$V^{L,M,1}=\bigoplus_{j=0}^{L-1}U^{j}W^{L,M}.$ (3.1)
### 3.1 Warmup Examples
#### General $L$, $M=2$, $K=1$.
The simplest situation is when $M=2$ and $K=1$. This means there are a single
$\phi_{3}$ field, a single $\phi_{2}$ field, and $L-2$ $\phi_{1}$ fields. A
natural basis for $W^{L,2}$ is given by $L-1$ states
$\ket{211\cdots 113},\ket{121\cdots 113},\dots,\ket{111\cdots 123}.$ (3.2)
In this sector the states clearly form a single Jordan block of length $L-1$,
as can be seen by acting with $H$ repeatedly on $\ket{211\cdots 113}$
$\ket{211\cdots 113}\xrightarrow{H}\ket{121\cdots
113}\xrightarrow{H}\cdots\xrightarrow{H}\ket{111\cdots 123}\xrightarrow{H}0.$
(3.3)
We will refer to any state of the form $\ket{2^{M-K}1^{L-M}3^{K}}$ as anti-
locked, and $\ket{1^{L-M}2^{M-K}3^{K}}$ as locked. Similarly for the spaces
$U^{j}W^{L,M}$, $j=1,\dots,L-1$ there is a single Jordan block of length
$L-1$. Therefore for $M=2$ and $K=1$ we have
$\text{JNF}_{L,2,1}=(L-1)^{L},$ (3.4)
meaning there are $L$ blocks of length $L-1$.
#### $L=7,M=3,K=1$.
The situation becomes more intricate with increasing $M$, which we illustrate
with the example $L=7,M=3,K=1$. In this sector there are 4 $\phi_{1}$ fields,
$2$ $\phi_{2}$ fields, and a single $\phi_{3}$ field. In $W^{7,3}$ there are
15 states. We use the important observation that the anti-locked state is
always a top state for the longest Jordan block
$\displaystyle\ket{2211113}$ $\displaystyle H^{0}$ (3.5)
$\displaystyle\rightarrow$ $\displaystyle\ket{2121113}$ $\displaystyle H^{1}$
$\displaystyle\rightarrow$ $\displaystyle\ket{2112113}+\ket{1221113}$
$\displaystyle H^{2}$ $\displaystyle\rightarrow$
$\displaystyle\ket{2111213}+2\ket{1212113}$ $\displaystyle H^{3}$
$\displaystyle\rightarrow$
$\displaystyle\ket{2111123}+3\ket{1211213}+2\ket{1122113}$ $\displaystyle
H^{4}$ $\displaystyle\rightarrow$ $\displaystyle
4\ket{1211123}+5\ket{1121213}$ $\displaystyle H^{5}$
$\displaystyle\rightarrow$ $\displaystyle 5\ket{1112213}+9\ket{1121123}$
$\displaystyle H^{6}$ $\displaystyle\rightarrow$ $\displaystyle
14\ket{1112123}$ $\displaystyle H^{7}$ $\displaystyle\rightarrow$
$\displaystyle 14\ket{1111223}$ $\displaystyle H^{8}$
$\displaystyle\rightarrow$ $\displaystyle 0,$ $\displaystyle H^{9}$
so we have identified a Jordan block of length 9, whose eigenstate is
proportional to the locked state $\ket{1111223}$. However since there are 15
states in the sector there must be additional Jordan blocks.
We note that each of the 15 elementary states appear in the tower of states
(3.5). We classify these 15 states by where in this state tower they appear,
by defining the level $S$ of a state. We give the anti-locked state
$\ket{2211113}$ $S=8$ and the locked state $\ket{1111223}$ $S=0$. In general,
if an elementary state appears in the row $H^{k}$ of (3.5), we give it
$S=8-k$. One notices that the $S$-value for a state is the total number of 1’s
to the right of each of the 2’s. Defining $W_{S}^{7,3}$ to be the vector
subspace of $W^{7,3}$ spanned by states with level $S$, we get
$W^{7,3}=\bigoplus_{S=0}^{8}W_{S}^{7,3},$ (3.6)
and it is clear that
$H:W_{S}^{7,3}\rightarrow W_{S-1}^{7,3},\qquad HW_{0}^{7,3}=0.$ (3.7)
In light of this, the next natural place to look for a top state of a Jordan
block is in $W^{7,3}_{6}$. This is because a single state from each
$W^{7,3}_{S}$ is already contained in the largest Jordan block, and
$W^{7,3}_{6}$ is the space with largest $S$ with dimension larger than 1. We
thus deduce that the top state for the next Jordan block must be of the form
$\alpha\ket{2112113}+\beta\ket{1221113}\in W^{7,3}_{6},$ (3.8)
where $\alpha\neq\beta$ as we want the state to be linearly independent from
the corresponding state in the length 9 block. We act repeatedly on this state
with $H$ until there is a possible choice for $\alpha$ and $\beta$ which makes
the state vanish
$\displaystyle\alpha\ket{1221113}+\beta\ket{2112113}$
$\displaystyle\rightarrow\beta\ket{2111213}+(\alpha+\beta)\ket{1212113}$
$\displaystyle\rightarrow\beta\ket{2111123}+(\alpha+2\beta)\ket{1211213}+(\alpha+\beta)\ket{1122113}$
$\displaystyle\rightarrow(\alpha+3\beta)\ket{1211123}+(2\alpha+3\beta)\ket{1121213}$
$\displaystyle\rightarrow(2\alpha+3\beta)\ket{1112213}+(3\alpha+6\beta)\ket{1121123}$
$\displaystyle\rightarrow(5\alpha+9\beta)\ket{1112123}.$
We see that this yields a zero vector if $5\alpha+9\beta=0$, for example
$\alpha=-9,\beta=5$. Therefore this chain of states determines a Jordan block
of length 5, with top state $5\ket{2112113}-9\ket{1221113}\in W^{7,3}_{6}$ and
eigenstate $-3\ket{1112213}+3\ket{1121123}\in W^{7,3}_{8-6}=W^{7,3}_{2}$.
There must be a single Jordan block of length 1 remaining, and by state
counting this must be contained in $W_{4}^{7,3}$, since this is the only space
with dimension greater than 2. We make the ansatz for the top state
$\alpha^{\prime}\ket{2111123}+\beta^{\prime}\ket{1211213}+\gamma^{\prime}\ket{1122113}\in
W_{4}^{7,3}.$ (3.9)
This is easily checked to be an eigenstate of $H$ for
$\alpha^{\prime}=-\beta^{\prime}=\gamma^{\prime}=1$ and thus determines a
Jordan block of length 1. The story is identical for the remaining spaces
$U^{j}W^{7,3}$, $j=1,\dots,6$, so the overall Jordan normal form for
$L=7,M=3,K=1$ is
$\text{JNF}_{7,3,1}=(9^{7},5^{7},1^{7}).$ (3.10)
Let us step back and look at the state tower (3.5), from which we can see the
dimensions
$\text{dim}\hskip 1.42271ptW_{S}^{7,3},\hskip 28.45274ptS=0,1,\dots,8$ (3.11)
by counting the number of elementary states in each row. We note that these
dimensions form a diamond, in that they start from 1 at $S=8$, increase to a
maximum of 3 at $S=4$, and decrease symmetrically to 1 at $S=0$. We encode
these dimensions in a generating function
$\bar{Z}_{7,3}(q)=\sum_{S=0}^{8}\text{dim}\hskip
1.42271ptW^{7,3}_{S}q^{S}=1+q+2q^{2}+2q^{3}+3q^{4}+2q^{5}+2q^{6}+q^{7}+q^{8}.$
(3.12)
Because of this diamond structure it is actually possible to deduce the Jordan
block structure in $W^{7,3}$ from the generating function, a purely
combinatorial object, up to some possible subtleties described in the next
section. Given the generating function (3.12) we identify the Jordan block of
length 9 by the degree of the polynomial plus 1. We then subtract
$1+q+q^{2}+\dots+q^{8}$ to represent the fact that there is one state at each
level in this largest block. We then normalise the resulting polynomial to
have lowest power $q^{0}$, and repeat the procedure:
$\displaystyle 1+q+2q^{2}+2q^{3}+3q^{4}+2q^{5}+2q^{6}+q^{7}+q^{8}$ (3.13)
$\displaystyle\rightarrow$ $\displaystyle 1+q+2q^{2}+q^{3}+q^{4}$
$\displaystyle\rightarrow$ $\displaystyle 1,$
from which we deduce the Jordan block spectrum $(9,5,1)$. Therefore in the
next section it will be our goal to generalise the arguments of this section
and compute the generating function $\bar{Z}_{L,M}(q)$ for arbitrary $L,M$.
### 3.2 Generating Function
For general $L,M$ we similarly grade the vector space in the static sector by
the action of $H$
$W^{L,M}=\bigoplus_{S=0}^{S_{\text{max}}}W_{S}^{L,M},$ (3.14)
$H:W_{S}^{L,M}\rightarrow W_{S-1}^{L,M},\qquad HW_{0}^{L,M}=0.$ (3.15)
We have in general $S_{\text{max}}=L_{1}M_{1}$, where $L_{1}\equiv L-M$ is the
number of 1’s in the sector and $M_{1}\equiv M-1$ is the number of 2’s. The
anti-locked state is $\ket{2^{M_{1}}1^{L_{1}}3}\in W_{S_{\text{max}}}^{L,M}$
and the locked state is $\ket{1^{L_{1}}2^{M_{1}}3}\in W_{0}^{L,M}$. In general
an elementary state takes the form
$\ket{n_{1},n_{2},\dots,n_{M_{1}}}\equiv|\underbrace{1\cdots
1}_{n_{0}}\mathbf{2}\underbrace{1\cdots
1}_{n_{1}}\mathbf{2}\underbrace{1\cdots
1}_{n_{2}}\cdots\mathbf{2}\underbrace{1\cdots
1}_{n_{M_{1}}}\mathbf{3}\rangle,$ (3.16)
where $n_{j}$ is the number of $1$’s between the $j^{th}$ and $(j+1)^{th}$ 2\.
Clearly they should satisfy
$\sum_{j=0}^{M_{1}}n_{j}=L-M=L_{1}.$ (3.17)
In this notation we can define the level $S$ for such a state which counts the
number of 1’s on the right hand side of each of the 2’s. Explicitly the state
$\ket{n_{1},n_{2},\dots,n_{M_{1}}}$ defined in (3.16) has
$S=\sum_{j=1}^{M_{1}}jn_{j}.$ (3.18)
As before we define $W_{S}^{L,M}$ to be spanned by elementary states with this
level $S$. The Hamiltonian acts on (3.16) as
$H:\ket{n_{1},n_{2},\dots,n_{M_{1}}}\quad\to\quad\sum_{j=1}^{M_{1}}\ket{n_{1},n_{2},\dots,n_{j-1}+1,n_{j}-1,\dots,n_{M_{1}}}.$
(3.19)
(3.18) and (3.19) make it clear that $H$ decreases $S$ to $S-1$.
We now consider the problem of determining the dimensions of the spaces
$W_{S}^{L,M}$. We would like to determine a generating function
$\bar{Z}_{L,M}(q)=\sum_{S=0}^{S_{\text{max}}}\text{dim}\hskip
1.42271ptW^{L,M}_{S}q^{S}.$ (3.20)
These dimensions $\text{dim}\hskip 1.42271ptW^{L,M}_{S}$ are given by the
number of partitions of the integer $S$ into at most $M_{1}$ parts, each less
than or equal to $L_{1}$. Expressing (3.18) as
$S=(n_{1}+n_{2}+\dots+n_{M_{1}})+(n_{2}+\dots+n_{M_{1}})+\dots+n_{M_{1}}$
(3.21)
one can notice that there is one-to-one correpondence between an elementary
vector in (3.16) and such a restricted partition of $S$ in (3.21). For
example, consider the case of the previous section, $L=7,M=3,K=1$. There were
3 elementary states in $W^{7,3}_{4}$:
$\displaystyle\ket{2111123},\qquad(n_{1}+n_{2},n_{2})=(4,0),$ (3.22)
$\displaystyle\ket{1211213},\qquad(n_{1}+n_{2},n_{2})=(3,1),$
$\displaystyle\ket{1122113},\qquad(n_{1}+n_{2},n_{2})=(2,2).$
These correspond to the partitions of the integer $4$ into at most $M_{1}=2$
parts, where each part is less than or equal to $L_{1}=4$. There are 3 such
partitions $4=4=3+1=2+2$.
Such restricted partitions described above can be generated by Gaussian (or
$q$-) binomial coefficients [27]
$\bar{Z}_{L,M}(q)=\sum_{S=0}^{S_{\text{max}}}\text{dim}\hskip
1.42271ptW^{L,M}_{S}q^{S}=\binom{L-1}{M-1}_{q}=\prod_{k=1}^{M-1}\frac{1-q^{L-k}}{1-q^{k}},$
(3.23)
which is always a polynomial in $q$. Note that if we send $q\rightarrow 1$,
the $q$-binomial reduces to the ordinary binomial coefficient and we have
$\sum_{S=0}^{S_{\text{max}}}\text{dim}\hskip
1.42271ptW^{L,M}_{S}=\binom{L-1}{M-1}=\text{dim}\hskip 1.42271ptW^{L,M},$
(3.24)
as expected because of (3.14). (3.23) generates a list of
dimensions555$\mathbf{d}_{S}=\mathbf{d}_{S}(L_{1},M_{1})$, we suppress the
$L_{1},M_{1}$ dependence for now. $\mathbf{d}_{S}\equiv\text{dim}\hskip
1.42271ptW^{L,M}_{S}$
$(\mathbf{d}_{S_{\rm max}},\mathbf{d}_{S_{\rm
max}-1},\dots,\mathbf{d}_{1},\mathbf{d}_{0})\qquad{\rm
with}\quad\mathbf{d}_{0}=\mathbf{d}_{S_{\rm max}}=1.$ (3.25)
Furthermore, from a property of the $q$-binomial coefficient, the dimensions
are increasing from the left to the right until the midpoint, and decreasing
after that, because of the symmetry
$\mathbf{d}_{S}=\mathbf{d}_{\tilde{S}},\qquad{\tilde{S}}\equiv{S_{\rm
max}-S}.$ (3.26)
For the space $W^{L,M}_{S_{\text{max}}}$, there is only one elementary state
$\psi_{0}\equiv\ket{2^{M_{1}}1^{L_{1}}3}$, the anti-locked state. By
successive action of $H$, a Jordan string of states is generated
$\psi_{0}\stackrel{{\scriptstyle
H}}{{\longrightarrow}}H\psi_{0}\stackrel{{\scriptstyle
H}}{{\longrightarrow}}H^{2}\psi_{0}\stackrel{{\scriptstyle
H}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
H}}{{\longrightarrow}}H^{S_{\rm max}}\psi_{0}\stackrel{{\scriptstyle
H}}{{\longrightarrow}}0.$ (3.27)
Therefore, this generates a Jordan block of size $S_{\rm max}+1$, the largest
one.
It turns out that the next dimension $\mathbf{d}_{S_{\rm max}-1}$ in (3.25) is
also one, as can be computed from (3.23). This means $W^{L,M}_{S_{\rm max}-1}$
is spanned by $H\psi_{0}$, the first descendant of the anti-locked state in
(3.27). Therefore there is no other independent vector in $W^{L,M}_{S_{\rm
max}-1}$ which can generate a new Jordan string.
The top state of the second Jordan block arises at the first level $S=S_{1}$
below $S_{\text{max}}$ whose dimension is bigger than $1$. We can form
$\mathbf{d}_{S_{1}}-1$ linearly independent potential top states in
$W_{S_{1}}^{L,M}$, which are linearly independent from the $H$-descendant of
the anti-locked state. We denote these states by $\psi^{(S_{1})}_{j}$
($j=1,\dots,\mathbf{d}_{S_{1}}-1$), and make the ansatz
$\psi^{(S_{1})}_{j}=\sum_{i=1}^{\mathbf{d}_{S_{1}}}\alpha_{j}^{(i)}e_{i}^{(S_{1})},$
(3.28)
where $e_{i}^{(S_{1})}$ are the elementary states in $W^{L,M}_{S_{1}}$.
$\alpha_{j}^{(i)}$ are constants which are determined by the condition that
each $\psi^{(S_{1})}_{j}$ constitutes a top state for a new Jordan block. Each
of these states generates a Jordan string
$\psi^{(S_{1})}_{j}\stackrel{{\scriptstyle
H}}{{\longrightarrow}}H\psi^{(S_{1})}_{j}\stackrel{{\scriptstyle
H}}{{\longrightarrow}}H^{2}\psi^{(S_{1})}_{j}\cdots\stackrel{{\scriptstyle
H}}{{\longrightarrow}}H^{S_{1}-{\tilde{S}}_{1}}\psi^{(S_{1})}_{j}\stackrel{{\scriptstyle
H}}{{\longrightarrow}}0,\quad j=1,\dots,\mathbf{d}_{S_{1}}-1.$ (3.29)
The condition
$H^{S_{1}-{\tilde{S}}_{1}}\psi^{(S_{1})}_{j}\stackrel{{\scriptstyle
H}}{{\longrightarrow}}0$ leads to a linear system of equations for the
$\alpha_{j}^{(i)}$ which can be solved to determine the $\mathbf{d}_{S_{1}}-1$
new top states. These new Jordan blocks each have size
$S_{1}-{\tilde{S}}_{1}+1$. The only possible subtlety is the potential for an
‘unexpected shortening’ of the Jordan block, that is the possibility for the
equation $H^{k}\psi^{(S_{1})}_{j}=0$ to admit a solution in the
$\alpha_{j}^{(i)}$ for some $k<S_{1}-\tilde{S}_{1}+1$. While we have not yet
been able to rigorously disprove shortening in full generality, we have
verified for a large number values of $L$ and $M$ that it does not happen. We
were able to perform these extensive tests thanks to a mathematically more
succinct reformulation of the problem, see appendix A for details. We will
assume that shortening cannot occur for the remainder of this paper.
The third set of Jordan blocks occurs at a level $S_{2}$, which is the largest
integer satisfying $\mathbf{d}_{S_{2}}>\mathbf{d}_{S_{1}}$. Then, as before,
we can form $\mathbf{d}_{S_{2}}-\mathbf{d}_{S_{1}}$ linearly independent
potential top states which are linearly independent from $H$-descendants of
the previous vectors, $\psi_{0}$ and $\psi^{(S_{1})}_{j}$. We make a similar
ansatz for these potential top states
$\psi^{(S_{2})}_{j}=\sum_{i=1}^{\mathbf{d}_{S_{2}}}\beta_{j}^{(i)}e_{i}^{(S_{2})},$
(3.30)
where $\beta_{j}^{(i)}$ are constants. These states create new Jordan strings
$\psi^{(S_{2})}_{j}\stackrel{{\scriptstyle
H}}{{\longrightarrow}}H\psi^{(S_{2})}_{j}\stackrel{{\scriptstyle
H}}{{\longrightarrow}}H^{2}\psi^{(S_{2})}_{j}\cdots\stackrel{{\scriptstyle
H}}{{\longrightarrow}}H^{S_{2}-{\tilde{S}}_{2}}\psi^{(S_{2})}_{j}\stackrel{{\scriptstyle
H}}{{\longrightarrow}}0,\quad
j=1,\dots,\mathbf{d}_{S_{2}}-\mathbf{d}_{S_{1}},$ (3.31)
and the final condition
$H^{S_{2}-{\tilde{S}}_{2}}\psi^{(S_{2})}_{j}\stackrel{{\scriptstyle
H}}{{\longrightarrow}}0$ is solved to determine the constants
$\beta_{j}^{(i)}$. This leads to $\mathbf{d}_{S_{2}}-\mathbf{d}_{S_{1}}$
Jordan blocks of size $S_{2}-{\tilde{S}}_{2}+1$. This procedure can be
continued until it reaches the maximum value of the dimension $\mathbf{d}_{S}$
which occurs at $S=[S_{\rm max}/2]$.
We note that for a given $L,M,$ the dimensions $\mathbf{d}_{S}$ are sufficient
to determine the sizes and multiplicities of the Jordan blocks. For example,
for $L=9,M=5$ we compute using (3.23)
$\displaystyle\bar{Z}_{9,5}(q)=$ $\displaystyle
1+q+2q^{2}+3q^{3}+5q^{4}+5q^{5}+7q^{6}+7q^{7}+8q^{8}$ (3.32)
$\displaystyle+7q^{9}+7q^{10}+5q^{11}+5q^{12}+3q^{13}+2q^{14}+q^{15}+q^{16},$
from which we can identify the Jordan normal form of $H$ in $W^{9,5}$ to be
$(17,13,11,9^{2},5^{2},1)$ using the same procedure666With some practice one
can easily and quickly ‘read off’ the Jordan normal form from the generating
function by visual inspection, i.e. this does not involve any calculations,
just a bit of bookkeeping. as (3.13). We can exhaust the Hilbert space by
application of $U^{j},j=1,\dots,8$, so that overall we have
$\text{JNF}_{9,5,1}=(17^{9},13^{9},11^{9},9^{18},5^{18},1^{9}).$ (3.33)
For higher $K$, see the next section 4, we find it necessary to work with a
slightly modified generating function for the dimensions of $W_{S}^{L,M}$ that
is symmetric under $q\rightarrow q^{-1}$:
$Z_{L,M}(q)=q^{-S_{\text{max}/2}}\bar{Z}_{L,M}(q)\equiv\genfrac{[}{]}{0.0pt}{0}{L-1}{M-1}_{q}.$
(3.34)
For example, (3.12) and (3.32) are modified to
$Z_{7,3}(q)=q^{-4}+q^{-3}+2q^{-2}+2q^{-1}+3+2q+2q^{2}+q^{3}+q^{4},$ (3.35)
$\displaystyle Z_{9,5}(q)=q^{-8}$
$\displaystyle+q^{-7}+2q^{-6}+3q^{-5}+5q^{-4}+5q^{-3}+7q^{-2}+7q^{-1}+8$
(3.36) $\displaystyle+7q+7q^{2}+5q^{3}+5q^{4}+3q^{5}+2q^{6}+q^{7}+q^{8}.$
The modified function also provides an elegant way to determine the sizes and
multiplicities of the Jordan blocks in a sector uniquely. We have
$Z_{L,M}(q)=\sum_{j}N_{j}[j]_{q}=N_{1}\,q^{0}+N_{2}\,\left(q^{-\frac{1}{2}}+q^{\frac{1}{2}}\right)+N_{3}\,\left(q^{-1}+q^{0}+q^{1}\right)+\ldots\,,$
(3.37)
where $N_{j}$ is the number of Jordan blocks of length $j$. $[j]_{q}$ is a
modified $q$-number
$[j]_{q}=\frac{q^{j/2}-q^{-j/2}}{q^{1/2}-q^{-1/2}}=\sum_{k=\frac{-j+1}{2}}^{\frac{j-1}{2}}q^{k}.$
(3.38)
For example $Z_{7,3}(q)$ and $Z_{9,5}(q)$ can also be written
$Z_{7,3}(q)=[1]_{q}+[5]_{q}+[9]_{q},$ (3.39)
$Z_{9,5}(q)=[1]_{q}+2[5]_{q}+2[9]_{q}+[11]_{q}+[13]_{q}+[17]_{q},$ (3.40)
reflecting the Jordan block structures $(9,5,1)$ and
$(17,13,11,9^{2},5^{2},1)$ respectively. A generating function which generates
the Jordan block spectrum of all of $V^{L,M,1}$ can be obtained as a trace
over the Hilbert space
$\mathcal{Z}_{L,M}(q)=\text{tr}\hskip
1.42271ptq^{\hat{S}-S_{\text{max}}/2}=L\genfrac{[}{]}{0.0pt}{0}{L-1}{M-1}_{q},$
(3.41)
where $\hat{S}$ acts on elementary states with well-defined values of $S$
$\hat{S}\ket{S}=S\ket{S},$ (3.42)
and is extended by linearity.
#### Cyclicity classes.
We note that instead of considering states in $U^{j}W^{L,M}$ where the
$\phi_{3}$ field is in a fixed position, we could have considered states in
any cyclicity class $k$. If we replaced the states $\ket{j_{1}j_{2}\cdots
j_{L-1}3}\rightarrow\mathcal{C}_{k}\ket{j_{1}j_{2}\cdots j_{L-1}3}$ for any
$k=0,1,\dots,L-1$ the arguments of this section are unchanged because
$[H,\mathcal{C}_{k}]=0$, where $\mathcal{C}_{k}$ is the unnormalised projector
defined in (2.14). Therefore the Jordan normal form of $H$ is the same in
$W^{L,M}$ and $V^{L,M,1}_{k}$ for any $k$.
## 4 General Hypereclectic
Here we discuss the extension of the previous section to sectors with many
walls, i.e. $K>1$. The main observation is that $K>1$ states behave
essentially like a tensor product of $K$ states with $K=1$. Any elementary
state $v\in V^{L,M,K}$ ending in a 3 can be written
$v=v_{1}\otimes v_{2}\otimes\cdots\otimes v_{K},$ (4.1)
where $v_{i}\in W^{\ell_{i}+m_{i}+1,m_{i}+1}$ are elementary states
themselves. We defined $W^{L,M}$ above (3.1). $\ell_{i}$ denotes the number of
1’s in $v_{i}$ and $m_{i}$ denotes the number of 2’s. The hypereclectic
Hamiltonian $H$ acts on states of the form (4.1) as
$Hv=Hv_{1}\otimes v_{2}\otimes\cdots\otimes v_{K}+v_{1}\otimes
Hv_{2}\otimes\cdots\otimes v_{K}+\cdots+v_{1}\otimes v_{2}\otimes\cdots\otimes
Hv_{K}.$ (4.2)
We define $\boldsymbol{\ell}\equiv(\ell_{1},\dots,\ell_{K})$ and
$\boldsymbol{m}\equiv(m_{1},\dots,m_{K})$, which should satisfy
$\sum_{i=1}^{K}\ell_{i}=L-M=L_{1},\qquad\sum_{i=1}^{K}m_{i}=M-K=M_{1}.$ (4.3)
We will denote the spaces $\bigotimes_{i=1}^{K}W^{\ell_{i}+m_{i}+1,m_{i}+1}$
as subsectors, and picking the vectors $\boldsymbol{\ell},\boldsymbol{m}$
corresponds to a choice of subsector. We consider subsectors
$(\boldsymbol{\ell},\boldsymbol{m})$ satisfying (4.3) which are unique up to
application of the translation operator $U^{j}$. In practise this means we
identify
$(\boldsymbol{\ell},\boldsymbol{m})\sim(\boldsymbol{\ell}^{\prime},\boldsymbol{m}^{\prime})$
if $\boldsymbol{\ell},\boldsymbol{\ell}^{\prime}$ and
$\boldsymbol{m},\boldsymbol{m}^{\prime}$ are related by the same cyclic
permutation $\sigma^{n}$
$(\boldsymbol{\ell},\boldsymbol{m})\sim(\boldsymbol{\ell}^{\prime},\boldsymbol{m}^{\prime})\quad\longleftrightarrow\quad(\boldsymbol{\ell}^{\prime},\boldsymbol{m}^{\prime})=(\sigma^{n}\boldsymbol{\ell},\sigma^{n}\boldsymbol{m}),$
(4.4)
$\sigma(\ell_{1},\ell_{2},\dots,\ell_{K})\equiv(\ell_{2},\dots,\ell_{K},\ell_{1}).$
(4.5)
In this way we can describe all the states in $V^{L,M,K}$ using the
translation operator $U$. Overall we have
$V^{L,M,K}=\bigoplus_{(\boldsymbol{\ell},\boldsymbol{m})/\sim}\bigoplus_{j=1}^{L/S_{\boldsymbol{l},\boldsymbol{m}}}U^{j}\bigotimes_{i=1}^{K}W^{\ell_{i}+m_{i}+1,m_{i}+1},$
(4.6)
where we introduced the symmetry factor for a subsector
$S_{\boldsymbol{l},\boldsymbol{m}}$. The symmetry factor reflects the fact
that some subsectors are especially symmetric with respect to cyclicity. This
occurs when there is an $n<K$ such that
$(\sigma^{n}\boldsymbol{\ell},\sigma^{n}\boldsymbol{m})=(\boldsymbol{\ell},\boldsymbol{m}),$
(4.7)
where $\sigma$ is the cyclic permutation defined in (4.5). In this case we
give the subsector a symmetry factor
$S_{\boldsymbol{\ell},\boldsymbol{m}}=K/n$. For example, let $L=14,M=8,K=4$
and take the subsector $\boldsymbol{\ell}=(2,1,2,1),\boldsymbol{m}=(1,1,1,1)$.
We have $\sigma^{2}\boldsymbol{\ell}=\boldsymbol{\ell}$ and
$\sigma^{2}\boldsymbol{m}=\boldsymbol{m}$ and so
$S_{\boldsymbol{\ell},\boldsymbol{m}}=4/2=2$ in this case.
### 4.1 Warmup Examples
#### $L=7,M=4,K=2$.
We begin with the simple example $L=7,M=4,K=2$. In this sector there are three
$\phi_{1}$ fields, two $\phi_{2}$ fields, two $\phi_{3}$ fields and
$\frac{7!}{3!2!2!}=210$ total states. In table 1 we show the 6 inequivalent
choices of $(\boldsymbol{\ell},\boldsymbol{m})$, which corresponds to the 6
ways to decompose the states into $K=1$ states, on which $H$ acts block
diagonally:
| Form of state | Number of states | $\quad\hskip 2.84544pt\boldsymbol{\ell},\boldsymbol{m}$ | JNF
---|---|---|---|---
$\boldsymbol{1}$ | $\ket{111223}\otimes\ket{3}$ | $10\times 1=10$ | $(3,0),(2,0)$ | $7\oplus 3$
$\boldsymbol{2}$ | $\ket{11123}\otimes\ket{23}$ | $4\times 1=4$ | $(3,0),(1,1)$ | 4
$\boldsymbol{3}$ | $\ket{11223}\otimes\ket{13}$ | $6\times 1=6$ | $(2,1),(2,0)$ | $5\oplus 1$
$\boldsymbol{4}$ | $\ket{1123}\otimes\ket{123}$ | $3\times 2=6$ | $(2,1),(1,1)$ | $3\otimes 2$
$\boldsymbol{5}$ | $\ket{1223}\otimes\ket{113}$ | $3\times 1=3$ | $(1,2),(2,0)$ | 3
$\boldsymbol{6}$ | $\ket{1113}\otimes\ket{223}$ | $1\times 1=1$ | $(3,0),(0,2)$ | 1
Table 1: Decomposition of $L=7,M=4,K=2$ states into $K=1$ states. The 3’s
should be regarded as fixed, whereas the 1’s and 2’s can be permuted within
their ket.
All subsectors except for $\boldsymbol{4}$ behave trivially as a single $K=1$
sector under the action of $H$. Their Jordan normal forms were determined in
the previous section and are listed in the table. We look at states of the
form $\boldsymbol{4}$ in a bit more detail. These states have the form of an
$L=4,M=2,K=1$ state and an $L=3,M=2,K=1$ state glued together, which have
Jordan blocks of size 3 and 2 respectively. The natural ‘anti-locked’ state
comes from gluing together the anti-locked states of the respective $K=1$
parts $\ket{2113213}$. We act successively on this state with $H$
$\displaystyle\ket{2113213}\rightarrow\ket{1213213}+\ket{2113123}$ (4.8)
$\displaystyle\rightarrow\ket{1123213}+2\ket{1213123}\rightarrow
3\ket{1123123}\rightarrow 0,$
which is a Jordan block of length 4. There is a further Jordan block of length
2 obtained by making the ansatz for a new top state
$\gamma_{1}\ket{1213213}+\gamma_{2}\ket{2113123},$ (4.9)
and similarly to the last section this gives a Jordan block of length 2 for
$\gamma_{1}=-1,\gamma_{2}=2$. Thus the Jordan decomposition of the subsector
$\boldsymbol{4}$ is $(4,2)$. Since the Jordan decompositions of the $K=1$
sectors are $(3)$ and $(2)$ respectively, we denote this as $3\otimes
2=4\oplus 2$.
At the level of generating functions, we can deduce the Jordan normal form of
the ‘tensor product’ sectors by multiplying the generating functions of the
corresponding $K=1$ sectors. For example, for the subsector $\boldsymbol{4}$
we have
$\displaystyle
Z_{7,4,2}^{\boldsymbol{4}}(q)=Z_{4,2}(q)Z_{3,2}(q)=(q^{-1}+1+q)(q^{-1/2}+q^{1/2})$
(4.10) $\displaystyle=q^{-3/2}+2q^{-1/2}+2q^{1/2}+q^{3/2},$
from which the Jordan normal form $(4,2)$ can be easily deduced using (3.37).
To obtain the full generating function for each of the subsectors in
$L=7,M=4,K=2$ we can simply add the generating functions for each of the
subsectors $\boldsymbol{1},\boldsymbol{2},\dots,\boldsymbol{6}$
$\displaystyle
Z_{7,4,2}(q)=\sum_{\boldsymbol{i}=\boldsymbol{1}}^{\boldsymbol{6}}Z_{7,4,2}^{\boldsymbol{i}}(q)$
$\displaystyle=q^{-3}+2q^{-2}+2q^{-3/2}+4q^{-1}+3q^{-1/2}+6+3q^{1/2}+4q+2q^{3/2}+2q^{2}+q^{3}.$
(4.11)
Using (3.37) leads to the following Jordan normal form:
$\text{JNF}_{7,4,2}=(7,5,4^{2},3^{2},2,1^{2}).$ (4.12)
In this sector there are no subtleties with cyclicity and the rest of the
Hilbert space can be exhausted by application of the translation operator
$U^{j}$, $j=1,\dots,6$. For each $j$ we have the same argument as before, so
the full Jordan block structure can be obtained as seven copies of (4.12)
$\text{JNF}^{\text{tot}}_{7,4,2}=(7^{7},5^{7},4^{14},3^{14},2^{7},1^{14}).$
(4.13)
At the level of the generating function this can be obtained by multiplying
(4.11) by $L=7$. However, there are cases where cyclic symmetry leads to some
subtleties, as we discuss next.
#### $L=8,M=4,K=2$.
Let us consider the case of $L=8,M=4,K=2$. There are $\frac{8!}{4!4!2!}=420$
states in this sector. Therein one finds an
($\boldsymbol{\ell},\boldsymbol{m}$) subsector that is symmetric with respect
to cyclicity. In table 2 we break the states into $K=1$ states as in the
previous section,
| Form of state | Number of states | $\quad\hskip 2.84544pt\boldsymbol{\ell},\boldsymbol{m}$ | Jordan decomposition
---|---|---|---|---
$\boldsymbol{1}$ | $\ket{1111223}\otimes\ket{3}$ | $15\times 1=15$ | $(4,0),(2,0)$ | $9\oplus 5\oplus 1$
$\boldsymbol{2}$ | $\ket{111223}\otimes\ket{13}$ | $10\times 1=10$ | $(3,1),(2,0)$ | $7\oplus 3$
$\boldsymbol{3}$ | $\ket{111123}\otimes\ket{23}$ | $5\times 1=5$ | $(4,0),(1,1)$ | $5$
$\boldsymbol{4}$ | $\ket{11123}\otimes\ket{123}$ | $4\times 2=8$ | $(3,1),(1,1)$ | $4\otimes 2=5\oplus 3$
$\boldsymbol{5}$ | $\ket{11223}\otimes\ket{113}$ | $6\times 1=6$ | $(2,2),(2,0)$ | $7\oplus 3$
$\boldsymbol{6}$ | $\ket{11113}\otimes\ket{223}$ | $1\times 1=1$ | $(4,0),(0,2)$ | 5
$\boldsymbol{7}$ | $\ket{1123}\otimes\ket{1123}$ | $3\times 3=9$ | $(2,2),(1,1)$ | $3\otimes 3=5\oplus 3\oplus 1$
$\boldsymbol{8}$ | $\ket{1223}\otimes\ket{1113}$ | $3\times 1=3$ | $(1,3),(2,0)$ | $3$
Table 2: Decomposition of $L=8,M=4,K=2$ states into $K=1$ states.
where we replaced $4\otimes 2=5\oplus 3$ and $3\otimes 3=5\oplus 3\oplus 1$ by
multiplying the appropriate $K=1$ generating functions and naively extracting
the resulting Jordan block structures using (3.37). We see that
$\boldsymbol{7}$ is the subsector where the issues with cyclicity emerge. For
the other subsectors we can exhaust the rest of the state space by acting with
$U^{j},j=1,\dots,7$. However for subsector $\boldsymbol{7}$ applying the
translation $U^{4}$ maps the states to a state in the same subsector, which
reflects the fact this subsector has a symmetry factor
$S_{\boldsymbol{\ell},\boldsymbol{m}}=2$. Therefore acting with
$U^{j},j=0,1,\dots,7$ leads to a double counting by a factor of 2. We can
realise this at the level of an overall generating function for the
$L=8,M=4,K=2$ sector by multiplying by
$1/S_{\boldsymbol{\ell},\boldsymbol{m}}=1/2$ for the subsector
$\boldsymbol{7}$
$\mathcal{Z}_{8,4,2}(q)=8(Z_{8,4,2}^{\boldsymbol{1}}+Z_{8,4,2}^{\boldsymbol{2}}+Z_{8,4,2}^{\boldsymbol{3}}+Z_{8,4,2}^{\boldsymbol{4}}+Z_{8,4,2}^{\boldsymbol{5}}+Z_{8,4,2}^{\boldsymbol{6}}+\frac{1}{2}Z_{8,4,2}^{\boldsymbol{7}}+Z_{8,4,2}^{\boldsymbol{8}}).$
(4.14)
We compute (4.14) to be
$\mathcal{Z}_{8,4,2}(q)=8q^{-4}+16q^{-3}+52q^{-2}+80q^{-1}+80q+52q^{2}+16q^{3}+8q^{4}.$
(4.15)
Using (3.37) we identify the Jordan normal form to be
$\text{JNF}^{\text{tot}}_{8,4,2}=(9^{8},7^{8},5^{36},3^{28},1^{28}).$ (4.16)
### 4.2 General $L,M,K$
Here we generalise the observations of the previous subsections to arbitrary
$L,M,K$ sectors. Given an $L,M,K$ sector we consider a subsector
$\bigotimes_{i=1}^{K}W^{\ell_{i}+m_{i}+1,m_{i}+1}$ defined by the vectors
$\boldsymbol{\ell},\boldsymbol{m}$. The anti-locked state takes the form
$\Omega=\ket{(2\cdots 21\cdots 1)_{1}\mathbf{3}(2\cdots 21\cdots
1)_{2}\mathbf{3}\cdots(2\cdots 21\cdots 1)_{K}\mathbf{3}},$ (4.17)
where $(\ell_{j},m_{j})$ are the numbers of $1$’s and $2$’s in the
$j^{\text{th}}$ bracket. Recall that we have
$\quad\sum_{j=1}^{K}\ell_{j}=L_{1}=L-M,\quad\sum_{j=1}^{K}m_{j}=M_{1}=M-K.$
(4.18)
As for $K=1$, we can grade the vector space by the action of $H$
$\bigotimes_{i=1}^{K}W^{\ell_{i}+m_{i}+1,m_{i}+1}=\bigoplus_{S=0}^{S_{\text{max}}}W^{\boldsymbol{\ell},\boldsymbol{m}}_{S},$
(4.19)
where $W^{\boldsymbol{\ell},\boldsymbol{m}}_{S_{\text{max}}}$ is spanned by
the anti-locked state and $H$ lowers the level $S\rightarrow S-1$. By acting
successively with $H$ on $\Omega$, we will arrive at the locked state
$|(1\cdots 12\cdots 2)_{1}\mathbf{3}(1\cdots 12\cdots
2)_{2}\mathbf{3}\cdots(1\cdots 12\cdots 2)_{K}\mathbf{3}\rangle.$ (4.20)
There will be many different configurations in the middle with lower values of
$S$. For the anti-locked state we have
$S=S_{\rm
max}=\boldsymbol{\ell}\cdot\boldsymbol{m}=\sum_{j=1}^{K}\ell_{j}m_{j},$ (4.21)
and so the size of the largest Jordan block in each subsector is $S_{\rm
max}+1$. If we define the number of actions of $H$ on the $j^{\text{th}}$
bracket as $n_{j}$, a general state has a level
$S=\sum_{j=1}^{K}s_{j}=S_{\rm max}-N,\qquad s_{j}=\ell_{j}m_{j}-n_{j},\quad
N=\sum_{j=1}^{K}n_{j},\quad{\rm with}\quad 0\leq n_{j}\leq\ell_{j}m_{j}.$
(4.22)
The anti-locked state has $S=S_{\rm max}$ (or $N=0$) and the locked state has
$S=0$ (or $N=S_{\rm max}$).
Now consider states obtained by acting with $H$ $N$-times on the anti-locked
state,
$H^{N}\Omega=\sum_{n_{1}=0}^{\ell_{1}m_{1}}\cdots\sum_{n_{K}=0}^{\ell_{K}m_{K}}\ket{H^{n_{1}}(2\cdots
21\cdots 1)\mathbf{3}H^{n_{2}}(2\cdots 21\cdots 1)\mathbf{3}\cdots
H^{n_{K}}(2\cdots 21\cdots 1)\mathbf{3}}.$ (4.23)
The number of elementary states generated by each $H^{n_{j}}(2\cdots 21\cdots
1)$ was found in section 3.2 to be
$\mathbf{d}_{l_{j}m_{j}-n_{j}}(\ell_{j},m_{j})$, which appeared as a
coefficient of the $q$-binomial $\binom{\ell_{j}+m_{j}}{m_{j}}_{q}$ as defined
in (3.23). Therefore we can compute the number of elementary states at each
level $S$ to be
$\mathbf{D}^{\boldsymbol{\ell},\boldsymbol{m}}_{S_{\text{max}}-N}\equiv\text{dim}\hskip
1.42271ptW^{\boldsymbol{\ell},\boldsymbol{m}}_{S_{\text{max}}-N}=\sum_{n_{1}=0}^{\ell_{1}m_{1}}\cdots\sum_{n_{K}=0}^{\ell_{K}m_{K}}\prod_{j=1}^{K}\mathbf{d}_{l_{j}m_{j}-n_{j}}(\ell_{j},m_{j}),\qquad{\rm
with}\quad\sum_{j=1}^{K}n_{j}=N.$ (4.24)
This can be recast into a generating function
$\displaystyle\bar{Z}^{\boldsymbol{\ell},\boldsymbol{m}}(q)$ $\displaystyle=$
$\displaystyle\sum_{N=0}^{S_{\text{max}}}\mathbf{D}^{\boldsymbol{\ell},\boldsymbol{m}}_{S_{\text{max}}-N}\,q^{N}=\sum_{N=0}^{S_{\text{max}}}\left[\sum_{n_{1}=0}^{\ell_{1}m_{1}}\cdots\sum_{n_{K}=0}^{\ell_{K}m_{K}}\delta_{N,\sum_{i=1}^{K}n_{i}}\prod_{j=1}^{K}\mathbf{d}_{l_{j}m_{j}-n_{j}}(\ell_{j},m_{j})\right]\,q^{N}$
(4.25) $\displaystyle=$
$\displaystyle\sum_{n_{1}=0}^{\ell_{1}m_{1}}\cdots\sum_{n_{K}=0}^{\ell_{K}m_{K}}\prod_{j=1}^{K}\,\left[\mathbf{d}_{l_{j}m_{j}-n_{j}}(\ell_{j},m_{j})q^{n_{j}}\right]=\prod_{j=1}^{K}\,\left[\prod_{k=1}^{m_{j}}\frac{1-q^{\ell_{j}+m_{j}+1-k}}{1-q^{k}}\right],$
using the expression for $K=1$ in (3.23). This may be expressed through
$q$-binomials as
$\bar{Z}^{\boldsymbol{\ell},\boldsymbol{m}}(q)=\prod_{j=1}^{K}\,\binom{l_{j}+m_{j}}{m_{j}}_{q}.$
(4.26)
It proves that the generating function for an
${\boldsymbol{\ell}},\boldsymbol{m}$ is simply a product of the corresponding
$K=1$ generating functions. For example, if we take $L=13,M=7,K=3$ and
consider the subsector $\boldsymbol{\ell}=(3,2,1),\boldsymbol{m}=(2,1,1)$ we
find
$\displaystyle\bar{Z}^{\boldsymbol{\ell},\boldsymbol{m}}(q)=\binom{5}{2}_{q}\binom{3}{1}_{q}\binom{2}{1}_{q}=\sum_{N=0}^{9}\mathbf{D}^{\boldsymbol{l},\boldsymbol{m}}_{9-N}q^{N}$
(4.27)
$\displaystyle=1+3q+6q^{2}+9q^{3}+11q^{4}+11q^{5}+9q^{6}+6q^{7}+3q^{8}+q^{9}.$
Analagously to the $K=1$ case, we can use (3.13) to determine the Jordan block
spectrum in this subsector
$\text{JNF}^{\boldsymbol{\ell},\boldsymbol{m}}_{13,7,3}=(2^{2},4^{3},6^{3},8^{2},10).$
(4.28)
Since the states belonging to a given partition
${\boldsymbol{\ell}},\boldsymbol{m}$ of $(L_{1},M_{1})$ are not mixed with
those in a different partition, the total Jordan block spectrum is just direct
sum of all the spectrum sets. One can sum over all inequivalent partitions
formally. For this purpose, it is necessary to use the modified $q$-binomial
coefficients defined in (3.34)
$Z^{\boldsymbol{\ell},\boldsymbol{m}}(q)=\prod_{j=1}^{K}\genfrac{[}{]}{0.0pt}{0}{\ell_{j}+m_{j}}{m_{j}}_{q}=\prod_{j=1}^{K}\,q^{-\ell_{j}m_{j}/2}\binom{l_{j}+m_{j}}{m_{j}}_{q}.$
(4.29)
For each $\boldsymbol{\ell},\boldsymbol{m}$ subsector we can exhaust the rest
of the state space by acting with the translation operator
$U^{j},j=1,\dots,L-1$. The arguments of this section do not change in these
cases, and so the overall generating function for a subsector can be obtained
by simply multiplying it by $L$. The only exception is
$\boldsymbol{\ell},\boldsymbol{m}$ subsectors which have a symmetry factor
$S_{\boldsymbol{\ell},\boldsymbol{m}}\neq 1$. Adjusting for this possibility,
we can define the generating function for a whole $L,M,K$ sector as a sum over
inequivalent partitions
$\mathcal{Z}_{L,M,K}(q)=\sum_{(\boldsymbol{\ell},\boldsymbol{m})/\sim}\frac{L}{S_{\boldsymbol{\ell},\boldsymbol{m}}}Z^{\boldsymbol{\ell},\boldsymbol{m}}(q).$
(4.30)
This total generating function gives the complete Jordan block spectrum, as in
(3.37):
$\mathcal{Z}_{L,M,K}(q)=\sum_{j}N_{j}[j]_{q}=N_{1}\,q^{0}+N_{2}\,\left(q^{-\frac{1}{2}}+q^{\frac{1}{2}}\right)+N_{3}\,\left(q^{-1}+q^{0}+q^{1}\right)+\ldots\,.$
(4.31)
As for the $K=1$ case, $\mathcal{Z}_{L,M,K}(q)$ can alternatively be computed
as a trace over the entire Hilbert space
$\mathcal{Z}_{L,M,K}(q)=\text{tr}\hskip
1.42271ptq^{\hat{S}-\hat{S}_{\text{max}}/2},$ (4.32)
where $\hat{S}$ measures the level $S$ of an elementary state and
$\hat{S}_{\text{max}}$ measures
$S_{\text{max}}=\boldsymbol{\ell}\cdot\boldsymbol{m}$ of a state in an
$\boldsymbol{\ell},\boldsymbol{m}$ subsector. Both operators are extended to
the full Hilbert space by linearity. We can define
$\hat{S}^{\prime}\equiv\hat{S}-\hat{S}_{\text{max}}/2$ for brevity.
#### Cyclicity classes.
The expression (4.32), which can also be expressed as (4.30), gives a
generating function that describes the Jordan block spectrum of the
hypereclectic model in an arbitrary sector of operators defined by $L,M,K$.
However, in certain circumstances it might be useful to compute the Jordan
block spectrum in a specific cyclicity class $k$, for example the cyclic
sector $k=0$ relevant to quantum field theory. In this case, remarkably the
formula (4.32) still applies
$\mathcal{Z}^{k}_{L,M,K}(q)=\text{tr}_{k}\hskip
1.42271ptq^{\hat{S}^{\prime}},$ (4.33)
where we take care to trace only over states of a fixed cyclicity $k$.
## 5 Eclectic Spin Chain and Universality
In the previous sections we described a method to find the full Jordan block
spectrum of the hypereclectic model, as opposed to the eclectic model (2.2)
which is our main interest. However, we claim a universality hypothesis: The
Jordan block spectrum of the eclectic model for generic couplings
$\xi_{1},\xi_{2},\xi_{3}$ is identical to that of the hypereclectic model,
provided $L,M,K$ satisfy
$L_{1}=L-M\geq K,\qquad M_{1}=M-K\geq K.$ (5.1)
(5.1) implies that the number of $\phi_{3}$ fields in the sector does not
exceed the number of $\phi_{1}$’s or $\phi_{2}$’s. Without loss of generality
we can further take
$L-M\geq M-K\geq K.$ (5.2)
Throughout this section we will consider (5.2) to be satisfied, otherwise we
can simply relabel the fields so that it is. It is possible to fine-tune the
couplings to break down the Jordan block structure in certain cyclicity
classes, as discussed in appendix C. Since the $\phi_{3}$ fields no longer act
as walls it is useful to work with states of a fixed cyclicity $k$, see
section 2.2. For definiteness in the following examples we will restrict to
the cyclic sector $k=0$, which in addition happens to be the case relevant to
quantum field theory.
### 5.1 Eclectic Spin Chain and Level $S$
Recall that for elementary states in $K=1$ sectors we defined a level $S$,
which corresponds to the total number of 1’s to the right of each of the 2’s
in a state. Here we work with cyclic states
$\ket{j_{1}j_{2}\cdots j_{L-1}3}_{0}\equiv\mathcal{C}_{0}\ket{j_{1}j_{2}\cdots
j_{L-1}3}=\sum_{j=0}^{L-1}U^{j}\ket{j_{1}j_{2}\cdots j_{L-1}3}.$ (5.3)
We define $S$ in an analogous manner for states of the form (5.3). For example
the state $\ket{1211213}_{0}$ has $S=4$. Let us define $V_{S}$ to be the
vector subspace of $V^{L,M,1}$ spanned by cyclic states with level $S$.777We
suppress the $L,M$ dependence of $V_{S}$. We saw previously that the
hypereclectic Hamiltonian maps states in $V_{S}$ to $V_{S-1}$
$H_{3}:V_{S}\rightarrow V_{S-1},\qquad H_{3}V_{0}=0.$ (5.4)
Let us investigate the action of the full eclectic Hamiltonian
$H_{\text{ec}}=H_{1}+H_{2}+H_{3}$ on the vector spaces $V_{S}$. We find that
$H_{1}:V_{S}\rightarrow V_{S-L_{1}},\qquad H_{2}:V_{S}\rightarrow
V_{S-M_{1}}.$ (5.5)
Since $L_{1}\geq M_{1}\geq 1$ (5.5) implies that $H_{1}$ and $H_{2}$ decrease
$S$ for a state by a greater than or equal amount to $H_{3}$. This already
makes plausible that they will not affect the Jordan normal form of $H_{3}$,
since $H_{2}$ and $H_{1}$ will annihilate states faster than $H_{3}$. For
example, consider the anti-locked state $\ket{221113}_{0}\in V_{6}$ for
$L=6,M=3,K=1$, so that $L_{1}=3,M_{1}=2$. Then
$\displaystyle H_{1}\ket{221113}_{0}=\ket{211123}_{0}\in V_{3},$ (5.6)
$\displaystyle H_{2}\ket{221113}_{0}=\ket{122113}_{0}\in V_{4},$
$\displaystyle H_{3}\ket{221113}_{0}=\ket{212113}_{0}\in V_{5}.$
### 5.2 Warmup Example
Let us consider the eclectic model for $L=7,M=3,K=1$. In the hypereclectic
model this sector has the Jordan block spectrum $(9,5,1)$ in $W^{9,5}$. Here
we show that the eclectic model has the same Jordan block spectrum in the
cyclic sector.
#### Top block.
The anti-locked state in the cyclic sector $\ket{2211113}_{0}\in V_{8}$ again
determines a Jordan block of length 9. The first descendant of the anti-locked
state is
$\displaystyle
H_{\text{ec}}\ket{2211113}_{0}=\xi_{1}\ket{2111123}_{0}+\xi_{2}\ket{1221113}_{0}+\xi_{3}\ket{2121113}_{0}.$
(5.7)
Note that the coefficients of $\xi_{1},\xi_{2},$ and $\xi_{3}$ are states with
$S=4,6,$ and $7$ respectively, which reflects equations (5.4) and (5.5). In
general acting with a power of $H_{\text{ec}}$ on $\ket{2211113}_{0}$ gives
$H_{\text{ec}}^{n}\ket{2211113}_{0}=H_{3}^{n}\ket{2211113}_{0}\hskip
5.69046pt+\hskip 5.69046pt\text{lower $S$ states}.$ (5.8)
It is then easy to see that $H^{9}\ket{2211113}_{0}=0$ and thus
$\ket{2211113}_{0}$ is the top state for a Jordan block of length 9, as
before.
#### Middle block.
In the hypereclectic case the top state of the next Jordan block is
$\psi^{(6)}=-9\ket{1221113}_{0}+5\ket{2112113}_{0}\in V_{6},$ (5.9)
which satisfies $H_{3}^{5}\psi^{(6)}=0$. Thus $\psi^{(6)}$ determines a Jordan
block of length 5 for $H_{3}$. However, in this case things are a bit trickier
in the eclectic model. We have
$H_{\text{ec}}^{5}\psi^{(6)}=15\xi_{2}\xi_{3}^{4}\ket{1111223}_{0}\neq 0.$
(5.10)
It is however possible to modify the top state (5.9) by adding states of lower
$S$, such that the residual term (5.10) vanishes. In this case it is
sufficient to add states with $S=5$ to $\psi^{(6)}$. Since $\text{dim}\hskip
1.42271ptV_{5}=2$ we can add 2 states, to arrive at a new top state
$\chi^{(6)}=\psi^{(6)}+\gamma_{1}\ket{1212113}_{0}+\gamma_{2}\ket{2111213}.$
(5.11)
This state satisfies
$H_{\text{ec}}^{5}\psi^{(6)}=(-15\xi_{2}+(5\gamma_{1}+4\gamma_{2})\xi_{3})\xi_{3}^{4}\ket{1111223}_{0},$
(5.12)
which is 0 for $5\gamma_{1}+4\gamma_{2}=15\xi_{2}/\xi_{3}$. Note that this
defines a one-parameter family of top states. Therefore the eclectic model
also has a Jordan block of length 5 in this sector, with a slightly modified
top state (5.11) which contains lower $S=5$ states.
#### Bottom block.
In the hypereclectic model the top state for the final Jordan block is
$\psi^{(4)}=\ket{2111123}_{0}-\ket{1211213}_{0}+\ket{1122113}_{0}\in V_{4},$
(5.13)
which satisfies $H_{3}\psi^{(4)}=0$ and thus determines a Jordan block of
length 1. The action of the eclectic Hamiltonian on this state gives a
residual
$H_{\text{ec}}\psi^{(4)}=-\xi_{1}\ket{1111223}_{0}-\xi_{2}\ket{1112213}_{0}-\xi_{2}\ket{1121123}_{0}\neq
0,$ (5.14)
which consists of states with $S=0$ and $S=2$. As before we can eliminate this
residual by adding states of lower $S$ to the top state (5.13). We first try
to add states with $S=3$, and since $\text{dim}\hskip 1.42271ptV_{3}=2$ we add
2 states
$\chi^{(4)}=\psi^{(4)}+\alpha_{1}\ket{1121213}_{0}+\alpha_{2}\ket{1211123}_{0}.$
(5.15)
We check that for $\alpha_{1}=\xi_{2}/\xi_{3},\alpha_{2}=-2\xi_{2}/\xi_{3}$
the $S=2$ states in the residual (5.14) vanish
$H_{\text{ec}}\chi^{(4)}=-\xi_{1}\ket{1111223}_{0}+\frac{\xi_{2}^{2}}{\xi_{3}}\ket{1112123}_{0},$
(5.16)
which is a new residual consisting of an $S=1$ and an $S=0$ state. These can
be removed by adding $S=2$ states into the top state ansatz
$\bar{\chi}^{(4)}=\chi^{(4)}+\beta_{1}\ket{1112213}_{0}+\beta_{2}\ket{1121123}_{0},$
(5.17)
and setting
$\beta_{1}=\xi_{1}/\xi_{2},\beta_{2}=-\xi_{1}/\xi_{2}-\xi_{2}^{2}/\xi_{3}^{2}$.
With these choices for $\alpha_{i}$ and $\beta_{i}$ we have
$H_{\text{ec}}\bar{\chi}^{(4)}=0,$ (5.18)
and so we have identified the Jordan block of length one in this sector of the
eclectic model. In summary, by taking a top state for the hypereclectic model
at a level $S$, we can manufacture a top state (of a Jordan block of the same
length) for the eclectic model by adding appropriate combinations of states
with lower values of $S$. We will argue that it is always possible to add
these states of lower $S$, thus rendering the Jordan block spectra of the
hypereclectic and eclectic models equivalent.
### 5.3 General Argument for $K=1$
Here we sketch a proof of universality for $K=1$, where the filling condition
(5.1) is trivially satisfied, if all three particles are present in the spin
chain state. We find it useful to first introduce the notion of supereclectic
models. These are intermediate models between the eclectic model
$H_{\text{ec}}$ and the hypereclectic model $H_{3}$, defined by setting only a
single coupling $\xi_{1}$ or $\xi_{2}$ equal to zero
$H_{\text{super},i}=H_{i}+H_{3},\qquad i=1,2.$ (5.19)
For both of these cases it is possible to prove rigorously that
$H_{\text{super},i}$ has the same Jordan normal form as $H_{3}$ for generic
couplings. The general strategy of the proof is reminiscent of the example
given in section 5.2. For the hypereclectic model, at a level $S$ satisfying
$\mathbf{d}_{S}>\mathbf{d}_{S+1}$ we can construct
$\mathbf{d}_{S}-\mathbf{d}_{S+1}$ top states
$\psi^{(S)}=\sum_{j=1}^{\mathbf{d}_{S}}\alpha_{j}^{(S)}e_{j}^{(S)},$ (5.20)
where $\alpha_{j}^{(S)}$ are known coefficients and $e_{j}^{(S)}$ are the
elementary states at level $S$. $\psi^{(S)}$ is the top state for a Jordan
block of length $S-\tilde{S}+1$
$H_{3}^{S-\tilde{S}+1}\psi^{(S)}=0,$ (5.21)
where we recall $\tilde{S}=S_{\text{max}}-S=(L-M)(M-1)-S$. We show that it is
always possible to modify the state by adding a linear combination of states
with lower $S$
$\psi^{(S)}\to\psi_{i}^{(S)}=\psi^{(S)}+\sum_{n=0}^{S-1}\varphi^{(n)}$ (5.22)
where $\varphi^{(n)}\in V_{n}$. The modified state is a top state for a Jordan
block of the same length in the supereclectic model $H_{\text{super},i}$
$H_{\text{super},i}^{S-\tilde{S}+1}\psi_{i}^{(S)}=0,$ (5.23)
which renders the Jordan normal forms of $H_{\text{super},i}$ and $H_{3}$
equivalent for generic couplings. More technical details of this proof are
given in appendix B. This argument can then be slightly modified to motivate
that the Jordan normal forms of $H_{\text{ec}}$ and $H_{3}$ are also
equivalent, see again B, even if we have not yet worked out all details of the
proof.
#### Universality for $K>1$.
It is even more complicated to show the universality for $K>1$. One main
difference from the $K=1$ case of the supereclectic models, as explained in
appendix B, is that the action of $h_{j}$ on $\varphi^{(S)}$ in general
generates several states with differing $S$-values. If we interpret
${\hat{S}}(h_{j}\varphi^{(S)})$ in (B.4) as the largest among these and
replace $L_{1}$ with the associated $\ell_{j}$, the same logic should be
valid, so that one can construct for the supereclectic models all subleading
states in (4.24).
For the eclectic model, however, a critical simplification used in (B.22) is
not valid. While we have extensive numerical evidence for general
universality, we are currently unable to provide a proof. We leave this for
future work.
## 6 Conclusions and Outlook
We introduced a generating function $\mathcal{Z}_{L,M,K}(q)$ that we
conjectured (and partially proved) to fully enumerate the Jordan block
spectrum of the hypereclectic model introduced in [2], for any sector of
particles labelled by $L,M,K$. Interestingly, it takes a form reminiscent of a
partition function, where one traces a certain kind of number operator over
the state space. It may also be expressed as a sum over products of (shifted)
$q$-binomial coefficients, which elegantly reduces to a single $q$-binomial
for the case of one wall, i.e. $K=1$. Furthermore, our approach for
determining this generating function yields an algorithmic method for
generating the states of the Jordan blocks. We also provided further strong
evidence and partial proofs for the validity of the universality hypothesis of
[2], i.e. the claim that the spectrum of the hypereclectic and eclectic models
agree for special filling conditions. This is important, as the hypereclectic
model is much easier to handle combinatorially in comparison with the eclectic
one. Apart from its intrinsic value as a new type of solution for a new type
of spin chain, our results appear to be an important starting point for an in-
depth analysis of the indecomposability properties of the dynamical fishnet
theory, cf. (1.1), an integrable logarithmic conformal field theory in four
dimensions. In this context, note that $q$-binomials are ubiquitous in the
analysis of two-dimensional logarithmic conformal field theories, see for
example [28].
There are a number of gaps in our derivations that call for further research.
Firstly, our combinatorial arguments do not rigorously exclude the possibility
of ‘unexpected shortening’ of Jordan blocks, as explained in appendix A.
Secondly, while we made some progress towards a proof of the universality
conjecture, a full proof is still missing. It is possible that the filling of
these two gaps will require entirely new methods.
In this context, note that our results for these integrable models have not
been derived by directly using integrability. Instead, they have been obtained
by linear algebra arguments combined with combinatorics. Still, note that we
were able to provide rather elegant formulas that clean up and organise to a
large degree the (at first sight) incredibly intricate Jordan block structure
of the models. One therefore wonders whether this, at least to us, rather
astonishing fact is not an indirect manifestation of the integrability of
these non-hermitian spin chain models. Understanding our findings from
integrability is not only an interesting intellectual challenge, but might
eventually allow to fill in the above mentioned gaps and incomplete proofs of
this paper. Also, using integrability might lead to more explicit formulas
than (4.30) for $\mathcal{Z}_{L,M,K}(q)$ for $K>1$.
There are numerous further directions for investigations to consider. An
interesting conceptual question is whether the Jordan block spectrum of other
non-diagonalisable spin chains, integrable or not, can also be described by
similar generating functions. Or else, is this something particular to the
(hyper)eclectic spin chain? There would be a few natural ways to test this.
For example, one could study the dilatation operator in other non-
diagonalisable sectors of (dynamical) fishnet theory. These sectors could
contain derivative fields/fermions, and would be more intricate to analyse.
There are also different strong twist limits of $\mathcal{N}=4$ SYM available,
which should contain new diagonalisable models, see [2]. One could also
consider the dilatation operator in the strong twist limit of ABJM theory
[29]. In this case the first quantum correction to the dilatation operator
appears at two loops, and we expect this would be a chiral version of the
alternating spin chain given in [30].
The results of this paper concern the dilatation operator at one-loop order.
It is natural to ask what might happen at higher loops. The dilatation
operator certainly continues to be nilpotent, and therefore is non-
diagonalisable. It would be interesting to see in detailed generality if and
how the dilatation operator at different loop orders refines the Jordan block
spectrum. And clearly if would be exciting to understand the structure of
Jordan blocks on the non-perturbative level. Note that the QSC approach does
not, in its current form, allow to even address the question [31].
### Acknowledgements
We are very thankful to Moritz Kade for helpful discussions, comments on the
draft, and for writing a very useful Mathematica program for numerically
obtaining the Jordan normal form of the hypereclectic spin chain. We would
like to express our sincere gratitude to the Brain Pool Program of the Korean
National Research Foundation (NRF) under grant 2-2019-1283-001-1 for generous
support of this research. MS thanks Ewha Womans University for hospitality in
this difficult period. This project has received funding from NRF grant (NRF-
2016R1D1A1B02007258) (CA) and from the European Union’s Horizon 2020 research
and innovation programme under the Marie Sklodowska-Curie grant agreement No.
764850 ‘SAGEX’ (LC, MS).
## Appendix A Unexpected Shortening
Here we succinctly reformulate the conditions for the unwanted ‘unexpected
shortening’ described in section 3.2. This might be helpful for eventually
finding a rigorous proof. In any case, it was very useful for the extensive
numerical checking of our conjecture: unexpected shortening cannot happen.
In a sector with general $L,M$, $K=1$, we argued for the existence of a top
state in $W^{L,M}_{S}$, where $S$ was such that
$\mathbf{d}_{S}>\mathbf{d}_{S+1}:$
$\psi^{(S)}=\sum_{i=1}^{\mathbf{d}_{S}}\alpha_{i}e_{i}^{(S)},$ (A.1)
where $\alpha_{i}$ are constants and $e_{i}^{(S)}$ are the elementary states
in $W^{L,M}_{S}$. Acting with a power of $H$ on this state gives
$H^{k}\psi^{(S)}=\sum_{i=1}^{\mathbf{d}_{S-k}}\sum_{j=1}^{\mathbf{d}_{S}}A^{(k)}_{ij}\alpha_{j}e_{i}^{(S-k)}=\sum_{i=1}^{\mathbf{d}_{S-k}}(A^{(k)}\alpha)_{i}e_{i}^{(S-k)},$
(A.2)
where $A^{(k)}$ is a $\mathbf{d}_{S-k}\times\mathbf{d}_{S}$ matrix, and
$\alpha$ is a vector of length $\mathbf{d}_{S}$ with entries $\alpha_{i}$. The
top state $\psi^{(S)}$ defines a Jordan block of length $k$ if
$H^{k}\psi^{(S)}=0$, or equivalently the homogeneous linear system
$A^{(k)}\alpha=0$ (A.3)
admits at least one nontrivial solution in $\alpha$. We claim that the rank of
$A^{(k)}$ is always maximal:
$\text{rank}(A^{(k)})=\text{min}(\mathbf{d}_{S-k},\mathbf{d}_{S}).$ (A.4)
In this case, it is well-known that (A.3) can only admit a nontrivial solution
in $\alpha$ if and only if rank$(A^{(k)})<\mathbf{d}_{S}$. Moreover, the
number of independent nontrivial solutions is
$\mathbf{d}_{S}-\text{rank}(A^{(k)})$. Therefore a nontrivial solution only
exists when $\mathbf{d}_{S-k}<\mathbf{d}_{S}$. This occurs precisely when
$k=S-\tilde{S}+1$, as can be deduced from (3.23). Therefore, if the rank of
$A^{(k)}$ is always maximal, the top state $\psi^{(S)}$ determines
$\mathbf{d}_{S}-\mathbf{d}_{S+1}$ Jordan blocks, each of length
$S-\tilde{S}+1$. We checked the rank of $A^{(k)}$ for all top states and for
all values of $k,$ up to $L=30,M=6,$ and always found it to be maximal, in
line with our conjecture.
## Appendix B Universality Details for $K=1$
In this section we prove that $H_{\text{super},i}$, defined in (5.19), has the
same Jordan block structure as the hypereclectic model $H_{3}$ for $K=1$,
under the assumption discussed in appendix A. Then we describe how to modify
these arguments to include the full eclectic Hamiltonian, and sketch a
possible universality proof for $K=1$.
#### Universality for $H_{\text{super},1}$.
We start with the first supereclectic model defined in (5.19),
$H_{\text{super},1}$. Consider a top vector $\psi^{(S)}$ for the hypereclectic
model at a level $S$. This vector determines a Jordan block of length
$n_{S}\equiv S-\tilde{S}+1$
$H_{3}^{n_{S}}\psi^{(S)}=0.$ (B.1)
We can expand $H_{{\rm super},1}^{n_{S}}$ as
$H_{{\rm
super},1}^{n_{S}}=\sum_{k=0}^{n_{S}}\binom{n_{S}}{k}H_{1}^{k}H_{3}^{n_{S}-k}=H_{3}^{n_{S}}+n_{S}H_{1}H_{3}^{n_{S}-1}+\frac{n_{S}(n_{S}-1)}{2}H_{1}^{2}H_{3}^{n_{S}-2}+\cdots,$
(B.2)
where we have used $[H_{1},H_{3}]=0$. We introduce a shorthand notation
$H_{{\rm super},1}^{n_{S}}=\sum_{j=0}^{n_{S}}h_{j},\quad
h_{0}=H_{3}^{n_{S}},\quad
h_{j}\equiv\binom{n_{S}}{k}H_{1}^{j}H_{3}^{n_{S}-j},\quad j=1,\dots,n_{S}.$
(B.3)
B͡ecause of (5.4) and (5.5) each $h_{j}$ lowers the $S$-value of a state by
$j(L_{1}-1)+n_{S}$. In other words, given a vector $\varphi^{(S)}\in V_{S}$ we
have
$\hat{S}(h_{j}\varphi^{(S)})=S-j(L_{1}-1)-n_{S}=(\tilde{S}-1)-j(L_{1}-1).$
(B.4)
In particular, $h_{j}\varphi^{(S)}=0$ if this value is negative. Now let us
consider the $\tilde{S}$ value of a top vector to be in an interval
$\ell(L_{1}-1)\leq\tilde{S}-1<(\ell+1)(L_{1}-1).$ (B.5)
In this case, all operators $h_{j}$ with $j>\ell$ will annihilate the top
vector and its descendants. Therefore, we may consider only operators
$h_{0},h_{1},\dots,h_{\ell}$ and disregard others in (B.3).
For this $S$ value of the top vector of the hypereclectic model $\psi^{(S)}$,
we claim that we can construct a corresponding top vector $\psi^{(S)}_{1}$ of
the supereclectic model $H_{\text{super},1}$, defined by
$H_{{\rm super},1}^{n_{S}}\psi^{(S)}_{1}=0,$ (B.6)
via the ansatz
$\psi^{(S)}_{1}=\varphi_{0}+\varphi_{1}+\cdots+\varphi_{\ell},\qquad\varphi_{0}=\psi^{(S)},$
(B.7)
if the top vector has ${\tilde{S}}$ which satisfies (B.5). The condition (B.6)
can be written as
$\displaystyle(h_{0}+h_{1}+h_{2}+\cdots+h_{\ell})(\varphi_{0}+\varphi_{1}+\cdots+\varphi_{\ell})$
$\displaystyle=$
$\displaystyle(h_{0}\varphi_{0})+(h_{0}\varphi_{1}+h_{1}\varphi_{0})+\cdots+(h_{0}\varphi_{\ell}+h_{1}\varphi_{\ell-1}+\cdots+h_{\ell}\varphi_{0})+\cdots=0,$
where we have grouped terms in a very particular way. The first term
$h_{0}\varphi_{0}$ in (B) vanishes due to (B.1). Now we want to find
$\varphi_{1}$ in the second bracket from the restriction that it vanishes
$h_{0}\varphi_{1}+h_{1}\varphi_{0}=0.$ (B.9)
Since $\hat{S}(h_{1}\varphi_{0})=(\tilde{S}-1)-(L_{1}-1)$ from (B.4), this
equation should be expressed by elementary vectors with this $S$ value. There
are $\mathbf{d}_{(\tilde{S}-1)-(L_{1}-1)}$ of them, which becomes the number
of constraints.888In fact, this is the maximum number of constraints since
some of the elementary vectors may not appear. This equation also determines
$\hat{S}(\varphi_{1})=\hat{S}(h_{1}\varphi_{0})+n_{S}=S-(L_{1}-1)$. Therefore,
$\varphi_{1}$ can be expressed as a linear combination of
$\mathbf{d}_{S-(L_{1}-1)}$ elementary states. Since
$\mathbf{d}_{S-(L_{1}-1)}=\mathbf{d}_{\tilde{S}+(L_{1}-1)}>\mathbf{d}_{(\tilde{S}-1)-(L_{1}-1)}$,
one can solve coefficients of the linear combination from (B.9) (not always
unique). This proves that we can always find the solution $\varphi_{1}$.
We require the next bracket in (B) to vanish:
$h_{0}\varphi_{2}+h_{1}\varphi_{1}+h_{2}\varphi_{0}=0.$ (B.10)
Again, one can find that
$\hat{S}(h_{1}\varphi_{1})=\hat{S}(h_{2}\varphi_{0})=(\tilde{S}-1)-2(L_{1}-1)$,
from which we determine $\hat{S}(\varphi_{2})=S-2(L_{1}-1)$. Since the maximum
number of constraints is smaller than that of the coefficients due to
$\mathbf{d}_{S-2(L_{1}-1)}>\mathbf{d}_{(\tilde{S}-1)-2(L_{1}-1)}$, one can
find $\varphi_{2}$ from the known vectors $\varphi_{1}$ and $\varphi_{0}$
using (B.10).
One can easily generalise this argument up to the $\ell$-th bracket in (B):
$h_{0}\varphi_{\ell}+h_{1}\varphi_{\ell-1}+\cdots+h_{\ell}\varphi_{0}=0,$
(B.11)
where the vectors $\varphi_{j},\ j=0,\dots,\ell-1$ have already been found in
previous steps. Since $\hat{S}(\varphi_{j})=S-j(L_{1}-1)$ we have
$\hat{S}(h_{j}\varphi_{\ell-j})=(\tilde{S}-1)-\ell(L_{1}-1)$ for
$j=1,\dots,\ell$. This determines $S$-value of the unknown vector
$\varphi_{\ell}$ to be $\hat{S}(\varphi_{\ell})=S-\ell(L_{1}-1)$. Again, the
maximum number of constraints in (B.11) is smaller than the number of
coefficients in the expansion of $\varphi_{\ell}$ in terms of elementary
states, which guarantees that we can always find its solution.
There are more terms which we did not include in the second line of (B), but
it is easy to show they all vanish. For example, the $(\ell+1)$-th bracket
would be
$h_{1}\varphi_{\ell}+\cdots+h_{\ell}\varphi_{1}.$ (B.12)
Their $S$-values should be $(\tilde{S}-1)-(\ell+1)(L_{1}-1)$, which is
negative due to (B.5). This means that all these vectors vanish.
This proves our universality conjecture for the supereclectic model
$H_{\text{super},1}$ by constructing the top vector explicitly as
$\psi^{(S)}_{1}=\psi^{(S)}+\varphi_{1}+\cdots+\varphi_{\ell},$ (B.13)
for $\tilde{S}$ in (B.5).
Because $\tilde{S}\leq S_{\rm max}/2$ ($\tilde{S}\leq S$ by definition), the
interval (B.5) is limited by the maximum value of $\ell$ which is
$\ell_{\rm max}=\left[\frac{L_{1}M_{1}}{2(L_{1}-1)}\right],$ (B.14)
where $[x]$ is the largest integer not exceeding $x$.
#### Universality for $H_{\text{super},2}$.
The second supereclectic model $H_{\text{super},2}$ defined in (5.19) can be
analysed in exactly the same way. Again, one can express
$H_{\text{super},2}^{n_{S}}=\sum_{m=0}^{n_{S}}g_{m},\quad g_{m}\equiv\
\binom{n_{S}}{m}H_{2}^{m}H_{3}^{n_{S}-m},\quad m=0,\dots,n_{S},\quad
g_{0}=h_{0}=H_{3}^{n_{S}}.$ (B.15)
Each $g_{m}$ lowers $S$-values as follows:
$\hat{S}(g_{m}\phi^{(S)})=S-m(M_{1}-1)-n_{S}.$ (B.16)
In the same way as before, a top vector with level $S$ (and corresponding
$\tilde{S}$) with
$m(M_{1}-1)\leq\tilde{S}-1<(m+1)(M_{1}-1),$ (B.17)
we only need to consider terms in (B.15) up to $g_{m}$.
The remaining procedure is identical to the previous case. One can always find
${\tilde{\varphi}}_{k}$ from
${\tilde{\varphi}}_{0},\dots,{\tilde{\varphi}}_{k-1}$ using
$g_{0}{\tilde{\varphi}}_{k}+g_{1}{\tilde{\varphi}}_{k-1}+\dots+g_{k}{\tilde{\varphi}}_{0}=0,\quad
k=1,\dots,m.$ (B.18)
This proves the universality conjecture for $H_{\text{super},2}$ by
constructing the top vector explicitly as
$\psi^{(S)}_{2}=\psi^{(S)}+{\tilde{\varphi}}_{1}+\cdots+{\tilde{\varphi}}_{m},$
(B.19)
for $\tilde{S}$ in (B.17), where $m$ should be limited by the maximum value
$m_{\rm max}=\left[\frac{L_{1}M_{1}}{2(M_{1}-1)}\right].$ (B.20)
#### Universality for General Eclectic Model.
Powers of $H_{\rm ec}$ can be written as
$H_{\rm ec}^{n_{S}}=\sum_{k=0}^{n_{S}}\
\binom{n_{S}}{k}\,(H_{1}+H_{2})^{k}H_{3}^{n_{S}-k}.$ (B.21)
This expression can be simplified greatly by observing that
$H_{1}H_{2}=H_{2}H_{1}=0$ in sectors where $K=1$. This can be seen by acting
with $H_{1}$ on any state
$\displaystyle H_{1}|\mathbf{2}1\cdots 1\mathbf{2}1\cdots
1\cdots\mathbf{2}1\cdots 1\mathbf{3}\rangle=|1\cdots 1\mathbf{2}1\cdots
1\cdots\mathbf{2}1\cdots 1\mathbf{2}\mathbf{3}\rangle.$ (B.22)
Then, $H_{2}$ will annihilate the resulting state since it cannot contain
$1\mathbf{3}$. Therefore we can remove any terms with both $H_{1}$ and $H_{2}$
in the expansion (B.21), which leads to
$H_{\rm
ec}^{n_{S}}=h_{0}+(g_{1}+g_{2}+\cdots+g_{n_{S}})+(h_{1}+h_{2}+\cdots+h_{n_{S}}).$
(B.23)
We can restrict the interval for $\tilde{S}$ by the two relations (B.5) and
(B.17). Since $L_{1}\geq M_{1}$, for a given $\ell$ we can find $m$ such that
$m(M_{1}-1)\leq\ell(L_{1}-1)<(m+1)(M_{1}-1).$ (B.24)
In this case, the intersection of the two intervals is given by
$m(M_{1}-1)\leq\ell(L_{1}-1)\leq\tilde{S}-1<(m+1)(M_{1}-1).$ (B.25)
For these values of $S$, the expansion of power of the eclectic Hamiltonian is
truncated to
$H_{\rm
ec}^{n_{S}}=h_{0}+(g_{1}+g_{2}+\cdots+g_{m})+(h_{1}+h_{2}+\cdots+h_{\ell}).$
(B.26)
We now claim that the top vector of the eclectic model can be always
constructed from the hypereclectic top state $\psi^{(S)}=\varphi_{0}$ as
follows:
$\psi^{(S)}_{\rm
ec}=\varphi_{0}+\sum_{i=1}^{m}{\tilde{\varphi}}_{i}+\sum_{j=1}^{\ell}\varphi_{j}.$
(B.27)
Let us provide the detailed proof for the simplest case $m=2,\ell=1$, with
$2(M_{1}-1)\leq(L_{1}-1)\leq\tilde{S}-1<3(M_{1}-1).$ (B.28)
We will show that the top vector for the eclectic model can be constructed as
$\psi^{(S)}_{\rm
ec}=\varphi_{0}+{\tilde{\varphi}}_{1}+{\tilde{\varphi}}_{2}+\varphi_{1}.$
(B.29)
One can expand $H_{\rm ec}^{n_{S}}\psi_{\text{ec}}^{(S)}=0$ as
$\displaystyle(h_{0}+g_{1}+g_{2}+h_{1})(\varphi_{0}+{\tilde{\varphi}}_{1}+{\tilde{\varphi}}_{2}+\varphi_{1})=(h_{0}\varphi_{0})+(g_{0}{\tilde{\varphi}}_{1}+g_{1}\varphi_{0})+$
(B.30) $\displaystyle+$
$\displaystyle(g_{0}{\tilde{\varphi}}_{2}+g_{1}{\tilde{\varphi}}_{1}+g_{2}\varphi_{0})+(g_{0}\varphi_{1}+g_{1}{\tilde{\varphi}}_{2}+g_{2}{\tilde{\varphi}}_{1}+h_{1}\varphi_{0})+\cdots=0.$
The first three brackets in (B.30) have already been solved for
$H_{\text{super},2}$, therefore we only need to consider the fourth term and
ellipsis. The $S$-values of each term have already been computed as
$\hat{S}(g_{1}{\tilde{\varphi}}_{2})=\hat{S}(g_{2}{\tilde{\varphi}}_{1})=({\tilde{S}}-1)-3(M_{1}-1)<\hat{S}(h_{1}\varphi_{0})=({\tilde{S}}-1)-(L_{1}-1).$
(B.31)
Therefore, $\varphi_{1}$ can be determined from $\varphi_{0}$ in the same way
as for $H_{\text{super},1}$ with additional subleading terms in $S$ from the
known ${\tilde{\varphi}}_{1},{\tilde{\varphi}}_{2}$. The terms in the ellipsis
in (B.30) are
$\cdots=g_{1}\varphi_{1}+g_{2}{\tilde{\varphi}}_{2}+h_{1}{\tilde{\varphi}}_{1}+g_{2}{\varphi}_{1}+h_{1}{\tilde{\varphi}}_{2}+h_{1}{\varphi}_{1}.$
(B.32)
The $S$-values for these vectors are given by
$\displaystyle\hat{S}(g_{i}\varphi_{j})=\hat{S}(h_{j}{\tilde{\varphi}}_{i})=({\tilde{S}}-1)-j(L_{1}-1)-i(M_{1}-1),$
$\displaystyle\hat{S}(h_{i}\varphi_{j})=({\tilde{S}}-1)-(i+j)(L_{1}-1),\quad\hat{S}(g_{i}{\tilde{\varphi}}_{j})=({\tilde{S}}-1)-(i+j)(M_{1}-1).$
(B.33)
It is not difficult to see from (B.28) that all these vectors should vanish
since their $S$-values are all negative.
This procedure can now be generalised in principle to any value of $(\ell,m)$,
although it is hard to give general, explicit expressions, since the mixed
interval depends closely on explicit vaues of $L_{1},M_{1}$. It would be
interesting to complete the details of this sketch of a proof of $K=1$
universality.
## Appendix C Fine Tuning and Cyclicity Classes
Although we have proven the universality hypothesis for generic values of the
couplings $\xi_{i}$ for $K=1$, it is possible to fine-tune the couplings to
destroy the Jordan block structures in a particular cyclicity class. We give a
simple example of this occurring, for the sector $L=5,M=3,K=1$. There are 30
states in this sector:
$\displaystyle\mathcal{C}_{k}\ket{22113},\quad\mathcal{C}_{k}\ket{21213},\quad\mathcal{C}_{k}\ket{12213},$
(C.1)
$\displaystyle\mathcal{C}_{k}\ket{21123},\quad\mathcal{C}_{k}\ket{12123},\quad\mathcal{C}_{k}\ket{11223},$
where $\mathcal{C}_{k}$ is the unnormalised projector defined in (2.14) and
$k=0,1,2,3,4$ labels the cyclicity class. In each cyclicity class $k$ the
hypereclectic model $H_{3}$ has Jordan decomposition $(5,1)$, so that the
overall Jordan decomposition is $(5^{5},1^{5})$. The other models related to
$H_{3}$ by permutations of the fields $H_{1}$ and $H_{2}$ have Jordan
decomposition (3,2,1) in each cyclicity class. For generic $\xi_{i}$ we have
argued that the eclectic Hamiltonian $H_{\text{ec}}=H_{1}+H_{2}+H_{3}$ also
has the Jordan decomposition $(5^{5},1^{5})$, since this sector satisfies the
filling conditions (5.2). Setting $\xi_{3}=0$ leads to a Jordan decomposition
$(3^{5},2^{5},1^{5})$. Interestingly, this decomposition can be further
refined by tuning $\xi_{1}$ and $\xi_{2}$. Let us act with
$H_{\text{ec}}|_{\xi_{3}=0}$ on the top state $\mathcal{C}_{k}\ket{22113}$:
$\displaystyle\mathcal{C}_{k}\ket{22113}$
$\displaystyle\rightarrow\omega^{k}\xi_{1}\mathcal{C}_{k}\ket{21123}+\omega^{-k}\xi_{2}\mathcal{C}_{k}\ket{12213}$
(C.2)
$\displaystyle\rightarrow(\omega^{2k}\xi_{1}^{2}+\omega^{-2k}\xi_{2}^{2})\mathcal{C}_{k}\ket{11223}\rightarrow
0,$
where $\omega=e^{2\pi i/5}$ and we used $\mathcal{C}_{k}U^{\pm
1}\psi=\omega^{\pm k}\mathcal{C}_{k}\psi,[H_{i},\mathcal{C}_{k}]=0$. For
generic couplings this gives a length 3 block in each cyclicity class.
However, if we tune the couplings such that
$\xi_{2}^{2}=-\omega^{4k}\xi_{1}^{2}$ the block splits into a 2-block and a
1-block in this cyclicity class $k$. There are two further top states in this
sector:
$\displaystyle\mathcal{C}_{k}\ket{21213}\rightarrow(\xi_{1}\omega^{k}+\xi_{2}\omega^{-k})\mathcal{C}_{k}\ket{12123}\rightarrow
0,$ (C.3)
$\displaystyle\xi_{2}\omega^{-k}\mathcal{C}_{k}\ket{21123}-\xi_{1}\omega^{k}\mathcal{C}_{k}\ket{12213}\rightarrow
0.$
The first of these is a 2-block, which can be broken into two 1-blocks in a
single cyclicity class if $\xi_{2}=-\omega^{2k}\xi_{1}$. The next of these is
always a 1-block. From this example we see explicitly that finer Jordan block
decompositions can be obtained in specific cyclicity classes by tuning the
couplings appropriately.
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* [29] J. Caetano, O. Gurdogan and V. Kazakov, “Chiral limit of N = 4 SYM and ABJM and integrable Feynman graphs”, arxiv:1612.05895.
* [30] J. Minahan and K. Zarembo, “The Bethe ansatz for superconformal Chern-Simons”, Journal of High Energy Physics 2008, 040 (2008), arxiv:0806.3951, http://dx.doi.org/10.1088/1126-6708/2008/09/040.
* [31] V. Kazakov, “Quantum Spectral Curve of $\gamma$-Twisted $\mathcal{N}$= 4 SYM Theory and Fishnet CFT”, Reviews in Mathematical Physics 30, 1840010 (2018), arxiv:1802.02160, http://dx.doi.org/10.1142/S0129055X1840010X.
|
-titleHadron Collider Physics symposium (HCP 2011)
11institutetext: Royal Holloway, University of London.
# A Search for Heavy Resonances in the Dilepton Channel
Daniel Hayden On behalf of the ATLAS Collaboration<EMAIL_ADDRESS>
###### Abstract
There are many extensions to the Standard Model of particle physics which
predict the addition of a U(1) symmetry, and/or extra spatial dimensions,
which give rise to new high mass resonances such as the Z′ and Randall-Sundrum
graviton. The LHC provides a unique opportunity to explore the TeV scale where
these phenomena may become apparent, and can be searched for using the
precision tracking and high energy resolution calorimetry of the ATLAS
detector. This poster presents the search for high mass resonances in the
dilepton channel, and was conducted with an integrated luminosity of 1.08/1.21
fb-1 in the dielectron/dimuon channel respectively, at a centre of mass energy
$\sqrt{s}$ = 7 TeV.
## 1 Introduction
There are many possible extensions to the Standard Model (SM) predicted at the
TeV energy scale which may be visible at the LHC. Many of these extensions
predict extra U(1) symmetry with an associated spin-1 particle Theory:ZP1
Theory:ZP2 . In its simplest form this U(1) symmetry can be arbitrarily added
to the existing SM gauge group, resulting in SU(3) $\times$ SU(2) $\times$
U(1) from the SM, and additionally U(1)′ for the Sequential Standard Model
Z${}^{\prime}_{SSM}$. More rigorously motivated models proceed via the
decomposition of Grand Unified Theories such as E6 $\rightarrow$
SO(10)$\times$U(1)ψ $\rightarrow$ SU(5) $\times$ U(1)χ $\times$ U(1)ψ leading
to Z′($\theta$) = Z${}^{\prime}_{\chi}$cos$\theta$ \+
Z${}^{\prime}_{\psi}$sin$\theta$, where the mixing angle $\theta$ determines
the coupling to fermions and results in various possible models with specific
Z′ states. Other extensions of the SM seek to answer questions such as the
hierarchy problem where the relative weakness of gravity compared to the other
forces of nature can be explained with the use of warped extra dimensions in
theories such as the Randall-Sundrum model Theory:G . A feature of this theory
would be a massive spin-2 particle called the graviton (G∗) which should be
observable at the LHC and have a mass/width that depends on the curvature of
the warped dimension, k, and the reduced Planck scale, $\overline{M}_{Pl}$,
leading to another parameter of interest, the coupling k/$\overline{M}_{Pl}$.
Both of the new particles mentioned would appear as resonances in the dilepton
invariant mass spectrum measured by the ATLAS detector ATLAS , and these
results comprise a search using the detector in this endeavor.
## 2 Dilepton Resonance Search
The search for dilepton resonances was conducted in both the electron and muon
channels separately, which were then combined to give the final result. To
identify candidate events from data, each analysis selected high energy
electron/muon pairs. The main background to a Z′/G∗ search in these channels
is from Drell-Yan, with smaller contributions from $t\bar{t}$, W+jets,
diboson, and QCD events 111QCD events here are defined as semi-leptonic decays
of b and c quarks in the dimuon sample, or at least one electron coming from
photon conversions, semi-leptonic heavy quark decays or a hadronic fake, in
the dielectron sample.. These SM background contributions were estimated using
Monte Carlo (MC) simulation, except for QCD which was estimated from data
using a reverse identification selection sample for electrons, and a non-
isolated sample for muons.
For both the dielectron and dimuon channel analyses, a data quality
requirement is made to ensure parts of the ATLAS detector important for
e/$\gamma$ or $\mu$ analysis respectively are working optimally. The events
are also required to have at least one primary vertex with greater than two
tracks, and pass a single electron trigger with a transverse energy (ET)
greater than 20 GeV or equivalently for the dimuon analysis, a muon trigger
with transverse momentum (pT) greater than 22 GeV.
For an event to be accepted by the analysis in the dielectron channel, an
event must contain at least two electron candidates with ET $>$ 25 GeV and
$|$$\eta$$|$ $<$ 2.47, also excluding the region between the barrel and endcap
calorimeters 1.37 $\leq$ $|$$\eta$$|$ $\leq$ 1.52. The electron candidates
that pass these criteria must have been reconstructed from electromagnetic
cells clusters with an associated charged particle track from the inner
detector. Shower shape variables and hadronic calorimeter leakage, along with
information from the inner detector is then used to strengthen the
identification of the electron candidates. A hit in the first layer of the
pixel detector is required to suppress background from photon conversions.
From the remaining electron candidates the highest ET pair is selected and the
higher ET electron required to pass an isolation threshold of less than 7 GeV
in a cone of 0.2 around the cluster ($\Delta$R =
$\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$) to reduce the QCD background.
Finally, the invariant mass of the selected pair must be greater than 70 GeV
to be accepted by the analysis, and no opposite charge requirement is made to
minimise the impact of possible charge mis-identification.
In the dimuon channel, two oppositely charged muons are required. Each muon
must have pT $>$ 25 GeV, and pass quality criteria from the inner detector as
well as having at least three hits in each of the inner, middle, and outer
layers of the muon spectrometer to improve momentum resolution. Muons are
discarded if they have hits in both the barrel and endcap regions because of
residual misalignment. To suppress the cosmic ray background the $z$ position
of the primary vertex is required to be less than 200 mm, and muon tracks must
have a transverse impact parameter $|$$d_{0}$$|$ $<$ 0.2 mm, also being within
1 mm of the primary vertex along the beam-line. To reduce the QCD background
in the muon channel, each muon is required to be isolated such that
$\sum{p_{T}}$($\Delta$R $<$ 0.3)/pT $<$ 0.05. The two highest $p_{T}$ muons
passing this selection form a pair and are required to have an invariant mass
greater than 70 GeV to be accepted by the dimuon analysis.
The dilepton analysis was performed with an integrated luminosity of 1.08 fb-1
in the electron channel, and 1.21 fb-1 in the muon channel. The results of
this analysis can be found in Exotics:EPS , and the main kinematic plots of
interest, namely the invariant mass spectrum for both the electron and muon
channel, are presented in Figure 1.
Figure 1: Invariant mass spectrum for the electron (top) and muon (bottom)
channel dilepton resonance search. Various possible Z${}^{\prime}_{SSM}$
signals are overlayed to show how an expected signal would manifest itself.
## 3 Statistical Analysis
Any excess in the observed data over the SM prediction can be quantified using
a Log Likelihood Ratio (LLR) test:
$LLR=-2ln\frac{{\cal L}(data|N_{sig}+N_{bkg})}{{\cal L}(data|N_{bkg})}$ (1)
In this dataset the greatest excesses give $p$-values of 54% and 24% for the
dielectron and dimuon channels respectively. Therefore as no significant
excess is observed, limits are set on the cross section times branching ratio
($\sigma$B) for the Z${}^{\prime}_{SSM}$ and G∗ decaying to leptons, at 95%
confidence level using the Bayesian Analysis Toolkit (BAT) BAT . BAT
constructs a binned likelihood, combining the electron and muon channel
searches and accounting for observed ($n$) and expected ($\mu$) events with
associated nuisance parameters ($\theta$) on a bin by bin basis:
${\cal L}(data|\sigma
B,\theta_{i})=\prod_{l=1}^{N_{channel}}\prod_{k=1}^{N_{bin}}\frac{\mu_{lk}^{n_{lk}}e^{-\mu_{lk}}}{n_{lk!}}\prod_{i=1}^{N_{sys}}G(\theta_{i},0,1)$
(2)
Employing Bayesian statistics (assuming a flat positive prior so that
$\pi$($\sigma$B) = 1) and treating the nuisance parameters as Gaussian priors,
Markov Chain Monte Carlo is used to reduce the likelihood (${\cal
L}^{\prime}$) and obtain the marginalised posterior probability, which is then
solved for ($\sigma$B)95:
$0.95=\frac{\int_{0}^{{\sigma B}_{95}}{\cal L}^{\prime}(\sigma B)\pi(\sigma
B)d(\sigma B)}{\int_{0}^{\infty}{\cal L}^{\prime}(\sigma B)\pi(\sigma
B)d(\sigma B)}$ (3)
The resulting limits on the Z′/G∗ $\sigma$B are converted into mass exclusion
limits using the theoretical dependence of $\sigma$B as a function of
resonance mass. The $\sigma$B limits are presented in Figure 2. Table 1
summarises the excluded mass values for the models considered. The results
presented here represent a large step forward in the search for heavy dilepton
resonances, exceeding previous experiments’ mass exclusion limits for Z′/G∗
resonances in the dilepton channel. With a total integrated luminosity of
$\sim$5 fb-1 recorded by the ATLAS detector in 2011, this search will soon be
updated probing even further into the TeV scale regime in search of new
physics beyond the current SM.
Figure 2: 95% confidence level $\sigma$B limits for various Z′ models (top),
and RS graviton k/Mpl couplings (bottom).
| E6 Z′ Models
---|---
Model | Z${}^{\prime}_{\psi}$ | Z${}^{\prime}_{N}$ | Z${}^{\prime}_{\eta}$ | Z${}^{\prime}_{I}$
Mass limit [TeV] | 1.49 | 1.52 | 1.54 | 1.56
| E6 Z′ Models |
---|---|---
Model | Z${}^{\prime}_{S}$ | Z${}^{\prime}_{\chi}$ | Z${}^{\prime}_{SSM}$
Mass limit [TeV] | 1.60 | 1.64 | 1.83
G∗ Coupling k/MPl | 0.01 | 0.03 | 0.05 | 0.10
---|---|---|---|---
Mass limit [TeV] | 0.71 | 1.03 | 1.33 | 1.63
Table 1: 95% confidence level lower mass exclusion limits for various Z′
models and RS graviton k/Mpl couplings, decaying to two leptons (dielectron or
dimuon).
## References
* (1) D. London and J. L. Rosner, Extra Gauge Bosons in E(6), Phys. Rev D34 (1986) 1530.
* (2) P. Langacker, The Physics of Heavy Z′ Gauge Bosons, Rev. Mod. Phys 81 (2009) 1199.
* (3) L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension, Phys. Rev. Lett 83 (1999) 3370.
* (4) ATLAS Collaboration, The ATLAS Experiment at the CERN Large Hadron Collider, JINST 3 S08003 (2008) .
* (5) The ATLAS Collaboration, Search for high-mass dilepton resonances in pp collisions at $\sqrt{s}$ = 7 TeV with the ATLAS detector, Phys. Rev. Lett 107, 272002 (2011) .
* (6) A Caldwell, D. Kollar, K. Kröninger, BAT - The Bayesian Analysis Toolkit, Computer Physics Communication 180 (2009) 2197.
|
# Classical results for alternating virtual links
Hans U. Boden Mathematics & Statistics, McMaster University, Hamilton,
Ontario<EMAIL_ADDRESS>and Homayun Karimi Mathematics & Statistics,
McMaster University, Hamilton, Ontario<EMAIL_ADDRESS>
###### Abstract.
We extend some classical results of Bankwitz, Crowell, and Murasugi to the
setting of virtual links. For instance, we show that an alternating virtual
link is split if and only if it is visibly split, and that the Alexander
polynomial of any almost classical alternating virtual link is alternating.
The first result is a consequence of an inequality relating the link
determinant and crossing number for any non-split alternating virtual link.
The second is a consequence of the matrix-tree theorem of Bott and Mayberry.
We extend the first result to semi-alternating virtual links. We discuss the
Tait conjectures for virtual and welded links and note that Tait’s second
conjecture is not true for alternating welded links.
###### Key words and phrases:
Alternating link, virtual link, split link, checkerboard coloring,
determinant, almost classical link, Alexander polynomial, welded link,
branched double cover, Tait conjectures
###### 2020 Mathematics Subject Classification:
Primary: 57K12
## §1. Introduction
In this paper, we establish conditions that are satisfied by invariants of
alternating virtual links, such as the link determinant and Alexander
polynomial. As an application, we deduce that a reduced alternating virtual
link diagram is split if and only if it is visibly split.
A link is said to be alternating if it admits an alternating diagram, and a
diagram is alternating if the crossings alternate between over and under-
crossing as one travels around any component. This applies to classical and
virtual links, with the proviso that virtual crossings are ignored.
In [Ban30], Bankwitz proved that $\det(L)\geq c(L)$ for any non-split
alternating link $L$, where $\det(L)$ denotes the link determinant and $c(L)$
the crossing number of $L$. In [Cro59a, Mur58], Crowell and Murasugi
independently proved that the Alexander polynomial of an alternating link is
alternating. Here, a Laurent polynomial $\Delta_{L}(t)=\sum c_{i}t^{i}$ is
said to be alternating if its coefficients satisfy $(-1)^{i+j}c_{i}c_{j}\geq
0$.
We extend the results of Bankwitz, Crowell, and Murasugi to alternating
virtual links. Virtual knots were introduced by Kauffman in [Kau99], and they
represent a natural generalization to knots in thickened surfaces up to
stabilization. Classical knots embed faithfully into virtual knot theory
[GPV00], and many invariants from classical knot theory extend in a natural
way to the virtual setting.
For example, the link determinant $\det(L)$ is defined in terms of the
coloring matrix and extends to checkerboard colorable virtual links (defined
below). The link $L$ admits a $p$-coloring if and only if $p$ divides
$\det(L)$. One of our main results is that $\det(L)\geq c(L)$ for any non-
split alternating virtual link $L,$ where $c(L)$ is the classical crossing
number of $L$. This result applies to show that a reduced alternating virtual
link diagram is split if and only if it is visibly split.
The Alexander polynomial $\Delta_{L}(t)$ is defined in terms of the Alexander
module of $L$, and it extends to almost classical virtual links (defined
below). Another one of our main results is that, for any reduced alternating
link $L$ that is almost classical, its Alexander polynomial $\Delta_{L}(t)$ is
alternating. To prove this result, we appeal to the Matrix-Tree Theorem. It
applies to show that many virtual knots cannot be represented by alternating
diagrams.
The link determinant and Alexander polynomial are both invariant under welded
equivalence. Therefore, our main results can be seen as providing restrictions
on a virtual link diagram for it to be welded equivalent to an alternating
virtual link. This is discussed at the end of the paper, where we state open
problems related to the Tait conjectures for welded links.
We provide a brief synopsis of the contents of the rest of this paper. In §2,
we review background material on links in thickened surfaces and virtual and
welded links, together with Cheng coloring and Alexander numbering for virtual
links. In §3, we review the link group and determinant. In §4, we recall the
Matrix-Tree Theorem, which is used to prove one of the main results. In §5, we
prove that split alternating virtual links are visibly split. In §6, we prove
analogous results for semi-alternating links, and in §7, we present a
discussion on the Tait conjectures for welded links and state some interesting
open problems.
## §2. Virtual links
In this section we review the basic properties of virtual links, including
Gauss diagrams, links in thickened surfaces, welded links, ribbon torus links,
alternating virtual links, virtual linking numbers, Cheng colorings, and
Alexander numberings.
Virtual link diagrams. Virtual links are defined as equivalence classes of
virtual link diagrams. Here, a virtual link diagram is an immersion of one or
more circles in the plane with only finitely many regular singularities, each
of which is a double point. Each double point is either classical (indicated
by over- and under-crossings) or virtual (indicated by a circle). Two diagrams
are said to be virtually equivalent if they can be related by planar isotopies
and a series of generalized Reidemeister moves ($r1$)–($r3$) and ($v1$)–($v4$)
depicted in Figure 1.
An orientation for a virtual link is obtained by choosing an orientation for
each component. For a diagram $D$, the orientation is usually indicated by
placing one arrow on each component of $D$.
$r1$$r2$$r3$$v1$$v2$$v3$$v4$$f1$ Figure 1. The generalized Reidemeister moves
($r1$)–($r3$), ($v1$)–($v4$) and the forbidden move ($f1$).
Gauss diagrams. Virtual links can also be defined as equivalence classes of
Gauss diagrams, which consist of one or more circles traversed
counterclockwise, together with directed chords on the circles representing
the classical crossings. The chords point from over-crossings to under-
crossings and are decorated with a sign ($+$ or $-$) to indicate whether the
crossing is positive or negative. Each virtual link diagram determines a Gauss
diagram, and vice versa, and this correspondence is well-defined up to moves
($v1$)–($v4$). The Reidemeister moves can be translated into moves between
Gauss diagrams, and in this way a virtual link can be regarded as an
equivalence class of Gauss diagrams. By convention, the core circles of a
Gauss diagram are oriented counterclockwise.
Notice that the Gauss diagram does not keep track of the virtual crossings. In
effect, the virtual crossings are not really there, rather they are an
inevitable consequence of trying to represent a non-planar virtual link
diagram by a diagram in the plane.
A virtual link diagram is said to be split if its associated Gauss diagram is
disconnected, and a virtual link is split if it can be represented by a split
diagram. For classical links, this agrees with the usual definition. For
virtual links, a diagram can be split and connected. However, any diagram that
is split can be transformed into a disconnected diagram using moves
($v1$)–($v4$).
Links in thickened surfaces. A third approach is to define virtual links as
stable equivalence classes of links in thickened surfaces, and we take a
moment to explain this.
Let $\Sigma$ be a closed, oriented surface and $I=[0,1]$. Consider a link
$\mathcal{L}\subset\Sigma\times I$ in the thickened surface, up to isotopy.
Let $p\colon\Sigma\times I\to\Sigma$ be the projection map.
Stabilization is the operation of adding a 1-handle to $\Sigma$, disjoint from
$p(\mathcal{L})$, and destabilization is the opposite procedure. Two links
$\mathcal{L}\subset\Sigma\times I$ and
$\mathcal{L}^{\prime}\subset\Sigma^{\prime}\times I$ are said to be stably
equivalent if one is obtained from the other by a finite sequence of
stabilizations, destablizations, and orientation-preserving diffeomorphisms of
the pairs $(\Sigma\times I,\Sigma\times\\{0\\})$ and $(\Sigma^{\prime}\times
I,\Sigma^{\prime}\times\\{0\\})$. In [CKS02], Carter, Kamada, and Saito show
there is a one-to-one correspondence between virtual links and stable
equivalence classes of links in thickened surfaces.
Thus, every virtual link can be represented as a link in a thickened surface.
Further, any such link itself can be represented as a link diagram on
$\Sigma.$ A link diagram $\mathcal{D}$ on $\Sigma$ is a tetravalent graph with
over= and under-crossing information drawn at each vertex in the usual way.
A link diagram $\mathcal{D}$ on $\Sigma$ is said to be a split diagram if it
is disconnected, and a link in $\Sigma\times I$ is said to be split if it can
be represented by a split diagram.
Welded links. Two virtual links are said to be welded equivalent if one can be
obtained from the other by a sequence of generalized Reidemeister moves and
the first forbidden move ($f1$) as depicted in Figure 1. In terms of Gauss
diagrams, the first forbidden move corresponds to exchanging two adjacent
arrow feet without changing their signs or arrowheads, see Figure 2.
Therefore, a welded link can also be viewed as an equivalence class of Gauss
diagrams.
Figure 2. The forbidden overpass $(f1)$ for Gauss diagrams.
Ribbon torus links. Every welded link determines a ribbon knotted surface in
$S^{4}$. This is based on a beautiful construction by Satoh [Sat00], which
associates to a welded link $L$ a ribbon torus link $\operatorname{Tube}(L)$
in $S^{4}$. In [Sat00], Satoh shows that every ribbon torus link occurs as
$\operatorname{Tube}(L)$ for some welded link, and that
$\pi_{1}(S^{4}\smallsetminus\operatorname{Tube}(L))$ is isomorphic to the link
group $G_{L}$, defined below.
The correspondence between welded links and ribbon torus links is not one-to-
one, see [Win09]. It is an open problem to determine necessary and sufficient
conditions for two welded knots to represent the same ribbon torus knot (cf.
[Aud16, Question 3.6]).
Alternating virtual links. A virtual link diagram $D$ is called alternating if
the classical crossings alternate between over-crossing and under-crossing as
we go around each component. A Gauss diagram is alternating if it alternates
between arrow heads and tails when going around each of the core circles. A
virtual or welded link is called alternating if it can be represented by an
alternating virtual link diagram. For example, consider the virtual links in
Figure 3. The virtual Borromean rings is alternating, but the virtual Hopf
link is not.
Virtual linking numbers. If $J$ and $K$ are oriented virtual knots, then the
virtual linking number $\operatorname{{\it v}\ell{\it k}}(J,K)$ is defined as
the sum of the writhe of the classical crossings where $J$ goes over $K$.
Using the same definition, we can define $\operatorname{{\it v}\ell{\it
k}}(J,K)$ more generally when $J$ and $K$ are oriented virtual links. Note
that $\operatorname{{\it v}\ell{\it k}}(J,K)$ is additive, namely if
$J=J^{\prime}\cup J^{\prime\prime}$, then $\operatorname{{\it v}\ell{\it
k}}(J,K)=\operatorname{{\it v}\ell{\it k}}(J^{\prime},K)+\operatorname{{\it
v}\ell{\it k}}(J^{\prime\prime},K)$, and likewise if $K=K^{\prime}\cup
K^{\prime\prime}.$ The virtual linking numbers are not symmetric, i.e., it is
not generally true that $\operatorname{{\it v}\ell{\it
k}}(J,K)=\operatorname{{\it v}\ell{\it k}}(K,J)$. For example, consider the
oriented virtual links in Figure 3. For the virtual Hopf link, we have
$\operatorname{{\it v}\ell{\it k}}(J,K)=1$ and $\operatorname{{\it v}\ell{\it
k}}(K,J)=0$, and for the virtual Borromean rings, we have
$\operatorname{{\it v}\ell{\it k}}(I,J)=\operatorname{{\it v}\ell{\it
k}}(J,K)=\operatorname{{\it v}\ell{\it k}}(K,I)=0,\;\operatorname{{\it
v}\ell{\it k}}(J,I)=\operatorname{{\it v}\ell{\it k}}(K,J)=1,\text{ and
}\operatorname{{\it v}\ell{\it k}}(I,K)=-1.$
Figure 3. The virtual Hopf link and the virtual Borromean rings.
Cheng colorings. Given a virtual link diagram $D$, a Cheng coloring of $D$ is
an assignment of integer labels to each arc of $D$ that satisfies the local
rules in Figure 4. An elementary exercise shows if $D$ and $D^{\prime}$ are
two virtual link diagrams that are related by virtual Reidemeister moves, then
$D$ admits a Cheng coloring if and only if $D^{\prime}$ does. A virtual link
$L$ is said to be Cheng colorable if it can be represented by a virtual link
diagram with a Cheng coloring.
Not all virtual links are Cheng colorable. For example, the virtual Hopf link
in Figure 3 is not Cheng colorable. More generally, given a virtual link
$L=K_{1}\cup\cdots\cup K_{m}$ with $m$ components, then an elementary argument
shows that $L$ admits a Cheng coloring if and only if it satisfies
$\operatorname{{\it v}\ell{\it k}}(K_{i},L\smallsetminus
K_{i})=\operatorname{{\it v}\ell{\it k}}(L\smallsetminus K_{i},K_{i})=0$
for each $i=1,\ldots,m.$
Figure 4. Local rules for a Cheng coloring at classical and virtual crossings.
Alexander numberings and almost classical links. Given a virtual link diagram
$D$, an Alexander numbering on $D$ is an assignment of integer labels to each
arc of $D$ that satisfies the local rules in Figure 5. If $D$ admits an
integer labeling that satisfies the local rules mod $p$, then $D$ is said to
be mod $p$ Alexander numberable.
Notice that if $D$ is Alexander numberable, then it is Cheng colorable.
Conversely, a virtual link diagram $D$ is Alexander numberable if and only if
it admits a Cheng coloring such that the arc labels satisfy $b=a-1$ at each
classical crossing. Likewise, a virtual link diagram $D$ is mod $p$ Alexander
numberable if and only if it admits a Cheng coloring such that the arc labels
satisfy $b\equiv a-1$ (mod $p$) at each classical crossing.
Figure 5. Local rules for an Alexander numbering at classical and virtual
crossings.
A virtual link is said to be almost classical if it admits an Alexander
numberable diagram, and it is said to be checkerboard colorable if it admits a
mod 2 Alexander numberable diagram.
Recall from [BGH+17, Theorem 6.1] that a virtual link is almost classical if
and only if it can be represented by a null-homologous link
$\mathcal{L}\subset\Sigma\times I$, or equivalently if $\mathcal{L}$ admits a
Seifert surface.
Recall also from [BCK21, Proposition 1.1] that a virtual link is checkerboard
colorable if and only if it can be represented by a ${\mathbb{Z}}/2$ null-
homologous link $\mathcal{L}\subset\Sigma\times I$, or equivalently if
$\mathcal{L}$ admits an unoriented spanning surface. This is the case if and
only if $\mathcal{L}$ can be represented by a checkerboard colorable diagram
on $\Sigma$.
If a virtual link admits a diagram which is Cheng colorable, then any diagram
for the same link is also Cheng colorable. The reason is that Cheng colorings
extend along generalized Reidemeister moves. The same thing is not true for
Alexander numberings. Indeed, one can easily find two virtual link diagrams
for the same link such that one of them is Alexander numberable and the other
is not. Thus, Alexander numberings of virtual links do not always extend along
generalized Reidemeister moves.
However, if two virtual knot diagrams are Alexander numberable and are related
by generalized Reidemeister moves, then one can arrange that they are related
through Alexander numberable diagrams. More precisely, suppose $D$ and
$D^{\prime}$ are two virtual knot diagrams and
(1) $D=D_{1}\sim D_{2}\sim\cdots\sim D_{r}=D^{\prime}\ $
is a chain of diagrams, where $D_{i+1}$ is obtained from $D_{i}$ by a single
generalized Reidemeister move. If $D$ and $D^{\prime}$ are Alexander
numberable, then there is a chain (1) such that each $D_{i}$ is Alexander
numberable. A similar result holds if $D$ and $D^{\prime}$ are assumed to be
mod $p$ Alexander numberable. These statements can be proved using parity
projection, see [Man10, Nik13]. Any minimal crossing diagram of an almost
classical link is Alexander numberable, and this can also be proved using
parity projection, see [Rus21, BR21].
The corresponding statements for welded knots and links are either not true,
or not known to be true.
## §3. Link group, Alexander module, and determinant
In this section, we introduce the link group and the Alexander module
associated to a virtual link. We also recall the definition of the link
determinant associated to a checkerboard colorable virtual link and show that
$\det(L)=0$ when $L$ is split. Finally, we discuss mod $p$ labelings of
virtual knots and show that $K$ admits a mod $p$ labeling if and only if
$\det(K)=0$ (mod $p$).
Link Group. For classical links, the link group is just the fundamental group
of the complement of the link. For a link $L$, this group is denoted $G_{L}$.
Thus, $G_{L}=\pi_{1}(X_{L})$ where $X_{L}$ is the result of removing an open
tubular neighborhood of $L$ from $S^{3}$.
As an invariant of classical knots, the knot group is an unknot detector,
indeed the only classical knot $K$ whose knot group is infinite cyclic is the
trivial knot. In fact, Waldhausen’s theorem implies that the knot group
together with its peripheral structure is a complete invariant of classical
knots, which is to say that two classical knots are equivalent if and only if
they have isomorphic knot groups with equivalent peripheral structures.
The link group generalizes in a natural way to give an invariant of virtual
links by means of Wirtinger presentations. In fact, the abstract group
together with its peripheral structure are invariant under the first forbidden
move and thus define invariants of the underlying welded link.
Given a virtual link diagram for $L$, we will describe the Wirtinger
presentation of $G_{L}$. Let $D$ be a regular projection of $L$, and suppose
it has $n$ classical crossings. We define a long arc of the diagram $D$ to be
one that goes from one classical under-crossing to the next, passing through
virtual crossings as it goes. Enumerate the long arcs of $D$ by
$x_{1},\ldots,x_{m}$ and the classical crossings by $c_{1},\ldots,c_{n}$.
Figure 6. Arc labels at a crossing.
For each crossing, labeled as in Figure 6, we have the relation
$r_{i}=x_{\ell}x_{j}^{-1}x_{k}^{-1}x_{j}$. The Wirtinger presentation for
$G_{L}$ is then:
(2) $G_{L}=\langle x_{1},\ldots,x_{m}\mid r_{1},\ldots,r_{n}\rangle.$
Alexander module. In order to define the Alexander module, we briefly recall
Fox differentiation. Let $F_{m}$ be the free group on $m$ generators, so
elements of $F_{m}$ are words in $x_{1},\ldots,x_{m}$. For $j=1,\ldots,m$, the
Fox derivative $\partial/\partial x_{j}$ is an endomorphism of
${\mathbb{Z}}[F_{m}]$, the group ring, defined so that $\partial/\partial
x_{j}(1)=0$ and
$\frac{\partial}{\partial x_{j}}(x_{i})=\begin{cases}1&\text{if $i=j$,}\\\
0&\text{otherwise.}\end{cases}$
Further, given words $w,z\in F_{m}$, the Fox derivative satisfies the Leibnitz
rule:
$\frac{\partial}{\partial x_{j}}(wz)=\frac{\partial}{\partial
x_{j}}(w)+w\frac{\partial}{\partial x_{j}}(z).$
These relations completely determine $\partial/\partial x_{j}$ on every word
$w\in F_{m},$ and it is extended linearly to the group ring
${\mathbb{Z}}[F_{m}]$.
We use this to describe the construction of the Alexander module associated to
a link $L$. Let $G_{L}^{\prime}=[G_{L},G_{L}]$ and
$G_{L}^{\prime\prime}=[G_{L}^{\prime},G_{L}^{\prime}]$ be the first and second
commutator subgroups, then the Alexander module is the quotient
$G_{L}^{\prime}/G_{L}^{\prime\prime}$. It is a finitely generated module over
${\mathbb{Z}}[t,t^{-1}]$, the ring of Laurent polynomials, and it is
determined by the Fox Jacobian matrix $A$ as follows. Here, $A$ is the
$n\times m$ matrix with $ij$ entry equal to $\left.\frac{\partial
r_{i}}{\partial x_{j}}\right|_{x_{1},\ldots,x_{m}=t}$. In particular, the Fox
Jacobian is obtained by Fox differentiating the relations $r_{i}$ with respect
to the generators $x_{j}$ and applying the abelianization map $x_{j}\mapsto t$
for $j=1,\ldots,m$. We define the $k$-th elementary ideal $\mathscr{E}_{k}$ as
the ideal of ${\mathbb{Z}}[t,t^{-1}]$ generated by all $(n-k)\times(n-k)$
minors of $A$.
The matrix $A$ depends on the choice of a presentation for $G_{L}$, but the
associated sequence of elementary ideals
$\\{0\\}=\mathscr{E}_{0}\subset\mathscr{E}_{1}\subset\ldots\subset\mathscr{E}_{n}={\mathbb{Z}}[t,t^{-1}]$
does not.
For any classical link $L$, the first elementary ideal $\mathscr{E}_{1}$ is a
principal ideal, and the Alexander polynomial $\Delta_{L}(t)$ is defined as
the generator of $\mathscr{E}_{1}$. The Alexander polynomial is well-defined
up to multiplication by $\pm t^{k}$ for $k\in{\mathbb{Z}}$. It is obtained by
taking the determinant of the Alexander matrix, which is the
$(n-1)\times(n-1)$ matrix obtained by removing a row and column from $A$.
For a virtual link $L$, one can mimic the construction of the Alexander module
by regarding the quotient $G_{L}^{\prime}/G_{L}^{\prime\prime}$ as a module
over ${\mathbb{Z}}[t,t^{-1}],$ This can be used to define elementary ideals
and the Alexander polynomial for virtual links. However, in contrast to the
case of classical links, the first elementary ideal may not be principal. One
way to remedy the situation is to replace the elementary ideals
$\mathscr{E}_{k}$ with the smallest principal ideal containing them. For
instance, this would suggest a way to define an Alexander polynomial for a
virtual link $L$ to be a generator of the principal ideal containing
$\mathscr{E}_{1}$. However, since the link group itself is only an invariant
of the associated welded link, the invariants one obtains in this way will not
be very refined. Indeed, for many virtual knots, the Alexander polynomial is
trivial.
Link determinant for checkerboard colorable virtual links. We review the
definition of the link determinant in terms of the coloring matrix and show
that it extends to an invariant of checkerboard colorable virtual links. We
also prove that the determinant of the coloring matrix is odd for any
checkerboard colorable virtual knot, and that a checkerboard colorable virtual
knot $K$ admits a mod $p$ labeling if and only if $p$ divides $\det(K)$.
Let $L$ be a virtual link that is represented by a checkerboard colorable
diagram $D$ with $n$ classical crossings $\\{c_{1},\ldots,c_{n}\\}$ and $m$
long arcs $\\{a_{1},\ldots,a_{m}\\}.$ If $D$ has $k$ connected components,
then $m=n+k-1$.
Define the $n\times m$ coloring matrix $B(D)$ so that its $ij$ entry is given
by
$\displaystyle b_{ij}(D)$ $\displaystyle=$
$\displaystyle\begin{cases}2,&\text{if $a_{j}$ is the over-crossing arc at
$c_{i}$},\\\ -1,&\text{if $a_{j}$ is one of the under-crossing arcs at
$c_{i}$},\\\ 0,&\text{otherwise}.\end{cases}$
In case $a_{j}$ is coincidentally the over-crossing arc and one of the under-
crossing arcs at $c_{i}$, then we set $b_{ij}(D)=1$. In that case, if $a_{k}$
is the other under-crossing arc at $c_{i}$, then we set $b_{ik}(D)=-1$.
Note that the matrix $B(D)$ is the one obtained by specializing the Fox
Jacobian matrix $A(D)$ at $t=-1$.111This is only true up to sign for any given
row. Here, $A(D)$ is defined in terms of taking Fox derivatives of the
Wirtinger presentation of the link group $G_{D}$ whose generators are given by
the arcs $a_{1},\ldots,a_{m}$ and relations are given by classical crossings
$c_{1},\ldots,c_{n}$ and applying the abelianization homomorphism
$G_{L}\to\left\langle t\right\rangle,\ a_{i}\mapsto t$. For details, see
[BGH+17, Section 5].
Notice that the entries in each row of $B(D)$ sum to zero, therefore, it has
rank at most $n-1$. The proof of the next result is similar to that of [BNW18,
Proposition 2.6].
###### Proposition 3.1.
Any two $(n-1)\times(n-1)$ minors of $B(D)$ are equal up to sign. The absolute
value of the minor is independent of the choice of checkerboard colorable
diagram $D$. It defines an invariant of checkerboard colorable links $L$
called the determinant of $L$ and denoted $\det(L)$.
###### Proof.
As previously noted, the columns of $B(D)$ always sum to zero, and we will use
checkerboard colorability to derive a linear relation among the rows. Recall
that the diagram $D$ is checkerboard colorable if and only if it admits a mod
2 Alexander numbering. For each crossing $c_{i}$ of $D$, let
$\gamma_{i}=(-1)^{\lambda_{i}},$ where $\lambda_{i}\in\\{0,1\\}$ is the
Alexander number on the incoming under-crossing at $c_{i}$. Then we claim that
one obtains a linear relation on the rows by multiplying the $i$-th row of
$B(D)$ by $\gamma_{i}$.
To see why this is true, notice that the columns of $B(D)$ correspond to arcs
of the diagram, and in any given column, there are nonzero entries for each
crossing the arc is involved in. The arc starts and ends with under-crossings,
and the associated column entries are both $-1$. Every time the arc crosses
over another arc, there is an associated column entry equal to 2. Since the
diagram is mod 2 Alexander numberable, the numbers on the transverse arcs
alternate between $0$ and $1$ as one travels along the arc. Consequently, the
coefficients $\gamma_{i}$ alternate in sign as one travels along the arc.
Therefore, after multiplying the $i$-th row by $\gamma_{i}$, this shows that
the entries in each column sum to zero. Furthermore, since each coefficient
$\gamma_{i}$ is a unit, every row of $B(D)$ is a linear combination of the
other rows. This shows that the $(n-1)\times(n-1)$ minors of $B(D)$ are all
equal up to sign. ∎
###### Proposition 3.2.
Suppose $L$ is a checkerboard colorable virtual link. If $L$ is split, then
$\det(L)=0$.
###### Proof.
Suppose $D=D_{1}\cup D_{2}$ is a split checkerboard colorable diagram for $L$.
In each row of the coloring matrix, the nonzero elements are either $2,-1,-1$
or $1,-1$. It follows the rows add up to zero. We consider a simple closed
curve in the plane which separates $D$ into two parts. It follows that the
coloring matrix $B=B(D)$ admits a $2\times 2$ block decomposition of the form
$B=\begin{bmatrix}B_{1}&0\\\ 0&B_{2}\end{bmatrix},$
where $B_{1}$ and $B_{2}$ are the coloring matrices for $D_{1}$ and $D_{2},$
respectively. Since $\det(B_{1})=0=\det(B_{2})$, it follows that the matrix
obtained by removing a row and column from $B$ also has determinant zero. ∎
Next, we define a mod $p$ labeling for a virtual knot diagram.
###### Definition 3.3.
Let $p$ be a prime number. A link diagram can be labeled mod $p$ if each long
arc can be labeled with an integer from $0$ to $p-1$ such that
* (i)
at each crossing the relation $2x-y-z=0\;\text{(mod $p$)}$ holds, where $x$ is
the label on the over-crossing and $y$ and $z$ the other two labels, and
* (ii)
at least two labels are distinct.
If a diagram has a mod $p$ labeling, then multiplying each label by a number
$m$ gives a mod $pm$ labeling, so we assume $p$ is always a prime number.
###### Remark 3.4.
If $p=2$, then the equation $2x-y-z=0\;\text{(mod $p$)}$ indicates at each
crossing the two under-crossings have the same label, hence all the labels are
equal. Therefore, a mod $2$ label for a knot diagram does not exist.
Given a knot diagram, label each long arc with a variable $x_{i}$. At each
crossing we define a relation $2x_{i}-x_{j}-x_{k}=0\;\text{(mod $p$)}$, if the
arc $x_{i}$ crosses over the arcs $x_{j}$ and $x_{k}$. Therefore, a knot can
be labeled mod $p$, if this system has a solution mod $p$ such that not all
$x_{i}$’s are equal to each other.
Fix a variable $x_{j}$. Since $x_{i}=1$ for all $i$, is a solution and adding
two solutions together forms a new solution, if there was a solution such that
not all $x_{i}$’s are equal, then there is a solution with $x_{j}=0$.
Conversely, a nontrivial solution with $x_{j}=0$ results in a labeling of the
knot. So we can delete the $j$-th column and look for the nontrivial solutions
of the resulting system.
Since we assume the knot diagram is checkerboard colorable, and we know for
such a diagram, a linear combination of rows of $B(D)$ is zero, so we can
delete the $j$-th row as well. The result is a square matrix, and a nontrivial
solution means the determinant should be zero mod $p$. The absolute value of
this determinant is $\det(K)$. So, we have the following.
###### Proposition 3.5.
We can mod $p$ label the knot $K$ if and only if $\det(K)=0\;\text{(mod
$p$)}$.
###### Corollary 3.6.
For a checkerboard colorable knot $K$, $\det(K)$ is an odd integer.
###### Proof.
Combining the Proposition 3.5 and Remark 3.4, the result follows. ∎
It would be interesting to compare the link determinant defined here with the
link determinants defined for checkerboard colorable virtual links in terms of
Goeritz matrices [ILL10].
## §4. The Matrix-Tree theorem and an application
In this section, we recall the matrix-tree theorem from [BM54] (cf. [BZH14,
Theorem 13.22]). Using it, we adapt Crowell’s proof [Cro59a] to show that the
Alexander polynomial of any almost classical alternating link is alternating.
Here, we say a Laurent polynomial is alternating if its coefficients alternate
in sign. Specifically, a polynomial $f(t)=\sum
c_{i}t^{i}\in{\mathbb{Z}}[t,t^{-1}]$ is alternating if its coefficients
satisfy $(-1)^{i+j}c_{i}c_{j}\geq 0.$
The spectacular results concerning the Jones polynomial of classical
alternating links are generally not true in the virtual case. For instance,
the span of the Jones polynomial is not equal to the crossing number. For
example, the knot $K=6.90101$ is alternating and has Jones polynomial
$V_{K}(t)=1$.
In [Thi87], Thistlethwaite proved that the Jones polynomial $V_{L}(t)$ of any
non-split, alternating classical link $L$ is alternating. This result does not
extend to virtual links. For example, the virtual knot $K=5.2426$ in Figure 7
is alternating and has Jones polynomial $V_{K}(t)=1/t^{2}+1/t^{3}-1/t^{5}.$
Since $V_{K}(t)$ is not alternating, Thistlethwaite’s result is not true for
virtual links.
Figure 7. A virtual knot diagram for $5.2426$.
Let $L$ be a virtual link. We define the link group $G_{L}$ as in §2. We use
Fox derivatives to define the Jacobian matrix $A$. For virtual knots, the
first elementary ideal $\mathscr{E}_{1}$ is not necessarily principal. We
define the Alexander polynomial $\Delta_{K}(t)$ to be the generator of the
smallest principal ideal containing $\mathscr{E}_{1}$. Since
${\mathbb{Z}}[t,t^{-1}]$ is a gcd domain, it is given by taking the gcd of all
the $(n-1)\times(n-1)$ minors of $A$ . If we remove the $i$-th row and $j$-th
column of $A$ we denote the corresponding minor by $A_{ij}.$
In [NNST12] and [BNW18], the authors showed for almost classical links,
$\mathscr{E}_{1}$ is principal, and the Alexander polynomial $\Delta_{L}(t)$
is given by taking the determinant of the $(n-1)\times(n-1)$ matrix obtained
by removing any row and any column from $A$.
###### Proposition 4.1.
For an almost classical link $L$, the determinant $\det(L)$ is equal to
$|\Delta_{L}(-1)|$.
###### Proof.
If $D$ is a diagram for $L$, the coloring matrix $B(D)$ is exactly the matrix
obtained from the Fox Jacobian matrix by replacing $t$ with $-1$. 222These
matrices are equal up to multiplication by $\pm 1$ in the rows. ∎
###### Remark 4.2.
Since any almost classical knot $K$ is checkerboard colorable, Corollary 3.6
shows that $\Delta_{K}(-1)$ is an odd number (see [BGH+17]).
Next, we state the Matrix-Tree theorem, as proved by Bott and Mayberry in
[BM54]. Tutte had given an earlier proof in [Tut48]. The result goes back to
even earlier work of Kirchhoff, to whom this theorem is usually
attributed.333See [Cro59b] for references to the early papers on the Matrix-
Tree theorem.
Let $\Gamma$ be a finite oriented graph with vertices $\\{c_{i}\mid 1\leq
i\leq n\\}$ and oriented edges $\\{u_{ij}^{\delta}\\}$, such that $c_{i}$ is
the initial point and $c_{j}$ the terminal point of $u_{ij}^{\delta}$. Notice
that $\delta$ enumerates the different edges from $c_{i}$ to $c_{j}$. By a
rooted tree (with root $c_{i}$) we mean a subgraph of $n-1$ edges such that
every point $c_{k}$ is terminal point of a path with initial point $c_{i}$.
Let $a_{ij}$ denote the number of edges with initial point $c_{i}$ and
terminal point $c_{j}$.
###### Theorem 4.3 (Matrix-Tree Theorem).
Let $\Gamma$ be a finite oriented graph without loops ($a_{ii}=0$). The
principal minor $H_{ii}$ of the graph matrix
$H(\Gamma)=\left[\begin{matrix}(\sum_{k\neq
1}a_{k1})&-a_{12}&-a_{13}&\cdots&-a_{1n}\\\ -a_{21}&(\sum_{k\neq
2}a_{k2})&-a_{23}&\cdots&-a_{2n}\\\ \vdots&\vdots&\vdots&&\vdots\\\
-a_{n1}&-a_{n2}&-a_{n3}&\cdots&(\sum_{k\neq n}a_{kn})\end{matrix}\right],$
is equal to the number of rooted trees with root $c_{i}$.
###### Corollary 4.4.
Let $\Gamma$ be a finite oriented loopless graph with a valuation
$f\colon\\{u_{ij}^{\delta}\\}\rightarrow\\{-1,1\\}$ on edges. Then the
principal minor $H_{ii}$ of the matrix $H=[b_{ij}]$, where
$b_{ij}=\begin{cases}\sum_{\delta}f(u_{ij}^{\delta}),\ \ \ i\neq j,\\\
-\sum_{k\neq i}b_{ki},\ \ i=j,\end{cases}$
satisfies the following equation:
$H_{ii}=\sum f(\operatorname{Tr}(i)),$
where the sum is to be taken over all $c_{i}$-rooted trees
$\operatorname{Tr}(i)$, and where
$f(\operatorname{Tr}(i))=\prod_{u_{kj}^{\delta}\in\operatorname{Tr}(i)}f(u_{kj}^{\delta}).$
For a virtual link diagram, there are (at least) two ways one can associate a
$4$-valent graph. One way is to consider the diagram $D$ itself. It has
vertices for the classical and virtual crossings and edges running from one
classical or virtual crossing to the next. This graph is planar. The other way
to associate a graph is to consider vertices only for classical crossings. The
key difference is that in general, this graph is not planar. For an
alternating diagram $D$, we describe this graph and an orientation on it as
follows:
Let $D$ have classical crossings $c_{1},\ldots,c_{n}$. The vertices of
$\Gamma$ are $c_{1},\ldots,c_{n}$. At each vertex consider two out-going edges
corresponding to the over-crossing arc, and two in-coming edges for the under-
crossing arcs (see Figure 8). This is called the source-sink orientation or
the alternate orientation. This orientation is possible because $D$ is
alternating, and an out-going edge at the vertex $c_{i}$, should be an in-
coming edge for the adjacent vertex.
###### Remark 4.5.
In general, any checkerboard colorable diagram $D$ admits a source-sink
orientation. In fact, a diagram is checkerboard colorable if and only if it
admits a source-sink orientation (see [KNS02, Proposition 6]).
Figure 8. The source-sink orientation.
###### Theorem 4.6.
If $L$ is a non-split, almost classical alternating link, then its Alexander
polynomial $\Delta_{L}(t)$ is alternating.
###### Proof.
For the unknot the result is obvious. Assume $D$ has $n\geq 1$ classical
crossings. Orient $D$ and enumerate the crossings by $c_{1},\ldots,c_{n}$.
Label the long arcs by $g_{1},\ldots,g_{n}$. At the crossing $c_{i}$, label
the over-crossing arc $g_{\nu(i)}$ and the under-crossing arcs
$g_{\lambda(i)}$ and $g_{\rho(i)}$ as in Figure 9. Define the relation
$r_{i}=g_{\lambda(i)}g_{\nu(i)}^{-1}g_{\rho(i)}^{-1}g_{\nu(i)}$.
Figure 9. Arc labels at the crossing $c_{i}$.
Now consider the graph $\Gamma$ associated with $D$, with the source-sink
orientation on it. Label the edges by $u_{ij}^{\delta}$. Define the valuation
$f$ as follows. At the crossing $c_{j}$, if $u_{ij}^{\delta}$ corresponds to
$g_{\lambda(j)}$, then $f(u_{ij}^{\delta})=1$, and if it corresponds to
$g_{\rho(j)}$, then $f(u_{ij}^{\delta})=-t$.
Define the matrix $H$ as in the Corollary 4.4. Notice that $D$ is alternating
and there is a one-to-one correspondence between the classical crossings of
$D$ and the set of over-crossing arcs. Therefore, we can choose to label over-
crossing arcs, such that $\nu(i)=i$. The matrix $H$ is the transpose of the
Jacobian matrix $A$. The Alexander polynomial $\Delta_{L}(t)=A_{ii}=H_{ii}$.
By Corollary 4.4,
$H_{ii}=\sum\prod_{u_{kj}^{\delta}\in\operatorname{Tr}(i)}f(u_{kj}^{\delta}).$
Since $f(u_{kj}^{\delta})=1,\text{or}\ -t$, the product
$\prod_{u_{kj}^{\delta}\in\operatorname{Tr}(i)}f(u_{kj}^{\delta})$ is of the
form $(-1)^{l}t^{l}$ and $H_{ii}$ is an alternating polynomial. Therefore,
$\Delta_{L}(t)$ is alternating. ∎
###### Example 4.7.
Up to $6$ classical crossings, the almost classical knots in Table 1 do not
have alternating Alexander polynomials. Therefore, by Theorem 4.6 they do not
admit alternating virtual knot diagrams. $\Diamond$
$K$ | $\Delta_{K}(t)$
---|---
5.2331 | $t^{2}-1+t^{-1}$
6.85091 | $1+t^{-1}-t^{-2}$
6.85774 | $t-1+t^{-2}$
6.87548 | $-t^{2}+2t+1-t^{-1}$
6.87875 | $t+1-2t^{-1}+t^{-2}$
6.89156 | $2t-1-t^{-1}+t^{-2}$
6.89812 | $t^{2}-2+2t^{-1}$
6.90099 | $t-t^{-1}+t^{-3}$
Table 1. Almost classical knots with non-alternating Alexander polynomials.
###### Remark 4.8.
Any integral polynomial $\Delta(t)$ of degree $2n$ satisfying $\Delta(1)=1$
and $\Delta(t)=t^{2n}\Delta(t^{-1})$ is the Alexander polynomial for some
classical knot, see [Sei35]. In [Fox62], Fox asked for a characterization of
Alexander polynomials of alternating knots. If $K$ is an alternating knot,
then $\Delta_{K}(t)=\sum_{j=0}^{2n}(-1)^{j}a_{j}t^{j}$. Fox conjectured that
the Alexander polynomial of any alternating knot satisfies the trapezoidal
inequalities:
$a_{0}<a_{1}<\cdots<a_{n-k}=\cdots=a_{n+k}>\cdots>a_{2n}.$
Fox verified this conjecture for alternating knots up to 11 crossings, and it
has been verified in many other cases [Jon09, HM13, AC21, Che21]. Despite this
progress, Fox’s trapezoidal conjecture remains an intriguing open problem.
Any integral polynomial $\Delta(t)$ satisfying $\Delta(1)=1$ is the Alexander
polynomial for some almost classical knot $K$, see [BCG20]. Is there a way to
characterize the Alexander polynomials of alternating almost classical knots?
Do they satisfy Fox’s trapezoidal inequalities? This has been verified for
almost classical knots up to six classical crossings.
## §5. Split alternating virtual links are visibly split
A classical result of Bankwitz [Ban30] implies that $\det(L)$ is nontrivial
for non-split alternating links. We extend this result to virtual alternating
links and apply it to show that an alternating virtual link $L$ is split if
and only if it is visibly split.
The weak form of the first Tait Conjecture, namely that every knot having a
reduced alternating diagram with at least one crossing is nontrivial, was
first proved by Bankwitz [Ban30] in 1930; and since then, Menasco and
Thistlethwaite [MT91] and Andersson [And95] published simpler proofs. Here we
outline the proof by Balister et al. [BBRS01] and generalize it to alternating
virtual links. This result was first proved for alternating virtual knots by
Cheng [Che15, Proposition 3.3].
Consider the graph $\Gamma$ with vertices $\\{c_{1},\ldots,c_{n}\\}$ as
before.
###### Definition 5.1.
The outdegree of the vertex $c_{i}$, denoted $d^{+}(c_{i})$, is the number of
edges of $\Gamma$ with initial point $c_{i}$. The indegree of the vertex
$c_{i}$, denoted $d^{-}(c_{i})$, is the number of edges of $\Gamma$ with
terminal point $c_{i}$. Therefore,
$d^{+}(c_{i})=\sum_{j=1}^{n}a_{ij}\ \ ,\ \ d^{-}(c_{i})=\sum_{j=1}^{n}a_{ji}.$
###### Definition 5.2.
A walk in a graph is an alternating sequence of vertices and edges, starting
with a vertex $c_{i}$ and ending with a vertex $c_{j}$. A walk is called a
trail if all the edges in that walk are distinct. A circuit is a trail which
starts and ends at a vertex $c_{i}$. An Eulerian circuit is a circuit which
contains all the edges of $\Gamma$. A graph $\Gamma$ is called Eulerian if it
has an Eulerian circuit.
An Eulerian graph is necessarily connected and has $d^{+}(c_{i})=d^{-}(c_{i})$
for every vertex. Let $t_{i}(\Gamma)$ be the number of rooted trees with root
$c_{i}$, then the BEST Theorem is as follows (see [vAEdB51] and [Bol98,
Theorem 13]).
###### Theorem 5.3.
Let $s(\Gamma)$ be the number of Eulerian circuits of $\Gamma$, then
$s(\Gamma)=t_{i}(\Gamma)\prod_{j=1}^{n}(d^{+}(c_{j})-1)!$
In particular, if $\Gamma$ is a two-in two-out oriented graph, i.e.,
$d^{+}(c_{i})=d^{-}(c_{i})=2$ for every $i$, then by Theorems 4.3 and 5.3,
$s(\Gamma)=t_{i}(\Gamma)=H_{ii},\ \ \text{for every}\ i.$
A vertex $c$ of a graph $\Gamma$ is an articulation vertex if $\Gamma$ is the
union of two nontrivial graphs with only the vertex $c$ in common. In [BBRS01]
Balister et al. proved the following result:
Figure 10. The oriented smoothing at $c$.
###### Theorem 5.4.
Let $\Gamma$ be a connected two-in two-out oriented graph with $n\geq 2$
vertices and with no articulation vertex. Then $s(\Gamma)\geq n$.
Given an oriented virtual link diagram $D$, recall that the oriented smoothing
at a crossing $c$ is the diagram with the crossing $c$ removed, see Figure 10.
Recall also that a self-crossing of $D$ is a crossing where one of the
components of the link crosses over itself.
###### Definition 5.5.
Let $D$ be an oriented non-split virtual link diagram. Then a self-crossing
$c$ is said to be nugatory if the oriented smoothing of $D$ at $c$ is a split
link diagram.
The diagram $D$ is said to be reduced if it does not contain any nugatory
crossings.
There is an equivalent definition of nugatory crossing for links in surfaces.
Let $c$ be a crossing of a connected link diagram $\mathcal{D}$ on a surface
$\Sigma$. Then $c$ is said to be nugatory if there is a simple closed curve on
$\Sigma$ that separates $\Sigma$ and intersects $\mathcal{D}$ exactly once at
$c$.
For classical links, nugatory crossings are always removable. For virtual
links, this is no longer true. Indeed, there are examples of virtual knots
that contain essential nugatory crossings, see [BKS20, Example 20]. For welded
links, nugatory crossings are once again always removable, see Remark 6.5
below.
Recall that associated with an alternating virtual link diagram $D$, there is
an oriented two-in two-out graph $\Gamma$. If $D$ has no nugatory crossings,
then $\Gamma$ has no articulation vertex.
###### Corollary 5.6.
Let $K$ be an almost classical knot and $D$ a reduced alternating diagram for
$K$. If $D$ has $n$ classical crossings, then
$|\Delta_{K}(-1)|\geq n.$
###### Proof.
By the proof of Theorem 4.6, $|\Delta_{K}(-1)|$ counts $t_{i}(\Gamma)$ the
number of rooted trees with root $c_{i}$ in the oriented graph $\Gamma$,
associated with the knot diagram $D$. By Theorem 5.3,
$t_{i}(\Gamma)=s(\Gamma)$, and the result follows from Theorem 5.4. ∎
###### Theorem 5.7.
Let $L$ be a non-split virtual link and $D$ a reduced alternating diagram for
$L$. If $D$ has $n$ classical crossings, then the determinant of $L$ satisfies
$\det(L)\geq n.$
###### Proof.
Since $D$ is alternating, we can repeat the proof of Theorem 4.6. By Corollary
4.4, the determinant of $L$ counts the number of spanning trees which is equal
to $s(\Gamma)$. The result follows from Theorem 5.4. ∎
###### Corollary 5.8.
Suppose $L$ is a virtual link which admits an alternating diagram $D$ without
nugatory crossings. Then $L$ is a split link if and only if $D$ is a split
diagram.
###### Proof.
Clearly, if $D$ is a split diagram, then $L$ is split. Suppose then that $D$
is a non-split alternating diagram with $n=n(D)>0$ classical crossings. (If
$n=0$, then $D$ has one component and is an unknot diagram.) Theorem 5.7
implies that $\det(L)\geq n$. Hence $\det(L)\neq 0$, and Proposition 3.2 shows
that $L$ is not split. ∎
## §6. Weakly alternating virtual links
In this section, we extend the results from the previous section to semi-
alternating virtual links, defined below. We also give a formula for the link
determinant of a connected sum, and we use it to show that a semi-alternating
virtual link is split if and only if it is visibly split.
We begin by reviewing the connected sum of virtual links. Suppose $D_{1}$ and
$D_{2}$ are virtual link diagrams and $p_{1},p_{2}$ are points on
$D_{1},D_{2}$ respectively, distinct from the crossings. The connected sum is
denoted $D_{1}\\#D_{2}$ and is formed by removing small arcs from $D_{1}$ and
$D_{2}$ near the basepoints and joining them with trivial unknotted arcs. If
$D_{1}$ and $D_{2}$ are oriented, then the arcs are required to preserve
orientations. The connected sum depends on the choice of diagrams and
basepoints. It does not lead to a well-defined operation on virtual links.
###### Definition 6.1.
A virtual link diagram $D$ is said to be semi-alternating if it can be written
$D=D_{1}\\#\cdots\\#D_{n}$, a connected sum of alternating virtual link
diagrams $D_{1},\ldots,D_{n}$.
The set of semi-alternating virtual links includes, as a proper subset, those
that can be represented as weakly alternating links in thickened surfaces, see
[BKS20, §5].
Every semi-alternating virtual link diagram is checkerboard colorable. This
follows from the fact that every alternating virtual link diagram is
checkerboard colorable (see [Kam02, Lemma 7]), and the observation that the
connected sum of two or more checkerboard colorable diagrams is checkerboard
colorable.
###### Definition 6.2.
A virtual link is said to be $w$-split if it is welded equivalent to a split
virtual link.
Clearly, any virtual link that is split is necessarily $w$-split, but there
are virtual links that are $w$-split but not split. For example, consider the
virtual link $L$ whose Gauss diagram appears on the left of Figure 11. Using
forbidden moves, it is seen to be welded equivalent to the split classical
link $8_{20}\cup\bigcirc$ shown on the right. Thus $L$ is $w$-split.
Let $L^{\prime}=8_{20}\cup\bigcirc$ be the split classical link shown on the
right of Figure 11. Using [Lic97, Definition 3.1 & Theorem 3.5], one can see
that its Jones polynomial satisfies
$\begin{split}(t^{-1/2}-t^{1/2})V(L^{\prime})&=(t-t^{-1})V(8_{20})\\\
&=t^{-6}-t^{-5}-t^{-3}+2t-t^{2}.\end{split}$
In particular, $(t^{-1/2}-t^{1/2})V(L^{\prime})$ lies in
${\mathbb{Z}}[t,t^{-1}].$ (This is true for any classical link with two
components.) On the other hand, direct computation shows that the Jones
polynomial of $L$ satisfies
$\begin{split}(t^{-1/2}-t^{1/2})V(L)=&-t^{-1/2}+3t^{-3/2}+2t^{-2}-3t^{-5/2}-3t^{-3}+2t^{-7/2}+\\\
&\quad 3t^{-4}-t^{-9/2}-3t^{-5}-t^{-11/2}+t^{-6}+t^{-13/2}.\end{split}$
Since $(t^{-1/2}-t^{1/2})V(L)$ does not lie in ${\mathbb{Z}}[t,t^{-1}]$, $L$
cannot be virtually equivalent to a classical link. On the other hand, if $L$
were split, then it would be virtually equivalent to $L^{\prime}$. Since that
is not the case, we see that $L$ is non-split.
Figure 11. The Gauss diagram for a non-split virtual link $L$ (left). Notice
that $L$ is $w$-split; in fact it is welded equivalent to the link
$8_{20}\cup\bigcirc$ (right).
###### Proposition 6.3.
If $L$ is $w$-split, then $\det(L)=0.$
###### Proof.
This follows directly from Proposition 3.2 and the fact that $\det(L)$ is an
invariant of welded links. ∎
There is a nice geometric interpretation of $\det(L)$ in terms of two-fold
branched covers. Let $T=\operatorname{Tube}(L)$ be the ribbon torus link
associated to $L$, and let $X$ be the two-fold cover of $S^{4}$ branched along
$T$. Using the isomorphism $\pi_{1}(S^{4}\smallsetminus T)\cong G_{L}$, one
can identify $\pi_{1}(X)$ with the quotient of $G_{L}$ under the relation
$x_{i}^{2}=1$ for each generator in (2). The coloring matrix is then a
presentation matrix for $H_{1}(X)$. Therefore, $\det(L)=|H_{1}(X)|$ if it is
finite, and $\det(L)=0$ if $H_{1}(X)$ is infinite. Here homology groups are
taken with ${\mathbb{Z}}$ coefficients. Note that, if $L$ is split, then
$H_{1}(X)$ is infinite. This gives an alternative explanation for Propositions
3.2 and 6.3.
###### Theorem 6.4.
If $D=D_{1}\\#D_{2}$ is a connected sum of two checkerboard colorable virtual
link diagrams, then $\det(D)=\det(D_{1})\det(D_{2}).$
###### Proof.
If $D_{1}$ or $D_{2}$ is split, then $D$ is split and
$\det(D)=0=\det(D_{1})\det(D_{2}).$ Therefore, we can assume that $D_{1}$ and
$D_{2}$ are non-split.
There is a proof which is direct and elementary but long. We present an
alternative proof that is shorter and makes use of the interpretation of
$\det(D)$ as the order of the first homology of the two-fold cover of $S^{4}$
branched along $\operatorname{Tube}(D)$. In the following, all homology groups
are taken with ${\mathbb{Z}}$ coefficients.
Let $D_{1}$ and $D_{2}$ be checkerboard colorable virtual link diagrams, and
let $X_{1},X_{2},$ and $X$ be the two-fold covers of $S^{4}$ branched along
$\operatorname{Tube}(D_{1}),\operatorname{Tube}(D_{2})$ and
$\operatorname{Tube}(D)$, respectively. We can then write $X_{1}=A_{1}\cup
B_{1},X_{2}=A_{2}\cup B_{2}$, and $X=A_{1}\cup A_{2}$. Here $A_{i}$ is the
double cover of $D^{4}$ branched along the knotted annulus which is part of
$\operatorname{Tube}(D_{i})$ for $i=1,2$, and $B_{i}$ is the double cover of
$D^{4}$ branched along trivial annulus. In particular,
$A_{i}=X_{i}\smallsetminus\operatorname{Int}(B_{i})$ for $i=1,2$. By [AK80,
Corollary 4.3], $B_{i}$ is diffeomorphic to $S^{2}\times D^{2}$, and
$H_{1}(B_{i})=0$ for $i=1,2.$
Let $M=A_{1}\cap B_{1}=A_{2}\cap B_{2}=A_{1}\cap A_{2}$. Then $M$ is the
3-manifold obtained as the double cover of $S^{3}$ branched along the two
component unlink. Thus $M$ can be identified with the boundary of $B_{1}$ (or
$B_{2}$) and is diffeomorphic to $S^{2}\times S^{1}$. Thus
$H_{1}(M)\cong{\mathbb{Z}}.$
Now consider the decompositions $X_{1}=A_{1}\cup B_{1}$, $X_{2}=A_{2}\cup
B_{2}$, and $X=A_{1}\cup A_{2}$, along with their Mayer-Vietoris sequences in
reduced homology:
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Note that $H_{1}(A_{1}\cap B_{1})\cong{\mathbb{Z}}\cong H_{1}(A_{2}\cap
B_{2})=H_{1}(A_{1}\cap A_{2})$.
We claim that the maps $\varphi_{1},\varphi_{2}$ and $\varphi$ are zero. We
prove this for $\varphi$; the argument for the other cases is similar. It
suffices to show that the maps $H_{1}(A_{1}\cap A_{2})\to H_{1}(A_{i})$
induced by inclusion are zero for $i=1,2$.
Take two points in $S^{3}$, one on each component of the unlink, and join them
by an arc in $S^{3}$ that is otherwise disjoint from the link. The arc lifts
to a loop in the double branched cover, and the loop is a generator of
$H_{1}(A_{1}\cap A_{2})$. However, when pushed into $A_{1}$, the loop does not
link the annulus in $D^{4}$, so it is trivial in $H_{1}(A_{1})$. A similar
argument shows it is also trivial in $H_{1}(A_{2})$. Therefore, the maps
$H_{1}(A_{1}\cap A_{2})\to H_{1}(A_{j})$ are zero for $j=1,2$, and it follows
that $\varphi=0.$
From the claim, it follows that $\psi_{1},\psi_{2},$ and $\psi$ are
isomorphisms. Using (3) and the fact that $H_{1}(B_{1})=0=H_{1}(B_{2})$, we
deduce that
$H_{1}(X_{1})\cong H_{1}(A_{1}),\;H_{1}(X_{2})\cong H_{1}(A_{2}),\text{ and
}H_{1}(X)\cong H_{1}(A_{1})\oplus H_{1}(A_{2}).$
Therefore,
$\begin{split}\det(D)&=|H_{1}(X)|,\\\ &=|H_{1}(A_{1})|\cdot|H_{1}(A_{2})|,\\\
&=|H_{1}(X_{1})|\cdot|H_{1}(X_{2})|,\\\ &=\det(D_{1})\det(D_{2}),\end{split}$
and this completes the proof. ∎
###### Remark 6.5.
We claim that, for welded links, nugatory crossings are always removable. Let
$D$ be a diagram with a nugatory crossing $c$ as in Figure 12 (left). Using
forbidden moves, we can transform $D$ by pulling the over-crossing arc off
$c$, as in Figure 12 (middle). Thus, $D$ is welded equivalent to the diagram
with $c$ removed. Alternatively, one can remove $c$ by making it virtual, as
in Figure 12 (right). At the level of Gauss diagrams, this is equivalent to
deleting the chord associated to $c$.
Figure 12. For welded links, nugatory crossing are removable.
###### Proposition 6.6.
Let $D$ be a virtual link diagram. If $D$ is non-split and alternating, then
$\det(D)\neq 0$.
###### Proof.
In general, if $D$ is a non-reduced virtual link diagram, then successively
removing all the nugatory crossings will produce a reduced diagram
$D^{\prime}$ welded equivalent to $D$. If $D$ is non-split, then $D^{\prime}$
will be too. If $D$ is alternating, then $D^{\prime}$ will be semi-
alternating.
Assume to the contrary that $\det(D)=0.$ Then $\det(D^{\prime})=0$ since
$\det(\cdot)$ is an invariant of welded type. Since $D$ is non-split and
alternating, it follows that $D^{\prime}$ is reduced, non-split, and semi-
alternating. Therefore, we can write
$D^{\prime}=D^{\prime}_{1}\\#\cdots\\#D^{\prime}_{n}$, where
$D^{\prime}_{1},\ldots,D^{\prime}_{n}$ are all reduced alternating diagrams.
By Theorem 6.4,
$0=\det(D^{\prime})=\det(D^{\prime}_{1})\cdots\det(D^{\prime}_{n}),$
thus $\det(D^{\prime}_{i})=0$ for some $1\leq i\leq n.$ Since $D^{\prime}_{i}$
is reduced and alternating, $\det(D^{\prime}_{i})=0$ implies that
$D^{\prime}_{i}$ is split. It follows that $D^{\prime}$ is split, which
implies that $D$ is split, giving the desired contradiction. ∎
###### Corollary 6.7.
Suppose $L$ is a virtual link which admits a semi-alternating diagram $D$,
possibly with nugatory crossings. Then $L$ is $w$-split if and only if $D$ is
a split diagram.
###### Proof.
Clearly if $D$ is split, then $L$ is split and also $w$-split.
On the other hand, suppose $D$ is non-split. Since $D$ is semi-alternating, we
can write $D=D_{1}\\#\cdots\\#D_{n}$, where $D_{1},\ldots,D_{n}$ are all non-
split, alternating diagrams. Proposition 6.6 implies that $\det(D_{i})\neq 0$
for $i=1,\ldots,n.$ Theorem 6.4 implies that
$\det(D)=\prod_{i=1}^{n}\det(D_{i})\neq 0.$ Therefore, $\det(L)\neq 0$, and by
Proposition 6.3, it follows that $L$ is not $w$-split. ∎
## §7. The Tait conjectures for welded links
In his early work on knot tabulation, Tait formulated three far-reaching
conjectures on reduced alternating classical link diagrams [Tai98]. (A link
diagram is reduced if it does not contain a nugatory crossing.) They assert
that, for a non-split link, any two reduced alternating diagrams have the same
crossing number, the same writhe, and are related by a sequence of flype
moves. The first two statements were famously solved by Kauffman, Murasugi,
and Thistlethwaite using the recently discovered Jones polynomial [Kau87,
Mur87, Thi87], and the third statement was subsequently proved by Menasco and
Thistlethwaite [MT93]. The three Tait conjectures lead to a simple and
effective algorithm for tabulating alternating knots and links that has been
implemented [RF04, RF06].
It is an interesting question whether similar results hold for virtual and/or
welded links. For example, analogues of the first and second Tait conjectures
have been established for virtual links using the Jones-Krushkal polynomial
and skein bracket, see [BK19, BKS20].
###### Problem 7.1.
Is the Tait flype conjecture true for alternating virtual links?
Figure 13. The flype move.
The flype move is depicted in Figure 13. By assumption, the tangle $T$ does
not contain any virtual crossings. Allowing the tangle to contain virtual
crossings results in a more general move called a virtual flype move. The
virtual flype move does not, in general, preserve the virtual link type, for
example, see [Kam17, ZJZ04].
It is unknown whether the Tait conjectures hold for welded links. More
generally, what conditions must the invariants of welded link satisfy in order
for it to be alternating?
Since $\det(L)$ is an invariant of welded links, any checkerboard colorable
virtual $L$ with $\det(L)\neq 1$ is nontrivial as a welded link. In
particular, Theorem 5.7 applies to show that any non-split virtual link
represented by a reduced, alternating diagram has $\det(L)\neq 1$ and
therefore, is nontrivial as a welded link.
The Alexander polynomial $\Delta_{L}(t)$ is also an invariant of the welded
type. Therefore, if $L$ is almost classical and $\Delta_{L}(t)$ is not
alternating, then $L$ is not welded equivalent to an alternating link.
Figure 14. Alternating welded knots with 3 and 4 classical crossings.
Figure 14 shows the five alternating welded knots with up to four classical
crossings. All the others can be ruled out using the consideration that
$\det(K)\geq n$, the crossing number.
###### Problem 7.2.
Is the first Tait conjecture true for alternating welded links?
One can find examples of virtual knots which are non-alternating but which
become alternating after adding one crossing. For example, consider Examples
19 and 20, [BKS20]. The first is non-alternating and has six crossings; the
second is alternating and is obtained by adding a nugatory crossing. The two
virtual knots are welded equivalent (see Figure 12). We conjecture that there
exist welded knots which are alternating, but every minimal crossing diagram
for them is non-alternating.
###### Problem 7.3.
Is it possible for an alternating welded knot to represent a non-alternating
classical knot?
Figure 15. Reduced alternating diagrams for the virtual knots $4.106$ and
$4.107$.
Interestingly, there are pairs of reduced alternating virtual knot diagrams
which are equivalent as welded knots but distinct as virtual knots. In
particular, this implies that Tait’s second conjecture is not true for welded
knots.
Figure 16. A sequence of moves on the Gauss diagrams, starting from a diagram
of $4.106$ ending in a diagram for $4.107$.
For example, consider the virtual knots $4.106$ and $4.107$ in Figure 15. Both
are reduced alternating diagrams, but the diagram for $4.106$ has writhe
$w=-2$ whereas the diagram for $4.107$ has writhe $w=0.$ Tait’s second
conjecture holds for reduced alternating virtual knot diagrams [BK19], and
thus comparing the writhes tells us these two are distinct as virtual knots.
However, these diagrams are equivalent as welded knots (see Figure 16 and
Figure 17). Since both diagrams are reduced and alternating, this shows that
the writhe of a reduced alternating diagram is not invariant under welded
equivalence.
This implies that the second Tait conjecture is not true in the welded
category. Since the Tait flype move preserves the writhe, this also shows that
the Tait’s third conjecture, if true, must take a different form in the
virtual and welded settings.
Figure 17. A sequence of moves on the virtual diagrams, starting from a
diagram of $4.106$ ending in a diagram for $4.107$. The fourth and fifth
diagrams are related by an $f1$ move, and this is seen by comparing Gauss
diagrams.
## Acknowledgements
This paper is based on several ideas in the Ph.D. thesis of the second author
[Kar18]. The authors would like to thank Robin Gaudreau, Andy Nicas, Will
Rushworth, and Adam Sikora for their helpful comments and feedback. They would
also like to thank the referee for their input. The first author gratefully
acknowledges grant funding from the Natural Sciences and Engineering Research
Council of Canada.
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|
# Inelastic x-ray scattering reveals the ergodic to nonergodic transition of
salol, a liquid with local order
L. Comez${}^{\textsf{\footnotesize{\mbox{?}}}}$, D.
Fioretto${}^{\textsf{\footnotesize{\mbox{?}}}}$, J.
Gapinski${}^{\textsf{\footnotesize{\mbox{?}}}}$, G.
Monaco${}^{\textsf{\footnotesize{\mbox{?}}}}$, A.
Patkowski${}^{\textsf{\footnotesize{\mbox{?}}}}$, W.
Steffen${}^{\textsf{\footnotesize{\mbox{?}}}}$
(Received June 25, 2019)
###### Abstract
We have studied the high-frequency dynamics of salol by inelastic x-ray
scattering over a wide temperature range between 50 and 450 K, across the
glass transition. We find that salol efficiently realizes the mechanism of
dynamical arrest described by the mode-coupling theory, as manifested by a
cusp singularity in the behaviour of the non-ergodicity parameter and a $Q$
dependence of the critical non-ergodicity parameter that is in phase with the
static structure factor. These results confront positively the mode-coupling
theory with liquids with local order.
Key words: glass transition, x-ray scattering, mode coupling theory
PACS: 64.70.Pf, 78.70.Ck, 61.20.Lc
###### Abstract
Ìè äîñëäèëè âèñîêîàñòîòíó äèíàìêó ñàëîëà íåïðóæíèì ðîçñþâàííÿì ðåíòãåíâñüêèõ
ïðîìåíâ â øèðîêé îáëàñò òåìïåðàòóð ìæ 50 450 K, ùî ïîêðèâà ïåðåõä â ñòàí ñêëà.
Ìè çíàéøëè, ùî ñàëîë åôåêòèâíî ðåàëçó ìåõàíçì äèíàìíîãî àðåøòó, ÿêèé îïèñóòüñÿ
òåîðþ âçàìîäþèõ ìîä, ùî ïðîÿâëÿòüñÿ ñèíãóëÿðíñòþ â ïîâåäíö ïàðàìåòðó
íååðãîäèíîñò òà â $Q$-çàëåæíîñò êðèòèíîãî ïàðàìåòðó íååðãîäèíîñò, ùî
óçãîäæóòüñÿ ç ñòàòèíèì ñòðóêòóðíèì ôàêòîðîì. Ö ðåçóëüòàòè ïîçèòèâíî
ñïâñòàâëÿþòü òåîðþ âçàìîäþèõ ìîä ç ðäèíàìè, ÿê ìàþòü ëîêàëüíå âïîðÿäêóâàííÿ.
Ключов слова: ïåðåõä â ñòàí ñêëà, ðîçñþâàííÿ ðåíòãåíâñüêèõ ïðîìåíâ, òåîðÿ
âçàìîäþèõ ìîä
## 1 Introduction
At the turn of the second and third millennium, a great deal of work, both
theoretical and experimental, has been devoted to the study of the physics of
disordered systems with particular emphasis on the processes connected with
the structural arrest and glass transition.
The mode coupling theory (MCT) was introduced to provide a self-consistent
treatment of the structural arrest in simple liquids [1]. The idealized
version of MCT describes the glass transition as an ergodic to non-ergodic
transition occurring at a critical temperature $T_{\text{c}}$, associated with
a singular behaviour of the long-time limit of the normalized density
correlator, the so-called non-ergodicity factor $f_{Q}$. The peculiar
temperature ($T$) and wave vector ($Q$) dependences expected for $f_{Q}$ are:
i) a square-root temperature behaviour below $T_{\text{c}}$,
$f_{Q}(T)=f_{Q}^{\text{c}}+h_{Q}\sqrt{(T-T_{\text{c}})/T_{\text{c}}}$, where
$f_{Q}^{\text{c}}$ is the critical non-ergodicity parameter and $h_{Q}$ is the
critical amplitude at a given wavevector $Q$; ii) a $Q$ dependence of
$f_{Q}^{\text{c}}$ and of $h_{Q}$, which are in phase and in antiphase with
the static structure factor $S(Q)$, respectively.
In the same years, the inelastic x-ray scattering (IXS) technique was
developed, capable of detecting the dynamic structure factor $S(Q,\omega)$ of
glasses [2] and glass forming liquids in the nm-1 $Q$-range [3]. IXS, together
with neutron scattering and MD simulations, became the method of choice for
the quantitative test of MCT predictions. Most of the tests were performed in
simple liquids, such as van der Waals molecular liquids, with relatively
simple relaxation patterns [4]. More complex systems were also analyzed, such
as polymers [5, 6], and a good coherence with MCT was found when the
contribution of the structural relaxation was singled out with respect to
those of secondary processes [7], complementing IXS with Brillouin light
scattering measurements [8]. An interesting class of liquids is that of
associated liquids, where local order extends over several neighboring
molecules, giving rise to a pre-peak in the low-$Q$ region of $S(Q)$. A
pioneering work performed by some of us on the associated liquid m-toluidine
provided experimental evidence that hydrogen bond clustering can coexist with
the signature of the ergodic to non-ergodic transition predicted by the MCT
[9, 10].
In the present work, we extend this investigation to salol, a widely studied
glass-forming system [11, 12, 13, 14].
## 2 Experiment
Salol (Phenyl salicylate, Fluka, purity $>98$%) was dried under vacuum for
three days at $95^{\circ}$C. To remove dust as potential sites of
heterogeneous nucleation, the sample was then filtered through a 0.22 µm
Durapore (Millipore company) filter into dust free vials which were sealed
until used in the actual experiment to fill the sample cells.
IXS experiments were performed at the very high energy resolution beamlines
ID16 and ID28 of the European Synchrotron Radiation Facility (ESRF), Grenoble.
The monochromator and analyzer crystals were operated at backscattering
configuration corresponding to an incident photon energy of 21.747 keV and a
total energy resolution of 1.5 meV. Spectra were taken at $Q$-values between 1
and 15 nm-1 in steps of 1 nm-1 at temperatures between 50 and 450 K. All
spectra were corrected for scattering of the empty cell and normalized to the
monitored incoming intensity.
Figure 1: Left-hand panel: IXS spectra of salol taken at $Q=2$ nm-1, at the
indicated temperatures. Right-hand panel: IXS spectra of salol taken at
$T=241$ K, at the indicated $Q$ values. The fitting curves (solid lines), the
quasi-elastic contributions (dash-dotted lines) and the inelastic
contributions (dashed lines) to the total fit are reported together with the
data points (open circles).
In figure 1 we report some IXS spectra for selected temperatures at the fixed
exchanged wave vector $Q=2$ nm-1 (left-hand panel), and some spectra obtained
for different values of $Q$ at $T=241$ K (right-hand panel).
## 3 Results and discussion
All measured spectra show quasielastic and inelastic contributions, whose
characteristic parameters were obtained by fitting the convolution of the
instrumental resolution function $R(\omega)$ with a model for $S(Q,\omega)$
including a delta function for the quasielastic line and a damped harmonic
oscillator for the two inelastic side peaks [15], which are due to the
Brillouin scattering of photons by longitudinal acoustic (LA) modes
propagating in salol. From this fitting procedure, the characteristic
frequency $(\Omega)$ and linewidth $(\Gamma)$ (FWHM) of the LA modes were
obtained, together with the intensities of the quasielastic and inelastic
contributions, $I_{\text{el}}(Q)$ and $I(Q)$.
Figure 2 shows the values of $\Omega$ and $\Gamma$ obtained by the fitting
procedure, as a function of $Q$ and for three selected temperatures. The
almost linear behaviour of $\Omega$ vs. $Q$ allows us to estimate the average
value for the velocity $(v)$ of the LA modes in the low $Q$ regime by fitting
to the first four data points the expression: $\Omega=v\cdot Q$. In the same
region, the almost quadratic behaviour of $\Gamma$ vs. $Q$ gives an estimate
of the unrelaxed value of the longitudinal kinematic viscosity $D_{\text{L}}$
and thermal diffusion $D_{\text{T}}$ through
$D_{\text{L}}+(\gamma-1)D_{\text{T}}=\Gamma/Q^{2}$ with $\gamma$ being the
ratio of specific heats.
Figure 2: Left-hand panels: Linear dependence on $Q$ of the longitudinal
acoustic frequency in the low-$Q$ range. Right-hand panels: In the $Q$ region,
where the acoustic dispersion relation is linear, the broadening of the
acoustic excitations shows a $T$-independent $Q^{2}$ behaviour.
Velocities and viscosities obtained by this method are reported in figure 3.
In the low temperature regime, the velocity of the LA modes obtained by IXS
agrees very well with the unrelaxed sound velocities $(v_{\infty})$ measured
at much lower $Q$ values by Brillouin light scattering (BLS) [16, 17] and
impulsive stimulated thermal scattering (ISTS) techniques [18]. For increasing
temperature, a change of the slope of $v(T)$ is visible close to the glass
transition of salol $T_{\text{g}}=220$ K, where a more pronounced temperature
dispersion starts to occur [19]. Interestingly, the temperature behaviour of
$v_{\text{IXS}}$ is also compatible [see linear extrapolations in figure 3
(a)] with the existence of a further transition in the liquid at
$T_{\text{A}}=348$ K. $T_{\text{A}}$ was proposed and shown as the transition
temperature of the $\alpha$-process from VFT to Arrhenius behaviour [20, 21].
Figure 3: (Colour online) Left-hand panel: Limiting high frequency
longitudinal sound velocity (squares) determined by a DHO $+$ delta function
analysis of IXS spectra compared with the literature data. Evidence is given
for two changes in the slope corresponding to $T_{\text{g}}$ and
$T_{\text{A}}$. Right-hand panel: Limiting high frequency longitudinal
kinematic viscosity.
At the highest investigated temperatures, the values of $v_{\text{BLS}}$ and
of $v_{\text{ISTS}}$ progressively approach those of the relaxed sound
velocity $v_{0}$. Conversely, the values of $v_{\text{IXS}}$ remain
considerably higher than $v_{0}$, suggesting that IXS probes the unrelaxed
sound velocity $v_{\infty}$ also in the liquid phase. This is also supported
by the temperature dependence of the linewidth of Brillouin peaks reported in
figure 3 (b). In fact, it can be seen that $\Gamma/Q^{2}$ at the two lowest
probed $Q$ values is almost constant in the whole investigated temperature
range suggesting that the mechanism responsible for the peak broadening in the
glassy phase continues to dominate also at temperatures higher than
$T_{\text{g}}$. This result can be explained by the fact that, at the probed
$Q$s, the phonon frequency is much higher than the rate of the structural
relaxation (unrelaxed condition) and, therefore, the sound waves probe the
system as “frozen” in the whole investigated temperature range. In this
regime, the broadening of the Brillouin peaks is due to the disordered
molecular structure rather than due to truly dynamic processes [22].
Figure 4: (Colour online) Temperature dependence of the effective non-
ergodicity factor $f_{Q}$ of salol for selected values of the exchanged
wavevector $Q$. The red line is a guide for eyes to indicate the critical
temperature $T_{\text{c}}=273\pm 5$ K. The solid lines are the best fits
obtained using the square-root function predicted by the mode coupling theory
[1].
An important consequence of the unrelaxed regime here probed by IXS is that
the contribution of the structural relaxation to $S(Q,\Omega)$ is all included
within the area of the quasielastic peak. In this condition, the relative
amplitude of the structural relaxation, i.e., the non-ergodicity parameter
$f_{Q}$, can be obtained from the IXS spectra as the ratio of the intensities
$f_{Q}=I_{\text{el}}(Q)/[I_{\text{el}}(Q)+I(Q)]$, and the analysis of the IXS
spectra becomes a powerful tool to test the predictions of MCT.
Figure 4 reports the temperature behaviour of $f_{Q}$ at four different values
of $Q$, showing clear evidence of the square-root singularity predicted by the
idealized MCT [1], even more clearly than in the previously reported
m-toluidine case [9], due to the improved signal-to-noise ratio of the present
IXS spectra. It is worth noting that the value of the critical temperature
$T_{\text{c}}=273\pm 5$ K is $Q$ independent within experimental error, and
that its value favourably agrees with previous estimates obtained from
dielectric spectroscopy [23] and from wide angle x-ray experiments combined
with MD simulations [21].
Figure 5: Left-hand panel: The non-ergodicity parameter $f_{Q}$ of salol at
different temperatures. Right-hand panel: $I(Q)$ obtained by wide angle x-ray
scattering measurements.
Figure 5 shows that $f_{Q}$ changes in phase with the static structure factor,
in agreement with previous observations [9]. For $T>T_{\text{c}}$, MCT
predicts a plateau in the temperature dependence of $f_{Q}$, i.e.,
$f_{Q}(T)=f_{Q}^{\text{c}}$, which is also clearly visible in figure 4. The
values of $f_{Q}^{\text{c}}$, on the whole, follow in phase the oscillations
of the static structure factor (figure 6), coherent with MCT calculations for
simple liquids.
Figure 6: (Colour online) The plateau of the non-ergodicity factor
$f^{\text{c}}_{Q}$ oscillates in phase with the static structure factor.
Based on these two evidences, consistent with the results previously obtained
in m-toluidine, we can infer that the predictions of MCT for the existence of
a square-root singularity at $T_{\text{c}}$ and for its $Q$ dependence are
robust enough to be also observed in clustering systems [24, 21].
A peculiar property of salol and m-toluidine which probably makes the cusp
behaviour of the non-ergodicity factor more visible than in other associated
liquids is their high degree of fragility. In fact, a phenomenological
analysis of IXS spectra of glass-forming systems has revealed that the
stronger is a glass former, the higher is the non-ergodicity factor around the
glass transition [25, 26]. The non-ergodicity factor can be also expressed as
the relative longitudinal modulus variation at $T_{\text{g}}$ [26]. For a
strong glass former, like SiO2, the value of $f_{Q}$ at $T_{\text{g}}$ is
close to unity, and thus the amplitude of the square-root cusp is hardly
visible within experimental error. Conversely, for fragile glass-formers such
as m-toluidine and salol, fragility favours the visibility of temperature and
$Q$-dependences of the non-ergodicity parameter, as shown here.
## 4 Conclusions
In conclusion, our findings corroborate the notion that the early stage of the
structural arrest, where the cage effect dominates the molecular dynamics of
simple liquids, shows a universal character — the square-root cusp of the non-
ergodicity factor at a critical temperature $T_{\text{c}}$ — which is also
shared by liquids with a local order [9]. This is a non-trivial result since
it suggests that the structural arrest which occurs in dense, simple liquids,
shares some common, if not universal, features with the structural arrest of
locally ordered liquids, where the coordination number is determined by local
symmetry constraints and where the mere significance of the cage effect is
questionable. To this respect, it is interesting to recall that an IXS
investigation of even more locally-structured glass formers, namely reactive
binary mixtures, has shown the same signatures of the ergodic to non-ergodic
transition predicted by the MCT [15]. In that system, in analogy to what is
found in supercooled simple liquids, the ergodicity breakdown is observed far
before the experimental glass transition, and the static structure evolving
during the covalent bond formation is found to consistently affect the wave
vector dependence of the non-ergodicity factor. It remains a current challenge
to identify the appropriate MCT model that describes the structural arrest in
presence of molecular association or even induced by chemical bonds.
## Acknowledgements
Special thanks to Giancarlo Ruocco, who largely inspired our research on
glasses.
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# Graph-Based Analysis and Visualisation of Mobility Data
Rafael Martínez Márquez, Giuseppe Patanè
CNR-IMATI, Italy
###### Abstract
Urban mobility forecast and analysis can be addressed through grid-based and
graph-based models. However, graph-based representations have the advantage of
more realistically depicting the mobility networks and being more robust since
they allow the implementation of Graph Theory machinery, enhancing the
analysis and visualisation of mobility flows. We define two types of mobility
graphs: Region Adjacency graphs and Origin-Destination graphs. Several node
centrality metrics of graphs are applied to identify the most relevant nodes
of the network in terms of graph connectivity. Additionally, the Perron vector
associated with a strongly connected graph is applied to define a circulation
function on the mobility graph. Such node values are visualised in the
geographically embedded graphs, showing clustering patterns within the
network. Since mobility graphs can be directed or undirected, we define
several Graph Laplacian for both cases and show that these matrices and their
spectral properties provide insightful information for network analysis. The
computation of node centrality metrics and Perron-induced circulation
functions for three different geographical regions demonstrate that basic
elements from Graph Theory applied to mobility networks can lead to structure
analysis for graphs of different connectivity, size, and orientation
properties.
Keywords: Network graphs, network structure, spatial analysis, visualisation,
gravity models
###### Contents
1. 1 Introduction
2. 2 Previous work on data-driven traffic models
1. 2.1 Grid-based traffic models
1. 2.1.1 ST-ResNet - Spatio-Temporal Residual Network
2. 2.1.2 STRN - Spatio-Temporal Relation Network
3. 2.1.3 Deep Gravity Mobility Model
2. 2.2 Graph-based traffic models
1. 2.2.1 ETGCN - Evolution Temporal Graph Convolutional Network
2. 2.2.2 GTA - Graph-based Temporal Attention Framework
3. 2.2.3 G-STARIMA - Graph-based Spatio-Temporal ARIMA Model
3. 2.3 Discussion and comparison
3. 3 “Generic” and traffic graphs: definition $\&$ metrics
1. 3.1 “Generic” graphs
2. 3.2 Centrality metrics for graphs
3. 3.3 Traffic graphs and metrics
4. 4 Graph matrices for network and traffic analysis
1. 4.1 Degree matrix
2. 4.2 Transition probability matrix
3. 4.3 Graph Laplacians
1. 4.3.1 Laplacians of undirected graphs
2. 4.3.2 Laplacians of directed graphs
3. 4.3.3 Properties and discussion/comparison
4. 4.4 Graph Laplacians and traffic models
5. 5 Conclusions, data sets and future work
## 1 Introduction
The mobility of individuals is a topic of critical importance for the
development and sustainability of cities. Forecasting the flows of crowds in a
city and foreseeing possible drawbacks in the transport system can boost
public decision-making in risk mitigation and optimising traffic flow, thus
leading to public safety and citizens’ well-being. Traffic flow is affected by
various factors, such as the spatial dependencies between the different
regions of a city, the temporal dependencies from near or distant past time
intervals, and the effect of external factors, such as weather, land and road
features.
The movement of people and vehicles within a city and region is increasingly
influenced by interdependence. In particular, the optimal use of different
transport infrastructures and networks in cities and their connection to
regional destinations and between neighbouring regions has a substantial
impact on various aspects, such as chemical and noise pollution, the
inefficiency of local public transport due to a general increase in traffic
volume, the risk of accidents, and high emergency management costs. Concerning
state-of-the-art in traffic and mobility data-driven modelling, we identify
two main approaches (Sect. 2): Euclidean models that use grid partitions over
a region for forecasting the flows within them (Zhang et al.,, 2017; Liang et
al.,, 2021) or to classify the available data into training and testing sets
(Simini et al.,, 2021), and Non-Euclidean models that use graph-based
structures (Zhang et al., 2021b, ; Zhang et al.,, 2022; Liu et al.,, 2021)
that would be more suitable and realistic for mobility modelling as the
spatial dependency is not necessarily between adjacent cells in a grid-based
partition.
For both Euclidean and non-Euclidean models, visualising the predicted flows
becomes an essential tool for traffic analysis because it provides additional
interpretations to the purely theoretical ones. For instance, visualised
traffic flows can help to localise and identify traffic patterns or areas with
the highest or lowest traffic values (e.g. speed, volume). In addition to the
visualisation of mobility flows and volumes, the node centrality metrics
(i.e., Closeness centrality, betweenness centrality, and Page rank centrality)
(Sect. 3) provide further analysis tools for the structure of the system,
particularly within a graph framework. For instance, the betweenness
centrality can be a measure of the vulnerability of a particular station to
traffic disruption (Dees et al.,, 2021; Mukherjee,, 2012); the closeness
centrality can be used to find the stations that are best connected to the
urban network in terms of minimising the total distance, travelling time, or
other road parameters. The Page rank centrality can identify the most popular
roads or stations regarding the dynamic mobility flow within the urban
network. By analysing both the traffic flow and the structure of the network,
it is also possible to better understand how the network structure affects the
dynamic of the traffic; other factors are operational interventions and
seasonal factors that affect the travel frequency on the network. We introduce
node centrality metrics analysis for two types of graphs: Region Adjacency
graphs and Origin-Destination graphs. The former type of graph is induced by a
given finite partition of a city or region, where the connections represent
the adjacency of a pair of partition units. The latter type of graph is
induced by an Origin-Destination matrix, a square matrix (possibly sparse)
whose entries are positive values representing a mobility counting between a
pair of sub-regions (not necessarily geographically adjacent). We compute,
visualise, and analyse the three centrality metrics for the Region Adjacency
graphs of Genova Province, United Kingdom and New York State. Additionally, we
perform the same visual analysis for the Origin-Destination graph for the
flows predicted by the Deep Gravity mobility model (Simini et al.,, 2021).
Furthermore, through some matrices associated with a graph, e.g. the Laplacian
matrix (Sect. 4), we can investigate demographic factors that affect mobility
flows such as the distribution of population surrounding each node of a
transport network (Dees et al.,, 2021). Also, the Laplacian spectrum is
fundamental for Graph Signal Processing tools that facilitate the analysis of
the correlation of time series of traffic volumes, thus identifying data
correlation structures over different locations and Origin-Destination pairs
in an urban network. The classical graph Laplacian is the Combinatorial
Laplacian, which satisfies important properties as symmetric and positive
semi-definite. However, these features hold only for undirected graphs because
they are based on the symmetry of the adjacency matrix. We introduce the
definitions of the Combinatorial Laplacian and Normalised Laplacian for
undirected graphs. Similarly, the Combinatorial Directed Laplacian,
Symmetrized Laplacian, Combinatorial Symmetrized Laplacian, and Diplacian are
defined for directed graphs. The Symmetrized Laplacian, Combinatorial
Symmetrized Laplacian, and Diplacian are defined only for strongly connected
graphs since they are based on the Perron vector associated with the
transition probability matrix, and their existence and uniqueness are
guaranteed when the graph has only one connected component. These definitions
for the Laplacian matrix grants their characteristic properties to hold
regardless of the graph’s connectivity or if it is directed or undirected.
Overall goals we aim to present a variety of data-driven traffic and mobility
models and develop a set of methods for analysing and visualising information
about the dynamics of mobility flows. The input data used for such models is
mainly consisting of trajectories of cars, bicycles, or people (Zhang et al.,,
2017; Liang et al.,, 2021), sensors and monitoring stations (Zhang et al.,
2021b, ; Zhang et al.,, 2022; Liu et al.,, 2021), and demographical data
(Simini et al.,, 2021). The primary research and development activities
include
* •
an overview of Grid-based and Graph-based data-driven traffic models;
* •
the visualisation of centrality metric values in mobility graphs to display an
overview of the network structure to more easily identify mobility flows of
interest;
* •
the definition and properties of distinct Laplacians matrices for undirected
and directed graphs.
These tools are essential to support autonomous decision-making for traffic
management in areas of interest, e.g., to propose suggested routes to
commuters entering a city to reduce traffic in certain areas based on traffic
forecasts. We draw several conclusions, indicate possible extents of this work
(Sect. 5), and describe the used data sets.
## 2 Previous work on data-driven traffic models
The traffic forecasting problem can be formulated as follows: given a
historical collection $\mathcal{X}$ of features, e.g., average speed, road
volume, and demand levels, during $P$ different time intervals, predict a
collection $\mathcal{Y}$ of features (possibly different) for the next $Q$
time intervals. This relation is expressed by $\mathcal{Y}=f(\mathcal{X})$,
where $f$ is an unknown function. Several data-driven traffic models depend on
the available data and the predicted variable. Also, these models can be
distinguished based on how they structure the data (e.g., in a regular or
irregular domain). Some grid-based approaches are presented where the data is
partitioned into squared cells, and the predicted value is the number of flows
within their inner sub-partitions. Similarly, we introduce some graph-based
models where data partition depends on the distribution of the road network or
data collectors (e.g., sensors, monitoring stations, road junctions). These
graph-based models can predict various traffic states (e.g., speed, flow,
occupancy). If the chosen model is grid-based, the city or area of interest is
partitioned into an $I\times J$ grid with $N=IJ$ cells. If the chosen model is
graph-based, then the road network over a city or area of interest is
represented by $N$ nodes and a set of edges connecting them. For both grid-
based and graph-based models, different techniques from Deep Learning can be
used, such as convolution networks and residual units (Zhang et al.,, 2017;
Liang et al.,, 2021), or graph attention mechanisms and spectral convolution
networks (Zhang et al., 2021a, ; Li et al.,, 2021).
### 2.1 Grid-based traffic models
Several flow forecasting models use different types of mobility data and
prediction techniques. Euclidean approaches for flow prediction in traffic
modelling consist of partitioning a city, state, or country into a square or
rectangular grid to assess the number of displacements between two elements of
the grid (Zhang et al.,, 2017; Liang et al.,, 2021), or dividing available
geographical data into training, and testing data for the use of a Deep
Learning model (Simini et al.,, 2021). When partitioning the city into an
$I\times J$ grid, urban flows, traffic volumes, and other mobility values,
data can be stored in matrices in $\mathbb{R}^{I\times J}$, or tensors in
$\mathbb{R}^{L\times I\times J}$ if the data consists of more than one type of
value, where $L$ is the number of different values that compose the data. The
forecasting problem can be stated as given historical observations of mobility
data $\\{\mathbf{X}_{1},\mathbf{X}_{2},\ldots,\mathbf{X}_{N}\\}$ during $N$
time intervals, predict $\mathbf{X}_{N+1}$. External factors may vary over
time and influence mobility in a city. Indeed, the crowd flows can be
represented by tensors in $\mathbb{R}^{K\times I\times J}$, where $K$ is the
number of external influences being considered. Suppose the $I\times J$ grid
is only used to classify the available mobility data. In that case, each of
the cells should be, in addition, sub-partitioned (possibly in irregular
units) to forecast the flows from one to another within the same cell.
#### 2.1.1 ST-ResNet - Spatio-Temporal Residual Network
In the ST-ResNet model (Spatio-temporal residual network) (Zhang et al.,,
2017), the subdivision of the city into regions of interest is performed by a
squared grid partition using the longitudes and latitudes. The predicted
traffic variables are the inflows and outflows among the cells of the squared
grid. Inflow is the total traffic crowds entering a cell in the grid from any
other cell during a given time interval. Outflow denotes the total traffic of
crowds leaving a cell to any other cell in the grid during a given time
interval. The input data for the ST-ResNet model consists of trajectories of
cars, bicycles, people, etc., during different constant length time intervals
and meteorological data as external factors that impact the traffic flow in a
city. Formally, the city, state, or country is split into an $I\times J$ mesh
with $N=IJ$ cells, and we can associate the inflow and outflow of crowds to
every cell $(i,j)$ at the time interval $t$.
The ST-ResNet model applies convolution-based residual networks for dealing
with the local and global spatial dependencies that affect the flows within a
city. There are three such networks, one for each type of temporal dependency:
closeness, period, and trend. Moreover, the ST-ResNet model uses a fully
connected network to include the effect of external factors that may impact
traffic flows, such as weather conditions and holidays. All these model
elements are dynamically aggregated to produce predicted inflow and outflow
matrices with the exact dimensions of the squared grid partition of the city.
#### 2.1.2 STRN - Spatio-Temporal Relation Network
The STRN model (Spatio-Temporal Relation Network) (Liang et al.,, 2021) is
another grid-based model for flow prediction. The subdivision of the city into
a squared grid partition consists of a considerably larger number of cells $N$
(of smaller size) than with the ST-ResNet model; this type of partition is
called fine-grained or high resolution. Moreover, a secondary irregular
partition of the city into $M$ regions, with $M$ considerably smaller than
$N$, is used to assess the effects of global spatial dependencies more
efficiently, namely, by analysing the diffusion within the connectivity
network that represents the irregular partition, which is computationally
faster than using a sequence of convolution layers as in the ST-ResNet model.
The input data for the STRN model consists of trajectories as in the ST-ResNet
model.
An assignment matrix $\mathbf{B}\in\mathbb{R}^{N\times M}$ is used to connect
the $N$ cells grid structure with the $M$ region’s network, where the entry
$b_{ij}$ denotes the likelihood that the cell $i$ belongs to the region $j$.
The irregular partition can be computationally generated by additional
available information, such as the road network, the administrative divisions
of a city, or census areas. The STRN model applies a Meta Learner that
converts the features related to external factors to a representation with the
same dimensionality as the Inflows and Outflows from the three temporal
dependencies, e.g., closeness, period, and trend. Then, these four feature
representations are passed to a Backbone Network consisting of squeeze-and-
excitation networks, and convolution layers, whose output is finally passed to
the Global Relation Module (GloNet), which converts the grid-based
representations to the network structure and applies Graph Convolution
Networks to assess the diffusion of the flows within the irregular regions.
Then, the Meta Learner converts the predictions to the original grid-based
structure.
#### 2.1.3 Deep Gravity Mobility Model
The Deep Gravity mobility model (Simini et al.,, 2021) is a grid-based model
for flow prediction without using the three-time dependencies paradigm (i.e.,
closeness, period, trend), which is fundamental in the ST-ResNet and STRN
models. To use the Deep Gravity model, partitioning a city, state, or country
starts with constructing a square grid covering the whole region. Additionally
to this handmade regular partition, it must be available a secondary finer
(and irregular) partition for which there is available population data, for
instance, the tracts or areas that constitute the units of a census. Then, the
barycentre of each irregular unit is allocated in one of the square cells.
Hence, the data of a cell consists of the data of the irregular units whose
barycentre belongs to that cell. The square cells are divided into training
and testing cells in a stratified manner based on the population data in the
cells so that the two groups have the same number of cells belonging to the
various population deciles. Newton’s law of universal gravitation inspires
this method, which is why its name is inspired. A generalised version of this
attraction law, by considering the deterrence function, is defined as
$y(l_{i},l_{j})=O_{i}\dfrac{m_{j}^{\beta_{1}}f(r_{ij})}{\sum_{k}m_{k}^{\beta_{1}}f(r_{ik})},$
(1)
where $y(l_{i},l_{j})$ represents the predicted flow from a location $l_{i}$
to a location $l_{j}$, and $r_{ij}$ is the distance between them; the $m_{i}$
are the populations, and $\beta_{1}$ is a real parameter. The relation in Eq.
(1) is called a singly-constrained gravity model since it requires knowing the
total outflow $O_{i}$ for each location in advance. The Deep Gravity model
applies feed-forward neural networks to predict the number of flows from a
given area from the irregular partition to any other of the areas within the
same cell. The choice of this type of network lies in the fact that feed-
forward neural networks generalise linear models as in Eq. (1). Unlike the ST-
ResNet and STRN models, the Deep Gravity model does not use the grid structure
to predict flows between them but to split the city into training and testing
regions.
The city’s subdivision into a squared grid partition splits the available data
sets into training and testing data since the predicted flows correspond to
the flows between additional irregular partitions within each cell provided by
census areas. The input data for the Deep Gravity model consists of population
values obtained from the official census and geographic features obtained from
OpenStreetMap. The intuition of this model is that the flow between two
locations is directly proportional to their population and inversely
proportional to the distance between them.
Given two irregular units $l_{i}$ and $l_{j}$ in a cell $C$, the components of
the input vector $x_{ij}$ used to predict the flow $y(l_{i},l_{j})$ from
$l_{i}$ to $l_{j}$ consist of the population in $l_{i}$ and $l_{j}$, the
distance between them, and eighteen geographical values extracted from
OpenStreetMap (land use areas, road lengths, counting of points of interest,
etc.). The input vector $x_{ij}$ passes through a multilayer feed-forward
neural network with an output layer of dimension $1$ (a scalar $y_{ij}$) in
the range $(-\infty,\infty)$. After computing $y_{ij}$ for $j=1,\ldots,M$,
where $M$ is the number of irregular units in the cell $C$, the $M$ values are
passed through a softmax layer to convert them into an $M$-dimensional vector
with non-negative components whose sum is $1$, namely, a probability vector
that represents the probability distribution of the flows starting in location
$l_{i}$ within the cell $C$. When training the model, the predicted values
$y_{ij}$ are compared to real flows $z_{ij}$ from location $l_{i}$ to location
$l_{j}$, for instance, provided by GPS trajectory data or commuting surveys. A
standard metric used to measure the performance of a flow prediction model is
the Common part of Commuters (CPC) defined by
$CPC=\dfrac{2\displaystyle\sum_{i,j}\min(y_{ij},z_{ij})}{\displaystyle\sum_{i,j}y_{ij}+\sum_{i,j}z_{ij}},$
where $y_{ij},z_{ij}$ refer to the predicted and real flow from location
$l_{i}$ to location $l_{j}$ respectively, and the indices $i,j$ run along all
the locations in a region of interest, for instance in one of the square cells
used in the Deep Gravity model. The CPC values are in the range
$\left[0,1\right]$ with $1$ indicating a perfect flow prediction. Suppose the
total number of outflows in a region of interest coincides with the predicted
flows (as with the Deep Gravity model used by (Simini et al.,, 2021)). In that
case, the CPC value is equal to the fraction of the flows correctly predicted
by the model.
### 2.2 Graph-based traffic models
Graph-based approaches have been used for metro traffic modelling (Dees et
al.,, 2021) and statistical analysis of road networks (Mukherjee,, 2012).
Graph theory provides additional theoretical elements, namely the weight of
the edges, which can be used to represent some features in traffic models,
such as the road length and road capacity (Tian et al., 2016a, ). Moreover,
the centrality properties of a graph can provide additional insight into the
distribution and importance of nodes and edges (Henry et al.,, 2019;
Boulmakoul et al.,, 2017).
There are several approaches to creating a graph structure for traffic
modelling depending on the available data type and the forecasted variable. In
a traffic graph (Jiang and Luo,, 2022), a graph signal
$\mathbf{X}_{t}\in\mathbb{R}^{N\times d}$ is defined at every time step $t$
where $N$ is the number of nodes and $d$ is the number of traffic elements
(e.g. speed, traffic state, traffic demand) that are measured. The majority of
graph-based spatio-temporal traffic problems belong to traffic state and
traffic demand prediction (Ye et al.,, 2022), which are modelled by various
graph-based deep learning architectures (e.g. Spectral Graph Convolution
(SGCN), Gated Recurrent Unit (GRU), Graph Attention Network (GTA)). The nodes
of a traffic graph can be defined as the intersections between roads or as the
sensors along a highway. The physical road connections between them usually
give the edges, and the predicted variables can be any traffic element for
each node, namely, a graph signal.
#### 2.2.1 ETGCN - Evolution Temporal Graph Convolutional Network
The Evolution Temporal Graph Convolutional Network (ETGCN) (Zhang et al.,
2021b, ) captures the spatial and temporal correlations among the nodes in a
Road graph. The road graph is a weighted graph whose nodes are traffic sensors
distributed over a city or region. The concept of adjacency matrix involves
the fusion of three types of information (Content Similarity Adjacency Matrix,
Graph Betweenness Adjacency Matrix, and Transportation Neighborhood Adjacency
Matrix) whose entries consider the geographical position of the sensors and
their connections in the original road network, and which fusion enhances
feature learning. Given a historical time series of the registered speed at
every node, the ETGCN model aims to predict the speed at every node in the
next timestep. The architecture of the ETGCN model combines Graph
Convolutional Networks and GRU to learn the sequence of spatial and temporal
features.
#### 2.2.2 GTA - Graph-based Temporal Attention Framework
The Graph-based Temporal Attention Framework (GTA) (Zhang et al.,, 2022) model
also considers a weighted sensor graph, but in contrast to the ETGCN model,
the weights of the edges are the road network distances between the sensors,
and the predicted variable is the traffic flow, which is defined as the number
of vehicles passing through the monitoring station over a given time interval.
The architecture of the GTA model introduces an attention mechanism to
adaptively identify the relations among three temporal dependencies: monthly
pattern, weekly pattern, and current pattern. A Long Short-Term Memory (LSTM)
network is employed to extract the temporal correlation for each dependency.
#### 2.2.3 G-STARIMA - Graph-based Spatio-Temporal ARIMA Model
The Graph-based spatio-temporal ARIMA (G-STARIMA) model (Liu et al.,, 2021) is
a graph-based framework built upon statistical methods. The road network is
described by a weighted undirected graph where the nodes are the traffic
intersections, and the edges are their road connections. The G-STARIMA model
uses a historical time series of graph signals at the network nodes (i.e.,
sensor observations such as speed, occupancy, or traffic flow) to predict the
traffic state at the next instant. Since the connectivity between traffic
intersections, represented by edge weights, may vary over time because of the
dynamics of urban traffic states, it is performed a dynamic estimation of a
weighted adjacency matrix which is found by solving a convex optimisation
problem.
### 2.3 Discussion and comparison
The ST-ResNet, STRN, and Deep Gravity models are grid-based models for flow
predictions within the units of a city, state, or country partition. The ST-
Resnet and the STRN models consider the CPT Paradigm (closeness, period, and
trend time dependencies) and predict the flow values as the next value of a
sequence of flows. In contrast, the Deep Gravity model only predicts a value
for the flows from a set of given features. Regarding the network
architectures that build the models, the Deep Gravity model has a simpler
architecture since feed-forward neural networks are the elementary Deep
Learning network. In contrast, the convolution layers and residual units of
the ST-ResNet and STRN models are more refined networks. Moreover, even if the
three models are based on a grid structure, they use the square cells
differently: the ST-ResNet and the STRN models use the cells as the units
where the flows will be predicted from historical values of the same flows. In
contrast, the Deep Gravity model uses the cells only to split the available
data into training and testing data.
Regarding the spatial dependencies, the Deep Gravity model performs flow
prediction only for irregular units within the same cell. Therefore, it does
not consider global spatial dependencies as the ST-ResNet and STRN models. The
three show significant improvements concerning other existing baseline models
for flow prediction. Still, it is necessary to make a performance comparison
between the Deep Gravity model and the other two grid-based models.
The grid-based models show interesting results in traffic forecasting.
However, some features could suggest modelling mobility in a city as a graph:
the structure of the roads as graph topology that connects different
locations, i.e., graph nodes, in the city, or the information associated with
movement between two locations as graph weights. The urban road network is a
typical spatial network because of its geographical factors (Tian et al.,
2016b, ; Ye et al.,, 2022), and usually, adjacent cells of the grid do not
have flows between them during some intervals of time because there are no
direct roads connecting them.
The ETGCN and GTA models are data-driven graph-based models for traffic
predictions that consider sensors or monitoring stations as the nodes of a
weighted graph. The GTA model uses the distance between the sensors as edge
weights and predicts the number of vehicles passing through every monitoring
station. In contrast, the ETGCN model considers a fusion of different types of
adjacency matrices that enhance the learning of features and predict the speed
at every network node. Another characteristic of the GTA model is the
implementation of three-time dependencies, which is analogous to the CPT
Paradigm of the ST-ResNet and STRN grid-based models. Regarding the Deep
Learning architecture of these graph-based models, the ETGCN uses GRU networks
to learn temporal features. In contrast, the GTA model combines LSTM networks
through an attention-based mechanism. In contrast to the ETGCN and the GTA
models, the G-STARIMA model is not a Deep Learning model but a graph-based
framework based on statistical methods. In addition, this model considers the
road intersections as network nodes instead of the monitoring stations and
predicts various traffic states (e.g., speed, traffic flow).
Both grid-based and graph-based traffic models show interesting results, and
the choice of a modelling framework should be based on the type and amount of
available data and the variables that will be predicted. However, graph-based
models have additional theoretical tools to boost traffic analysis, i.e. the
centrality metrics (Sect. 3.2). For instance, the betweenness centrality was
used in (Dees et al.,, 2021) to analyse the robustness of the metro network in
London since it facilitates the identification of metro stops that might
significantly impact the whole network during a disruption. Similarly, the
betweenness metric was used in (Mukherjee,, 2012) to identify potential
congestion points in the Indian highway network. In Ye et al., (2022), various
approaches are proposed to construct a graph-based framework, different
mobility data sets, and Deep Learning networks for graph-based models.
## 3 “Generic” and traffic graphs: definition $\&$ metrics
The graph-based approaches for traffic and mobility modelling certainly build
upon the fundamental concepts from Graph Theory. Basic definitions such as the
direction and weight of an edge enable the construction of different types of
graphs. This leads to the implementation and interpretation of diverse multi-
agent systems modelling, where the connection between the agents (the nodes)
stores significant information. The sequences of consecutive edges motivate
the definitions of paths, which induces the concept of connected components.
Furthermore, all these structures combined enable the introduction of several
quantitative scores for each node according to its importance in the
connectivity and influence to the rest of the network, which is characterised
by the centrality metrics. In traffic and mobility modelling, the centrality
metrics provide valuable information for the structure analysis of the road
and geographical networks.
### 3.1 “Generic” graphs
To represent a graph $\mathcal{G}$ with nodes set $V=\\{1,\ldots,N\\}$ and
edges set $E$, we use the notation $\mathcal{G}=(V,E)$. Undirected graphs
represent only connections between nodes, so $(u,v)=(v,u)$ for every $u,v\in
V$. A graph is a directed graph (or digraph) if the couples that define the
edges in $E$ cannot be arbitrarily ordered, namely, if in general
$(u,v)\neq(v,u)$. The positive value $w_{uv}$ is the weight of the edge
$(u,v)$, and $E$ is a weighted graph. If no values are associated with a
graph’s edges, it is called unweighted. The weights of the edges of a graph
with $N$ nodes can be stored in an $N\times N$ matrix called the adjacency
matrix of $\mathcal{G}$.
We can associate positive weights to every edge of a graph with $N$ nodes and
store them in the weighted adjacency matrix $\mathbf{A}\in\mathbb{R}^{N\times
N}$, namely
$A_{ij}=\left\\{\begin{array}[]{cl}w_{ij}&\text{if }(i,j)\in E;\\\
0&\text{otherwise}.\end{array}\right.$
If $\mathcal{G}$ is an unweighted graph; then we have binary weights, i.e.,
$1$ for existing connections between nodes, and $0$ otherwise. For undirected
graphs, since $(u,v)$ and $(v,u)$ refer to the same edge, then the weight
values satisfy $\omega_{uv}=\omega_{vu}$, the corresponding adjacency matrix
is symmetric. Knowing the adjacency matrix of a graph, we know the weights of
the edges and the connections between the nodes. Indeed, we write
$\mathcal{G}=(V,E,\mathbf{A})$ to make more explicit the relationship between
the graph representation of $\mathcal{G}$ and the matrix representation given
by the adjacency matrix $\mathbf{A}$.
##### Connected and strongly connected graphs
An undirected graph is connected if, for every pair of nodes $u,v\in V$ there
exists a path $w$ from $u$ to $v$, otherwise $\mathcal{G}$ is called
disconnected. A directed graph $\mathcal{G}$ is strongly connected if for any
pair $(p_{i},p_{j})$ of $\mathcal{G}$ there exists a path (i.e., a sequence of
edges) $(p_{i},p_{l_{1}})$, $(p_{l_{2}},p_{l_{3}})$, $\ldots$,
$(p_{l_{r-1}},p_{j})$, which leads from $p_{i}$ to $p_{j}$. In this case, the
path has length $r$. The associated directed graph $\mathcal{G}(\mathbf{A})$
of a $n\times n$ matrix $\mathbf{A}$ consists of $n$ nodes
$p_{1},\ldots,p_{n}$, where an edge leads from $p_{i}$ to $p_{j}$ if and only
if $A_{ij}\neq 0$. As main properties, we recall that a matrix $\mathbf{A}$ is
irreducible if and only if the associated directed graph
$\mathcal{G}(\mathbf{A})$ is strongly connected. Otherwise, the matrix is
called _irreducible_. A matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ is said
to be _reducible_ if there exists a perturbation matrix $\mathbf{P}$ such that
$\mathbf{C}=\mathbf{P}\mathbf{A}\mathbf{P}^{\top}=\left[\begin{array}[]{cc}\mathbf{A}_{11}&\mathbf{A}_{12}\\\
\mathbf{0}&\mathbf{A}_{22}\end{array},\right]$
$\mathbf{A}_{11}\in\mathbb{R}^{n\times r}$,
$\mathbf{A}_{12}\in\mathbb{R}^{(n-r)\times(n-r)}$, and
$\mathbf{A}_{22}\in\mathbb{R}^{n\times(n-r)}$. Let $\mathbf{A}$ be an
irreducible matrix. Then, (i) $\mathbf{A}$ has a positive real eigenvalue
equal to its spectral radius $\rho(\mathbf{A})$; (ii) $\rho(\mathbf{A})$ is a
single eigenvalue of $\mathbf{A}$ and its corresponding eigenvector $\phi>0$;
(iii) $\rho(\mathbf{A})$ increases when any entry of $\mathbf{A}$ increases;
and (iv) there is no other non-negative eigenvector of $\mathbf{A}$ different
from $\phi$. Finally, a digraph is called weakly connected if its underlying
undirected graph is connected. A disconnected graph can be decomposed into
smaller subgraphs connected as an independent graph, called connected
components.
##### Graph paths
A path $w$ in a graph (or a walk) (Chung,, 2005) of length $m$ is a sequence
of different $m$ nodes in $V$, namely, $w=\\{v_{1},\ldots,v_{m}\\}$ such that
the couple $(v_{i},v_{i+1})$ belongs to the edges set $E$ for every
$i=1,\ldots,m-1$. Given the nodes $a$ and $b$ of a graph (undirected or
directed), then $w=(v_{1},\ldots,v_{m})$ is a shortest path from $a$ to $b$ if
$v_{1}=a,v_{m}=b$ and the value $d_{w}(a,b):=\sum_{i=1}^{m-1}A_{v_{i}v_{i+1}}$
attains its minimum among all the paths starting in $a$ and ending in $b$.
Since there may be more than one shortest path from a node $a$ to a node $b$,
we define the distance from $a$ to $b$ as $d(a,b)=\min\\{d_{w}(a,b)\colon
w\text{ is a path from }a\text{ to }b\\}$, which is a unique value. If for a
couple of nodes $a,b\in V$ of a disconnected graph, there is no path starting
in $a$ and ending in $b$ then we define their distance as $d(a,b)=\infty$. The
distance between two nodes is not precisely a distance in the context of
metric spaces when the graph is directed because $d(a,b)$ is not necessarily
the same as $d(b,a)$.
For instance, the Region Adjacency graphs (Sect. 3.3) are undirected graphs
that are highly sparse since they represent the geometric partition of a
geographical region where most of the units have just a few neighbours.
However, there can be some areas where the sub-regions are clustered, which
produces square sub-matrices along the diagonal of the sparsity matrix (Fig.
1).
Undirected Region Adjacency graphs
---
| |
Region Adjacency graph
| |
Sparsity matrix
a) GOA Province | b) UK | c) NY State
Figure 1: Region Adjacency graphs and sparse adjacency matrices. a) Genova
Province (GOA Province) partitioned into 137 zones b) United Kingdom (UK)
partitioned into 344 local authority districts c) New York State (NY State)
partitioned into 5410 census tracts.
### 3.2 Centrality metrics for graphs
Different centrality metrics can be defined for the nodes in a directed graph
$\mathcal{G}=(V,E)$ with $N$ nodes. The main centrality metrics are degree
centrality, closeness centrality, harmonic centrality, betweenness centrality,
and Page rank centrality. The closeness and harmonic centralities measure the
likelihood of a node reaching the rest of the nodes, which can be extended to
the concept of being reached if the graph is directed. The betweenness
centrality measures the importance of a node to create connections between the
rest of the nodes, and it is computed in the same way for undirected and
directed graphs. Finally, PageRank centrality is a metric specifically for
directed graphs since the influence of a node depends on the influence of the
incoming connections.
| |
---|---|---
(a) GOA Province | (b) UK | (c) NY
Figure 2: Normalised node centrality metrics. The normalised betweenness
centrality is the smallest for the three Region Adjacency graphs regardless of
the number of nodes.
##### Degree centrality:
The out-degree centrality and in-degree centrality of a node $v$ in a directed
graph are defined by
$D^{+}(v)=\sum_{j=1}^{N}A_{vj},\qquad D^{-}(v)=\sum_{j=1}^{N}A_{jv},$
respectively. This centrality metric represents how big that node’s outgoing
or incoming flow is compared to the corresponding flow in the rest of the
nodes.
##### Closeness centrality:
The out-closeness centrality and in-closeness centrality of a node $v$ are
defined by
$C^{+}(v)=\dfrac{N-1}{\displaystyle\sum_{i\neq v}d(v,i)},\qquad
C^{-}(v)=\dfrac{N-1}{\displaystyle\sum_{i\neq v}d(i,v)}.$
The two types of closeness centrality defined on a directed graph measure the
likelihood of a node reaching the other nodes or being reached by them. For an
undirected graph, the closeness centrality measures the likelihood of a node
being connected to the rest of the nodes in a graph. If for some nodes $v,b$
there is no path from $v$ to $b$, then we would have $C(v)=0$. The bigger the
distances from $v\in V$ to the rest of the nodes are, the smaller the value of
the closeness centrality $C(v)$ is, the nodes that are closer to the rest of
the graph have higher closeness centrality values. The closeness centrality
for a directed graph may have more nodes with zero values, compared to the
undirected case, since more cases of not symmetric paths arise, for instance,
in a not strongly connected graph.
##### Harmonic centrality:
The out-harmonic centrality and in-harmonic centrality of a node $v$ are
defined by
$H^{+}(v)=\dfrac{1}{N-1}\sum_{i\neq v}\frac{1}{d(v,i)},\qquad
H^{-}(v)=\dfrac{1}{{N-1}}\sum_{i\neq v}\frac{1}{d(i,v)}.$
The harmonic centrality in a directed graph reduces zero closeness centrality
values when the graph is not strongly connected. It simply excludes the not-
reachable nodes to provide positive values for most nodes. A node will have a
zero out-harmonic centrality value if and only if its out-degree is zero. A
node will have a zero in-harmonic centrality value if and only if its in-
degree is zero. From the inequality between the arithmetic mean and the
harmonic mean of real values, it follows $H^{+}(v)\geq C^{+}(v)$ and
$H^{-}(v)\geq C^{-}(v)$. For any undirected graph $\mathcal{G}$ and any node
$v\in V$, from the inequality between the arithmetic mean and the harmonic
mean of real values it follows $H(v)\geq C(v)$ and the harmonic centrality is
just a variation of the closeness centrality to handle the infinite values
caused by disconnected graphs since the existence of a distance
$d(v,b)=\infty$ would vanish the value of $C(v)$ discarding the possibility of
further analysis for the node $v$.
##### Betweenness centrality:
Similarly to the undirected case, the betweenness centrality of a node $v$ is
defined by
$B(v)=\underset{(a,b)\in\mathcal{P}_{v}}{\sum}\dfrac{|S_{v}(a,b)|}{|S(a,b)|},$
where $S(a,b)$ is the set of shortest paths from $a$ to $b$,
$S_{v}(a,b)=\\{w\in S(a,b)\colon v\in w\\}$ is the set of shortest paths from
$a$ to $b$ with $v$ as an intermediate node, $\mathcal{P}_{v}=\\{(a,b)\in
V\times V\colon a,b\neq v,a\neq b,S(a,b)\neq\varnothing\\}\\}$ is the set of
node couples that have $v$ as part of a shortest path, and $v\in
w=(v_{1},\ldots,v_{m})$ means $v=v_{i}$ for some $i=2,\ldots,m-1$. The
notation $|X|$ represents the cardinality of a set $X$, in other words, the
number of elements of $X$. The betweenness centrality values for the nodes in
a directed graph are generally smaller than the values in the undirected case
because $S(a,b)\neq\varnothing$ does not imply that $S(b,a)\neq\varnothing$
and therefore there might be fewer elements in the sum.
For directed graphs, the betweenness centrality estimates the importance of a
node $v$ as a connection between the rest of the graph. Namely, it measures
the proportion of shortest paths between any other couple of nodes passing
through $v$. The higher the betweenness centrality of a node is, the more
significant the proportion of shortest paths between other nodes passing
through it. A node with a high betweenness centrality value can be interpreted
as a node whose removal would considerably affect the graph since some nodes
may remain without an optimal path between them, and some nodes may even
become disconnected. If no short paths between any couple of nodes pass
through a given (and different) node, then its betweenness centrality is zero.
If for every $a,b\in V$ it holds $S(a,b)=\varnothing$, then the graph consists
of isolated nodes, and the betweenness centrality values can be set to zero
for each node.
##### PageRank centrality:
This centrality metric is particularly used with directed graphs; it measures
the influence of a node $v\in V$ depending on the influence of every other
node $u\in V$ such that $(u,v)\in E$ (incoming edges to $V$). Intuitively, if
a node has incoming edges from a relevant node, its influence in the graph
would be bigger than if those edges were from a less relevant node. The
PageRank centrality of a node $v$ originally defined in (Page et al.,, 1999),
is defined in a recursive way through
$PR(v)=(1-c)+c\left(\dfrac{PR(v_{1})}{D^{+}(v_{1})}+\ldots\dfrac{PR(v_{m})}{D^{+}(v_{m})}\right),$
where the $v_{i}$ are the nodes in $\mathcal{G}$ such that there is an edge
from $v_{i}$ to $v$, for $i=1,\ldots,m$. Moreover, $c\in(0,1)$ is a value
called the damping factor that helps to deal with dead-end nodes (without
outgoing edges). Usually, it is set to $c=0.85$. The PageRank values can be
seen as the principal eigenvector of a matrix $\mathbf{\hat{P}}^{\top}$ given
by $\mathbf{\hat{P}}=c(\mathbf{P}+\delta\cdot b^{\top})+(1-c)\mathbf{E}$,
where $\mathbf{P}$ is the transition probability matrix $\mathbf{P}$ (Sect.
4.2), $\mathbf{b}\in\mathbb{R}^{N}$ is a distribution probability called
teleportation vector, whose component $i$ denotes the probability to move
arbitrarily to the node $i$ while moving along the edges of the graph (with
direction), and $\mathbf{E}:=(1,\ldots,1)\mathbf{b}^{\top}$ (Berkhin,, 2005)).
The vector $\delta\in\mathbb{R}^{N}$ has components
$\delta_{i}:=\delta(d_{i},0)$ (Kronecker delta) for $i=1,\ldots,N$, where
$d_{i}$ is the degree of the node $i$.
Table 1: Summary of centrality metrics for the nodes of a graph Metric and definition | Undirected graphs | Directed graphs
---|---|---
| Degree centrality
---
$D(v)=\sum_{j=1}^{N}A_{vj}$
Yes | Out-degree and in-degree variations
| Closeness centrality
---
$C(v)=\dfrac{N-1}{\displaystyle\sum_{i\neq v}d(v,i)}.$
Yes | Outgoing and incoming variations
| Harmonic centrality
---
$H(v)=\dfrac{\displaystyle\sum_{i\neq v}\frac{1}{d(v,i)}}{N-1}$
Yes | Outgoing and incoming variations
| Betweenness centrality
---
$B(v)=\underset{(a,b)\in\mathcal{P}_{v}}{\sum}\dfrac{|S_{v}(a,b)|}{|S(a,b)|}$
Yes | Yes
| PageRank centrality
---
$PR(v)=(1-c)+c\displaystyle\sum_{u\rightarrow v}\dfrac{PR(u)}{d(u)}$
No | Yes
For instance, after normalising each centrality metric for the three Region
Adjacency graphs from Fig. 1, a trend in common is shown for all of them (Fig.
2), i.e. the closeness centrality has invariably the largest values. In
contrast, the betweenness centrality has the smallest ones because the maximum
value of the betweenness centrality is larger than the maximum value of the
other two centralities. Moreover, for the NY State graph, the majority of
betweenness centrality values are localised in the first decile of its
distribution as the number of nodes in the NY State graph is considerably
larger than in the other two graphs, which results in larger betweenness
centralities because every node is part of the shortest path between more node
pairs.
##### Flow and circulation of a directed graph
In a graph $\mathcal{G}$, we consider a function
$F:\mathcal{E}(\mathcal{G})\rightarrow\mathbb{R}^{+}$ that assigns to each
directed edge $(i,j)$ a non-negative value $F_{ij}$, which is said to be a
_circulation_ if at each node $i$ we have that $\sum_{j,\,j\rightarrow
i}F_{ji}=\sum_{j,\,i\rightarrow j}F_{ij}$. A circulation is said to be
_invertible_ if $F_{ij}=F_{ji}$. For a strongly connected directed graph
$\mathcal{G}$, the eigenvector $\phi$ of the transition probability matrix
$\mathbf{P}$ with eigenvalue $1$ is associated with a circulation
$F^{\phi}_{ij}:=\phi_{i}P_{ij}.$ (2)
The average node circulation of each node is defined by
$\tilde{F^{\phi}_{i}}=\sum_{j}F^{\phi}_{ij}/D^{+}(i)$. For instance, as the
Region-Adjacency graphs are connected (after possibly removing the ”islands”,
i.e. nodes with node degree equal to $1$), it is possible to define a
circulation associated to their Perron vector by using Eq. 2. The trend of the
average node circulation (Fig. 3) has an inverse behaviour with respect to the
closeness centrality values in Fig. 5. Namely, the nodes with higher average
circulation have the lowest closeness centrality values, i.e., the nodes with
less accessibility to the rest of the network. This inverse behaviour is more
evident for the GOA Province from the central subregions to the eastern ones.
The inverse behaviour is clearer for the UK from the central subregions to the
southern ones. Since the number of nodes of the NY State Region Adjacency
graph is considerably larger than the other two graphs, the average node
circulation values variation is also larger, so its visualisation does not
require a quartile partition.
Undirected Region Adjacency graphs
---
| |
(a) GOA Province | (b) UK | (c) NY State
Figure 3: Average node circulation values for Region Adjacency graphs. The
values for a) GOA Province and b) UK correspond to the quartiles partition to
enhance the visualisation.
### 3.3 Traffic graphs and metrics
##### Region Adjacency and Origin-Destination graphs
We now introduce two types of traffic analysis and simulation graphs: the
Region Adjacency graph and the Origin-Destination graph. Given a finite non-
empty compact partition $\mathcal{P}$ of a region $K\subset\mathbb{R}^{2}$,
i.e. a collection $\\{R_{1},\ldots,R_{N}\\}$ of non-empty compact subsets of
$\mathbb{R}^{2}$ such that $\bigcup_{i}R_{i}=K$, we can identify each element
$R_{i}$ of $\mathcal{P}$ with its centroid
$c_{i}=(x_{i},y_{i})\in\mathbb{R}^{2}$. The Region Adjacency graph associated
to $\mathcal{P}$ consists of the undirected unweighted graph with $N$ nodes
embedded in $V=\\{c_{1},\ldots,c_{N}\\}$, and edges
$E=\\{(c_{i},c_{j})\colon\partial R_{i}\cap\partial R_{j}\neq\varnothing\\}$,
where $\partial X$ represents the boundary of the set $X$.
Given an Origin-Destination matrix $M\in\mathbb{R}^{N\times N}$, the Origin-
Destination graph associated to $M$ consists of the directed weighted graph
with weighted adjacency matrix $M$. The $N$ nodes can be embedded in some
$\mathbb{R}^{2}$ representation associated to $M$. For instance, we can
construct an Origin-Destination graph for the New York State using the
predicted flows generated by the Deep Gravity mobility model (Sect. 2.1.3).
The entry $M_{ij}$ of the weighted adjacency matrix $M$ is the predicted flow
from the location indexed by $i$ to the location indexed by $j$. The Origin-
Destination graph $\mathcal{G}$ associated to $M$ (Fig. 4) consists of $2836$
nodes and $939888$ edges, and it is not connected since the Deep Gravity model
predicts flows only for the locations in the same regular partition, i.e. the
same square cell.
The adjacency matrices for the Region Adjacency graphs are highly sparse (Fig.
1). In contrast, the Origin-Destination graph shows more clusters with a more
significant number of connections since the Deep Gravity mobility model
predicts flows for some regions that are not adjacent geographically (Fig.
4a), thus its adjacency matrix (Fig. 4b) has many more square sub-matrices
that are related to the number of different strongly connected components
since the Deep Gravity model predicts flows only for the irregular units
within the same squared cell.
Directed Origin-Destination graph
---
|
(a) NY State Origin-Destination graph | (b) Sparsity matrix
Figure 4: Origin-Destination digraph associated to the predicted OD matrix for
NY State using the Deep Gravity mobility model.
##### Metrics on the Region Adjacency graph
The closeness centrality values of the nodes in a Region Adjacency graph can
have a clustered behaviour showing the subregions that have higher levels of
accessibility to the rest of the subregions in a given planar partition. For
instance, there is a trend of lower levels of accessibility from the
peripherical subregions. In contrast, the central subregions have higher
closeness centrality values (Fig. 5), except for NY State, which has a
significant higher values cluster in the Southeast area, which could be due to
its geometrical shape and the number of subregions in that area. The
betweenness centrality values in the same Region Adjacency graph are, in
general, low for most of the network (Fig. 5), so there are no clusters of
subregions with higher connectivity importance that serve to link other
subregions through the shortest path. Nevertheless, in the NY State Region
Adjacency graph, there are a few nodes with the highest betweenness centrality
values around the ”bottleneck” that connects the Southeast area to the rest of
the network, which means that their removal may have an impact on the shortest
paths between the subregions in the Southeast area and the ones in the rest of
the NY State. Indeed, this difference between the maximum betweenness
centrality values and the values in the rest of the nodes are the ones that
cause the low trend for the normalised betweenness centrality (Fig. 2).
Undirected Region Adjacency graphs
---
| |
Closeness centrality
| |
Betweenness centrality
(a) GOA Province | (b) UK | (c) NY State
Figure 5: Closeness and Betweenness centralities for Region Adjacency graphs.
The closeness values show significant changes depending on their geographical
position, while the betweenness values are nearly constant.
##### Metrics on Origin-Destination graphs
As with the Region Adjacency graphs, the centrality metrics values of the
nodes in an Origin-Destination graph can also reveal some information
regarding the connectivity of the subregions in a given planar partition, and
their flows represented in an Origin-Destination matrix. However, it is
possible that the available flows are not sufficient to generate a strongly
connected graph. Indeed, the Deep Gravity mobility model predicts flows only
for the irregular subregions within the same squared cell, creating as many
strongly connected components as cells exist. The higher values of the
outcloseness and betweenness centralities of the Origin-Destination graph for
New York State (Fig. 6) are localised approximately in the same subregions as
the nodes with higher closeness centrality values in the Region Adjacency
graph of New York State (Fig. 5). This trend occurs because the Origin-
Destination graph of New York State has significantly more nodes in this area,
creating a strongly connected component with a larger number of nodes and
consequently with more accessible nodes and connections, which increases
outcloseness and betweenness centralities, respectively. The Page rank
centrality exhibits a more randomised trend because several strongly connected
components have various nodes. The original values were classified in
quartiles to visualise their distribution better.
Directed Origin-Destination graph
---
| |
(a) Outcloseness centrality | (b) Betweenness centrality | (c) Page rank centrality
Figure 6: Quartiles visualisation of Centrality metrics for the NY State
Origin-Destination graph. The edges are not shown for enhancing the
visualisation of node centrality values.
## 4 Graph matrices for network and traffic analysis
The node degree centrality motivates the definition of the degree matrix,
which associates a value to each node based on the number of its neighbours.
The transition probability matrix represents the connection’s relevance for
each node’s neighbours, as its entries are values that measure the proportion
of information flow across the edges of the graph. Furthermore, the graph
Laplacian allows us to study structure properties, such as connectedness
through eigenvalue problems, and there are different types of Laplacians on a
graph $\mathcal{G}$, depending on the model represented by $\mathcal{G}$, and
also whether the graph is undirected or directed. For connected graphs and
strongly connected graphs, the transition probability matrix has an associated
Perron vector that is used to define several Laplacian matrices and a
circulation function that provides another sense of information flow between
any couple of nodes of the graph.
### 4.1 Degree matrix
If $\mathcal{G}$ is a graph with $N$ nodes (weighted or unweighted, directed
or undirected), then the degree of $v\in V$ is defined as the sum of the
elements of the $v$-th row of the adjacency matrix. Namely, the degree $d_{v}$
of a node $v\in V$ is defined by $d_{v}=\sum_{j=1}^{N}A_{vj}$. Indeed, the
degree of a node is the sum of the weights of the edges joining that node for
undirected graphs and the sum of the weights of the edges outgoing from that
node for digraphs. The degree matrix $\mathbf{D}$ is the diagonal matrix
$\mathbf{D}\in\mathbb{R}^{N\times N}$ given by $D_{ij}=d_{i}$ if $i=j$ and $0$
otherwise.
### 4.2 Transition probability matrix
Let $\mathcal{G}$ be a weighted graph and let $i\in V$ be a node with
$d_{i}>0$ outgoing edges (arriving to the nodes $v_{1},\ldots,v_{d_{i}}$). The
proportion of the flow at node $i$ that will continue through node $k$ is
given by $p_{ik}=A_{ik}/d_{i}$, for $k=v_{1},\ldots,v_{d_{i}}$. On the other
hand, if $d_{i}=0$ then the proportion of outgoing flows from $i$ to $k$ is
simply zero for every $k\in V$. This discussion motivates the definition of
the transition probability matrix $\mathbf{P}$, which defines a Markov chain
associated with random walks on the graph $\mathcal{G}$ (Li and Zhang,, 2012),
where the entry $P_{ij}$ denotes the probability of moving from node $i$ to
node $j$, for $i,j=1,\ldots,N$. However, it is required that there are no
nodes without any edge for undirected graphs or nodes without outgoing edges
(called dead-end nodes, possibly with incoming edges, for directed graphs).
The reason is that if $i\in V$ is a node with such conditions. The proportion
of the flow going from $i$ to $j$ will be zero for any node $j\in V$;
therefore, the whole $i$-th row would be zero. Still, a transition matrix of a
Markov chain must satisfy having the sum of each column equal to $1$ (row
stochastic).
For the sake of theoretical completeness in Markov chains (Li and Zhang,,
2012), we assume that a graph is strongly connected to define its transition
probability matrix. The transition probability matrix $\mathbf{P}$ of the
graph $\mathcal{G}$ is the $N\times N$ matrix that satisfies the relation
$\mathbf{P}=\mathbf{D}^{-1}\mathbf{A}$, where $\mathbf{A}$ and $\mathbf{D}$
are the adjacency and degree matrices, respectively. The entries $p_{ij}$
represent the probability of the flow moving from node $i$ to node $j$ (or the
proportion of flows that moves from $i$ to $j$), for $i,j=1,\ldots,N$. The
matrix $\mathbf{P}$ is sometimes called the normalised adjacency matrix of
$\mathcal{G}$ (Veerman and Lyons,, 2020). From the definition of the degree
matrix $\mathbf{D}$ it follows that
$\sum_{j=1}^{N}P_{ij}=\sum_{j=1}^{N}(\mathbf{D}^{-1}\mathbf{A})_{ij}=\sum_{j=1}^{N}\sum_{k=1}^{N}D^{-1}_{ik}A_{kj}=\sum_{j=1}^{N}\dfrac{1}{d_{i}}A_{ij}=\dfrac{1}{d_{i}}\sum_{j=1}^{N}A_{ij}=1,$
for every $i=1,\ldots,N$, in other words, the sum of all the entries of a row
equals $1$, for every row in $\mathbf{P}$. Indeed, the matrix $\mathbf{P}$ is
row stochastic, and there is a Markov chain associated with random walks on
the graph $\mathcal{G}$ defined by $\mathbf{P}$.
##### Perron vector
For a strongly connected directed graph, the transition probability matrix
$\mathbf{P}$ has a unique left eigenvector $\phi$ with positive components,
namely, $\phi^{\top}\mathbf{P}=\rho\phi^{\top}$. The vector $\phi$ is called
the Perron vector of $\mathbf{P}$, and in fact, it can be easily proven that
$\rho=1$. We notice that $\phi$ is the Perron vector of the transition
probability matrix $\mathbf{P}:=\mathbf{D}^{-1}\mathbf{A}$ if and only if
$\widetilde{\phi}:=\mathbf{D}^{-1}\phi$ is the generalised eigenvector of the
couple $(\mathbf{A},\mathbf{D})$ associated with the eigenvalue $1$, i.e.,
$\mathbf{A}\widetilde{\phi}=\mathbf{D}\widetilde{\phi}$. Indeed, we solve the
generalised eigenproblem and then compute
$\phi:=\mathbf{D}^{-1}\widetilde{\phi}$. Since the eigenvector associated with
the eigenvalue $1$ is unique (as the graph is strongly connected), its entries
are all positive or negative; if negative, we change their sign to guarantee
that the entries of $\phi$ are all positive. Since
$\mathbf{P}\mathbf{1}=\mathbf{1}$, we get that $\rho=1$ is an eigenvalue of
$\mathbf{P}$ and all the eigenvalues of $\mathbf{P}$ are lower than $1$. In
particular, we normalise the entries of $\phi$ such that
$\sum_{i=1}^{n}\phi(i)=1$.
### 4.3 Graph Laplacians
An essential property of the graph Laplacians of an undirected graph is the
symmetry, which follows from the symmetry of the adjacency graph and
guarantees that all its eigenvalues are real numbers. The Combinatorial
Laplacian and the Normalised Laplacian are two possible definitions when the
graph is undirected, being the latter a transformation of the former that
results in an upper bound for its real eigenvalues. For a directed graph, it
is possible to define the Combinatorial Directed Laplacian, Symmetrized
Laplacian, and the Combinatorial Symmetrized Laplacian, which have real
eigenvalues even if the adjacency matrix is not symmetric. The Diplacian is
another viable definition that involves obtaining complex eigenvalues, which,
however, coincides with the Normalised Laplacian when the graph is undirected.
#### 4.3.1 Laplacians of undirected graphs
For an undirected graph $\mathcal{G}$ with $N$ nodes the Combinatorial
Laplacian (Veerman and Lyons,, 2020), is the $N\times N$ matrix defined by
$\mathbf{L}=\mathbf{D}-\mathbf{A}.$ (3)
Since the adjacency matrix of an undirected graph is symmetric, the
Combinatorial Laplacian is also symmetric. Moreover, $\mathbf{D}$ is positive
definite, and it is possible to define a scalar product in $\mathbb{R}^{N}$ by
letting
$\langle\mathbf{x},\mathbf{y}\rangle_{\mathbf{D}}:=\mathbf{x}^{\top}\mathbf{Dy}$,
for $\mathbf{x},\mathbf{y}\in\mathbb{R}^{N}$. The Combinatorial Laplacian is
also called the Kirchhoff matrix of the graph (Caughman and Veerman,, 2006).
(Zhang et al., 2021a, ) defines the normalised graph Laplacian
$\mathbf{\hat{L}}$ by
$\mathbf{\hat{L}}=\mathbf{I}-\mathbf{D}^{-1/2}\mathbf{A}\mathbf{D}^{-1/2}$,
where $\mathbf{I}$ is the $N\times N$ identity matrix, being $N$ the number of
nodes in the graph. The normalised graph Laplacian of an undirected graph is
also symmetric, and it is also positive semidefinite since $\mathbf{D}$ is an
invertible diagonal matrix with positive entries. Furthermore,
$\mathbf{\hat{L}}=\mathbf{D}^{-1/2}\mathbf{LD}^{-1/2}$.
#### 4.3.2 Laplacians of directed graphs
Since the Combinatorial Laplacian of a directed graph is not necessarily
symmetric, (Hasanzadeh et al.,, 2017) defines the Combinatorial Directed
Laplacian of a digraph $\mathcal{G}=(V,E,\mathbf{A})$ by
$\mathbf{L_{G}}=\dfrac{1}{2}(\mathbf{D}_{out}+\mathbf{D}_{in}-\mathbf{A}-\mathbf{A}^{\top}),\quad\mathbf{D}_{out}=\sum_{j=1}^{N}A_{ij},\quad\mathbf{D}_{in}=\sum_{j=1}^{N}A_{ji},$
where $\mathbf{D}_{out},\mathbf{D}_{in}$ are the out-degree and in-degree and
$\mathbf{A}$ is the adjacency matrix. The Combinatorial Directed Laplacian is
symmetric regardless of $\mathcal{G}$ being directed or not. Moreover, if
$\mathcal{G}$ is undirected then $\mathbf{L}=\mathbf{L_{G}}$. The
Combinatorial Directed Laplacian is also positive semi-definite since it can
be seen as the Combinatorial Laplacian of an undirected with adjacency matrix
$\tilde{\mathbf{A}}=(\mathbf{A}+\mathbf{A}^{\top})/2$. Assuming that the input
graph is strongly connected, the unique stationary probability distribution
$\mathbf{\phi}$ of the transition probability matrix $\mathbf{P}$ is defined
as the unique vector $\phi\in\mathbb{R}^{N}$ with strictly positive components
such that $\phi^{\top}\mathbf{P}=\phi^{\top}$. In (Chung,, 2005), the
Symmetrized Laplacian of a strongly connected graph $\mathcal{G}$ is
$\mathbf{\mathcal{L}}=\mathbf{I}-\dfrac{\mathbf{\Phi}^{1/2}\mathbf{P}\mathbf{\Phi}^{-1/2}+\mathbf{\Phi}^{-1/2}\mathbf{P}^{\top}\mathbf{\Phi}^{1/2}}{2},$
where $\mathbf{\Phi}=\textrm{diag}(\phi_{i})$. The Symmetrised Laplacian is
indeed symmetric. (Chung,, 2005) also defines the Combinatorial Symmetrized
Laplacian by
$\mathbf{\mathcal{L}_{G}}=\mathbf{\Phi}-\dfrac{\mathbf{\Phi
P}+\mathbf{P}^{\top}\mathbf{\Phi}}{2},$
which is symmetric and positive semi-definite since it can be written as the
Combinatorial Laplacian of an undirected graph
$\tilde{G}=(V,\tilde{E},\tilde{A})$ with adjacency matrix
$\tilde{\mathbf{A}}=(\mathbf{\Phi P}+\mathbf{P^{\top}\Phi})/2$. The
Combinatorial Symmetrised Laplacian coincides with the Combinatorial Laplacian
defined in (3) when the graph is undirected. However, $\mathbf{\mathcal{L}}$
does not capture the unique characteristic of random walks on digraphs, since
different directed graphs can have the same $\mathbf{\mathcal{L}}$. To
overcome this problem, (Li and Zhang,, 2012) defines the Diplacian
$\mathbf{\Gamma}$ though
$\mathbf{\Gamma}=\mathbf{\Phi}^{1/2}(\mathbf{I}-\mathbf{P})\mathbf{\Phi}^{-1/2},$
for which the strongly connected assumption for the graph $\mathcal{G}$ still
holds since the stationary probabilities are required.
#### 4.3.3 Properties and discussion/comparison
The Laplacian matrix associated with a graph is an essential operator for
network models because it constitutes the foundation for Deep Learning
techniques on graph structures. The symmetry of the Laplacian is a desired
feature even if the graph is directed. Consequently, it is necessary to define
more Laplacian matrices in addition to the classic one in Eq. (3). Some
matrices, such as the Combinatorial Directed Laplacian, the Symmetrised
Laplacian, and the Combinatorial Symmetrised Laplacian, are always symmetric
independently of whether the graph is directed or undirected. Moreover, these
Laplacians are also positive semi-definite operators because they could be
seen as the Combinatorial Laplacian of an undirected graph
$\tilde{G}=(V,\tilde{E})$ with an appropriate choice for the adjacency matrix
$\tilde{\mathbf{A}}$, so the positive semi-definite property of $\mathbf{L}$
is inherited to the new matrices.
Table 2: Summary of the properties of different Laplacians Graph Laplacian | Real Eigenvalues | Positive Semidefinite
---|---|---
Undirected graph
| Combinatorial Laplacian
---
$\mathbf{L}=\mathbf{D}-\mathbf{A}$
Yes | Yes
| Normalised Laplacian
---
$\hat{\mathbf{L}}=\mathbf{I}-\mathbf{D}^{-1/2}\mathbf{AD}^{-1/2}$
Yes | Yes
Directed graph
| Combinatorial Directed Laplacian
---
$\mathbf{L}_{G}=\frac{1}{2}(\mathbf{D}_{out}+\mathbf{D}_{in}-\mathbf{A}-\mathbf{A}^{\top})$
Yes | Yes
| Symmetrized Laplacian
---
$\mathbf{\mathcal{L}}=\mathbf{I}-\dfrac{\mathbf{\Phi}^{1/2}\mathbf{P\Phi}^{-1/2}+\mathbf{\Phi}^{-1/2}\mathbf{P}^{\top}\mathbf{\Phi}^{1/2}}{2}$
Yes | Yes
| Combinatorial Symmetrized Laplacian
---
$\mathbf{\mathcal{L}}_{G}=\mathbf{\Phi}-\dfrac{\mathbf{\Phi}\mathbf{P}+\mathbf{P}^{\top}\mathbf{\Phi}}{2}$
Yes | Yes
| Diplacian
---
$\mathbf{\Gamma}=\mathbf{\Phi}^{1/2}(\mathbf{I}-\mathbf{P})\mathbf{\Phi}^{-1/2}$
No | -
The Combinatorial Laplacian $\mathbf{L}$ of an undirected graph is symmetric
since $\mathbf{D}$ and $\mathbf{A}$ are symmetric. $\mathbf{L}$ is also self-
adjoint since $\mathbf{L}$ is symmetric, in fact, if
$\mathbf{x},\mathbf{y}\in\mathbb{R}^{N}$ then
$\langle\mathbf{x},\mathbf{Ly}\rangle=\langle\mathbf{Lx},\mathbf{y}\rangle\Leftrightarrow\mathbf{x}^{\top}\mathbf{Ly}=(\mathbf{Lx})^{\top}\mathbf{y}=\mathbf{x}^{\top}\mathbf{L}^{\top}\mathbf{y}$.
Additionally, since $\mathbf{L}$ is positive semidefinite, then the Normalised
Laplacian $\hat{\mathbf{L}}$ of an undirected graph is also positive
semidefinite because it can be written as
$\hat{\mathbf{L}}=\mathbf{D}^{-1/2}\mathbf{LD}^{-1/2}$. Rewriting a Laplacian
matrix of a directed graph $\mathcal{G}=(V,E,\mathbf{A})$ as
$\tilde{\mathbf{D}}-\tilde{\mathbf{A}}$, namely, a diagonal matrix
$\tilde{\mathbf{D}}$ whose $i$-th element equals the sum of the $i$-th row of
a non-negative matrix $\tilde{\mathbf{A}}$, then the positive semidefinite
property follows from considering the Laplacian as the one corresponding to an
undirected graph. For instance, for the Combinatorial Directed Laplacian
$\mathbf{L}_{G}$ we can set
$\tilde{\mathbf{A}}=(\mathbf{A}+\mathbf{A}^{\top})/2$ and
$\tilde{\mathbf{D}}=(\mathbf{D}_{out}+\mathbf{D}_{in})/2$, which satisfies
that the sum of the $i$-th row equals $\tilde{D}_{ii}$. Similarly, for the
Combinatorial Symmetrised Laplacian $\mathcal{L}_{G}$ we have that
$\tilde{\mathbf{A}}=(\mathbf{\Phi}\mathbf{P}+\mathbf{P}^{\top}\mathbf{\Phi})/2$
and $\tilde{\mathbf{D}}=\mathbf{\Phi}$. The sum of the $i$-th row of
$\tilde{\mathbf{A}}$ equals $\Phi_{ii}=\phi_{i}$, in fact,
$2\sum_{j=1}^{N}\tilde{A}_{ij}=\sum_{j=1}^{N}(\Phi\mathbf{P})_{ij}+\sum_{j=1}^{N}(\mathbf{P}^{\top}\Phi)_{ij}=\sum_{j=1}^{N}\phi_{i}P_{ij}+\sum_{j=1}^{N}P_{ji}\phi_{j}=2\phi_{i},$
since $\mathbf{P}$ is row stochastic, and by definition the Perron vector
$\mathbf{\phi}$ satisfies the relation
$\mathbf{\phi}^{\top}=\mathbf{\phi}^{\top}\mathbf{P}$. The positive
semidefinite property for the Symmetrised Laplacian follows from the relation
$\mathcal{L}=\mathbf{\Phi}^{-1/2}\mathcal{L}_{G}\mathbf{\Phi}^{-1/2}$. The
graph Laplacians for directed graphs, except for the Diplacian, are generally
symmetric. Consequently, all their eigenvalues are real. Also, they are
bounded for the Symmetrised and the Combinatorial Symmetrised Laplacians (Fig.
7).
Undirected Region Adjacency graphs
---
| |
(a) GOA Province | (b) UK | (c) NY State
Figure 7: Laplacian eigenvalues of Region Adjacency Graphs. The eigenvalues of
the Combinatorial Laplacian are not bounded. The largest eigenvalue (spectral
radius) increases as the number of nodes increases in the Region Adjacency
graph. In contrast, the eigenvalues of the Normalised Laplacian are bounded
regardless of the number of nodes in the graph.
The Diplacian $\mathbf{\Gamma}$ of a directed graph is symmetric if and only
if its adjacency matrix is symmetric. Indeed, we have
$\mathbf{\Gamma}=\mathbf{I}-\mathbf{\Phi}^{1/2}\mathbf{P\Phi}^{-1/2}$.
Assuming $\mathbf{\Gamma}=\mathbf{\Gamma}^{\top}$, then
$\mathbf{\Phi}^{1/2}\mathbf{P\Phi}^{-1/2}=\mathbf{\Phi}^{-1/2}\mathbf{P}^{\top}\mathbf{\Phi}^{1/2}$.
Recalling that $\mathbf{P}=\mathbf{D}^{-1}\mathbf{A}$, $\mathbf{D}$ and
$\mathbf{\Phi}$ are diagonal matrices, using their multiplication
commutativity, and multiplying by $\mathbf{\Phi}^{1/2}$ from the left and from
the right, we obtain
$\mathbf{\Phi}\mathbf{D}^{-1}\mathbf{A}=\mathbf{A}^{\top}\mathbf{D}^{-1}\mathbf{\Phi}=\mathbf{\Phi}\mathbf{D}^{-1}\mathbf{A}^{\top}$,
Since $\mathbf{D}^{-1}$ and $\mathbf{\Phi}$ have an inverse, we get
$\mathbf{A}=\mathbf{A}^{\top}$ as a necessary condition for the symmetry of
the Diplacian. Conversely, if $\mathbf{A}=\mathbf{A}^{\top}$ then
$\mathbf{\Gamma}$ is symmetric. If $\mathbf{A}$ is symmetric, then the
Diplacian reduces to
$\mathbf{I}-\mathbf{P}=\mathbf{I}-\mathbf{D}^{-1/2}\mathbf{AD}^{-1/2}=\hat{\mathbf{L}}$,
i.e. the Normalised Laplacian.
### 4.4 Graph Laplacians and traffic models
The Graph Laplacian can be used to estimate the population of the sub-regions
in a given area for which mobility or traffic flows are known. In (Dees et
al.,, 2021) it is proposed a way to estimate the population distribution
$\phi$ surrounding the stations of the London underground network, by using
Fick’s law of diffusion $\mathbf{q}=-k\nabla\mathbf{\phi}$ which states that
the flux $\mathbf{q}$ flows from regions of high concentration to regions of
low concentration, with a magnitude proportional to the concentration gradient
$\nabla\mathbf{\phi}$, and coefficient of diffusivity $k$. Because of the
graph structure and the discrete nature of the stations, an algebraic
manipulation of the Fick’s law gives the relation
$\mathbf{q}=-k\mathbf{L\phi}$, where $\mathbf{L}$ is the combinatorial
Laplacian of the network. The flux $\mathbf{q}$ was estimated using the
average daily flow of passengers obtained from Transport for London data.
Using the pseudo-inverse matrix $\mathbf{L}^{+}$, the estimation of the
population surrounding each station is obtained through
$\hat{\mathbf{\phi}}=-(1/k)\mathbf{L}^{+}\mathbf{q}$.
## 5 Conclusions, data sets and future work
The data for defining the traffic features include sensors, GPS, rail-hailing,
and transaction data sets. Sensors can be implemented on roads and highways
and collect traffic measurements, such as speed. GPS trajectories are
generated by taxis or rented vehicles during a period. Rail-haling records
include some transport services demand, such as rented bicycles or mobility as
a service option. Transaction data sets are generated by automatic systems,
such as ticket machines in metro stations, which count the number of departing
or arriving passengers.
Graph-based structures allow the analysis of the interplay between the
different elements of the system, as well as the identification of the most
relevant or vulnerable nodes within the whole network. This type of method is
even applicable for some grid-based structures in traffic and mobility
modelling as long as Origin-Destination matrices are available, as in the case
of the Deep Gravity model. Furthermore, node centrality metrics are an
instrument of structure analysis of the network even if there are no available
mobility flows, as in the case of the Region Adjacency graphs. Similarly, the
Laplacian matrix can provide additional reasoning about external information,
e.g., the distribution of the population surrounding the nodes of an urban
network. For connected and strongly connected graphs, the existence and
uniqueness of the Perron vector associated with the transition probability
matrix enables the definition of several Laplacian matrices and constructs a
circulation function in the graph, giving more quantitative information for
analysis and visualisation.
In future work, we plan to address the monitoring of mobility data to
reconstruct traffic flows in urban, regional and inter-regional contexts
through the analysis of origin-destination data, i.e., of mobility
trajectories, considering heterogeneous, partial and uncertain data, and
integrating them with meteorological and pollution data. In particular, it
will be possible to analyse the analysis of short-term mobility patterns to
classify the mobility habits of users, e.g., assessing whether a particular
route/access is regular or sporadic. Furthermore, an implementation of a
graph-based Deep Gravity mobility model with a strongly connected structure is
desired to implement the circulations induced by the Perron vector of the
transition probability matrix, as well as the exploration of alternatives for
the circulation function when the graph is not connected or strongly
connected, for instance, based in node centrality metrics.
Data availability. The Python code, the Region Adjacency data and the
predicted flows using the Deep Gravity mobility model (Simini et al.,, 2021)
for New York State are freely available at github.com/scikit-
mobility/DeepGravity. The Region Adjacency data for the United Kingdom and
Genova Province are freely available at
https://census.ukdataservice.ac.uk/use-data/guides/boundary-data, and
https://smart.comune.genova.it/opendata, respectively.
Declarations of interest. None.
Funding. Rafael Martínez Márquez has been supported by a REACT-EU PhD fellow.
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|
# Pólya’s conjecture for thin products
Xiang He, Zuoqin Wang School of Mathematical Sciences
University of Science and Technology of China
Hefei, 230026
P.R. China
<EMAIL_ADDRESS>School of Mathematical Sciences
University of Science and Technology of China
Hefei, 230026
P.R. China
<EMAIL_ADDRESS>
###### Abstract.
Let $\Omega\subset\mathbb{R}^{d}$ be a bounded Euclidean domain. According to
the famous Weyl law, both its Dirichlet eigenvalue $\lambda_{k}(\Omega)$ and
its Neumann eigenvalue $\mu_{k}(\Omega)$ have the same leading asymptotics
$w_{k}(\Omega)=C(d,\Omega)k^{2/d}$ as $k\to\infty$. G. Pólya conjectured in
1954 that each Dirichlet eigenvalue $\lambda_{k}(\Omega)$ is greater than
$w_{k}(\Omega)$, while each Neumann eigenvalue $\mu_{k}(\Omega)$ is no more
than $w_{k}(\Omega)$. In this paper we prove Pólya’s conjecture for thin
products, i.e. domains of the form $(a\Omega_{1})\times\Omega_{2}$, where
$\Omega_{1},\Omega_{2}$ are Euclidean domains, and $a$ is small enough. We
also prove that the same inequalities hold if $\Omega_{2}$ is replaced by a
Riemannian manifold, and thus get Pólya’s conjecture for a class of “thin”
Riemannian manifolds with boundary.
Partially supported by National Key R and D Program of China 2020YFA0713100,
and by NSFC no. 12171446.
## 1\. Introduction
Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain. Then the Dirichlet
Laplacian on $\Omega$ has discrete spectrum which forms an increasing sequence
of positive numbers (each with finite multiplicity) that tend to infinity,
$0<\lambda_{1}(\Omega)\leq\lambda_{2}(\Omega)\leq\lambda_{3}(\Omega)\leq\cdots\nearrow+\infty,$
and the Neumann Laplacian on $\Omega$ has a similar discrete spectrum (under
suitable boundary regularity assumptions, which we always assume below without
further mentioning)
$0=\mu_{0}(\Omega)\leq\mu_{1}(\Omega)\leq\mu_{2}(\Omega)\leq\cdots\nearrow+\infty.$
Moreover, by a simple variational argument one has
$\mu_{k-1}(\Omega)<\lambda_{k}(\Omega)$ for all $k$, which was strengthened to
(1.1) $\mu_{k}(\Omega)<\lambda_{k}(\Omega),\quad\forall k$
by L. Friedlander in [15] (See also N. Filonov [6]), answering a conjecture of
L. E. Payne [27].
Starting from H. Weyl ([35]), the asymptotic behavior of the eigenvalues
$\lambda_{k}(\Omega)$ and $\mu_{k}(\Omega)$ as $k\to\infty$ has attracted a
lot of attention. In fact, both $\lambda_{k}(\Omega)$ and $\mu_{k}(\Omega)$
admit the same leading term asymptotics
$\lambda_{k}(\Omega)\sim\frac{4\pi^{2}}{(\omega_{d}|\Omega|)^{\frac{2}{d}}}k^{\frac{2}{d}}\quad\text{and}\quad\mu_{k}(\Omega)\sim\frac{4\pi^{2}}{(\omega_{d}|\Omega|)^{\frac{2}{d}}}k^{\frac{2}{d}},$
where $|\Omega|$ represents the volume of $\Omega$, and $\omega_{d}$ is the
volume of the unit ball in $\mathbb{R}^{d}$.
In his classical book [29], G. Pólya conjectured (in a slightly weaker form
for the Neumann case) that for each $k$, the $k^{\mathrm{th}}$ Dirichlet
eigenvalue
(1.2)
$\lambda_{k}(\Omega)\geq\frac{4\pi^{2}}{(\omega_{d}|\Omega|)^{\frac{2}{d}}}k^{\frac{2}{d}}$
while the $k^{\mathrm{th}}$ positive Neumann eigenvalue
(1.3)
$\mu_{k}(\Omega)\leq\frac{4\pi^{2}}{(\omega_{d}|\Omega|)^{\frac{2}{d}}}k^{\frac{2}{d}}.$
As observed by G. Pólya, these conjectured inequalities hold for all
rectangles. As for arbitrary domain, the conjecture holds for $k=1$ (the
Faber-Krahn inequality ([5], [19]) for the Dirichlet eigenvalue, and the
Szegö-Weinberger inequality ([33], [34]) for the Neumann case) and $k=2$ (the
Krahn-Szegö inequality ([20]) for the Dirichlet case, and recently proved by
D. Bucur and A. Henrot in [4] for the Neumann case).
The first major progress on the conjecture was made by G. Pólya himself in
1961 ([30]), in which he presented an elegant proof of his conjecture for
planar tiling domains (in fact G. Pólya’s proof for the Neumann eigenvalue
case was a bit complicated and assumed the tiling to be regular. The
regularity assumption was removed and the proof was simplified in 1966 by R.
Kellner [17]). Very recently, N. Filonov, M. Levitin, I. Polterovich and D.
Sher ([7]) proved that Pólya’s conjecture holds for planar disks (and for
Euclidean balls of all dimensions for the Dirichlet case), and thus gave the
first non-tiling planar domain for which Pólya’s conjecture is known to be
true.
For an arbitrary Euclidean domain $\Omega\subset\mathbb{R}^{d}$, P. Li and
S.T. Yau proved in [23] that
(1.4)
$\sum_{j=1}^{k}\lambda_{k}(\Omega)\geq\frac{d}{d+2}\frac{4\pi^{2}}{(\omega_{d}|\Omega|)^{\frac{2}{d}}}k^{\frac{d+2}{d}},$
and as a consequence, got a weaker version of Pólya’s inequality for all
Dirichlet eigenvalues,
(1.5)
$\lambda_{k}(\Omega)\geq\frac{d}{d+2}\frac{4\pi^{2}}{(\omega_{d}|\Omega|)^{\frac{2}{d}}}k^{\frac{2}{d}}.$
P. Kröger proved in [18] a similar weaker upper bound for the Neumann
eigenvalues of any Euclidean domain with piecewise smooth boundary.
Another important class of domains satisfying Pólya’s conjecture was obtained
by A. Laptev [21], in which he proved that if Pólya’s conjecture (1.2) holds
for $\Omega_{1}\subset\mathbb{R}^{d_{1}}$, where $d_{1}\geq 2$, then Pólya’s
conjecture (1.2) also holds for any domain of the form
$\Omega=\Omega_{1}\times\Omega_{2}$. One key ingredient in his proof is the
following inequality (which is a special case of Berezin-Lieb inequality ([3],
[24]) and is equivalent to Li-Yau’s inequality (1.4) above) for the Riesz
mean,
(1.6)
$\sum_{\lambda_{k}(\Omega)<\lambda}(\lambda-\lambda_{k}(\Omega))^{\gamma}\leq
L_{\gamma,d}|\Omega|\lambda^{\gamma+\frac{d}{2}},$
where $\gamma\geq 1$, and
(1.7)
$L_{\gamma,d}=\frac{\Gamma(\gamma+1)}{(4\pi)^{\frac{d}{2}}\Gamma(\gamma+1+\frac{d}{2})}.$
For Neumann eigenvalues, A. Laptev also got a similar inequality
(1.8) $\sum_{\mu_{k}(\Omega)<\lambda}(\lambda-\mu_{k}(\Omega))^{\gamma}\geq
L_{\gamma,d}|\Omega|\lambda^{\gamma+\frac{d}{2}}$
using which one can get Pólya’s conjecture (1.3) for
$\Omega=\Omega_{1}\times\Omega_{2}$ provided $\Omega_{1}$ satisfies (1.3) and
has dimension $d_{1}\geq 2$. For other recent progresses concerning Pólya’s
conjecture, we refer to [8],[11], [12], [13], [25] etc.
In this paper we will prove Pólya’s conjecture for domains of product type
that are “thin” in one component, namely regions of the form
$\Omega=a\Omega_{1}\times\Omega_{2}$
for $a$ small enough, without assuming that $\Omega_{1}$ or $\Omega_{2}$
satisfies Pólya’s conjecture. We first prove
###### Theorem 1.1.
Let $\Omega_{1}\subset\mathbb{R}^{d_{1}}$ and
$\Omega_{2}\subset\mathbb{R}^{d_{2}}$ be bounded Euclidean domains, where
$d_{1},d_{2}\geq 2$, and $\Omega_{2}$ has piecewise smooth boundary.
1. (1)
If $\Omega_{1}$ has Lipschitz boundary, then there exists $a_{0}>0$ (depends
on $\Omega_{1}$ and $\Omega_{2}$) such that for any $0<a<a_{0}$, the product
$\Omega=a\Omega_{1}\times\Omega_{2}$ satisfies the Dirichlet Pólya’s
conjecture (1.2).
2. (2)
If $\Omega_{1}$ has $C^{1}$ boundary, then there exists $a_{0}>0$ (depends on
$\Omega_{1}$ and $\Omega_{2}$) such that for any $0<a<a_{0}$, the product
$\Omega=a\Omega_{1}\times\Omega_{2}$ satisfies the Neumann Pólya’s conjecture
(1.3).
Here is the strategy of proof: Following Laptev’s argument [21], we write the
eigenvalue counting function of $a\Omega_{1}\times\Omega_{2}$ as the sum of
many eigenvalue counting functions of $\Omega_{2}$. Although we don’t have
Pólya’s inequality for $\Omega_{2}$, we do have weaker inequalities (See (2.6)
and (2.7) below) that follow from Seeley’s version of the two-term Weyl law
(which only requires $\Omega_{2}$ to have piecewise smooth boundary). Now
instead of applying Laptev’s inequalities on Riesz mean above, we apply
stronger two-term inequalities on Riesz mean, namely (2.8) obtained by R.
Frank and S. Larson in [10] (for $\Omega_{1}$ to have Lipschitz boundary) and
its Neumann analogue, (2.9), obtained by R. Frank and L. Geisinger in [9] (for
$\Omega_{1}$ to have $C^{1}$ boundary) to control the sum of both terms in
Seeley’s inequalities. By comparing what we lose from Seeley’s two-term bound
and what we gain from these two-term Riesz mean bound, we are able to prove
that for $a$ small enough, Pólya’s inequalities hold for $\lambda$ large
enough (which depends on $a$). For smaller $\lambda$, we use a simple
observation that Laptev’s inequalities above are in fact strict (See §2.2
below), and thus (by taking $a$ even smaller) give us the demanded gap to
prove Pólya’s inequality. This argument works perfectly well for $d_{2}\geq
3$, but the first part fails for $d_{2}=2$ since in this case we can’t apply
Laptev-type inequality on Riesz mean (which requires
$\gamma=\frac{d_{2}-1}{2}\geq 1$) to control the sum of the second term of
Seeley’s inequality. Fortunately, we can overcome this problem by using Li-
Yau’s estimate (1.5) above and an explicit integral computation. The Neumann
case is a bit simpler, since we only need Weyl’s law to control the second
term in Seeley’s inequality for large $\lambda$, and thus we don’t need to
distinguish the case $d_{2}=2$ with $d_{2}\geq 3$.
Note that Laptev’s argument does not work for the case $d_{1}=1$, since the
inequalities (1.6) and (1.8) require $\gamma\geq 1$. Even though the interval
$(0,1)$ tiles $\mathbb{R}$, it is still not known whether $(0,1)\times\Omega$
satisfies Pólya’s conjecture for general $\Omega$. In the second part of this
paper, we turn to study Pólya’s conjecture for thin products
$(0,a)\times\Omega$. Instead of writing the eigenvalue counting function of
$(0,a)\times\Omega$ as the sum of many eigenvalue counting functions of
$(0,a)$ (which is a tiling domain) that we have a nice control, we will write
it as the sum of many eigenvalue counting functions of $\Omega$ and apply
Seeley’s two-term inequalities. By carefully analyzing the two sums, we shall
prove that in this case, all thin products satisfy Pólya’s conjecture:
###### Theorem 1.2.
Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with piecewise smooth
boundary, then there exists $a_{0}>0$ (depends on $\Omega$) such that for any
$0<a<a_{0}$, $(0,a)\times\Omega$ satisfies Pólya’s conjecture (1.2) and (1.3).
Since scaling will not affect Pólya’s inequalities, we immediately see that
for any bounded Euclidean domain $\Omega$, there exists a constant $C>0$ such
that for all $A>C$, $(0,1)\times A\Omega$ satisfies Pólya’s conjecture.
Unfortunately we still can’t prove Pólya’s conjecture for products of the form
$(0,1)\times a\Omega$ for small $a$, which obviously implies Pólya’s
conjecture for $(0,1)\times\Omega$.
The last part of this paper devoted to Pólya’s inequalities for Riemannian
manifolds with boundary. Although the original conjecture was only proposed
for Euclidean domains, people do study the analogous problem in the more
general Riemannian setting. For example, P. Bérard and G. Besson proved in [2]
that for a 2-dimensional hemisphere (or a quarter of a sphere, or even an
octant of a sphere), both Dirichlet eigenvalues and Neumann eigenvalues
satisfy Pólya’s inequalities above. Recently in [14], P. Freitas, J. Mao and
I. Salavessa studied the problem for hemispheres in arbitrary dimension. They
showed that (1.3) holds for Neumann eigenvalues of hemispheres in any
dimension, while (1.2) fails for Dirichlet eigenvalues when $d>2$, and they
derived sharp inequality for Dirichlet eigenvalues by adding a correction
term.
It is thus a natural problem to find out more Riemannian manifolds with
boundary satisfying Pólya’s inequalities. Note that in the proof of Theorem
1.1 and Theorem 1.2, for $\Omega_{2}$ and $\Omega$ we mainly used Seeley’s
two-term Weyl’s inequality. As a result, by literally repeating the proof one
can easily see that for any closed Riemannian manifold $M$, the Neumann
eigenvalues of the product $a\Omega\times M$ satisfy Pólya conjecture (1.3) as
long as $a$ is small enough. For the Dirichlet case, there will be one extra
term (since $0$ is an eigenvalue of $M$) in the eigenvalue counting function
of the product, namely the number of eigenvalues of $\Omega$ that is less than
$a^{2}\lambda$, which can be explicitly calculated if $d_{1}=\dim\Omega=1$ and
can be controlled via Li-Yau’s estimate (1.5) if $d_{2}\geq 2$. As a result,
we are able to prove that Pólya’s conjecture holds for such Riemannian
manifolds with boundary:
###### Theorem 1.3.
Let $\Omega\subset\mathbb{R}^{d_{1}}$ be a bounded domain with $C^{1}$
boundary and $(M,g)$ be a closed Riemannian manifold of dimension $d_{2}\geq
2$. Then there exists $a_{0}>0$ (depends on $\Omega$ and $M$) such that for
any $0<a<a_{0}$, $a\Omega\times M$ satisfies Pólya’s conjecture (1.2) and
(1.3).
The arrangement of this paper is as follows. In Section 2 we will list the
two-term inequalities for the eigenvalues counting functions and for the Riesz
means that will be used later, and also prove strict version of Laptev’s
inequalities for completeness. In Section 3 we will prove Theorem 1.1, and in
Section 4 we will prove Theorem 1.2. In Section 5 we will turn to the
Riemannian manifold setting and prove Theorem 1.3. Moreover we will explain
how to get similar results for a larger class of eigenvalue problems. Finally
in Section 6 we will give an explicit non-tiling planar domain $\Omega$ and
explicitly calculate the constant involved in the proof, and as a result, show
that the Dirichlet eigenvalues of $[0,\frac{1}{4\pi}]\times\Omega$ for that
$\Omega$ satisfies (1.2). Similarly in the Riemannian setting we show that
$(0,a)\times S^{2}$ satisfies Pólya’s conjecture for $a\leq\frac{\pi}{24}$,
but will break both Pólya’s inequalities if $a$ is large.
## 2\. Some preparations
### 2.1. Two term inequalities for the eigenvalue counting functions and the
Riesz means
For any bounded domain $\Omega\subset\mathbb{R}^{d}$, we denote the Dirichlet
eigenvalue counting function by
$\mathcal{N}^{D}_{\Omega}(\lambda):=\\#\\{n:\ \lambda_{n}(\Omega)<\lambda\\},$
and the Neumann eigenvalue counting function by
$\mathcal{N}^{N}_{\Omega}(\lambda):=\\#\\{n:\ \mu_{n}(\Omega)<\lambda\\}.$
Then the inequality (1.1) implies
$\mathcal{N}^{D}_{\Omega}(\lambda)\leq\mathcal{N}^{N}_{\Omega}(\lambda),\quad\forall\lambda>0,$
while Pólya’s conjectures (1.2) and (1.3) can be restated as
(2.1) $\mathcal{N}^{D}_{\Omega}(\lambda)\leq
C_{d}|\Omega|\lambda^{\frac{d}{2}},\qquad\forall\lambda>0,$
for all bounded domains, and
(2.2) $\mathcal{N}^{N}_{\Omega}(\lambda)\geq
C_{d}|\Omega|\lambda^{\frac{d}{2}},\qquad\forall\lambda>0,$
for all bounded domains with suitable boundary regularity, where the constant
(2.3)
$C_{d}=\frac{\omega_{d}}{(2\pi)^{d}}=\frac{1}{(4\pi)^{\frac{d}{2}}\Gamma(\frac{d}{2}+1)}=L_{0,d}.$
Since the unit balls satisfy $B^{d_{1}+d_{2}}\subset B^{d_{1}}\times
B^{d_{2}}$, one has $\omega_{d_{1}+d_{2}}<\omega_{d_{1}}\cdot\omega_{d_{2}}$
and thus
(2.4) $C_{d_{1}+d_{2}}<C_{d_{1}}\cdot C_{d_{2}}.$
It was first obtained by H. Weyl ([35]) that both eigenvalue counting
functions $\mathcal{N}^{D}_{\Omega}(\lambda)$ and
$\mathcal{N}^{N}_{\Omega}(\lambda)$ have the same leading asymptotics
(2.5)
$\mathcal{N}^{D/N}_{\Omega}(\lambda)=C_{d}|\Omega|\lambda^{\frac{d}{2}}+o(\lambda^{\frac{d}{2}})$
as $\lambda\to\infty$, and the famous Weyl’s conjecture, proven by V. Ivrii
([16]) and R. Melrose ([26]) under extra assumptions on the behavior of
billiard dynamics, claims that for $\Omega\subset\mathbb{R}^{d}$ with
piecewise smooth boundary,
$\mathcal{N}^{D}_{\Omega}(\lambda)=C_{d}|\Omega|\lambda^{\frac{d}{2}}-\frac{1}{4}C_{d-1}|\partial\Omega|\lambda^{\frac{d-1}{2}}+o(\lambda^{\frac{d-1}{2}})$
while
$\mathcal{N}^{N}_{\Omega}(\lambda)=C_{d}|\Omega|\lambda^{\frac{d}{2}}+\frac{1}{4}C_{d-1}|\partial\Omega|\lambda^{\frac{d-1}{2}}+o(\lambda^{\frac{d-1}{2}}),$
where $|\partial\Omega|$ is the surface area of $\partial\Omega$.
Although Weyl’s conjecture was not proven in its full generality, R. Seeley
([31], [32]) proved a weaker version, namely both eigenvalue counting
functions satisfy
$\mathcal{N}^{D/N}_{\Omega}(\lambda)=C_{d}|\Omega|\lambda^{\frac{d}{2}}+\mathrm{O}(\lambda^{\frac{d-1}{2}}),\qquad\text{as\
}\lambda\to\infty,$
for all bounded domains in $\mathbb{R}^{d}$ with piecewise smooth boundary. In
view of the facts $\lambda_{1}(\Omega)>0$ and $\mu_{0}(\Omega)=0$, we see that
there exists a positive constant $C(\Omega)$ such that for any $\lambda>0$,
(2.6) $\mathcal{N}^{D}_{\Omega}(\lambda)\leq
C_{d}|\Omega|\lambda^{\frac{d}{2}}+C(\Omega)\lambda^{\frac{d-1}{2}}$
and
(2.7) $\mathcal{N}^{N}_{\Omega}(\lambda)\geq
C_{d}|\Omega|\lambda^{\frac{d}{2}}-C(\Omega)\lambda^{\frac{d-1}{2}}.$
These two-term inequalities sharpen Weyl’s leading estimates and will play a
crucial role below.
We also need two-term inequalities for the Riesz mean that sharpen Laptev’s
inequalities (1.6) and (1.8). For the Dirichlet case, R. L. Frank and S.
Larson ([10]) proved that for any bounded domain $\Omega$ in $\mathbb{R}^{d}$
($d\geq 2$) with Lipschitz boundary and any $\gamma\geq 1$,
$\sum_{\lambda_{k}(\Omega)<\lambda}(\lambda-\lambda_{k}(\Omega))^{\gamma}=L_{\gamma,d}|\Omega|\lambda^{\gamma+\frac{d}{2}}-\frac{1}{4}L_{\gamma,d-1}|\partial\Omega|\lambda^{\gamma+\frac{d-1}{2}}+\mathrm{o}(\lambda^{\gamma+\frac{d-1}{2}})$
as $\lambda\to\infty$. As a consequence, for fixed $\gamma$, there exists a
positive constant $C_{1}(\Omega)$ such that if $\lambda>C_{1}(\Omega)$, one
has
(2.8)
$\sum_{\lambda_{k}(\Omega)<\lambda}(\lambda-\lambda_{k}(\Omega))^{\gamma}\leq
L_{\gamma,d}|\Omega|\lambda^{\gamma+\frac{d}{2}}-\frac{1}{5}L_{\gamma,d-1}|\partial\Omega|\lambda^{\gamma+\frac{d-1}{2}}.$
For the Neumann case, R. L. Frank and L. Geisinger ([9]) proved that for any
bounded domain $\Omega$ in $\mathbb{R}^{d}$ ($d\geq 2$) with $C^{1}$ boundary
and any $\gamma\geq 1$, one has
$\sum_{\mu_{k}(\Omega)<\lambda}(\lambda-\mu_{k}(\Omega))^{\gamma}=L_{\gamma,d}|\Omega|\lambda^{\gamma+\frac{d}{2}}+\frac{1}{4}L_{\gamma,d-1}|\partial\Omega|\lambda^{\gamma+\frac{d-1}{2}}+\mathrm{o}(\lambda^{\gamma+\frac{d-1}{2}})$
as $\lambda\to\infty$. As a consequence, for fixed $\gamma$, there exists a
positive constant $C_{2}(\Omega)$ such that if $\lambda>C_{2}(\Omega)$, one
has
(2.9) $\sum_{\mu_{k}(\Omega)<\lambda}(\lambda-\mu_{k}(\Omega))^{\gamma}\geq
L_{\gamma,d}|\Omega|\lambda^{\gamma+\frac{d}{2}}+\frac{1}{5}L_{\gamma,d-1}|\partial\Omega|\lambda^{\gamma+\frac{d-1}{2}}.$
### 2.2. Strict inequalities for Riesz means
Another ingredient in our proof is the inequality (1.6) of A. Laptev, which
follows from
$\sum_{\lambda_{k}(\Omega)<\lambda}(\lambda-\lambda_{k}(\Omega))\leq
L_{1,d}|\Omega|\lambda^{\frac{d+2}{2}}.$
By carefully analyzing his proof, one can see that the inequality is strict.
We include the proof for reader’s convenience.
###### Proposition 2.1.
Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$, then for any
$\lambda>0$, one has
$\sum_{\lambda_{k}(\Omega)<\lambda}(\lambda-\lambda_{k}(\Omega))<L_{1,d}|\Omega|\lambda^{\frac{d+2}{2}}.$
###### Proof.
We first sketch Laptev’s proof ([21, Theorem 2.1]). Let
$\\{\varphi_{k}\\}_{k=1}^{\infty}$ be the $L^{2}$ normalized eigenfunctions
associated to $\\{\lambda_{k}(\Omega)\\}_{k=1}^{\infty}$. Then
$\sum_{\lambda_{k}(\Omega)<\lambda}(\lambda-\lambda_{k}(\Omega))\leq(2\pi)^{-d}\int_{\mathbb{R}^{d}}(\lambda-|\xi|^{2})_{+}\cdot\sum_{\lambda_{k}(\Omega)<\lambda}|\hat{\varphi}_{k}(\xi)|^{2}\mathrm{d}\xi.$
Let
$e_{\xi}(x)=\begin{cases}e^{-ix\cdot\xi},&\qquad x\in\Omega,\\\ 0,&\qquad
x\notin\Omega.\end{cases}$
Then
$\sum_{\lambda_{k}(\Omega)<\lambda}|\hat{\varphi}_{k}(\xi)|^{2}=\sum_{\lambda_{k}(\Omega)<\lambda}\langle
e_{\xi},\varphi_{k}\rangle^{2}_{L^{2}(\Omega)}\\\
\leq\|e_{\xi}\|^{2}_{L^{2}(\Omega)}=|\Omega|.$
So
$\sum_{\lambda_{k}(\Omega)<\lambda}(\lambda-\lambda_{k}(\Omega))\leq(2\pi)^{-d}|\Omega|\int_{\mathbb{R}^{d}}(\lambda-|\xi|^{2})_{+}\mathrm{d}\xi=L_{1,d}|\Omega|\lambda^{\frac{d+2}{2}}.$
Now assume by contradiction that
$\sum_{\lambda_{k}(\Omega)<\lambda}(\lambda-\lambda_{k}(\Omega))=L_{1,d}|\Omega|\lambda^{\frac{d+2}{2}}$
for some $\lambda$, then
$\sum_{\lambda_{k}(\Omega)<\lambda}\langle
e_{\xi},\varphi_{k}\rangle^{2}_{L^{2}(\Omega)}=\|e_{\xi}\|^{2}_{L^{2}(\Omega)},\qquad\forall|\xi|^{2}<\lambda.$
Take $N$ such that $\lambda_{N}(\Omega)<\lambda\leq\lambda_{N+1}(\Omega)$,
then one has $e_{\xi}\in\mathrm{span}\\{\varphi_{1},\cdots,\varphi_{N}\\}$ for
any $\xi$ with $|\xi|^{2}<\lambda$. So if we take $N+1$ different points
$\\{\xi_{1},\cdots,\xi_{N+1}\\}$ with $|\xi_{i}|^{2}<\lambda$, $\forall 1\leq
i\leq N+1$, then there exists $(c_{1},\cdots,c_{N+1})\neq(0,\cdots,0)$ such
that
$\sum_{i=1}^{N+1}c_{i}e_{\xi_{i}}(x)=0,\qquad\forall x\in\Omega.$
By the analyticity of $e^{-ix\cdot\xi}$, one has
$\sum_{i=1}^{N+1}c_{i}e^{-ix\cdot\xi_{i}}=0,\qquad\forall x\in\mathbb{R}^{d},$
which is a contradiction. ∎
Together with the fact
(2.10)
$\int^{\infty}_{0}(z-\lambda-t)_{+}t^{\gamma-2}\mathrm{d}t=\frac{(z-\lambda)^{\gamma}_{+}}{\gamma(\gamma-1)},\qquad\forall\gamma>1,$
we immediately get
###### Corollary 2.2.
Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$, then for any $\lambda>0$
and $\gamma\geq 1$, one has
$\sum_{\lambda_{k}(\Omega)<\lambda}(\lambda-\lambda_{k}(\Omega))^{\gamma}<L_{\gamma,d}|\Omega|\lambda^{\gamma+\frac{d}{2}}.$
In particular, for any $0<a<b$, one has
(2.11) $K(a,b,\Omega):=\inf_{a\leq\lambda\leq
b}\frac{L_{\gamma,d}|\Omega|\lambda^{\gamma+\frac{d}{2}}-\sum_{\lambda_{k}(\Omega)<\lambda}(\lambda-\lambda_{k}(\Omega))^{\gamma}}{\lambda^{\gamma+\frac{d-1}{2}}}>0.$
We also need the Neumann version of the same (strict) inequality, which is
essentially proved by A. Laptev (c.f. [21], Theorem 3.1):
###### Proposition 2.3.
Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ with discrete Neumann
spectrum. Then for any $\lambda\geq 0$, one has
$\sum_{\mu_{k}(\Omega)<\lambda}(\lambda-\mu_{k}(\Omega))>L_{1,d}|\Omega|\lambda^{\frac{d+2}{2}}.$
###### Proof.
Again we first sketch Laptev’s proof of the corresponding (non-strict)
inequality in [21]. Let $\\{\psi_{k}\\}_{k=0}^{\infty}$ be the $L^{2}$
normalized eigenfunctions associated to $\\{\mu_{k}(\Omega)\\}_{k=0}^{\infty}$
and denote $\varphi_{\lambda}(x)=(\lambda-x)_{+}$. Then
$\sum_{\mu_{k}(\Omega)<\lambda}(\lambda-\mu_{k}(\Omega))=(2\pi)^{-d}|\Omega|\int_{\mathbb{R}^{d}}\sum_{k=0}^{\infty}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}\cdot\varphi_{\lambda}(\mu_{k}(\Omega))\mathrm{d}\xi.$
Using the facts
$\sum_{k=0}^{\infty}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}=1\quad\text{and}\quad\sum_{k=0}^{\infty}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}\cdot\mu_{k}=|\xi|^{2}$
and the Jensen inequality (since $\varphi_{\lambda}$ is convex), Laptev proved
$\sum_{k=0}^{\infty}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}\cdot\varphi_{\lambda}(\mu_{k}(\Omega))\geq\varphi_{\lambda}(|\xi|^{2})$
and thus
$\sum_{\mu_{k}(\Omega)<\lambda}(\lambda-\mu_{k}(\Omega))\geq
L_{1,d}|\Omega|\lambda^{\frac{d+2}{2}}.$
To show that the inequality is strict, again we fix $\lambda$ and let $N$ be
the integer with $\mu_{N}(\Omega)\leq\lambda<\mu_{N+1}(\Omega)$. Take $\xi$
with $|\xi|^{2}<\lambda$ such that
$\langle e_{\xi},1\rangle_{L^{2}(\Omega)}\neq 0,\text{ and
}e_{\xi}\notin\mathrm{span}\\{\psi_{1},\cdots,\psi_{N}\\},$
then for such a $\xi$, Jensen’s inequality is strict:
$\displaystyle\sum_{k=0}^{\infty}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}\cdot\varphi_{\lambda}(\mu_{k}(\Omega))$
$\displaystyle=$
$\displaystyle\sum_{\mu_{k}(\Omega)<\lambda}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}\cdot\varphi_{\lambda}(\mu_{k}(\Omega))+\sum_{\mu_{k}(\Omega)>\lambda}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}\cdot\varphi_{\lambda}(\mu_{k}(\Omega))$
$\displaystyle=$
$\displaystyle(\sum_{\mu_{k}(\Omega)<\lambda}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)})\cdot\varphi_{\lambda}(\frac{\sum_{\mu_{k}(\Omega)<\lambda}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}\cdot\mu_{k}(\Omega)}{\sum_{\mu_{k}(\Omega)<\lambda}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}})+$
$\displaystyle(\sum_{\mu_{k}(\Omega)>\lambda}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)})\cdot\varphi_{\lambda}(\frac{\sum_{\mu_{k}(\Omega)>\lambda}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}\cdot\mu_{k}(\Omega)}{\sum_{\mu_{k}(\Omega)>\lambda}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}})$ $\displaystyle>$
$\displaystyle\varphi_{\lambda}(\sum_{\mu_{k}(\Omega)<\lambda}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}\cdot\mu_{k}(\Omega)+\sum_{\mu_{k}(\Omega)>\lambda}\frac{1}{|\Omega|}\langle
e_{\xi},\psi_{k}\rangle^{2}_{L^{2}(\Omega)}\cdot\mu_{k}(\Omega))$
$\displaystyle=$ $\displaystyle\varphi_{\lambda}(|\xi|^{2}).$
This completes the proof. ∎
As a consequence,
###### Corollary 2.4.
Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ with discrete Neumann
spectrum. Then for any $\lambda\geq 0$ and $\gamma\geq 1$, one has
$\sum_{\mu_{k}(\Omega)<\lambda}(\lambda-\mu_{k}(\Omega))^{\gamma}>L_{\gamma,d}|\Omega|\lambda^{\gamma+\frac{d}{2}}.$
In particular, for any $0<a<b$, one has
$K_{1}(a,b,\Omega):=\inf_{a\leq\lambda\leq
b}\frac{\sum_{\mu_{k}(\Omega)<\lambda}(\lambda-\mu_{k}(\Omega))^{\gamma}-L_{\gamma,d}|\Omega|\lambda^{\gamma+\frac{d}{2}}}{\lambda^{\gamma+\frac{d-1}{2}}}>0.$
## 3\. Proof of Theorem 1.1
As observed by P. Freitas, J. Lagace and J. Payette in [12, Proposition 3.1],
it is enough to assume that both $\Omega_{1}$ and $\Omega_{2}$ are connected.
We divide the proof of Theorem 1.1 into three parts: the Dirichlet case with
$d_{2}\geq 3$, the Dirichlet case with $d_{2}=2$, and the Neumann case.
For the Dirichlet case, the eigenvalues of $a\Omega_{1}\times\Omega_{2}$ are
$a^{-2}\lambda_{l}(\Omega_{1})+\lambda_{k}(\Omega_{2}),\qquad\forall
l,k\in\mathbb{Z}_{>0}$
and thus
$\mathcal{N}^{D}_{a\Omega_{1}\times\Omega_{2}}(\lambda)=\sum_{l=1}^{Z^{\lambda}_{a}}\mathcal{N}^{D}_{\Omega_{2}}(\lambda-a^{-2}\lambda_{l}(\Omega_{1})),$
where
$Z^{\lambda}_{a}=\mathcal{N}^{D}_{\Omega_{1}}(a^{2}\lambda).$
By inequality (2.6), there exists a constant $C(\Omega_{2})>0$ such that
$\mathcal{N}^{D}_{\Omega_{2}}(\lambda)\leq
C_{d_{2}}|\Omega_{2}|\lambda^{\frac{d_{2}}{2}}+C(\Omega_{2})\lambda^{\frac{d_{2}-1}{2}},\qquad\forall\lambda>0.$
So we get
(3.1)
$\displaystyle\sum_{l=1}^{Z^{\lambda}_{a}}\mathcal{N}^{D}_{\Omega_{2}}(\lambda-a^{-2}\lambda_{l}(\Omega_{1}))$
$\displaystyle\leq$ $\displaystyle
C_{d_{2}}|\Omega_{2}|\sum_{l=1}^{Z^{\lambda}_{a}}(\lambda-a^{-2}\lambda_{l}(\Omega_{1}))^{\frac{d_{2}}{2}}+C(\Omega_{2})\sum_{l=1}^{Z^{\lambda}_{a}}(\lambda-a^{-2}\lambda_{l}(\Omega_{1}))^{\frac{d_{2}-1}{2}}$
$\displaystyle=$ $\displaystyle
C_{d_{2}}|\Omega_{2}|a^{-d_{2}}\sum_{l=1}^{Z^{\lambda}_{a}}(a^{2}\lambda-\lambda_{l}(\Omega_{1}))^{\frac{d_{2}}{2}}+C(\Omega_{2})a^{1-d_{2}}\sum_{l=1}^{Z^{\lambda}_{a}}(a^{2}\lambda-\lambda_{l}(\Omega_{1}))^{\frac{d_{2}-1}{2}}.$
By inequality (2.8), there exists a constant $C(\Omega_{1})>0$ such that if
$a^{2}\lambda>C(\Omega_{1})$, then
(3.2)
$\sum_{l=1}^{Z^{\lambda}_{a}}(a^{2}\lambda-\lambda_{l}(\Omega_{1}))^{\frac{d_{2}}{2}}\leq
L_{\frac{d_{2}}{2},d_{1}}|\Omega_{1}|a^{d_{1}+d_{2}}\lambda^{\frac{d_{1}+d_{2}}{2}}-\frac{1}{5}L_{\frac{d_{2}}{2},d_{1}-1}|\partial\Omega_{1}|a^{d_{1}+d_{2}-1}\lambda^{\frac{d_{1}+d_{2}-1}{2}}.$
### 3.1. The Dirichlet case with $d_{2}\geq 3$
By Corollary 2.2, one has
(3.3)
$\sum_{l=1}^{Z^{\lambda}_{a}}(a^{2}\lambda-\lambda_{l}(\Omega_{1}))^{\frac{d_{2}-1}{2}}<L_{\frac{d_{2}-1}{2},d_{1}}|\Omega_{1}|a^{d_{1}+d_{2}-1}\lambda^{\frac{d_{1}+d_{2}-1}{2}}.$
So by (3.1), (3.2), (3.3) and the fact
$C_{d_{2}}L_{\frac{d_{2}}{2},d_{1}}=C_{d_{1}+d_{2}},$
one has that if $a^{2}\lambda>C(\Omega_{1})$, then
$\displaystyle\sum_{l=1}^{Z^{\lambda}_{a}}\mathcal{N}^{D}_{\Omega_{2}}(\lambda-a^{-2}\lambda_{l}(\Omega_{1}))$
$\displaystyle\leq$ $\displaystyle
C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+d_{2}}{2}}-\frac{1}{5}C_{d_{1}+d_{2}-1}|\partial\Omega_{1}||\Omega_{2}|a^{d_{1}-1}\lambda^{\frac{d_{1}+d_{2}-1}{2}}$
$\displaystyle+L_{\frac{d_{2}-1}{2},d_{1}}|\Omega_{1}|C(\Omega_{2})a^{d_{1}}\lambda^{\frac{d_{1}+d_{2}-1}{2}}.$
Thus if we assume
$a<\frac{C_{d_{1}+d_{2}-1}|\partial\Omega_{1}||\Omega_{2}|}{5L_{\frac{d_{2}-1}{2},d_{1}}|\Omega_{1}|C(\Omega_{2})}=\frac{C_{d_{2}-1}|\partial\Omega_{1}||\Omega_{2}|}{5|\Omega_{1}|C(\Omega_{2})},$
then for any $\lambda>a^{-2}C(\Omega_{1})$, we will get the demanded
inequality
$\mathcal{N}^{D}_{a\Omega_{1}\times\Omega_{2}}(\lambda)\leq
C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+d_{2}}{2}}.$
Note that if $0<\lambda<a^{-2}\lambda_{1}(\Omega_{1})$, then we automatically
have
$\mathcal{N}^{D}_{a\Omega_{1}\times\Omega_{2}}(\lambda)=0<C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+d_{2}}{2}}.$
So it remains to consider the case
$a^{-2}\lambda_{1}(\Omega_{1})\leq\lambda\leq a^{-2}C(\Omega_{1})$ assuming
$C(\Omega_{1})>\lambda_{1}(\Omega_{1})$. Let $\mu=a^{2}\lambda$, then by
Corollary 2.2, one has
$K(\Omega_{1})=\inf_{\lambda_{1}(\Omega_{1})\leq\mu\leq
C(\Omega_{1})}\frac{L_{\frac{d_{2}}{2},d_{1}}|\Omega_{1}|\mu^{\frac{d_{1}+d_{2}}{2}}-\sum_{\lambda_{l}(\Omega_{1})<\mu}(\mu-\lambda_{l}(\Omega_{1}))^{\frac{d_{2}}{2}}}{\mu^{\frac{d_{1}+d_{2}-1}{2}}}>0$
and thus
(3.4)
$\sum_{\lambda_{l}(\Omega_{1})<\mu}(\mu-\lambda_{l}(\Omega_{1}))^{\frac{d_{2}}{2}}\leq
L_{\frac{d_{2}}{2},d_{1}}|\Omega_{1}|\mu^{\frac{d_{1}+d_{2}}{2}}-K(\Omega_{1})\mu^{\frac{d_{1}+d_{2}-1}{2}}$
for all $\lambda_{1}(\Omega_{1})\leq\mu\leq C(\Omega_{1})$. Thus by (3.1),
(3.3) and (3.4), one has that if $a^{-2}\lambda_{1}(\Omega_{1})\leq\lambda\leq
a^{-2}C(\Omega_{1})$, then
$\displaystyle\sum_{l=1}^{Z^{\lambda}_{a}}\mathcal{N}^{D}_{\Omega_{2}}(\lambda-a^{-2}\lambda_{l}(\Omega_{1}))$
$\displaystyle\leq$ $\displaystyle
C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+d_{2}}{2}}-K(\Omega_{1})C_{d_{2}}|\Omega_{2}|a^{d_{1}-1}\lambda^{\frac{d_{1}+d_{2}-1}{2}}$
$\displaystyle+L_{\frac{d_{2}-1}{2},d_{1}}|\Omega_{1}|C(\Omega_{2})a^{d_{1}}\lambda^{\frac{d_{1}+d_{2}-1}{2}}.$
So if we assume
$a<\frac{K(\Omega_{1})C_{d_{2}}|\Omega_{2}|}{L_{\frac{d_{2}-1}{2},d_{1}}|\Omega_{1}|C(\Omega_{2})}$,
then for any $a^{-2}\lambda_{1}(\Omega_{1})\leq\lambda\leq
a^{-2}C(\Omega_{1})$,
$\mathcal{N}^{D}_{a\Omega_{1}\times\Omega_{2}}(\lambda)\leq
C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+d_{2}}{2}}.$
Combining all discussions above, one has that if
$a<\min\bigg{(}\frac{C_{d_{2}-1}|\partial\Omega_{1}||\Omega_{2}|}{5|\Omega_{1}|C(\Omega_{2})},\frac{K(\Omega_{1})C_{d_{2}}|\Omega_{2}|}{L_{\frac{d_{2}-1}{2},d_{1}}|\Omega_{1}|C(\Omega_{2})}\bigg{)},$
then all Dirichlet eigenvalues of $a\Omega_{1}\times\Omega_{2}$ satisfy
Pólya’s conjecture (2.1). This complete the proof of Theorem 1.1, part (1),
for the case of $d_{2}\geq 3$.
### 3.2. The Dirichlet case with $d_{2}=2$
Since $C_{2}=\frac{1}{4\pi}$, the inequality (3.1) becomes
(3.5)
$\displaystyle\sum_{l=1}^{Z^{\lambda}_{a}}\mathcal{N}^{D}_{\Omega_{2}}(\lambda-a^{-2}\lambda_{l}(\Omega_{1}))$
$\displaystyle\leq$
$\displaystyle(4\pi)^{-1}|\Omega_{2}|a^{-2}\sum_{l=1}^{Z^{\lambda}_{a}}(a^{2}\lambda-\lambda_{l}(\Omega_{1}))+C(\Omega_{2})a^{-1}\sum_{l=1}^{Z^{\lambda}_{a}}(a^{2}\lambda-\lambda_{l}(\Omega_{1}))^{\frac{1}{2}}$
and the inequality (3.2) gives, for $a^{2}\lambda>C(\Omega_{1})$,
(3.6) $\sum_{l=1}^{Z^{\lambda}_{a}}(a^{2}\lambda-\lambda_{l}(\Omega_{1}))\leq
L_{1,d_{1}}|\Omega_{1}|a^{d_{1}+2}\lambda^{\frac{d_{1}+2}{2}}-\frac{1}{5}L_{1,d_{1}-1}|\partial\Omega_{1}|a^{d_{1}+1}\lambda^{\frac{d_{1}+1}{2}}.$
To estimate the second term in (3.5), we use Li-Yau’s lower bound (1.5),
namely
$\lambda_{l}(\Omega_{1})\geq\frac{d_{1}}{d_{1}+2}l^{\frac{2}{d_{1}}}(C_{d_{1}}|\Omega_{1}|)^{-\frac{2}{d_{1}}},$
to get
(3.7)
$\displaystyle\sum_{l=1}^{Z^{\lambda}_{a}}(a^{2}\lambda-\lambda_{l}(\Omega_{1}))^{\frac{1}{2}}\leq$
$\displaystyle\sum_{l=1}^{Z^{\lambda}_{a}}\big{(}a^{2}\lambda-\frac{d_{1}}{d_{1}+2}l^{\frac{2}{d_{1}}}(C_{d_{1}}|\Omega_{1}|)^{-\frac{2}{d_{1}}}\big{)}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\infty}\big{(}a^{2}\lambda-\frac{d_{1}}{d_{1}+2}x^{\frac{2}{d_{1}}}(C_{d_{1}}|\Omega_{1}|)^{-\frac{2}{d_{1}}}\big{)}_{+}^{\frac{1}{2}}\mathrm{d}x$
$\displaystyle=$
$\displaystyle(\frac{d_{1}+2}{d_{1}})^{\frac{d_{1}}{2}}C_{d_{1}}|\Omega_{1}|\frac{d_{1}}{2}a^{d_{1}+1}\lambda^{\frac{d_{1}+1}{2}}\int_{0}^{1}(1-s)^{\frac{1}{2}}s^{\frac{d_{1}}{2}-1}\mathrm{d}s$
$\displaystyle=$
$\displaystyle(\frac{d_{1}+2}{d_{1}})^{\frac{d_{1}}{2}}L_{\frac{1}{2},d_{1}}|\Omega_{1}|a^{d_{1}+1}\lambda^{\frac{d_{1}+1}{2}}.$
Then by (3.5), (3.6) and (3.7), one has that if $a^{2}\lambda>C(\Omega_{1})$,
then
$\displaystyle\sum_{l=1}^{Z^{\lambda}_{a}}\mathcal{N}^{D}_{\Omega_{2}}(\lambda-a^{-2}\lambda_{l}(\Omega_{1}))$
$\displaystyle\leq$ $\displaystyle
C_{d_{1}+2}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+2}{2}}-\frac{1}{5}C_{d_{1}+1}|\Omega_{2}||\partial\Omega_{1}|a^{d_{1}-1}\lambda^{\frac{d_{1}+1}{2}}+$
$\displaystyle(\frac{d_{1}+2}{d_{1}})^{\frac{d_{1}}{2}}C(\Omega_{2})L_{\frac{1}{2},d_{1}}|\Omega_{1}|a^{d_{1}}\lambda^{\frac{d_{1}+1}{2}}.$
So if we assume
$a<\frac{C_{d_{1}+1}|\Omega_{2}||\partial\Omega_{1}|}{5(\frac{d_{1}+2}{d_{1}})^{\frac{d_{1}}{2}}C(\Omega_{2})L_{\frac{1}{2},d_{1}}|\Omega_{1}|}=\frac{1}{5\pi}\big{(}\frac{d_{1}}{d_{1}+2}\big{)}^{\frac{d_{1}}{2}}\frac{|\Omega_{2}||\partial\Omega_{1}|}{C(\Omega_{2})|\Omega_{1}|},$
then for any $\lambda>a^{-2}C(\Omega_{1})$, one also gets the demanded
inequality
$\mathcal{N}^{D}_{a\Omega_{1}\times\Omega_{2}}(\lambda)\leq
C_{d_{1}+2}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+2}{2}}.$
For $\lambda<a^{-2}C(\Omega_{1})$, one just repeat the corresponding part of
the proof of the Dirichlet case with $d_{2}\geq 3$, so we omit it. This
completes the proof of Theorem 1.1, (1).
### 3.3. The Neumann case
Since the Neumann eigenvalues of $a\Omega_{1}\times\Omega_{2}$ are
$a^{-2}\mu_{l}(\Omega_{1})+\mu_{k}(\Omega_{2}),\qquad\forall
l,k\in\mathbb{Z}_{\geq 0},$
one has
$\mathcal{N}^{N}_{a\Omega_{1}\times\Omega_{2}}(\lambda)=\sum_{l=0}^{Y_{a}^{\lambda}}\mathcal{N}^{N}_{\Omega_{2}}(\lambda-a^{-2}\mu_{l}(\Omega_{1}))$
where
$Y_{a}^{\lambda}=\mathcal{N}^{N}_{\Omega_{1}}(a^{2}\lambda)-1.$
By inequality (2.7), there exists a constant $C_{1}(\Omega_{2})>0$ such that
$\mathcal{N}^{N}_{\Omega_{2}}(\lambda)\geq
C_{d_{2}}|\Omega_{2}|\lambda^{\frac{d_{2}}{2}}-C_{1}(\Omega_{2})\lambda^{\frac{d_{2}-1}{2}},\qquad\forall\lambda>0.$
So we get
(3.8)
$\displaystyle\sum_{l=0}^{Y_{a}^{\lambda}}\mathcal{N}^{N}_{\Omega_{2}}(\lambda-a^{-2}\mu_{l}(\Omega_{1}))$
$\displaystyle\geq$ $\displaystyle
C_{d_{2}}|\Omega_{2}|\sum_{l=0}^{Y_{a}^{\lambda}}(\lambda-a^{-2}\mu_{l}(\Omega_{1}))^{\frac{d_{2}}{2}}-C_{1}(\Omega_{2})\sum_{l=0}^{Y_{a}^{\lambda}}(\lambda-a^{-2}\mu_{l}(\Omega_{1}))^{\frac{d_{2}-1}{2}}$
$\displaystyle=$ $\displaystyle
C_{d_{2}}|\Omega_{2}|a^{-d_{2}}\sum_{l=0}^{Y_{a}^{\lambda}}(a^{2}\lambda-\mu_{l}(\Omega_{1}))^{\frac{d_{2}}{2}}-C_{1}(\Omega_{2})a^{1-d_{2}}\sum_{l=0}^{Y_{a}^{\lambda}}(a^{2}\lambda-\mu_{l}(\Omega_{1}))^{\frac{d_{2}-1}{2}}.$
For the first term, by (2.9), there exists a constant $C_{1}(\Omega_{1})>0$
such that if $a^{2}\lambda>C_{1}(\Omega_{1})$, then
(3.9)
$\sum_{l=0}^{Y_{a}^{\lambda}}(a^{2}\lambda-\mu_{l}(\Omega_{1}))^{\frac{d_{2}}{2}}\geq
L_{\frac{d_{2}}{2},d_{1}}|\Omega_{1}|a^{d_{1}+d_{2}}\lambda^{\frac{d_{1}+d_{2}}{2}}+\frac{1}{5}L_{\frac{d_{2}}{2},d_{1}-1}|\partial\Omega_{1}|a^{d_{1}+d_{2}-1}\lambda^{\frac{d_{1}+d_{2}-1}{2}}.$
To estimate the second term, we use (2.5) to get $L=L(\Omega_{1})>0$ such that
$\mu_{l}(\Omega_{1})\geq\frac{1}{2}l^{\frac{2}{d_{1}}}(C_{d_{1}}|\Omega_{1}|)^{-\frac{2}{d_{1}}},\text{
if }l\geq L.$
Note that if $a^{2}\lambda$ is large enough, one has
$\sum_{l=L}^{3L-1}\big{(}a^{2}\lambda-\frac{1}{2}l^{\frac{2}{d_{1}}}(C_{d_{1}}|\Omega_{1}|)^{-\frac{2}{d_{1}}}\big{)}^{\frac{d_{2}-1}{2}}\geq
L(a^{2}\lambda)^{\frac{d_{2}-1}{2}}\geq\sum_{l=0}^{L-1}(a^{2}\lambda-\mu_{l}(\Omega_{1}))^{\frac{d_{2}-1}{2}}.$
So there exists a constant $C_{2}(\Omega_{1})>0$ such that if
$a^{2}\lambda>C_{2}(\Omega_{1})$, then
(3.10)
$\displaystyle\sum_{l=0}^{Y_{a}^{\lambda}}(a^{2}\lambda-\mu_{l}(\Omega_{1}))^{\frac{d_{2}-1}{2}}$
$\displaystyle\leq$ $\displaystyle
2\sum_{l=0}^{Y_{a}^{\lambda}}\big{(}a^{2}\lambda-\frac{1}{2}l^{\frac{2}{d_{1}}}(C_{d_{1}}|\Omega_{1}|)^{-\frac{2}{d_{1}}}\big{)}^{\frac{d_{2}-1}{2}}$
$\displaystyle=$ $\displaystyle
2(a^{2}\lambda)^{\frac{d_{2}-1}{2}}+2\sum_{l=1}^{Y_{a}^{\lambda}}\big{(}a^{2}\lambda-\frac{1}{2}l^{\frac{2}{d_{1}}}(C_{d_{1}}|\Omega_{1}|)^{-\frac{2}{d_{1}}}\big{)}^{\frac{d_{2}-1}{2}}$
$\displaystyle\leq$ $\displaystyle
4\int_{0}^{\infty}\big{(}a^{2}\lambda-\frac{1}{2}x^{\frac{2}{d_{1}}}(C_{d_{1}}|\Omega_{1}|)^{-\frac{2}{d_{1}}}\big{)}_{+}^{\frac{d_{2}-1}{2}}\mathrm{d}x$
$\displaystyle=$ $\displaystyle
2^{\frac{d_{1}}{2}+2}C_{d_{1}}|\Omega_{1}|\frac{d_{1}}{2}a^{d_{1}+d_{2}-1}\lambda^{\frac{d_{1}+d_{2}-1}{2}}\int_{0}^{1}(1-s)^{\frac{d_{2}-1}{2}}s^{\frac{d_{1}}{2}-1}\mathrm{d}s$
$\displaystyle=$ $\displaystyle
2^{\frac{d_{1}}{2}+1}C_{d_{1}}B(\frac{d_{1}}{2},\frac{d_{2}+1}{2})|\Omega_{1}|d_{1}a^{d_{1}+d_{2}-1}\lambda^{\frac{d_{1}+d_{2}-1}{2}}.$
Thus by (3.8), (3.9) and (3.10), one has that if
$a^{2}\lambda>\max(C_{1}(\Omega_{1}),C_{2}(\Omega_{1}))$, then
$\displaystyle\sum_{l=0}^{Y^{\lambda}_{a}}\mathcal{N}^{N}_{\Omega_{2}}(\lambda-a^{-2}\mu_{l}(\Omega_{1}))$
$\displaystyle\geq$ $\displaystyle
C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+d_{2}}{2}}+\frac{1}{5}C_{d_{1}+d_{2}-1}|\Omega_{2}||\partial\Omega_{1}|a^{d_{1}-1}\lambda^{\frac{d_{1}+d_{2}-1}{2}}-$
$\displaystyle
C_{1}(\Omega_{2})2^{\frac{d_{1}}{2}+1}C_{d_{1}}B(\frac{d_{1}}{2},\frac{d_{2}+1}{2})|\Omega_{1}|d_{1}a^{d_{1}}\lambda^{\frac{d_{1}+d_{2}-1}{2}}.$
So if we require
$a<\frac{C_{d_{1}+d_{2}-1}|\Omega_{2}||\partial\Omega_{1}|}{5C_{1}(\Omega_{2})2^{\frac{d_{1}}{2}+1}C_{d_{1}}B(\frac{d_{1}}{2},\frac{d_{2}+1}{2})|\Omega_{1}|d_{1}}=\frac{C_{d_{2}-1}|\Omega_{2}||\partial\Omega_{1}|}{5\cdot
2^{\frac{d_{1}}{2}+2}C_{1}(\Omega_{2})|\Omega_{1}|},$
then for any $\lambda>a^{-2}\max(C_{1}(\Omega_{1}),C_{2}(\Omega_{1}))$, one
gets the demanded
$\mathcal{N}^{N}_{a\Omega_{1}\times\Omega_{2}}(\lambda)\geq
C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+d_{2}}{2}}.$
Next, we consider $0<\lambda<a^{-2}\mu_{1}(\Omega_{1})$, in which case
$\mathcal{N}^{N}_{a\Omega_{1}\times\Omega_{2}}(\lambda)=\mathcal{N}^{N}_{\Omega_{2}}(\lambda).$
By Szegö-Weinberger inequality ([33], [34]), one has
$\mu_{1}(\Omega_{1})\leq(C_{d_{1}}|\Omega_{1}|)^{-\frac{2}{d_{1}}}$
which implies that for $0<\lambda<a^{-2}\mu_{1}(\Omega_{1})$,
$\displaystyle
C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+d_{2}}{2}}$
$\displaystyle<C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|(a^{-2}\mu_{1}(\Omega_{1}))^{\frac{d_{1}}{2}}\lambda^{\frac{d_{2}}{2}}$
$\displaystyle\leq\frac{C_{d_{1}+d_{2}}}{C_{d_{1}}}|\Omega_{2}|\lambda^{\frac{d_{2}}{2}}.$
On the other hand, by (2.4) one has
$\frac{C_{d_{1}+d_{2}}}{C_{d_{1}}}<C_{d_{2}}$. So by (2.5), there exists a
constant $C_{2}(\Omega_{2})>0$ such that for $\lambda>C_{2}(\Omega_{2})$,
$\mathcal{N}^{N}_{\Omega_{2}}(\lambda)>\frac{C_{d_{1}+d_{2}}}{C_{d_{1}}}|\Omega_{2}|\lambda^{\frac{d_{2}}{2}}.$
Thus if $a^{-2}\mu_{1}(\Omega_{1})>\lambda>C_{2}(\Omega_{2})$, one gets
$\mathcal{N}^{N}_{a\Omega_{1}\times\Omega_{2}}(\lambda)=\mathcal{N}^{N}_{\Omega_{2}}(\lambda)>\frac{C_{d_{1}+d_{2}}}{C_{d_{1}}}|\Omega_{2}|\lambda^{\frac{d_{2}}{2}}\geq
C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+d_{2}}{2}}.$
For $0<\lambda\leq C_{2}(\Omega_{2})$, we only need to require
$a<\big{(}C_{d_{1}+d_{2}}|\Omega_{1}||\Omega_{2}|C_{2}(\Omega_{2})^{\frac{d_{1}+d_{2}}{2}}\big{)}^{-\frac{1}{d_{1}}}=:C(\Omega_{1},\Omega_{2}),$
to get
$C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+d_{2}}{2}}<1\leq\mathcal{N}^{N}_{a\Omega_{1}\times\Omega_{2}}(\lambda).$
It remains to consider the case $a^{-2}\mu_{1}(\Omega_{1})\leq\lambda\leq
a^{-2}\max(C_{1}(\Omega_{1}),C_{2}(\Omega_{1}))$ assuming
$\max(C_{1}(\Omega_{1}),C_{2}(\Omega_{1}))>\mu_{1}(\Omega_{1})$. Let
$\mu=a^{2}\lambda$, then by Corollary 2.4, one has
(3.11)
$K_{1}(\Omega_{1})=\inf_{\mu_{1}(\Omega_{1})\leq\mu\leq\max(C_{1}(\Omega_{1}),C_{2}(\Omega_{1}))}\frac{\underset{\mu_{l}(\Omega_{1})<\mu}{\sum}(\mu-\mu_{l}(\Omega_{1}))^{\frac{d_{2}}{2}}-L_{\frac{d_{2}}{2},d_{1}}|\Omega_{1}|\mu^{\frac{d_{1}+d_{2}}{2}}}{\mu^{\frac{d_{1}+d_{2}-1}{2}}}>0.$
Let
(3.12)
$K_{2}(\Omega_{1}):=\sup_{\mu_{1}(\Omega_{1})\leq\mu\leq\max(C_{1}(\Omega_{1}),C_{2}(\Omega_{1}))}\frac{\underset{\mu_{l}(\Omega_{1})<\mu}{\sum}(\mu-\mu_{l}(\Omega_{1}))^{\frac{d_{2}-1}{2}}}{\mu^{\frac{d_{1}+d_{2}-1}{2}}}>0.$
Then by (3.8) (3.11) and (3.12), for $a^{-2}\mu_{1}(\Omega_{1})\leq\lambda\leq
a^{-2}\max(C_{1}(\Omega_{1}),C_{2}(\Omega_{1}))$ one has
$\displaystyle\sum_{l=0}^{Y_{a}^{\lambda}}\mathcal{N}^{N}_{\Omega_{2}}(\lambda-a^{-2}\mu_{l}(\Omega_{1}))$
$\displaystyle\geq$ $\displaystyle
C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+d_{2}}{2}}+C_{d_{2}}|\Omega_{2}|K_{1}(\Omega_{1})a^{d_{1}-1}\lambda^{\frac{d_{1}+d_{2}-1}{2}}-$
$\displaystyle
C_{1}(\Omega_{2})K_{2}(\Omega_{1})a^{d_{1}}\lambda^{\frac{d_{1}+d_{2}-1}{2}}.$
Thus if we assume
$a<\frac{C_{d_{2}}|\Omega_{2}|K_{1}(\Omega_{1})}{C_{1}(\Omega_{2})K_{2}(\Omega_{1})}$
then for any $a^{-2}\mu_{1}(\Omega_{1})\leq\lambda\leq
a^{-2}\max(C_{1}(\Omega_{1}),C_{2}(\Omega_{1}))$, one gets the demanded
inequality
$\mathcal{N}^{N}_{a\Omega_{1}\times\Omega_{2}}(\lambda)\geq
C_{d_{1}+d_{2}}a^{d_{1}}|\Omega_{1}||\Omega_{2}|\lambda^{\frac{d_{1}+d_{2}}{2}}.$
Thus we conclude that for
$\displaystyle
a<\min\bigg{(}\frac{C_{d_{2}-1}|\Omega_{2}||\partial\Omega_{1}|}{5\cdot
2^{\frac{d_{1}}{2}+2}C_{1}(\Omega_{2})|\Omega_{1}|},\frac{C_{d_{2}}|\Omega_{2}|K_{1}(\Omega_{1})}{C_{1}(\Omega_{2})K_{2}(\Omega_{1})},C(\Omega_{1},\Omega_{2})\bigg{)},$
all Neumann eigenvalues of $a\Omega_{1}\times\Omega_{2}$ satisfies Pólya’s
conjecture (2.2). $\hfill\square$
## 4\. Proof of Theorem 1.2
### 4.1. Two elementary lemmas
Before proving Theorem 1.2, we give two elementary lemmas that will play
important roles later.
###### Lemma 4.1.
Let $f_{d}(x)=(\lambda-\frac{x^{2}\pi^{2}}{a^{2}})^{\frac{d}{2}}$, then
1. $\mathrm{(1)}$
$f_{d}$ is decreasing on $(0,\frac{a\sqrt{\lambda}}{\pi})$.
2. $\mathrm{(2)}$
If $d\geq 3$, $f_{d}$ is concave on
$(0,\sqrt{\frac{\lambda}{d-1}}\frac{a}{\pi})$ and is convex on
$(\sqrt{\frac{\lambda}{d-1}}\frac{a}{\pi},\frac{a\sqrt{\lambda}}{\pi})$.
3. $\mathrm{(3)}$
$\int_{0}^{\frac{a\sqrt{\lambda}}{\pi}}f_{d}(x)\mathrm{d}x=a\cdot\frac{C_{d+1}}{C_{d}}\lambda^{\frac{d+1}{2}}$.
###### Proof.
(1) is trivial. (2) follows from
$f_{d}^{\prime\prime}(x)=\frac{\pi^{2}d}{a^{2}}(\lambda-\frac{x^{2}\pi^{2}}{a^{2}})^{\frac{d}{2}-2}((d-1)\frac{\pi^{2}x^{2}}{a^{2}}-\lambda),$
and (3) is also elementary:
$\displaystyle\int_{0}^{\frac{a\sqrt{\lambda}}{\pi}}f_{d}(x)\mathrm{d}x=\lambda^{\frac{d+1}{2}}\frac{a}{\pi}\int_{0}^{1}(1-t^{2})^{\frac{d}{2}}\mathrm{d}t$
$\displaystyle=$
$\displaystyle\lambda^{\frac{d+1}{2}}\frac{a}{\pi}\int_{0}^{\frac{\pi}{2}}(\cos\theta)^{d+1}\mathrm{d}\theta=\lambda^{\frac{d+1}{2}}\frac{a}{\pi}\cdot\frac{\Gamma(\frac{1}{2})\Gamma(\frac{d}{2}+1)}{2\Gamma(\frac{d+1}{2}+1)}=a\cdot\frac{C_{d+1}}{C_{d}}\lambda^{\frac{d+1}{2}}.$
∎
The second lemma is
###### Lemma 4.2.
Let
(4.1) $M_{a}^{\lambda}=\lfloor{\frac{a\sqrt{\lambda}}{\pi}}\rfloor,$
then for $\lambda\geq\frac{\pi^{2}}{a^{2}}$ (i.e. $M_{a}^{\lambda}\geq 1$), we
have
(4.2)
$\sum_{l=1}^{M_{a}^{\lambda}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})\leq\frac{2a\lambda^{\frac{3}{2}}}{3\pi}-\frac{\lambda}{8}-\frac{\sqrt{\lambda}\pi}{12a}$
and
(4.3)
$\sum_{l=0}^{M_{a}^{\lambda}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})\geq\frac{2a}{3\pi}\lambda^{\frac{3}{2}}+\frac{1}{12}\lambda.$
###### Proof.
We have
$\sum_{l=1}^{M_{a}^{\lambda}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})=\lambda
M_{a}^{\lambda}-\frac{\pi^{2}}{3a^{2}}(M_{a}^{\lambda})^{3}-\frac{\pi^{2}}{2a^{2}}(M_{a}^{\lambda})^{2}-\frac{\pi^{2}}{6a^{2}}M_{a}^{\lambda}.$
Let
(4.4) $g(x)=\lambda x-\frac{\pi^{2}}{3a^{2}}x^{3},$
then $g^{\prime}(x)=\lambda-\frac{\pi^{2}}{a^{2}}x^{2}$ which is positive if
$x\in(0,\frac{a\sqrt{\lambda}}{\pi})$. So
$g(M^{\lambda}_{a})\leq
g(\frac{a\sqrt{\lambda}}{\pi})=\frac{2a\lambda^{\frac{3}{2}}}{3\pi}.$
Combining with the fact
$\lfloor{x}\rfloor\geq\frac{x}{2},\qquad\forall x\geq 1,$
one gets (4.2).
Similarly,
$\sum_{l=0}^{M_{a}^{\lambda}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})=\lambda+\lambda
M_{a}^{\lambda}-\frac{\pi^{2}}{3a^{2}}(M_{a}^{\lambda})^{3}-\frac{\pi^{2}}{a^{2}}\cdot\frac{3(M_{a}^{\lambda})^{2}+M_{a}^{\lambda}}{6}.$
Again consider the function $g(x)$ defined in (4.4). Since $g^{\prime}(x)$ is
positive and monotonically decreasing on $(0,\frac{a\sqrt{\lambda}}{\pi})$,
one has
$\frac{2a\lambda^{\frac{3}{2}}}{3\pi}-(\lambda
M_{a}^{\lambda}-\frac{\pi^{2}}{3a^{2}}(M_{a}^{\lambda})^{3})=g(\frac{a\sqrt{\lambda}}{\pi})-g(M_{a}^{\lambda})\leq
g^{\prime}(M_{a}^{\lambda})=\lambda-\frac{\pi^{2}}{a^{2}}(M_{a}^{\lambda})^{2},$
which implies (4.3). ∎
Now we start to prove Theorem 1.2. Again by [12, Proposition 3.1], it is
enough to assume that $\Omega$ is connected. Since any rectangle in
$\mathbb{R}^{2}$ satisfies Pólya’s conjecture, we can assume that the
dimension $d$ of $\Omega$ is at least 2. Again we argue by treating $d\geq 3$
and $d=2$ separately, and by treating Dirichlet case and Neumann case
separately.
### 4.2. The Dirichlet case with $d=2$
The Dirichlet eigenvalues of $(0,a)\times\Omega$ are
$\frac{l^{2}\pi^{2}}{a^{2}}+\lambda_{k}(\Omega),\qquad l,k\in\mathbb{Z}_{>0},$
and thus
(4.5)
$\mathcal{N}^{D}_{(0,a)\times\Omega}(\lambda)=\sum_{l=1}^{M_{a}^{\lambda}}\mathcal{N}^{D}_{\Omega}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}}).$
Note that if $0<\lambda<\frac{\pi^{2}}{a^{2}}$, then
$\mathcal{N}^{D}_{(0,a)\times\Omega}(\lambda)=0<C_{3}a|\Omega|\lambda^{\frac{3}{2}}.$
So one only need to consider the case $\lambda\geq\frac{\pi^{2}}{a^{2}}$, i.e.
$M_{a}^{\lambda}\geq 1$. By inequality (2.6), for any $\lambda>0$, there
exists a constant $C(\Omega)>0$ such that
$\mathcal{N}^{D}_{\Omega}(\lambda)\leq\frac{|\Omega|}{4\pi}\lambda+C(\Omega)\lambda^{\frac{1}{2}}.$
In view of (4.2) and the fact
$\sum_{l=1}^{M_{a}^{\lambda}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})^{\frac{1}{2}}\leq\int_{0}^{\frac{a\sqrt{\lambda}}{\pi}}(\lambda-\frac{x^{2}\pi^{2}}{a^{2}})^{\frac{1}{2}}\mathrm{d}x=\frac{a}{4}\lambda$
we get
$\displaystyle\sum_{l=1}^{M_{a}^{\lambda}}\mathcal{N}^{D}_{\Omega}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})\leq$
$\displaystyle\frac{|\Omega|}{4\pi}\sum_{l=1}^{M_{a}^{\lambda}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})+C(\Omega)\sum_{l=1}^{M_{a}^{\lambda}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\frac{a|\Omega|\lambda^{\frac{3}{2}}}{6\pi^{2}}-\frac{|\Omega|\lambda}{32\pi}+\frac{C(\Omega)a}{4}\lambda.$
Thus if we assume $a<\frac{|\Omega|}{8\pi C(\Omega)}$, then
$\mathcal{N}^{D}_{(0,a)\times\Omega}(\lambda)\leq\frac{a|\Omega|\lambda^{\frac{3}{2}}}{6\pi^{2}}=C_{3}a|\Omega|\lambda^{\frac{3}{2}}.$
This completes the proof of the Dirichlet case with $d=2$.
### 4.3. The Neumann case with $d=2$
For the Neumann case, the eigenvalues of $(0,a)\times\Omega$ are
$\frac{l^{2}\pi^{2}}{a^{2}}+\mu_{k}(\Omega),\qquad l,k\in\mathbb{Z}_{\geq 0},$
thus
(4.6)
$\mathcal{N}^{N}_{(0,a)\times\Omega}(\lambda)=\sum_{l=0}^{M_{a}^{\lambda}}\mathcal{N}^{N}_{\Omega}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}}).$
By inequality (2.6), for any $\lambda>0$, there exists $C(\Omega)>0$ such that
$\mathcal{N}^{N}_{\Omega}(\lambda)\geq
C_{d}|\Omega|\lambda-C(\Omega)\lambda^{\frac{1}{2}}.$
In view of (4.3) and the fact
$\sum_{l=0}^{M_{a}^{\lambda}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})^{\frac{1}{2}}\leq\lambda^{\frac{1}{2}}+\int_{0}^{\frac{a\sqrt{\lambda}}{\pi}}(\lambda-\frac{x^{2}\pi^{2}}{a^{2}})^{\frac{1}{2}}\mathrm{d}x=\lambda^{\frac{1}{2}}+\frac{a}{4}\lambda$
we get, for $\lambda\geq\frac{\pi^{2}}{a^{2}}$,
$\displaystyle\sum_{l=0}^{M_{a}^{\lambda}}\mathcal{N}^{N}_{\Omega}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})\geq$
$\displaystyle\frac{|\Omega|}{4\pi}\sum_{l=0}^{M_{a}^{\lambda}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})-C(\Omega)\sum_{l=0}^{M_{a}^{\lambda}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})^{\frac{1}{2}}$
$\displaystyle\geq$
$\displaystyle\frac{a|\Omega|\lambda^{\frac{3}{2}}}{6\pi^{2}}+\frac{|\Omega|\lambda}{48\pi}-C(\Omega)(\lambda^{\frac{1}{2}}+\frac{a\lambda}{4}).$
So if we assume $a\leq\frac{|\Omega|}{96C(\Omega)}$, then
$\lambda\geq\frac{\pi^{2}}{a^{2}}\geq(\frac{96\pi C(\Omega)}{|\Omega|})^{2}$
and thus
$\frac{C(\Omega)a\lambda}{4}\leq\frac{|\Omega|\lambda}{4\cdot
96}<\frac{|\Omega|\lambda}{96\pi}\quad\text{and}\quad
C(\Omega)\lambda^{\frac{1}{2}}\leq\frac{|\Omega|\lambda}{96\pi}.$
In other words, if we assume $a\leq\frac{|\Omega|}{96C(\Omega)}$, then for any
$\lambda\geq\frac{\pi^{2}}{a^{2}}$,
$\mathcal{N}^{N}_{(0,a)\times\Omega}(\lambda)\geq\frac{a|\Omega|\lambda^{\frac{3}{2}}}{6\pi^{2}}=C_{3}a|\Omega|\lambda^{\frac{3}{2}}.$
Next, if $0<\lambda<\frac{\pi^{2}}{a^{2}}$, then
$\mathcal{N}^{N}_{(0,a)\times\Omega}(\lambda)=\mathcal{N}^{N}_{\Omega}(\lambda),\quad\text{
and }\quad C_{3}a|\Omega|\lambda^{\frac{3}{2}}<C_{3}\pi|\Omega|\lambda.$
By (2.4), one has $C_{3}\pi<C_{2}$. So by (2.5), there exists a constant
$C_{1}(\Omega)>0$, such that if $\lambda\geq C_{1}(\Omega)$, then
$\mathcal{N}^{N}_{\Omega}(\lambda)>C_{3}\pi|\Omega|\lambda.$
Thus for $C_{1}(\Omega)\leq\lambda<\frac{\pi^{2}}{a^{2}}$, one gets
$\mathcal{N}^{N}_{(0,a)\times\Omega}(\lambda)=\mathcal{N}^{N}_{\Omega}(\lambda)>C_{3}\pi|\Omega|\lambda>C_{3}a|\Omega|\lambda^{\frac{3}{2}}.$
Finally for $0<\lambda\leq C_{1}(\Omega)$, we may require
$a\leq(C_{3}|\Omega|)^{-1}C_{1}(\Omega)^{-\frac{3}{2}}$ to get
$C_{3}a|\Omega|\lambda^{\frac{3}{2}}\leq
1\leq\mathcal{N}^{N}_{(0,a)\times\Omega}(\lambda).$
Combining all discussions above, one has that if
(4.7)
$a<\min\bigg{(}\frac{|\Omega|}{96C(\Omega)},(C_{3}|\Omega|)^{-1}C_{1}(\Omega)^{-\frac{3}{2}}\bigg{)},$
then all Neumann eigenvalues of $(0,a)\times\Omega$ satisfy Pólya’s conjecture
(1.3). Thus we complete the proof of Theorem 1.2 with $d=2$.
### 4.4. The Dirichlet case with $d\geq 3$
Again we have
$\mathcal{N}^{D}_{(0,a)\times\Omega}(\lambda)=\sum_{l=1}^{M_{a}^{\lambda}}\mathcal{N}^{D}_{\Omega}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})$
and there exists a constant $C(\Omega)>0$ such that
$\mathcal{N}^{D}_{\Omega}(\lambda)\leq
C_{d}|\Omega|\lambda^{\frac{d}{2}}+C(\Omega)\lambda^{\frac{d-1}{2}}.$
So we get
(4.8) $\mathcal{N}^{D}_{(0,a)\times\Omega}(\lambda)\leq
C_{d}|\Omega|\sum_{l=1}^{M_{a}^{\lambda}}f_{d}(l)+C(\Omega)\sum_{l=1}^{M_{a}^{\lambda}}f_{d-1}(l),$
where $f_{d}$ is defined in Lemma 4.1. We split the first sum into two parts.
Denote
$N_{a}^{\lambda}=\lfloor\sqrt{\frac{\lambda}{d-1}}\frac{a}{\pi}\rfloor.$
By concavity of $f_{d}$ (see (2) of Lemma 4.1), one has
$\int_{0}^{N_{a}^{\lambda}}f_{d}(x)\mathrm{d}x-\sum_{l=1}^{N_{a}^{\lambda}}f_{d}(l)\geq\sum_{l=0}^{N_{a}^{\lambda}-1}f_{d}(l)-\int_{0}^{N_{a}^{\lambda}}f_{d}(x)\mathrm{d}x$
which implies
$\sum_{l=1}^{N_{a}^{\lambda}}f_{d}(l)\leq\int_{0}^{N_{a}^{\lambda}}f_{d}(x)\mathrm{d}x-\frac{1}{2}\bigg{(}\lambda^{\frac{d}{2}}-f_{d}(N_{a}^{\lambda})\bigg{)}.$
First consider $\lambda\geq\frac{(d-1)\pi^{2}}{a^{2}}$, in which case
$N_{a}^{\lambda}\geq\frac{1}{2}\frac{a}{\pi}\sqrt{\frac{\lambda}{d-1}}$, and
thus
$f_{d}(N_{a}^{\lambda})\leq
f_{d}(\frac{1}{2}\frac{a}{\pi}\sqrt{\frac{\lambda}{d-1}})=(\frac{4d-5}{4d-4})^{\frac{d}{2}}\lambda^{\frac{d}{2}}.$
For simplicity, we denote
$A_{d}=\frac{1}{2}\bigg{(}1-\big{(}\frac{4d-5}{4d-4}\big{)}^{\frac{d}{2}}\bigg{)}C_{d}|\Omega|.$
Then we get, for $\lambda\geq\frac{(d-1)\pi^{2}}{a^{2}}$,
$\displaystyle\mathcal{N}^{D}_{(0,a)\times\Omega}(\lambda)\leq$ $\displaystyle
C_{d}|\Omega|\sum_{l=1}^{N_{a}^{\lambda}}f_{d}(l)+C_{d}|\Omega|\sum_{l=N_{a}^{\lambda}+1}^{M_{a}^{\lambda}}f_{d}(l)+C(\Omega)\sum_{l=1}^{M_{a}^{\lambda}}f_{d-1}(l)$
$\displaystyle\leq$ $\displaystyle
C_{d}|\Omega|\int_{0}^{{\frac{a\sqrt{\lambda}}{\pi}}}f_{d}(x)\mathrm{d}x-A_{d}\lambda^{\frac{d}{2}}+C(\Omega)\int_{0}^{{\frac{a\sqrt{\lambda}}{\pi}}}f_{d-1}(x)\mathrm{d}x$
$\displaystyle=$ $\displaystyle
C_{d+1}a|\Omega|\lambda^{\frac{d+1}{2}}-A_{d}\lambda^{\frac{d}{2}}+C(\Omega)\frac{C_{d}a}{C_{d-1}}\lambda^{\frac{d}{2}}.$
So if we assume $a\leq\frac{A_{d}\cdot C_{d-1}}{C(\Omega)\cdot C_{d}}$, then
for any $\lambda\geq\frac{(d-1)\pi^{2}}{a^{2}_{1}}$,
(4.9) $\mathcal{N}^{D}_{(0,a)\times\Omega}(\lambda)\leq
C_{d+1}a|\Omega|\lambda^{\frac{d+1}{2}}.$
Note that if $0<\lambda<\frac{\pi^{2}}{a^{2}}$, then we automatically have
$\mathcal{N}^{D}_{(0,a)\times\Omega}(\lambda)=0<C_{d+1}a|\Omega|\lambda^{\frac{d+1}{2}},$
so it remains to consider the case
$\frac{\pi^{2}}{a^{2}}\leq\lambda<\frac{(d-1)\pi^{2}}{a^{2}}$. Let
$\mu=\frac{\lambda a^{2}}{\pi^{2}}$, then $1\leq\mu<d-1$. Let
$H_{1}:=\inf_{1\leq\mu<d-1}\frac{\int^{\sqrt{\mu}}_{0}(\mu-x^{2})^{\frac{d}{2}}\mathrm{d}x-\sum_{0<l^{2}<\mu}(\mu-l^{2})^{\frac{d}{2}}}{\mu^{\frac{d}{2}}}>0,$
then
$\displaystyle\mathcal{N}^{D}_{(0,a)\times\Omega}(\lambda)\leq$ $\displaystyle
C_{d}|\Omega|\int_{0}^{{\frac{a\sqrt{\lambda}}{\pi}}}f_{d}(x)\mathrm{d}x-C_{d}|\Omega|H_{1}\lambda^{\frac{d}{2}}+C(\Omega)\int_{0}^{{\frac{a\sqrt{\lambda}}{\pi}}}f_{d-1}(x)\mathrm{d}x$
$\displaystyle=$ $\displaystyle
C_{d+1}a|\Omega|\lambda^{\frac{d+1}{2}}-C_{d}|\Omega|H_{1}\lambda^{\frac{d}{2}}+C(\Omega)\frac{C_{d}a}{C_{d-1}}\lambda^{\frac{d}{2}}.$
So if we assume $a\leq\frac{C_{d-1}|\Omega|H_{1}}{C(\Omega)}$, then for any
$\frac{\pi^{2}}{a^{2}}\leq\lambda<\frac{(d-1)\pi^{2}}{a^{2}}$,
$\mathcal{N}^{D}_{(0,a)\times\Omega}(\lambda)\leq
C_{d+1}a|\Omega|\lambda^{\frac{d+1}{2}}.$
Combining with (4.9), one gets that if
$a\leq\min\bigg{(}\frac{A_{d}\cdot C_{d-1}}{C(\Omega)\cdot
C_{d}},\frac{C_{d-1}|\Omega|H_{1}}{C(\Omega)}\bigg{)},$
then all Dirichlet eigenvalues of $(0,a)\times\Omega$ satisfy Pólya’s
conjecture (1.2).
### 4.5. The Neumann case with $d\geq 3$
Again we have
$\mathcal{N}^{N}_{(0,a)\times\Omega}(\lambda)=\sum_{l=0}^{M_{a}^{\lambda}}\mathcal{N}^{N}_{\Omega}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})$
and there exists a constant $C(\Omega)>0$ such that
$\mathcal{N}^{N}_{\Omega}(\lambda)\geq
C_{d}|\Omega|\lambda^{\frac{d}{2}}-C(\Omega)\lambda^{\frac{d-1}{2}},\qquad\forall\lambda>0.$
So we get
(4.10) $\mathcal{N}^{N}_{(0,a)\times\Omega}(\lambda)\geq
C_{d}|\Omega|\sum_{l=0}^{M_{a}^{\lambda}}f_{d}(l)-C(\Omega)\sum_{l=0}^{M_{a}^{\lambda}}f_{d-1}(l).$
By convexity of $f_{d}$ (see (2) of Lemma 4.1), one has
$\sum_{l=N_{a}^{\lambda}+1}^{M_{a}^{\lambda}}f_{d}(l)-\int_{N_{a}^{\lambda}+1}^{\frac{a\sqrt{\lambda}}{\pi}}f_{d}(x)\mathrm{d}x\geq\int_{N_{a}^{\lambda}+1}^{\frac{a\sqrt{\lambda}}{\pi}}f_{d}(x)\mathrm{d}x-\sum_{l=N_{a}^{\lambda}+2}^{M_{a}^{\lambda}}f_{d}(l)$
which implies
$\sum_{l=N_{a}^{\lambda}+1}^{M_{a}^{\lambda}}f_{d}(l)\geq\int_{N_{a}^{\lambda}+1}^{\frac{a\sqrt{\lambda}}{\pi}}f_{d}(x)\mathrm{d}x+\frac{1}{2}f_{d}(N_{a}^{\lambda}+1).$
If $\lambda\geq\frac{9\pi^{2}(d-1)}{a^{2}}$, then $N_{a}^{\lambda}\geq 3$ and
thus $N_{a}^{\lambda}+1\leq\frac{4a}{3\pi}\sqrt{\frac{\lambda}{d-1}}$, which
implies
$f_{d}(N_{a}^{\lambda}+1)\geq
f_{d}(\frac{4a}{3\pi}\sqrt{\frac{\lambda}{d-1}})\geq
3^{-{d}}\lambda^{\frac{d}{2}},$
where we used $d\geq 3$. For simplicity, we denote
$B_{d}=\frac{1}{2}3^{-d}C_{d}|\Omega|.$
Then for $\lambda\geq\frac{9\pi^{2}(d-1)}{a^{2}}$,
$\displaystyle\mathcal{N}^{N}_{(0,a)\times\Omega}(\lambda)\geq$ $\displaystyle
C_{d}|\Omega|\sum_{l=0}^{N_{a}^{\lambda}}f_{d}(l)+C_{d}|\Omega|\sum_{l=N_{a}^{\lambda}+1}^{M_{a}^{\lambda}}f_{d}(l)-C(\Omega)\sum_{l=0}^{M_{a}^{\lambda}}f_{d-1}(l)$
$\displaystyle\geq$ $\displaystyle
C_{d}|\Omega|\int_{0}^{\frac{a\sqrt{\lambda}}{\pi}}\\!\\!\\!f_{d}(x)\mathrm{d}x+B_{d}\lambda^{\frac{d}{2}}-C(\Omega)\big{(}\lambda^{\frac{d-1}{2}}+\int_{0}^{{\frac{a\sqrt{\lambda}}{\pi}}}\\!\\!\\!f_{d-1}(x)\mathrm{d}x\big{)}$
$\displaystyle=$ $\displaystyle
C_{d+1}a|\Omega|\lambda^{\frac{d+1}{2}}+B_{d}\lambda^{\frac{d}{2}}-C(\Omega)\big{(}\lambda^{\frac{d-1}{2}}+\frac{C_{d}a}{C_{d-1}}\lambda^{\frac{d}{2}}\big{)}.$
So if we assume
$a\leq\min\bigg{(}\frac{B_{d}C_{d-1}}{2C(\Omega)C_{d}},\frac{B_{d}3\pi\sqrt{d-1}}{2C(\Omega)}\bigg{)},$
then
$\lambda\geq\frac{9\pi^{2}(d-1)}{a^{2}}\geq\frac{4C(\Omega)^{2}}{B_{d}^{2}}$
and thus
$\frac{C(\Omega)C_{d}a}{C_{d-1}}\lambda^{\frac{d}{2}}\leq\frac{1}{2}B_{d}\lambda^{\frac{d}{2}}\quad\text{and}\quad
C(\Omega)\lambda^{\frac{d-1}{2}}\leq\frac{1}{2}B_{d}\lambda^{\frac{d}{2}}.$
Thus for any $\lambda\geq\frac{9\pi^{2}(d-1)}{a^{2}}$, one gets the demanded
inequality
$\mathcal{N}^{N}_{(0,a)\times\Omega}(\lambda)\geq
C_{d+1}a|\Omega|\lambda^{\frac{d+1}{2}}.$
Next if $\frac{\pi^{2}}{a^{2}}\leq\lambda\leq\frac{9\pi^{2}(d-1)}{a^{2}}$, let
$\mu=\frac{\lambda a^{2}}{\pi^{2}}$, then $1\leq\mu\leq 9(d-1)$. Let
$H_{2}=\inf_{1\leq\mu\leq 9(d-1)}\frac{\sum_{0\leq
l^{2}<\mu}(\mu-l^{2})^{\frac{d}{2}}-\int_{0}^{\sqrt{\mu}}(\mu-x^{2})^{\frac{d}{2}}\mathrm{d}x}{\mu^{\frac{d}{2}}}>0,$
then
$\displaystyle\mathcal{N}^{N}_{(0,a)\times\Omega}(\lambda)\geq$ $\displaystyle
C_{d}|\Omega|\int_{0}^{\frac{a\sqrt{\lambda}}{\pi}}\\!\\!\\!f_{d}(x)\mathrm{d}x+C_{d}|\Omega|H_{2}\lambda^{\frac{d}{2}}-C(\Omega)\big{(}\lambda^{\frac{d-1}{2}}+\int_{0}^{{\frac{a\sqrt{\lambda}}{\pi}}}\\!\\!\\!f_{d-1}(x)\mathrm{d}x\big{)}$
$\displaystyle=$ $\displaystyle
C_{d+1}a|\Omega|\lambda^{\frac{d+1}{2}}+C_{d}|\Omega|H_{2}\lambda^{\frac{d}{2}}-C(\Omega)\big{(}\lambda^{\frac{d-1}{2}}+\frac{C_{d}a}{C_{d-1}}\lambda^{\frac{d}{2}}\big{)}.$
Similar to the case $\lambda\geq\frac{9\pi^{2}(d-1)}{a^{2}}$, one can prove
that if we take $a\leq\min\big{(}\frac{|\Omega|H_{2}}{2C_{d-1}},\frac{\pi
C_{d}|\Omega|H_{2}}{2C(\Omega)}\big{)}$, then for any
$\frac{\pi^{2}}{a^{2}}\leq\lambda\leq\frac{9\pi^{2}(d-1)}{a^{2}}$, one gets
$\mathcal{N}^{N}_{(0,a)\times\Omega}(\lambda)\geq
C_{d+1}a|\Omega|\lambda^{\frac{d+1}{2}}.$
Finally for $0<\lambda<\frac{\pi^{2}}{a^{2}}$, we just repeat the
corresponding part in the proof of the Neumann case with $d=2$. In conclusion,
we get: if
$a<\min\bigg{(}\frac{B_{d}C_{d-1}}{2C(\Omega)C_{d}},\frac{B_{d}3\pi\sqrt{d-1}}{2C(\Omega)},\frac{|\Omega|H_{2}}{2C_{d-1}},\frac{\pi
C_{d}|\Omega|H_{2}}{2C(\Omega)},(C_{d+1}|\Omega|)^{-1}C_{1}(\Omega)^{-\frac{d+1}{2}}\bigg{)},$
then all Neumann eigenvalues of $(0,a)\times\Omega$ satisfy Pólya’s conjecture
(1.3). So we complete the proof of Theorem 1.2. $\hfill\square$
## 5\. Proof of Theorem 1.3
Again by [12, Proposition 3.1], it is enough to assume that both $\Omega$ and
$M$ are connected. Let the eigenvalues of $M$ be
$0=\lambda_{0}(M)<\lambda_{1}(M)\leq\cdots\nearrow\infty,$
and the counting functions for the eigenvalues of $M$ be
$\mathcal{N}_{M}(\lambda)=\\#\\{n|\ \lambda_{n}(M)<\lambda\\}.$
B. M. Levitan ([22]) and V. G. Avakumović ([1]) proved that
$\mathcal{N}_{M}(\lambda)=C_{d_{2}}|M|\lambda^{\frac{d_{2}}{2}}+\mathrm{O}(\lambda^{\frac{d-1}{2}}),\text{
as }\lambda\to\infty.$
So there exists a constant $C(M)>0$ such that
(5.1) $\mathcal{N}_{M}(\lambda)\geq
C_{d_{2}}|M|\lambda^{\frac{d_{2}}{2}}-C(M)\lambda^{\frac{d_{2}-1}{2}},\
\forall\lambda>0.$
Repeating the proof of the Neumann case of Theorem 1.1 and Theorem 1.2 word by
word, one can easily prove the Neumann case of Theorem 1.3.
For the Dirichlet case of Theorem 1.3, since 0 is an eigenvalue of $M$, one
can only get that there exists a constant $C_{1}(M)>0$ such that
(5.2) $\mathcal{N}_{M}(\lambda)\leq
C_{d_{2}}|M|\lambda^{\frac{d_{2}}{2}}+C_{1}(M)\lambda^{\frac{d_{2}-1}{2}}+1,\
\forall\lambda>0.$
So to prove the Dirichlet case of Theorem 1.3, one need to carefully handle
this extra number. Again we divided the proof into three parts: the Dirichlet
case with $d_{1}=1$ and $d_{2}=2$, the Dirichlet case with $d_{1}=1$ and
$d_{2}\geq 3$, and the Dirichlet case with $d_{1}\geq 2$.
### 5.1. The Dirichlet case with $d_{1}=1$ and $d_{2}=2$
When $d_{1}=1$, we can assume $\Omega=(0,1)$ for simplicity. The Dirichlet
eigenvalues of $(0,a)\times M$ are
$\frac{l^{2}\pi^{2}}{a^{2}}+\lambda_{k}(M),\qquad l\in\mathbb{Z}_{>0},\
k\in\mathbb{Z}_{\geq 0}.$
If $0<\lambda<\frac{\pi^{2}}{a^{2}}$, then $\mathcal{N}^{D}_{(0,a)\times
M}(\lambda)=0<C_{3}a|M|\lambda^{\frac{3}{2}}$. For
$\lambda\geq\frac{\pi^{2}}{a^{2}}$, by (4.2) we get
$\displaystyle\mathcal{N}^{D}_{(0,a)\times M}(\lambda)$
$\displaystyle=\sum_{l=1}^{M^{\lambda}_{a}}\mathcal{N}_{M}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})$
$\displaystyle\leq\frac{|M|}{4\pi}\sum_{l=1}^{M^{\lambda}_{a}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})+C_{1}(M)\sum_{l=1}^{M^{\lambda}_{a}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})^{\frac{1}{2}}+M^{\lambda}_{a}$
$\displaystyle\leq\frac{|M|}{4\pi}(\frac{2a\lambda^{\frac{3}{2}}}{3\pi}-\frac{\lambda}{8}-\frac{\sqrt{\lambda}\pi}{12a})+\frac{C_{1}(M)a}{4}\lambda+\frac{a\sqrt{\lambda}}{\pi}.$
Note that if $a\leq\sqrt{\frac{|M|\pi}{48}}$, then
$\frac{|M|}{4\pi}(\frac{2a\lambda^{\frac{3}{2}}}{3\pi}-\frac{\lambda}{8}-\frac{\sqrt{\lambda}\pi}{12a})+\frac{C_{1}(M)a}{4}\lambda+\frac{a\sqrt{\lambda}}{\pi}\leq\frac{a|M|\lambda^{\frac{3}{2}}}{6\pi^{2}}-\frac{|M|\lambda}{32\pi}+\frac{C_{1}(M)a}{4}\lambda.$
Thus we proved: if
(5.3) $a\leq\min\big{(}\sqrt{\frac{|M|\pi}{48}},\frac{|M|}{8\pi
C_{1}(M)}\big{)},$
then all Dirichlet eigenvalues of $(0,a)\times M$ satisfy Pólya’s conjecture
(1.2).
### 5.2. The Dirichlet case with $d_{1}=1$ and $d_{2}\geq 3$
We still have
$\displaystyle\mathcal{N}^{D}_{(0,a)\times M}(\lambda)$
$\displaystyle=\sum_{l=1}^{M^{\lambda}_{a}}\mathcal{N}_{M}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})$
$\displaystyle\leq
C_{d_{2}}|M|\sum_{l=1}^{M^{\lambda}_{a}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})^{\frac{d_{2}}{2}}+C_{1}(M)\sum_{l=1}^{M^{\lambda}_{a}}(\lambda-\frac{l^{2}\pi^{2}}{a^{2}})^{\frac{d_{2}-1}{2}}+M^{\lambda}_{a}.$
As in §4.4, if $\lambda\geq\frac{(d_{2}-1)\pi^{2}}{a^{2}}$, one has
$\mathcal{N}^{D}_{(0,a)\times M}(\lambda)\leq
C_{d_{2}+1}a|M|\lambda^{\frac{d_{2}+1}{2}}-A_{d_{2}}\lambda^{\frac{d_{2}}{2}}+C_{1}(M)\frac{C_{d_{2}}a}{C_{d_{2}-1}}\lambda^{\frac{d_{2}}{2}}+\frac{a\sqrt{\lambda}}{\pi},$
where
$A_{d_{2}}=\frac{1}{2}(1-(\frac{4d_{2}-5}{4d_{2}-4})^{\frac{d_{2}}{2}})C_{d_{2}}|M|$.
To control the last term, we require
$a<\big{(}\frac{\pi}{2}A_{d_{2}}\big{)}^{\frac{1}{d_{2}}}\big{(}\pi^{2}(d_{2}-1)\big{)}^{\frac{d_{2}-1}{2d_{2}}}$
to get
$\frac{a\sqrt{\lambda}}{\pi}\leq\frac{1}{2}A_{d_{2}}\lambda^{\frac{d_{2}}{2}}$
for all $\lambda\geq\frac{(d_{2}-1)\pi^{2}}{a^{2}}$. Repeating §4.4, we will
get: if
$a<\min\bigg{(}\pi\big{(}\frac{1}{2}A_{d_{2}}\big{)}^{\frac{1}{d_{2}}}(d_{2}-1)^{\frac{d_{2}-1}{2d_{2}}},\frac{A_{d_{2}}C_{d_{2}-1}}{2C_{1}(M)C_{d_{2}}}\bigg{)},$
then for any $\lambda\geq\frac{(d_{2}-1)\pi^{2}}{a^{2}}$,
$\mathcal{N}^{D}_{(0,a)\times M}(\lambda)\leq
C_{d_{2}+1}a|M|\lambda^{\frac{d_{2}+1}{2}}.$
For $\lambda<\frac{(d_{2}-1)\pi^{2}}{a^{2}}$, again one only needs to consider
$\frac{\pi^{2}}{a^{2}}\leq\lambda<\frac{(d_{2}-1)\pi^{2}}{a^{2}}$. As in §4.4,
in this case one has
$\mathcal{N}^{D}_{(0,a)\times M}(\lambda)\leq
C_{d_{2}+1}a|M|\lambda^{\frac{d_{2}+1}{2}}-C_{d_{2}}|M|H_{1}\lambda^{\frac{d_{2}}{2}}+C_{1}(M)\frac{C_{d_{2}}a}{C_{d_{2}-1}}\lambda^{\frac{d_{2}}{2}}+\frac{a\sqrt{\lambda}}{\pi},$
where
$H_{1}:=\inf_{1\leq\mu\leq
d_{2}-1}\frac{\int^{\sqrt{\mu}}_{0}(\mu-x^{2})^{\frac{d_{2}}{2}}\mathrm{d}x-\sum_{0<l^{2}<\mu}(\mu-l^{2})^{\frac{d_{2}}{2}}}{\mu^{\frac{d_{2}}{2}}}>0.$
So if we assume
$a<\min\bigg{(}\pi\big{(}\frac{1}{2}C_{d_{2}}|M|H_{1}\big{)}^{\frac{1}{d_{2}}},\frac{C_{d_{2}-1}|M|H_{1}}{2C_{1}(M)}\bigg{)},$
then for all
$\frac{\pi^{2}}{a^{2}}\leq\lambda<\frac{(d_{2}-1)\pi^{2}}{a^{2}}$, one has
$\mathcal{N}^{D}_{(0,a)\times M}(\lambda)\leq
C_{d_{2}+1}a|M|\lambda^{\frac{d_{2}+1}{2}}.$
Thus if $d_{2}\geq 3$ and
$a<\min\bigg{(}\pi\big{(}\frac{1}{2}A_{d_{2}}\big{)}^{\frac{1}{d_{2}}}(d_{2}-1)^{\frac{d_{2}-1}{2d_{2}}},\frac{A_{d_{2}}C_{d_{2}-1}}{2C_{1}(M)C_{d_{2}}},\pi\big{(}\frac{1}{2}C_{d_{2}}|M|H_{1}\big{)}^{\frac{1}{d_{2}}},\frac{C_{d_{2}-1}|M|H_{1}}{2C_{1}(M)}\bigg{)},$
then all Dirichlet eigenvalues of $(0,a)\times M$ satisfy Pólya’s Conjecture
(1.2).
### 5.3. The Dirichlet case with $d_{1}\geq 2$
The Dirichlet eigenvalues of $a\Omega\times M$ are
$a^{-2}\lambda_{l}(\Omega)+\lambda_{k}(M),\qquad l\in\mathbb{Z}_{>0},\
k\in\mathbb{Z}_{\geq 0},$
which implies
$\mathcal{N}^{D}_{a\Omega\times M}(\lambda)\leq
C_{d_{2}}|M|a^{-d_{2}}\sum_{l=1}^{Z^{\lambda}_{a}}(a^{2}\lambda-\lambda_{l}(\Omega))^{\frac{d_{2}}{2}}+C_{1}(M)a^{1-d_{2}}\sum_{l=1}^{Z^{\lambda}_{a}}(a^{2}\lambda-\lambda_{l}(\Omega))^{\frac{d_{2}-1}{2}}+Z^{\lambda}_{a}$
where $Z^{\lambda}_{a}=\mathcal{N}^{D}_{\Omega}(a^{2}\lambda)$. To control the
extra $Z^{\lambda}_{a}$, we use Li-Yau’s estimate (1.5) to get
$\mathcal{N}^{D}_{\Omega}(\lambda)\leq\big{(}\frac{d_{1}+2}{d_{1}}\big{)}^{\frac{d_{1}}{2}}C_{d_{1}}|\Omega|\lambda^{\frac{d_{1}}{2}},\qquad\forall\lambda>0.$
Thus for all $a>0$ and $\lambda>0$,
$Z^{\lambda}_{a}=\mathcal{N}^{D}_{\Omega}(a^{2}\lambda)\leq\big{(}\frac{d_{1}+2}{d_{1}}\big{)}^{\frac{d_{1}}{2}}C_{d_{1}}|\Omega|a^{d_{1}}\lambda^{\frac{d_{1}}{2}}.$
If $d_{2}\geq 3$, then as in §3.1, there exists a constant $C(\Omega)>0$ such
that for $a^{2}\lambda>C(\Omega)$,
$\displaystyle\mathcal{N}^{D}_{a\Omega\times M}(\lambda)\leq$ $\displaystyle
C_{d_{1}+d_{2}}a^{d_{1}}|\Omega||M|\lambda^{\frac{d_{1}+d_{2}}{2}}-\frac{1}{5}C_{d_{1}+d_{2}-1}|\partial\Omega||M|a^{d_{1}-1}\lambda^{\frac{d_{1}+d_{2}-1}{2}}+$
$\displaystyle
L_{\frac{d_{2}-1}{2},d_{1}}|\Omega|C_{1}(M)a^{d_{1}}\lambda^{\frac{d_{1}+d_{2}-1}{2}}+(\frac{d_{1}+2}{d_{1}})^{\frac{d_{1}}{2}}C_{d_{1}}|\Omega|a^{d_{1}}\lambda^{\frac{d_{1}}{2}}.$
So if we assume
$a<\big{(}\frac{C_{d_{1}+d_{2}-1}|\partial\Omega||M|}{10C_{d_{1}}|\Omega|}\big{)}^{\frac{1}{d_{2}}}\big{(}\frac{d_{1}+2}{d_{1}}\big{)}^{-\frac{1}{2}}C(\Omega)^{\frac{d_{2}-1}{2d_{2}}},$
then for any $\lambda>a^{-2}C(\Omega)$, the extra term is controlled by
$(\frac{d_{1}+2}{d_{1}})^{\frac{d_{1}}{2}}C_{d_{1}}|\Omega|a^{d_{1}}\lambda^{\frac{d_{1}}{2}}\leq\frac{1}{10}C_{d_{1}+d_{2}-1}|\partial\Omega||M|a^{d_{1}-1}\lambda^{\frac{d_{1}+d_{2}-1}{2}}.$
If $d_{2}=2$, then as in §3.2, there exists a constant $C(\Omega)>0$ such that
if $a^{2}\lambda>C(\Omega)$, one has
$\displaystyle\mathcal{N}^{D}_{a\Omega\times M}(\lambda)\leq$ $\displaystyle
C_{d_{1}+2}a^{d_{1}}|\Omega||M|\lambda^{\frac{d_{1}+2}{2}}-\frac{1}{5}C_{d_{1}+1}|M||\partial\Omega|a^{d_{1}-1}\lambda^{\frac{d_{1}+1}{2}}+$
$\displaystyle\big{(}\frac{d_{1}+2}{d_{1}}\big{)}^{\frac{d_{1}}{2}}C_{1}(M)L_{\frac{1}{2},d_{1}}|\Omega|a^{d_{1}}\lambda^{\frac{d_{1}+1}{2}}+(\frac{d_{1}+2}{d_{1}})^{\frac{d_{1}}{2}}C_{d_{1}}|\Omega|a^{d_{1}}\lambda^{\frac{d_{1}}{2}}.$
and similarly we can control the last term.
The rest of the proof for both cases are identically the same as before, and
thus will be omitted. $\hfill\square$
### 5.4. An abstract extension
As we have seen, although the upper bound given by (5.2) is a bit weaker than
(2.6), the extra term 1 can be controlled. Of course one may replace 1 by
other number.
More generally, one may start with two increasing sequence
$0<s_{1}\leq s_{2}\leq\cdots+\infty,\qquad t_{1}\leq
t_{2}\leq\cdots\to+\infty$
and study the new increasing sequence
$\\{\nu_{k}(a)\\}_{k=1}^{\infty}=\\{a^{-2}s_{m}+t_{n}\\}$. As usual we will
denote
$\mathcal{N}_{(s_{k})}(\lambda)=\\#\\{k|s_{k}\leq\lambda\\}$
and likewise for $\mathcal{N}_{(t_{k})}(\lambda)$. By using the same idea and
modifying the proof above slightly, it is easy to prove
###### Theorem 5.1.
Suppose there exist constants $V_{t},B_{1},B_{2}>0,d\geq 2$ such that
(5.4) $\mathcal{N}_{(t_{k})}(\lambda)\leq
V_{t}C_{d}\lambda^{\frac{d}{2}}+B_{1}\lambda^{\frac{d-1}{2}}+B_{2},\qquad\forall\lambda,$
and suppose either $s_{k}=\pi^{2}k^{2}(k\geq 1)$ (in which case we take
$V_{s}=1$, $d^{\prime}=1$ below), or there exist $V_{s}>0$ and $d^{\prime}\geq
2$ such that
$\sum_{s_{k}<\lambda}(\lambda-
s_{k})<L_{1,{d^{\prime}}}V_{s}\lambda^{\frac{d^{\prime}}{2}+1},\qquad\forall\lambda>0,$
and there exist $C^{\prime}>0$ and $C_{s}>0$ such that for all
$\lambda>C_{s}$,
$\sum_{s_{k}<\lambda}(\lambda-s_{k})\leq
L_{1,d^{\prime}}V_{s}\lambda^{\frac{d^{\prime}}{2}+1}-C^{\prime}\lambda^{\frac{d^{\prime}+1}{2}},$
then there exists $a_{0}>0$ such that for any $0<a<a_{0}$,
$\nu_{k}(a)\geq\frac{4\pi^{2}}{(\omega_{d+d^{\prime}}a^{d^{\prime}}V_{s}V_{t})^{\frac{2}{d+d^{\prime}}}}k^{\frac{2}{d+d^{\prime}}},\qquad\forall
k\geq 1.$
Similarly one may write down an abstract version that extends the results for
the Neumann eigenvalues above, in which case one may relax the condition on
$\mathcal{N}_{(t_{k})}(\lambda)$ to
(5.5) $\mathcal{N}_{(t_{k})}(\lambda)\geq
V_{t}C_{d}\lambda^{\frac{d}{2}}-B_{1}\lambda^{\frac{d-1}{2}},\qquad\forall\lambda>0,$
and pose suitable conditions on $(s_{k})$ (including a Szegö-Weinberger type
condition on $s_{1}$).
As a consequence, we could get a bunch of eigenvalue problems that satisfies
Pólya inequalities. For example, let $(M,g)$ be a compact Riemannian manifold
of dimension $d\geq 2$, with piecewise smooth boundary $\partial M$. Let $(H)$
be certain boundary condition so that the Laplace-Beltrami operator on $(M,g)$
has discrete spectrum. As usual we denote the corresponding eigenvalue
counting function by $\mathcal{N}^{(H)}_{M}(\lambda)$. Then we have
###### Theorem 5.2.
Let $\Omega\subset\mathbb{R}^{d_{1}}$ be a bounded domain with $C^{1}$
boundary and consider the product manifold $a\Omega\times M$.
1. (1)
If $\mathcal{N}^{(H)}_{M}(\lambda)$ satisfies (5.4), then there exists
$a_{0}>0$ (depends on $\Omega$ and $M$) such that for any $0<a<a_{0}$, the
eigenvalues of the Laplace-Beltrami operator on $a\Omega\times M$ with the
following mixed boundary condition
$\text{ Dirichlet condition on }\partial(a\Omega)\times M,\quad\text{
condition (H) on }a\Omega\times\partial M$
satisfy Pólya’s conjecture (1.2),
2. (2)
If $\mathcal{N}^{(H)}_{M}(\lambda)$ satisfies (5.5), then there exists
$a_{0}>0$ (depends on $\Omega$ and $M$) such that for any $0<a<a_{0}$, the
eigenvalues of the Laplace-Beltrami operator on $a\Omega\times M$ with the
following mixed boundary condition
$\text{ Neumann condition on }\partial(a\Omega)\times M,\quad\text{ condition
(H) on }a\Omega\times\partial M$
satisfy Pólya’s conjecture (1.3).
For example, one may take the condition (H) to be either Dirichlet boundary
condition or Neumann boundary condition or Robin ($\frac{\partial
f}{\partial\nu}=\rho f$, with bounded $\rho$) boundary condition, and in all
these cases the inequalities (5.4) and (5.5) hold. Thus one get many
eigenvalue problems whose eigenvalues satisfy Pólya’s conjecture.
## 6\. Two examples with explicit constants
In this section, we give two examples for which one can calculate the
constants involved in the proof, and thus give explicit domains/manifolds for
which Pólya’s conjecture holds.
We first construct a planar domain $\Omega$ for which we can calculate the
constant $C(\Omega)$ in (2.6) explicitly, and thus find out the number $a_{0}$
in Theorem 1.2 for $\Omega$.
Let $S$ be a square with side length 10 and $T$ be an equilateral triangle
with side length 1. The domain $\Omega$ is constructed by placing $T$ at the
center of one side of $S$, as shown by the picture below:
Note that the angle $\theta=\frac{2\pi}{3}$, which implies that $\Omega$
cannot tile $\mathbb{R}^{2}$. In what follows we prove
###### Proposition 6.1.
For any $a\leq\frac{1}{4\pi}$, the Dirichlet eigenvalues of
$(0,a)\times\Omega$ satisfies Pólya’s conjecture (1.2).
###### Proof.
By Faber–Krahn’s inequality ([5], [19]),
$\lambda_{1}(\Omega)\geq\frac{4\pi^{2}}{\omega_{2}|\Omega|}>10^{-1}.$
So $\mathcal{N}^{D}_{\Omega}(\lambda)=0$ for $\lambda\leq 10^{-1}$.
Now suppose $\lambda>10^{-1}$. For the square $S$ we have
$\displaystyle\mathcal{N}^{N}_{S}(\lambda)$
$\displaystyle=\\#\\{(m,n)\in\mathbb{Z}_{\geq 0}^{2}|\
m^{2}+n^{2}<\frac{100\lambda}{\pi^{2}}\\}$
$\displaystyle\leq\frac{100\lambda}{4\pi}+\frac{20}{\pi}\sqrt{\lambda}+2$
$\displaystyle<\frac{100\lambda}{4\pi}+20\sqrt{\lambda}.$
For the triangle $T$, let
$P=\\{(x,2x)\\}_{x\in\mathbb{R}}\cup\\{(2x,x)\\}_{x\in\mathbb{R}}\cup\\{(x,-x)\\}_{x\in\mathbb{R}}$,
then by [28, Proposition 3], one has
$\displaystyle\mathcal{N}^{N}_{T}(\lambda)=$
$\displaystyle\frac{1}{6}\\#\\{(m,n)\in\mathbb{Z}^{2}|\ (m,n)\notin
P,3|(m+n),\frac{16\pi^{2}}{27}(m^{2}+n^{2}-mn)<\lambda\\}+$
$\displaystyle\frac{1}{3}\\#\\{(m,n)\in\mathbb{Z}^{2}|\ (m,n)\in
P,\frac{16\pi^{2}}{27}(m^{2}+n^{2}-mn)<\lambda\\}+\frac{2}{3}.$
Since
$\displaystyle\\#\\{(m,n)\in\mathbb{Z}^{2}|\ (m,n)\notin
P,3|(m+n),\frac{16\pi^{2}}{27}(m^{2}+n^{2}-mn)<\lambda\\}$ $\displaystyle=$
$\displaystyle\\#\\{(m,n)\in\mathbb{Z}^{2}|\ (m,n)\notin
P,3|(m+n),(m-\frac{n}{2})^{2}+\frac{3n^{2}}{4}<\frac{27\lambda}{16\pi^{2}}\\}$
$\displaystyle\leq$ $\displaystyle\frac{1}{3}\\#\\{(m,n)\in\mathbb{Z}^{2}|\
m^{2}+\frac{3n^{2}}{4}<\frac{27\lambda}{16\pi^{2}}\\}+2(\frac{3\sqrt{3}\sqrt{\lambda}}{4\pi}+\frac{3\sqrt{\lambda}}{2\pi})+4$
$\displaystyle<$ $\displaystyle\frac{3\sqrt{3}\lambda}{8\pi}+60\sqrt{\lambda}$
and
$\displaystyle\\#\\{(m,n)\in\mathbb{Z}^{2}|\ (m,n)\in
P,\frac{16\pi^{2}}{27}(m^{2}+n^{2}-mn)<\lambda\\}$ $\displaystyle\leq$
$\displaystyle 3\\#\\{k\in\mathbb{Z}|\
k^{2}<\frac{9\lambda}{16\pi^{2}}\\}<\frac{9\sqrt{\lambda}}{4\pi}+3<30\sqrt{\lambda},$
we get
$\mathcal{N}^{N}_{T}(\lambda)<\frac{\sqrt{3}\lambda}{16\pi}+30\sqrt{\lambda}.$
So we get
$\mathcal{N}^{D}_{\Omega}(\lambda)<\mathcal{N}^{N}_{\Omega}(\lambda)\leq\mathcal{N}^{N}_{S}(\lambda)+\mathcal{N}^{N}_{T}(\lambda)\leq\frac{1}{4\pi}(100+\frac{\sqrt{3}}{4})\lambda+50\sqrt{\lambda}.$
In other words, one may take $C(\Omega)=50$ in (2.6). It follows from the
proof in §4.2 that for any $a\leq\frac{1}{4\pi}$, all Dirichlet eigenvalues of
$(0,a)\times\Omega$ satisfy Pólya’s Conjecture (1.2). ∎
Finally we turn to the Riemannian manifold setting and consider the standard
two-sphere $M=S^{2}$. It is well known that the eigenvalues of $(S^{2},g_{0})$
are $k(k+1)$, with multiplicity $2k+1$ for all $k\in\mathbb{Z}_{\geq 0}$. It
follows that
$\mathcal{N}_{S^{2}}\big{(}k(k+1)\big{)}=k^{2},\qquad\mathcal{N}_{S^{2}}\big{(}k(k+1)+\varepsilon\big{)}=(k+1)^{2}.$
In other words, one can choose $C(S^{2})$ in (5.1) to be 1 and $C_{1}(S^{2})$
in (5.2) to be 1. Plugging into (5.3) and (4.7), we get
###### Proposition 6.2.
For any $a\leq\frac{\pi}{24}$, the manifold $(0,a)\times S^{2}$ satisfy
Pólya’s conjecture (1.2) and (1.3).
Note that in this example, if we take $a$ large,
* •
If we take $a>\sqrt{\frac{2}{3}}\pi$, then the first Dirichlet eigenvalue of
$(0,a)\times S^{2}$
$\lambda_{1}\big{(}(0,a)\times
S^{2}\big{)}=\frac{\pi^{2}}{a^{2}}<\frac{4\pi^{2}}{({4\pi\omega_{3}}a)^{\frac{2}{3}}},$
* •
If we take $\frac{\pi}{\sqrt{2}}\leq a<\sqrt{\frac{2}{3}}\pi$, then the first
nonzero Neumann eigenvalue of $(0,a)\times S^{2}$
$\mu_{1}\big{(}(0,a)\times
S^{2}\big{)}=\frac{\pi^{2}}{a^{2}}>\frac{4\pi^{2}}{({4\pi\omega_{3}}a)^{\frac{2}{3}}},$
so Pólya’s inequalities (1.2) and (1.3) will not hold for $(0,a)\times S^{2}$
when $a$ is large.
###### Remark 6.3.
We remark that in [13, Example 2.D], P. Freitas and I. Salavessa had already
observed (from a very simple tiling argument) that $(0,a)\times S^{2}$
satisfies Pólya’s inequalities for $a$ small enough but fails to satisfy
Pólya’s inequalities for $a$ large. Our method is more complicated but has the
advantage that we could give explicit estimates of $a$ for Pólya’s
inequalities to hold. We would like to thank the authors for pointing out this
fact to us.
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# Multi-band Extension of the Wideband Timing Technique
Avinash Kumar Paladi1,2, Churchil Dwivedi2, Prerna Rana3, Nobleson K.4,
Abhimanyu Susobhanan5, Bhal Chandra Joshi6,7, Pratik Tarafdar8, Debabrata
Deb8, Swetha Arumugam10, A. Gopakumar3, M. A. Krishnakumar11,12, Neelam Dhanda
Batra13, Jyotijwal Debnath8,9, Fazal Kareem14,15, Paramasivan Arumugam7,
Manjari Bagchi8,9, Adarsh Bathula16, Subhajit Dandapat3, Shantanu Desai17,
Yashwant Gupta6, Shinnosuke Hisano18, Divyansh Kharbanda17, Tomonosuke
Kikunaga18, Neel Kolhe19, Yogesh Maan6, P. K. Manoharan20, Jaikhomba Singha7,
Aman Srivastava17, Mayuresh Surnis21, Keitaro Takahashi22,23
1 Joint Astronomy Programme, Indian Institute of Science, Bengaluru,
Karnataka, 560012, India
2 Department of Earth and Space Sciences, Indian Institute of Space Science
and Technology, Valiamala, Thiruvananthapuram 695547, Kerala, India
3 Department of Astronomy and Astrophysics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Navy Nagar, Colaba, Mumbai 400005, India
4 Department of Physics, BITS Pilani Hyderabad Campus, Hyderabad 500078,
Telangana, India
5 Center for Gravitation Cosmology and Astrophysics, University of Wisconsin-
Milwaukee, Milwaukee, WI 53211, USA
6 National Centre for Radio Astrophysics, Pune University Campus, Pune 411007,
India
7 Department of Physics, Indian Institute of Technology Roorkee,
Roorkee-247667, India
8 The Institute of Mathematical Sciences, C. I. T. Campus, Taramani, Chennai
600113, India
9 Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400094, India
10 Department of Electrical Engineering, IIT Hyderabad, Kandi, Telangana
502284, India
11 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn,
Germany
12 Fakultät für Physik, Universität Bielefeld, Postfach 100131, 33501
Bielefeld, Germany
13 Department of Physics and Astrophysics, University of Delhi, Delhi
14 Department of Physical Sciences, Indian Institute of Science Education and
Research Kolkata, West Bengal, 741246, India
15 Center of Excellence in Space Sciences India, Indian Institute of Science
Education and Research Kolkata, West Bengal, 741246, India
16 Indian Institute of Science Education and Research, Mohali - 140306,
Punjab, India
17 Department of Physics, IIT Hyderabad, Kandi, Telangana 502284, India
18 Kumamoto University, Graduate School of Science and Technology, Kumamoto,
860-8555, Japan
19 Department of Physics, St. Xavier’s College (Autonomous), Mumbai 400001,
Maharashtra, India
20 Arecibo Observatory, University of Central Florida, Arecibo 00612, USA
21 Department of Physics, IISER Bhopal, Bhauri Bypass Road, Bhopal 462066,
India
22 Faculty of Advanced Science and Technology, Kumamoto University, 2-39-1
Kurokami, Kumamoto 860-8555, Japan
23 International Research Organization for Advanced Science and Technology,
Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555, Japan E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
The wideband timing technique enables the high-precision simultaneous
estimation of pulsar Times of Arrival (ToAs) and Dispersion Measures (DMs)
while effectively modeling frequency-dependent profile evolution. We present
two novel independent methods that extend the standard wideband technique to
handle simultaneous multi-band pulsar data incorporating profile evolution
over a larger frequency span to estimate DMs and ToAs with enhanced precision.
We implement the wideband likelihood using the libstempo python interface to
perform wideband timing in the tempo2 framework. We present the application of
these techniques to the dataset of fourteen millisecond pulsars observed
simultaneously in Band 3 ($300-500$ MHz) and Band 5 ($1260-1460$ MHz) of the
upgraded Giant Metrewave Radio Telescope (uGMRT) with a large band gap of 760
MHz as a part of the Indian Pulsar Timing Array (InPTA) campaign. We achieve
increased ToA and DM precision and sub-microsecond root mean square post-fit
timing residuals by combining simultaneous multi-band pulsar observations done
in non-contiguous bands for the first time using our novel techniques.
###### keywords:
pulsars: general — galaxies: ISM — gravitational waves — methods: data
analysis
††pubyear: 2023††pagerange: Multi-band Extension of the Wideband Timing
Technique–B
## 1 Introduction
Pulsars are rotating neutron stars emitting broadband electromagnetic
radiation that is observed as periodic pulses. The rotation of a pulsar can be
tracked accurately by measuring the times of arrival (ToAs) of its pulses, and
this technique is known as pulsar timing (Hobbs et al., 2006; Edwards et al.,
2006). The pulsar signal is dispersed while propagating through the ionized
interstellar medium (IISM) by an amount that is proportional to the integrated
free electron column density along the line of sight, usually referred to as
the dispersion measure (DM), and inversely proportional to the square of the
observing frequency $\nu$ (Lorimer & Kramer, 2012). Conventionally, the rough
measurement of DM for a pulsar used to be done by splitting the data into
multiple sub-bands and correcting for the DM induced delay for each sub-band
and then adding the dispersed bands again (Lorimer & Kramer, 2012). In recent
days, many sophisticated techniques have been proposed, which not only
provides more accurate values of DM but also provide epoch to epoch variations
of DM (e.g. Ahuja et al., 2005).
Pulsar timing has traditionally been done by splitting the data into multiple
sub-bands with negligible dispersion smear and independently measuring the ToA
from each sub-band, known as narrowband timing (Taylor, 1992). The improvement
in telescope sensitivity, the advent of wideband receivers and backends (e.g.
Johnston et al., 2021; Hobbs et al., 2020; Reddy et al., 2017; Gupta et al.,
2017), and decades-long pulsar timing campaigns such as Pulsar Timing Arrays
(PTAs: Foster & Backer, 1990) have presented significant challenges to the
narrowband approach. These challenges include inadequate modeling of the pulse
profile variability as a function of frequency, difficulty in correcting for
interstellar scattering, and large data sizes. The wideband timing technique
seeks to address these issues by treating the pulse profile as a two-
dimensional entity in frequency and pulse phase (usually referred to as a
portrait) and simultaneously measuring one ToA and one DM per observation
using the full bandwidth (Pennucci et al., 2014; Pennucci, 2019).
PTA experiments, such as the Parkes Pulsar Timing Array (PPTA: Hobbs, 2013),
the European Pulsar Timing Array (EPTA: Kramer & Champion, 2013), the North
American Nanohertz Observatory for Gravitational Waves (NANOGrav: McLaughlin,
2013), the Indian Pulsar Timing Array (InPTA: Joshi et al., 2018), the Chinese
Pulsar Timing Array (CPTA: Lee, 2016), MeerKat Pulsar Timing Array (MPTA:
Miles et al., 2023), and the International Pulsar Timing Array (IPTA: Hobbs et
al., 2010; Perera et al., 2019; Verbiest et al., 2016) consortium which
combines the data and resources from various PTAs, aim to detect nanohertz
gravitational waves using an ensemble of millisecond pulsars (MSPs) as
celestial clocks. Recent wideband timing studies across a wide range of
observing frequencies have demonstrated significant improvements in ToA and DM
measurement precision (Liu et al., 2014; Fonseca et al., 2021; Nobleson et
al., 2022), and PTAs and other high-precision pulsar timing campaigns are now
increasingly adopting wideband techniques due to their advantages in dealing
with broadband observations (e.g. Alam et al., 2021; Tarafdar et al., 2022;
Curyło et al., 2023).
The InPTA experiment complements the international PTA efforts by employing
the unique features of the upgraded Giant Metrewave Radio Telescope (uGMRT:
Gupta et al., 2017). The high sensitivity of the uGMRT at low frequencies,
combined with its ability to perform simultaneous multi-band observations,
makes it an ideal instrument to characterize interstellar medium effects,
which are stronger at low frequencies (Krishnakumar et al., 2021). The
application of the wideband technique to the uGMRT data and the ToA and DM
precision improvements therefrom were demonstrated in Nobleson et al. (2022).
The recently published first data release of the InPTA (InPTA DR1: Tarafdar et
al., 2022) built on the work of Krishnakumar et al. (2021) and Nobleson et al.
(2022), has presented the results of narrowband and wideband timing of 14
pulsars observed over a time span of 3.5 years. This work included some of the
most precise DM measurements to date, estimated using both the narrowband and
the wideband techniques. Recently the InPTA collaboration has completed Single
Pulsar Noise Analysis on the DR1 pulsars using narrowband data (Srivastava et
al., 2023).
The InPTA observes pulsars in two uGMRT bands, namely the Band 3 ($300-500$
MHz) and the Band 5 ($1260-1460$ MHz). Although Nobleson et al. (2022) and
Tarafdar et al. (2022) only used Band 3 data for the wideband timing, the DM
precision achieved therein was comparable to the combined Band 3+5 narrowband
DM estimates. This raises the exciting possibility of attaining further
improvements in DM precision by combining the two uGMRT bands in the wideband
paradigm.
In this work, we develop two novel methods to combine simultaneous
observations of the same pulsar obtained at multiple bands using the wideband
technique to obtain a single ToA and DM combination per epoch across these
multiple bands. We then demonstrate the ToA and DM precision improvement
achieved with these techniques using the InPTA observations of 14 MSPs,
simultaneously observed at Band 3 and Band 5 seperated by a large band gap of
about 760 MHz, which were selected for the InPTA first data release (Tarafdar
et al., 2022). These techniques provide significant improvements in the DM
precision estimation that can be achieved using existing and future telescopes
which can perform simultaneous or quasi-simultaneous multi-frequency
observations such as the Square Kilometer Array (SKA) (Kramer & Stappers,
2015; Janssen et al., 2015). For the timing analysis, we extend the
traditional timing methodology to incorporate the wideband timing likelihood
function (Appendix B of Alam et al. (2021)) in tempo2 using libstempo.
This paper is structured as follows. We begin by providing a brief overview of
the wideband timing technique in subsection 2.1. In subsection 2.2, we present
two novel independent methods for applying the wideband technique to two
simultaneous band observations of a pulsar taken at different radio
frequencies, which can be easily extended to multiple bands. We apply our
methods to the case of PSR J1909$-$3744 and show the validation scheme and
comparisons against each other as well as against the single band (Band 3)
results in Section 3. We present the application of our novel methods to the
InPTA dataset of 14 MSPs in Section 4. We summarize our results in Section 5
and discuss avenues for future improvements and extensions in Section 6. Our
implementation of the wideband likelihood function using tempo2 and libstempo
is briefly described in Appendix B.
## 2 Multi-band extension of the Wideband Timing technique
### 2.1 Brief overview of the Wideband Technique
We begin by briefly summarising the wideband technique developed in Pennucci
et al. (2014), Pennucci (2019), and Alam et al. (2021). The total intensity
integrated pulse profile of a pulsar can be expressed as a two-dimensional
object $D(\nu,\varphi)$ which is a function of the observing frequency $\nu$
and the pulse phase $\varphi$, and is referred to as a pulse portrait. Given a
model for the observed portrait $P(\nu,\varphi)$, referred to as the template
portrait or the model portrait, $D(\nu,\varphi)$ can be written as:
$D(\nu,\varphi)=B(\nu)+a(\nu)P(\nu,\varphi-\phi(\nu))+N(\nu,\varphi)\,,$ (1)
where $B(\nu)$ is the DC offset in each frequency channel, $a(\nu)$ is an
amplitude that arises from the intrinsic power spectral density of the pulsar
emission and interstellar scintillation and also depends on the receiver
bandpass, and $N(\nu,\varphi)$ is an additive noise that is usually assumed to
be Gaussian and uncorrelated in the absence of radio frequency interference
(RFI). In practice, $D(\nu,\varphi)$ and $P(\nu,\varphi)$ are discretised in
both $\nu$ and $\varphi$, i.e., $D_{nj}\equiv D(\nu_{n},\varphi_{j})$ such
that $n$ denotes the frequency channels and $j$ corresponds to the phase bins.
The frequency dependence of the phase shift $\phi(\nu)$ arises primarily due
to the interstellar dispersion and is given by:
$\phi(\nu)=\phi_{\text{ref}}+\frac{K\times\text{DM}}{P_{s}}\left(\frac{1}{\nu^{2}}-\frac{1}{\nu_{\text{ref}}^{2}}\right)\,,$
(2)
where $\phi_{\text{ref}}$ is the achromatic phase shift, $K$ is the Dispersion
constant, $P_{s}$ is the apparent spin period of the pulsar at the epoch of
observation, and $\nu_{\text{ref}}$ is a Barycentric reference frequency.
Given $\phi_{\text{ref}}$, the ToA can be computed as
$t=t_{f}+{P_{s}\phi_{\text{ref}}}/{2\pi}$ where $t_{f}$ is the timestamp
corresponding to a fiducial phase point in the data portrait111In practice,
$t_{f}$ may be affected by instrumental delays such as those encountered in
Tarafdar et al. (2022), and one must correct for them.. $\phi_{\text{ref}}$
can be understood as the difference between the fiducial phases of the data
portrait and the template portrait.
Computing the discrete Fourier transform of equation (1) along the $\phi$
axis, applying the discrete Fourier shift theorem, and excluding the DC term,
we have:
$\widetilde{D}_{nk}=a_{n}\widetilde{P}_{nk}\;e^{2\pi
ik\phi_{n}}+\widetilde{N}_{nk}\,.$ (3)
where $\widetilde{D}_{nk}$ and $\widetilde{P}_{nk}$ denote the discrete
Fourier transform of the data portrait and the template portrait respectively,
and $\phi_{n}=\phi(\nu_{n})$. The quantities of interest $\phi_{\text{ref}}$
and DM can then be estimated by minimizing the weighted least-squares
statistic:
$\chi^{2}(\phi_{\text{ref}},\text{DM},a_{n})=\sum_{n,k}\frac{\left|\widetilde{D}_{nk}-a_{n}\widetilde{P}_{nk}e^{2\pi
ik\phi_{n}}\right|^{2}}{{\sigma_{n}^{\prime}}^{2}}\,,$ (4)
where ${\sigma_{n}^{\prime}}^{2}$ denotes the noise variance of the Fourier
coefficients $\widetilde{D}_{nk}$. It turns out that
$\chi^{2}(\phi_{\text{ref}},\text{DM},a_{n})$ can be analytically minimised
over the amplitudes $a_{n}$, and this leads to
$\chi^{2}(\phi_{\text{ref}},\text{DM})=S-\sum_{n}\frac{C_{n}^{2}}{T_{n}}\,,$
(5)
where
$\displaystyle S$
$\displaystyle=\sum_{n,k}\frac{|\widetilde{D}_{nk}|^{2}}{{\sigma_{n}^{\prime}}^{2}}\,,$
(6a) $\displaystyle T_{n}$
$\displaystyle=\sum_{k}\frac{|\widetilde{P}_{nk}|^{2}}{{\sigma_{n}^{\prime}}^{2}}\,,$
(6b) $\displaystyle C_{n}$
$\displaystyle=\Re\left\\{\sum_{k}\frac{\widetilde{D}_{nk}\widetilde{P}_{nk}^{*}\;e^{2\pi
ik\phi_{n}}}{{\sigma_{n}^{\prime}}^{2}}\right\\}\,.$ (6c)
Choosing $\nu_{\text{ref}}$ such that the covariance between
$\phi_{\text{ref}}$ and DM vanishes (see the Appendix of Pennucci et al.
(2014)), $\phi_{\text{ref}}$ and DM can be estimated by numerically minimizing
$\chi^{2}(\phi_{\text{ref}},\text{DM})$.
The template portrait $P(\nu,\varphi)$ is usually obtained from a single high
signal-to-noise ratio (S/N) portrait or an averaged portrait generated from
many observations. The mean-subtracted template portrait is decomposed into
many ‘eigenprofiles’ using principal component analysis (PCA). A smoothed
template portrait is then reconstructed from a small number of significant
eigenprofiles by spline-interpolating them (Pennucci, 2019). By linearly
combining the $n_{\text{eig}}$ significant eigenprofiles $\hat{e}_{i}$ using
the spline coefficients $B_{i}$ and adding it to the mean profile
$\widetilde{p}$, a template profile $T(\nu)$ at any frequency $\nu$ can be
created as
$T(\nu)=\sum_{i=1}^{n_{\text{eig}}}B_{i}(\nu)\;\hat{e}_{i}+\widetilde{p}\,.$
(7)
Note that the DMs estimated from the wideband technique are not derived from
ToAs unlike in the narrowband case, but rather measured simultaneously with
each ToA. Therefore, DM measurements should be treated as data points on an
equal footing with the ToAs while computing the likelihood function. In the
simple case of a pulsar with timing model parameters ($\boldsymbol{\theta}$),
ToAs ($t_{i}$), timing residuals ($r_{i}$), ToA uncertainties ($\sigma_{i}$),
DM measurements ($\text{DM}_{i}$), DM uncertainties ($\varsigma_{i}$), and DM
model ($d(t)$), the wideband log-likelihood can be written as
$\ln
L(\boldsymbol{r},\boldsymbol{\sigma},\boldsymbol{\text{DM}},\boldsymbol{\varsigma}|\boldsymbol{\theta})=\ln
L_{0}-\frac{1}{2}\sum_{i}\left[\left(\frac{r_{i}}{\sigma_{i}}\right)^{2}-\left(\frac{\text{DM}_{i}-d(t_{i})}{\varsigma_{i}}\right)^{2}\right]\,,$
(8)
where the first term is a normalization term, the second term is the usual
narrowband likelihood, and the third term is the likelihood function of the DM
measurements. A more general version of the above equation, applicable to more
rigorous noise models can be found in Appendix B of Alam et al. (2021).
### 2.2 Extending the Wideband technique for multiple bands
The standard wideband timing technique, summarized in section 2.1, has been
applied to various single band observations across a wide range of observing
frequencies (Fonseca et al., 2021; Alam et al., 2021; Nobleson et al., 2022;
Tarafdar et al., 2022; Curyło et al., 2023). In this section, we present and
demonstrate two novel independent methods namely the Combined Portrait (CP)
method and Combined Chi-squared (CC) method to combine simultaneous
observations performed in two non-contiguous frequency bands within the
paradigm of wideband technique to estimate a single ToA and DM combination per
epoch covering the entire frequency range of these bands. It is
straightforward to extend these techniques to multiple bands with simultaneous
observations, which will be part of a future work.
#### 2.2.1 The Combined-portrait (CP) method
In this method, we begin by time-collapsing the frequency-resolved profiles
obtained simultaneously in the two frequency bands using the pam command of
PSRCHIVE (Hotan et al., 2004). We then combine the profiles in the two bands
along the frequency axis using the psradd command of PSRCHIVE. This requires
both profiles to have the same number of phase bins; hence, we phase-collapse
the higher-phase resolution profile to match the lower-phase resolution one
using the pam command before appending them using psradd. The profiles of each
frequency band are also collapsed in frequency to an appropriate number of
sub-bands such that there is a reasonable signal-to-noise ratio (S/N) in each
sub-band, and there are also enough sub-bands to obtain a 2-D template
containing information of profile evolution across the band. Since the exact
start time of the observation in each band may not be identical, the profiles
are aligned by the psradd command by rotating them in phase using the pulsar
ephemeris used for folding. For generating a noise-free template portrait, we
use an epoch with high-S/N in both the bands. We first excise frequency
channels with any residual RFI from both the bands for the template epoch
using the pazi command and then obtain a combined data profile using psradd
covering the frequency of the two bands. Finally, a template portrait is
generated from this combined data profile using the ppalign and ppspline
modules of PulsePortraiture (Pennucci et al., 2014; Pennucci, 2019). Here, the
spline model is interpolated over the large frequency gap in between the two
bands. For accurate modeling of the profile evolution across the two bands, we
choose the required number of eigenprofiles and tolerance values for the
template portrait using the procedure described in section 4.2 of Tarafdar et
al. (2022). A single wideband ToA and the corresponding DM for the combined
observation of each epoch are then estimated using the ppToAs module of
PulsePortraiture.
#### 2.2.2 The Combined Chi-squared (CC) method
In this method, we treat the data portraits and the corresponding templates
for each band in their native phase resolution (without phase-collapsing or
combining them along the frequency axis) and bandwidths. We use the time-
collapsed data of two bands and partially collapse the frequency channels in
each band to maintain a reasonable S/N in each sub-band. The noise-free
templates are generated for each band separately using a high-S/N epoch after
RFI excision. Here, there is no interpolation of spline model over the large
frequency gap in between two bands, as both bands are treated separately.
While generating the templates, we take care of the phase offset between
multiple bands by rotating them appropriately. We estimate a single ToA and DM
pair for multiple bands in each epoch by minimizing a combined chi-squared
statistic defined as
$\chi^{2}(\phi_{\text{ref}},\text{DM})=\sum_{b}\left\\{S_{b}-\sum_{n}\frac{C_{bn}^{2}}{T_{bn}}\right\\}\,,$
(9)
where the index $b$ labels the different bands, and $S_{b}$, $C_{bn}$ and
$T_{bn}$ are defined by equations (6) using the data portrait $D_{bnj}$ and
the template portrait $P_{bnj}$ for each band $b$. Since the timestamp
$t_{fb}$ corresponding to the fiducial phase for different bands need not be
the same, equation (2) should be modified as follows:
$\phi_{bn}=\phi_{\text{ref}}+\frac{K\times\text{DM}}{P_{s}}\left(\frac{1}{\nu_{n}^{2}}-\frac{1}{\nu_{\text{ref}}^{2}}\right)-\delta_{b}\,,$
(10)
where
$\delta_{b}=\frac{t_{fb}-t_{f0}}{P_{s}}\,,$ (11)
and we have arbitrarily chosen the band labeled $b=0$ as the reference and
$P_{s}$ is the pulsar spin period222In this work, we are considering the
period from the center of the observation. The frequency $\nu_{\text{ref}}$ is
chosen such that the covariance between $\phi_{\text{ref}}$ and the DM implied
by equation (9) vanishes. Note that this method preserves the full phase
resolution available in each band since the number of phase bins need not be
equal for the different bands in equations (6) and (9)-(11).
#### 2.2.3 Wideband Timing with tempo2 using libstempo
The wideband likelihood was previously only available in tempo (Nice et al.,
2015) and PINT (Luo et al., 2021). In this work, We implement the wideband
likelihood using the libstempo (Vallisneri, 2020) python interface to perform
wideband timing in the tempo2 framework (refer Appendix B for details). We
considered DMEFAC and T2EFAC333DMEFAC and T2EFAC are white noise parameters
used to scale the DM and ToA uncertainties, respectively to account for the
radiometer noise contribution to the DM and ToA uncertainties, respectively.
These are estimated via a $\chi^{2}$-implementation done with libstempo and
the optimum fit parameters for various pulsars were chosen as per the InPTA
DR1 Narrowband timing (Tarafdar et al., 2022). The DMEFAC and T2EFAC values
were estimated such that the reduced $\chi^{2}$ obtained by iteratively
fitting the timing parameters is close to unity along with the post-fit
weighted RMS to be of the order of a few 100s of ns to a few $\mu$s, for each
of the Band 3, CC, and CP ToAs. In this way, ToAs obtained from the
combination of data from two non-contiguous frequency bands are timed for the
first time within the paradigm of the wideband technique.
## 3 Application on PSR J1909–3744
PSR J1909$-$3744 is a binary MSP with a rotational period $P_{s}\sim$2.95 ms.
It was discovered using the Parkes 64-m Radio Telescope in the Swinburne High
Latitude Pulsar Survey (Jacoby et al., 2003). It is one of the best pulsars
for PTA studies (Verbiest et al., 2016; Perera et al., 2019) due to its sharp
pulse profile, low-profile evolution with the radio frequency, and well-
studied timing model (Liu et al., 2020). Here, we demonstrate and validate the
CC and CP methods (§2.2) using the uGMRT Band 3 and Band 5 data of PSR
J1909$-$3744 from Cycles 37-40 (MJDs $58781-59496$), with 200 MHz bandwidth
(BW), obtained as a part of the InPTA campaign (Tarafdar et al., 2022). We
used MJD $59630$ as the template epoch obtained from InPTA observations of
Cycle 41 of the uGMRT. The details of observations and data reduction
procedures for these datasets can be found in Susobhanan et al. (2021) and
Tarafdar et al. (2022).
### 3.1 Combined Portrait (CP) method
As discussed in subsection 2.2.1, the CP method requires the phase resolution
of two bands to be the same for combining the data. The Band 5 uGMRT data of
the InPTA campaign is configured to be recorded with a smaller time resolution
than the Band 3 data, which leads to a smaller number of phase bins in Band 5
than in Band 3 when the data is folded. Hence, we phase-collapsed the Band 3
data to the same number of phase bins as those of Band 5 before appending the
two bands using psradd.
A comparison of wideband DM time series of Band 3+5 (CP) and Band 3 is shown
in figure 1, where the Band 3 DM time series is obtained while preserving the
original phase resolution. The Kendall Tau correlation coefficient (Kendall,
1938) of value 0.7188 and $p$-value $\sim\times 10^{-12}$ indicates a good
agreement between the two DM time series. We also see a slight offset between
Band 3 and Band 3+5 CP method DM time series (refer section 4.2 for a
discussion). In figure 2, we compare the DM (left panel) and ToA (right panel)
precisions of Band 3+5 (CP) and Band 3 time series. The points lying below the
dashed diagonal line indicate an improved DM or ToA precision with the CP
method compared to Band 3 results and vice versa.
We see in figure 2 that all epochs do not show an improved DM precision, and
most of the epochs show a worsened ToA precision i.e., we found a decrement in
the median precision or an increase in the median uncertainties values
($\sigma_{\text{DM}}$ and $\sigma_{\text{ToA}}$) of Band 3+5 (CP) results
compared to Band 3. This is primarily due to the decreased phase resolution of
Band 3 data used in the CP method. Hence, for combining bands, the CP method
has a disadvantage, especially for MSPs like J1909$-$3744, wherein the pulse
profile is sharp with minimal features, leading to only a few phase bins in
the pulse region of the profile upon toning down the phase resolution which
leads to a loss of information content, and thereby leading to poor template
construction as well as bad DM and ToA estimates.
Figure 1: Wideband DMs obtained from the Combined Portrait (CP) method for
Band 3+5 data along with traditional Band 3 (single-band) results for PSR
J1909$-$3744\. The top panel shows the DM time series for Band 3+5 CP (green
points) and Band 3 (blue points) overlaid for comparison. The Kendall Tau
coefficient on top right shows the correlation between the two time series
with the mentioned $p$-value. The bottom panel shows the DM differences
($\delta$DM in units of $10^{-5}$ pc cm-3) on subtracting both time series
(Band 3+5 $-$ Band 3) where the median value is shown by the dashed line and
the dash-dotted lines representing the MAD-band (equivalent to $3\sigma$
contour). Figure 2: A comparison of Wideband DM (left panel) and ToA (right
panel) uncertainties obtained from the CP method for Band 3+5 data is shown
against those obtained from the traditional Band 3 data for PSR J1909$-$3744\.
The Band 3+5 uncertainties estimated with the CP method are shown on the
vertical axis and Band 3 uncertainties on the horizontal axis (in units of
$10^{-5}\;\text{pc}\;\text{cm}^{-3}$ for DM uncertainties, and in units of
$\mu\text{s}$ for ToA uncertainties). The diagonal dash-dotted line shows the
$y=x$ curve. Points lying below the diagonal line show that the obtained
uncertainties with the CP method are lower compared to Band 3 results and vice
versa.
### 3.2 Combined Chi-squared (CC) method
The CC method preserves the native phase resolution of Band 3 and Band 5 data
as well as the template portraits, since it incorporates them within a
combined Fourier domain $\chi^{2}$-statistic as described in subsection 2.2.2.
Figure 3 shows the Band 3+5 DM time series obtained using the CC method in
comparison with the Band 3 DM time series, wherein we can see that the Band
3+5 DMs bear a high positive correlation with the Band 3 DMs, showing a good
agreement between the two. Figure 4 shows the Band 3+5 (CC) DM (left panel)
and ToA (right panel) uncertainties in comparison with the Band 3 results. We
can see that the Band 3+5 $\sigma_{\text{DM}}$ values are smaller than those
of Band 3 (all lying below the $y=x$ curve), hence showing a universal
improvement in the median DM precision after band-combination. The Band 3+5
$\sigma_{\text{ToA}}$ values are also slightly less compared to Band 3 leading
to an improvement in the median ToA precision as well.
Overall, the CC method provides significant improvements for Band 3+5 compared
to Band 3 results, especially because of preserving the native Band 3 phase
resolution, unlike the CP method. The templates are also more effectively
modelled because of applying PCA separately on each band without the need for
interpolating over a large frequency gap of $760$ MHz between Band 3 and Band
5. Similarly, when we apply PCA to model the template for the CP method on the
Band 3+5 data obtained using psradd, there is a possibility that the PCA
method may not be interpolating the profile evolution accurately due to the
wide band separation ($\sim 760$ MHz) between Band 3 and Band 5 data.
Therefore, CC method comes out as a more robust method for the band
combination.
Figure 3: Wideband DMs obtained from the Combined Chi-squared (CC) method for
Band 3+5 data along with traditional Band 3 (single-band) results for PSR
J1909$-$3744\. The top panel shows the DM time series for Band 3+5 CC (magenta
points) and Band 3 (blue points) overlaid for comparison. The Kendall Tau
coefficient on top right shows the correlation between the two time series
with the mentioned p-value. The bottom panel shows the DM differences
($\delta$DM in units of $10^{-5}\;\text{pc}\;\text{cm}^{-3}$) on subtracting
both time series (Band 3+5 $-$ Band 3), where the median value is shown by the
dashed line and the dash-dotted lines representing the MAD-band (equivalent to
$3\sigma$ contour). Figure 4: A comparison of Wideband DM (left panel) and ToA
(right panel) uncertainties obtained from the CC method for Band 3+5 data is
shown against those obtained from the traditional Band 3 data for PSR
J1909$-$3744\. The Band 3+5 uncertainties estimated with the CC method are
shown on the vertical axis and Band 3 uncertainties on the horizontal axis (in
units of $10^{-5}\;\text{pc}\;\text{cm}^{-3}$ for DM uncertainties, and in
units of $\mu\text{s}$ for ToA uncertainties). The diagonal dash-dotted line
shows the $y=x$ curve. Points lying below the diagonal line show that the
obtained uncertainties with the CC method are lower compared to Band 3 results
and vice versa.
### 3.3 Split-band test for the CC method
To validate the application of our novel CC method to combine the data of two
bands for estimating wideband DMs and ToAs, we perform a split-band test. In
this test, we consider one of the 200 MHz BW data (Band 3 is selected as it
has higher S/N than Band 5) and split it into two sub-bands each with a
bandwidth of 100 MHz using the psrsplit command of psrchive. We then estimate
the DM time series obtained by applying the CC method on these two sub-bands
and compare it with the wideband DM estimates obtained for the full 200 MHz BW
data. The split-band test results for PSR J1909-3744 are shown in figure 5,
where we can see that the DM values are in close agreement with Kendall Tau
value $\tau\sim 0.94$ and $p\sim 2\times 10^{-20}$ implying strong (positive)
correlation with the single band result for Band 3. The strong (positive)
correlation with negligible offsets indicates that the CC method for combining
bands, within the regime of the wideband technique, is working well. Hence,
the split-band test serves as a litmus test for validating the new technique.
Figure 5: The split-band test results for PSR J1909$-$3744\. The top panel
shows the wideband DM time series for Band 3 alone (blue points), and by
splitting it into two 100 MHz bands and using the CC method on the two sub-
bands (yellow points) overlaid for comparison. The Kendall Tau coefficient on
top right shows the correlation between the two time series with the mentioned
p-value. The bottom panel shows the DM differences ($\delta$DM in units of
$10^{-5}\;\text{pc}\;\text{cm}^{-3}$) on subtracting both time series (Band 3
split $-$ Band 3 single-band) with the median value shown by the dashed line
and the dash-dotted lines representing the MAD-band (equivalent to $3\sigma$
contour).
### 3.4 Wideband Timing results for PSR J1909$-$3744
The wideband timing results for PSR J1909$-$3744 obtained by implementing the
wideband likelihood with tempo2 using libstempo are shown in figure 6. The
DMEFAC and T2EFAC values are estimated for each of the Band 3, CC, and CP
ToAs. We then incorporate the T2EFAC and DMEFAC values to generate a global
timing solution. The post-fit timing residuals obtained from this procedure
are shown in figure 6. The post-fit weighted RMS ToA residual values for the
Band 3, CC, and CP timing residuals are obtained to be 0.235 $\mu$s, 0.326
$\mu$s, and 0.471 $\mu$s respectively, and are consistent with each other. We
fit the same parameters as fitted in the InPTA DR1 (Tarafdar et al., 2022)
narrowband timing method, which are F0 and F1 for the case of J1909$-$3744.
Figure 6: Timing results for PSR J1909-3744 obtained by implementing the wideband likelihood inside tempo2 using libstempo. The DMEFAC and T2EFAC values are estimated separately for the cases of Band 3 (blue points), CC (magenta points) and CP (green points) ToAs with only $200$ MHz data. The error bars shown in the plot have T2EFAC values incorporated. The post-fit weighted RMS ToA residual values for the Band 3, CC, and CP timing residuals are shown in the top right corner and the dashed horizontal line corresponds to zero residual level. | Parameter
---
Name (unit)
| Band 3
---
| Value | Uncertainty
---|---
| Band 3+5 CC
---
| Value | Uncertainty
---|---
| Band 3+5 CP
---
| Value | Uncertainty
---|---
F0 ($\text{s}^{-1}$) | | $339.31568666042$ | $3.1\times 10^{-11}$
---|---
| $339.31568666042$ | $2.6\times 10^{-11}$
---|---
| $339.31568666042$ | $1.8\times 10^{-11}$
---|---
F1 ($\text{s}^{-2}$) | | $-1.615\times 10^{-15}$ | $1.3\times 10^{-18}$
---|---
| $-1.615\times 10^{-15}$ | $1.1\times 10^{-18}$
---|---
| $-1.6158\times 10^{-15}$ | $8.1\times 10^{-19}$
---|---
Table 1: Table of fitted parameters obtained from the wideband timing of PSR
J1909$-$3744 using tempo2. The first column specifies the fitted parameters.
The second, third, and fourth columns list the Band 3, Band 3+5 CC, and Band
3+5 CP timing results respectively, enlisting the parameter values and their
uncertainties.
The post-fit RMS values for Band 3 and Band 3+5 (both CC and CP) are obtained
to be very close, while the fitted parameters estimated using the Band 3+5
methods are having better precision compared to the former, as shown in table
1. This clearly highlights the improvement in timing precision attained with
Band 3+5 wideband timing, especially the CC method.
## 4 Application on InPTA Data
We now present the results obtained by implementing the CP and CC methods, and
the wideband timing technique using tempo2 on the InPTA dual-band data (Band
3: 300$-$500 MHz and Band 5: 1260$-$1460 MHz) of 14 MSPs. The same data was
used for the first data release of InPTA (InPTA DR1: Tarafdar et al. (2022)).
| | DM uncertainties ($\text{pc}\;\text{cm}^{-3}$) |
---|---|---|---
| Pulsar
---
Name
| Band 3
---
| Median | Minimum
---|---
| Band 3+5 CC
---
| Median | Minimum
---|---
| Band 3+5 CP
---
| Median | Minimum
---|---
J0437$-$4715 | | $1.8\times 10^{-4}$ | $1.2\times 10^{-4}$
---|---
| $4.2\times 10^{-5}$ | $3.7\times 10^{-5}$
---|---
| $4.1\times 10^{-5}$ | $3.7\times 10^{-5}$
---|---
J0613$-$0200 | | $7.8\times 10^{-5}$ | $3.2\times 10^{-5}$
---|---
| $5.6\times 10^{-5}$ | $2.9\times 10^{-5}$
---|---
| $6.0\times 10^{-5}$ | $3.1\times 10^{-5}$
---|---
J0751$+$1807 | | $4.1\times 10^{-4}$ | $1.9\times 10^{-4}$
---|---
| $2.0\times 10^{-4}$ | $1.0\times 10^{-4}$
---|---
| $2.1\times 10^{-4}$ | $1.0\times 10^{-4}$
---|---
J1012$+$5307 | | $5.9\times 10^{-5}$ | $1.9\times 10^{-5}$
---|---
| $3.8\times 10^{-5}$ | $1.8\times 10^{-5}$
---|---
| $4.0\times 10^{-5}$ | $2.4\times 10^{-5}$
---|---
J1022$+$1001 | | $1.1\times 10^{-4}$ | $4.8\times 10^{-5}$
---|---
| $9.8\times 10^{-5}$ | $4.8\times 10^{-5}$
---|---
| $1.0\times 10^{-4}$ | $4.8\times 10^{-5}$
---|---
J1600$-$3053 | | $2.1\times 10^{-4}$ | $1.3\times 10^{-4}$
---|---
| $7.8\times 10^{-5}$ | $6.1\times 10^{-5}$
---|---
| $8.4\times 10^{-5}$ | $5.7\times 10^{-5}$
---|---
J1643$-$1224 | | $1.2\times 10^{-4}$ | $6.4\times 10^{-5}$
---|---
| $6.2\times 10^{-5}$ | $3.6\times 10^{-5}$
---|---
| $6.3\times 10^{-5}$ | $3.4\times 10^{-5}$
---|---
J1713$+$0747 | | $7.3\times 10^{-5}$ | $2.8\times 10^{-5}$
---|---
| $3.2\times 10^{-5}$ | $1.8\times 10^{-5}$
---|---
| $4.1\times 10^{-5}$ | $2.2\times 10^{-5}$
---|---
J1744$-$1134 | | $2.6\times 10^{-5}$ | $1.5\times 10^{-5}$
---|---
| $1.9\times 10^{-5}$ | $8.7\times 10^{-6}$
---|---
| $2.7\times 10^{-5}$ | $1.2\times 10^{-5}$
---|---
J1857$+$0943 | | $2.0\times 10^{-4}$ | $7.2\times 10^{-5}$
---|---
| $8.7\times 10^{-5}$ | $3.5\times 10^{-5}$
---|---
| $9.1\times 10^{-5}$ | $3.6\times 10^{-5}$
---|---
J1909$-$3744 | | $1.6\times 10^{-5}$ | $6.7\times 10^{-6}$
---|---
| $1.3\times 10^{-5}$ | $6.2\times 10^{-6}$
---|---
| $1.8\times 10^{-5}$ | $1.2\times 10^{-5}$
---|---
J1939$+$2134 | | $2.8\times 10^{-6}$ | $1.1\times 10^{-6}$
---|---
| $2.7\times 10^{-6}$ | $1.1\times 10^{-6}$
---|---
| $1.7\times 10^{-5}$ | $2.6\times 10^{-6}$
---|---
J2124$-$3358 | | $1.3\times 10^{-4}$ | $2.0\times 10^{-5}$
---|---
| $1.1\times 10^{-4}$ | $2.0\times 10^{-5}$
---|---
| $1.3\times 10^{-4}$ | $2.2\times 10^{-5}$
---|---
J2145$-$0750 | | $3.3\times 10^{-5}$ | $1.0\times 10^{-5}$
---|---
| $2.5\times 10^{-5}$ | $1.0\times 10^{-5}$
---|---
| $2.5\times 10^{-5}$ | $1.0\times 10^{-5}$
---|---
Table 2: Table of DM uncertainties (in units of $\text{pc}\;\text{cm}^{-3}$).
The first column specifies the pulsar names. The second column lists the
median and minimum errors in the DM estimation using the standard Wideband
technique on Band 3 (single-band). The third and fourth columns enlist the
median and minimum DM errors using the wideband CC and CP methods
respectively. All listed values are calculated by including both 100 MHz and
200 MHz bandwidth data.
### 4.1 Description of InPTA DR1
The InPTA DR1 (Tarafdar et al., 2022) constitutes 3.5 years of data
corresponding to the observations of 14 MSPs obtained using the uGMRT Gupta et
al. (2017). The data spans from 2018 to 2021 and has a typical cadence of two
weeks, carried out during uGMRT observing cycles 34$-$35 and 37$-$40\. These
observations were performed by dividing the 30 uGMRT antennae into multiple
phased subarrays which were used to observe the same source in multiple
frequency bands simultaneously. The data were recorded in total intensity mode
(Joshi et al., 2022). The GMRT Wideband Backend (GWB: Reddy et al., 2017) was
used to record the channelized time series data in binary format, and then
RFI-mitigated and reduced to PSRFITS archives using the pinta pipeline
Susobhanan et al. (2021). During cycles 34$-$35 we observed MSPs
simultaneously in Band 3 (400$-$500 MHz), Band 4 (650$-$750 MHz) and Band 5
(1360$-$1460 MHz) of uGMRT with 100 MHz bandwidth in each band. During cycles
37$-$40, we performed simultaneous observations only in Band 3 (300$-$500 MHz)
and Band 5 (1260$-$1460 MHz) with 200 MHz bandwidth. The Band 3 data in all
cycles as well as the Band 5 data in cycles 34$-$35 (except observations
between Oct. 20, 2018 and Nov. 14, 2018) were coherently dedispersed using a
real-time pipeline (De & Gupta, 2016) to the known DM of each pulsar. uGMRT
can perform coherent dedispersion on a total bandwidth of 200 MHz only, so in
cycles 34$-$35, observations were made with 100 MHz bandwidth in each band so
that both Band 3 and Band 5 data can be coherently dedispersed (Tarafdar et
al., 2022).
The Global Positioning System (GPS) was used to measure the narrowband ToAs
and the hydrogen maser clock at the uGMRT provided a local topocentric
frequency standard. The narrowband timing residuals in the InPTA DR1 were
obtained using tempo2 Hobbs et al. (2006). The timing residuals were also
generated from the wideband likelihood method Pennucci et al. (2014); Pennucci
(2019); Alam et al. (2021); Nobleson et al. (2022) using TEMPO Nice et al.
(2015) for Band 3 data only. The epoch-wise DM corrections were introduced in
the fit. The DMX parameters were calculated from the DM time series estimated
using DMcalc (Krishnakumar et al., 2021) for the narrowband timing from low-
frequency uGMRT data obtained in Band 3 and Band 5 simultaneously. Similarly,
DMX parameters were estimated using the wideband likelihood method for
wideband timing from Band 3 data of the uGMRT.
### 4.2 DM time series
Figure 7: Consolidated DM time series for 14 InPTA DR1 pulsars. The vertical
axes depict the difference ($\Delta\text{DM}$ in units of
$10^{-4}\;\text{pc}\;\text{cm}^{-3}$) between the fiducial DM and the
corresponding estimated DMs for each pulsar estimated by applying the standard
wideband technique on Band 3 data (blue points), using the wideband CC method
(magenta points) and the wideband CP method (green points) on Band 3+5 data.
The horizontal axes depict epochs in terms of the Modified Julian Date. The
estimated DM precision of 200 MHz bandwidth data is higher than that of 100
MHz bandwidth data, hence the horizontal axes are split into two parts at MJD
58600 with dashed vertical lines, with 100 MHz bandwidth epochs on the left
side and 200 MHz bandwidth epochs on the right side of the dashed line. The
vertical axes in each panel are scaled differently for 100 MHz (left axis) and
200 MHz (right axis) epochs such that the DM variations are clearly visible.
Pulsar names and their respective fiducial DM values are mentioned at the
bottom of each respective panel.
In this section, we present the wideband DM time series obtained for 14 InPTA
DR1 pulsars using the CC and CP methods, described in sections 2.2.1 and
2.2.2, for the combination of Band 3 and Band 5 InPTA data. We compare our
Band 3+5 combination results with the Band 3 (single-band) DM time series of
these pulsars estimated using the standard wideband method (described in
section 2.1). The template epochs used for ToA and DM estimation are the same
as those used in the InPTA DR1 analysis for all pulsars, which are high-S/N
epochs selected from Cycle 41 of the uGMRT for the respective pulsars. We also
keep the template epoch to be the same for Band 3 alone, the CC, and the CP
analysis to maintain consistency. In table 2, we have listed the median and
minimum uncertainties in DMs estimated for (i) Band 3, (ii) Band 3+5 CC, and
(iii) Band 3+5 CP, including both 100 MHz and 200 MHz data. It is evident from
the listed uncertainty values that there is a significant improvement in the
DM precision when Band 3 and Band 5 data is combined using the CC method.
However, for the CP method we find that the median DM precision goes slightly
down compared to Band 3 for PSRs J1744$-$1134, J1909$-$3744, and J1939$+$2134,
while remains same for PSRs J2124$-$3358\. Overall, there is nearly two times
increment in median DM precision using CC method for most of the pulsars.
A consolidated DM time-series plot illustrating the epoch-by-epoch DM
variations for all 14 InPTA DR1 pulsars is presented in figure 7. The plot
shows the Band 3, CC, and CP method results, where the vertical axes in both
panels depict the differences between the estimated DMs and the fiducial DMs.
The DM precisions estimated from 100 MHz bandwidth (BW) data are lower as
compared to those obtained from the 200 MHz bandwidth (BW) data, hence the
scaling of the vertical axes is made separately for these two cases to make
the DM variations over both 100 MHz and 200 MHz bandwidth epochs clearly
visible. The epochs having these two different bandwidths are separated along
the horizontal axis with a vertical dashed line at MJD 58600. The fiducial DM
value for each pulsar is mentioned inside the respective panel of the figure.
We have taken fiducial DM from the InPTA DR1 analysis. Refer (Tarafdar et al.,
2022) for more details.
In both CC and CP methods, we are combining over a large gap in frequency that
can cause differences in template portrait computations which are reflected as
systematic DC offsets in DM time-series of Band3, CC, and CP methods, as seen
in figure 7. A similar DM offset was also seen in the InPTA DR1 DM time-series
estimated from the narrowband and wideband analysis, which was found to be
caused by different templates used in the techniques (Tarafdar et al., 2022).
Here, in the CP method, we first psradd Band 3 and Band 5 data and then create
an analytic template using the standard wideband technique, which means that
the spline interpolation is done over a band-gap of $\sim$760 MHz. Whereas in
the CC method, we supply separate analytic wideband templates of Band 3 and
Band 5 which are internally used within the combined chi-square metric to
estimate DMs and ToAs. This leads to the selection of different number of
eigenprofiles and tolerance values (Pennucci et al., 2014; Pennucci, 2019) in
the CC and CP methods, leading to different analytical templates.
We have provided a series of plots for 14 InPTA DR1 pulsars in appendix A to
show a comparison between DM uncertainties estimated for Band 3 alone and Band
3+5 data with CC and CP methods. As the data of 100 and 200 MHz bandwidth have
different sensitivities, they have different scales of corresponding
uncertainties, hence we have presented them in different panels for each
pulsar. In the case of 100 MHz bandwidth data, we see a significant
improvement in the median DM precision for all pulsars with Band 3+5 data
using both CC and CP methods compared to Band 3 alone results. For the 200 MHz
bandwidth data, the CC method shows much higher improvement in the median DM
precision than the CP method for all pulsars except J0437$-$4715\. For PSRs
J1744$-$1134, J1909$-$3744, and J1939$+$2134, we find a decrement in the
median DM precision using the CP method compared to Band 3 alone for 200 MHz
data, whereas the CC method shows improvement for these pulsars also. Such
decrement in DM precision using CP method is expected due to reduced phase
resolution in Band 3 which affects pulsars with sharp pulse profiles as
explained in subsection 3.1. There is also a frequency gap of $\sim 750$ MHz
between Band 3 and Band 5 data which affects the modeling of profile evolution
across band edges in CP method, hence altering the results of pulsars with
high profile evolution with radio frequency. For PSR J1643$-$1224, we observe
that the trend in DM timeseries is not in agreement betweeen Band 3 and CC or
CP methods. This effect can be explained in terms of scattering variations.
PSR J1643-1224 has a highly scattered profile, especially at low radio
frequencies. At widely separated radio frequencies, scattered pulses sample
different path lengths through the ISM, which manifests as distinct variations
in DMs (McKee et al., 2018; Singha et al., 2023; Cordes et al., 2016;
Krishnakumar et al., 2019) estimated for Band 3 and combination of Band 3+5
using CC or CP methods as seen in figure 7.
### 4.3 ToAs and Timing residuals
We show a comparison of ToA uncertainties estimated for Band 3 alone and Band
3+5 data with CC and CP methods in a series of plots for 14 InPTA DR1 pulsars
in appendix A. Similar to DM precision, we see a significant improvement in
the median ToA precision for 100 MHz bandwidth data of all pulsars with Band
3+5 data using both CC and CP methods compared to Band 3 alone data. In the
case of 200 MHz bandwidth data, there is improvement in median ToA precision
using the CC method for PSRs J0751$+$1807, J1012$+$5307, J1600$-$3053,
J1643$-$1224, J1713$+$0747, J1744$-$1134, J1857$+$0943 and J2145$-$0750 while
it stays at par with Band 3 results for other pulsars. As the ToA precision
depends on the S/N, and as Band 5 S/N is comparatively lesser than Band 3 S/N,
therefore ToAs obtained after band combination, i.e. CC or CP ToAs, are not
able to achieve a significant improvement in ToA uncertainty for 200 MHz data
of all pulsars. The CP method shows improvement in median ToA precision than
Band 3 alone for PSRs J0751$+$1807, J1012$+$5307, J1022$+$1001, J1600$-$3053,
J1643$-$1224, and J2145$-$0750, whereas it decreases ToA precision for all
other pulsars which can be attributed to the aforementioned reasons.
A consolidated wideband timing residual plot obtained from Band 3, CC, and CP
ToAs for all the 14 InPTA DR1 pulsars is shown in figure 8. The timing
procedure that we followed is the same as that described in sections 2.2.3 and
3.4. The DMEFAC and T2EFAC values are estimated separately for $100$ MHz and
$200$ MHz BW data (as they have different sensitivities) for each of the Band
3, CC, and CP ToAs. We then incorporate the T2EFAC and DMEFAC values along
with combining the 100 MHz and 200 MHz BW data ToAs to generate a global
timing solution. The details of timing parameter estimates obtained after
wideband timing using Band 3, CC, and CP method ToAs for all 14 pulsars are
mentioned in table 3, where the fit parameters are chosen as per InPTA DR1
narrowband timing (Tarafdar et al., 2022). We find that the precision of the
fitted parameters are improved when the timing is done on Band 3+5 data using
both CC and CP ToAs for most of the pulsars.
Figure 8: Consolidated wideband timing residuals for 14 InPTA DR1 pulsars. The
post-fit wideband timing residuals (in units of $\mu\text{s}$) obtained from
Band 3 ToAs (blue points), and using the wideband CC (magenta points) and CP
(green points) methods on the Band 3+5 combination ToAs by implementing
wideband timing technique in tempo2, are plotted against the corresponding
epochs. Pulsar names and their respective post-fit weighted RMS residuals for
Band 3, CC, and CP methods (Band 3+5) are mentioned at the bottom of the
respective panels. The vertical axes in each panel are scaled differently for
100 MHz (left axis) and 200 MHz (right axis) epochs and a vertical dashed line
is added such that the residual variations are clearly visible. Epochs are
depicted in terms of the Modified Julian Date on the consolidated horizontal
axes.
Pulsar Method Timing Parameters RA (hh:mm:ss) DEC ($\deg$:mm:ss) PMRA (mas/yr)
A1 (lt s) F0 ($\mathrm{s}^{-1}$) F1 ($\mathrm{s}^{-2}$) PB (s) Value Error
($\mathrm{s}$) Value Error ($\mathrm{s}$) Value Error ((mas/yr)) Value Error
($\mathrm{lt}\ \mathrm{s}$) Value Error ($\mathrm{s}^{-1}$) Value Error
($\mathrm{s}^{-2}$) Value Error ($\mathrm{s}$) J0437$-$4715 Band 3
$4:37:16.0432$ $5.9\times 10^{-4}$ $-47:15:09.991$ $9.5\times 10^{-3}$ —— ——
$3.36673$ $1.2\times 10^{-5}$ $173.6879451409$ $3.7\times 10^{-10}$ —— ——
$5.741045808$ $5.0\times 10^{-9}$ CC $4:37:16.04316$ $8.3\times 10^{-5}$
$-47:15:09.991$ $1.1\times 10^{-3}$ —— —— $3.366759$ $1.6\times 10^{-6}$
$173.68794514023$ $4.2\times 10^{-11}$ —— —— $5.7410458046$ $8.1\times
10^{-10}$ CP $4:37:16.04316$ $9.1\times 10^{-5}$ $-47:15:09.991$ $1.2\times
10^{-3}$ —— —— $3.366759$ $1.8\times 10^{-6}$ $173.68794514020$ $4.5\times
10^{-11}$ —— —— $5.74104580486$ $8.8\times 10^{-10}$ J0613$-$0200 —— —— —— ——
—— —— —— —— —— —— —— —— —— —— —— J0751$+$1807 —— —— —— —— —— —— —— —— —— —— ——
—— —— —— —— J1012$+$5307 Band 3 —— —— —— —— —— —— —— —— $190.26783422728$
$1.0\times 10^{-11}$ —— —— —— —— CC —— —— —— —— —— —— —— —— $190.267834227285$
$4.8\times 10^{-12}$ —— —— —— —— CP —— —— —— —— —— —— —— —— $190.267834227289$
$4.8\times 10^{-12}$ —— —— —— —— J1022+1001 —— —— —— —— —— —— —— —— —— —— ——
—— —— —— —— J1600$-$3053 Band 3 —— —— —— —— —— —— —— —— $277.937706735932$
$6.8\times 10^{-11}$ —— —— —— —— CC —— —— —— —— —— —— —— ——
$277.9377067359737$ $8.9\times 10^{-12}$ —— —— —— —— CP —— —— —— —— —— —— ——
—— $277.937706735978$ $8.8\times 10^{-12}$ —— —— —— —— J1643$-$1224 Band 3 ——
—— —— —— $5.5$ $1.0$ $25.072598$ $1.7\times 10^{-6}$ $216.37333684399$
$1.2\times 10^{-11}$ $-8.599\times 10^{-16}$ $8.2\times 10^{-19}$
$147.01739775$ $4.0\times 10^{-8}$ CC —— —— —— —— $7.1$ $0.4$ $25.0725981$
$8.9\times 10^{-7}$ $216.373336843944$ $6.3\times 10^{-12}$ $-8.637\times
10^{-16}$ $3.2\times 10^{-19}$ $147.01739786$ $2.1\times 10^{-8}$ CP —— —— ——
—— $7.2$ $0.4$ $25.0725978$ $9.3\times 10^{-7}$ $216.373336843935$ $6.5\times
10^{-12}$ $-8.635\times 10^{-16}$ $3.3\times 10^{-19}$ $147.01739786$
$2.3\times 10^{-8}$ J1713$+$0747 Band 3 —— —— —— —— —— —— $32.3424310$
$5.7\times 10^{-7}$ —— —— —— —— $67.825130884$ $2.5\times 10^{-9}$ CC —— —— ——
—— —— —— $32.3424298$ $2.1\times 10^{-7}$ —— —— —— —— $67.8251308817$
$8.9\times 10^{-10}$ CP —— —— —— —— —— —— $32.3424297$ $2.5\times 10^{-7}$ ——
—— —— —— $67.825130881$ $1.0\times 10^{-9}$ J1744$-$1134 Band 3 —— —— —— —— ——
—— —— —— $245.42611950378$ $3.0\times 10^{-11}$ —— —— —— —— CC —— —— —— —— ——
—— —— —— $245.42611950381$ $2.6\times 10^{-11}$ —— —— —— —— CP —— —— —— —— ——
—— —— —— $245.42611950381$ $1.6\times 10^{-11}$ —— —— —— —— J1857$+$0943 Band
3 —— —— —— —— —— —— —— —— $186.49407816357$ $1.5\times 10^{-11}$ $-6.22\times
10^{-16}$ $1.2\times 10^{-18}$ —— —— CC —— —— —— —— —— —— —— ——
$186.494078163546$ $3.6\times 10^{-12}$ $-6.205\times 10^{-16}$ $1.6\times
10^{-19}$ —— —— CP —— —— —— —— —— —— —— —— $186.494078163547$ $2.8\times
10^{-12}$ $-6.204\times 10^{-16}$ $1.4\times 10^{-19}$ —— —— J1909$-$3744 Band
3 —— —— —— —— —— —— —— —— $339.31568666040$ $1.0\times 10^{-11}$
$-1.6142\times 10^{-15}$ $4.4\times 10^{-19}$ —— —— CC —— —— —— —— —— —— —— ——
$339.315686660410$ $2.7\times 10^{-12}$ $-1.6149\times 10^{-15}$ $1.7\times
10^{-19}$ —— —— CP —— —— —— —— —— —— —— —— $339.315686660412$ $2.3\times
10^{-12}$ $-1.6152\times 10^{-15}$ $1.4\times 10^{-19}$ —— —— J1939$+$2134
Band 3 —— —— —— —— —— —— —— —— $641.92820961498$ $1.2\times 10^{-11}$
$-4.33096\times 10^{-14}$ $7.5\times 10^{-19}$ —— —— CC —— —— —— —— —— —— ——
—— $641.928209615009$ $6.7\times 10^{-12}$ $-4.33091\times 10^{-14}$
$4.1\times 10^{-19}$ —— —— CP —— —— —— —— —— —— —— —— $641.928209615108$
$6.7\times 10^{-12}$ $-4.33058\times 10^{-14}$ $2.6\times 10^{-19}$ —— ——
J2124-3358 —— —— —— —— —— —— —— —— —— —— —— —— —— —— —— J2145$-$0750 Band 3 ——
—— —— —— —— —— —— —— $62.295887797432$ $1.7\times 10^{-12}$ $-1.155\times
10^{-16}$ $1.0\times 10^{-19}$ $6.8389026151$ $1.0\times 10^{-10}$ CC —— —— ——
—— —— —— —— —— $62.2958877974360$ $9.1\times 10^{-13}$ $-1.1565\times
10^{-16}$ $4.5\times 10^{-20}$ $6.838902615$ $5.5\times 10^{-11}$ CP —— —— ——
—— —— —— —— —— $62.29588779744$ $1.0\times 10^{-12}$ $-1.1547\times 10^{-16}$
$4.2\times 10^{-20}$ $6.8389026152$ $4.4\times 10^{-11}$
Table 3: Table of timing parameters for 14 InPTA DR1 pulsars. The first column
lists the pulsar names. The second column lists the methodology used to obtain
ToAs which are then used for estimating the timing parameters in separate
rows, namely the Band 3, CC, and CP methods. Columns three to nine represent
various fitted pulsar timing parameters, their units, and their uncertainties.
The choice of the fitted parameters for each pulsar is consistent with the
timing analysis of Tarafdar et al. (2022), where no timing parameters are fit
for PSRs J0613$-$0200, J0751+1807, J1022+1001, and J2124$-$3358.
## 5 Summary and Conclusions
Table of wRMS and $\chi^{2}$
---
| Pulsar Name | Method | ToA $\chi^{2}$ | DM $\chi^{2}$ | DOF | Total red. $\chi^{2}$ | wRMS
---|---|---|---|---|---|---
J0437$-$4715 | B3 | 0.087 | 10.196 | 9 | 1.143 | 0.201
CC | 0.631 | 9.777 | 9 | 1.156 | 0.217
CP | 0.585 | 8.928 | 9 | 1.057 | 0.225
J0613$-$0200 | B3 | 7.034 | 17.621 | 21 | 1.174 | 0.963
CC | 2.729 | 31.737 | 21 | 1.641 | 0.555
CP | 7.271 | 21.607 | 21 | 1.375 | 0.972
J0751$+$1807 | B3 | 0.845 | 22.384 | 23 | 1.01 | 0.654
CC | 0.785 | 27.978 | 23 | 1.251 | 0.348
CP | 1.582 | 35.636 | 23 | 1.618 | 0.302
J1012$+$5307 | B3 | 6.088 | 28.914 | 22 | 1.591 | 0.77
CC | 6.631 | 24.177 | 22 | 1.4 | 0.53
CP | 2.558 | 21.554 | 22 | 1.096 | 0.306
J1022$+$1001 | B3 | 0.398 | 21.283 | 24 | 0.903 | 0.264
CC | 0.448 | 20.798 | 24 | 0.885 | 0.214
CP | 0.734 | 32.475 | 24 | 1.384 | 0.286
J1600$-$3053 | B3 | 1.72 | 19.029 | 21 | 0.988 | 1.018
CC | 9.953 | 16.848 | 21 | 1.276 | 1.804
CP | 13.781 | 13.766 | 21 | 1.312 | 2.178
J1643$-$1224 | B3 | 9.984 | 164.109 | 63 | 2.763 | 0.869
CC | 4.317 | 63.795 | 63 | 1.081 | 0.257
CP | 3.799 | 63.037 | 63 | 1.061 | 0.269
J1713$+$0747 | B3 | 5.203 | 49.358 | 43 | 1.269 | 0.399
CC | 18.088 | 48.758 | 43 | 1.555 | 0.417
CP | 25.567 | 111.641 | 43 | 3.191 | 0.516
J1744$-$1134 | B3 | 3.859 | 11.284 | 14 | 1.082 | 0.49
CC | 3.413 | 11.203 | 14 | 1.044 | 0.434
CP | 2.348 | 10.309 | 14 | 0.904 | 0.255
J1857$+$0943 | B3 | 1.315 | 31.023 | 34 | 0.951 | 0.323
CC | 6.395 | 38.517 | 34 | 1.321 | 0.352
CP | 11.967 | 33.109 | 34 | 1.326 | 0.589
J1909$-$3744 | B3 | 5.944 | 50.301 | 56 | 1.004 | 0.235
CC | 11.395 | 45.588 | 56 | 1.018 | 0.326
CP | 19.304 | 35.852 | 56 | 0.985 | 0.471
J1939$+$2134 | B3 | 2.827 | 61.999 | 63 | 1.029 | 0.148
CC | 9.856 | 69.861 | 63 | 1.265 | 0.253
CP | 23.987 | 74.188 | 63 | 1.558 | 0.329
J2124$-$3358 | B3 | 4.094 | 63.051 | 39 | 1.722 | 0.515
CC | 4.697 | 67.579 | 39 | 1.853 | 0.324
CP | 5.501 | 73.437 | 39 | 2.024 | 0.54
J2145$-$0750 | B3 | 2.284 | 40.514 | 44 | 0.973 | 0.231
CC | 11.141 | 60.259 | 44 | 1.623 | 0.335
CP | 19.252 | 92.857 | 44 | 2.548 | 0.472
Table 4: Table of reduced chi-squares of 14 InPTA DR1 pulsars. The first
column lists the pulsar names. The second column lists the methodology used to
obtain ToAs and DM, namely the Band 3, CC, and CP methods. Columns three to
seven represent ToA component of the chi-square, DM component of the chi-
square, degrees of freedom (DOF), the total reduced chi-square and the wRMS of
the timing residuals.
In this work, we have developed two independent novel techniques, namely the
Combined Portrait (CP) and Combined Chi-squared (CC) methods, to combine data
simultaneously recorded in two non-contiguous frequency bands within the
paradigm of wideband technique (Pennucci et al., 2014; Pennucci, 2019) to
obtain a single DM and ToA per epoch encapsulating information contained in
both the bands. In the CP method, we create an auxiliary dataset by combining
the data of two frequency bands to create a single 2-dimensional analytic
template containing the information on pulse profile evolution with frequency.
This template is then used for cross-correlation with other epochs to obtain
wideband DMs and ToAs. In the CC method, we create separate 2-dimensional
analytic templates for both the bands, and these are integrated within a
combined Fourier-domain $\chi^{2}$-statistic and perform a global fit over the
whole frequency space to generate a single wideband DM and ToA per epoch. We
have applied these two techniques to 14 millisecond pulsars observed under the
InPTA campaign using uGMRT in Band 3 and Band 5 frequency bands
simultaneously, and they are included in the first data release of the InPTA
(Tarafdar et al., 2022).
We obtained high-precision DMs and ToAs for Band 3+5 data using these
techniques. We observe that combining the data having 100 MHz bandwidth in
each band showed consistent improvement in DM and ToA precision for all 14
pulsars and that both CC and CP are performing equally well. However, the
combination of data having 200 MHz bandwidth in each band shows
inconsistencies using the CP method. This is due to the reduction in the
number of phase bins in Band 3 which is essential to combine it with Band 5
data to create a single analytic template of Band 3+5. Another caveat of the
CP method is the band gap of $\sim 760$ MHz, which needs to be interpolated
over, between two bands leading to probable imperfections in the modeling of
profile evolution with frequency across the bands. The combination of data
having 200 MHz bandwidth in each band using the CC method shows much higher
improvement in DM and ToA precision than the CP method and Band 3 alone. We
plan to extend these techniques further to combine simultaneously recorded
data of multiple non-contiguous bands in future work.
We have also incorporated the wideband likelihood in tempo2 using libstempo
for the first time. We perform the wideband timing analysis on ToAs obtained
from the CC and CP methods along with Band 3 ToAs for comparison. We achieved
the weighted RMS ToA residuals in the range of 214 ns to 1.8 $\mu$s for ToAs
obtained from the CC method, while in the range of 225 ns to 2.1 $\mu$s for
ToAs obtained from the CP method for the whole spectrum of InPTA DR1 pulsars.
We observe an improvement in the precision of fitted timing parameters with
Band 3+5 combination compared to Band 3 alone for all pulsars. Since we are
combining data of multiple frequency bands, we may require frequency-dependent
parameters to obtain a better fit for our timing solutions. This will be
explored in future work.
We observe that the DM chi-square, obtained from the DM part of the
likelihood, is larger than the ToA chi-square (see Table 4). We suspect that
this could be related to the way we estimate DMEFAC and T2EFAC parameters. We
plan to investigate this further in future work where we will apply Bayesian
methods to estimate optimum DMEFAC and T2EFAC parameter values.
## 6 Discussion and Future directions
The extension of the wideband technique to multiple non-contiguous frequency
bands demonstrated in this work, is likely to be useful in largely removing
chromatic noise sources, such as variations in the pulse profile, DM and
scattering, in precision timing experiments like pulsar timing arrays. This
technique not only improves the ToA precision significantly by accumulating
the signal over the entire frequency range of combined bands, it also takes
care of DM noise across the bands by incorporating DM-chromatic noise
measurements in the timing likelihood naturally (see Appendix B). This
restricts the noise analysis of PTA data to just the time-independent and
time-correlated achromatic and scattering noise sources, greatly simplifying
and constraining these noise models. This has implications both for the
computational needs as well as the sensitivity of PTA data for a GW search.
Other precision timing experiments targeted at measuring timing noise,
parameters of relativistic binary systems and tests of General Theory of
Relativity are also likely to benefit from this extension of the standard
wideband technique. With large upcoming and future telescopes, such as the SKA
(Kramer & Stappers, 2015; Janssen et al., 2015) and DSA (Hallinan et al.,
2019), likely to employ simultaneous observations over multiple bands with
frequency coverage as large as 5 GHz, we expect this extended technique or its
variants to be widely used in the future.
## Software
RFIclean (Maan et al., 2021), DSPSR (Straten & Bailes, 2011), PSRCHIVE (Hotan
et al., 2004), pinta (Susobhanan et al., 2021), PulsePortraiture (Pennucci et
al., 2014; Pennucci, 2019), tempo2 (Hobbs et al., 2006; Edwards et al., 2006),
libstempo (Vallisneri, 2020), tempo (Nice et al., 2015), numpy (Harris et al.,
2020), scipy (Virtanen et al., 2020), matplotlib (Hunter, 2007)
## Acknowledgements
We thank the staff of the GMRT who made our observations possible. GMRT is
operated by the National Centre for Radio Astrophysics of the Tata Institute
of Fundamental Research. AKP is supported by CSIR fellowship Grant number
$09/0079(15784)/2022$-EMR-I. BCJ acknowledges support from Raja Ramanna Chair
(Track - I) grant from the Department of Atomic Energy, Government of India.
KN is supported by the Birla Institute of Technology and Science Institute
fellowship. AS is supported by the NANOGrav NSF Physics Frontiers Center
(awards 1430284 and 2020265). DD acknowledges the support from the Department
of Atomic Energy, Government of India through ‘Apex Project - Advance Research
and Education in Mathematical Sciences at IMSc’. MB acknowledges the support
from the Department of Atomic Energy, Government of India through ‘Apex
Project - Advance Research and Education in Mathematical Sciences at IMSc’. YG
and BCJ acknowledges support from the Department of Atomic Energy, Government
of India, under project number 12-R&D-TFR-5.02-0700. TK is partially supported
by the JSPS Overseas Challenge Program for Young Researchers. AmS is supported
by CSIR fellowship Grant number $09/1001(12656)/2021$-EMR-I and DST-ICPS
T-641. KT is partially supported by JSPS KAKENHI Grant Numbers 20H00180,
21H01130, and 21H04467 and the ISM Cooperative Research Program
(2023-ISMCRP-2046). We thank Scott Ransom for his suggestions that improved
the manuscript.
## Data Availability
The python scripts used for the analysis are available in
https://github.com/AvinashKumarPaladi/Multiband-extension-of-Wideband-Timing-
Technique. The data underlying this article will be shared on reasonable
request to the corresponding author.
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## Appendix A DM and ToA Uncertainty Comparison Plots
The DM and ToA uncertainty ($\sigma_{\text{DM}}$ and $\sigma_{\text{ToA}}$)
comparison between Band 3 and Band 3+5 data – CC and CP methods is presented
here for all the 14 InPTA DR1 pulsars. Since the sensitivity of both
bandwidths is different, it is not visually feasible to plot all of them on
the same scale, hence we have shown them in different panels.
Figure 9: A comparison of Wideband DM (left panel) and ToA (right panel)
uncertainties obtained from the CC and CP methods for Band 3+5 data is shown
against those obtained from the traditional Band 3 data for PSR J0437$-$4715\.
This pulsar has only 100MHz InPTA data before Cycle 40 of the uGMRT. The Band
3+5 uncertainties estimated with the CC (magenta points) and CP (green points)
methods are shown on the vertical axis and Band 3 uncertainties on the
horizontal axis (in units of $10^{-5}\;\text{pc}\;\text{cm}^{-3}$ for DM
uncertainties, and in units of $\mu\text{s}$ for ToA uncertainties). The
diagonal dashed-dotted line shows the $y=x$ while the magenta and green dashed
lines indicate median DM (left panel) and ToA (right panel) uncertainties for
CC and CP method respectively. Points lying below the diagonal line indicate
that the obtained precision with Band 3+5 combination is better than Band 3
(single-band) results and vice versa. Hence, we obtain an overall increase in
the DM and ToA precisions from both CP and CC methods for PSR J0437$-$4715.
Figure 10: A comparison of Wideband DM (left panel) and ToA (right panel)
uncertainties obtained from the CC and CP methods for Band 3+5 data is shown
against those obtained from the traditional Band 3 data for PSR J0613$-$0200\.
The results are shown for both 100 MHz and 200 MHz bandwidth data. The curves
and colors displayed in the figure conform to the details mentioned in the
legend at the bottom. Figure 11: A comparison of Wideband DM (left panel) and
ToA (right panel) uncertainties obtained from the CC and CP methods for Band
3+5 data is shown against those obtained from the traditional Band 3 data for
PSR J0751$+$1807\. The results are shown for both 100 MHz and 200 MHz
bandwidth data. The curves and colors displayed in the figure conform to the
details mentioned in the legend at the bottom. Figure 12: A comparison of
Wideband DM (left panel) and ToA (right panel) uncertainties obtained from the
CC and CP methods for Band 3+5 data is shown against those obtained from the
traditional Band 3 data for PSR J1012$+$5307\. The results are shown for both
100 MHz and 200 MHz bandwidth data. The curves and colors displayed in the
figure conform to the details mentioned in the legend at the bottom. Figure
13: A comparison of Wideband DM (left panel) and ToA (right panel)
uncertainties obtained from the CC and CP methods for Band 3+5 data is shown
against those obtained from the traditional Band 3 data for PSR J1022$+$1001\.
The results are shown for both 100 MHz and 200 MHz bandwidth data. The curves
and colors displayed in the figure conform to the details mentioned in the
legend at the bottom. Figure 14: A comparison of Wideband DM (left panel) and
ToA (right panel) uncertainties obtained from the CC and CP methods for Band
3+5 data is shown against those obtained from the traditional Band 3 data for
PSR J1600$-$3053\. The results are shown for both 100 MHz and 200 MHz
bandwidth data. The curves and colors displayed in the figure conform to the
details mentioned in the legend at the bottom. Figure 15: A comparison of
Wideband DM (left panel) and ToA (right panel) uncertainties obtained from the
CC and CP methods for Band 3+5 data is shown against those obtained from the
traditional Band 3 data for PSR J1643$-$1224\. The results are shown for both
100 MHz and 200 MHz bandwidth data. The curves and colors displayed in the
figure conform to the details mentioned in the legend at the bottom. Figure
16: A comparison of Wideband DM (left panel) and ToA (right panel)
uncertainties obtained from the CC and CP methods for Band 3+5 data is shown
against those obtained from the traditional Band 3 data for PSR J1713$+$0747\.
The results are shown for both 100 MHz and 200 MHz bandwidth data. The curves
and colors displayed in the figure conform to the details mentioned in the
legend at the bottom. Figure 17: A comparison of Wideband DM (left panel) and
ToA (right panel) uncertainties obtained from the CC and CP methods for Band
3+5 data is shown against those obtained from the traditional Band 3 data for
PSR J1744$-$1134 with only 200 MHz data as it wasn’t observed in earlier
cycles. The curves and colors displayed in the figure conform to the details
mentioned in the legend at the bottom. Figure 18: A comparison of Wideband DM
(left panel) and ToA (right panel) uncertainties obtained from the CC and CP
methods for Band 3+5 data is shown against those obtained from the traditional
Band 3 data for PSR J1857$+$0943\. The results are shown for both 100 MHz and
200 MHz bandwidth data. The curves and colors displayed in the figure conform
to the details mentioned in the legend at the bottom. Figure 19: A comparison
of Wideband DM (left panel) and ToA (right panel) uncertainties obtained from
the CC and CP methods for Band 3+5 data is shown against those obtained from
the traditional Band 3 data for PSR J1909$-$3744\. The results are shown for
both 100 MHz and 200 MHz bandwidth data. The curves and colors displayed in
the figure conform to the details mentioned in the legend at the bottom.
Figure 20: A comparison of Wideband DM (left panel) and ToA (right panel)
uncertainties obtained from the CC and CP methods for Band 3+5 data is shown
against those obtained from the traditional Band 3 data for PSR J1939$+$2134\.
The results are shown for both 100 MHz and 200 MHz bandwidth data. The curves
and colors displayed in the figure conform to the details mentioned in the
legend at the bottom. Figure 21: A comparison of Wideband DM (left panel) and
ToA (right panel) uncertainties obtained from the CC and CP methods for Band
3+5 data is shown against those obtained from the traditional Band 3 data for
PSR J2124$-$3358\. The results are shown for both 100 MHz and 200 MHz
bandwidth data. The curves and colors displayed in the figure conform to the
details mentioned in the legend at the bottom. Figure 22: A comparison of
Wideband DM (left panel) and ToA (right panel) uncertainties obtained from the
CC and CP methods for Band 3+5 data is shown against those obtained from the
traditional Band 3 data for PSR J2145$-$0750\. The results are shown for both
100 MHz and 200 MHz bandwidth data. The curves and colors displayed in the
figure conform to the details mentioned in the legend at the bottom.
## Appendix B Implementing the wideband likelihood using tempo2 and libstempo
The wideband timing residuals $\delta t$ can be modeled as
$\delta t=M\epsilon+r$ (12)
The product of the timing model design matrix $M$ with small offsets in the
timing model parameters $\epsilon$ describes the systematic residuals from
subtracting the timing model. $r$ represents the uncorrelated noise in the
residuals.
The Narrowband likelihood for the timing residuals is given by
$p\left(\delta
t|\epsilon,\phi\right)=\frac{\exp\left(-\frac{1}{2}r^{T}N^{-1}r\right)}{\sqrt{|2\pi
N|}}$ (13) $N_{ij}=(E_{k(i)}^{2}\sigma_{i}^{2}+Q_{k(i)}^{2})\delta_{ij}$ (14)
where $\phi$ comprises of EFAC $E_{k(i)}$ and EQUAD $Q_{k(i)}$. $\sigma_{i}$
are the uncertainties in ToAs.
In wideband timing we have an additional likelihood term that includes the DMX
priors,
$p\left(\epsilon^{DMX}\;\delta
D,E^{DM}\right)=\frac{e^{-\frac{1}{2}\left(\left(\epsilon^{DMX}-\delta
D\right)^{T}N^{DM^{-1}}\left(\epsilon^{DMX}-\delta
D\right)\right)}}{\sqrt{|2\pi N^{DM}|}}$ (15)
$N_{ij}^{DM}=(E_{k(i)}^{DM}\sigma_{i}^{DM})^{2}\delta_{ij}$ (16)
where $E^{DM}$ is the DM EFAC and $\sigma_{i}^{DM}$ is the DM error.
$\epsilon^{DMX}$ represents subset of timing model offsets $\epsilon$ that
describe the piece-wise constant DMX model. $\delta D$ is the vector
containing difference of DM measurements with respect to the fiducial dm.
The complete wideband timing likelihood is given by the product of both
narrowband likelihood and likelihood containing DM priors, Alam et al. (2021)
$p\left(\epsilon,\phi,E^{DM}|\delta t,\delta D\right)\propto p\left(\delta
t|\epsilon,\phi\right)\times p\left(\epsilon^{DMX}|\delta D,E^{DM}\right)\ $
(17)
This wideband likelihood is implemented in python using libstempo, a python
wrapper for tempo2. We obtained the design matrix $M$ from libstempo by giving
the par file and ToAs for a particular pulsar as inputs. This design matrix is
then extended to account for $\delta D$ and $\epsilon^{DMX}$. Using the
extended design matrix, we estimate the timing residuals and DMX parameters
from equation 17 using the Generalized Least Squares (GLS) method.
|
0}\int_{0}^{T}\|f^{\delta}(\varphi^{\delta})\|_{L^{2}(\Gamma(t))}^{2}\leq C,$
and hence $\varphi(\varphi)\in L^{2}_{L^{2}}$. Now by the uniqueness of weak
limits and a suitable variant of the dominated convergence theorem for
evolving surfaces (see [18], Theorem B.2), one finds that
$f(\varphi)=\tilde{f}$.
## 6 Proof of uniqueness
### 6.1 Uniqueness for the regular potential
In this section we prove the uniqueness of solutions to (4.1)-(4.3). As a
preliminary result, we note that by elliptic regularity theory one has that
$\displaystyle\int_{0}^{T}\|\varphi\|_{H^{2}(\Gamma(t))}^{2}\leq
C\int_{0}^{T}\left(\|\mu\|_{L^{2}(\Gamma(t))}^{2}+\|F^{\prime}(\varphi)\|_{L^{2}(\Gamma(t))}^{2}\right).$
(6.1)
This $L^{2}_{H^{2}}$ regularity of $\varphi$ is invaluable for proving the
uniqueness of solutions, as it allows one to eliminate $\mu$ from (4.1). To
see this we notice that for almost all $t\in[0,T]$, and all
$\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)$
$\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi})=\int_{\Gamma(t)}\mu\nabla_{\Gamma}\varphi\cdot\boldsymbol{\phi}=\int_{\Gamma(t)}\left(-\varepsilon\Delta_{\Gamma}\varphi+\frac{1}{\varepsilon}F^{\prime}(\varphi)\right)\nabla_{\Gamma}\varphi\cdot\boldsymbol{\phi}.$
Now, as observed in the derivation, one formally calculates that
$-\varepsilon\Delta_{\Gamma}\varphi\nabla_{\Gamma}\varphi=-\varepsilon\nabla_{\Gamma}\cdot(\nabla_{\Gamma}\varphi\otimes\nabla_{\Gamma}\varphi)+\frac{\varepsilon}{2}\nabla_{\Gamma}|\nabla_{\Gamma}\varphi|^{2}-\varepsilon\left(\nabla_{\Gamma}\varphi\cdot\mathbb{H}\nabla_{\Gamma}\varphi\right)\boldsymbol{\nu},$
almost everywhere on $\Gamma(t)$. Thus one finds
$\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi})=\int_{\Gamma(t)}\nabla_{\Gamma}\left(\frac{\varepsilon}{2}|\nabla_{\Gamma}\varphi|^{2}+\frac{1}{\varepsilon}F(\varphi)\right)\cdot\boldsymbol{\phi}-\varepsilon\nabla_{\Gamma}\cdot(\nabla_{\Gamma}\varphi\otimes\nabla_{\Gamma}\varphi)\cdot\boldsymbol{\phi},$
where the normal term has vanished as $\boldsymbol{\phi}$ is tangential. Now
using integration by parts, and the fact that $\boldsymbol{\phi}$ is
solenoidal, it is clear that
$\displaystyle\int_{\Gamma(t)}\nabla_{\Gamma}\left(\frac{\varepsilon}{2}|\nabla_{\Gamma}\varphi|^{2}+\frac{1}{\varepsilon}F(\varphi)\right)\cdot\boldsymbol{\phi}=\int_{\Gamma(t)}\nabla_{\Gamma}\cdot\left(\frac{\varepsilon}{2}|\nabla_{\Gamma}\varphi|^{2}\boldsymbol{\phi}+\frac{1}{\varepsilon}F(\varphi)\boldsymbol{\phi}\right)-\int_{\Gamma(t)}\left(\frac{\varepsilon}{2}|\nabla_{\Gamma}\varphi|^{2}+\frac{1}{\varepsilon}F(\varphi)\right)\nabla_{\Gamma}\cdot\boldsymbol{\phi}=0,$
$\displaystyle-\varepsilon\int_{\Gamma(t)}\nabla_{\Gamma}\cdot(\nabla_{\Gamma}\varphi\otimes\nabla_{\Gamma}\varphi))\cdot\boldsymbol{\phi}=\varepsilon\int_{\Gamma(t)}(\nabla_{\Gamma}\varphi\otimes\nabla_{\Gamma}\varphi):\nabla_{\Gamma}\boldsymbol{\phi}.$
Hence we find
$\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi})=\varepsilon\mathbf{c}_{3}(\varphi,\varphi,\boldsymbol{\phi}),$
where the trilinear form $\mathbf{c}_{3}$ is defined as
$\mathbf{c}_{3}(t;\phi,\psi,\boldsymbol{\chi}):=\int_{\Gamma(t)}(\nabla_{\Gamma}\phi\otimes\nabla_{\Gamma}\psi):\nabla_{\Gamma}\boldsymbol{\chi}.$
The structure of this proof is similar to that in [29], with relevant
modifications for an evolving surface - as discussed in Appendix B.
###### Theorem 6.1.
Let $\Gamma(t),F$ be such that the assumptions in Theorem 4.1 hold. Moreover,
assume $F_{2}^{\prime}$ is Lipschitz continuous. Then the solution triple,
$(\varphi,\mu,\mathbf{u}_{T})$ solving (4.1)-(4.3) is unique.
The first step is to observe that if we have two solution triples,
$(\varphi^{i},\mu^{i},\mathbf{u}_{T}^{i})$, $i=1,2$, and defining
$\displaystyle\bar{\varphi}:=\varphi^{1}-\varphi^{2},$
$\displaystyle\bar{\mu}:=\mu^{1}-\mu^{2},$
$\displaystyle\bar{\mathbf{u}_{T}}:=\mathbf{u}_{T}^{1}-\mathbf{u}_{T}^{2},$
then these solve the system
$\displaystyle\begin{split}\mathbf{m}_{*}\left(\partial^{\circ}{\bar{\mathbf{u}_{T}}},\boldsymbol{\phi}\right)+\hat{\mathbf{a}}(\eta(\varphi^{1}),\mathbf{u}_{T}^{1},\boldsymbol{\phi})-\hat{\mathbf{a}}(\eta(\varphi^{2}),\mathbf{u}_{T}^{2},\boldsymbol{\phi})+\mathbf{c}_{1}(\mathbf{u}_{T}^{1},\mathbf{u}_{T}^{1},\boldsymbol{\phi})-\mathbf{c}_{1}(\mathbf{u}_{T}^{2},\mathbf{u}_{T}^{2},\boldsymbol{\phi})+\mathbf{l}(\bar{\mathbf{u}_{T}},\boldsymbol{\phi})+\mathbf{d}_{1}(\bar{\mathbf{u}_{T}},\boldsymbol{\phi})\\\
+\mathbf{d}_{2}(\eta(\varphi^{1})-\eta(\varphi^{2}),\boldsymbol{\phi})=\varepsilon\mathbf{c}_{3}(\varphi^{1},\varphi^{1},\boldsymbol{\phi})-\varepsilon\mathbf{c}_{3}(\varphi^{2},\varphi^{2},\boldsymbol{\phi})\end{split},$
(6.2) $\displaystyle
m_{*}\left(\partial^{\circ}{\bar{\varphi}},\phi\right)+a(\bar{\mu},\phi)+\mathbf{c}_{2}(\phi,\varphi^{1},\mathbf{u}_{T}^{1})-\mathbf{c}_{2}(\phi,\varphi^{2},\mathbf{u}_{T}^{2})+\mathbf{c}_{2}(\phi,\bar{\varphi},\widetilde{\mathbf{u}_{T}})=0,$
(6.3) $\displaystyle m(\bar{\mu},\phi)=\varepsilon
a(\bar{\varphi},\phi)+\frac{1}{\varepsilon}m(F^{\prime}(\varphi^{1})-F^{\prime}(\varphi^{2}),\phi),$
(6.4)
for almost all $t\in[0,T]$, and all $\phi\in
H^{1}(\Gamma(t)),\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)$. This proof
firstly requires obtaining bounds for
$\bar{\varphi},P_{\mathcal{K}}\bar{\mathbf{u}_{T}},P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}$
in appropriate norms. We refer to Appendix B for definitions and properties of
the operators $P_{\mathcal{K}},P_{\mathcal{K}^{\perp}}$.
The proof relies on proving the following differential inequalities.
###### Lemma 6.2.
$P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}$ is such that
$\frac{1}{2}\frac{d}{dt}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\frac{\eta_{*}}{2}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}\leq
C\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\frac{\varepsilon}{4}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}\\\
+K_{1}(t)\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+C_{\log}(\bar{\varphi},\mathbf{u}_{T}^{1})\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp},$
(6.5)
where
$K_{1}(t)=C\left(\|\mathbb{E}(\mathbf{u}_{T}^{1})\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))}^{4}+\|\mathbf{u}_{T}^{2}\|_{\mathbf{L}^{4}(\Gamma(t))}^{4}+\|\varphi^{1}\|_{H^{1,4}(\Gamma(t))}^{4}+\|\varphi^{2}\|_{H^{1,4}(\Gamma(t))}^{4}\right),$
and
$C_{\log}(\bar{\varphi},\mathbf{u}_{T}^{1})=C\log\left(\frac{C_{2}(\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}+\|\bar{\varphi}\|_{H^{2}(\Gamma(t))})}{\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}}\right)^{\frac{1}{2}}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}\|\mathbb{E}(\mathbf{u}_{T}^{1})\|_{\mathbf{L}^{2}(\Gamma(t))}.$
$P_{\mathcal{K}}\bar{\mathbf{u}_{T}}$ is such that
$\displaystyle\frac{1}{2}\frac{d}{dt}\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}\leq\frac{\eta_{*}}{4}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\frac{\varepsilon}{4}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}+K_{2}(t)\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2},$
(6.6)
where
$K_{2}(t)=C\left(1+\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{4}(\Gamma(t))}^{2}+\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))}^{2}+\|\mathbf{u}_{T}^{2}\|_{\mathbf{L}^{4}(\Gamma(t))}^{2}+\|\varphi^{1}\|_{H^{1,4}(\Gamma(t))}^{4}+\|\varphi^{2}\|_{H^{1,4}(\Gamma(t))}^{4}\right).$
Lastly, $\bar{\varphi}$ is such that
$\frac{1}{2}\frac{d}{dt}\|\bar{\varphi}\|_{-1}^{2}+\varepsilon\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}\leq\frac{\varepsilon}{4}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}+\frac{\eta_{*}}{4}\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\frac{\eta_{*}}{4}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}\\\
+K_{3}(t)\|\bar{\varphi}\|_{-1}^{2},$ (6.7)
where
$K_{3}(t)=C\left(1+\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{\infty}(\Gamma(t))}^{2}+\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))}^{2}+\|\varphi^{2}\|_{L^{\infty}(\Gamma(t))}^{2}\right).$
###### Proof of uniqueness.
With these bounds we are now in a position to show uniqueness. Taking the sum
of (6.5), (6.6), (6.7) one finds
$\frac{1}{2}\frac{d}{dt}\left(\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\|\bar{\varphi}\|_{-1}^{2}\right)+\frac{\varepsilon}{4}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}\\\
\leq
K(t)\left(\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\|\bar{\varphi}\|_{-1}^{2}\right)\\\
+C_{\log}(\bar{\varphi},\mathbf{u}_{T}^{1})\left(\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\|\bar{\varphi}\|_{-1}^{2}\right)^{\frac{1}{2}},$
where
$K(t)=K_{1}(t)+K_{2}(t)+K_{3}(t).$
Now recall that we have $\mathbf{u}_{T}^{i}\in L^{\infty}_{\mathbf{L}^{2}}\cap
L^{2}_{\mathbf{H}^{1}},\varphi^{i}\in L^{\infty}_{H^{1}}\cap L^{2}_{H^{2}}$,
and in particular this implies $\varphi^{i}\in L^{4}_{H^{1,4}}$ as
$\displaystyle\int_{0}^{T}\|\varphi^{i}\|_{H^{1,4}(\Gamma(t))}^{4}$
$\displaystyle\leq
C\int_{0}^{T}\|\varphi^{i}\|_{H^{1}(\Gamma(t))}^{2}\|\varphi^{i}\|_{H^{2}(\Gamma(t))}^{2}$
$\displaystyle\leq
C\left(\sup_{t\in[0,T]}\|\varphi^{i}\|_{H^{1}(\Gamma(t))}^{2}\right)\int_{0}^{T}\|\varphi^{i}\|_{H^{2}(\Gamma(t))}^{2}<\infty.$
Moreover
$\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}(0)\|_{\perp}^{2}+\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}(0)\|_{\mathbf{L}^{2}(\Gamma(t))}+\|\bar{\varphi}(0)\|_{-1}^{2}=0$
by definition, we see that we may use Lemma 3.10 to see that
$\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp},\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))},\|\bar{\varphi}\|_{-1}$
all vanish on $[0,T]$. In particular, since $\|\cdot\|_{\perp}$ and
$\|\cdot\|_{-1}$ are norms on $\mathcal{K}^{\perp}$ and $\\{z\in
H^{-1}(\Gamma(t))\mid m_{*}(z,1)=0\\}$ respectively we see that
$P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}=0,\quad
P_{\mathcal{K}}\bar{\mathbf{u}_{T}}=0,\quad\bar{\varphi}=0,$
and as
$\bar{\mathbf{u}_{T}}=P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}+P_{\mathcal{K}}\bar{\mathbf{u}_{T}}$
we see $\bar{\mathbf{u}_{T}}$ vanishes on $[0,T]$ also. From this one can
readily show that $\bar{\mu}=0$ on $[0,T]$ and hence determine uniqueness of
weak solutions. ∎
#### 6.1.1 Proof of Lemma 6.2
###### Proof.
Firstly we establish the most complicated inequality, (6.5). Testing (6.2)
with $\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}$, as defined in Appendix B, and
rewriting terms in a suitable way we find that
$\mathbf{m}_{*}\left(\partial^{\circ}{\bar{\mathbf{u}_{T}}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\right)+\hat{\mathbf{a}}(\eta(\varphi^{1})-\eta(\varphi^{2}),\mathbf{u}_{T}^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})+\hat{\mathbf{a}}(\eta(\varphi^{2}),\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})+\mathbf{c}_{1}(\bar{\mathbf{u}_{T}},\mathbf{u}_{T}^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})+\mathbf{c}_{1}(\mathbf{u}_{T}^{2},\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})\\\
+\mathbf{l}(\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})+\mathbf{d}_{1}(\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})+\mathbf{d}_{2}(\eta(\varphi^{1})-\eta(\varphi^{2}),\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})=\varepsilon\mathbf{c}_{3}(\bar{\varphi},\varphi^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})+\varepsilon\mathbf{c}_{3}(\varphi^{2},\bar{\varphi},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}).$
(6.8)
Firstly, we claim that
$\mathbf{m}_{*}\left(\partial^{\circ}{\bar{\mathbf{u}_{T}}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\right)=\frac{1}{2}\frac{d}{dt}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\frac{1}{2}\mathbf{b}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})-\mathbf{m}(P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}HV_{N},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})\\\
+\sum_{i=1}^{N_{K}}\mathbf{m}(\bar{\mathbf{u}_{T}},\boldsymbol{\kappa}_{i})\mathbf{m}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\partial^{\circ}\boldsymbol{\kappa}_{i}),$
(6.9)
where we are using the notation from Appendix B. To see this we write
$\mathbf{m}_{*}\left(\partial^{\circ}{\bar{\mathbf{u}_{T}}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\right)=\frac{d}{dt}\mathbf{m}(\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})-\mathbf{m}\left(\bar{\mathbf{u}_{T}},\partial^{\circ}{\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}}\right)-\mathbf{m}\left(\bar{\mathbf{u}_{T}}HV_{N},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\right).$
Since $\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\in\mathcal{K}^{\perp}(t)$ we
see that
$\mathbf{m}(\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})=\mathbf{m}(P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})$,
and hence by the definition of $\mathcal{S}^{\perp}$
$\frac{d}{dt}\mathbf{m}(\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})=\frac{d}{dt}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2},$
and so all that remains is to consider
$\mathbf{m}\left(\bar{\mathbf{u}_{T}},\partial^{\circ}{\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}}\right)$.
Firstly, we split this term into two parts
$\mathbf{m}\left(\bar{\mathbf{u}_{T}},\partial^{\circ}{\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}}\right)=\mathbf{m}\left(P_{\mathcal{K}}\bar{\mathbf{u}_{T}},\partial^{\circ}{\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}}\right)+\mathbf{m}\left(P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}},\partial^{\circ}{\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}}\right).$
We write this first term as
$\mathbf{m}\left(P_{\mathcal{K}}\bar{\mathbf{u}_{T}},\partial^{\circ}{\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}}\right)=\frac{d}{dt}\mathbf{m}\left(P_{\mathcal{K}}\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\right)-\mathbf{m}\left(P_{\mathcal{K}}\bar{\mathbf{u}_{T}}HV_{N},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\right)-\mathbf{m}_{*}\left(\partial^{\circ}P_{\mathcal{K}}\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\right),$
where clearly
$\mathbf{m}\left(P_{\mathcal{K}}\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\right)=0$.
Next we use (B.5) to see that
$\mathbf{m}_{*}\left(\partial^{\circ}P_{\mathcal{K}}\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\right)=\mathbf{m}_{*}\left(\partial^{\circ}\bar{\mathbf{u}_{T}},P_{\mathcal{K}}\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\right)\\\
+\sum_{i=1}^{N_{K}}\left[\mathbf{m}(\bar{\mathbf{u}_{T}},\partial^{\circ}\boldsymbol{\kappa}_{i})+\mathbf{m}(\bar{\mathbf{u}_{T}}HV_{N},\boldsymbol{\kappa}_{i})\right]\mathbf{m}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\boldsymbol{\kappa}_{i})+\mathbf{m}(\bar{\mathbf{u}_{T}},\boldsymbol{\kappa}_{i})\mathbf{m}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\partial^{\circ}\boldsymbol{\kappa}_{i}),$
which simplifies to
$\mathbf{m}_{*}\left(\partial^{\circ}P_{\mathcal{K}}\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\right)=\sum_{i=1}^{N_{K}}\mathbf{m}(\bar{\mathbf{u}_{T}},\boldsymbol{\kappa}_{i})\mathbf{m}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\partial^{\circ}\boldsymbol{\kappa}_{i}).$
Next we write
$\mathbf{m}\left(P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}},\partial^{\circ}{\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}}\right)=\mathbf{a}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\partial^{\circ}\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})=\frac{1}{2}\frac{d}{dt}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}-\frac{1}{2}\mathbf{b}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}).$
Combining all of this together one obtains (6.9).
Next we rewrite the second $\hat{\mathbf{a}}$ term in (6.8) by using
integration by parts. To do this we note that
$\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\in L^{2}_{H^{2}}$ and we have the
bound (B.4). Integration by parts yields
$\hat{\mathbf{a}}(\eta(\varphi^{2}),\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})=\int_{\Gamma(t)}\eta(\varphi^{2})\mathbb{E}(\bar{\mathbf{u}_{T}}):\mathbb{E}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})=-\int_{\Gamma(t)}\bar{\mathbf{u}_{T}}\cdot\mathbb{P}\nabla_{\Gamma}\cdot\left(\eta(\varphi^{2})\mathbb{E}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})\right)\\\
=-\int_{\Gamma(t)}\eta^{\prime}(\varphi^{2})\bar{\mathbf{u}_{T}}\cdot\mathbb{E}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})\nabla_{\Gamma}\varphi^{2}+\int_{\Gamma(t)}\eta(\varphi^{2})\bar{\mathbf{u}_{T}}\cdot
P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}},$
where we have used the fact that
$-\mathbb{P}\nabla_{\Gamma}\cdot\mathbb{E}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})=P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}$
a.e. on $\Gamma(t)$. Hence one obtains
$\hat{\mathbf{a}}(\eta(\varphi^{2}),\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})=-\int_{\Gamma(t)}\eta^{\prime}(\varphi^{2})\bar{\mathbf{u}_{T}}\cdot\mathbb{E}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})\nabla_{\Gamma}\varphi^{2}+\int_{\Gamma(t)}\eta(\varphi^{2})P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\cdot
P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\\\
+\int_{\Gamma(t)}\eta(\varphi^{2})P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\cdot
P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}.$ (6.10)
Next we bound the other $\hat{\mathbf{a}}$ term. To do this we write
$\hat{\mathbf{a}}(\eta(\varphi^{1})-\eta(\varphi^{2}),\mathbf{u}_{T}^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})=\int_{\Gamma(t)}\left(\int_{0}^{1}\eta^{\prime}(su^{1}+(1-s)\varphi^{2})\bar{\varphi}\,ds\right)\mathbb{E}(\mathbf{u}_{T}^{1}):\mathbb{E}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}),$
and noting the boundedness of $\eta^{\prime}$ we obtain the bound
$\hat{\mathbf{a}}(\eta(\varphi^{1})-\eta(\varphi^{2}),\mathbf{u}_{T}^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})\leq
C\|\bar{\varphi}\|_{L^{\infty}(\Gamma(t))}\|\mathbb{E}(\mathbf{u}_{T}^{1})\|_{\mathbf{L}^{2}(\Gamma(t))}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}.$
Now we use Lemma 3.11, and Poincaré’s inequality, to see that
$\|\bar{\varphi}\|_{L^{\infty}(\Gamma(t))}\leq
C\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}\left(1+\log\left(1+\frac{C\|\bar{\varphi}\|_{H^{2}(\Gamma(t))}}{\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}}\right)^{\frac{1}{2}}\right),$
where we note from (6.1) that $\bar{\varphi}$ is sufficiently smooth for this.
Now from the $L^{\infty}_{H^{1}}$ bounds for $\varphi_{1},\varphi_{2}$ we see
that there is a constant, $C_{1}$, such that
$\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}\leq C_{1}$ for almost all
$t\in[0,T]$. Hence one can find a sufficiently large constant $C_{2}$ so that
$\log\left(1+\frac{C\|\bar{\varphi}\|_{H^{2}(\Gamma(t))}}{\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}}\right)\leq\log\left(\frac{C_{2}(\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma)}+\|\bar{\varphi}\|_{H^{2}(\Gamma(t))})}{\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}}\right).$
Moreover for $C_{2}>1$ this logarithmic term is positive, and we will
ultimately be able to apply Lemma 3.10. All in all this gives us the bound
$|\hat{\mathbf{a}}(\eta(\varphi^{1})-\eta(\varphi^{2}),\mathbf{u}_{T}^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|\leq\frac{\varepsilon}{8}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}+C\|\mathbb{E}(\mathbf{u}_{T}^{1})\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}\\\
+C\log\left(\frac{C_{2}(\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}+\|\bar{\varphi}\|_{H^{2}(\Gamma(t))})}{\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}}\right)^{\frac{1}{2}}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}\|\mathbb{E}(\mathbf{u}_{T}^{1})\|_{\mathbf{L}^{2}(\Gamma(t))}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp},$
(6.11)
where we have used Young’s inequality where appropriate.
Now we bound the $\mathbf{c}_{1}$ terms. Firstly we note that
$|\mathbf{c}_{1}(\bar{\mathbf{u}_{T}},\mathbf{u}_{T}^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|=|\mathbf{c}_{1}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\mathbf{u}_{T}^{1},\bar{\mathbf{u}_{T}})|\leq\|\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\|_{\mathbf{H}^{1,4}(\Gamma(t))}\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}.$
We recall the interpolation inequality777See [10], and note that a $C^{3}$
surface is sufficiently smooth for this to hold. This can be extended to
evolving surfaces with a time independent constant as in [45] Lemma 3.4.,
$\|\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\|_{\mathbf{H}^{1,4}(\Gamma(t))}\leq
C\|\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\|_{\mathbf{H}^{1}(\Gamma(t))}^{\frac{1}{2}}\|\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\|_{\mathbf{H}^{2}(\Gamma(t))}^{\frac{1}{2}},$
and use (B.1), (B.4) to see that
$\|\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\|_{\mathbf{H}^{1,4}(\Gamma(t))}\leq
C\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{\frac{1}{2}}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{\frac{1}{2}}\leq
C\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{\frac{1}{2}}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{\frac{1}{2}}.$
Hence we observe that
$|\mathbf{c}_{1}(\bar{\mathbf{u}_{T}},\mathbf{u}_{T}^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|\leq
C\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{\frac{1}{2}}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{\frac{3}{2}},$
and hence Young’s inequality yields
$\displaystyle|\mathbf{c}_{1}(\bar{\mathbf{u}_{T}},\mathbf{u}_{T}^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|\leq
C\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))}^{4}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\frac{\eta_{*}}{14}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}.$
(6.12)
An identical argument yields
$\displaystyle|\mathbf{c}_{1}(\mathbf{u}_{T}^{2},\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|\leq
C\|\mathbf{u}_{T}^{2}\|_{\mathbf{L}^{4}(\Gamma(t))}^{4}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\frac{\eta_{*}}{14}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}.$
(6.13)
We now turn to the contributions from the evolution of the surface, that is
the terms involving $\mathbf{l},\mathbf{d}_{1},\mathbf{d}_{2}$, which would
vanish for a stationary surface. The simplest of these terms is
$\displaystyle\mathbf{l}(\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})\leq
C\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}\|\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}\leq\frac{\eta_{*}}{14}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+C\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2},$
(6.14)
where we have use Young’s inequality and (B.1). Next we look at
$\displaystyle\mathbf{d}_{2}(\eta(\varphi^{1})-\eta(\varphi^{2}),\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})\leq
C\|\bar{\varphi}\|_{L^{2}(\Gamma(t))}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}\leq\frac{\varepsilon}{12}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}+C\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2},$
(6.15)
which follows similarly to the above inequality, but we have also use the
Lipschitz continuity of $\eta(\cdot)$, Poincaré’s inequality, and the uniform
bounds on $\widetilde{\mathbf{u}_{T}}$. Finally to bound the $\mathbf{d}_{1}$
term we see
$\mathbf{d}_{1}(\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})=\mathbf{c}_{1}(\bar{\mathbf{u}_{T}},\widetilde{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})+\mathbf{c}_{1}(\widetilde{\mathbf{u}_{T}},\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})\\\
=-\mathbf{c}_{1}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\widetilde{\mathbf{u}_{T}},\bar{\mathbf{u}_{T}})-\mathbf{c}_{1}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\bar{\mathbf{u}_{T}},\widetilde{\mathbf{u}_{T}})+2\mathbf{m}(\bar{\mathbf{u}_{T}}HV_{N},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}),$
where we have used the antisymmetry of $\mathbf{c}_{1}$ (for solenoidal
functions), and the extra term comes from the fact that
$\nabla_{\Gamma}\cdot\widetilde{\mathbf{u}_{T}}=-HV_{N}$. From this one
readily sees that
$|\mathbf{d}_{1}(\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|\leq
C\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{\infty}(\Gamma(t))}+C\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp},$
where we have used (B.1). This clearly yields
$\displaystyle|\mathbf{d}_{1}(\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|\leq
C\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{\infty}(\Gamma(t))}^{2}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\frac{\eta_{*}}{14}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}.$
(6.16)
The last terms for us to bound are the $\mathbf{c}_{3}$ contributions, from
which one readily sees from Young’s inequality that
$\displaystyle\varepsilon|\mathbf{c}_{3}(\bar{\varphi},\varphi^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|+\varepsilon|\mathbf{c}_{3}(\varphi^{2},\bar{\varphi},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|\leq\frac{\varepsilon}{12}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}+C\left(\|\varphi^{1}\|_{H^{1,4}(\Gamma(t))}^{2}+\|\varphi^{2}\|_{H^{1,4}(\Gamma(t))}^{2}\right)\|\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\|_{\mathbf{H}^{1,4}(\Gamma(t))}^{2}.$
Again using the above interpolation inequality we see that
$\|\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}\|_{\mathbf{H}^{1,4}(\Gamma(t))}^{2}\leq
C\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}.$
Hence using Young’s inequality we find
$\varepsilon|\mathbf{c}_{3}(\bar{\varphi},\varphi^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|+\varepsilon|\mathbf{c}_{3}(\varphi^{2},\bar{\varphi},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|\leq\frac{\varepsilon}{12}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}+C\left(\|\varphi^{1}\|_{H^{1,4}(\Gamma(t))}^{4}+\|\varphi^{2}\|_{H^{1,4}(\Gamma(t))}^{4}\right)\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}\\\
+\frac{\eta_{*}}{14}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2},$
(6.17)
where we note that $\varphi^{i}\in L^{4}_{H^{1,4}}$ as shown above.
Now we use (6.9)-(6.17) in (6.8) to see that
$\frac{1}{2}\frac{d}{dt}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\int_{\Gamma(t)}\eta(\varphi^{2})P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\cdot
P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\leq\left|\int_{\Gamma(t)}\eta^{\prime}(\varphi^{2})\bar{\mathbf{u}_{T}}\cdot\mathbb{E}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})\nabla_{\Gamma}\varphi^{2}\right|+\left|\int_{\Gamma(t)}\eta(\varphi^{2})P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\cdot
P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\right|\\\
+\frac{\varepsilon}{4}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}+C\|\mathbb{E}(\mathbf{u}_{T}^{1})\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+C\left(\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))}^{4}+\|\mathbf{u}_{T}^{2}\|_{\mathbf{L}^{4}(\Gamma(t))}^{4}\right)\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\frac{6\eta_{*}}{14}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}\\\
+\frac{\eta_{*}}{14}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+C\left(\|\varphi^{1}\|_{H^{1,4}(\Gamma(t))}^{4}+\|\varphi^{2}\|_{H^{1,4}(\Gamma(t))}^{4}\right)\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}\\\
+C\log\left(\frac{C_{2}(\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}+\|\bar{\varphi}\|_{H^{2}(\Gamma(t))})}{\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}}\right)^{\frac{1}{2}}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}\|\mathbb{E}(\mathbf{u}_{T}^{1})\|_{\mathbf{L}^{2}(\Gamma(t))}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp},$
where we have also bounded the extra terms in (6.9) as
$\frac{1}{2}|\mathbf{b}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|+|\mathbf{m}(P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}HV_{N},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})|+\sum_{i=1}^{N_{K}}|\mathbf{m}(\bar{\mathbf{u}_{T}},\boldsymbol{\kappa}_{i})\mathbf{m}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\partial^{\circ}\boldsymbol{\kappa}_{i})|\leq
C\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}\\\
+\frac{\eta_{*}}{14}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\frac{\eta_{*}}{14}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2},$
where we use (B.1) and the uniform bounds on
$H,V_{N},\boldsymbol{\kappa}_{i},\partial^{\circ}\boldsymbol{\kappa}_{i}$
appropriately. Now recalling that $\eta_{*}\leq\eta(\cdot)\leq\eta^{*}$,
$\eta^{\prime}$ is bounded, and that by construction
$\displaystyle\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}=\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2},$
(6.18)
one can use Young’s inequality to arrive at (6.5).
Now we establish (6.6). Testing (6.2) with
$P_{\mathcal{K}}\bar{\mathbf{u}_{T}}$, we see that the $\hat{\mathbf{a}}$ and
$\mathbf{d}_{2}$ terms vanish and one has
$\mathbf{m}_{*}\left(\partial^{\circ}{\bar{\mathbf{u}_{T}}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\right)+\mathbf{c}_{1}(\bar{\mathbf{u}_{T}},\mathbf{u}_{T}^{1},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})+\mathbf{c}_{1}(\mathbf{u}_{T}^{2},\bar{\mathbf{u}_{T}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})+\mathbf{l}(\bar{\mathbf{u}_{T}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})+\mathbf{d}_{1}(\bar{\mathbf{u}_{T}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})\\\
=\varepsilon\mathbf{c}_{3}(\bar{\varphi},\varphi^{1},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})+\varepsilon\mathbf{c}_{3}(\varphi^{2},\bar{\varphi},P_{\mathcal{K}}\bar{\mathbf{u}_{T}}),$
(6.19)
where we have rewritten the nonlinear terms in a more appropriate way. As
before we pull out the derivative term so that
$\mathbf{m}_{*}\left(\partial^{\circ}{\bar{\mathbf{u}_{T}}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\right)=\frac{1}{2}\frac{d}{dt}\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}-\mathbf{m}(\bar{\mathbf{u}_{T}}HV_{N},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})+\frac{1}{2}\mathbf{m}(P_{\mathcal{K}}\bar{\mathbf{u}_{T}}HV_{N},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})\\\
-\sum_{i=1}^{N_{K}}\mathbf{m}(\bar{\mathbf{u}_{T}},\boldsymbol{\kappa}_{i})\mathbf{m}(P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}},\partial^{\circ}\boldsymbol{\kappa}_{i}).$
(6.20)
To see this, we see that
$\mathbf{m}_{*}\left(\partial^{\circ}\bar{\mathbf{u}_{T}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\right)=\frac{d}{dt}\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}-\mathbf{m}(\bar{\mathbf{u}_{T}}HV_{N},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})-\mathbf{m}_{*}\left(\partial^{\circ}P_{\mathcal{K}}\bar{\mathbf{u}_{T}},\bar{\mathbf{u}_{T}}\right).$
Then one writes
$\mathbf{m}_{*}\left(\partial^{\circ}P_{\mathcal{K}}\bar{\mathbf{u}_{T}},\bar{\mathbf{u}_{T}}\right)=\mathbf{m}_{*}\left(\partial^{\circ}P_{\mathcal{K}}\bar{\mathbf{u}_{T}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\right)+\mathbf{m}_{*}\left(\partial^{\circ}P_{\mathcal{K}}\bar{\mathbf{u}_{T}},P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\right),$
where the transport theorem lets us see that
$\mathbf{m}_{*}\left(\partial^{\circ}P_{\mathcal{K}}\bar{\mathbf{u}_{T}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\right)=\frac{1}{2}\frac{d}{dt}\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}-\frac{1}{2}\mathbf{m}(P_{\mathcal{K}}\bar{\mathbf{u}_{T}}HV_{N},P_{\mathcal{K}}\bar{\mathbf{u}_{T}}).$
The term involving $P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}$ is then dealt
with by using (B.5) so that
$\displaystyle\mathbf{m}_{*}\left(\partial^{\circ}P_{\mathcal{K}}\bar{\mathbf{u}_{T}},P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\right)=\sum_{i=1}^{N_{K}}\mathbf{m}(\bar{\mathbf{u}_{T}},\boldsymbol{\kappa}_{i})\mathbf{m}(P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}},\partial^{\circ}\boldsymbol{\kappa}_{i}),$
and combining all this together yields (6.20).
Next we consider the $\mathbf{c}_{1}$ terms. We notice that
$|\mathbf{c}_{1}(\bar{\mathbf{u}_{T}},\mathbf{u}_{T}^{1},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})|=|\mathbf{c}_{1}(P_{\mathcal{K}}\bar{\mathbf{u}_{T}},\mathbf{u}_{T}^{1},\bar{\mathbf{u}_{T}})|\leq\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{H}^{1,4}(\Gamma(t))}\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))},$
and by using (B.3) one can deduce
$|\mathbf{c}_{1}(\bar{\mathbf{u}_{T}},\mathbf{u}_{T}^{1},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})|\leq
C\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}.$
The same calculations for the other $\mathbf{c}_{1}$ term and Young’s
inequality yields
$\displaystyle|\mathbf{c}_{1}(\bar{\mathbf{u}_{T}},\mathbf{u}_{T}^{1},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})|+|\mathbf{c}_{1}(\mathbf{u}_{T}^{2},\bar{\mathbf{u}_{T}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})|\leq\frac{\eta_{*}}{12}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+C\left(\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))}^{2}+\|\mathbf{u}_{T}^{2}\|_{\mathbf{L}^{4}(\Gamma(t))}^{2}\right)\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}.$
(6.21)
We then turn to the terms involving $\mathbf{l},\mathbf{d}_{1}$. As seen
before, it is straightforward to see that
$\displaystyle\mathbf{l}(\bar{\mathbf{u}_{T}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})\leq\frac{\eta_{*}}{12}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t)}^{2}+C\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}.$
(6.22)
We then express the $\mathbf{d}_{1}$ term as
$\mathbf{d}_{1}(\bar{\mathbf{u}_{T}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})=\mathbf{c}_{1}(\bar{\mathbf{u}_{T}},\widetilde{\mathbf{u}_{T}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})+\mathbf{c}_{1}(\widetilde{\mathbf{u}_{T}},\bar{\mathbf{u}_{T}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})\\\
=-\mathbf{c}_{1}(P_{\mathcal{K}}\bar{\mathbf{u}_{T}},\widetilde{\mathbf{u}_{T}},\bar{\mathbf{u}_{T}})-\mathbf{c}_{1}(P_{\mathcal{K}}\bar{\mathbf{u}_{T}},\bar{\mathbf{u}_{T}},\widetilde{\mathbf{u}_{T}})+2\mathbf{m}(\bar{\mathbf{u}_{T}}HV_{N},P_{\mathcal{K}}\bar{\mathbf{u}_{T}}),$
and so arguing as we did for the previous $\mathbf{d}_{1}$ term we find
$\displaystyle\mathbf{d}_{1}(\bar{\mathbf{u}_{T}},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})\leq\frac{\eta_{*}}{12}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+C\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{4}(\Gamma(t))}^{2}\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}.$
(6.23)
Lastly we bound the $\mathbf{c}_{3}$ terms as before, where we find
$|\mathbf{c}_{3}(\bar{\varphi},\varphi^{1},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})|\leq\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}\|\varphi^{1}\|_{H^{1,4}(\Gamma(t))}\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{H}^{1,4}(\Gamma(t))}\leq
C\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma)}\|\varphi^{1}\|_{H^{1,4}(\Gamma(t))}\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))},$
where we have used (B.3) in the last inequality. From this, using Young’s
inequality it is straightforward to see
$\varepsilon|\mathbf{c}_{3}(\bar{\varphi},\varphi^{1},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})|+\varepsilon|\mathbf{c}_{3}(\varphi^{2},\bar{\varphi},P_{\mathcal{K}}\bar{\mathbf{u}_{T}})|\leq\frac{\varepsilon}{4}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}\\\
+C\left(\|\varphi^{1}\|_{H^{1,4}(\Gamma(t))}^{2}+\|\varphi^{2}\|_{H^{1,4}(\Gamma(t))}^{2}\right)\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}.$
(6.24)
Hence using (6.20)-(6.24) and (6.18) in (6.19) yields (6.6).
The final inequality to show is (6.7). To do this we test (6.3) with
$\mathcal{G}\bar{\varphi}$, which we note is well defined as
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}\bar{\varphi}=0$ for almost all
$t\in[0,T]$, which yields
$\displaystyle
m_{*}\left(\partial^{\circ}{\bar{\varphi}},\mathcal{G}\bar{\varphi}\right)+a(\bar{\mu},\mathcal{G}\bar{\varphi})+\mathbf{c}_{2}(\mathcal{G}\bar{\varphi},\bar{\varphi},\mathbf{u}_{T}^{1})+\mathbf{c}_{2}(\mathcal{G}\bar{\varphi},\varphi^{2},\bar{\mathbf{u}_{T}})+\mathbf{c}_{2}(\mathcal{G}\bar{\varphi},\bar{\varphi},\widetilde{\mathbf{u}_{T}})=0.$
(6.25)
The first term can be expressed as
$\displaystyle
m_{*}\left(\partial^{\circ}{\bar{\varphi}},\mathcal{G}\bar{\varphi}\right)=\frac{1}{2}\frac{d}{dt}\|\bar{\varphi}\|_{-1}^{2}-m(\bar{\varphi}HV_{N},\mathcal{G}\bar{\varphi})+\frac{1}{2}b(\mathcal{G}\bar{\varphi},\mathcal{G}\bar{\varphi}).$
(6.26)
To see this we express this term as
$m_{*}\left(\partial^{\circ}{\bar{\varphi}},\mathcal{G}\bar{\varphi}\right)=\frac{d}{dt}m\left(\bar{\varphi},\mathcal{G}\bar{\varphi}\right)-m\left(\bar{\varphi},\partial^{\circ}{\mathcal{G}\bar{\varphi}}\right)-m(\bar{\varphi}HV_{N},\mathcal{G}\bar{\varphi}),$
and note
$m(\bar{\varphi},\mathcal{G}\bar{\varphi})=\|\bar{\varphi}\|_{-1}^{2}$, and
that
$m\left(\bar{\varphi},\partial^{\circ}{\mathcal{G}\bar{\varphi}}\right)=a\left(\mathcal{G}\bar{\varphi},\partial^{\circ}{\mathcal{G}\bar{\varphi}}\right)=\frac{1}{2}\frac{d}{dt}\|\bar{\varphi}\|_{-1}^{2}-\frac{1}{2}b(\mathcal{G}\bar{\varphi},\mathcal{G}\bar{\varphi}).$
To bound the second term, we see from the definition of the inverse Laplacian
that
$a(\bar{\varphi},\mathcal{G}\bar{\varphi})=m(\bar{\varphi},\bar{\varphi})$,
and hence testing (6.3) with $\bar{\varphi}$ we see
$a(\bar{\mu},\mathcal{G}\bar{\varphi})=\varepsilon
a(\bar{\varphi},\bar{\varphi})+\frac{1}{\varepsilon}m(F^{\prime}(\varphi^{1})-F^{\prime}(\varphi^{2}),\bar{\varphi}).$
We recall that $F=F_{1}+F_{2}$ where $F_{1}$ is convex, so that
$m(F^{\prime}(\varphi^{1})-F^{\prime}(\varphi^{2}),\bar{\varphi})\geq
m(F_{2}^{\prime}(\varphi^{1})-F_{2}^{\prime}(\varphi^{2}),\bar{\varphi}).$
By using the Lipschitz continuity of $F_{2}^{\prime}$, and the definition of
the inverse Laplacian, one readily sees that
$\displaystyle|m(F_{2}^{\prime}(\varphi^{1})-F_{2}^{\prime}(\varphi^{2}),\bar{\varphi})|\leq
C\|\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}=Ca(\bar{\varphi},\mathcal{G}\bar{\varphi})\leq\frac{\varepsilon}{16}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}+C\|\bar{\varphi}\|_{-1}^{2}.$
(6.27)
It remains to bound the various $\mathbf{c}_{2}$ terms. Firstly we find that
$|\mathbf{c}_{2}(\mathcal{G}\bar{\varphi},\bar{\varphi},\mathbf{u}_{T}^{1})|\leq\|\mathcal{G}\bar{\varphi}\|_{L^{4}(\Gamma(t))}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))},$
and by using the embedding $H^{1}(\Gamma(t))\hookrightarrow L^{4}(\Gamma(t))$,
Poincaré’s inequality and Young’s inequality we find that
$\displaystyle|\mathbf{c}_{2}(\mathcal{G}\bar{\varphi},\bar{\varphi},\mathbf{u}_{T}^{1})|\leq\frac{\varepsilon}{16}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}+C\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))}^{2}\|\bar{\varphi}\|_{-1}^{2}.$
(6.28)
The other $\mathbf{c}_{2}$ term is similar, but now we use the antisymmetry of
$\mathbf{c}_{2}$ in the first two arguments so that
$|\mathbf{c}_{2}(\mathcal{G}\bar{\varphi},\varphi^{2},\bar{\mathbf{u}_{T}})|=|\mathbf{c}_{2}(\varphi^{2},\mathcal{G}\bar{\varphi},\bar{\mathbf{u}_{T}})|\leq\|\varphi^{2}\|_{L^{\infty}(\Gamma(t))}\|\bar{\varphi}\|_{-1}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}.$
It is then clear from Young’s inequality that
$\displaystyle|\mathbf{c}_{2}(\mathcal{G}\bar{\varphi},\varphi^{2},\bar{\mathbf{u}_{T}})|\leq\frac{\eta_{*}}{4}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+C\|\varphi^{2}\|_{L^{\infty}(\Gamma(t))}^{2}\|\bar{\varphi}\|_{-1}^{2}.$
(6.29)
Finally we bound the term involving $\widetilde{\mathbf{u}_{T}}$. To do this,
we observe that
$|\mathbf{c}_{2}(\mathcal{G}\bar{\varphi},\bar{\varphi},\widetilde{\mathbf{u}_{T}})|\leq|\mathbf{c}_{2}(\bar{\varphi},\mathcal{G}\bar{\varphi},\widetilde{\mathbf{u}_{T}})|+|m(\bar{\varphi}HV_{N},\mathcal{G}\bar{\varphi})|,$
so that by similar arguments to the above
$\displaystyle|\mathbf{c}_{2}(\mathcal{G}\bar{\varphi},\bar{\varphi},\widetilde{\mathbf{u}_{T}})|\leq\frac{\varepsilon}{16}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}+C\left(1+\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{\infty}(\Gamma(t))}^{2}\right)\|\bar{\varphi}\|_{-1}^{2}.$
(6.30)
Hence using (6.26)-(6.30) and (6.18) in (6.25) one obtains (6.7), where we
have also bounded the extra terms in (6.26) as
$|m(\bar{\varphi}HV_{N},\mathcal{G}\bar{\varphi})|+|b(\mathcal{G}\bar{\varphi},\mathcal{G}\bar{\varphi})|\leq\frac{\varepsilon}{16}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}+C\|\bar{\varphi}\|_{-1}^{2}.$
∎
Next we show a stability result for the case of constant viscosity. This also
provides a simpler proof for uniqueness in this special case, where we no
longer require Lemma 3.11.
###### Proposition 6.3.
Let $(\varphi^{i},\mu^{i},\mathbf{u}_{T}^{i})$ denote the solution triple
corresponding to some choice of initial data $\varphi_{0}^{i}\in
H^{1}(\Gamma_{0}),\mathbf{u}_{T,0}^{i}\in\mathbf{H}_{\sigma}$, for $i=1,2$,
where $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma_{0}}\varphi_{0}^{1}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma_{0}}\varphi_{2}^{1}$. Then, under the same
assumptions as the preceding theorem, we have
$\|P_{\mathcal{K}^{\perp}}(\mathbf{u}_{T}^{1}(t)-\mathbf{u}_{T}^{2}(t))\|_{\perp}^{2}+\|P_{\mathcal{K}}(\mathbf{u}_{T}^{1}(t)-\mathbf{u}_{T}^{2}(t))\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\|\varphi^{1}(t)-\varphi^{2}(t)\|_{-1}^{2}\\\
\leq
C\left(\|P_{\mathcal{K}^{\perp}}(\mathbf{u}_{T,0}^{1}-\mathbf{u}_{T,0}^{2})\|_{\perp}^{2}+\|P_{\mathcal{K}}(\mathbf{u}_{T,0}^{1}-\mathbf{u}_{T,0}^{2})\|_{\mathbf{L}^{2}(\Gamma_{0})}^{2}+\|\varphi_{0}^{1}-\varphi_{0}^{2}\|_{-1}^{2}\right),$
(6.31)
for a constant $C$ which depends on $t,\Gamma$ and the initial data.
###### Proof.
This proof is largely the same as that of the previous theorem, where we have
some minor modifications. We use the same notation as before, except now we
denote $(\varphi^{i},\mu^{i},\mathbf{u}_{T}^{i})$ as the solution
corresponding to some choice of initial data $\varphi_{0}^{i}\in
H^{1}(\Gamma),\mathbf{u}_{T,0}^{i}\in\mathbf{H}_{\sigma}.$ We define
$(\bar{\varphi},\bar{\mu},\bar{\mathbf{u}_{T}})$ as before, and note that
instead of (6.2) we find that $(\bar{\varphi},\bar{\mu},\bar{\mathbf{u}_{T}})$
solves
$\mathbf{m}_{*}\left(\partial^{\circ}{\bar{\mathbf{u}_{T}}},\boldsymbol{\phi}\right)+\eta\mathbf{a}(\bar{\mathbf{u}_{T}},\boldsymbol{\phi})+\mathbf{c}_{1}(\mathbf{u}_{T}^{1},\mathbf{u}_{T}^{1},\boldsymbol{\phi})-\mathbf{c}_{1}(\mathbf{u}_{T}^{2},\mathbf{u}_{T}^{2},\boldsymbol{\phi})+\mathbf{l}(\bar{\mathbf{u}_{T}},\boldsymbol{\phi})+\mathbf{d}_{1}(\bar{\mathbf{u}_{T}},\boldsymbol{\phi})\\\
=\varepsilon\mathbf{c}_{3}(\varphi^{1},\varphi^{1},\boldsymbol{\phi})-\varepsilon\mathbf{c}_{3}(\varphi^{2},\varphi^{2},\boldsymbol{\phi}),$
(6.32)
but (6.3) and (6.4) are still satisfied. As before we test (6.32) with
$\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}}$ to see
$\frac{1}{2}\frac{d}{dt}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\eta\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\mathbf{c}_{1}(\bar{\mathbf{u}_{T}},\mathbf{u}_{T}^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})+\mathbf{c}_{1}(\mathbf{u}_{T}^{2},\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})+\mathbf{l}(\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})+\mathbf{d}_{1}(\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})\\\
=\varepsilon\mathbf{c}_{3}(\bar{\varphi},\varphi^{1},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})+\varepsilon\mathbf{c}_{3}(\varphi^{2},\bar{\varphi},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})-\frac{1}{2}\mathbf{b}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})+\mathbf{m}(P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}HV_{N},\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}})-\sum_{i=1}^{N_{K}}\mathbf{m}(\bar{\mathbf{u}_{T}},\boldsymbol{\kappa}_{i})\mathbf{m}(\mathcal{S}^{\perp}\bar{\mathbf{u}_{T}},\partial^{\circ}\boldsymbol{\kappa}_{i})$
where we have used (6.9) and the definition of $\mathcal{S}^{\perp}$. Now by
arguing as we did for (6.12)-(6.17) it is straightforward to see that
$\displaystyle\frac{1}{2}\frac{d}{dt}\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\eta\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}\leq\frac{\eta}{2}\|\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\frac{\varepsilon}{4}\|\nabla_{\Gamma}\bar{\varphi}\|_{L^{2}(\Gamma(t))}^{2}+K_{1}(t)\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2},$
(6.33)
where
$K_{1}(t)=C\left(\|\mathbf{u}_{T}^{1}\|_{\mathbf{L}^{4}(\Gamma(t))}^{4}+\|\mathbf{u}_{T}^{2}\|_{\mathbf{L}^{4}(\Gamma(t))}^{4}+\|\varphi^{1}\|_{H^{1,4}(\Gamma(t))}^{4}+\|\varphi^{2}\|_{H^{1,4}(\Gamma(t))}^{4}+\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{\infty}(\Gamma(t))}^{2}\right).$
Notice that the requirement $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma}\varphi_{0}^{1}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma}\varphi_{0}^{2}$ allows us to define
$\mathcal{G}\bar{\varphi}$, and so related calculations from the preceding
theorem still hold. By summing (6.6), (6.7), and (6.33) one finds
$\frac{1}{2}\frac{d}{dt}\left(\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\|\bar{\varphi}\|_{-1}^{2}\right)\leq
K(t)\left(\|P_{\mathcal{K}^{\perp}}\bar{\mathbf{u}_{T}}\|_{\perp}^{2}+\|P_{\mathcal{K}}\bar{\mathbf{u}_{T}}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\|\bar{\varphi}\|_{-1}^{2}\right),$
where $K\in L^{1}([0,T])$. An application of Grönwall’s inequality then yields
(6.31). ∎
###### Remark 6.4.
One may notice that this proof can be somewhat simplified by defining
$\mathcal{S}\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)$, where
$\boldsymbol{\phi}\in\mathbf{L}^{2}(\Gamma(t))$, to be the unique solution of
$\mathbf{a}(\mathcal{S}\boldsymbol{\phi},\boldsymbol{\psi})+\mathbf{m}(\mathcal{S}\boldsymbol{\phi},\boldsymbol{\psi})=\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\psi}),$
for all $\boldsymbol{\psi}\in\mathbf{L}^{2}(\Gamma(t))$. From (3.9) it is
clear that this operator is coercive, and so one does not have to quotient out
a kernel as we did above. Proceeding as before one defines a corresponding
norm, and the proof should follow but without the Killing vector splitting. We
have not chosen to do this, as this obscures the fact that the Killing vector
component of $\mathbf{u}_{T}$ has better stability properties than the non-
Killing component. Indeed, by using (B.3) and the
$\mathcal{K}-\mathcal{K}^{\perp}$ splitting as in our proof one observes that
we have $\mathbf{H}^{1,p}-\mathbf{V}_{\sigma}^{\prime}$ stability, whereas
this simplified proof would only give $\mathbf{V}_{\sigma}^{\prime}$ stability
for both components. Similarly if one considers a surface with boundary and
prescribes Dirichlet boundary conditions on $\mathbf{u}_{T}$ then one can
define the inverse Stokes operator in a way that doesn’t involve this Killing
vector splitting.
### 6.2 Uniqueness for the logarithmic potential
It is clear that (6.1) still holds, that is $\varphi\in L^{2}_{H^{2}}$, since
we know $\mu,f(\varphi)\in L^{2}_{L^{2}},$ and so we may use the
$\mathbf{c}_{3}$ bilinear form as before. With this at hand, the proofs of
Theorem 6.1 and Proposition 6.3 follow. That is, for $\Gamma(t)$ be a $C^{3}$
evolving surface, and initial data
$\varphi_{0}\in\mathcal{I}_{0},\mathbf{u}_{T,0}\in\mathbf{H}_{\sigma}(0)$ the
solution triple $(\varphi,\mu,\mathbf{u}_{T})$ solving (4.4)-(4.6) is unique.
Moreover, for a constant viscosity, and $\varphi_{0}^{i}\in\mathcal{I}_{0}$
such that where $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma_{0}}\varphi_{0}^{1}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma_{0}}\varphi_{0}^{2}$, then we have
stability bound similar to (6.31)
###### Remark 6.5.
The results for the logarithmic potential extend to more general singular
potentials of the form $F(r)=F_{1}(r)-\frac{\theta}{2}r^{2}$, with $F_{1}\in
C^{2}((a,b))\cap C^{0}([a,b])$ for some $a,b\in\mathbb{R}$ under some
necessary assumptions we do not expand upon. Potentials of this form are
treated on a Euclidean domain in [1, 29], but here we have only covered the
thermodynamically relevant logarithmic potential - which is still illustrative
of the general case.
## 7 Reintroducing the surface pressure
We end our discussion by reintroducing the surface pressure and the correct
divergence condition. We now consider the mixed formulation, with a regular
potential, where one finds a solution $(\varphi,\mu,\mathbf{u}_{T},p)$, with
$\varphi\in H^{1}_{H^{-1}}\cap L^{2}_{H^{1}},\mu\in
L^{2}_{H^{1}},\mathbf{u}_{T}\in H^{1}_{\mathbf{H}^{-1}}\cap
L^{2}_{\mathbf{H}^{1}},p\in L^{2}_{L^{2}}$, solving
$\displaystyle\begin{split}\langle\partial^{\circ}\mathbf{u}_{T},\boldsymbol{\phi}\rangle_{\mathbf{H}^{-1}(\Gamma(t)),\mathbf{H}^{1}(\Gamma(t))}+\hat{\mathbf{a}}(\eta(\varphi),\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{c}_{1}(\mathbf{u}_{T},\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{l}(\mathbf{u}_{T},\boldsymbol{\phi})+m(p,\nabla_{\Gamma}\cdot\boldsymbol{\phi})+\mathbf{d}_{1}(\mathbf{u}_{T},\boldsymbol{\phi})\\\
+\mathbf{d}_{2}(\eta(\varphi),\boldsymbol{\phi})=\mathbf{m}(\mathbf{B},\boldsymbol{\phi})+\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi})\end{split},$
(7.1) $\displaystyle m(q,\nabla_{\Gamma}\cdot\mathbf{u}_{T})=0,$ (7.2)
$\displaystyle
m_{*}(\partial^{\circ}\varphi,\phi)+a(\mu,\phi)+\mathbf{c}_{2}(\phi,\varphi,\mathbf{u}_{T})+\mathbf{c}_{2}(\phi,\varphi,\widetilde{\mathbf{u}_{T}})=0,$
(7.3) $\displaystyle m(\mu,\phi)=\varepsilon
a(\varphi,\phi)+\frac{1}{\varepsilon}m(F^{\prime}(\varphi),\phi),$ (7.4)
for all $q\in L^{2}(\Gamma(t)),\phi\in
H^{1}(\Gamma(t)),\boldsymbol{\phi}\in\mathbf{H}^{1}(\Gamma(t))$ for almost all
$t\in[0,T]$. Here the initial data is $\varphi_{0}\in
H^{1}(\Gamma_{0}),\mathbf{u}_{T,0}\in\mathbf{H}_{\sigma}(0)$, so one has
$\varphi(0)=\varphi_{0},\mathbf{u}_{T}(0)=\mathbf{u}_{T,0}$ almost everywhere
on $\Gamma_{0}$.
Before proving the existence and uniqueness of this system, we recall the
uniform inf-sup condition of [45].
###### Lemma 7.1 ([45], Lemma 3.3).
There exists a constant, $C$, independent of time such that for all $q\in
L_{0}^{2}(\Gamma(t)):=\left\\{\phi\in
L^{2}(\Gamma(t))\mid\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}\phi=0\right\\}$,
$\displaystyle\|\nabla_{\Gamma}q\|_{\mathbf{H}^{-1}(\Gamma(t))}:=\sup_{\boldsymbol{\phi}\in\mathbf{H}^{1}(\Gamma(t))\setminus\\{0\\}}\frac{\int_{\Gamma(t)}q\nabla_{\Gamma}\cdot\boldsymbol{\phi}}{\|\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}}\geq
C\|q\|_{L^{2}(\Gamma(t))}.$ (7.5)
###### Theorem 7.2.
There exists a unique solution, $(\varphi,\mu,\mathbf{u}_{T},p)$, with
$\varphi\in H^{1}_{H^{-1}}\cap L^{2}_{H^{1}},\mu\in
L^{2}_{H^{1}},\mathbf{u}_{T}\in H^{1}_{\mathbf{H}^{-1}}\cap
L^{2}_{\mathbf{H}^{1}},p\in L^{2}_{L^{2}}$, of (7.1)-(7.4).
###### Proof.
To begin, let $(\varphi,\mu,\mathbf{u}_{T})$ be the unique solution of
(4.1)-(4.3), and define $\mathcal{F}(t)$ by
$\langle\mathcal{F}(t),\boldsymbol{\phi}\rangle_{\mathbf{H}^{-1}(\Gamma(t)),\mathbf{H}^{1}(\Gamma(t))}:=\langle\partial^{\circ}\mathbf{u}_{T},\boldsymbol{\phi}\rangle_{\mathbf{H}^{-1}(\Gamma(t)),\mathbf{H}^{1}(\Gamma(t))}+\hat{\mathbf{a}}(\eta(\varphi),\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{c}_{1}(\mathbf{u}_{T},\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{l}(\mathbf{u}_{T},\boldsymbol{\phi})+m(p,\nabla_{\Gamma}\cdot\boldsymbol{\phi})\\\
+\mathbf{d}_{1}(\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{d}_{2}(\eta(\varphi),\boldsymbol{\phi})-\mathbf{m}(\mathbf{B},\boldsymbol{\phi})-\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi}),$
for $\boldsymbol{\phi}\in\mathbf{H}^{1}(\Gamma(t))$. We claim that
$\mathcal{F}\in L^{2}_{\mathbf{H}^{-1}}$. To see this, we recall the
equivalence,
$\partial^{\circ}\boldsymbol{\phi}\in
L^{2}_{\mathbf{V}_{\sigma}^{\prime}}\Leftrightarrow\partial^{\circ}\boldsymbol{\phi}\in
L^{2}_{\mathbf{H}^{-1}},\ \boldsymbol{\phi}\in L^{2}_{\mathbf{V}_{\sigma}}$
from [45], and repeat various estimates we have used throughout. We elaborate
on the estimates for
$\mathbf{c}_{1}(\mathbf{u}_{T},\mathbf{u}_{T},\boldsymbol{\phi}),\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi})$,
but skip further calculations. As
$\boldsymbol{\phi}\in\mathbf{H}^{1}(\Gamma(t))$, and not necessarily
$\mathbf{V}_{\sigma}(t)$, we cannot use the properties,
$\mathbf{c}_{1}(\mathbf{u}_{T},\mathbf{u}_{T},\boldsymbol{\phi})=-\mathbf{c}_{1}(\boldsymbol{\phi},\mathbf{u}_{T},\mathbf{u}_{T})$
and
$\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi})=-\mathbf{c}_{2}(\varphi,\mu,\boldsymbol{\phi})$.
However, by using the divergence theorem and the fact that
$\partial\Gamma(t)=\emptyset$, one finds
$\displaystyle
c_{1}(\mathbf{u}_{T},\mathbf{u}_{T},\boldsymbol{\phi})=-c_{1}(\boldsymbol{\phi},\mathbf{u}_{T},\mathbf{u}_{T})-\int_{\Gamma(t)}(\mathbf{u}_{T}\cdot\mathbf{u}_{T})\nabla_{\Gamma}\cdot\boldsymbol{\phi},$
$\displaystyle
c_{2}(\mu,\varphi,\boldsymbol{\phi})=-c_{2}(\varphi,\mu,\boldsymbol{\phi})-\int_{\Gamma(t)}\varphi\mu\nabla_{\Gamma}\cdot\boldsymbol{\phi},$
and hence
$\displaystyle\int_{0}^{T}|c_{1}(\mathbf{u}_{T},\mathbf{u}_{T},\boldsymbol{\phi})|\leq
C\sup_{t\in[0,T]}\|\mathbf{u}_{T}\|_{\mathbf{L}^{2}(\Gamma(t))}\left(\int_{0}^{T}\|\mathbf{u}_{T}\|_{\mathbf{H}^{1}(\Gamma(t))}\right)^{\frac{1}{2}}\left(\int_{0}^{T}\|\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}\right)^{\frac{1}{2}},$
$\displaystyle\int_{0}^{T}|c_{2}(\mu,\varphi,\boldsymbol{\phi})|\leq
C\sup_{t\in[0,T]}\|\varphi\|_{H^{1}(\Gamma(t))}\left(\int_{0}^{T}\|\mu\|_{H^{1}(\Gamma(t))}^{2}\right)^{\frac{1}{2}}\left(\int_{0}^{T}\|\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}^{2}\right)^{\frac{1}{2}},$
where we have used Sobolev embeddings and (3.8) as appropriate. The other
terms follow similar, but simpler, arguments.
We now observe that from the inf-sup condition (7.5) that the distributional
divergence,
$\nabla_{\Gamma}:L_{0}^{2}(\Gamma(t))\rightarrow\mathbf{H}^{-1}(\Gamma(t)),$
has a closed range $R(\nabla_{\Gamma})\subset\mathbf{H}^{-1}(\Gamma(t))$. This
follows from (7.5) and continuity of $\nabla_{\Gamma}$ as an operator. Now by
the closed range theorem, see for example [52] VII.5, we find that
$R(\nabla_{\Gamma})=\ker(\nabla_{\Gamma}^{*})^{\perp},\text{ where
}\ker(\nabla_{\Gamma}^{*})=\mathbf{V}_{\sigma}(t),$
where $\nabla_{\Gamma}^{*}$ is the adjoint of $\nabla_{\Gamma}$.
Since $(\varphi,\mu,\mathbf{u}_{T})$ solves (4.1)-(4.3) for almost all
$t\in[0,T]$, we see that
$\langle\mathcal{F}(t),\boldsymbol{\phi}\rangle_{\mathbf{H}^{-1}(\Gamma(t)),\mathbf{H}^{1}(\Gamma(t))}=0$
for all $\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t).$ Hence from the above we
see that $\mathcal{F}(t)\in R(\nabla_{\Gamma})$ for almost all $t\in[0,T]$.
Thus there exists some $p\in L^{2}_{0}(\Gamma(t))$ such that
$\nabla_{\Gamma}p=\mathcal{F}(t)$ in the distributional sense. The map
$t\mapsto\|p\|_{L^{2}(\Gamma(t))}$ is measurable by the same logic as in the
proof of [45], Theorem 4.2. Moreover, by using (7.5) we see that $p$ is unique
and one has
$\int_{0}^{T}\|p\|_{L^{2}(\Gamma(t))}^{2}\leq
C\int_{0}^{T}\|\mathcal{F}\|_{\mathbf{H}^{-1}(\Gamma(t))}^{2},$
where the latter term can be expressed in terms of
$\varphi,\mu,\mathbf{u}_{T}$. ∎
Lastly we want to return to the setting of non-solenoidal vectors. Letting
$(\varphi,\mu,\widehat{\mathbf{u}_{T}},p)$ be the solution from the previous
theorem, then by our construction of $\widetilde{\mathbf{u}_{T}}$ and the
bilinear forms $\mathbf{d}_{1},\mathbf{d}_{2}$, it is clear that
$\mathbf{u}_{T}:=\widehat{\mathbf{u}_{T}}-\widetilde{\mathbf{u}_{T}}$ is such
that
$\displaystyle\langle\partial^{\circ}\mathbf{u}_{T},\boldsymbol{\phi}\rangle_{\mathbf{H}^{-1}(\Gamma(t)),\mathbf{H}^{1}(\Gamma(t))}+\hat{\mathbf{a}}(\eta(\varphi),\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{c}_{1}(\mathbf{u}_{T},\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{l}(\mathbf{u}_{T},\boldsymbol{\phi})+m(p,\nabla_{\Gamma}\cdot\boldsymbol{\phi})=\mathbf{m}(\mathbf{F}_{T},\boldsymbol{\phi})+\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi}),$
$\displaystyle m(q,\nabla_{\Gamma}\cdot\mathbf{u}_{T})=-m(q,HV_{N}),$
$\displaystyle
m_{*}(\partial^{\circ}\varphi,\phi)+a(\mu,\phi)+\mathbf{c}_{2}(\phi,\varphi,\mathbf{u}_{T})=0,$
$\displaystyle m(\mu,\phi)=\varepsilon
a(\varphi,\phi)+\frac{1}{\varepsilon}m(F^{\prime}(\varphi),\phi),$
for all $q\in L^{2}(\Gamma(t)),\phi\in
H^{1}(\Gamma(t)),\boldsymbol{\phi}\in\mathbf{H}^{1}(\Gamma(t))$ for almost all
$t\in[0,T]$. Moreover we find that $\varphi\in H^{1}_{H^{-1}}\cap
L^{2}_{H^{1}},\mu\in L^{2}_{H^{1}},\mathbf{u}_{T}\in
H^{1}_{\mathbf{H}^{-1}}\cap L^{2}_{\mathbf{H}^{1}},p\in L^{2}_{L^{2}}$.
The initial condition for $\varphi$ is unchanged, but the initial condition
for $\mathbf{u}_{T}$ is required to be such that
$\mathbf{u}_{T}(0)=\mathbf{u}_{T,0}\in\widetilde{\mathbf{u}_{T}}+\mathbf{H}_{\sigma}(0)$.
One deduces the appropriate regularity for $\mathbf{u}_{T}$ from the
regularity of $\widehat{\mathbf{u}_{T}},\widetilde{\mathbf{u}_{T}}$. The above
arguments also work for the logarithmic potential, but we omit further
details.
###### Remark 7.3.
In this section we have not discriminated between the pressure, $p$, and the
modified pressure, $\tilde{p}$, as it is largely beside the point - that is
the existence of some Lagrange multiplier enforcing the divergence condition.
The distinction between these two pressures is discussed in Section 2.
Moreover, it is straightforward to establish that $p\in
L^{2}_{L^{2}}\Leftrightarrow\tilde{p}\in L^{2}_{L^{2}}$.
## 8 Concluding remarks
We have derived a system coupling the Navier-Stokes equations with the Cahn-
Hilliard equations on an evolving surface, and shown the well-posedness for a
prescribed, sufficiently smooth normal evolution. There is still much work to
be done on this topic, which we expound upon here.
Firstly, for the (evolving surface) Cahn-Hilliard equations with a logarithmic
potential one observes a “separation from the pure phases” where after some
small time the solution, $\varphi$, is such that $|\varphi|<1-\xi$ for some
small $\xi$ \- as was shown in [19]. This has been established for a Navier-
Stokes-Cahn-Hilliard system on a stationary domain in [29], and so it seems
reasonable it would extend to our setting.
If one does not prescribe the normal component of the velocity then the system
(1.1)-(1.4) also contains a geometric evolution equation, (2.8), which one
must solve. Unlike more standard geometric evolution equations, for example
mean curvature flow, this flow is essentially second order in time as one
considers the material derivative of the normal velocity. Indeed, even if one
ignores the Cahn-Hilliard component of (1.1)-(1.4) there are, to the authors’
knowledge, no results on the well-posedness of the evolving surface Navier-
Stokes equations (with unknown normal component) as discussed in [16, 45].
Moreover, the model we have considered is a diffuse interface model - and
depends strongly on the choice of the interface width, $\varepsilon$. It is
known that, in the sharp interface limit, $\varepsilon\rightarrow 0$, the
zero-level set of the solution of the Cahn-Hilliard equation (with a constant
mobility) converges in a suitably weak sense to the Mullins-Sekerka system,
also known as the Hele-Shaw system - see [5]. Likewise, it is known that the
analogous zero-level set from the Navier-Stokes-Cahn-Hilliard system converges
to a coupled Navier-Stokes-Mullins-Sekerka system - see for instance [3, 4].
However such results, or even formal asymptotics, have not been obtained for
the corresponding systems on an evolving surface - or even on a stationary
surface, to our knowledge. In particular, it would be interesting to study the
sharp interface limit of (1.1)-(1.4), as the limiting system should consist of
a coupling been a Navier-Stokes type equation for the surface velocity coupled
with the Mullins-Sekerka problem.
Lastly, there is interest in the numerical simulation of the system we have
considered (with or without a prescribed normal velocity). There has recently
(see [43]) been some numerical analysis of the tangential Navier-Stokes
equations, where the authors discretise by using the TraceFEM method - but
this has not yet been considered for the system (1.5)-(1.8). It therefore
would be interesting to see how existing results for a stationary domain, for
instance [48], adapt to an evolving surface.
### Acknowledgments
Thomas Sales is supported by the Warwick Mathematics Institute Centre for
Doctoral Training, and gratefully acknowledges funding from the University of
Warwick and the UK Engineering and Physical Sciences Research Council (Grant
number: EP/TS1794X/1). For the purpose of open access, the author has applied
a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising from this submission. The authors would also like
to thank Achilles Mavrakis for discussions which resulted in Remark 6.4.
## Appendix A Laplace’s equation on an evolving surface
In this appendix we consider the regularity the solution of Laplace’s equation
on an evolving domain. For $t\in[0,T]$ we define $\Psi(t)$ to be the unique
weak solution of
$-\Delta_{\Gamma}\Psi(t)=H(t)V_{N}(t),$
on $\Gamma(t)$, subject to the constraint
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}\Psi=0$. We note that this is well
defined since
$\int_{\Gamma(t)}H(t)V_{N}(t)=-2\int_{\Gamma(t)}\partial^{\circ}1=0.$
Here we denote the normal pushforward map as
$\Phi_{t}^{n}:\Gamma_{0}\rightarrow\Gamma(t),$ and $\Phi_{-t}^{n}$ denotes its
inverse. As these are $C^{2}$ diffeomorphisms the differentials
$D\Phi_{t}^{n}(p):T_{p}\Gamma_{0}\rightarrow T_{\Phi_{t}^{n}(p)}\Gamma(t)$ are
invertible. We recall the notation $J(p,t)=\det(D\Phi_{t}^{n}(p))$,
$J^{-1}(x,t)=\det(D\Phi_{-t}^{n}(x))=J(t,\Phi_{-t}^{n}x)^{-1}$,
$\mathbb{D}(p,t)=D\Phi_{t}^{n}(p))\mathbb{P}(p,0)$, and
$\mathbb{D}^{-1}(x,t)=D\Phi_{-t}^{n}(x))\mathbb{P}(x,t)$. These matrices are
such that $\mathbb{D}\mathbb{D}^{-1}=\mathbb{D}^{-1}\mathbb{D}=\mathbb{P}.$
###### Lemma A.1.
Let $\Psi$ be as above, and $\Gamma(t)$ be a $C^{3}$ evolving surface. Then
$\Psi\in C^{0}_{H^{3,p}}\cap C^{1}_{H^{1,p}},$ for all $p\in[1,\infty).$
###### Proof.
Let $\chi\in H^{1}(\Gamma_{0})$, then by the weak formulation of the above PDE
and the compatibility of $(H^{1}(\Gamma(t)),\Phi_{t}^{n})$ we find that
$\int_{\Gamma(t)}\nabla_{\Gamma}\Psi(t)\cdot\nabla_{\Gamma}\Phi_{t}^{n}\chi=\int_{\Gamma{t}}H(t)V_{N}(t)\Phi_{t}^{n}\chi.$
Hence by pulling back the integrals onto $\Gamma_{0}$ we see that
$\int_{\Gamma(t)}J(t)\mathbb{D}\nabla_{\Gamma}\Phi_{-t}^{n}\Psi(t)\cdot\mathbb{D}\nabla_{\Gamma}\chi=\int_{\Gamma_{0}}J(t)\Phi_{-t}^{n}(HV_{N})\chi,$
where the operators now are $\nabla_{\Gamma_{0}}.$ Similarly, the mean value
condition transforms as
$0=\int_{\Gamma(t)}\Psi(t)=\int_{\Gamma_{0}}J(t)\Phi_{-t}^{n}\Psi(t),$
and as such we focus on the function $\psi(t):=J(t)\Phi_{-t}^{n}\Psi(t)$,
where we see $\psi(t)\in H^{1}(\Gamma_{0})$ for all $t\in[0,T]$. We similarly
write $f(t):=J(t)\Phi_{-t}^{n}(HV_{n})\in H^{1}(\Gamma_{0}).$ It is then clear
that $\psi(t)$ solves the PDE
$\displaystyle\int_{\Gamma_{0}}\tilde{\mathbb{D}}(t)\nabla_{\Gamma}\psi(t)\cdot\nabla_{\Gamma}\chi+\psi(t)\boldsymbol{\omega}(t)\cdot\nabla_{\Gamma}\chi=\int_{\Gamma_{0}}f(t)\chi,$
(A.1)
for all $\chi\in H^{1}(\Gamma_{0})$, where
$\tilde{\mathbb{D}}=\mathbb{D}^{T}\mathbb{D},\qquad\boldsymbol{\omega}=J(t)\tilde{\mathbb{D}}\nabla_{\Gamma}(J(t)^{-1}).$
We note that clearly $\tilde{\mathbb{D}}$ is positive definite, and the
uniqueness of $\Psi$ implies uniqueness of $\psi$.
Then our assumptions on $\Phi_{t}^{n}$ imply we have sufficient smoothness so
that we may apply elliptic regularity theory to see that
$\|\psi(t)\|_{H^{3,p}(\Gamma_{0})}\leq C\|f(t)\|_{H^{1,p}(\Gamma_{0})},$
for $p\in[1,\infty)$ and $C$ depends on
$p,\Gamma_{0},\tilde{\mathbb{D}}(t),\boldsymbol{\omega}(t)$. It is
straightforward to see that by considering (A.1) at two times $t,s\in[0,T]$,
and noting that $\tilde{\mathbb{D}},\boldsymbol{\omega}$ are $C^{2}$ in $t$
and $f$ is $C^{1}$ in $t$, that the map
$t\mapsto\|\psi(t)\|_{H^{3,p}(\Gamma_{0})}$ is continuous on $[0,T]$. We omit
further details on this calculation.
Next we show that $\psi$ has a strong derivative. By considering (A.1) at
times $t\in[0,T)$ and $t+h$ for some small $h>0$ so that $t+h\in(0,T)$ we find
that
$\frac{1}{h}\int_{\Gamma_{0}}\left(\tilde{\mathbb{D}}(t+h)\nabla_{\Gamma}\psi(t+h)-\tilde{\mathbb{D}}(t)\nabla_{\Gamma}\psi(t)\right)\cdot\nabla_{\Gamma}\chi+\left(\psi(t+h)\boldsymbol{\omega}(t+h)-\psi(t)\boldsymbol{\omega}(t)\right)\cdot\nabla_{\Gamma}\chi\\\
=\frac{1}{h}\int_{\Gamma_{0}}\left(f(t+h)-f(t)\right)\chi,$
for all $\chi\in H^{1}(\Gamma_{0})$. We write this in terms of difference
quotients as
$\int_{\Gamma_{0}}\left(\Delta_{h}\tilde{\mathbb{D}}(t)\nabla_{\Gamma}\psi(t+h)+\tilde{\mathbb{D}}(t)\nabla_{\Gamma}\Delta_{h}\psi(t)\right)\cdot\nabla_{\Gamma}\chi+\left(\psi(t+h)\Delta_{h}\boldsymbol{\omega}(t)+\Delta_{h}\psi(t)\boldsymbol{\omega}(t)\right)\cdot\nabla_{\Gamma}\chi=\int_{\Gamma_{0}}\Delta_{h}f(t)\chi,$
where $\Delta_{h}X(t)=\frac{X(t+h)-X(t)}{h}$ for some quantity $X$. Now by
letting $h,h^{\prime}>0$ be sufficiently small one readily finds that
$\displaystyle\langle
L(t)(\Delta_{h}\psi(t)-\Delta_{h^{\prime}}\psi(t)),\Delta_{h}\psi(t)-\Delta_{h^{\prime}}\psi(t)\rangle$
$\displaystyle\leq
C\|\Delta_{h^{\prime}}\tilde{\mathbb{D}}(t)\nabla_{\Gamma}\psi(t+h^{\prime})-\Delta_{h}\tilde{\mathbb{D}}(t)\nabla_{\Gamma}\psi(t+h)\|_{L^{2}(\Gamma_{0})}^{2}$
$\displaystyle+C\|\psi(t+h^{\prime})\Delta_{h^{\prime}}\boldsymbol{\omega}(t)-\psi(t+h)\Delta_{h}\boldsymbol{\omega}(t)\|_{\mathbf{L}^{2}(\Gamma_{0})}^{2}$
$\displaystyle+C\|\Delta_{h}f(t)-\Delta_{h^{\prime}}f(t)\|_{L^{2}(\Gamma_{0})}^{2}+\gamma\|\nabla_{\Gamma}(\Delta_{h}\psi(t)-\Delta_{h^{\prime}}\psi(t))\|_{L^{2}(\Gamma_{0})}^{2},$
for some small $\gamma$ to be determined. Here
$L(t)\in\mathcal{L}(H^{1}(\Gamma_{0})\cap
L_{0}^{2}(\Gamma_{0}),(H^{1}(\Gamma_{0})\cap L_{0}^{2}(\Gamma_{0}))^{\prime})$
is the operator defined so that
$\langle
L(t)\zeta,\chi\rangle=\int_{\Gamma_{0}}\tilde{\mathbb{D}}(t)\nabla_{\Gamma}\zeta\cdot\nabla_{\Gamma}\chi+\zeta\boldsymbol{\omega}(t)\cdot\nabla_{\Gamma}\chi,$
where $L^{2}_{0}(\Gamma_{0})$ is the subspace of $L^{2}(\Gamma_{0})$
containing elements such that $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma_{0}}\phi=0$. By pushing the integral
forward onto $\Gamma(t)$, in the reverse to the beginning of the proof, we can
observe $L(t)$ is elliptic by the ellipticity of $-\Delta_{\Gamma(t)}$ on
$H^{1}(\Gamma(t))\cap L_{0}^{2}(\Gamma(t)),$ and moreover the ellipticity
constant is independent of $t$. Thus there exists some constant $\kappa$ such
that
$\langle
L(t)(\Delta_{h}\psi(t)-\Delta_{h^{\prime}}\psi(t)),\Delta_{h}\psi(t)-\Delta_{h^{\prime}}\psi(t)\rangle\geq\kappa\|\nabla_{\Gamma}(\Delta_{h}\psi(t)-\Delta_{h^{\prime}}\psi(t))\|_{L^{2}(\Gamma_{0})}^{2},$
and hence choosing $\gamma=\frac{\kappa}{2}$, and using Poincaré’s inequality
on $\Gamma_{0}$ we see
$\displaystyle\|\Delta_{h}\psi(t)-\Delta_{h^{\prime}}\psi(t)\|_{H^{1}(\Gamma_{0})}^{2}$
$\displaystyle\leq
C\|\Delta_{h^{\prime}}\tilde{\mathbb{D}}(t)\nabla_{\Gamma}\psi(t+h^{\prime})-\Delta_{h}\tilde{\mathbb{D}}(t)\nabla_{\Gamma}\psi(t+h)\|_{L^{2}(\Gamma_{0})}^{2}$
$\displaystyle+C\|\psi(t+h^{\prime})\Delta_{h^{\prime}}\boldsymbol{\omega}(t)-\psi(t+h)\Delta_{h}\boldsymbol{\omega}(t)\|_{\mathbf{L}^{2}(\Gamma_{0})}^{2}$
$\displaystyle+C\|\Delta_{h}f(t)-\Delta_{h^{\prime}}f(t)\|_{L^{2}(\Gamma_{0})}^{2},$
for some constants $C(t)$ depending on $\Gamma_{0}$, and the ellipticity of
$L(t)$. Now, by the differentiability of
$\tilde{\mathbb{D}},\boldsymbol{\omega},f$, and the continuity of $\psi$ it is
clear that by taking $h,h^{\prime}$ sufficiently small that we can make
$\|\Delta_{h}\psi(t)-\Delta_{h^{\prime}}\psi(t)\|_{H^{1}(\Gamma_{0})}$
arbitrarily small. Thus $\Delta_{h}\psi(t)$ is a Cauchy sequence in
$H^{1}(\Gamma_{0})$ and a right time derivative of $\psi$ exists at
$t\in[0,T).$ A similar calculation verifies that a left time derivative exists
too.
Differentiating (A.1) in time we find
$\displaystyle\int_{\Gamma_{0}}\frac{\partial\tilde{\mathbb{D}}}{\partial
t}(t)\nabla_{\Gamma}\psi(t)\cdot\nabla_{\Gamma}\chi+\tilde{\mathbb{D}}(t)\nabla_{\Gamma}\frac{\partial\psi}{\partial
t}(t)\cdot\nabla_{\Gamma}\chi+\frac{\partial\psi}{\partial
t}(t)\boldsymbol{\omega}(t)\cdot\nabla_{\Gamma}\chi+\psi(t)\frac{\partial\boldsymbol{\omega}}{\partial
t}(t)\cdot\nabla_{\Gamma}\chi=\int_{\Gamma_{0}}\frac{\partial f}{\partial
t}(t)\chi,$
for all $\chi\in H^{1}(\Gamma_{0})$, $t\in[0,T]$. As above, by noting that
$\tilde{\mathbb{D}},\boldsymbol{\omega}$ are $C^{2}$ in $t$ and $f$ is $C^{1}$
in $t$, one can now readily observe that the map
$t\mapsto\|\frac{\partial\psi}{\partial t}(t)\|_{H^{1}(\Gamma_{0})}$ is
continuous on $[0,T]$. Applying elliptic regularity theory we find that
$\psi\in C^{0}([0,T];H^{3,p}(\Gamma_{0}))\cap
C^{1}([0,T];H^{1,p}(\Gamma_{0}))$, and using the hence the compatibility of
$(H^{3,p}(\Gamma(t)),\Phi_{t}^{n})$ (and uniform bounds on $J(t)$ where
needed) it follows that $\Psi\in C^{0}_{H^{1,p}}\cap C^{1}_{H^{1,p}}.$ ∎
## Appendix B Killing Vectors and the inverse Stokes operator
In this appendix we discuss a solution operator related to the surface Stokes
equation. We refer the reader to [15, 34] for further details. For
$t\in[0,T]$, and a given $\boldsymbol{\phi}\in\mathbf{H}_{\sigma}(t)$ we are
interested in finding a solution
$\mathcal{S}\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)$ solving
$-\mathbb{P}\nabla_{\Gamma}\cdot(2\mathbb{E}(\mathcal{S}\boldsymbol{\phi}))=\boldsymbol{\phi},\text{
on }\Gamma(t),$
in a weak sense. However, for general
$\boldsymbol{\phi}\in\mathbf{H}_{\sigma}(t)$ there is not a unique solution as
$\mathbf{E}(\cdot)$ has a non-trivial kernel. However one can obtain a unique
solution by filtering out Killing vectors, that is
$\boldsymbol{\psi}\in\mathcal{K}(t):=\\{\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)\mid\mathbb{E}(\boldsymbol{\phi})=0\\}.$
This is analogous to considering the subspace $L^{2}_{0}(\Gamma(t)\leq
L^{2}(\Gamma(t))$ when solving the Laplace equation. Clearly we have that
$\mathcal{K}(t)$ is a closed subspace of $\mathbf{V}_{\sigma}(t)$, and it is
known to have dimension $\leq 3$ (see [47], Proposition III.6.5). We then
define an orthogonal space $\mathcal{K}^{\perp}$, with respect to the
$\mathbf{L}^{2}$ inner product, so that
$\mathbf{H}_{\sigma}(t)=\mathcal{K}(t)\oplus\mathcal{K}^{\perp}(t)$.
For $\boldsymbol{\phi}\in\mathcal{K}^{\perp}(t)$ we now define the inverse
Stokes’ operator to be the unique solution,
$\mathcal{S}\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)\cap\mathcal{K}^{\perp}(t)$,
of
$\mathbf{a}(\mathcal{S}\boldsymbol{\phi},\boldsymbol{\psi})=\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\psi}),$
for all $\boldsymbol{\psi}\in\mathbf{V}_{\sigma}(t)$. The well-posedness of
this owes to the following Korn-type inequality.
###### Lemma B.1 ([34], Lemma 4.1).
There exists a constant, $C$, independent of $t$, such that for
$\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)\cap\mathcal{K}^{\perp}(t)$ we have
$\displaystyle\|\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}\leq
C\|\mathbb{E}(\boldsymbol{\phi})\|_{\mathbf{L}^{2}(\Gamma(t))}.$ (B.1)
The fact this constant is independent of $t$ follows from the same logic as in
the proof of [45] Lemma 3.2. One also obtains an inequality for
$\boldsymbol{\phi}\in\mathcal{K}(t)$,
$\displaystyle\|\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}\leq
C\|\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))},$ (B.2)
which follows from (3.9) and the definition of $\mathcal{K}(t)$. In fact,
using (3.10), the Sobolev embedding
$\mathbf{H}^{1}(\Gamma(t))\hookrightarrow\mathbf{L}^{p}(\Gamma(t))$, and the
above inequality one finds
$\displaystyle\|\boldsymbol{\phi}\|_{\mathbf{H}^{1,p}(\Gamma(t))}\leq
C\|\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))},$ (B.3)
for $p\in[1,\infty)$.
As in [15], we have a sufficiently smooth surface, $\Gamma(t)$, so that one
has improved regularity
$\displaystyle\|\mathcal{S}\boldsymbol{\phi}\|_{\mathbf{H}^{2}(\Gamma(t))}\leq
C\|\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t)},$ (B.4)
and the constant $C$ is independent of $t$ by the usual arguments.
With this inverse Stokes operator, and (B.1), one can define a norm on
$\mathcal{K}^{\perp}(t)$ given by
$\|\boldsymbol{\phi}\|_{\perp}^{2}:=\mathbf{a}(\mathcal{S}\boldsymbol{\phi},\mathcal{S}\boldsymbol{\phi})=\mathbf{m}(\boldsymbol{\phi},\mathcal{S}\boldsymbol{\phi}),$
and it is straightforward to see that $\|\boldsymbol{\phi}\|_{\perp}\leq
C\|\boldsymbol{\phi}\|_{L^{2}(\Gamma(t))},$ for the constant $C$ in (B.1).
To extend $\mathcal{S}$ to general elements of $\mathbf{H}_{\sigma}(t)$ we use
projections. For $\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)$ we define the
projections
$P_{\mathcal{K}}\boldsymbol{\phi}\in\mathcal{K}(t),P_{\mathcal{K}^{\perp}}\boldsymbol{\phi}\in\mathcal{K}^{\perp}(t)$
to be the unique solutions of
$\displaystyle\mathbf{m}(P_{\mathcal{K}}\boldsymbol{\phi},\boldsymbol{\psi})=\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\psi}),\text{
for all }\boldsymbol{\psi}\in\mathcal{K}(t),$
$\displaystyle\mathbf{m}(P_{\mathcal{K}^{\perp}}\boldsymbol{\phi},\boldsymbol{\psi})=\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\psi}),\text{
for all }\boldsymbol{\psi}\in\mathcal{K}^{\perp}(t),$
respectively. It is clear that
$\|P_{\mathcal{K}}\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))}\leq\|\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))}$
and
$\|P_{\mathcal{K}^{\perp}}\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))}\leq\|\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))}$.
Using this we now extend $\mathcal{S}$ to $\mathbf{V}_{\sigma}(t)$ as follows.
For $\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)$ we define
$\mathcal{S}^{\perp}\boldsymbol{\phi}\in\mathcal{K}^{\perp}(t)$ to be the
unique solution of
$\mathbf{a}(\mathcal{S}^{\perp}\boldsymbol{\phi},\boldsymbol{\psi})=\mathbf{m}(P_{\mathcal{K}^{\perp}}\boldsymbol{\phi},\boldsymbol{\psi}),$
for all $\boldsymbol{\psi}\in\mathbf{V}_{\sigma}(t)$. This is just the
composition $\mathcal{S}^{\perp}=\mathcal{S}\circ P_{\mathcal{K}^{\perp}}$,
but it is convenient to have a shorthand notation for this. It is clear that
from the above properties of the projections that
$\|P_{\mathcal{K}^{\perp}}\boldsymbol{\phi}\|_{\perp}\leq
C\|\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))}.$
These operators arise in our proof of uniqueness, Theorem 6.1. In order to
discuss uniqueness, we need to consider the differentiability of
$P_{\mathcal{K}}\boldsymbol{\phi},P_{\mathcal{K}^{\perp}}\boldsymbol{\phi}$ in
time. We denote the killing vectors by
$\boldsymbol{\kappa}_{1},...,\boldsymbol{\kappa}_{N_{K}}$ which are determined
by $\Gamma(t)$, where we have used the notation $N_{K}:=\dim\mathcal{K}(t)\leq
3.$ We note that $N_{K}$ is independent of time, as the surfaces $\Gamma(t)$
are all diffeomorphic. We choose Killing vectors, $\boldsymbol{\kappa}_{i}$,
such that they are orthonormal with respect to the $\mathbf{L^{2}}(\Gamma(t))$
inner product. The projection $P_{\mathcal{K}}$ is then seen to be such that
$P_{\mathcal{K}}\boldsymbol{\phi}=\sum_{i=1}^{N_{K}}\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\kappa}_{i})\boldsymbol{\kappa}_{i}.$
Thus, heuristically, one expects that for sufficiently smooth
$\boldsymbol{\phi}$,
$\partial^{\circ}P_{\mathcal{K}}\boldsymbol{\phi}=P_{\mathcal{K}}\partial^{\circ}\boldsymbol{\phi}+\text{lower
order terms in }\boldsymbol{\phi},$
and likewise an analogous result for
$P_{\mathcal{K}^{\perp}}=\text{id}-P_{\mathcal{K}}$.
We will show this in a later lemma, but first need to see that we can map from
$\mathcal{K}(0)\rightarrow\mathcal{K}(t)$ in a sufficiently smooth way.
###### Lemma B.2.
There exists a $C^{1}_{\mathbf{H}^{1}}$ mapping from $\mathcal{K}(0)$ onto
$\mathcal{K}(t)$.
###### Proof.
Firstly we note that our definition of a Killing vector is equivalent to the
more standard definition of a Killing vector on a Riemannian manifold, defined
in terms of the Lie derivative. That is for some
$\boldsymbol{\kappa}\in\mathcal{K}(t),$
$\mathbb{E}(\boldsymbol{\kappa})=0\text{ on
}\Gamma(t)\Leftrightarrow\mathcal{L}_{\boldsymbol{\kappa}}g(t)=0,$
where $g(t)$ is the associated metric for $\Gamma(t)$. To see this we refer to
reader to [20], Appendix A. As $\Gamma(t)$ is an embedded surface in
$\mathbb{R}^{3}$, the natural metric on $\Gamma(t)$ is given by
$g(t)=x(t)^{*}\mathfrak{e},$ the pullback of the Euclidean metric,
$\mathfrak{e}$, on $\mathbb{R}^{3}$ by a $C^{3}$ embedding
$x(t):\Gamma_{0}\rightarrow\Gamma(t)$. Here, we consider $\Gamma(t)$ to be
parametrised over $\Gamma_{0}$, with the embedding
$x:\Gamma_{0}\times[0,T]\rightarrow\mathcal{G}_{T}$ given by the unique
solution of
$\frac{dx}{dt}=V_{N}(x(t),t),\qquad x(0)=x_{0},$
for all points $x_{0}\in\Gamma_{0}$. In particular this lets us see that
$g(t)=x(t)^{*}g(0)$, and hence
$x(t)^{*}g(0)=x(t)^{*}(x(0)^{*}\mathfrak{e})=(x(t)\circ
x(0))^{*}\mathfrak{e}=x(t)^{*}\mathfrak{e}=g(t).$
Now it is known that the Lie derivative commutes with pullbacks, as can be
seen from the fact that the exterior derivative commutes with pullbacks and
writing the Lie derivative in terms of the exterior derivative - see for
example [35], Theorem 2.3.4. Hence for $\boldsymbol{\kappa}\in\mathcal{K}(0)$
we see $\mathcal{L}_{\boldsymbol{\kappa}}g(0)=0$, and so
$0=x(t)^{*}\left(\mathcal{L}_{\boldsymbol{\kappa}}g(0)\right)=\mathcal{L}_{x(t)^{*}\boldsymbol{\kappa}}x(t)^{*}g(0)=\mathcal{L}_{x(t)^{*}\boldsymbol{\kappa}}g(t).$
Hence the map $\boldsymbol{\kappa}\mapsto x(t)^{*}\boldsymbol{\kappa}$ is such
that $\mathcal{K}(0)\rightarrow\mathcal{K}(t)$. Moreover we see that the
choice of embedding we used that
$x(t)^{*}\boldsymbol{\kappa}=\Phi_{t}^{n}\boldsymbol{\kappa}$, and
$\Phi_{t}^{n}\boldsymbol{\kappa}$ clearly has a strong material time
derivative (which vanishes by definition). ∎
Given an orthonormal basis (with respect to $\mathbf{L}^{2}(\Gamma_{0})$),
$\boldsymbol{\kappa}_{1},...,\boldsymbol{\kappa}_{N_{K}}$, of
$\mathcal{K}(0)$, the above calculations let us see that
$\Phi_{t}^{n}\boldsymbol{\kappa}_{1},...,\Phi_{t}^{n}\boldsymbol{\kappa}_{N_{K}}$
form a basis of $\mathcal{K}(t)$. However this new basis is not necessarily
orthonormal with respect to $\mathbf{L}^{2}(\Gamma(t))$, but using the Gram-
Schmidt procedure on
$\Phi_{t}^{n}\boldsymbol{\kappa}_{1},...,\Phi_{t}^{n}\boldsymbol{\kappa}_{N_{K}}$,
we obtain an orthonormal basis which we label
$\boldsymbol{\kappa}_{1}(t),...,\boldsymbol{\kappa}_{N_{K}}(t)$. In using the
Gram-Schmidt process we no longer retain the fact that the material time
derivative vanishes, but it is straightforward to show that
$\partial^{\circ}\boldsymbol{\kappa}_{i}\in C^{0}_{\mathbf{H}^{1}}$.
We now show the desired differentiability of the Killing projections.
###### Lemma B.3.
For $\Gamma(t)$ a $C^{3}$ evolving surface we have the following.
1. 1.
If $\boldsymbol{\phi}\in L^{2}_{\mathbf{V}_{\sigma}}\cap
H^{1}_{\mathbf{V}_{\sigma}^{\prime}}$ then
$P_{\mathcal{K}}\boldsymbol{\phi}\in L^{2}_{\mathbf{V}_{\sigma}}\cap
H^{1}_{\mathbf{V}_{\sigma}^{\prime}}$ with
$\displaystyle\int_{0}^{T}\left\|\partial^{\circ}P_{\mathcal{K}}\boldsymbol{\phi}\right\|_{\mathbf{V}_{\sigma}(t)^{\prime}}^{2}$
$\displaystyle\leq
C\int_{0}^{T}\left\|\partial^{\circ}\boldsymbol{\phi}\right\|_{\mathbf{V}_{\sigma}(t)^{\prime}}^{2}+C\int_{0}^{T}\|\boldsymbol{\phi}\|_{L^{2}(\Gamma(t)}^{2},$
$\displaystyle\int_{0}^{T}\left\|\partial^{\circ}P_{\mathcal{K}^{\perp}}\boldsymbol{\phi}\right\|_{\mathbf{V}_{\sigma}(t)^{\prime}}^{2}$
$\displaystyle\leq
C\int_{0}^{T}\left\|\partial^{\circ}\boldsymbol{\phi}\right\|_{\mathbf{V}_{\sigma}(t)^{\prime}}^{2}+C\int_{0}^{T}\|\boldsymbol{\phi}\|_{L^{2}(\Gamma(t)}^{2}$
2. 2.
If $\boldsymbol{\phi}\in L^{2}_{\mathbf{V}_{\sigma}}\cap
H^{1}_{\mathbf{V}_{\sigma}^{\prime}}$ is such that
$\boldsymbol{\phi}(t)\in\mathcal{K}^{\perp}$ for almost all $t\in[0,T]$ then
$\mathcal{S}\boldsymbol{\phi}\in H^{1}_{\mathbf{V}_{\sigma}}$ with
$\int_{0}^{T}\left\|\partial^{\circ}\mathcal{S}\boldsymbol{\phi}\right\|_{\mathbf{H}^{1}(\Gamma(t))}^{2}\leq
C\int_{0}^{T}\left(\|\mathcal{S}\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}^{2}+\|\partial^{\circ}\boldsymbol{\phi}\|_{\mathbf{V}_{\sigma}(t)^{\prime}}^{2}+\|\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}\right).$
###### Proof.
For the first part of this one considers
$\boldsymbol{\phi},\boldsymbol{\psi}\in H^{1}_{\mathbf{V}_{\sigma}}$ and by
using the definition of $P_{\mathcal{K}}$ we see
$\mathbf{m}(P_{\mathcal{K}}\boldsymbol{\phi},\boldsymbol{\psi})=\sum_{i=1}^{N_{K}}\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\kappa}_{i})\mathbf{m}(\boldsymbol{\psi},\boldsymbol{\kappa}_{i}),$
where $\boldsymbol{\kappa}_{i}$ are an orthonormal basis of $\mathcal{K}(t)$
with respect to the $\mathbf{L}^{2}(\Gamma(t))$ inner product. Differentiating
the above yields
$\mathbf{m}(\partial^{\circ}P_{\mathcal{K}}\boldsymbol{\phi},\boldsymbol{\psi})+\mathbf{m}(P_{\mathcal{K}}\boldsymbol{\phi},\partial^{\circ}\boldsymbol{\psi})+\mathbf{m}(P_{\mathcal{K}}\boldsymbol{\phi},\boldsymbol{\psi}HV_{N})=\sum_{i=1}^{N_{K}}\left[\mathbf{m}(\partial^{\circ}\boldsymbol{\phi},\boldsymbol{\kappa}_{i})+\mathbf{m}(\boldsymbol{\phi},\partial^{\circ}\boldsymbol{\kappa}_{i})+\mathbf{m}(\boldsymbol{\phi}HV_{N},\boldsymbol{\kappa}_{i})\right]\mathbf{m}(\boldsymbol{\psi},\boldsymbol{\kappa}_{i})\\\
+\sum_{i=1}^{N_{K}}\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\kappa}_{i})\left[\mathbf{m}(\partial^{\circ}\boldsymbol{\psi},\boldsymbol{\kappa}_{i})+\mathbf{m}(\boldsymbol{\psi},\partial^{\circ}\boldsymbol{\kappa}_{i})+\mathbf{m}(\boldsymbol{\psi}HV_{N},\boldsymbol{\kappa}_{i})\right],$
from which, by expanding terms on the left similarly, one finds
$\displaystyle\mathbf{m}(\partial^{\circ}P_{\mathcal{K}}\boldsymbol{\phi},\boldsymbol{\psi})=\mathbf{m}(\partial^{\circ}\boldsymbol{\phi},P_{\mathcal{K}}\boldsymbol{\psi})+\sum_{i=1}^{N_{K}}\left[\mathbf{m}(\boldsymbol{\phi},\partial^{\circ}\boldsymbol{\kappa}_{i})+\mathbf{m}(\boldsymbol{\phi}HV_{N},\boldsymbol{\kappa}_{i})\right]\mathbf{m}(\boldsymbol{\psi},\boldsymbol{\kappa}_{i})+\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\kappa}_{i})\mathbf{m}(\boldsymbol{\psi},\partial^{\circ}\boldsymbol{\kappa}_{i}).$
From this one can use an approximation argument to extend to
$\boldsymbol{\phi}\in L^{2}_{\mathbf{V}_{\sigma}}\cap
H^{1}_{\mathbf{V}_{\sigma}^{\prime}},\boldsymbol{\psi}\in
L^{2}_{\mathbf{V}_{\sigma}}$, and see that more generally
$\displaystyle\mathbf{m}_{*}(\partial^{\circ}P_{\mathcal{K}}\boldsymbol{\phi},\boldsymbol{\psi})=\mathbf{m}_{*}(\partial^{\circ}\boldsymbol{\phi},P_{\mathcal{K}}\boldsymbol{\psi})+\sum_{i=1}^{N_{K}}\left[\mathbf{m}(\boldsymbol{\phi},\partial^{\circ}\boldsymbol{\kappa}_{i})+\mathbf{m}(\boldsymbol{\phi}HV_{N},\boldsymbol{\kappa}_{i})\right]\mathbf{m}(\boldsymbol{\psi},\boldsymbol{\kappa}_{i})+\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\kappa}_{i})\mathbf{m}(\boldsymbol{\psi},\partial^{\circ}\boldsymbol{\kappa}_{i}),$
(B.5)
from which one obtains
$|\mathbf{m}_{*}(\partial^{\circ}P_{\mathcal{K}}\boldsymbol{\phi},\boldsymbol{\psi})|=\|\partial^{\circ}\boldsymbol{\phi}\|_{\mathbf{V}_{\sigma}^{\prime}}\|P_{\mathcal{K}}\boldsymbol{\psi}\|_{\mathbf{H}^{1}(\Gamma(t))}\\\
+\|\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t)}\|\boldsymbol{\psi}\|_{\mathbf{L}^{2}(\Gamma(t))}\sum_{i=1}^{N_{K}}\left[2\|\partial^{\circ}\boldsymbol{\kappa}_{i}\|_{\mathbf{L}^{2}(\Gamma(t))}\|\boldsymbol{\kappa}_{i}\|_{\mathbf{L}^{2}(\Gamma(t))}+\|HV_{N}\|_{L^{\infty}(\Gamma(t))}\|\boldsymbol{\kappa}_{i}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}\right].$
Now by using (B.2) we see
$\|P_{\mathcal{K}}\boldsymbol{\psi}\|_{\mathbf{H}^{1}(\Gamma(t))}\leq
C\|\boldsymbol{\psi}\|_{\mathbf{H}^{1}(\Gamma(t))}$, from which one can now
readily deduce the $L^{2}_{\mathbf{V}_{\sigma}^{\prime}}$ bound for
$\partial^{\circ}P_{\mathcal{K}}\phi$ by integrating over $[0,T]$. The
argument for bounding
$\partial^{\circ}P_{\mathcal{K}^{\perp}}\boldsymbol{\phi}$ is similar, where
we exploit the fact that
$P_{\mathcal{K}^{\perp}}\boldsymbol{\phi}=\boldsymbol{\phi}-P_{\mathcal{K}}\boldsymbol{\phi}$.
We omit the details.
For the second part of this result, one differentiates the equation
$\mathbf{a}(\mathcal{S}\boldsymbol{\phi},\boldsymbol{\psi})=\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\psi}),$
where we take $\boldsymbol{\psi}\in H^{1}_{\mathbf{V}_{\sigma}}$. This yields
$\mathbf{a}\left(\partial^{\circ}\mathcal{S}\boldsymbol{\phi},\boldsymbol{\psi}\right)+\mathbf{a}\left(\mathcal{S}\boldsymbol{\phi},\partial^{\circ}\boldsymbol{\psi}\right)+\mathbf{b}(\mathcal{S}\boldsymbol{\phi},\boldsymbol{\psi})=\mathbf{m}_{*}\left(\partial^{\circ}\boldsymbol{\phi},\boldsymbol{\psi}\right)+\mathbf{m}\left(\boldsymbol{\phi},\partial^{\circ}\boldsymbol{\psi}\right)+\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\psi}HV_{N}),$
and we note that by definition
$\mathbf{a}\left(\mathcal{S}\boldsymbol{\phi},\partial^{\circ}\boldsymbol{\psi}\right)=\mathbf{m}\left(\boldsymbol{\phi},\partial^{\circ}\boldsymbol{\psi}\right).$
Hence we find $\partial^{\circ}\mathcal{S}\boldsymbol{\phi}$ solves
$\displaystyle\mathbf{a}\left(\partial^{\circ}\mathcal{S}\boldsymbol{\phi},\boldsymbol{\psi}\right)=\mathbf{m}_{*}\left(\partial^{\circ}\boldsymbol{\phi},\boldsymbol{\psi}\right)+\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\psi}HV_{N})-\mathbf{b}(\mathcal{S}\boldsymbol{\phi},\boldsymbol{\psi}),$
(B.6)
for all $\boldsymbol{\psi}\in L^{2}_{\mathbf{V}_{\sigma}}$, where we have
extended to $L^{2}_{\mathbf{V}_{\sigma}}$ by density. One can verify that for
a fixed $t$, say $t^{*}$, and $\boldsymbol{\psi}\in\mathcal{K}(t^{*})$ that
$\mathbf{m}_{*}\left(\partial^{\circ}\boldsymbol{\phi},\boldsymbol{\psi}\right)+\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\psi}HV_{N})-\mathbf{b}(\mathcal{S}\boldsymbol{\phi},\boldsymbol{\psi})=0,$
by considering the function $\Phi_{t}^{n}\Phi_{-t^{*}}^{n}\boldsymbol{\psi}$
which has vanishing strong material derivative so that
$\mathbf{m}_{*}\left(\partial^{\circ}\boldsymbol{\phi},\boldsymbol{\psi}\right)+\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\psi}HV_{N})=\frac{d}{dt}\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\psi})=0.$
Likewise $\mathbf{b}(\mathcal{S}\boldsymbol{\phi},\boldsymbol{\psi})=0$ as
$\mathbb{E}(\boldsymbol{\psi})=0$. Hence one observes that the right-hand side
of (B.7) is a bounded linear functional acting on $\mathbf{V}_{\sigma}(t)$,
which vanishes on $\mathcal{K}(t)$. Hence using the coercivity result (B.1)
and the Lax-Milgram theorem one finds there is a unique solution of (B.6) in
$\mathcal{K}^{\perp}(t)$, and from the above we see this must be
$P_{\mathcal{K}^{\perp}}\partial^{\circ}\mathcal{S}\boldsymbol{\phi}$.
Moreover we see that this is bounded by
$\displaystyle\|P_{\mathcal{K}^{\perp}}\partial^{\circ}\mathcal{S}\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}\leq
C\left(\|\mathcal{S}\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}^{2}+\|\partial^{\circ}\boldsymbol{\phi}\|_{\mathbf{V}_{\sigma}(t)^{\prime}}^{2}+\|\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}\right).$
(B.7)
It remains to bound
$P_{\mathcal{K}}\partial^{\circ}\mathcal{S}\boldsymbol{\phi}$. Firstly, we use
a duality argument to bound
$P_{\mathcal{K}}\partial^{\circ}\mathcal{S}\boldsymbol{\phi}$, writing
$\|P_{\mathcal{K}}\partial^{\circ}\mathcal{S}\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))}=\sup_{\boldsymbol{\psi}\in\mathbf{L}^{2}(\Gamma(t))\setminus\\{0\\}}\frac{|\mathbf{m}(P_{\mathcal{K}}\partial^{\circ}\mathcal{S}\boldsymbol{\phi},\boldsymbol{\psi})|}{\|\boldsymbol{\psi}\|_{\mathbf{L}^{2}(\Gamma(t))}},$
and from (B.5) one finds
$\mathbf{m}(P_{\mathcal{K}}\partial^{\circ}\mathcal{S}\boldsymbol{\phi},\boldsymbol{\psi})=-\sum_{i=1}^{N_{K}}\left[\mathbf{m}(\mathcal{S}\boldsymbol{\phi},\partial^{\circ}\boldsymbol{\kappa}_{i})+\mathbf{m}(\mathcal{S}\boldsymbol{\phi}HV_{N},\boldsymbol{\kappa}_{i})\right]\mathbf{m}(\boldsymbol{\psi},\boldsymbol{\kappa}_{i})-\mathbf{m}(\mathcal{S}\boldsymbol{\phi},\boldsymbol{\kappa}_{i})\mathbf{m}(\boldsymbol{\psi},\partial^{\circ}\boldsymbol{\kappa}_{i}),$
where we have used the fact that
$P_{\mathcal{K}}\mathcal{S}\boldsymbol{\phi}=0$. Combining these facts, we see
$\|P_{\mathcal{K}}\partial^{\circ}\mathcal{S}\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))}\leq
C\|\mathcal{S}\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))},$
for a constant $C$ which depends on the relevant geometric quantities. Then by
using (B.2) it follows that we have in fact that
$\displaystyle\|P_{\mathcal{K}}\partial^{\circ}\mathcal{S}\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}\leq
C\|P_{\mathcal{K}}\partial^{\circ}\mathcal{S}\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))}\leq
C\|\mathcal{S}\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))},$ (B.8)
and combining (B.7), (B.8) yields the result. ∎
From this one also deduces $\mathcal{S}^{\perp}\boldsymbol{\phi}\in
H^{1}_{\mathbf{V}_{\sigma}}$ for $\boldsymbol{\phi}\in
L^{2}_{\mathbf{V}_{\sigma}}\cap H^{1}_{\mathbf{V}_{\sigma}^{\prime}}$, which
we use in our uniqueness result.
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|
from the instances fenerated from SYNTCOMP benchmark suite are listed in Table
LABEL:table:FPvsOddfairzlk. On the Odd-fair instances with $50\%-$liveness
generated from the SYNTCOMP benchmark suite, there are 204 instances where
neither of the algorithms OF-FP, OF-ZL, N-FP or N-ZL timed out. On these
instances, OF-ZL gives an average computation time of $4.6$ seconds while OF-
FP took $122.7$ seconds on average. On the same examples, N-ZL takes on
average $3.6$ seconds to compute while N-FP gives an average of $45.2$
seconds. For the PGSolver dataset OF-FP timed out on all generated instances,
whereas OF-ZL took $24.9$ seconds on average to terminate.
#### A.4.2 Sensitivity
To monitor the sensitivity of OF-ZL to the change in number of priorities as
well as the percentage of live edges in the game, we picked $12$ parity game
instances from the SYNTCOMP dataset which did not timeout (after one hour).
With priorities $3-4-5-6$ and liveness degrees 0$\%$777regular parity
game-30$\%$-50$\%$-80$\%$ we get 192 different Odd-fair parity instances. Fig.
5 shows the runtime of OF-ZL on these instances.
We can see that the runtimes of instances with different priority and liveness
percentages are distributed in a seemingly random manner. This tells us that
Odd-fair Zielonka’s algorithm is highly insensitive to a change in the
percentage of live edges and the number of priorities. This observation is
inline with the known insensitivity of Zielonka’s algorithm for the number of
priorities.
Figure 5: Runtime of OF-ZL on the 192 Odd-fair parity instances generated from
12 fixed parity examples through changing their priorities and liveness
degrees. Different shapes indicate the number of prioirities an instance has,
and the $x-$axis denotes their liveness percentages. At each coloumn we view
48 different instances of the 12 examples with varying colours.
#### A.4.3 Comparative Evaluation
In order to validate the computational advantage of OF-ZL over OF-FP, we have
run both algorithms on all 50$\%$-liveness instances generated from the
SYNTCOMP benchmark dataset. On 58 of these instances, both algorithms time
out. The run-times for all other instances are depicted in Fig. 6 (right), 7
(right) and 8 (right). The left plots in Fig. 6-8 show the same comparison for
the ?normal? parity algorithms N-ZL and N-FP. In both cases, Fig. 7 shows the
zoomed-in version of the respective plot in Fig. 6. Fig. 8 shows the data-
points from the respective plot in Fig. 7 as a scatter plot in log-scale. The
examples on which only x-FP times out, can be seen as the dots on the ceiling
of the plots in Fig. 6. In all plots, points above the diagonal correspond to
instances where Zielonka’s algorithm outperforms the fixed-point algorithm.
We clearly see in Fig. 6-8 that Zielonka’s algorithm performs significantly
better than the fixed-point version, both in the Odd-fair (right) and in the
normal (left) case. More importantly, the overall performance comparison
between OF-ZL over OF-FP (right plots) mimics the comparison between N-ZL over
N-FP. This allows us to conclude that our new Odd-fair Zielonka’s algorithm
retains the computational advantages of Zielonka’s algorithm.
In addition, Table LABEL:table:FPvsOddfairzlk shows that OF-ZL results in
almost the same run-time as N-ZL, showing that our changes in the algorithm
incur almost no computational disadvantages over the original algorithm. This
allows us to handle transition fairness for almost free in practice.
Figure 6: (Zoomed out version) A comparison of N-FP vs. N-ZL in regular parity
games (left), and OF-FP vs. OF-ZL on fair parity games (right)
Figure 7: (Zoomed in version) A comparison of N-FP vs. N-ZL in regular parity
games (left), and OF-FP vs. OF-ZL on fair parity games (right)
Figure 8: A comparison of N-FP vs. N-ZL in regular parity games (left), and
OF-FP vs. OF-ZL on fair parity games (right) in terms of log-scale plots where
the timeouts are removed.
Figure 9: A comparison of N-ZL vs. OF-ZL over examples that do not timeout on
both. Right hand side plot visualizes the same data in logscale.
Conclusion: The results show that Zielonka’s algorithm is significantly faster
in solving Odd-fair parity games compared to the calculation performed by the
fixed-point algorithm, as is the case in normal parity games. The fixed-point
algorithm started timing out as soon as the examples became more complex,
being especially sensitive to the increase in the number of priorities.
Whereas, Zielonka’s algorithm preserves its performance considerably in the
face of the increase in the same parameters. These outcomes match the known
comparison results between the naive fixed-point calculation versus Zielonka’s
algorithm, on normal parity games.
Table 1: Detailed run-time comparison of N-FP and N-ZL on the original parity game instances (yellow rows) with OF-FP and OF-ZL on their respective $30\%$\- and $50\%$-liveness Odd-fair parity game instances (white rows). The instance name is taken from the original benchmark suite. Name | $\\#$ | $\\#$ | $\\#$ | FP | ZL
---|---|---|---|---|---
| nodes | edges | priorities | (sec.) | (sec.)
EscalatorCountingInit | 99 | 148 | 3 | 0.064 | 0.012
$30\%$-EscalatorCountingInit | 99 | 148 | 3 | 0.075 | 0.018
$50\%$-EscalatorCountingInit | 99 | 148 | 3 | 0.072 | 0.02
KitchenTimerV1 | 80 | 124 | 3 | 0.055 | 0.008
$30\%$-KitchenTimerV1 | 80 | 124 | 3 | 0.068 | 0.012
$50\%$-KitchenTimerV1 | 80 | 124 | 3 | 0.21 | 0.009
KitchenTimerV6 | 4099 | 6560 | 3 | 87 | 11
$30\%$-KitchenTimerV6 | 4099 | 6560 | 3 | 88 | 11
$50\%$-KitchenTimerV6 | 4099 | 6560 | 3 | 352 | 18
MusicAppSimple | 344 | 562 | 3 | 0.488 | 0.073
$30\%$-MusicAppSimple | 344 | 562 | 3 | 0.496 | 0.082
$50\%$-MusicAppSimple | 344 | 562 | 3 | 0.799 | 0.089
TwoCountersRefinedRefined | 1933 | 3140 | 3 | 14.9 | 2.5
$30\%$-TwoCountersRefinedRefined | 1933 | 3140 | 3 | 15 | 1.2
$50\%$-TwoCountersRefinedRefined | 1933 | 3140 | 3 | 74 | 3.72
Zoo5 | 479 | 768 | 3 | 0.96 | 0.135
$30\%$-Zoo5 | 479 | 768 | 3 | 0.981 | 0.152
$50\%$-Zoo5 | 479 | 768 | 3 | 1.57 | 0.172
amba$\\_$decomposed$\\_$lock$\\_$3 | 1558 | 2336 | 3 | 72 | 1.5
$30\%$-amba$\\_$decomposed$\\_$lock$\\_$3 | 1558 | 2336 | 3 | 73 | 1.5
$50\%$-amba$\\_$decomposed$\\_$lock$\\_$3 | 1558 | 2336 | 3 | 56 | 2.9
full$\\_$arbiter$\\_$2 | 204 | 324 | 3 | 0.59 | 0.049
$30\%$-full$\\_$arbiter$\\_$2 | 204 | 324 | 3 | 0.602 | 0.047
$50\%$-full$\\_$arbiter$\\_$2 | 204 | 324 | 3 | 5 | 0.059
full$\\_$arbiter$\\_$3 | 1403 | 2396 | 3 | 21.18 | 2
$30\%$-full$\\_$arbiter$\\_$3 | 1403 | 2396 | 3 | 21.5 | 2
$50\%$-full$\\_$arbiter$\\_$3 | 1403 | 2396 | 3 | 93 | 3.46
lilydemo06 | 369 | 548 | 3 | 8.1 | 0.18
$30\%$-lilydemo06 | 369 | 548 | 3 | 8.13 | 0.206
$50\%$-lilydemo06 | 369 | 548 | 3 | 18 | 0.212
lilydemo07 | 78 | 108 | 3 | 0.27 | 0.01
$30\%$-lilydemo07 | 78 | 108 | 3 | 0.284 | 0.017
$50\%$-lilydemo07 | 78 | 108 | 3 | 0.33 | 0.008
simple$\\_$arbiter$\\_$unreal1 | 2178 | 3676 | 3 | 22.8 | 3
$30\%$-simple$\\_$arbiter$\\_$unreal1 | 2178 | 3676 | 3 | 23 | 3
$50\%$-simple$\\_$arbiter$\\_$unreal1 | 2178 | 3676 | 3 | 254 | 7
amba$\\_$decomposed$\\_$arbiter$\\_$2 | 141 | 212 | 4 | 0.72 | 0.03
$30\%$-amba$\\_$decomposed$\\_$arbiter$\\_$2 | 141 | 212 | 4 | 0.73 | 0.06
$50\%$-amba$\\_$decomposed$\\_$arbiter$\\_$2 | 141 | 212 | 4 | 1 | 0.035
loadfull3 | 1159 | 2030 | 4 | 5.62 | 0.609
$30\%$-loadfull3 | 1159 | 2030 | 4 | 5 | 0.614
$50\%$-loadfull3 | 1159 | 2030 | 4 | 5 | 0.754
ltl2dba01 | 101 | 152 | 4 | 0.074 | 0.031
$30\%$-ltl2dba01 | 101 | 152 | 4 | 0.075 | 0.030
$50\%$-ltl2dba01 | 101 | 152 | 4 | 1.4 | 0.028
ltl2dba14 | 97 | 144 | 4 | 0.18 | 0.016
$30\%$-ltl2dba14 | 97 | 144 | 4 | 0.181 | 0.013
$50\%$-ltl2dba14 | 97 | 144 | 4 | 0.574 | 0.012
ltl2dba22 | 21 | 30 | 4 | 0.037 | 0.002
$30\%$-ltl2dba22 | 21 | 30 | 4 | 0.036 | 0.002
$50\%$-ltl2dba22 | 21 | 30 | 4 | 0.03 | 0.0009
prioritized$\\_$arbiter$\\_$unreal2 | 851 | 1412 | 4 | 15.8 | 0.73
$30\%$-prioritized$\\_$arbiter$\\_$unreal2 | 851 | 1412 | 4 | 16 | 0.759
$50\%$-prioritized$\\_$arbiter$\\_$unreal2 | 851 | 1412 | 4 | 126 | 1.2
lilydemo17 | 3102 | 5334 | 7 | 1237 | 41
$30\%$-lilydemo17 | 3102 | 5334 | 7 | Timeout | 41
$50\%$-lilydemo17 | 3102 | 5334 | 7 | Timeout | 24
lilydemo18 | 449 | 728 | 9 | 220 | 0.6
$30\%$-lilydemo18 | 449 | 728 | 9 | 224 | 0.621
$50\%$-lilydemo18 | 449 | 728 | 9 | Timeout | 0.552
### A.5 Additional material for Ex. 4.3
Below we present an extended version of the fixed-point calculation in (8),
$\displaystyle Y_{4}^{0}=\emptyset$ $\displaystyle\quad X_{3}^{0,0}=V$
$\displaystyle\quad\quad Y_{2}^{0,0,0}=\emptyset$
$\displaystyle\quad\quad\quad X_{1}^{0,0,0,0}=V$ $\displaystyle\quad\quad\quad
X_{1}^{0,0,0,1}=\Phi^{Y_{4}^{0},X_{3}^{0,0},Y_{2}^{0,0,0},X_{1}^{0,0,0,0}}=C_{3}\cup
C_{1}$ $\displaystyle\quad\quad\quad
X_{1}^{0,0,0,2}=\Phi^{Y_{4}^{0},X_{3}^{0,0},Y_{2}^{0,0,0},X_{1}^{0,0,0,1}}=C_{3}\cup(C_{1}\cap\mathsf{Npre}(Y_{2}^{0,0,0},X_{1}^{0,0,0,1}))=C_{3}$
$\displaystyle\quad\quad\quad
X_{1}^{0,0,0,3}=\Phi^{Y_{4}^{0},X_{3}^{0,0},Y_{2}^{0,0,0},X_{1}^{0,0,0,1}}=C_{3}\cup(C_{1}\cap\mathsf{Npre}(Y_{2}^{0,0,0},X_{1}^{0,0,0,2}))=C_{3}$
$\displaystyle\quad\quad Y_{2}^{0,0,1}=X_{1}^{0,0,0,\infty}=C_{3}$
$\displaystyle\quad\quad\quad X_{1}^{0,0,1,0}=V$ $\displaystyle\quad\quad\quad
X_{1}^{0,0,1,1}=\Phi^{Y_{4}^{0},X_{3}^{0,0},Y_{2}^{0,0,1},X_{1}^{0,0,0,0}}=C_{3}\cup
C_{1}\cup\\{2b\\}$ $\displaystyle\quad\quad\quad
X_{1}^{0,0,1,2}=\Phi^{Y_{4}^{0},X_{3}^{0,0},Y_{2}^{0,0,1},X_{1}^{0,0,0,1}}=C_{3}\cup\\{2b\\}$
$\displaystyle\quad\quad\quad
X_{1}^{0,0,1,3}=\Phi^{Y_{4}^{0},X_{3}^{0,0},Y_{2}^{0,0,1},X_{1}^{0,0,0,2}}=C_{3}\cup\\{2b\\}$
$\displaystyle\quad\quad Y_{2}^{0,0,2}=X_{1}^{0,0,1,\infty}=C_{3}\cup\\{2b\\}$
$\displaystyle\quad\quad\quad X_{1}^{0,0,2,0}=V$ $\displaystyle\quad\quad\quad
X_{1}^{0,0,2,1}=\Phi^{Y_{4}^{0},X_{3}^{0,0},Y_{2}^{0,0,2},X_{1}^{0,0,0,0}}=C_{3}\cup
C_{1}\cup\\{2b,2c\\}$ $\displaystyle\quad\quad\quad
X_{1}^{0,0,2,2}=\Phi^{Y_{4}^{0},X_{3}^{0,0},Y_{2}^{0,0,2},X_{1}^{0,0,0,1}}=C_{3}\cup\\{2b,2c\\}$
$\displaystyle\quad\quad\quad
X_{1}^{0,0,2,3}=\Phi^{Y_{4}^{0},X_{3}^{0,0},Y_{2}^{0,0,2},X_{1}^{0,0,0,2}}=C_{3}\cup\\{2b,2c\\}$
$\displaystyle\quad\quad
Y_{2}^{0,0,3}=X_{1}^{0,0,2,\infty}=C_{3}\cup\\{2b,2c\\}$
$\displaystyle\quad\quad\ldots$ $\displaystyle\quad\quad
Y_{2}^{0,0,4}=X_{1}^{0,0,3,\infty}=C_{3}\cup\\{2b,2c\\}$ $\displaystyle\quad
X_{3}^{0,1}=Y_{2}^{0,0,\infty}=C_{3}\cup\\{2b,2c\\}$ $\displaystyle\quad\quad
Y_{2}^{0,1,0}=\emptyset\quad$ $\displaystyle\quad\quad
Y_{2}^{0,1,1}=X_{1}^{0,1,0,\infty}=\\{3b\\}\quad$ $\displaystyle\quad\quad
Y_{2}^{0,1,2}=X_{1}^{0,1,1,\infty}=\\{2b,3b\\}\ \quad$
$\displaystyle\quad\quad
Y_{2}^{0,1,3}=Y_{2}^{0,1,4}=X_{1}^{0,1,2,\infty}=X_{1}^{0,1,3,\infty}=\\{2b,2c,3b\\}$
$\displaystyle\quad X_{3}^{0,2}=Y_{2}^{0,1,\infty}=\\{2b,2c,3b\\}$
$\displaystyle\quad\ldots$ $\displaystyle\quad
X_{3}^{0,3}=Y_{2}^{0,2,\infty}=\\{2b,2c,3b\\}$ $\displaystyle
Y_{4}^{1}=X_{3}^{0,\infty}=\\{2b,2c,3b\\}$ $\displaystyle\quad X_{3}^{1,0}=V$
$\displaystyle\quad\quad Y_{2}^{1,0,0}=\emptyset$ $\displaystyle\quad\quad
Y_{2}^{1,0,1}=X_{1}^{1,0,0,\infty}=C_{3}\cup C_{4}$ $\displaystyle\quad\quad
Y_{2}^{1,0,2}=X_{1}^{1,0,1,\infty}=C_{3}\cup C_{4}\cup\\{2b\\}$
$\displaystyle\quad\quad Y_{2}^{1,0,4}=Y_{2}^{1,0,3}=C_{1}\cup C_{3}\cup
C_{4}\cup\\{2b,2c\\}$ $\displaystyle\quad
X_{3}^{1,1}=Y_{2}^{1,0,\infty}=C_{1}\cup C_{3}\cup C_{4}\cup\\{2b,2c\\}$
$\displaystyle\quad\quad Y_{2}^{1,1,0}=\emptyset$ $\displaystyle\quad\quad
Y_{2}^{1,1,1}=C_{3}\cup C_{4}$ $\displaystyle\quad\quad
Y_{2}^{1,1,2}=C_{3}\cup C_{4}\cup\\{2b\\}$ $\displaystyle\quad\quad
Y_{2}^{1,1,3}=Y_{2}^{1,1,4}=C_{1}\cup C_{3}\cup C_{4}\cup\\{2b,2c\\}$
$\displaystyle\quad X_{3}^{1,2}=Y_{2}^{1,1,\infty}=C_{1}\cup C_{3}\cup
C_{4}\cup\\{2b,2c\\}$ $\displaystyle Y_{4}^{2}=X_{3}^{1,\infty}=C_{1}\cup
C_{3}\cup C_{4}\cup\\{2b,2c\\}$ $\displaystyle\ldots$ $\displaystyle
Y_{4}^{3}=C_{1}\cup C_{3}\cup C_{4}\cup\\{2b,2c\\}$
And finally,
$\mathcal{W}_{Odd}=Y_{4}^{\infty}=C_{1}\cup C_{3}\cup
C_{4}\cup\\{2b,2c\\}=V\setminus\\{2a\\}$
|
1,2,3,4]Salvatore Capozziello<EMAIL_ADDRESS>5,2]Maurizio Capriolo<EMAIL_ADDRESS>5,6]Gaetano Lambiase<EMAIL_ADDRESS>
[1]Dipartimento di Fisica "E. Pancini", Università di Napoli “Federico II”, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy
[2]Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy
[3]Scuola Superiore Meridionale, Largo S. Marcellino 10, I-80138, Napoli, Italy
[4]Department of Mathematics, Faculty of Civil Engineering,VSB-Technical University of Ostrava, Ludvika Podeste 1875/17, 708 00 Ostrava-Poruba,
Czech Republic
[5]Dipartimento di Fisica "E. R. Caianiello", Università degli Studi di Salerno, via Giovanni Paolo II, 132 I-84084 Fisciano, Salerno, Italy
[6]Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Gruppo Collegato di Salerno, via Giovanni Paolo II, 132 I-84084 Fisciano, Salerno, Italy
Energy-Momentum Complex in Higher Order Curvature-Based Local Gravity
An unambiguous definition of gravitational energy remains one of the unresolved issues of physics today. This problem is related to the non-localization of gravitational energy density. In General Relativity, there have been many proposals for defining the gravitational energy density, notably those proposed by Einstein, Tolman, Landau and Lifshitz, Papapetrou, Møller, and Weinberg. In this review, we firstly explored the energy–momentum complex in an $n^{th}$ order gravitational Lagrangian $L=L\left(g_{\mu\nu}, g_{\mu\nu,i_{1}}, g_{\mu\nu,i_{1}i_{2}},g_{\mu\nu,i_{1}i_{2}i_{3}},\cdots, g_{\mu\nu,i_{1}i_{2}i_{3}\cdots i_{n}}\right)$ and then in a gravitational Lagrangian as . Its gravitational part was obtained by invariance of gravitational action under infinitesimal rigid translations using Noether's theorem. We also showed that this tensor, in general, is not a covariant object but only an affine object, that is, a pseudo-tensor. Therefore, the pseudo-tensor $\tau^{\eta}_{\alpha}$ becomes the one introduced by Einstein if we limit ourselves to General Relativity and its extended corrections have been explicitly indicated. The same method was used to derive the energy–momentum complex in $ f\left (R \right) $ gravity both in Palatini and metric approaches. Moreover, in the weak field approximation the pseudo-tensor $\tau^{\eta}_{\alpha}$ to lowest order in the metric perturbation $h$ was calculated. As a practical application, the power per unit solid angle $\Omega$ emitted by a localized source carried by a gravitational wave in a direction $\hat{x}$ for a fixed wave number $\mathbf{k}$ under a suitable gauge was obtained, through the average value of the pseudo-tensor over a suitable spacetime domain and the local conservation of the pseudo-tensor. As a cosmological application, in a flat Friedmann–Lemaître–Robertson–Walker spacetime, the gravitational and matter energy density in $f(R)$ gravity both in Palatini and metric formalism was proposed. The gravitational energy–momentum pseudo-tensor could be a useful tool to investigate further modes of gravitational radiation beyond two standard modes required by General Relativity and to deal with non-local theories of gravity involving $\Box^{-k}$ terms.
Keywords: Energy–Momentum Complex; Pseudo-Tensor; Gravitational Energy
§ INTRODUCTION
A widely accepted definition of gravitational energy density and its localization in curved spacetime are serious problems that afflict the general relativity. Several prescriptions for gravitational contribution to energy–momentum density and more generally for energy–momentum complex have been suggested by Einstein, Tolman, Landau and Lifshitz, Papapetrou, Møller, and Weinberg [8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. These attempts are based on the introduction of a super-potential or through the expansion of the Ricci tensor in the metric perturbation $h$. Thus, the gravitational part of the energy–momentum density transforms as an affine tensor not as a covariant tensor, and for this reason, it is not really a tensor but a pseudo-tensor. This affine property of the gravitational stress–energy tensor
makes the gravitational energy–momentum density not localizable. However, integrating the density over a suitable spatial region at a certain time such as over an asymptotically flat spacetime, viable for isolated systems, the gravitational energy–momentum becomes a four-vector, as meaning that changes in right way under asymptotically flat coordinate transformations. Over all space it becomes quasi independent of the coordinate system, that is, the gravitational energy–momentum of the spacetime exists, but it cannot be localized. In this review a generalization of Einstein's pseudo-tensor to Extended Theories of Gravity [18, 19] is proposed by imposing the invariance of the higher order gravitational Lagrangian under an infinitesimal rigid translation and by using Noether's theorem. Then, thanks to a continuity equation, a Noether current and a Noether charge were derived that correspond to a gravitational energy–momentum pseudo-tensor and gravitational energy–momentum, respectively, both locally conserved.
By weakly perturbing the metric tensor around the Minkowskian metric, a weak-field limit, in a suitable gauge, the gravitational energy–momentum pseudo-tensor for a Lagrangian of $n^{th}$ order appears an object easier to handle. Then, by averaging of the pseudo-tensor over a suitable spacetime domain, it is possible to calculate the power emitted by some localized astrophysical source carried away by the gravitational waves. This approach could be relevant for searching for polarization states of gravitational waves in addition to the two standards of general relativity [20, 21]. Finally, after deriving the gravitational energy–momentum pseudo-tensor in $F(R)$ gravity formulated in Palatini and metric formalism, some cosmological applications were discussed, wherein a flat FLRW metric the total energy density was obtained in both approaches [22, 23].
For more details on the issue of energy–momentum localization in modified theories of gravity such as $f(R)$, ${f(R,\Box R,\dots, \Box^{k} R)}$ [23, 24], teleparallel gravity and its extended version $f(T)$, see Ref. [25]. Meanwhile, for a study of wavelike solutions of modified teleparallel gravity necessary for future applications of the pseudo-tensor, see references [29, 30].
The review is organized as follows. Firstly in Sec. <ref> some definitions of gravitational pseudo-tensors in general relativity are listed. In Sec. <ref> we derived the gravitational energy–momentum pseudo-tensor for a general Lagrangian of $n^{th}$ order through two procedures: the first method uses a variational principle under rigid transformations via Noether's theorem and the second adopts the Landau–Lifshitz procedure [8] without the introduction of the super-potential. Hence, in Sec. <ref>, we proved that a stress–energy object is transformed in the correct manner under linear transformations but not under diffeomorphisms and, therefore, is a pseudo-tensor and not a covariant tensor. In Sec. <ref>, we calculated the Euler–Lagrange equations and the gravitational energy–momentum pseudo-tensor for $f(R)$ gravity, always using Noether's theorem applied to a particular one-parameter group of diffeomorphisms given by rigid translations. Therefore, in all models of gravity we obtained the continuity equation for an energy–momentum complex. In Sec. <ref>, we derived the gravitational energy–momentum pseudo-tensor of a gravitation field for a particular Lagrangian $L_{g}=(\overline{R}+a_{0}R^{2}+\sum_{k=1}^{p} a_{k}R\Box^{k}R)\sqrt{-g}$. Sec. <ref>, is devoted to the weak-field limit of the gravitational stress–energy pseudo-tensor expanded to lowest order in a small perturbation $h$, i.e., up to $h^2$ order, and we have shown two simple cases where the index $p$ is equal to zero and one. Hence, in Sec. <ref>, we averaged the pseudo-tensor over an suitable region containing the isolated sources and then we found the emitted power carried by the gravitational radiation. Afterward, in Sec. <ref>, in Palatini $f({\cal R})$ gravity, related field equations and related gravitational energy–momentum pseudo-tensor were obtained. Therefore in Sec. <ref>, by adopting a flat FLRW spacetime, an explicit calculus of an energy density complex for power law cosmological solutions was performed, also in the metric formalism of $f(R)$. Conclusions are summarized in Sec. <ref>. Finally in Appendix <ref>, we proved that the additive terms related to the symmetries of $g_{\mu\nu}$ and its derivatives yield a mean of zero, i.e., $\langle\left(A_{p}\right)_{\alpha}^{\eta}\rangle=\langle\left(B_{p}\right)_{\alpha}^{\eta}\rangle=0$. While in Appendix <ref>, we explicitly showed the six polarization tensors associated with the gravitational waves present in higher-order theories.
§ SEVERAL DEFINITIONS OF GRAVITATIONAL ENERGY–MOMENTUM PSEUDO-TENSOR IN GENERAL RELATIVITY
Here are some of the most important definitions of gravitational energy–momentum pseudo-tensor in general relativity in the scientific literature, for details see [31].
§.§ Einstein energy–momentum complex
In special relativity the law of conservation of energy and momentum is given by
\begin{equation}\label{conserveinongrav}
\frac{\partial T^{\mu\nu}}{\partial x^{\mu}}=0\ ,
\end{equation}
with $T^{\mu\nu}$ the energy–momentum tensor of matter and non-gravitational fields. In general relativity this principle becomes for general covariance
\begin{equation}\label{conserveigrav}
\nabla_{\mu}T^{\mu\nu}=0\ ,
\end{equation}
which does not correspond to any law of conservation of physical quantities. Einstein therefore formulated the conservation law in the following way
\begin{equation}\label{conserveigravespl}
\frac{\partial \theta_{\mu}^{\phantom{\mu}\nu}}{\partial x^{\nu}}=\frac{\partial}{\partial x^{\nu}}\left(\sqrt{-g}\left(T_{\mu}^{\phantom{\mu}\nu}+t_{\mu}^{\phantom{\mu}\nu}\right)\right)=0\ ,
\end{equation}
where $t_{\mu}^{\phantom{\mu}\nu}$ is an pseudo-tensor. So what is conserved is not only the tensor of non-gravitational fields and matter $T_{\mu}^{\phantom{\mu}\nu}$ but a pseudo-tensor $t_{\mu}^{\phantom{\mu}\nu}$ must be added to it. This pseudo-tensor added can be interpreted as associated with the gravitational field and the energy due to the sum of the contributions of the gravitational fields plus those due to the matter is conserved. However, the pseudo-tensoriality behaviour of $t_{\mu}^{\phantom{\mu}\nu}$ makes it dependent on coordinates and the gravitational energy becomes non localizable. In order to write the Eq. (<ref>) in the form of an ordinary divergence equation Eq. (<ref>), Einstein starting from the following Lagrangian density which is a non-covariant scalar density
\begin{equation}\label{LagrEinst}
L=\sqrt{-g} g^{\mu\nu}\left(\Gamma^{\sigma}_{\mu\nu}\Gamma^{\rho}_{\sigma\rho}-\Gamma^{\sigma}_{\mu\rho}\Gamma^{\rho}_{\nu\sigma}\right)\ ,
\end{equation}
introduced a pseudo-tensor defined by the relation
\begin{equation}
\sqrt{-g}t_{\mu}^{\phantom{\mu}\nu}=\frac{1}{16\pi}\left(\delta^{\nu}_{\mu}L-\frac{\partial L}{\partial g^{\rho\sigma}_{\phantom{\rho\sigma},\nu}}g^{\rho\sigma}_{\phantom{\rho\sigma},\mu}\right)\ .
\end{equation}
§.§ Landau–Lifshitz energy–momentum pseudo-tensor
The gravitational energy–momentum pseudo-tensor defined by Landau–Lifshitz has the great advantage of being symmetric unlike Einstein's, which in general is not.This allows defining the angular momentum and therefore the related conservation law. We adopt a system of geodetic coordinates where the first derivatives of the metric tensor $g^{\mu\nu}$ vanish. Then, the Eq. (<ref>) is reduced to (<ref>) which can be written in terms of the following antisymmetric quantity in the last two indices $\eta^{\mu\nu\sigma}=-\eta^{\mu\sigma\nu}$
\begin{equation}
T^{\mu\nu}=\frac{\partial \eta^{\mu\nu\sigma}}{\partial x^{\sigma}}\ .
\end{equation}
Since the Levi–Civita connection $\Gamma$ vanishes at one point, in such coordinate system it is possible using Einstein’s equations in the presence of matter written in such coordinates, to express the stress–energy tensor of matter $T^{\mu\nu}$ as
\begin{equation}
T^{\mu\nu}=\frac{1}{\left(-g\right)}\frac{\partial}{\partial x^{\sigma}}\left\{\frac{1}{16\pi}\frac{\partial}{\partial x^{\rho}}\left[\left(-g\right)\left(g^{\mu\nu}g^{\sigma\rho}-g^{\mu\sigma}g^{\nu\rho}\right)\right]\right\}\ ,
\end{equation}
where indicating the term in braces with the antisymmetric quantity in the last two indices
$h^{\mu\nu\sigma}=-h^{\mu\sigma\nu}$, we get
\begin{equation}
\frac{\partial h^{\mu\nu\sigma}}{\partial x^{\sigma}}-\left(-g\right)T^{\mu\nu}=0\ .
\end{equation}
Returning to an arbitrary coordinate system the previous difference does not cancel anymore so we can indicate it with $\left(-g\right)t^{\mu\nu}$ or
\begin{equation}\label{LLECCG}
\left(-g\right)\left(T^{\mu\nu}+t^{\mu\nu}\right)=\frac{\partial h^{\mu\nu\sigma}}{\partial x^{\sigma}}\ .
\end{equation}
Quantities $t^{\mu\nu}$ are symmetric but are not the components of a covariant tensor but affine. Using Einstein’s field equations again it is possible from Eq. (<ref>) to get an explicit expression of $t^{\mu\nu}$, defined as the energy–momentum pseudo-tensor of the gravitational field, by means of the derivatives of the components of the metric tensor, that is
\begin{equation}\label{LLEMPT}
\begin{split}
16\pi\left(-g\right)t^{\mu\nu}=&\mathfrak{g}^{\mu\nu}_{\phantom{\mu\nu},\rho}\mathfrak{g}^{\rho\sigma}_{\phantom{\rho\sigma},\sigma}-\mathfrak{g}^{\mu\rho}_{\phantom{\mu\rho},\rho}\mathfrak{g}^{\nu\sigma}_{\phantom{\nu\sigma},\sigma}+\frac{1}{2}g^{\mu\nu}g_{\rho\sigma}\mathfrak{g}^{\rho\alpha}_{\phantom{\rho n},\beta}\mathfrak{g}^{\beta\sigma}_{\phantom{\beta\sigma},\alpha}\\
&-\left(g^{\mu\rho}g_{\sigma\alpha}\mathfrak{g}^{\nu\alpha}_{\phantom{\nu n},\beta}\mathfrak{g}^{\sigma\beta}_{\phantom{\sigma\beta},\rho}+g^{\nu\rho}g_{\sigma \alpha}\mathfrak{g}^{\mu\alpha}_{\phantom{\mu\alpha},\beta}\mathfrak{g}^{\sigma\beta}_{\phantom{\sigma\beta},\rho}\right)+g_{\rho\sigma}g^{\alpha\beta}\mathfrak{g}^{\mu\rho}_{\phantom{\mu\rho},\alpha}\mathfrak{g}^{\nu\sigma}_{\phantom{\nu m},\beta}\\
&+\frac{1}{8}\left(2g^{\mu\rho}g^{\nu\sigma}-g^{\mu\nu}g^{\rho\sigma}\right)\left(2g_{\alpha\beta}g_{\gamma\lambda}-g_{\beta \gamma}g_{\alpha\lambda}\right)\mathfrak{g}^{\alpha\lambda}_{\phantom{\alpha\lambda},\rho}\mathfrak{g}^{\beta\gamma}_{\phantom{\beta\gamma},\sigma}\ ,
\end{split}
\end{equation}
where $\mathfrak{g}^{\mu\nu}=\sqrt{-g}g^{\mu\nu}$.
§.§ Møller energy–momentum complex
The energy–momentum pseudo-tensors $t^{\mu\nu}$ of both Einstein and Landau–Lifshitz besides having the flaw of being tensors only affine and not covariant also depend on the choice of coordinates. Then, Møller looked for an expression for energy and gravitational momentum independent of the particular coordinate system. To do this Møller exploited the fact that the pseudo-tensor including matter plus gravity $\theta^{\mu\nu}=T^{\mu\nu}+t^{\mu\nu}$ can be defined at less than a magnitude $S^{\mu\nu}$ at zero divergence $\partial_{\mu} S^{\mu\nu}=0$. In 1958 Møller proposed the following complex tensor of energy–momentum complex $\mathcal{T}_{\mu}^{\phantom{\mu}\nu}=\theta_{\mu}^{\phantom{\mu}\nu}+S_{\mu}^{\phantom{\mu}\nu}$ looking for the $S_{\mu}^{\phantom{\mu}\nu}$ such that $\mathcal{T}_{\mu}^{\phantom{\mu}\nu}$ transformed as a tensor for only spatial transformations
\begin{equation}\label{EMTGM}
\mathcal{T}_{\mu}^{\phantom{\mu}\nu}=\frac{1}{8\pi}\partial_{\rho}\left[\sqrt{-g}\left(g_{\mu\sigma,\lambda}-g_{\mu\lambda,\sigma}\right)g^{\lambda\nu}g^{\sigma\rho}\right]\ ,
\end{equation}
where the expression in square brackets is the antisymmetric super-potential $U_{\mu}^{\phantom{\mu}\nu\rho}=-U_{\mu}^{\phantom{\mu}\rho\nu}$ such that
\begin{equation}
\partial_{\nu}\mathcal{T}_{\mu}^{\phantom{\mu}\nu}=0\ .
\end{equation}
§.§ Papapetrou energy–momentum pseudo-tensor
Papapetrou in 1948 used the generalized Belifante method to derive his pseudo-tensor due to the complex of energy–momentum contributions using Tolman’s expression of Einstein's total pseudo-tensor $\theta_{\mu}^{\phantom{\mu}\nu}$ (<ref>) i.e.
\begin{equation}
\theta_{\mu}^{\phantom{\mu}\nu}=\frac{1}{8\pi}\frac{\partial}{\partial x^{\rho}}\left(-\mathfrak{g}^{\nu\sigma}\frac{\partial L}{\partial \mathfrak{g}^{\mu\sigma}_{\phantom{\mu\sigma},\rho}}+\frac{1}{2}\delta_{\mu}^{\nu}\mathfrak{g}^{\alpha\beta}\frac{\partial L}{\partial \mathfrak{g}^{\alpha\beta}_{\phantom{\alpha\beta},\rho}}\right)\ ,
\end{equation}
where $L$ is Einstein Lagrangian give by (<ref>) and $\mathfrak{g}^{\nu\sigma}$ have been defined in Eq. (<ref>). Belifante’s method consists in finding a symmetric quantity $\Omega^{\mu\nu}=\Omega^{\nu\mu}$ divergence free which differs by $\eta^{\mu\rho}\theta_{\rho}^{\phantom{\rho}\nu}$ only for an antisymmetric quantity divergence in the first two indices $B^{\mu\nu\rho}=-B^{\mu\nu\rho}$ i.e.
\begin{equation}
\Omega^{\mu\nu}=\eta^{\mu\rho}\theta_{\rho}^{\phantom{\rho}\nu}+\frac{\partial}{\partial x^{\rho}}B^{\mu\nu\rho}\ ,
\end{equation}
such that
\begin{equation}\label{coserlocpapa}
\frac{\partial}{\partial x^{\nu}}\Omega^{\mu\nu}=0\ ,
\end{equation}
with $\eta_{\mu\nu}=diag\left(1,-1,-1,-1\right)$. Expressing $B^{\mu\nu\rho}$ in terms of of the field spin density $S^{\mu\nu\rho}$
\begin{equation}
B^{\mu\nu\rho}=-\frac{1}{2}\left(S^{\mu\nu\rho}+S^{\rho\mu\nu}+S^{\rho\nu\mu}\right)\ ,
\end{equation}
you get after a few counts the expression for the total pseudo-tensor $\Omega^{\mu\nu}$
\begin{equation}
\Omega^{\mu\nu}=\frac{1}{16\pi}\frac{\partial^{2}}{\partial x^{\rho}x^{\sigma}}\left[\sqrt{-g}\left(g^{\mu\nu}\eta^{\rho\sigma}-g^{\mu\rho}\eta^{\nu\sigma}-g^{\rho\sigma}\eta{\mu\nu}-g^{\nu\sigma}\eta^{\mu\rho}\right)\right]\ .
\end{equation}
This geometric object is symmetric with respect to the first two indices$\mu$ e $\nu$.
§.§ Weinberg gravitational energy–momentum pseudo-tensor
Weinberg [32] derived the gravitational energy–momentum pseudo-tensor by adopting a quasi-minkowskian coordinate system. In this system the metric tensor $g_{\mu\nu}$ tends to that of Minkowski $\eta_{\mu\nu}$ at great distances from a localized material system. We write the metric $g_{\mu\nu}$ as the sum of the metric of Minkowski $\eta_{\mu\nu}$ plus $h_{\mu\nu}$ that goes to zero to infinity
\begin{equation}
g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\ .
\end{equation}
We linearize Einstein equations $G_{\mu\nu}=-8\pi G T_{\mu\nu}$, expanding Ricci tensor $R_{\mu\nu}$ in terms of powers of $h_{\mu\nu}$ as
\begin{equation}\label{EELFIRST}
R^{\left(1\right)}_{\phantom{\left(1\right)}\mu\nu}-\frac{1}{2}\eta_{\mu\nu}R^{\left(1\right)}=-8\pi G\left[T_{\mu\nu}+t_{\mu\nu}\right]\ ,
\end{equation}
\begin{equation}
t_{\mu\nu}=\frac{1}{8\pi G}\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-R^{\left(1\right)}_{\phantom{\left(1\right)}\mu\nu}+\frac{1}{2}\eta_{\mu\nu}R^{\left(1\right)}\right)\ ,
\end{equation}
is the gravitational energy–momentum pseudo-tensor. So in the Eq. (<ref>) you see that reading the equation from right to left, $t_{\mu\nu}$ assumes the meaning of the source of the linearized curvature together with the tensor of the non-gravitational fields and of the matter $T_{\mu\nu}$. From the linearized Bianchi law to which quantity $R^{\left(1\right)}_{\phantom{\left(1\right)}\mu\nu}$ must satisfy, we get the following local conservation law
\begin{equation}
\frac{\partial}{\partial x^{\nu}}\left(T^{\mu\nu}+t^{\mu\nu}\right)=0\ .
\end{equation}
The pseudo-tensor $t_{\mu\nu}$ to the second order in $h$ is
\begin{equation}
t_{\mu\nu}=\frac{1}{8\pi G}\left(-\frac{1}{2}h_{\mu\nu}R^{\left(1\right)}+\frac{1}{2}\eta_{\mu\nu}\eta^{\rho\sigma}R^{\left(1\right)}_{\phantom{\left(1\right)}\rho\sigma}+R^{\left(2\right)}_{\phantom{\left(2\right)}\mu\nu}-\frac{1}{2}\eta_{\mu\nu}\eta^{\rho\sigma}R^{\left(2\right)}_{\phantom{\left(2\right)}\rho\sigma}\right)+\mathcal{O}\left(h^{3}\right)\ ,
\end{equation}
where $R_{\mu\nu}$ to first order in $h$ is
\begin{equation}
R^{\left(1\right)}_{\phantom{\left(1\right)}\mu\nu}=\frac{1}{2}\left(\frac{\partial^{2}h^{\lambda}_{\phantom{\lambda}\lambda}}{\partial x^{\mu}\partial x^{\nu}}-\frac{\partial^{2}h^{\lambda}_{\phantom{\lambda}\mu}}{\partial x^{\lambda}\partial x^{\nu}}-\frac{\partial^{2}h^{\lambda}_{\phantom{\lambda}\nu}}{\partial x^{\lambda}\partial x^{\mu}}+\frac{\partial^{2}h_{\mu\nu}}{\partial x^{\lambda}\partial x_{\lambda}}\right)\ ,
\end{equation}
while to second order $h$ becomes
\begin{equation}
\begin{split}
R^{\left(2\right)}_{\phantom{\left(2\right)}\mu\nu}=&-\frac{1}{2}h^{\lambda\rho}\left(\frac{\partial^{2}h_{\lambda\rho}}{\partial x^{\nu}\partial x^{\mu}}-\frac{\partial^{2}h_{\mu\rho}}{\partial x^{\nu}\partial x^{\lambda}}-\frac{\partial^{2}h_{\lambda\nu}}{\partial x^{\rho}\partial x^{\mu}}+\frac{\partial^{2}h_{\mu\nu}}{\partial x^{\rho}\partial x^{\lambda}}\right)\\
&+\frac{1}{4}\left(2\frac{\partial h^{\rho}_{\phantom{\rho}\sigma}}{\partial x^{\rho}}-\frac{\partial h^{\rho}_{\phantom{\rho}\rho}}{\partial x^{\sigma}}\right)\left(\frac{\partial h^{\sigma}_{\phantom{\sigma}\mu}}{\partial x^{\nu}}+\frac{\partial h^{\sigma}_{\phantom{\sigma}\nu}}{\partial x^{\mu}}-\frac{\partial h_{\mu\nu}}{\partial x_{\sigma}}\right)\\
&-\frac{1}{4}\left(\frac{\partial h_{\sigma\nu}}{\partial x^{\lambda}}+\frac{\partial h_{\sigma\lambda}}{\partial x^{\nu}}-\frac{\partial h_{\lambda\nu}}{\partial x^{\sigma}}\right)\left(\frac{\partial h^{\sigma}_{\phantom{\sigma}\mu}}{\partial x_{\lambda}}+\frac{\partial h^{\sigma\lambda}}{\partial x^{\mu}}-\frac{\partial h^{\lambda}_{\phantom{\lambda}\mu}}{\partial x_{\sigma}}\right)
\end{split}\ .
\end{equation}
§ ENERGY–MOMENTUM COMPLEX IN CURVATURE BASED GRAVITY
§.§ The gravitational energy–momentum ”tensor” of $n^{th}$ order Lagrangian
Let us examine the energy–momentum complex for a fourth order gravitational Lagrangian, that is, which depends up to fourth derivatives of the metric tensor $g_{\mu\nu}$ as $L=L(g_{\mu\nu}, g_{\mu\nu,\rho}, g_{\mu\nu,\rho\lambda},g_{\mu\nu,\rho\lambda\xi}, g_{\mu\nu,\rho\lambda\xi\sigma})$, whose field equations, in general, are of eighth order in metric formalism (see also [33, 34]). In this manner we include all possible curvature invariants, not only $\Box$ operators, into the gravitational action. Then, we will generalize the approach to a gravitational Lagrangian of $n$-th order, i.e., which depends up to $n^{th}$ derivatives of metric tensor. We will derive the energy–momentum tensor using the Noether's theorem, imposing that gravitational action is invariant under global translations [8]. In this review the metric signature of $g_{\mu\nu}$ adopted is $(+\ \ , -\ \ , -\ \ , -)$, while Ricci tensor is defined as
$R_{\mu\nu}=R_{\ \ \mu\rho\nu}^{\rho}$ and Riemann tensor as $R_{\ \ \beta\mu\nu}^{\alpha}=\Gamma_{\beta\nu,\mu}^{\alpha}+\ldots$.
Let us vary the gravitational action with respect to metric $g_{\mu\nu}$ and coordinates $x^{\mu}$ [18, 35, 36]
\begin{equation}\label{1}
I=\int_{\Omega}d^{4}x L \rightarrow \tilde{\delta} I=\int_{\Omega^{\prime}} d^{4}x^{\prime} L^{\prime}-\int_{\Omega}d^{4}x L=\int_{\Omega}d^{4}x \left[{\delta}L+\partial_{\mu}\left(L\delta x^{\mu}\right)\right]\ ,
\end{equation}
where $\tilde{\delta}$ stands for the local variation while $\delta$ means the total variation, that is, keeping the value of coordinate $x$ fixed. By infinitesimal transformations as
\begin{equation}\label{2}
x^{\prime\mu}=x^{\mu}+\epsilon^{\mu}\left(x\right)\ ,
\end{equation}
the total variation of the metric tensor reads
\begin{equation}\label{3}
\delta g_{\mu\nu}=g^{\prime}_{\mu\nu}\left(x\right)-g_{\mu\nu}\left(x\right)=-\epsilon^{\alpha}\partial_{\alpha}g_{\mu\nu}-g_{\mu\alpha}\partial_{\nu}\epsilon^{\alpha}-g_{\nu\alpha}\partial_{\mu}\epsilon^{\alpha}\ .
\end{equation}
Under global transformation, $\partial_{\lambda}\epsilon^{\mu}=0$, the functional variation of the metric becomes $\delta g_{\mu\nu}=-\epsilon^{\alpha}\partial_{\alpha}g_{\mu\nu}$. If we also require that the action to be invariant under this transformation, that is, $\tilde{\delta I} =0$, from arbitrariness of domain of integration $\Omega$, we have
\begin{equation}\label{4}
\begin{split}
0=\delta L +\partial_{\mu}\left(L\delta x^{\mu}\right)=\biggl(\frac{\partial L}{\partial g_{\mu\nu}}-\partial_{\rho}\frac{\partial L}{\partial g_{\mu\nu,\rho}}+\partial_{\rho}\partial_{\lambda}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda}}-\partial_{\rho}\partial_{\lambda}\partial_{\xi}
\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi}}\\+\partial_{\rho}\partial_{\lambda}\partial_{\xi}\partial_{\sigma}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi\sigma}}\biggr)\delta g_{\mu\nu}+\partial_{\eta}\left(2\chi\sqrt{-g}\tau_{\alpha}^{\eta}\right)\epsilon^{\alpha}\ ,
\end{split}
\end{equation}
where the explicit expression of gravitational energy–momentum tensor, that we will see being a pseudo-tensor or affine tensor, is
\begin{multline}\label{7}
\tau_{\alpha}^{\eta}=\frac{1}{2\chi\sqrt{-g}}\biggl[\left(\frac{\partial L}{\partial g_{\mu\nu,\eta}}-\partial_{\lambda}\frac{\partial L}{\partial g_{\mu\nu,\eta\lambda}}+\partial_{\lambda}\partial_{\xi}\frac{\partial L}{\partial g_{\mu\nu,\eta\lambda\xi}}-\partial_{\lambda}\partial_{\xi}\partial_{\sigma}\frac{\partial L}{\partial g_{\mu\nu,\eta\lambda\xi\sigma}}\right)g_{\mu\nu,\alpha}\\
+\left(\frac{\partial L}{\partial g_{\mu\nu,\rho\eta}}-\partial_{\xi}\frac{\partial L}{\partial g_{\mu\nu,\rho\eta\xi}}+\partial_{\xi}\partial_{\sigma}\frac{\partial L}{\partial g_{\mu\nu,\rho\eta\xi\sigma}}\right)g_{\mu\nu,\alpha\rho}
+\left(\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\eta}}-\partial_{\sigma}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\eta\sigma}}\right)g_{\mu\nu,\rho\lambda\alpha}\\
+\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\eta\sigma}}g_{\mu\nu,\rho\lambda\xi\alpha}-\delta^{\eta}_{\alpha}L\biggr]\ .
\end{multline}
If the metric tensor $g_{\mu\nu}$ satisfies the Euler–Lagrange equations for our gravitational Lagrangian
\begin{equation}\label{5}
\frac{\delta L}{\delta g_{\mu\nu}}=\frac{\partial L}{\partial g_{\mu\nu}}-\partial_{\rho}\frac{\partial L}{\partial g_{\mu\nu,\rho}}+\partial_{\rho}\partial_{\lambda}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda}}-\partial_{\rho}\partial_{\lambda}\partial_{\xi}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi}}+\partial_{\rho}\partial_{\lambda}\partial_{\xi}\partial_{\sigma}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi\sigma}}=0\ ,
\end{equation}
for an arbitrary $\epsilon^{\alpha}$, we get a local continuity equation for our Noether current
\begin{equation}\label{6}
\partial_{\eta}\left(\sqrt{-g} \tau_{\alpha}^{\eta}\right)=0\ .
\end{equation}
In a more compact form, the gravitational energy–momentum tensor takes the following form
\begin{multline}\label{tensem4}
\tau_{\alpha}^{\eta}=\frac{1}{2\chi\sqrt{-g}}\biggl[\sum_{m=0}^{3}\left(-1\right)^{m}\left(
\frac{\partial L}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{m}}g_{\mu\nu,\alpha}
\\ +\sum_{j=0}^{2}\sum_{m=j+1}^{3}\left(-1\right)^{j}\left(
\frac{\partial L}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{j}}g_{\mu\nu,i_{j+1}\cdots i_{m}\alpha}-\delta_{\alpha}^{\eta}L\biggr]\ ,
\end{multline}
where we used the following notation
\begin{equation*}
\left(\right)_{,i_{0}}=\mathbb{I} ; \qquad \left(\right)_{,i_{0}\cdots i_{m}}=
\begin{cases}
\left(\right)_{,i_{1}}& \quad \text{if} \quad m=1\\
\left(\right)_{,i_{1}i_{2}}& \quad \text{if} \quad m=2\\
\left(\right)_{,i_{1}i_{2}i_{3}}& \quad \text{if} \quad m=3\\
\text{and so on}&
\end{cases}
;\qquad \left(\right)_{,i_{k}\.i_{k}}=\left(\right)_{,i_{k}}
\end{equation*}
Let us now generalize our approach considering a general Lagrangian density depending up to $n^{th}$ derivative of $g_{\mu\nu}$, that is, $L=L\left(g_{\mu\nu}, g_{\mu\nu,i_{1}}, g_{\mu\nu,i_{1}i_{2}},g_{\mu\nu,i_{1}i_{2}i_{3}},\cdots, g_{\mu\nu,i_{1}i_{2}i_{3}\cdots i_{n}}\right)$. Total variation of Lagrangian $L$ and its Euler–Lagrange equations yield
\begin{equation}\label{8}
\delta L=\sum_{m=0}^{n}\frac{\partial L}{\partial g_{\mu\nu,i_{0}\cdots i_{m}}}\delta g_{\mu\nu,i_{0}\cdots i_{m}}=\sum_{m=0}^{n}\frac{\partial L}{\partial g_{\mu\nu,i_{0}\cdots i_{m}}}\partial_{i_{0}\cdots i_{m}}\delta g_{\mu\nu}\ ,
\end{equation}
\begin{equation}\label{9}
\frac{\delta L}{\delta g_{\mu\nu}}=\sum_{m=0}^{n}\left(-1\right)^{m}\partial_{i_{0}\cdots i_{m}}\frac{\partial L}{\partial g_{\mu\nu,i_{0}\cdots i_{m}}}=0\ ,
\end{equation}
where $\delta/\delta g_{\mu\nu}$ is the functional derivative, while it is possible to exchange the variation $\delta$ with the derivatives $\delta g_{\mu\nu,i_{0}\cdots i_{m}}=\partial_{i_{0}\cdots i_{m}}\delta g_{\mu\nu}$, because we are varying keeping $x$ fixed. So, we can find a most general local continuity equation which allows us to define the energy–momentum pseudo-tensor (which is an affine tensor as it will be proved later) for the gravitational field of $2n^{th}$ order gravity
\begin{multline}\label{tensemN}
\tau_{\alpha}^{\eta}=\frac{1}{2\chi\sqrt{-g}}\biggl[\sum_{m=0}^{n-1}\left(-1\right)^{m}\left(
\frac{\partial L}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{m}}g_{\mu\nu,\alpha}\\+\Theta_{\left[2,+\infty\right[}\left(n\right)\sum_{j=0}^{n-2}\sum_{m=j+1}^{n-1}\left(-1\right)^{j}\left(
\frac{\partial L}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{j}}g_{\mu\nu,i_{j+1}\cdots i_{m}\alpha}-\delta_{\alpha}^{\eta}L\biggr]\ ,
\end{multline}
where $\Theta$ is the Heaviside function
\begin{equation}
\Theta_{\left[a,+\infty\right[}\left(n\right)=
\begin{cases}
1& \quad \text{if} \quad n\in \left[a,+\infty\right[\\
0& \quad \text{otherwise}
\end{cases}\ .
\end{equation}
If fields and its derivatives vanish on boundary of our spatial region or rapidly decreasing to the spatial infinite on an infinity spacelike hypersurface, the gravitational energy–momentum tensor is totally conserved and satisfies a more general conservation law.
An alternative way to obtain the tensor (<ref>) is the procedure developed by Landau [8]. For example, we start by deriving the tensor (<ref>), because its generalization to higher order Lagrangians is the same. First of all, let us impose the stationary condition and vary the action with respect to the metric to find the field equations under the hypothesis that both $\delta g_{\mu\nu}$ and the variation of derivative $\delta \partial^{n}g$ vanish on the boundary of integration domain, canceling the surface integrals. Hence, the following occurs:
\begin{gather}\label{15}
\delta I=\delta\int_{\Omega}d^{4}x L\left(g_{\mu\nu}, g_{\mu\nu,\rho}, g_{\mu\nu,\rho\lambda},g_{\mu\nu,\rho\lambda\xi}, g_{\mu\nu,\rho\lambda\xi\sigma}\right) =0\ , \\
\updownarrow\nonumber\\
\frac{\partial L}{\partial g_{\mu\nu}}-\partial_{\rho}\frac{\partial L}{\partial g_{\mu\nu,\rho}}+\partial_{\rho}\partial_{\lambda}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda}}-\partial_{\rho}\partial_{\lambda}\partial_{\xi}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi}}+\partial_{\rho}\partial_{\lambda}\partial_{\xi}\partial_{\sigma}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi\sigma}}=0\ .
\end{gather}
Now, we perform the derivative of Lagrangian respect to metric tensor and then we put it into the field equations (<ref>). We obtain
\begin{multline}\label{16}
\frac{\partial L}{\partial x^{\alpha}} =\frac{\partial L}{\partial g_{\mu\nu}}\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}+\frac{\partial L}{\partial g_{\mu\nu,\rho}}\frac{\partial g_{\mu\nu,\rho}}{\partial x^{\alpha}}+\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda}}\frac{\partial g_{\mu\nu,\rho\lambda}}{\partial x^{\alpha}}\\
+\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi}}\frac{\partial g_{\mu\nu,\rho\lambda\xi}}{\partial x^{\alpha}}+\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi\sigma}}\frac{\partial g_{\mu\nu,\rho\lambda\xi\sigma}}{\partial x^{\alpha}}\\
=\partial_{\rho} \frac{\partial L}{\partial g_{\mu\nu,\rho}}g_{\mu\nu,\alpha}-\partial_{\rho}\partial_{\lambda}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda}}g_{\mu\nu,\alpha}+\partial_{\rho}\partial_{\lambda}\partial_{\xi}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi}}g_{\mu\nu,\alpha}-\partial_{\rho}\partial_{\lambda}\partial_{\xi}\partial_{\sigma}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi\sigma}}g_{\mu\nu,\alpha}\\
+\frac{\partial L}{\partial g_{\mu\nu,\rho}} g_{\mu\nu,\rho\alpha}+\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda}}g_{\mu\nu,\rho\lambda\alpha}+\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi}}g_{\mu\nu,\rho\lambda\xi\alpha}+\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi\sigma}} g_{\mu\nu,\rho\lambda\xi\sigma\alpha}\\
=\partial_{\rho}\left(\frac{\partial L}{\partial g_{\mu\nu,\rho}}g_{\mu\nu,\alpha}\right)-\partial_{\rho}\left(\partial_{\lambda}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda}}g_{\mu\nu,\alpha}\right)+\partial_{\lambda}\left(\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda}}g_{\mu\nu,\rho\alpha}\right)\\
+\partial_{\rho}\left(\partial_{\lambda}\partial_{\xi}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi}}g_{\mu\nu,\alpha}\right)
+\partial_{\lambda}\left(\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi}}g_{\mu\nu,\rho\xi\alpha}\right)\\
-\partial_{\xi}\left(\partial_{\lambda}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi}}g_{\mu\nu,\alpha\rho}\right)-\partial_{\rho}\left(\partial_{\lambda}\partial_{\xi}\partial_{\sigma}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi\sigma}}g_{\mu\nu,\alpha}\right)\\
+\partial_{\lambda}\left(\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi\sigma}}g_{\mu\nu,\rho\xi\sigma\alpha}\right)-\partial_{\xi}\left(\partial_{\lambda}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi\sigma}}g_{\mu\nu,\rho\sigma\alpha}\right)
\\+\partial_{\sigma}\left(\partial_{\xi}\partial_{\lambda}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi\sigma}}g_{\mu\nu,\rho\alpha}\right)\ .
\end{multline}
Grouping together terms and renaming dumb indices, we obtain
\begin{equation}\label{17}
\partial_{\eta}\left(\sqrt{-g}\tau^{\eta}_{\alpha}\right)=0\ ,
\end{equation}
that is, the pseudo-tensor is locally conserved, where $\tau^{\eta}_{\alpha}$ is the tensor defined in (<ref>).
The energy–momentum complex, instead, can be derived considering the material Lagrangian $L_{m}=2\chi\sqrt{-g}\mathcal{L}_{m}$ with stress–energy tensor given by
\begin{equation}\label{18}
T^{\eta\alpha}=\frac{2}{\sqrt{-g}}\frac{\delta \left(\sqrt{-g}\mathcal{L}_{m}\right)}{\delta g_{\eta\alpha}}\ .
\end{equation}
Thus, we use the field equations in presence of matter, namely
\begin{equation}\label{19}
P^{\eta\alpha}=\chi T^{\eta\alpha}\ ,
\end{equation}
\begin{equation}\label{20}
P^{\eta\alpha}=-\frac{1}{\sqrt{-g}}\frac{\delta L_{g}}{\delta g_{\eta\alpha}}\,\qquad \mbox{with the coupling} \quad \chi=\frac{8\pi G}{c^{4}}\ .
\end{equation}
By field equations (<ref>), we obtain
\begin{multline}\label{21}
\left(2\chi\sqrt{-g}\tau^{\eta}_{\alpha}\right)_{,\eta}=-\sqrt{-g}P^{\rho\sigma}g_{\rho\sigma,\alpha}=-\chi\sqrt{-g}T^{\rho\sigma}g_{\rho\sigma,\alpha}\\
=2\chi\sqrt{-g} T^{\eta}_{\alpha;\eta}-\left(2\chi\sqrt{-g} T^{\eta}_{\alpha}\right)_{,\eta}\ ,
\end{multline}
\begin{equation}\label{22}
\partial_{\eta}\left[\sqrt{-g}\left(\tau^{\eta}_{\alpha}+ T^{\eta}_{\alpha}\right)\right]=\sqrt{-g}T^{\eta}_{\alpha;\eta}\ ,
\end{equation}
\begin{multline}\label{23}
\delta L +\partial_{\mu}\left(L\delta x^{\mu}\right)=-P^{\mu\nu}\sqrt{-g}\delta g_{\mu\nu}+\partial_{\eta}\left(2\chi\sqrt{-g}\tau^{\eta}_{\alpha}\right)\epsilon^{\alpha}\\
=\left[\sqrt{-g}P^{\mu\nu}g_{\mu\nu,\alpha}+\partial_{\eta}\left(2\chi\sqrt{-g}\tau^{\eta}_{\alpha}\right)\right]\epsilon^{\alpha}=0\ ,
\end{multline}
and because from symmetry of tensor $T^{\eta}_{\alpha}$, one gets
\begin{equation}\label{24}
\sqrt{-g}T^{\eta}_{\alpha;\eta}=\left(\sqrt{-g}T^{\eta}_{\alpha}\right)_{,\eta}-\frac{1}{2}g_{\rho\sigma,\alpha}T^{\rho\sigma}\sqrt{-g}\ .
\end{equation}
The relation (<ref>) tells us that the conservation law of the energy–momentum complex, i.e., the sum of two stress–energy tensors due to matter plus gravitational fields, is related to the covariant derivative of the only matter part. From contracted Bianchi identities we get the total conservation law and conversely
\begin{equation}\label{eqcontgeneralizzata}
G^{\eta\alpha}_{;\eta}=0 \leftrightarrow P^{\eta\alpha}_{;\eta}=0 \leftrightarrow T^{\eta\alpha}_{;\eta}=0 \leftrightarrow \partial_{\eta}\left[\sqrt{-g}\left(\tau^{\eta}_{\alpha}+ T^{\eta}_{\alpha}\right)\right]=0\ ,
\end{equation}
where ${\displaystyle G^{\eta\alpha}=R^{\eta\alpha}-\frac{1}{2}g^{\eta\alpha}R}$ is the Einstein tensor and the locally conserved energy–momentum complex is given by
\begin{equation}
\mathcal{T}_{\alpha}^{\eta}=\sqrt{-g}\left(\tau^{\eta}_{\alpha}+ T^{\eta}_{\alpha}\right)\ .
\end{equation}
In a nutshell, the contracted Bianchi identities lead to the local conservation of energy–momentum complex or, viceversa, the local conservation of matter and gravitational fields involves the contracted Bianchi identities (see also [37] for a detailed discussion in modified gravity).
From the local continuity equation (<ref>), it is possible to derive some conserved quantities, Noether charges, such as the total 4-momentum of matter plus gravitational field. If we require that the metric tensor derivatives up to the $n^{th}$ order vanish on the 3-dimensional space-domain $\Sigma$, the surface integrals are zero over the boundary $\partial\Sigma$, that is
\begin{equation}\label{25}
\partial_{0}\int_{\Sigma}d^{3}x \sqrt{-g}\left(T^{\mu0}+\tau^{\mu0}\right)=-\int_{\partial\Sigma}d\sigma_{i} \sqrt{-g}\left(T^{\mu i}+\tau^{\mu i}\right)=0\ ,
\end{equation}
where $\Sigma$ is a slice of 4-dimensional manifold of spacetime at $t$ fixed and $\partial\Sigma$ its boundary. Such conditions are fulfilled by when we are in the presence of localized objects, where we can take a spatial domain that becomes flat to infinity, i.e, a asymptotically flat spacetime. So, the energy and total momentum conserved become Ref. [38]
\begin{equation}\label{26}
P^{\mu}=\int_{\Sigma}d^{3}x \sqrt{-g}\left(T^{\mu0}+\tau^{\mu0}\right)\ .
\end{equation}
These quantities are very useful in astrophysical applications [39].
§.§ Non-covariance of gravitational energy–momentum tensor
We will prove that the tensor $\tau^{\eta}_{\alpha}$ is not a covariant object but affine, that is, it is changes like a tensor under affine transformations [40], i.e., a pseudo-tensor. We will limit ourselves first to a particular case, $n=2$, where the tensor (<ref>) reads
\begin{equation}\label{27}
\tau^{\eta}_{\alpha}=\frac{1}{2\chi\sqrt{-g}}\left[\left(\frac{\partial L}{\partial g_{\mu\nu,\eta}}-\partial_{\lambda}\frac{\partial L}{\partial g_{\mu\nu,\eta\lambda}}\right)g_{\mu\nu,\alpha}+\frac{\partial L}{\partial g_{\mu\nu,\eta\xi}}g_{\mu\nu,\xi\alpha}-\delta^{\eta}_{\alpha} L\right]\ ,
\end{equation}
We will show that, while under a general diffeomorphism transformation $x^{\prime}=x^{\prime}\left(x\right)$, the tensor changes as
\begin{equation}\label{28}
\tau^{\prime\eta}_{\ \alpha}\left(x^{\prime}\right) \neq \text{J}^{\eta}_{\sigma}\text{J}^{-1\tau}_{\ \ \ \alpha}\tau^{\sigma}_{\tau}\left(x\right)\ ,
\end{equation}
with Jacobian matrix and determinant defined as
\begin{equation}\label{29}
\text{J}^{\eta}_{\sigma}=\frac{\partial x^{\prime\eta}}{\partial x^{\sigma}}\qquad \text{J}^{-1\tau}_{\ \ \ \alpha}=\frac{\partial x^{\tau}}{\partial x^{\prime\alpha}}\qquad \text{det}\left(\text{J}^{\alpha}_{\beta}\right)=\vert J \vert =\frac{1}{\text{J}^{-1}}\ ,
\end{equation}
under the following affine transformations
\begin{equation}\label{30}
x^{\prime\mu}=\Lambda^{\mu}_{\nu}x^{\nu}\qquad \text{J}^{\mu}_{\nu}=\Lambda^{\mu}_{\nu} \qquad \vert \Lambda \vert \neq 0\ ,
\end{equation}
the tensor is transformed as
\begin{equation}\label{31}
\tau^{\prime\eta}_{\ \alpha}\left(x^{\prime}\right)=\Lambda^{\eta}_{\sigma}\Lambda^{-1\tau}_{\ \ \ \alpha}\tau^{\sigma}_{\tau}\left(x\right)\ .
\end{equation}
Generally, following identities occur
\begin{equation*}
\begin{split}
\sqrt{-g^{\prime}}&=\sqrt{-g}\qquad\qquad \ \ \qquad\quad \, \,\text{where $g$ is a scalar density of weight $w=-2$ }\ ,\\
L^{\prime}&=\text{J}^{-1}L\qquad\qquad \ \qquad \qquad \text{where $L$ is a scalar density of weight $w=-1$ }\ ,\\
g^{\prime}_{\mu\nu,\alpha}\left(x^{\prime}\right)&=\text{J}^{-1 a}_{\ \ \ \mu}\text{J}^{-1 b}_{\ \ \ \nu}\text{J}^{-1 c}_{\ \ \ \alpha}g_{ab,c}\left(x\right)+\partial^{\prime}_{\alpha}\left[\text{J}^{-1 a}_{\ \ \ \mu}\text{J}^{-1 b}_{\ \ \ \nu}\right]g_{ab}\left(x\right)\ ,\\
\frac{\partial g_{\gamma\rho,\tau}}{\partial g^{\prime}_{\mu\nu,\eta}}&=\frac{1}{2}\left[\left(\delta_{a}^{\mu}\delta_{b}^{\nu}+\delta_{a}^{\nu}\delta_{b}^{\mu}\right)\delta_{c}^{\eta}\right]\text{J}_{\gamma}^{a}\text{J}_{\rho}^{b}\text{J}_{\tau}^{c}=\text{J}^{(\mu}_{\gamma}\text{J}^{\nu)}_{\rho}\text{J}^{\eta}_{\tau}\ ,\\
\frac{\partial L^{\prime}}{\partial g^{\prime}_{\mu\nu,\eta}}&=\text{J}^{-1}\text{J}^{(\mu}_{\gamma}\text{J}^{\nu)}_{\rho}\text{J}^{\eta}_{\tau}\frac{\partial L}{\partial g_{\gamma\rho,\tau}}=\text{J}^{-1}\text{J}^{\mu}_{\gamma}\text{J}^{\nu}_{\rho}\text{J}^{\eta}_{\tau}\frac{\partial L}{\partial g_{\gamma\rho,\tau}}\ \\
& \qquad\qquad\qquad\qquad\qquad\qquad \text{tensorial density (3,0) of weight $w=-1$}\ ,\\
g^{\prime}_{\mu\nu,\xi\alpha}\left(x^{\prime}\right)&=\text{J}^{-1 a}_{\ \ \ \mu}\text{J}^{-1 b}_{\ \ \ \nu}\text{J}^{-1 c}_{\ \ \ \alpha}\text{J}^{-1 d}_{\ \ \ \xi}g_{ab,cd}\left(x\right)+\partial^{\prime 2}_{\xi\alpha}\left[\text{J}^{-1 a}_{\ \ \ \mu}\text{J}^{-1 b}_{\ \ \ \nu}\right]g_{ab}\left(x\right)\\
&+\partial^{\prime}_{\alpha}\left[\text{J}^{-1 a}_{\ \ \ \mu}\text{J}^{-1 b}_{\ \ \ \nu}\right]\text{J}^{-1 d}_{\ \ \ \xi}g_{ab,d}\left(x\right)+\partial^{\prime}_{\xi}\left[\text{J}^{-1 a}_{\ \ \ \mu}\text{J}^{-1 b}_{\ \ \ \nu}\text{J}^{-1 c}_{\ \ \ \alpha}\right]g_{ab,c}\left(x\right)\ ,\\
\frac{\partial g_{\gamma\rho,\tau\epsilon}}{\partial g_{\mu\nu,\eta\xi}^{\prime}}&=\left(\delta_{a}^{(\mu}\delta_{b}^{\nu)}\delta_{c}^{(\eta}\delta_{d}^{\xi)}\right)\text{J}_{\gamma}^{a}\text{J}_{\rho}^{b}\text{J}_{\tau}^{c}\text{J}_{\epsilon}^{d}=\text{J}_{\gamma}^{(\mu}\text{J}_{\rho}^{\nu)}\text{J}_{\tau}^{(\eta}\text{J}_{\epsilon}^{\xi)}\ ,\\
\frac{\partial L^{\prime}}{\partial g^{\prime}_{\mu\nu,\eta\xi}}&=\text{J}^{-1}\text{J}^{(\mu}_{\gamma}\text{J}^{\nu)}_{\rho}\text{J}^{(\eta}_{\tau}\text{J}^{\xi)}_{\epsilon}\frac{\partial L}{\partial g_{\gamma\rho,\tau\epsilon}}=\text{J}^{-1}\text{J}^{\mu}_{\gamma}\text{J}^{\nu}_{\rho}\text{J}^{\eta}_{\tau}\text{J}^{\xi}_{\epsilon}\frac{\partial L}{\partial g_{\gamma\rho,\tau\epsilon}}\\
& \qquad\qquad\qquad\qquad\qquad\qquad\text{tensorial density (4,0) of weight $w=-1$}\ ,\\
\partial^{\prime}_{\lambda}\frac{\partial L^{\prime}}{\partial g^{\prime}_{\mu\nu,\eta\lambda}}&=\text{J}^{-1}\text{J}^{\mu}_{\gamma}\text{J}^{\nu}_{\rho}\text{J}^{\eta}_{\tau}\text{J}^{\lambda}_{\epsilon}\text{J}^{-1 \sigma}_{\ \ \ \lambda}\partial_{\sigma}\frac{\partial L}{\partial g_{\gamma\rho,\tau\epsilon}}+\partial^{\prime}_{\lambda}\left[\text{J}^{-1}\text{J}^{\mu}_{\gamma}\text{J}^{\nu}_{\rho}\text{J}^{\eta}_{\tau}\text{J}^{\lambda}_{\epsilon}\right]\frac{\partial L}{\partial g_{\gamma\rho,\tau\epsilon}}\ ,
\end{split}
\end{equation*}
and by symmetry of $B_{\alpha\beta}$, i.e., $B_{\alpha\beta}=B_{\beta\alpha}$ follows that $A^{(\alpha\beta)}B_{\alpha\beta}=A^{\alpha\beta}B_{\alpha\beta}$.
Then we have
\begin{equation*}
\frac{\partial L^{\prime}}{\partial g^{\prime}_{\mu\nu,\eta}}g^{\prime}_{\mu\nu,\alpha}=\text{J}^{-1}\text{J}^{\eta}_{\tau}\text{J}^{-1\pi}_{\ \ \ \alpha}\frac{\partial L}{\partial g_{\gamma\rho,\tau}}g_{\gamma\rho,\pi}\left(x\right)+\frac{\partial}{\partial x^{\prime\alpha}}\left[\text{J}^{-1a}_{\ \ \ \mu}\text{J}^{-1b}_{\ \ \ \nu}\right]g_{ab}\left(x\right)\text{J}^{-1}\text{J}^{\mu}_{\gamma}\text{J}^{\nu}_{\rho}\text{J}^{\eta}_{\tau}\frac{\partial L}{\partial g_{\gamma\rho,\tau}}\ ,\\
\end{equation*}
\begin{multline*}
\partial^{\prime}_{\lambda}\frac{\partial L^{\prime}}{\partial g^{\prime}_{\mu\nu,\eta\lambda}}g^{\prime}_{\mu\nu,\alpha}\left(x^{\prime}\right)=\text{J}^{-1}\text{J}^{\eta}_{\tau}\text{J}^{-1c}_{\ \ \ \alpha}\partial_{\sigma}\frac{\partial L}{\partial g_{ab,\tau\sigma}}g_{ab,c}+\partial^{\prime}_{\lambda}\left[\text{J}^{-1}\text{J}^{\mu}_{\gamma}\text{J}^{\nu}_{\rho}\text{J}^{\eta}_{\tau}\text{J}^{\lambda}_{\epsilon}\right]
\partial^{\prime}_{\alpha}\left[\text{J}^{-1a}_{\ \ \ \mu}\text{J}^{-1b}_{\ \ \ \nu}\right]g_{ab}\left(x\right)\frac{\partial L}{\partial g_{\gamma\rho,\tau\epsilon}}\\
+\text{J}^{-1}\text{J}^{\mu}_{\gamma}\text{J}^{\nu}_{\rho}\text{J}^{\eta}_{\tau}\partial_{\sigma}\frac{\partial L}{\partial g_{\gamma\rho,\tau\sigma}}\partial^{\prime}_{\alpha}\left[\text{J}^{-1a}_{\ \ \ \mu}\text{J}^{-1b}_{\ \ \ \nu}\right]g_{ab}+\partial^{\prime}_{\lambda}\left[\text{J}^{-1}\text{J}^{\mu}_{\gamma}\text{J}^{\nu}_{\rho}\text{J}^{\eta}_{\tau}\text{J}^{\lambda}_{\epsilon}\right]\text{J}^{-1a}_{\ \ \ \mu}\text{J}^{-1b}_{\ \ \ \nu}\text{J}^{-1c}_{\ \ \ \alpha}\frac{\partial L}{\partial g_{\gamma\rho,\tau\epsilon}}g_{ab,c} \ ,
\end{multline*}
\begin{multline*}
\frac{\partial L^{\prime}}{\partial g^{\prime}_{\mu\nu,\eta\xi}}g^{\prime}_{\mu\nu,\xi\alpha}\left(x^{\prime}\right)=\text{J}^{-1}\text{J}^{\eta}_{\tau}\text{J}^{-1\omega}_{\ \ \ \alpha}\frac{\partial L}{\partial g_{\gamma\rho,\tau\epsilon}}g_{\gamma\rho,\omega\epsilon}\left(x\right)+\text{J}^{-1}\partial^{\prime 2}_{\xi\alpha}\left[\text{J}^{-1a}_{\ \ \ \mu}\text{J}^{-1b}_{\ \ \ \nu}\right]g_{ab}\left(x\right)\text{J}^{\mu}_{\gamma}\text{J}^{\nu}_{\rho}\text{J}^{\eta}_{\tau}\text{J}^{\xi}_{\epsilon}\frac{\partial L}{\partial g_{\gamma\rho,\tau\epsilon}}\\
+\text{J}^{-1}\partial^{\prime}_{\alpha}\left[\text{J}^{-1a}_{\ \ \ \mu}\text{J}^{-1b}_{\ \ \ \nu}\right]g_{ab,d}\left(x\right)\text{J}^{\mu}_{\gamma}\text{J}^{\nu}_{\rho}\text{J}^{\eta}_{\tau}\frac{\partial L}{\partial g_{\gamma\rho,\tau d}}+\text{J}^{-1}\text{J}^{\mu}_{\gamma}\text{J}^{\nu}_{\rho}\text{J}^{\eta}_{\tau}\text{J}^{\xi}_{\epsilon}\partial^{\prime}_{\xi}\left[\text{J}^{-1a}_{\ \ \ \mu}\text{J}^{-1b}_{\ \ \ \nu}\text{J}^{-1c}_{\ \ \ \alpha}\right]g_{ab,c}\left(x\right)\frac{\partial L}{\partial g_{\gamma\rho,\tau\epsilon}},
\end{multline*}
Finally, taking into account previous relations we get
\begin{equation}\label{affinitatensore}
\tau^{\prime\eta}_{\ \alpha}\left(x^{\prime}\right)=\text{J}^{\eta}_{\sigma}\text{J}^{-1\tau}_{\ \ \ \alpha}\tau^{\sigma}_{\tau}\left(x\right)+\left\{\text{terms containing }\frac{\partial^{2} x}{\partial x^{\prime 2}},\frac{\partial^{3} x}{\partial x^{\prime 3}} \right\}\ .
\end{equation}
Extra terms that include derivatives of order equal to or greater than two vanish for each non-singular affine transformation but not for generic diffeomorphisms. This proves that gravitational stress–energy tensor is non-covariant but affine, that is, it is invariant under affine transformations due to non-covariance of the derivatives of the metric tensor $g_{\mu\nu}$, that make it at least affine.
Generalizing to $n$-th order Lagrangian, metric tensor derivatives change as
\begin{multline}\label{32}
g^{\prime}_{\mu\nu,i_{1}\cdots i_{m}\alpha}\left(x^{\prime}\right)=\text{J}^{-1\alpha}_{\ \ \ \mu}\text{J}^{-1\beta}_{\ \ \ \nu}\text{J}^{-1 j_{1}}_{\ \ \ i_{1}}\cdots\text{J}^{-1 j_{m}}_{\ \ \ i_{m}}\text{J}^{-1\tau}_{\ \ \ \alpha}g_{\alpha\beta,j_{1}\cdots j_{m}\tau}\left(x\right)\\
+\left\{\text{containing terms}\;\frac{\partial^{2} x}{\partial x^{\prime 2}},\cdots,\frac{\partial^{m+2}x}{\partial x^{\prime m+2}} \right\},
\end{multline}
and derivatives of Lagrangian as
\begin{equation*}
\frac{\partial L^{\prime}}{\partial g^{\prime}_{\mu\nu,\eta i_{0}\cdots i_{m}}}=\text{J}^{-1}\text{J}^{\mu}_{\gamma}\text{J}^{\nu}_{\rho}\text{J}^{\eta}_{\tau}\text{J}^{i_{1}}_{j_{1}}\cdots \text{J}^{i_{m}}_{j_{m}} \frac{\partial L}{\partial g_{\gamma\rho,\tau j_{1}\cdots j_{m}}}\quad\text{tensorial density (m+3,0) of weight $w=-1$}\ ,
\end{equation*}
so that the non covariance of tensor $\tau^{\eta}_{\alpha}$ appears. Otherwise, we obtain for affine transformations
\begin{equation*}
\frac{\partial^{2}x}{\partial x^{\prime 2}}=\cdots=\frac{\partial^{m+2}x}{\partial x^{\prime m+2}}=0\ ,
\end{equation*}
\begin{equation*}
\tau^{\prime\eta}_{\ \alpha}\left(x^{\prime}\right)=\Lambda^{\eta}_{\sigma}\Lambda^{-1\tau}_{\ \ \ \alpha}\tau^{\sigma}_{\tau}\left(x\right)\ ,
\end{equation*}
that is, the energy–momentum tensor of gravitational field is a pseudo-tensor. This result generalize to Extended Theories of Gravity the result in [8]. The affine character of the stress–energy tensor $\tau ^ {\eta} _ {\alpha}$ is a exhibition of the non localizability of gravitational energy density. Specifically, the gravitational energy in a finite-dimensional space, at a given time, depends on the choice of coordinate system [41, 38].
It is worth highlighting that the existence of particular Lagrangians for which extra terms in Eq. (<ref>) vanish cannot be excluded a priori. This is because terms depending on derivatives in the bracket (<ref>) such as ${\frac{\partial^{2} x}{\partial x^{\prime 2}},\cdots,\frac{\partial^{m+2}x}{\partial x^{\prime m+2}}}$, can cancel each other out. Consequently, the energy–momentum pseudo-tensor ${\tau_{\alpha}^{\eta}}$ become a covariant tensor. However, due to the structure of (<ref>), in general, ${\tau_{\alpha}^{\eta}}$ is a pseudo-tensor.
§.§ The gravitational energy–momentum pseudo-tensor of $f\left(R\right)$ gravity
Let us examine the gravitational stress–energy tensor in the $f\left(R\right)$ gravity. Now, the gravitational action is given by
\begin{equation}\label{33}
\mathcal{S}_{f\left(R\right)}=\frac{1}{2\kappa^{2}}\int_{\Omega}d^{4}x\sqrt{-g}f\left(R\right)\,
\end{equation}
with the coupling $\kappa^{2}=8\pi G/c^{4}$. We perform the variation $\tilde{\delta}$ with respect to the metric $g^{\mu\nu}$ and coordinates $x^{\mu}$ for a generic infinitesimal transformation
\begin{equation}\label{34}
x^{\prime\mu}=x^{\mu}+\delta x^{\mu}\,\qquad g^{\prime\mu\nu}\left(x^{\prime}\right)= g^{\mu\nu}\left(x\right)+\tilde{\delta}g^{\mu\nu}\,\qquad g^{\prime\mu\nu}\left(x\right)= g^{\mu\nu}\left(x\right)+\delta g^{\mu\nu}\,
\end{equation}
\begin{equation}\label{varloc}
\tilde{\delta}\mathcal{S}_{f\left(R\right)}=\frac{1}{2\kappa^{2}}\int_{\Omega}d^{4}x\left[\delta\left(\sqrt{-g}f\left(R\right)\right)+\partial_{\mu}\left(\sqrt{-g}f\left(R\right) \delta x^{\mu}\right)\right]\,
\end{equation}
where $\delta$ is the global variation keeping $x$ fixed.
Thus, we get [18, 37, 42, 43, 44, 45]
\begin{multline}\label{varlocfR}
\tilde{\delta}\mathcal{S}_{f\left(R\right)}=\frac{1}{2\kappa^{2}}\int_{\Omega}d^{4}x\sqrt{-g}\Biggl[f'\left(R\right)R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}f\left(R\right)-\nabla_{\mu}\nabla_{\nu}f'\left(R\right)\\
+g_{\mu\nu}\Box f'\left(R\right)\Biggr]\delta g^{\mu\nu}+\int_{\Omega}d^{4}x\partial_{\alpha}\Biggl\{\frac{\sqrt{-g}}{2\kappa^{2}}\biggl[\partial_{\beta}f'\left(R\right)\left(g^{\eta\rho}g^{\alpha\beta}-g^{\alpha\eta}g^{\rho\beta}\right)\delta g_{\eta\rho}\\
+f'\left(R\right)\Bigl[\left(\stackrel{\circ}\Gamma{}^{\rho\eta\alpha}-\stackrel{\circ}\Gamma{}^{\eta\sigma}{}_{\sigma}g^{\alpha\rho}\right)\delta g_{\eta\rho}+\left(g^{\alpha\eta}g^{\tau\rho}-g^{\eta\rho}g^{\alpha\tau}\right)\delta g_{\eta\rho,\tau}\Bigr]+f\left(R\right)\delta_{\lambda}^{\alpha}\delta x^{\lambda}\biggr]\Biggr\}\,
\end{multline}
where $f'\left(R\right)=\partial f/\partial R$. By the condition of stationarity of the action at $x$ fixed, that is, $\delta\mathcal{S}_{f\left(R\right)}=0$, in a given domain $\Omega$ where the total variation of both metric and its first derivatives are zero on the boundary, that is, $\delta g_{\mu\nu}\vert_{\partial\Omega}=0$ and $\delta \left(\partial_{\alpha}g_{\mu\nu}\right)\vert_{\partial\Omega}=0$, the field equations in vacuum become
\begin{multline}\label{ECFRV}
P^{f\left(R\right)}_{\mu\nu}=\frac{2\kappa^{2}}{\sqrt{-g}}\frac{\delta L_{f\left(R\right)}}{\delta g^{\mu\nu}}\\
=f'\left(R\right)R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}f\left(R\right)-\nabla_{\mu}\nabla_{\nu}f'\left(R\right)+g_{\mu\nu}\Box f'\left(R\right)=0\,
\end{multline}
where $2\kappa^{2}L_{f\left(R\right)}=\sqrt{-g}f\left(R\right)$. For an infinitesimal transformation such as a rigid translation, one gets
\begin{equation}\label{TrRig}
x^{\prime\mu}=x^{\mu}+\epsilon^{\mu}\Rightarrow \delta g_{\mu\nu}=-\epsilon^{\lambda}g_{\mu\nu,\lambda}\,
\end{equation}
because $\partial_{\mu} \epsilon^{\mu}=0$. When the local variation of the action vanishes and the field $g_{\mu\nu}$ fulfils the field equations, we obtain the continuity equation
\begin{equation}\label{35}
\tilde{\delta}\mathcal{S}_{f\left(R\right)}=0 \Rightarrow \partial_{\sigma}\left(\sqrt{-g}\tau^{\sigma}_{\phantom{\sigma}{\lambda | f\left(R\right)}}\right)=0\,
\end{equation}
where the gravitational energy–momentum pseudo-tensor of $f\left(R\right)$ gravity is defined as
\begin{multline}\label{36}
2\kappa^{2}\tau^{\sigma}{}_{\lambda | f\left(R\right)}=2\partial_{\beta}f'\left(R\right)g^{\eta[\rho}g^{\sigma]\beta}g_{\eta\rho,\lambda}\\+f'\left(R\right)\Bigl[\bigl(\stackrel{\circ}{\Gamma}{}^{\rho\eta\sigma}
-\stackrel{\circ}{\Gamma}{}^{\eta\alpha}_{\phantom{\eta\alpha}{\alpha}}g^{\sigma\rho}\bigr)g_{\eta\rho,\lambda}+2g^{\sigma[\eta}g^{\tau]\rho}g_{\eta\rho,\tau\lambda}\Bigr]-f\left(R\right)\delta_{\lambda}^{\sigma}\ ,
\end{multline}
with $\stackrel{\circ}{\Gamma}{}^{\rho\eta\sigma}=g^{\eta\epsilon}g^{\sigma\varphi}\stackrel{\circ}{\Gamma}{}^{\rho}_{\phantom{\eta}{\epsilon\varphi}}$, and $\stackrel{\circ}{\Gamma}{}^{\eta\alpha}_{\phantom{\eta\alpha}{\alpha}}=g^{\alpha\epsilon}\stackrel{\circ}{\Gamma}{}^{\eta}_{\phantom{\eta}{\epsilon\alpha}}$.
Now to derive an equation of continuity for energy–momentum complex, we must also include matter fields, as in matter action
\begin{equation}\label{37}
\mathcal{S}_{m}=\int_{\Omega}d^{4}x L_{m}\,
\end{equation}
where $L_{m}$ depends, at most, on first derivatives of metric $g_{\mu\nu}$. Varying the matter action (<ref>), it gets
\begin{equation}\label{38}
\delta\mathcal{S}_{m}=\int_{\Omega}d^{4}x \frac{\delta L_{m}}{\delta g^{\mu\nu}}\delta g^{\mu\nu}=\int_{\Omega}d^{4}x \left(\frac{\sqrt{-g}}{2}\right)T^{\left(m\right)}_{\mu\nu}\delta g^{\mu\nu}\,
\end{equation}
where the energy–momentum tensor of matter fields $T^{\left(m\right)}_{\mu\nu}$ is defined as
\begin{equation}\label{39}
T^{\left(m\right)}_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta L_{m}}{\delta g^{\mu\nu}}\,.
\end{equation}
So minimizing the total action $\mathcal{S_{T}}=\mathcal{S}_{f\left(R\right)}+\mathcal{S}_{m}$ and imposing suitable boundary conditions, field equations in presence of matter take the following form
\begin{equation}\label{ECFRM}
\end{equation}
According to contracted Bianchi identities and following formula
\begin{equation}\label{trinabla}
\nabla^{\nu}\nabla_{\mu}\nabla_{\nu} f\left(R\right)=R^{\alpha}_{\phantom{\alpha}\mu}\nabla_{\alpha}f\left(R\right)+\nabla_{\mu}\Box f\left(R\right)
\end{equation}
we derive equivalences
\begin{equation}\label{IBFR}
\nabla^{\nu}G_{\mu\nu}=0\leftrightarrow\nabla^{\nu} \left({}^{f\left(R\right)}P_{\mu\nu}\right)=0\leftrightarrow\nabla^{\nu}T^{\left(m\right)}_{\mu\nu}=0\,
\end{equation}
The variation (<ref>) of gravitational action, the rigid translation (<ref>) and the matter field equations (<ref>) give
\begin{align}\label{41}
\delta L_{f\left(R\right)}+\partial_{\sigma}\left(L_{f\left(R\right)}\delta x^{\sigma}\right)&=\frac{\sqrt{-g}}{2\kappa^{2}}P_{{f\left(R\right)}}^{\mu\nu}\delta g_{\mu\nu}-\partial_{\sigma}\left(\sqrt{-g}\tau^{\sigma}_{\phantom{\sigma}{\lambda}}\right)\epsilon^{\lambda}\nonumber\\&=\left[-\frac{1}{2}\sqrt{-g}T_{\left(m\right)}^{\mu\nu}g_{\mu\nu,\lambda}-\partial_{\sigma}\left(\sqrt{-g}\tau^{\sigma}_{\phantom{\sigma}{\lambda}}\right)\right]\epsilon^{\lambda}\,.
\end{align}
Taking into account Eq. (<ref>), the expression Eq. (<ref>) yields
\begin{equation}\label{42}
\delta L_{f\left(R\right)}+\partial_{\sigma}\left(L_{f\left(R\right)}\delta x^{\sigma}\right)=\left[-\partial_{\sigma}\left(\sqrt{-g}T^{\sigma}_{\phantom{\sigma}{\lambda}}\right)+\sqrt{-g}T^{\sigma}_{\phantom{\sigma}{\lambda;\sigma}}-\partial_{\sigma}\left(\sqrt{-g}\tau^{\sigma}_{\phantom{\sigma}{\lambda}}\right)\right]\epsilon^{\lambda}\,.
\end{equation}
Imposing the local variation to zero, under rigid translations, we have
\begin{equation}\label{43}
\delta L_{f\left(R\right)}+\partial_{\sigma}\left(L_{f\left(R\right)}\delta x^{\sigma}\right)=0 \rightarrow \partial_{\sigma}\left[\sqrt{-g}\left(\tau^{\sigma}_{\ \lambda}+T^{\sigma}{}_{\lambda}\right)\right]=\sqrt{-g}\nabla_{\sigma}T^{\sigma}{}_{\lambda}\,.
\end{equation}
From the contracted Bianchi identities (<ref>), we derive local conservation law for the energy–momentum complex $\mathcal{T}^{\sigma}_{\phantom{\sigma}\lambda}$ in $f(R)$ gravity
\begin{equation}\label{44}
\partial_{\sigma}\left[\sqrt{-g}\left(\tau^{\sigma}_{\phantom{\sigma}{\lambda | f\left(R\right)}}+T^{\sigma}_{\phantom{\sigma}{\lambda}}\right)\right]=0\,.
\end{equation}
\begin{equation}\label{45}
\mathcal{T}^{\sigma}_{\phantom{\sigma}\lambda}=\sqrt{-g}\left(\tau^{\sigma}_{\phantom{\sigma}{\lambda | f\left(R\right)}}+T^{\sigma}_{\phantom{\sigma}{\lambda}}\right)\ .
\end{equation}
§.§ The gravitational energy–momentum pseudo-tensor of higher order gravity
Let us now address theories of gravity of order higher than fourth, where terms containing $\Box$ operators occur in the action up to $n$ times. In supergravity and, more broadly, in gauge theories concerning with gravity [46, 47, 48], these theories are not only effective field theories, but also fundamental theories. Actually, there is at least a subclass of local higher derivative theories, the so called Lee–Wick theories, that are unitary and super-renormalizable or finite at quantum level as demonstrated in [49, 50]. Then, we consider the linear and quadratic part of the Ricci scalar $R$, the first $\overline{R}$ depends only on first derivative of metric tensor $g_{\mu\nu}$ and the second $R^{\star}$ depends linearly on second derivative of metric tensor, as follows [8, 40, 32]
\begin{equation}\label{46}
R=R^{\star}+\overline{R}\ ,
\end{equation}
\begin{equation}\label{47}
R^{\star}=g^{\mu\nu}\left(\Gamma^{\rho}_{\mu\nu,\rho}-\Gamma^{\rho}_{\mu\rho,\nu}\right)\ ,
\end{equation}
\begin{equation}\label{48}
\overline{R}=g^{\mu\nu}\left(\Gamma_{\mu\nu}^{\sigma}\Gamma_{\sigma\rho}^{\rho}-\Gamma_{\mu\sigma}^{\rho}\Gamma_{\nu\rho}^{\sigma}\right)\ .
\end{equation}
Hence, we want to derive the energy–momentum pseudo-tensor $\tau_{\alpha}^{\eta}$ for a gravitational Lagrangian given by
\begin{equation}
\label{higher}
L_{g}=(\overline{R}+a_{0}R^{2}+\sum_{k=1}^{p} a_{k}R\Box^{k}R)\sqrt{-g}\ ,
\end{equation}
that has been first considered in [51].
Therefore, for the purpose of derive the pseudo-tensor $\tau^{\eta}_{\alpha}$, we have first to calculate derivatives present into the Eq. (<ref>), namely
\begin{align}
\frac{\partial L}{\partial g_{\mu\nu,\eta}}&=\sqrt{-g}\left[\frac{\partial\overline{R}}{\partial g_{\mu\nu,\eta}}+\left(2a_{0}R+\sum_{k=1}^{p}a_{k}\Box^{k}R\right)\frac{\partial R}{\partial g_{\mu\nu,\eta}}+\sum_{k=1}^{p}a_{k}R\frac{\partial\Box^{k}R}{\partial g_{\mu\nu,\eta}}\right]\ ,\\
-\partial_{\lambda}\left(\frac{\partial L}{\partial g_{\mu\nu,\eta\lambda}}\right)&=-\partial_{\lambda}\left(\sqrt{-g}\left[\left(2a_{0}R+\sum_{k=1}^{p}a_{k}\Box^{k}R\right)\frac{\partial R}{\partial g_{\mu\nu,\eta\lambda}}+\sum_{k=1}^{p}a_{k}R\frac{\partial\Box^{k}R}{\partial g_{\mu\nu,\eta\lambda}}\right]\right)\ ,
\end{align}
\begin{multline}\bar{49}
\sum_{m=2}^{n-1}\left(-1\right)^{m}\left(\frac{\partial L}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{m}}=\sum_{m=2}^{n-1}\sum_{k=1}^{p}\left(-1\right)^{m}\partial_{i_{0}\cdots i_{m}}\left[\sqrt{-g}a_{k}R\frac{\partial\Box^{k}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right]\\
=\sum_{k=1}^{p}\sum_{m=2}^{2p+3}\left(-1\right)^{m}\partial_{i_{0}\cdots i_{m}}\left[\sqrt{-g}a_{k}R\frac{\partial\Box^{k}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right]\\
=\sum_{k=1}^{p}\sum_{m=2}^{2k+1}\left(-1\right)^{m}\partial_{i_{0}\cdots i_{m}}\left[\sqrt{-g}a_{k}R\frac{\partial\Box^{k}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right]\ ,
\end{multline}
where $\lambda=i_{1}$, $n=2p+4$ and
\begin{equation}\label{50}
\frac{\partial\Box^{k}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}=0 \qquad \text{if}\quad m>2k+1\ .
\end{equation}
Then, after algebraic manipulations, one have
\begin{multline}\label{51}
\sum_{j=0}^{n-2}\sum_{m=j+1}^{n-1}\left(-1\right)^{j}\left(\frac{\partial L}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{j}}\\
=\sum_{h=1}^{p}\sum_{j=0}^{2p+2}\sum_{m=j+1}^{2p+3}\left(-1\right)^{j}\left(\sqrt{-g}a_{h}R\frac{\partial\Box^{h}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{j}}\ .
\end{multline}
Thereby, after observing that $j+1\leq m \leq 2h+1$ $\rightarrow$ $j\leq 2h$, we finally get
\begin{equation*}
\sum_{j=0}^{n-2}\sum_{m=j+1}^{n-1}\left(-1\right)^{j}\left(\frac{\partial L}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{j}}=\sum_{h=1}^{p}\sum_{j=0}^{2h}\sum_{m=j+1}^{2h+1}\left(-1\right)^{j}\left(\sqrt{-g}a_{h}R\frac{\partial\Box^{h}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{j}}\ .
\end{equation*}
By inserting these expressions into (<ref>), we obtain the gravitational energy–momentum pseudo-tensor for the Lagrangian (<ref>)
\begin{equation}\label{fulltensor}%MDPI: We removed the box foramt in the Equation (87), please confirm.
\begin{split}
\tau_{\alpha}^{\eta}=\tau_{\alpha\vert GR}^{\eta}+&\frac{1}{2\chi\sqrt{-g}}\Biggl\{\sqrt{-g}\left(2a_{0}R+\sum_{k=1}^{p}a_{k}\Box^{k}R\right)\left[\frac{\partial R}{\partial g_{\mu\nu,\eta}}g_{\mu\nu,\alpha}+\frac{\partial R}{\partial g_{\mu\nu,\eta\lambda}}g_{\mu\nu,\lambda\alpha}\right]\\
&-\partial_{\lambda}\left[\sqrt{-g}\left(2a_{0}R+\sum_{k=1}^{p}a_{k}\Box^{k}R\right)\frac{\partial R}{\partial g_{\mu\nu,\eta\lambda}}\right]g_{\mu\nu,\alpha}\\
&+\Theta_{\left[1,+\infty\right[}\left(p\right)\sum_{h=1}^{p}\Biggl\{\sum_{q=0}^{2h+1}\left(-1\right)^{q}\partial_{i_{0}\cdots i_{q}}\biggl[\sqrt{-g}a_{h}R\frac{\partial \Box^{h}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{q}}}\biggl]g_{\mu\nu,\alpha}\\
&+\sum_{j=0}^{2h}\sum_{m=j+1}^{2h+1}\left(-1\right)^{j}\partial_{i_{0}\cdots i_{j}}\biggl[\sqrt{-g}a_{h}R\frac{\partial \Box^{h}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\biggl]g_{\mu\nu,i_{j+1}\cdots i_{m}\alpha}\Biggr\}\\
&-\delta_{\alpha}^{\eta}\left(a_{0}R^{2}+\sum_{k=1}^{p} a_{k}R\Box^{k}R\right)\sqrt{-g}\Biggr\}
\end{split}
\end{equation}
where the notation
\partial_{i_{0}}=\mathbb{I}
$ is the identity operator and $\tau_{\alpha\vert GR}^{\eta}$ indicates the energy–momentum pseudo-tensor of general relativity [41] defined as
\begin{equation}\label{tensoreGR}
\tau_{\alpha\vert GR}^{\eta}=\frac{1}{2\chi}\left(\frac{\partial \overline{R}}{\partial g_{\mu\nu,\eta}}g_{\mu\nu,\alpha}-\delta^{\eta}_{\alpha} \overline{R}\right)\ .
\end{equation}
Given that only $\overline{R}$ contributes to the field equations we can replace scalar density $\sqrt{-g}R$ with $\sqrt{-g}\overline{R}$,which is not a scalar density. This makes the gravitational pseudo-tensor easier to manipulate and for a straightforward generalization of results see in Ref. [35].
An important extension of local Lagrangian (<ref>) to non-local Lagrangian is possible allowing $p \rightarrow \infty$.
Let $D^{p}$ be a linear differential operator defined by
\begin{equation}\label{52}
\end{equation}
If the weak or strong convergence is guaranteed under suitable assumptions for the coefficients $a_{k}$ (e.g. $\sum_{k=0}^{\infty}\vert a_{k} \vert<\infty$ ) and for the domain of the operator $ D^{p}$, we obtain the following non-local operator $F\left(\Box\right)$
\begin{equation}
\lim_{p\rightarrow\infty}\sum_{k=0}^{p}a_{k}\Box^{k}=F\left(\Box\right)\,
\end{equation}
and also our local action becomes non local, i.e.
\begin{equation}
\end{equation}
Accordingly integral operator acts as
\begin{equation}
\Phi\left(x\right)=\int_{\Omega}d^{4}yF\left(x-y\right)R\left(x\right)=F\left(\Box\right)R\left(x\right)\ .
\end{equation}
Let us carry out now the limit $n\rightarrow \infty$ for the energy–momentum pseudo-tensor of $n$-order Lagrangian (<ref>), we may obtain the non-local pseudo-tensor, that is
\begin{equation}\label{limitinftens}
\lim_{n\rightarrow\infty}\tau_{\alpha}^{\eta}\left(x\right)=\overline{\tau}_{\alpha}^{\eta}\left(x\right)\ .
\end{equation}
Whereas $\tau_{\alpha}^{\eta}\left(x\right)$ transforms as an affine tensor, we could show that also its limit for $n\rightarrow\infty$, i.e., $\overline{\tau}_{\alpha}^{\eta}\left(x\right)$, is an affine tensor. For an linear transformation
\begin{equation}
x^{\prime\mu}=\Lambda^{\mu}_{\nu}x^{\nu}\qquad \vert \Lambda \vert \neq 0
\end{equation}
the following affine pseudo-tensor changes as
\begin{equation}\label{trasfafftens}
\tau^{\eta}_{\alpha}\left(x\right)=\Lambda^{-1\eta}_{\ \ \ \sigma}\Lambda^{\tau}_{\alpha}\tau^{\prime\sigma}_{\tau}\left(x^{\prime}\right)\,.
\end{equation}
Substituting (<ref>) in (<ref>), we have
\begin{equation}
\overline{\tau}_{\alpha}^{\eta}\left(x\right)=\lim_{n\rightarrow\infty}\Lambda^{-1\eta}_{\ \ \ \sigma}\Lambda^{\tau}_{\alpha}\tau^{\prime\sigma}_{\tau}\left(x^{\prime}\right)=\Lambda^{-1\eta}_{\ \ \ \sigma}\Lambda^{\tau}_{\alpha}\lim_{n\rightarrow\infty}\tau^{\prime\sigma}_{\tau}\left(x^{\prime}\right)=\Lambda^{-1\eta}_{\ \ \ \sigma}\Lambda^{\tau}_{\alpha}\overline{\tau}^{\prime\sigma}_{\tau}\left(x^{\prime}\right)
\end{equation}
which implies that $\overline{\tau}^{\sigma}_{\tau}\left(x\right)$ transforms as an affine object also in the limit $n\rightarrow\infty$.
§.§ The weak-field limit of energy–momentum pseudo-tensor
The gravitational energy–momentum pseudo-tensor (<ref>) related to Lagrangian (<ref>) in weak field approximation can be performed perturbing weakly spacetime metric around the Minkowski metric $\eta_{\mu\nu}$ as
\begin{equation}\label{53}
g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\qquad\mbox{being}\quad |h_{\mu\nu}|\ll 1\ ,
\end{equation}
where $h=\eta^{\mu\nu}h_{\mu\nu}$ is the trace of perturbation.
Thus, we expand the energy–momentum pseudo-tensor to lower order in $h$, namely, retaining terms up to $h^2$. Let's see what becomes the weakly perturbed pseudo-tensor (<ref>) in harmonic coordinates where $g^{\mu\nu}\Gamma^{\sigma}_{\mu\nu}=0$. The quadratic part of the Ricci scalar $\overline{R}$ yields
\begin{equation}\label{54}
\overline{R}=-g^{\mu\nu}\left(\Gamma^{\rho}_{\mu\sigma}\Gamma^{\sigma}_{\nu\rho}\right)\ ,
\end{equation}
that is
\begin{equation}\label{55}
\overline{R}=-\frac{1}{4}g^{\mu\nu}g^{\sigma\lambda}g^{\rho\epsilon}\left(g_{\epsilon\mu,\sigma}+g_{\epsilon\sigma,\mu}-g_{\mu\sigma,\epsilon}\right)\left(g_{\lambda\nu,\rho}+g_{\lambda\rho,\nu}-g_{\nu\rho,\lambda}\right)\ .
\end{equation}
Keeping terms up to second order in $h^{2}$, we get
\begin{equation}\label{56}
\left(\frac{\partial\overline{R}}{\partial g_{\alpha\beta,\gamma}}\right)^{\left(1\right)}\left(g_{\alpha\beta,\delta}\right)^{\left(1\right)}\stackrel{h^{2}}=\left(\frac{1}{2}h^{\alpha\beta\ \gamma}_{\ \ ,}h_{\alpha\beta,\delta}-h^{\gamma\alpha\ \beta}_{\ \ ,}h_{\alpha\beta,\delta}\right)\ ,
\end{equation}
according to
\begin{multline}\label{57}
\frac{\partial\overline{R}}{\partial g_{\alpha\beta,\gamma}}g_{\alpha\beta,\delta}=-\frac{1}{4}\biggl\{\left(g^{\mu\beta}g^{\sigma\alpha}g^{\epsilon\gamma}+g^{\mu\gamma}g^{\sigma\alpha}g^{\beta\epsilon}-g^{\mu\alpha}g^{\sigma\gamma}g^{\beta\epsilon}\right)\left(g_{\epsilon\mu,\sigma}+g_{\epsilon\sigma,\mu}-g_{\sigma\mu,\epsilon}\right)\\
+\left(g^{\beta\nu}g^{\gamma\lambda}g^{\rho\alpha}+g^{\gamma\nu}g^{\beta\lambda}g^{\rho\alpha}-g^{\alpha\lambda}g^{\beta\nu}g^{\rho\gamma}\right)\left(g_{\lambda\nu,\rho}+g_{\lambda\rho,\nu}-g_{\nu\rho,\lambda}\right)\biggr\}g_{\alpha\beta,\delta}\ ,
%\end{split}\ ,
\end{multline}
and also
\begin{equation}\label{58}
\overline{R}^{\left(2\right)}=-\frac{1}{4}\left(h^{\sigma\lambda}_{\ \ ,\rho}h_{\lambda\sigma,}^{\ \ \ \rho}-2h^{\sigma\lambda}_{\ \ ,\rho}h^{\rho}_{\ \lambda,\sigma}\right)\ .
\end{equation}
Hence, when we put these terms into (<ref>) , the stress–energy pseudo-tensor in general relativity up to order $h^{2}$ takes the form
\begin{equation}\label{59}
\tau_{\alpha\vert GR}^{\eta}=\frac{1}{2\chi}\left[\frac{1}{2}h^{\mu\nu,\eta}h_{\mu\nu,\alpha}-h^{\eta\mu,\nu}h_{\mu\nu,\alpha}-\frac{1}{4}\delta_{\alpha}^{\eta}\left(h^{\sigma\lambda}_{\ \ ,\rho}h_{\lambda\sigma}^{\ \ ,\rho}-2h^{\sigma\lambda}_{\ \ ,\rho}h^{\rho}_{\ \lambda,\sigma}\right)\right]\ .
\end{equation}
Now, we have to expand to second order in $h$ the corrections of the pseudo-tensor (<ref>) due to extended gravity terms. To lower order in $h$ we consider the following expansions
\begin{multline}\label{60}
\left(\frac{\partial R}{\partial g_{\mu\nu,\eta\lambda}}\right)^{\left(0\right)}=\frac{1}{2}\left(g^{\mu\eta}g^{\nu\lambda}+g^{\mu\lambda}g^{\nu\eta}-2g^{\mu\nu}g^{\eta\lambda}\right)^{\left(0\right)}\\
=\frac{1}{2}\left(\eta^{\mu\eta}\eta^{\nu\lambda}+\eta^{\mu\lambda}\eta^{\nu\eta}-2\eta^{\mu\nu}\eta^{\eta\lambda}\right)\ ,
\end{multline}
\begin{equation}\label{61}
\left(\frac{\partial R}{\partial g_{\mu\nu,\eta\lambda}}\right)^{\left(0\right)}\left(g_{\mu\nu,\lambda\alpha}\right)^{\left(1\right)}=\left(h^{\lambda\eta}_{\ \ ,\lambda\alpha}-h^{,\eta}_{\ \ \alpha}\right)=\left(h^{\lambda\eta}-\eta^{\eta\lambda}h\right)_{,\lambda\alpha}\stackrel{\text{h.g.}}{=}-\frac{1}{2}h^{,\eta}_{\ \ \alpha}\ ,
\end{equation}
\begin{equation}\label{62}
\left(\frac{\partial R}{\partial g_{\mu\nu,\eta\lambda}}\right)^{\left(0\right)}\left(g_{\mu\nu,\alpha}\right)^{\left(1\right)}=\left(h^{\lambda\eta}-\eta^{\eta\lambda}h\right)_{,\alpha}\ ,
\end{equation}
\begin{multline}\label{derivatsupnonsimm}
\left(\frac{\partial\Box^{h}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)^{\left(0\right)}=\left(\frac{\partial\Box^{h}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{q}}}\right)^{\left(0\right)}=\left(\frac{\partial\Box^{h}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{2h+1}}}\right)^{\left(0\right)}\\=\eta^{i_{2}i_{3}}\cdots \eta^{i_{2h}i_{2h+1}}\left(\eta^{\mu i_{1}}\eta^{\nu\eta}-\eta^{\mu\nu}\eta^{\eta i_{1}}\right)+\cdots\ .
\end{multline}
Then, we take into account only the terms up to $h^{2}$ in harmonic gauge, as
\begin{equation}\label{primoterm}
\left(2a_{0}R+\sum_{k=1}^{p}a_{k}\Box^{k}R\right)\frac{\partial R}{g_{\mu\nu,\eta\lambda}}g_{\mu\nu,\lambda\alpha}\stackrel{h^{2}}{\stackrel{\text{h.g.}}{=}}\frac{1}{4}\left(\sum_{k=0}^{p}a_{k}\Box^{k+1}h\right)h^{,\eta}_{\ \ \alpha}+\frac{1}{4}a_{0}h^{,\eta}_{\ \ \alpha}\Box h\ ,
\end{equation}
\begin{multline}\label{secondoterm}
-\partial_{\lambda}\left[\sqrt{-g}\left(2a_{0}R+\sum_{k=1}^{p}a_{k}\Box^{k}R\right)\frac{\partial R}{\partial g_{\mu\nu,\eta\lambda}}\right]g_{\mu\nu,\alpha}\stackrel{h^{2}}{\stackrel{\text{h.g.}}{=}}a_{0}\Box h_{,\lambda}\left(h^{\lambda\eta}-\eta^{\eta\lambda}h\right)_{,\alpha}\\
+\frac{1}{2}\sum_{k=1}^{p}a_{k}\Box^{k+1}h_{,\lambda}\left(h^{\lambda\eta}-\eta^{\lambda\eta}h\right)_{,\alpha}\ ,
\end{multline}
\begin{multline}\label{formuladermax}
\sum_{h=1}^{p}\sum_{q=0}^{2h+1}\left(-1\right)^{q}\partial_{i_{0}\cdots i_{q}}\left[\sqrt{-g}a_{h}R\frac{\partial\Box^{h}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{q}}}\right]g_{\mu\nu,\alpha}\\\stackrel{h^{2}}{\stackrel{\text{h.g.}}{=}}\frac{1}{2}\sum_{h=1}^{p}a_{h}\Box^{h+1}h_{,\lambda}\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,\alpha}+\left(A_{p}\right)_{\alpha}^{\eta}\ ,
\end{multline}
\begin{multline}\label{formuladerivatesup}
\sum_{h=1}^{p}\sum_{j=0}^{2h}\sum_{m=j+1}^{2h+1}\left(-1\right)^{j}\partial_{i_{0}\cdots i_{j}}\left[\sqrt{-g}a_{h}R\frac{\partial\Box^{h}R}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right]g_{\mu\nu,i_{j+1}\cdots i_{m}\alpha}\stackrel{h^{2}}{\stackrel{\text{h.g.}}{=}}\frac{1}{4}\sum_{h=1}^{p}a_{h}\Box h \Box^{h} h^{,\eta}_{\ \ \alpha}\\
+\frac{1}{2}\sum_{h=0}^{1}\sum_{j=h}^{p-1+h}\sum_{m=j+1-h}^{p}\left(-1\right)^{h}a_{m}\Box^{m-j}\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,i_{h}\alpha}\Box^{j+1-h}h_{,\lambda}^{\ \ i_{h}}+\left(B_{p}\right)_{\alpha}^{\eta}\ .
\end{multline}
In Eqs. (<ref>), (<ref>) and (<ref>), we have disregarded the index permutations ($\mu\nu$) and $\left(\eta i_{1}\cdots i_{2h+1}\right)$ because $\left(A_{p}\right)_{\alpha}^{\eta}$ and $\left(B_{p}\right)_{\alpha}^{\eta}$ terms, averaged on a suitable spacetime region, vanish, according to Appendix (<ref>). Hence we calculated only the term deriving from (<ref>) without considering the index permutations ($\mu\nu$) and $\left(\eta i_{1}\cdots i_{2h+1}\right)$. This because, taking into account terms obtained from permutations in $\left(A_{p}\right)_{\alpha}^{\eta}$ and $\left(B_{p}\right)_{\alpha}^{\eta}$, averaged on a suitable spacetime region, we obtain that are equal to zero as we will see below in Appendix <ref>. This mathematical trick is essential to calculated the averaged gravitational energy–momentum pseudo-tensor and the power emitted by a source.
So, by inserting equalities (<ref>), (<ref>), (<ref>) and (<ref>) into (<ref>), we find the extra term of pseudo-tensor $\tau^{\eta}_{\alpha}$ to second order owing to extension of general relativity , that we call $\tilde{\tau}^{\eta}_{\alpha}$, that is
\begin{multline}\label{total}
\tilde{\tau}_{\alpha}^{\eta}\stackrel{h^{2}}{=}\frac{1}{2\chi}\Biggl\{\frac{1}{4}\left(\sum_{k=0}^{p}a_{k}\Box^{k+1}h\right)h^{,\eta}_{\ \ \alpha}+\frac{1}{2}\sum_{t=0}^{p}a_{t}\Box^{t+1}h_{,\lambda}\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,\alpha}\\
+\frac{1}{2}\sum_{h=0}^{1}\sum_{j=h}^{p}\sum_{m=j}^{p}\left(-1\right)^{h}a_{m}\Box^{m-j}\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,\alpha i_{h}}\Box^{j+1-h}h_{,\lambda}^{\ \ i_{h}}\\
+\frac{1}{4}\sum_{l=0}^{p} a_{l}\Box^{l}\left(h^{,\eta}_{\ \ \alpha}-\Box h\delta_{\alpha}^{\eta}\right)\Box h+\Theta_{\left[1,+\infty\right[}\left(p\right)\left[\left(A_{p}\right)_{\alpha}^{\eta}+\left(B_{p}\right)_{\alpha}^{\eta}\right]\Biggr\}\ ,
\end{multline}
where conventions used are
\begin{equation*}
\left(\right)_{,\alpha i_{0}}=\left(\right)_{,\alpha} \qquad h_{,\lambda}^{\ \ i_{0}}=h_{,\lambda}\ .
\end{equation*}
In summary, we can split the gravitational energy–momentum pseudo-tensor in the general relativity part and in the Extended Gravity part, that is
\begin{equation}\label{63}
\tau_{\alpha}^{\eta}\stackrel{h^{2}}=\tau_{\alpha\vert GR}^{\eta}+\tilde{\tau}_{\alpha}^{\eta}\ .
\end{equation}
Now in the particular case when $p$ is equal to $0$ and $1$, extended corrections of the pseudo-tensor $\tilde{\tau}^{\eta}_{\alpha}$ was derived. Then, for $p=0$, that is, $L_{g}=\left(\overline{R}+a_{0}R^{2}\right)\sqrt{-g}$ as in the case discussed in [35], we obtain
\begin{equation*}
\tau_{\alpha}^{\eta}\stackrel{h^{2}}=\tau_{\alpha\vert GR}^{\eta}+\tilde{\tau}_{\alpha}^{\eta}\ ,
\end{equation*}
\begin{equation}\label{64}
\tilde{\tau}_{\alpha}^{\eta}\stackrel{h^{2}}=\frac{a_{0}}{2\chi}\left(\frac{1}{2}h^{,\eta}_{\ \ \alpha}\Box h+ h^{\eta}_{\ \lambda,\alpha}\Box h^{,\lambda}-h_{,\alpha}\Box h^{,\eta}-\frac{1}{4}\left(\Box h\right)^{2}\delta_{\alpha}^{\eta}\right)\ .
\end{equation}
While for $p=1$, that is $L_{g}=\left(\overline{R}+a_{0}R^{2}+a_{1}R\Box R\right)\sqrt{-g}$, one has
\begin{equation*}
\tau_{\alpha}^{\eta}\stackrel{h^{2}}=\tau_{\alpha\vert GR}^{\eta}+\tilde{\tau}_{\alpha}^{\eta}\ ,
\end{equation*}
where extended corrections to pseudo-tensor are
\begin{multline}\label{65}
\tilde{\tau}_{\alpha}^{\eta}\stackrel{h^{2}}=\frac{1}{2\chi}\Biggl\{\frac{1}{4}\left(2a_{0}\Box h+a_{1}\Box^{2}h\right)h^{,\eta}_{\ \ \alpha}+\frac{1}{2 }\left(2a_{0}\Box h_{,\lambda}+a_{1}\Box^{2}h_{,\lambda}\right)\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,\alpha}\\
+\frac{1}{2}a_{1}\Box\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,\alpha}\Box h_{,\lambda}+\frac{1}{2}a_{1}\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,\alpha}\Box^{2}h_{,\lambda}-\frac{1}{2}a_{1}\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,\sigma\alpha}\Box h_{,\lambda}^{\ \ \sigma}\\
+\frac{1}{4}a_{1}\Box h^{,\eta}_{\ \ \alpha}\Box h-\frac{1}{4}\delta_{\alpha}^{\eta}\left[a_{0}\left(\Box h\right)+a_{1}\left(\Box^{2} h\right)\right]\Box h+\left(A_{1}\right)_{\alpha}^{\eta}+\left(B_{1}\right)_{\alpha}^{\eta}\Biggr\}\ .
\end{multline}
The iteration can be performed to every $p$ introducing new contributions into dynamics.
§ POWER EMITTED CARRIED BY A GRAVITATIONAL WAVE
We wish to calculate the power emitted in the form of gravitational waves by an isolated massive system considering the local conservation of the energy–momentum pseudo-tensor (<ref>).
§.§ The average of the energy–momentum pseudo-tensor
Let us now regard the wavelike solutions of the linearized field equations in vacuum associated with Lagrangian (<ref>), for details see Ref. [52]. Gravitational waves solutions can be expressed as
\begin{equation}
\label{wave}
h_{\mu\nu}\left(x\right)=\sum_{m=1}^{p+2}\int_{\Omega}\frac{d^{3}\mathbf{k}}{\left(2\pi\right)^{3}}\left(B_{m}\right)_{\mu\nu}\left(\mathbf{k}\right)e^{i\left(k_{m}\right)_{\alpha}x^{\alpha}}+c.c.\ ,
\end{equation}
\begin{equation}\label{66}
\left(B_{m}\right)_{\mu\nu}\left(\mathbf{k}\right)=
\begin{cases}
C_{\mu\nu}\left(\mathbf{k}\right)& \quad \text{for}\quad m=1 \\
\frac{1}{3}\left[\frac{\eta_{\mu\nu}}{2}+\frac{\left(k_{m}\right)_{\mu}\left(k_{m}\right)_{\nu}}{k_{\left(m\right)}^{2}}\right]\text{A}_{m}\left(\mathbf{k}\right)&\quad \text{for} \quad m\geq2
\end{cases}\ ,
\end{equation}
with $C_{\mu\nu}\left(\mathbf{k}\right)$ related to transverse-traceless polarization tensor typical of general relativity and $\text{A}_{m}\left(\mathbf{k}\right)$ the amplitude of wave at $\mathbf{k}$ fixed.
Here "c.c." stands for the complex conjugate.
The trace of tensor (<ref>) is
\begin{equation}\label{67}
\left(B_{m}\right)_{\lambda}^{\lambda}\left(\mathbf{k}\right)=
\begin{cases}
C_{\lambda}^{\lambda}\left(\mathbf{k}\right)&\quad \text{for}\quad m=1 \\
\text{A}_{m}\left(\mathbf{k}\right)&\quad \text{for} \quad m\geq2
\end{cases}\ ,
\end{equation}
and the $k_{m}^{\mu}=\left(\omega_{m}, \mathbf{k}\right)$ is the wave vector with $k_{m}^{2}=\omega_{m}^{2}-\vert \mathbf{k} \vert ^{2}=\text{M}^{2}$ where $k_{1}^{2}=0$ and $k_{m}^{2}\neq 0$ for $m\geq 2$.
Keeping $\mathbf{k}$ fixed, we derive the following relations
\begin{align}\label{68}
h^{\ \ \eta}_{,\alpha}=&2Re\left\{\sum_{j=1}^{p+2}\left(-1\right)\left(k_{j}\right)_{\alpha}\left(k_{j}\right)^{\eta}A_{j}e^{i k_{j}x}\right\}\ ,\\
\Box^{m}h_{,\lambda}=&2 Re\left\{\left(-1\right)^{m}i\sum_{j=1}^{p+2}\left(k_{j}\right)_{\lambda}\left(k_{j}^{2}\right)^{m}A_{j}e^{ik_{j}x}\right\} \ ,\\
\Box^{q}\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,\alpha}=&2Re\left\{\left(-1\right)^{q}i\sum_{l=1}^{p+2}\left(k_{l}\right)_{\alpha}\left(k_{l}^{2}\right)^{q}\left[\left(B_{l}\right)^{\eta\lambda}-\eta^{\eta\lambda}\left(B_{l}\right)_{\rho}^{\rho}\right]e^{ik_{l}x}\right\}\ ,\\
\Box^{m}h_{,\lambda}^{\ \ \sigma}=&2 Re\left\{\left(-1\right)^{m+1}\sum_{j=1}^{p+2}\left(k_{j}\right)_{\lambda}\left(k_{j}\right)^{\sigma}\left(k_{j}^{2}\right)^{m}A_{j}e^{ik_{j}x}\right\}\ , \\
\Box^{q}\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,\sigma\alpha}=&2Re\left\{\left(-1\right)^{q+1}\sum_{l=1}^{p+2}\left(k_{l}\right)_{\sigma}\left(k_{l}\right)_{\alpha}\left(k_{l}^{2}\right)^{q}\left[\left(B_{l}\right)^{\eta\lambda}-\eta^{\eta\lambda}\left(B_{l}\right)_{\rho}^{\rho}\right]e^{ik_{l}x}\right\}\ , \\
\Box^{n}h=&2Re\left\{\left(-1\right)^{n}\sum_{r=2}^{p+2}\left(k_{r}^{2}\right)^{n}A_{r}e^{ik_{r}x}\right\}\ .
\end{align}
Now, we choose a domain of the spacetime $\Omega$ such that $\vert \Omega \vert \gg \frac{1}{\vert k\vert}$ [32]. Then, we can perform the average of the gravitational energy–momentum pseudo-tensor $\tau_{\alpha}^{\eta}$ over our region and all integrals, including terms such as $e^{i\left(k_{i}-k_{j}\right)_{\alpha}x^{\alpha}}$, tend to zero, by means of following identities
\begin{equation}\label{69}
Re\{f\}Re\{g\}=\frac{1}{2}Re\{fg\}+\frac{1}{2}Re\{f\bar{g}\}\ ,
\end{equation}
\begin{equation}\label{70}
\left(k_{l}\right)_{\lambda}\left[\left(B_{l}\right)^{\eta\lambda}-\eta^{\eta\lambda}\left(B_{l}\right)_{\rho}^{\rho}\right]=-\frac{\left(k_{l}\right)^{\eta}}{2}A_{l}\ .
\end{equation}
In the harmonic gauge, after averaging and some algebraic manipulations, we find (see Appendix <ref>)
\begin{align}\label{medievarie}
\left\langle\Box^{m}h_{,\lambda}\Box^{q}\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,\alpha}\right\rangle=&\left(-1\right)^{m+q+1}\sum_{l=2}^{p+2}\left(k_{l}\right)_{\alpha}\left(k_{l}\right)^{\eta} \left(k_{l}^{2}\right)^{\left(m+q\right)}\vert A_{l}\vert^{2}\ , \nonumber\\
\left\langle\Box^{m}h_{,\lambda}^{\ \sigma}\Box^{q}\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,\sigma\alpha}\right\rangle=&\left(-1\right)^{m+q+1}\sum_{l=2}^{p+2}\left(k_{l}\right)_{\alpha}\left(k_{l}\right)^{\eta} \left(k_{l}^{2}\right)^{\left(m+q\right)+1}\vert A_{l}\vert^{2}\ ,\nonumber\\
\left\langle \Box^{q}h_{\ \alpha}^{,\eta}\Box^{m}h\right\rangle=&2\left(-1\right)^{m+q+1}\sum_{r=2}^{p+2}\left(k_{r}\right)_{\alpha}\left(k_{r}\right)^{\eta} \left(k_{r}^{2}\right)^{\left(m+q\right)}\vert A_{r}\vert^{2}\ ,\nonumber\\
\left\langle\Box^{m}h\Box h\right\rangle=&2\left(-1\right)^{m+1}\sum_{j=2}^{p+2}\left(k_{j}^{2}\right)^{m+1}\vert A_{j}\vert^{2}\ ,\nonumber\\
\langle\left(A_{p}\right)_{\alpha}^{\eta}\rangle=&\langle\left(B_{p}\right)_{\alpha}^{\eta}\rangle=0\ .
\end{align}
A set of polarization tensors forming a basis for the linearized solutions $h_{\mu\nu}$ is given in Appendix <ref>.
According to equalities (<ref>), we can calculate the average value of the energy–momentum pseudo-tensor as
\begin{multline}\label{MEMT}
\left\langle\tau_{\alpha}^{\eta}\right\rangle=\frac{1}{2\chi}\left[\left(k_{1}\right)^{\eta}\left(k_{1}\right)_{\alpha}\left(C^{\mu\nu}C_{\mu\nu}^{*}-\frac{1}{2}\vert C_{\lambda}^{\lambda}\vert^{2}\right)\right]\\
+\frac{1}{2\chi}\left[\left(-\frac{1}{6}\right)\sum_{j=2}^{p+2}\left(\left(k_{j}\right)^{\eta}\left(k_{j}\right)_{\alpha}-\frac{1}{2}k_{j}^{2}\delta_{\alpha}^{\eta}\right)\vert A_{j}\vert^{2}\right]\\
+\frac{1}{2\chi}\Biggl\{\Biggl[\sum_{l=0}^{p}\left(l+2\right)\left(-1\right)^{l}a_{l}\sum_{j=2}^{p+2}\left(k_{j}\right)^{\eta}\left(k_{j}\right)_{\alpha}\left(k_{j}^{2}\right)^{l+1}\vert A_{j}\vert^{2}\Biggr]\\
-\frac{1}{2}\sum_{l=0}^{p}\left(-1\right)^{l}a_{l}\sum_{j=2}^{p+2}\left(k_{j}^{2}\right)^{l+2}\vert A_{j}\vert^{2}\delta_{\alpha}^{\eta}\Biggr\}\ ,
\end{multline}
with gravitational coupling ${\chi=\frac{8\pi G}{c^{4}}}$.
In TT gauge for the first mode associated with $k_{1}$ and only in harmonic gauge for residual modes $k_{m}$, in the momentum space, it gets
\begin{equation}\label{71}
\begin{cases}
\left(k_{1}\right)_{\mu}C^{\mu\nu}=0 \quad \land \quad C_{\lambda}^{\lambda}=0&\quad \text{if}\quad m=1\\
\left(k_{m}\right)_{\mu}\left(B_{m}\right)^{\mu\nu}=\frac{1}{2}\left(B_{m}\right)_{\lambda}^{\lambda}k^{\nu}& \quad \text{if} \quad m\geq2
\end{cases}\ .
\end{equation}
We now explore a gravitational wave propagating in the $+z$-direction at $\mathbf{k}$ fixed, with 4-wave vector given by $k^{\mu}=\left(\omega,0,0,k_{z}\right)$ where $\omega_{1}^{2}=k_{z}^{2}$ if $k_{1}^{2}=0$ and $ k_{m}^{2}=m^{2}=\omega_{m}^{2}-k_{z}^{2}$ otherwise with $k_{z}>0$. Accordingly the averaged time-space tensorial component which can be seen as flux of gravitational energy along the $z$ axis through the surface that delimits our domain $\Omega$, reads
\begin{multline}\label{72}
\left\langle\tau_{0}^{3}\right\rangle=\frac{c^{4}}{8\pi G}\omega_{1}^{2}\left(C_{11}^{2}+C_{12}^{2}\right)+\frac{c^{4}}{16\pi G}\Biggl[\left(-\frac{1}{6}\right)\sum_{j=2}^{p+2}\omega_{j}k_{z}\vert A_{j}\vert^{2}\\
+\sum_{l=0}^{p}\left(l+2\right)\left(-1\right)^{l}a_{l}\sum_{j=2}^{p+2}\omega_{j}k_{z}m_{j}^{2\left(l+1\right)}\vert A_{j}\vert^{2}\Biggr]\ .
\end{multline}
Finally, we can calculate the emitted power per unit solid angle $\Omega$, radiated by the localized sources, in a direction $\hat{x}$ at $\mathbf{k}$ fixed. By choosing of the suitable gauge, for the local conservation of the energy–momentum pseudo-tensor (<ref>), the power is given by
\begin{equation}\label{73}
\frac{dP}{d\Omega}=r^2\hat{x}^{i}\left\langle\tau_{0}^{i}\right\rangle\ .
\end{equation}
By ranging the index $p$ of the pseudo-tensor (<ref>) over $\{0,1,2\}$, we obtain the following three cases
for p=0
\begin{gather}\label{74}
\left\langle\tau_{0}^{3}\right\rangle=\frac{c^4\omega_{1}^{2}}{8\pi G}\left[C_{11}^{2}+C_{12}^{2}\right]+\frac{c^{4}}{16\pi G}\biggl\{\left(-\frac{1}{6}\right)\omega_{2}\vert A_{2}\vert^{2}k_{z}+2a_{0}\omega_{2}m_{2}^{2}\vert A_{2}\vert^{2}k_{z}\biggr\}\ ,
\end{gather}
for p=1
\begin{multline}\label{75}
\left\langle\tau_{0}^{3}\right\rangle=\frac{c^4\omega_{1}^{2}}{8\pi G}\left[C_{11}^{2}+C_{12}^{2}\right]+\frac{c^{4}}{16\pi G}\biggl\{\left(-\frac{1}{6}\right)\left(\omega_{2}\vert A_{2}\vert^{2}+\omega_{3}\vert A_{3}\vert^{3}\right)k_{z}\\
+2a_{0}\left[\left(\omega_{2}m_{2}^{2}\vert A_{2}\vert^{2}+
\omega_{3}m_{3}^{2}\vert A_{3}\vert^{2}\vert^{2}\right)k_{z}\right] -3a_{1}\left[\left(\omega_{2}m_{2}^{4}\vert A_{2}\vert^{2}+\omega_{3}m_{3}^{4}\vert A_{3}\vert^{2}\right)k_{z}\right]\biggr\}\ ,
\end{multline}
and for p=2
\begin{multline}\label{76}
\left\langle\tau_{0}^{3}\right\rangle=\frac{c^4\omega_{1}^{2}}{8\pi G}\left[C_{11}^{2}+C_{12}^{2}\right]+\frac{c^{4}}{16\pi G}\biggl\{\left(-\frac{1}{6}\right)\left(\omega_{2}\vert A_{2}\vert^{2}+\omega_{3}\vert A_{3}\vert^{3}+\omega_{4}\vert A_{4}\vert^{2}\right)k_{z}\\+2a_{0}\left[\left(\omega_{2}m_{2}^{2}\vert A_{2}\vert^{2}+\omega_{3}m_{3}^{2}\vert A_{3}\vert^{2}+\omega_{4}m_{4}^{2}\vert A_{4}\vert^{2}\right)k_{z}\right]\\
-3a_{1}\left[\left(\omega_{2}m_{2}^{4}\vert A_{2}\vert^{2}+\omega_{3}m_{3}^{4}\vert A_{3}\vert^{2}+\omega_{4}m_{4}^{4}\vert A_{4}\vert^{2}\right)k_{z}\right]\\
+4a_{2}\left[\left(\omega_{2}m_{2}^{6}\vert A_{2}\vert^{2}+\omega_{3}m_{3}^{6}\vert A_{3}\vert^{2}+\omega_{4}m_{4}^{6}\vert A_{4}\vert^{2}\right)\right]\biggr\}\ ,
\end{multline}
where the gravitational coupling $\chi$ has been explicitly indicated. By formulas (<ref>), (<ref>) and (<ref>) it is obvious that the first term comes out of general relativity and the corrections strongly depends on $p$. In any context where corrections to general relativity can be investigated, this approach could constitute a paradigm to search for higher order effects.
§ ENERGY–MOMENTUM COMPLEX OF $F({\CAL R})$ GRAVITY IN PALATINI APPROACH.
§.§ The gravitational pseudo-tensor of $f({\cal R})$ gravity in Palatini formulation.
In Palatini approach the metric tensor $g_{\mu\nu}$ and the connection $\Gamma^\alpha_{\mu\nu}$ are independent, that means that we do not assume any relation between the metric and the connection, and Riemann and Ricci tensors are, in general, defined as
\begin{align}\label{78}
{\cal R}_{\mu\nu}(\Gamma)=& \partial_\alpha\Gamma^\alpha_{\mu\nu}-\partial_\nu\Gamma^\alpha_{\mu\alpha}+\Gamma^\alpha_{\mu\nu}\,\Gamma^\sigma_{\alpha\sigma} -\Gamma^\alpha_{\nu\lambda} \, \Gamma^\lambda_{\mu\alpha},
\\
{\cal R}(g,\Gamma) =& {\cal R}_{\mu\nu}(\Gamma)\, g^{\mu \nu}.
\end{align}
So, the Palatini gravitational action of $f({\cal R})$ appears as [53]
\begin{align} \label{actionf}
{\cal S} = \frac{1}{2 \kappa^2} \int {\rm d}^4x \,\sqrt{-g}\, f({\cal R}),
\end{align}
with the coupling $\kappa^2=8\pi G /c^4$ and $g$ the determinant of metric tensor $g_{\mu\nu}$. By varying the metric $g^{\mu\nu}$ and the connection $\Gamma^\alpha_{\mu\nu}$, for a general infinitesimal transformation coordinate $x^\mu$ it gets
\begin{align}
x^{\prime \mu} =& x^\mu + \delta x^\mu,
\\
g^{\prime \mu\nu}(x^\prime) =& g^{\mu\nu}(x) + \tilde{\delta}g^{\mu\nu} ,&
g^{\prime \mu\nu}(x) =& g^{\mu\nu}(x) + \delta g^{\mu\nu},
\\
\Gamma^{\prime \alpha}_{\mu\nu}(x^\prime) =& \Gamma^\alpha_{\mu\nu}(x) + \tilde{\delta} \Gamma^\alpha_{\mu\nu},&
\Gamma^{\prime\alpha}_{\mu\nu}(x) =& \Gamma^\alpha_{\mu\nu}(x) + \delta \Gamma^\alpha_{\mu\nu},
\end{align}
where $\tilde{\delta}$ is the local variation and $\delta$ is the variation that keeps the coordinates $x$ fixed.
The variation of the gravitational action with respect to the metric $g^{\mu\nu}$ and the connection $\Gamma^{\alpha}_{\beta\gamma}$ yield
\begin{multline}\label{79}
\tilde{\delta} {\cal S} = \frac{1}{2\kappa^2} \int {\rm d}^4x \Bigg\{ \sqrt{-g}\biggl[\left(f_{\cal R} {\cal R}_{\mu\nu}-\frac{1}{2} g_{\mu\nu} f\right) \, \delta g^{\mu\nu} \\
+ f_{\cal R} g^{\mu\nu} \, \delta {\cal R}_{\mu\nu}\biggl] +\partial_\mu \left(\sqrt{-g} f \, \delta x^\mu\right) \Bigg\},
\end{multline}
where $f_{\cal R}:={\rm d} f({\cal R})/{\rm d}{\cal R}$. According to the following Palatini identity
\begin{align}\label{80}
\delta {\cal R}_{\mu\nu} = \nabla_\alpha\left(\delta \Gamma^\alpha_{\mu\nu}\right)-\nabla_\nu\left(\delta\Gamma^\alpha_{\alpha\mu}\right).
\end{align}
the action (<ref>) takes the form
\begin{multline} \label{action1}
\tilde{\delta} {\cal S} = \frac{1}{2\kappa^2 } \int {\rm d}^4x \, \Bigg\{ \sqrt{-g}\left(f_{\cal R} {\cal R}_{\mu\nu}-\frac{1}{2} g_{\mu\nu} f\right) \, \delta g^{\mu\nu} \\
+\delta \Gamma^\lambda_{\phantom{\lambda}\nu\mu}\Bigl[-\nabla_\lambda\left(\sqrt{-g}g^{\nu\mu}f_{\cal R}\right)+\nabla_\alpha\left(\sqrt{-g}g^{\mu\alpha} \delta^\nu_\lambda f_{\cal R}\right)\Bigr]
\\
\partial_\lambda\left[\sqrt{-g}f_{\cal R} \left(g^{\mu\nu} \delta^\lambda_\alpha-g^{\mu\lambda} \delta^{\nu}_\alpha \right) \delta \Gamma^\alpha_{\phantom{\alpha}\mu\nu} +\sqrt{-g} f \, \delta x^\lambda \right]
\Bigg\}.
\end{multline}
By the principle of least action or stationary action (<ref>), by imposing that the variation of metric and its derivatives vanish at the boundary, we obtain field equations for the metric tensor and the connection in vacuum, i.e.,
\begin{align}\label{FieldEq1}
f_{\cal R} {\cal R}_{(\mu\nu)}-\frac{1}{2} g_{\mu\nu} f =& 0,
\\ \label{FieldEq2}
\nabla_\lambda\left(\sqrt{-g}g^{\nu\mu}f_{\cal R}\right) =& 0.
\end{align}
Given that we adopting an arbitrary non-compatible connection, the symmetric part of the Ricci tensor, ${\cal R}_{(\mu\nu)}$, enter in the Eq. (<ref>) and then the Ricci tensor is non symmetric, that is
\begin{equation}\label{81}
{\cal R}_{\mu\nu}={\cal R}_{\nu\mu}+{\cal R}^{\lambda}_{\phantom{\lambda}\lambda\mu\nu}\ ,
\end{equation}
being Riemann tensor ${\cal R}^{\sigma}_{\phantom{\sigma}\lambda\mu\nu}$ no longer antisymmetric on its first two indices, i.e., the term ${\cal R}^{\lambda}_{\phantom{\lambda}\lambda\mu\nu}$ does not vanishes. For a generic infinitesimal transformation, the metric tensor and the connection change as
\begin{align}\label{82}
x^{\prime \mu} =& \, x^\mu + \xi^\mu,
\\
g^{\prime \mu\nu}(x^\lambda) \simeq& \, g^{\mu\nu}(x^\lambda) - \xi^\lambda \frac{\partial g^{\mu\nu}}{\partial x^\lambda} + g^{\mu\alpha} \frac{\partial \xi^\nu}{\partial x^\alpha} + g^{\nu\alpha} \frac{\partial \xi^\mu}{\partial x^\alpha},
\\
\Gamma^{\prime \alpha}_{\phantom{\alpha}\mu\nu} (x^\lambda) \simeq& \, \Gamma^{\alpha}_{\phantom{\alpha}\mu\nu} (x^\lambda) - \xi^\lambda \frac{\partial \Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}}{\partial x^\lambda} + \Gamma^{\rho}_{\phantom{\rho}\mu\nu} \frac{\partial \xi^\alpha}{\partial x^\rho} -\Gamma^{\alpha}_{\phantom{\alpha}\sigma\nu} \frac{\partial \xi^\sigma}{\partial x^\mu} -\Gamma^{\alpha}_{\phantom{\alpha}\mu\sigma} \frac{\partial \xi^\sigma}{\partial x^\nu} - \frac{\partial^2 \xi^\alpha}{\partial x^\mu \, \partial x^\nu},
\end{align}
where we have neglected terms of higher order in $\xi^\mu$ in the series expansion.
Under a rigid infinitesimal translation, that is, $\partial_\mu \xi^\nu =0$, we obtain
\begin{align}
g^{\prime \mu\nu}(x^\lambda) \simeq \, & g^{\mu\nu}(x^\lambda) - \xi^\lambda \frac{\partial g^{\mu\nu}}{\partial x^\lambda},
\\
\Gamma^{\prime \alpha}_{\phantom{\alpha}\mu\nu} (x^\lambda) \simeq \, & \Gamma^{\alpha}_{\phantom{\alpha}\mu\nu} (x^\lambda) - \xi^\lambda \frac{\partial \Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}}{\partial x^\lambda} .
\end{align}
Therefore, the Palatini action (<ref>) becomes
\begin{multline}\label{83}
\tilde{\delta} {\cal S}_{\rm g} = \frac{1}{2\kappa^2} \int {\rm d}^4x \Bigg\{ - \sqrt{-g}\left(f_{\cal R} {\cal R}_{\mu\nu}-\frac{1}{2} g_{\mu\nu} f\right) \, \xi^\lambda \, g^{\mu\nu}_{\phantom{\mu\nu},\lambda}\\
-\xi^\lambda \, \Gamma^\beta_{\phantom{\beta}\nu\mu,\lambda}\left[-\nabla_\beta\left(\sqrt{-g}g^{\nu\mu}f_{\cal R}\right)+\nabla_\alpha\left(\sqrt{-g}g^{\mu\alpha} \delta^\nu_\beta f_{\cal R}\right)\right]
\\
\partial_\lambda\left[-\sqrt{-g}f_{\cal R} \left(g^{\mu\nu} \delta^\lambda_\alpha-g^{\mu\lambda} \delta^\nu_\alpha \right) \, \xi^\beta \, \Gamma^\alpha_{\phantom{\alpha}\mu\nu,\beta} +\sqrt{-g} f \, \xi^\lambda \right]
\Bigg\}.
\end{multline}
If the metric $g^{\mu\nu}$ and the Palatini connection $\Gamma^{\alpha}_{\beta\gamma}$ are solution of equations (<ref>) and (<ref>), the stationary of the local variation of the action (<ref>), gives the local conservation of gravitational energy–momentum pseudo-tensor $\tau^\lambda_{\phantom{\lambda}\beta}$ of Palatini $f({\cal R})$ gravity, namely
\begin{align}\label{84}
\partial_\lambda\left(\sqrt{-g} \, \tau^\lambda_{\phantom{\lambda}\beta} \right) = 0,
\end{align}
where $\tau^\lambda_{\phantom{\lambda}\beta}$ is defined as
\begin{align}\label{85}
\tau^\lambda_{\phantom{\lambda}\beta}=\frac{1}{2\kappa^{2}} \left[f\left({\cal R}\right) \, \delta^\lambda_\beta -f_{\cal R}\left({\cal R}\right)\, \left(g^{\mu\nu} \delta^\lambda_\alpha-g^{\mu\lambda} \delta^\nu_\alpha \right) \, \Gamma^\alpha_{\phantom{\alpha}\mu\nu,\beta} \right].
\end{align}
It is worth noting that the pseudo-tensor defined in Eq. (<ref>) has the opposite sign of the one defined above. In order to derive the energy–momentum complex, let us analyze the action containing the matter part, that is
\begin{align}\label{86}
{\cal S}_{\rm m} = \int {\rm d}^4x \, \sqrt{-g}\,{\cal L}_{\rm m}.
\end{align}
Generally, the matter Lagrangian ${\cal L}_{\rm m}$ depends on the connection as, for example, occurs in presence of fermion fields.
Here, we consider only material Lagrangian which does not depend on the affine connection $\Gamma$. Then, the matter energy–momentum tensor is defined as in (<ref>).
Hence, field equations for metric and connection, i.e., Eqs. (<ref>) and (<ref>), in presence of matter yield
\begin{align}\label{Einstein and Palatini Eqs.}
f_{\cal R} {\cal R}_{(\mu\nu)}-\frac{1}{2} g_{\mu\nu} f =& \kappa^2 T_{\mu\nu},
\\
\nabla_\lambda\left(\sqrt{-g}g^{\nu\mu}f_{\cal R}\right) =& 0 \label{bimetric}.
\end{align}
As already pointed out above the connection can be non compatible with the metric $g_{\mu\nu}$, i.e., $\nabla_{\lambda}g_{\mu\nu}\neq 0$. In compact form, we can define a new metric, conformally related to the metric $g_{\mu\nu}$, as
\begin{align}\label{87}
h_{\mu\nu}:=f_{\cal R}g_{\mu\nu}.
\end{align}
so that Eq. (<ref>) becomes
\begin{align}\label{88}
\nabla_\lambda \left(\sqrt{h}h^{\mu\nu}\right)=0.
\end{align}
Thus the Palatini connection $\Gamma^\alpha_{\phantom{\alpha}\mu\nu} $ appears as the Christoffel connection for the new metric $h_{\mu\nu}$, i.e.,
\begin{align}\label{PalatiniConnection}
\Gamma^\alpha_{\phantom{\alpha}\mu\nu}=
\frac{1}{2\, f_{\cal R}\left({\cal R}\right)}g^{\alpha \beta} \left[\partial_\mu\left( f_{\cal R}\left({\cal R}\right) g_{\nu\beta}\right)+\partial_\nu\left( f_{\cal R}\left({\cal R}\right) g_{\mu\beta}\right)-\partial_\beta\left( f_{\cal R}\left({\cal R}\right) g_{\mu\nu}\right)\right]\ .
\end{align}
The Palatini connection $\Gamma^\alpha_{\phantom{\alpha}\mu\nu} $ and Levi–Civita connection $\stackrel{\circ}{\Gamma}{}^\alpha_{\phantom{\alpha}\mu\nu} $ are related as
\begin{equation}\label{89}
\Gamma^\alpha_{\phantom{\alpha}\mu\nu}=\,\stackrel{\circ}{\Gamma^{\alpha}}_{\mu\nu}+\delta^{\alpha}_{\mu}A_{\nu}+\delta^{\alpha}_{\nu}A_{\mu}-g_{\mu\nu}A^{\alpha}\ ,
\end{equation}
where the four-vector $A_{\mu}$ is defined as
\begin{equation}\label{91}
A_{\mu}:=\frac{1}{2f_{\cal R}}\nabla_{\mu}f_{\cal R}\ .
\end{equation}
For $f({\cal R})={\cal R}$, we recover the Christoffel symbols constructed by the metric $g_{\mu\nu}$, that is
\begin{align}\label{92}
\Gamma^\alpha_{\phantom{\alpha}\mu\nu}=\stackrel{\circ}{\Gamma^\alpha}_{\phantom{\alpha}\mu\nu} = \frac{1}{2}g^{\alpha\beta}\left( g_{\beta\mu,\nu}+g_{\beta\nu,\mu}-g_{\mu\nu,\beta}\right)\,
\end{align}
this means that in general relativity no difference results in metric and Palatini formalism. The Ricci tensor ${\cal R}_{\mu\nu}$ in Palatini formalism and that in metric formalism $R_{\mu\nu}$, are related as follows
\begin{multline} \label{Rconf}
{\cal R}_{\mu\nu} = R_{\mu\nu} +\frac{3}{2} \frac{1}{\left(f_{\cal R}({\cal R})\right)^2} \left(\stackrel{\circ}{\nabla}_\mu f_{\cal R}({\cal R}) \right) \, \left(\stackrel{\circ}{\nabla}_\mu f_{\cal R}({\cal R})\right) \\
- \frac{1}{ f_{\cal R}({\cal R})} \left( \stackrel{\circ}{\nabla}_\mu \stackrel{\circ}{\nabla}_\nu-\frac{1}{2}g_{\mu\nu} \stackrel{\circ}{\square}\right) f_{\cal R}({\cal R}),
\end{multline}
where $\stackrel{\circ}{\square} := \stackrel{\circ}{\nabla}{}^\mu \stackrel{\circ}{\nabla}_\mu$ and $\stackrel{\circ}{\nabla}$ denotes the covariant derivative associated with the Levi–Civita connection. Contracting tensorial equality (<ref>) with $g^{\mu\nu}$, we obtain the relation between ${\cal R}$ and $R$, that is, the Ricci scalar in both approach
\begin{align}\label{93}
{\cal R} = R + \frac{3}{2\left(f_{\cal R}({\cal R})\right)^2} \left(\stackrel{\circ}{\nabla}_\mu f_{\cal R}({\cal R}) \right)\; \left(\stackrel{\circ}{\nabla}{ }^\mu f_{\cal R}({\cal R})\right) + \frac{3}{ f_{\cal R}({\cal R})} \stackrel{\circ}{\square} f_{\cal R}({\cal R}).
\end{align}
Adopting the Palatini connection $\Gamma^\alpha_{\phantom{\alpha}\mu\nu}$ (<ref>), the symmetry of Ricci tensor is restored on account of the relation
\begin{equation}\label{94}
\Gamma_{\lambda}=\frac{\partial_{\lambda}{\left(f_{\cal R}^{2}\sqrt{-g}\right)}}{f_{\cal R}^{2}\sqrt{-g}},
\end{equation}
which implies
\begin{equation}\label{95}
{\cal R}_{[\mu\nu]}=\partial_{[\mu}\Gamma_{\nu]}=0.
\end{equation}
Furthermore the connection is non compatible with metric $g_{\mu\nu}$ being
\begin{equation}\label{96}
\nabla_{\lambda}g_{\mu\nu}=-\frac{g_{\mu\nu}}{f_{\cal R}}\nabla_{\lambda}f_{\cal R}.
\end{equation}
Despite this, the covariant derivatives associated with Palatini connection commute each other, as displayed below
\begin{equation}\label{97}
\left[\nabla_{\rho},\nabla_{\lambda}\right]g_{\mu\nu}=0\,.
\end{equation}
Thus, we restore the antisymmetry on the first two indices of Riemann tensor, namely
\begin{equation}\label{99}
{\cal R}_{\mu\nu\lambda\rho}=-{\cal R}_{\nu\mu\lambda\rho}.
\end{equation}
by the definition of Riemann tensor for an arbitrary tensor $J_{\mu\nu}$
\begin{equation}\label{98}
\left[\nabla_{\rho},\nabla_{\lambda}\right]J_{\mu\nu}=-{\cal R}^{\alpha}_{\phantom{\alpha}\mu\rho\lambda}J_{\alpha\nu}-{\cal R}^{\alpha}_{\phantom{\alpha}\nu\rho\lambda}J_{\mu\alpha}.
\end{equation}
In addition, the contracted Bianchi identities are fulfilled, that is
\begin{equation}\label{100}
\nabla_{\mu}\left({\cal R}^{\mu\nu}-\frac{1}{2}g^{\mu\nu}{\cal R}\right)=0.
\end{equation}
According to the Palatini connection Eq. (<ref>) and from the symmetry of energy–momentum tensor $T_{\mu\nu}$, taking into account that for the new metric $h_{\mu\nu}$ we have
\begin{align}\label{101}
\Gamma_{\lambda}=\frac{\partial_{\lambda}{\sqrt{-h}}}{\sqrt{-h}}\ ,
\end{align}
\begin{align}\label{102}
\Gamma_{\mu\nu\lambda}+\Gamma_{\nu\mu\lambda}=\frac{1}{f_{{\cal R}}}\partial_{\lambda}h_{\mu\nu},
\end{align}
so we derive the following useful expression
\begin{align}\label{103}
\sqrt{-h}\nabla_{\sigma}T^{\sigma}_{\phantom{\sigma}\nu}=\partial_{\sigma}\left(\sqrt{-h}T^{\sigma}_{\phantom{\sigma}\nu}\right)-\frac{1}{2f_{\cal R}}T^{\lambda\rho}\partial_{\nu}h_{\lambda\rho}\sqrt{-h}.
\end{align}
Field equations in matter (<ref>) lead to
\begin{align}\label{104}
0=\frac{\sqrt{-h}}{2f_{\cal R}^{2}}T^{\mu\nu}g_{\mu\nu,\beta}\xi^{\beta}+\partial_{\lambda}\left\{\sqrt{-g}\frac{1}{2\kappa^{2}}\left[%\frac{1}{f_{\cal R}\left(\cal R\right)}\,
f\left({\cal R}\right) \, \delta^\lambda_\beta -f_{\cal R}\left(g^{\mu\nu} \delta^\lambda_\alpha-g^{\mu\lambda} \delta^\nu_\alpha \right) \, \Gamma^\alpha_{\phantom{\alpha}\mu\nu,\beta} \right]\xi^{\beta}\right\},
\end{align}
and from Eq. (<ref>), after some algebraic manipulations, we get the following 4-divergence of energy–momentum complex not vanishing
\begin{align}\label{105}
\partial_{\sigma}\left[\sqrt{-g}\left(T^{\sigma}_{\phantom{\sigma}\beta}+t^{\sigma}_{\phantom{\sigma}\beta}\right)\right]=\frac{\sqrt{-h}}{f_{\cal R}^2} \, \nabla_{\lambda}T^{\lambda}_{\phantom{\lambda}\beta}+\frac{2\sqrt{-h}}{f_{\cal R}^{3}}T^{\lambda}_{\phantom{\lambda}\beta} \, \nabla_{\lambda}f_{\cal R}-\frac{\sqrt{-h}}{2f_{\cal R}^{3}}T \, \nabla_{\beta}f_{\cal R}.
\end{align}
From contracted Bianchi identities and the field equations, the following relations are satisfied
\begin{equation}\label{106}
\left[\nabla_{\mu},\nabla_{\nu}\right]\nabla^{\mu}f_{\cal R}={\cal R}^{\alpha}_{\phantom{\alpha}\nu}\nabla_{\alpha}f_{\cal R},
\end{equation}
\begin{equation}\label{107}
\kappa^{2}\nabla_{\mu}T^{\mu}_{\phantom{\mu}\nu}={\cal R}^{\alpha}_{\phantom{\alpha}\nu}\nabla_{\alpha}f_{\cal R}=\left[\nabla_{\mu},\nabla_{\nu}\right]\nabla^{\mu}f_{\cal R}\ .
\end{equation}
The trace of Eqs. (<ref>) gives the so called structural equation of space-time [54], that is
\begin{equation}\label{108}
T=\frac{1}{\kappa^{2}}\left[f_{\cal R}{\cal R}-2f\left({\cal R}\right)\right],
\end{equation}
where $T=T_{\mu\nu}g^{\mu\nu}$. For a given $f({\cal R})$, we can, in principle, solve this equation and get a relation ${\cal R}={\cal R}(T)$. Thanks to Eq. (<ref>), considering $T=0$, the theory reduces to GR with a cosmological constant. Substituting Eqs. (<ref>) and (<ref>) into Eq. (<ref>), we get
\begin{equation} \label{109}
\partial_{\sigma}\left[\sqrt{-g}\left(T^{\sigma}_{\phantom{\sigma}\beta}+\tau^{\sigma}_{\phantom{\sigma}\beta}\right)\right]=-\frac{\sqrt{-g}}{\kappa^{2}}G^{\lambda}_{\phantom{\lambda}\beta}\nabla_{\lambda}f_{\cal R},
\end{equation}
where $G^{\lambda}_{\phantom{\lambda}\beta}$ is the Einstein tensor.
After some algebraic manipulations, we find the following expression
\begin{equation}\label{110}
G^{\lambda}_{\phantom{\lambda}\beta}\nabla_{\lambda}f_{\cal R}=-\kappa^{2}\stackrel{\circ}{\nabla}_{\mu}T^{\mu}_{\phantom{\mu}\beta}\ .
\end{equation}
The right hand side of Eq. (<ref>) vanishes [55, 56, 57] and then, according to Eqs. (<ref>) and (<ref>), the energy–momentum complex for Palatini $f({\cal R})$ gravity, $\mathcal{T}^{\sigma}_{\phantom{\sigma}\beta}$, is locally conserved, namely
\begin{equation}
\partial_{\sigma}\left[\sqrt{-g}\left(T^{\sigma}_{\phantom{\sigma}\beta}+\tau^{\sigma}_{\phantom{\sigma}\beta}\right)\right]=0,
\end{equation}
\begin{equation}\label{111}
\mathcal{T}^{\sigma}_{\phantom{\sigma}\beta}=\sqrt{-g}\left(T^{\sigma}_{\phantom{\sigma}\beta}+\tau^{\sigma}_{\phantom{\sigma}\beta}\right)\ .
\end{equation}
§ COSMOLOGICAL APPLICATIONS BOTH IN PALATINI AND METRIC APPROACH IN $F(R)$ GRAVITY
§.§ Palatini formalism
We consider a flat FLRW spacetime whose metric is
\begin{align}
{\rm d}s^2= -{\rm d}t^2+ a^2(t)\, \left({\rm d}x^2+{\rm d}y^2+{\rm d}z^2\right),
\end{align}
with scale factor $a(t)$ and cosmic time $t$. From the relation (<ref>) and the field equations (<ref>), we obtain
\begin{align}
2\kappa^2 T^0_{\phantom{0}0} = & -f +6f_{\cal R} \left(\dot{H}+H^2\right) +\ddot{f}_{\cal R} -3\frac{\dot{f}_{\cal R}^2}{f_{\cal R}} -3H\dot{f}_{\cal R},
\end{align}
where $H=\dot{a}/a$ is the Hubble parameter and dots stands for derivatives with respect to the cosmic time $t$. The gravitational energy density $\tau^0_{\phantom{0}0}$ is defined as
\begin{align}
2\kappa^2 \tau^0_{\phantom{0}0} = & f -6f_{\cal R} \left(\dot{H}+H^2\right) -3\ddot{f}_{\cal R} +3\frac{\dot{f}_{\cal R}^2}{f_{\cal R}} -3H\dot{f}_{\cal R}.
\end{align}
So, the energy density complex is
\begin{align}
\kappa^2(\tau^0_{\phantom{0}0}+T^0_{\phantom{0}0}) = -\ddot{f}_{\cal R} -3H \dot{f}_{\cal R}.
\end{align}
In general relativity , i.e., $f({\cal R})={\cal R}$, we obtain a null energy density complex
\begin{align}
\left(t^0_{\phantom{0}0}+T^0_{\phantom{0}0}\right)\Big|_{\rm GR}=0\,.
\end{align}
We postulate that perfect fluids including radiation and non-relativistic dust describe the matter and that the components of the energy–momentum tensor are
\begin{align}
\left(T^\mu_{\phantom{\mu}\nu} \right)_{\rm r} =& {\rm diag}\left(-\rho_{\rm r},p_{\rm r},p_{\rm r},p_{\rm r} \right) & \mbox{ with equation of state}& &p_{\rm r}=&\frac{1}{3} \rho_{\rm r},
\\
\left(T^\mu_{\phantom{\mu}\nu} \right)_{\rm m} =& {\rm diag}\left(-\rho_{\rm m},p_{\rm m},p_{\rm m},p_{\rm m} \right), & \mbox{ with equation of state }& &p_{\rm m}=&0,
\end{align}
where $\rho_{\rm i}$ and $p_{\rm i}$ are the energy density and pressure of each fluid component.
From the conservation of energy–momentum tensor, we obtain, respectively,
\begin{align} \label{c1}
\dot{\rho}_{\rm r}+4 H \rho_{\rm r}=&0,
\\ \label{c2}
\dot{\rho}_{\rm m}+3 H \rho_{\rm m}=&0.
\end{align}
Choosing a form for $f({\cal R})$, we can solve the structure equation Eq. (<ref>) and then explicit ${\cal R}$ as a function of $T$. Now, let us assume a polynomial form as $f({\cal R}) = {\cal R}+\alpha {\cal R}^2$, which is a model extensively studied in Palatini formalism, see for example [58, 59]. Thus, the solution of structural equation (<ref>) becomes
\begin{align}
{\cal R} = -\kappa^2 T.
\end{align}
This model implies power law cosmological solutions [60] as
\begin{align} \label{powerlaw}
a(t) = a_0 \, t^m,
\end{align}
where $m>0$ is a real number. From Eqs. (<ref>) and (<ref>), we get
\begin{align}
\rho_{\rm tot}(t) = \rho_{\rm m}(t)+\rho_{\rm r}(t)=\rho_{\rm m0}t^{-3m}+\rho_{\rm r0}t^{-4m}\,
\end{align}
with $\rho_{\rm m0}$ and $\rho_{\rm r0}$ initial values.
Therefore, we obtain the gravitational energy density
\begin{multline}
2 \kappa^2 \, \tau^0_{\phantom{0}0} = \frac{6m(1-m)}{t^2} +\kappa^2 \rho_{\rm m0}t^{-3m} +6m (5-2m) \alpha \kappa^2 \rho_{\rm m0} t^{-3m-2} \\
+\frac{108m^2 \alpha^2 \kappa^4 \rho_{\rm m0}t^{-6m-2}}{1+2m^2\kappa^2\rho_{\rm m0} t^{-3m}},
\end{multline}
and the energy density complex
\begin{align}
\tau^0_{\phantom{0}0}+T^0_{\phantom{0}0} =-6\alpha m \rho_{\rm m0} t^{-(3m+2)}.
\end{align}
The total energy density of gravitational and non-gravitational fields is then
\begin{equation}\label{300}
\sqrt{-g}(\tau^0_{\phantom{0}0}+T^0_{\phantom{0}0})=-6\alpha m \rho_{\rm m0} t^{-2},
\end{equation}
that tends to zero as the inverse square of cosmic time.
§.§ Metric approach
We consider also in this case a flat FLRW spacetime but in metric formalism. We can explicitly write the time-time components of the gravitational energy–momentum $\tau^\mu_{\phantom{\mu}\nu}$ and the matter energy–momentum $T^\mu_{\phantom{\mu}\nu}$, respectively, as follows
\begin{align}
\kappa^2\, \tau^0_{\phantom{0}0} =& \frac{1}{2}f(R)-3\left(H^2+\dot{H}\right)\, f_R(R) +3H\dot{R} \, f_{RR}(R),
\\ \label{f1}
\kappa^2 T^0_{\phantom{0}0} =& - \frac{1}{2}f(R)+3\left(H^2+\dot{H}\right)\, f_R(R) - 3H\dot{R} \, f_{RR}(R).
\end{align}
Subsequently, the total energy of the gravitation and matter vanishes for FLRW spacetime, i.e.
\begin{align}
\tau^0_{\phantom{0}0}+T^0_{\phantom{0}0} =0,
\end{align}
unlike Palatini approach where energy complex does not vanish Eq.(<ref>). Now, we can assume a power-law evolution for matter and radiation fluids such as Eq. (<ref>). We have
\begin{align} \tau^0_{\phantom{0}0} =&\rho_{\rm m}(t)+\rho_{\rm r}(t)\nonumber \\ =&\rho_{\rm m0} t^{-3m}+\rho_{\rm r0}t^{-4m}\,.
\end{align}
The Ricci curvature scalar, in this case, reads
\begin{equation}
R = 12H^2+6\dot{H} = 6m(2m-1) t^{-2}\,
\end{equation}
while the Friedman equation is reduced to
\begin{multline}\label{200}
\frac{f_{RR}\, R^2}{(2m-1)}+\frac{m-1}{2(2m-1)} f_R \, R -\frac{1}{2} f +\kappa^2 \rho_{\rm m0} \left( \frac{R}{6m(2m-1)}\right)^{\frac{3}{2}m}\\
+\kappa^2 \rho_{\rm r0} \left( \frac{R}{6m(2m-1)}\right)^{2m} =0.
\end{multline}
From this equation (<ref>), we get the explicit form of $f(R)$ that shows a power law behaviour, that is
\begin{align} \label{f}
f(R) =& -\frac{4\kappa^2 \rho_{\rm m0} (2m-1)}{12m-11} \left( \frac{R}{6m(2m-1)}\right)^{\frac{3}{2}m} -\frac{2\kappa^2 \rho_{\rm r0} (2m-1)}{10m^2-8m+1} \left( \frac{R}{6m(2m-1)}\right)^{2m}
\nonumber \\
&+C_1 R^{\frac{3}{4}-\frac{m}{4}-\frac{1}{4}\sqrt{m^2+10m+1}} +C_2 R^{\frac{3}{4}-\frac{m}{4}+\frac{1}{4}\sqrt{m^2+10m+1}}.
\end{align}
When $m=2/3$ and $\rho_{\rm r0}/\rho_{\rm m0} \ll 1$, occurs $f(R)\sim R$ and GR is restored.
§ CONCLUSIONS
Attempts to extend the general relativity through corrections to the Hilbert–Einstein Lagrangian, by introducing curvature, torsion and non-metricity invariants, both local and non-local, have increased in recent years. All of this is to address gravitational divergences at ultraviolet and infrared scales, and more generally to deal with cosmological and astrophysical issues such as current and early cosmic acceleration or the structure formation, without introducing exotic components such as dark energy and dark matter. For a detailed discussion on infinite derivative theories, see Ref. [61, 62, 63, 64, 65, 66, 67], while for non-local wavelike solutions, see Ref. [68, 69, 70]. However, most of the main features of general relativity should be retained to obtain self-consistent theories. In particular, a thorough study of the properties of the gravitational energy–momentum pseudo-tensor are indispensable in view of both the foundation and applications of any gravitational theory.
This review is devoted to generalizing the gravitational energy–momentum pseudo-tensor $\tau^{\eta}_{\alpha}$ to general $n^{th}$ order Lagrangian of the form
\begin{equation*}
L=L(g_{\mu\nu}, g_{\mu\nu,i_{1}}, g_{\mu\nu,i_{1}i_{2}},g_{\mu\nu,i_{1}i_{2}i_{3}}, \ldots, g_{\mu\nu,i_{1}i_{2}i_{3}\cdots i_{n}})\ ,
\end{equation*}
showing that in this model gravity a local conservation of energy–momentum complex is fulfilled.
Specifically, we considered Lagrangians such as $L_{g}=(\overline{R}+a_{0}R^{2}+\sum_{k=1}^{p} a_{k}R\Box^{k}R)\sqrt{-g}$ and $L=F(R)$, both in Palatini and metric approach. It has been shown that $\tau^{\eta}_{\alpha}$ is an affine and not covariant object because it changes as a tensor, under linear transformations but not under general coordinate transformations. The pseudo-tensor of higher order gravity has been weakly perturbed up to the order $h^2$, thus obtaining the weak field limit of the gravitational energy–momentum density. After averaging this object over a suitable four-dimensional domain under suitable gauge, by local conservation of pseudo-tensor, the power emitted by a gravitational source was found. Hence, the gravitational wave (<ref>) associated with higher order Lagrangian express, under the chosen gauge for a wave propagating along the $+z$-direction, in terms of six polarization tensors (see Appendix <ref>) reads as
\begin{multline}
\label{GW1}
+\text{A}^{\left(1\right)}\left(t-v_{G_{m}}z\right)\epsilon_{\mu\nu}^{\left(1\right)}+\text{A}^{\left(L\right)}\left(t-v_{G_{m}}z\right)\epsilon_{\mu\nu}^{\left(L\right)}\ ,
\end{multline}
where $v_{G_{m}}$ is the group velocity of the $m_{th}$ massive mode (see also [20, 33]). Thanks to these solutions, it was possible to derive an expression of the power emitted in terms of amplitudes of the waves $\text{A}_{j}\left(\mathbf{k}\right)$, $C_{11}\left(\mathbf{k}\right)$ and $C_{22}\left(\mathbf{k}\right)$, and the free parameters $a_{m}$. Three special cases for $p$ equal to $0,1,$ and $2$ have been shown where the extended corrections to the power are clearly visible. It was given a cosmological application of the pseudo-tensor in $f(R)$ gravity in both Palatini and metric formulation. Therefore, in a flat FLRW spacetime, we have derived that while the energy density complex vanishes in the metric formalism, in general, it does not vanish in the Palatini approach.
The analysis of gravitational waves and gravitational energy–momentum pseudo-tensor
are two indispensable tools for finding the viable theory of gravitation. Indeed, by wavelike solutions of linearized theory of gravity and by the locally conserved pseudo-tensor, it is possible to calculate the emitted power by isolated system. Then, from the local conservation of the energy–momentum complex, it is also possible to take into account the energy–momentum content of the source, which, through a multipole expansion, could also allow us to derive a generalized formula of the quadrupole formula. This procedure could lead us to fix the order of theory [51, 71], to investigate additional polarization states of gravitational wave and to establish the range of the masses $m_{j}$ of modes.
§ ACKNOWLEDGMENTS
S.C., M.C. and G.L. acknowledgment the Istituto Nazionale di Fisica Nucleare (INFN) Sez. di Napoli, Iniziative Specifiche QGSKY, and the Istituto Nazionale di Alta Matematica (INdAM), gruppo GNFM, for the support.
§ APPENDIX
§.§ The average of $\langle\left(A_{p}\right)_{\alpha}^{\eta}\rangle$ and $\langle\left(B_{p}\right)_{\alpha}^{\eta}\rangle$ terms
Let us now demonstrate the last two relations in (<ref>), that is $\langle\left(A_{p}\right)_{\alpha}^{\eta}\rangle=\langle\left(B_{p}\right)_{\alpha}^{\eta}\rangle=0$.
The general formula for $\Box^{h} R$ -derivative, according to symmetries of $g_{\mu\nu}$ and its derivatives, is [71]:
\begin{multline}\label{derivordsupsimm}
\frac{\partial \Box^{h}R}{\partial g_{\mu\nu,\eta i_{1}\cdots i_{2h+1}}}=g^{j_{2}j_{3}}\cdots g^{j_{2h}j_{2h+1}}g^{ab}g^{cd}\biggl\{\delta_{a}^{(\mu}\delta_{d}^{\nu)}\delta_{c}^{(\eta}\delta_{b}^{i_{1}}\delta_{j_{2}}^{i_{2}}\cdots\delta_{j_{2h}}^{i_{2h}}\delta_{j_{2h+1}}^{i_{2h+1})}\\
\end{multline}
We have to verify that $\langle \left(B_{p}\right)_{\alpha}^{\eta}\rangle=0$ holds. Inserting Eq. (<ref>) in the l.h.s. of Eq. (<ref>) that, in the weak field limit up to the order $h^{2}$ becomes
\begin{multline}\label{bassen}
\sum_{h=1}^{p}\sum_{j=0}^{2h}\sum_{m=j+1}^{2h+1}\left(-1\right)^{j}\partial_{i_{0}\cdots i_{j}}\left[\sqrt{-g}a_{h}R\frac{\partial\Box^{h}R}{\partial g_{\mu\nu,\eta i_{1}\cdots i_{m}}}\right]g_{\mu\nu,i_{j+1}\cdots i_{m}\alpha}\\
\stackrel{h^{2}} =\sum_{h=1}^{p}\sum_{j=0}^{2h}\left(-1\right)^{j}\sqrt{-g}^{\left(0\right)}a_{h}\partial_{i_{0}\cdots i_{j}}R^{\left(1\right)}\eta^{j_{2}j_{3}}\cdots \eta^{j_{2h}j_{2h+1}}\eta^{ab}\eta^{cd}\biggl\{\delta_{a}^{(\mu}\delta_{d}^{\nu)}\delta_{c}^{(\eta}\delta_{b}^{i_{1}}\delta_{j_{2}}^{i_{2}}\cdots\delta_{j_{2h}}^{i_{2h}}\delta_{j_{2h+1}}^{i_{2h+1})}\\
-\delta_{a}^{(\mu}\delta_{b}^{\nu)}\delta_{c}^{(\eta}\delta_{d}^{i_{1}}\delta_{j_{2}}^{i_{2}}\cdots\delta_{j_{2h}}^{i_{2h}}\delta_{j_{2h+1}}^{i_{2h+1})}\biggr\}h_{\mu\nu,i_{j+1}\cdots i_{2h+1}\alpha}\\
=\sum_{h=1}^{p}\sum_{j=0}^{2h}\left(-1\right)^{j}a_{h}\partial_{i_{0}\cdots i_{j}}R^{\left(1\right)}Q_{\left(\mu\nu\right)}^{\ \ \ \left(\eta i_{1}\cdots i_{2h+1}\right)}h^{\mu\nu}_{\ \ ,i_{j+1}\cdots i_{2h+1}\alpha}
\end{multline}
\begin{equation*}
Q_{\left(\mu\nu\right)}^{\ \ \ \left(\eta i_{1}\cdots i_{2h+1}\right)}=\frac{1}{2!\left(2h+2\right)!}\sum_{ \substack{\mu\nu\in \sigma\left({\mu\nu}\right) \\ \eta i_{1}\cdots i_{2h+1}\in\sigma\left(\eta i_{1}\cdots i_{2h+1}\right)}}Q_{\mu\nu}^{\ \ \ \eta i_{1}\cdots i_{2h+1}}
\end{equation*}
\begin{equation*}
Q_{\left(\mu\nu\right)}^{\ \ \ \left(\eta i_{1}\cdots i_{2h+1}\right)}=\delta_{(\mu}^{(\eta}\delta_{\nu)}^{i_{1}}\eta^{i_{2}i_{3}}\cdots\eta^{i_{2h}i_{2h+1})}-\eta_{(\mu\nu)}\eta^{(\eta i_{1}}\eta^{i_{2}i_{3}}\cdots\eta^{i_{2h}i_{2h+1})}
\end{equation*}
where $\sigma{\left(\mu\nu\right)}$ and $\sigma{\left(\eta i_{1}\cdots i_{2h+1}\right)}$ represent the set of index permutations in the brackets. Averaging Eq. (<ref>) by fixing $\mathbf{k}$ over a suitable spacetime region adopting a harmonic gauge, we get
\begin{multline}\label{media2}
\langle\sum_{h=1}^{p}\sum_{j=0}^{2h}\left(-1\right)^{j}a_{h}\partial_{i_{0}\cdots i_{j}}R^{\left(1\right)}Q_{\left(\mu\nu\right)}^{\ \ \ \left(\eta i_{1}\cdots i_{2h+1}\right)}h^{\mu\nu}_{\ \ ,i_{j+1}\cdots i_{2h+1}\alpha}\rangle\\
=\sum_{h=1}^{p}\sum_{j=0}^{2h}\frac{1}{2!\left(2h+2\right)!}\left(-1\right)^{j}a_{h}\sum_{ \substack{\mu\nu\in \sigma\left({\mu\nu}\right) \\ \eta i_{1}\cdots i_{2h+1}\in\sigma\left(\eta i_{1}\cdots i_{2h+1}\right)}}\langle\partial_{i_{0}\cdots i_{j}}R^{\left(1\right)}Q_{\mu\nu}^{\ \ \ \eta i_{1}\cdots i_{2h+1}}h^{\mu\nu}_{\ \ ,i_{j+1}\cdots i_{2h+1}\alpha}\rangle
\end{multline}
The average of Eq. (<ref>) is independent of index permutations in the lower and upper cases of $Q_{\mu\nu}^{\ \ \ \eta i_{1}\cdots i_{2h+1}}$, that is
\begin{equation}\label{media3}
\langle\partial_{i_{0}\cdots i_{j}}\left(-\frac{1}{2}\Box h\right)Q_{\mu\nu}^{\ \ \ \eta i_{1}\cdots i_{2h+1}}h^{\mu\nu}_{\ \ ,i_{j+1}\cdots i_{2h+1}\alpha}\rangle=\frac{1}{2}\sum_{m=2}^{p+2}\left(-1\right)^{j+h}\left(k_{m}^{2}\right)^{h+1}\left(k_{m}\right)^{\eta}\left(k_{m}\right)_{\alpha}\vert A_{m}\vert^{2}
\end{equation}
By substituting Eq. (<ref>) in Eq. (<ref>), we get
\begin{multline}\label{media4}
\langle\sum_{h=1}^{p}\sum_{j=0}^{2h}\sum_{m=j+1}^{2h+1}\left(-1\right)^{j}\partial_{i_{0}\cdots i_{j}}\left[\sqrt{-g}a_{h}R\frac{\partial\Box^{h}R}{\partial g_{\mu\nu,\eta i_{1}\cdots i_{m}}}\right]g_{\mu\nu,i_{j+1}\cdots i_{m}\alpha}\rangle\\
\stackrel{h^{2}}=\sum_{h=1}^{p}\sum_{j=0}^{2h}\left(-1\right)^{j}a_{h}\sum_{m=2}^{p+2}\left(-1\right)^{j+h}\left(k_{m}^{2}\right)^{h+1}\left(k_{m}\right)^{\eta}\left(k_{m}\right)_{\alpha}\vert A_{m}\vert^{2}\\
=\sum_{h=1}^{p}\sum_{m=2}^{p+2}\left(h+\frac{1}{2}\right)a_{h}\left(-1\right)^{h}\left(k_{m}^{2}\right)^{h+1}\left(k_{m}\right)^{\eta}\left(k_{m}\right)_{\alpha}\vert A_{m}\vert^{2}
\end{multline}
Averaging the right term in Eq. (<ref>), we have
\begin{multline}\label{media5}
\langle\frac{1}{4}\sum_{h=1}^{p}a_{h}\Box h \Box^{h} h^{,\eta}_{\ \ \alpha}+\frac{1}{2}\sum_{h=0}^{1}\sum_{j=h}^{p-1+h}\sum_{m=j+1-h}^{p}\left(-1\right)^{h}a_{m}\Box^{m-j}\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,i_{h}\alpha}\Box^{j+1-h}h_{,\lambda}^{\ \ i_{h}}\rangle\\
=\sum_{h=1}^{p}\sum_{m=2}^{p+2}\left(h+\frac{1}{2}\right)a_{h}\left(-1\right)^{h}\left(k_{m}^{2}\right)^{h+1}\left(k_{m}\right)^{\eta}\left(k_{m}\right)_{\alpha}\vert A_{m}\vert^{2}
\end{multline}
Finally, by averaging in the weak field limit Eq. (<ref>) and from Eqs. (<ref>) and (<ref>), we obtain:
\begin{equation}
\langle \left(B_{p}\right)_{\alpha}^{\eta}\rangle=0
\end{equation}
A similar argument gives $\langle \left(A_{p}\right)_{\alpha}^{\eta}\rangle=0$. It is
\begin{multline}
\langle\sum_{h=1}^{p}\sum_{q=0}^{2h+1}\left(-1\right)^{q}\partial_{i_{0}\cdots i_{q}}\left[\sqrt{-g}a_{h}R\frac{\partial\Box^{h}R}{\partial g_{\mu\nu,\eta i_{1}\cdots i_{q}}}\right]g_{\mu\nu,\alpha}\rangle\\
\stackrel{h^{2}}=\frac{1}{2}\sum_{h=1}^{p}\sum_{m=2}^{p+2}a_{h}\left(-1\right)^{h+1}\left(k_{m}^{2}\right)^{h+1}\left(k_{m}\right)^{\eta}\left(k_{m}\right)_{\alpha}\vert A_{m}\vert^{2}
\end{multline}
\begin{multline}
\langle\frac{1}{2}\sum_{h=1}^{p}a_{h}\Box^{h+1}h_{,\lambda}\left(h^{\eta\lambda}-\eta^{\eta\lambda}h\right)_{,\alpha}\rangle\stackrel{h^{2}}=\frac{1}{2}\sum_{h=1}^{p}\sum_{m=2}^{p+2}a_{h}\left(-1\right)^{h+1}\left(k_{m}^{2}\right)^{h+1}\left(k_{m}\right)^{\eta}\left(k_{m}\right)_{\alpha}\vert A_{m}\vert^{2}
\end{multline}
and then averaging Eq. (<ref>) on the l.h.s. and r.h.s., in the weak field limit, we have
\begin{equation}
\langle \left(A_{p}\right)_{\alpha}^{\eta}\rangle=0
\end{equation}
that completes our demonstration.
§.§ The polarizations of gravitational waves
The six polarizations in the solution (<ref>) can be defined in a suitable matrix base. That is [52]
\begin{align*}
\epsilon_{\mu\nu}^{\left(+\right)}&=\frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & -1 & 0\\0 & 0 & 0 & 0
\end{pmatrix}&
\epsilon_{\mu\nu}^{\left(\times\right)}&=\frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & 0 & 0 & 0\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\0 & 0 & 0 & 0
\end{pmatrix}\\
\epsilon_{\mu\nu}^{\left(\text{TT}\right)}&=\qquad
\begin{pmatrix}
1 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0
\end{pmatrix}&
\epsilon_{\mu\nu}^{\left(\text{TS}\right)}&=\frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & 0 & 0 & 1\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\1 & 0 & 0 & 0
\end{pmatrix}\\
\epsilon_{\mu\nu}^{\left(1\right)}&=\frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 0
\end{pmatrix}&
\epsilon_{\mu\nu}^{\left(L\right)}&=\qquad
\begin{pmatrix}
0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 1
\end{pmatrix}
\end{align*}
The $+$ and $\times$ are the two standard of general relativity. The other are related to the position of non-null terms with respect to the trace (T).
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|
# Deep learning of experimental electrochemistry for battery cathodes across
diverse compositions
Peichen Zhong<EMAIL_ADDRESS>Department of Materials Science and
Engineering, University of California, Berkeley, California 94720, United
States Materials Sciences Division, Lawrence Berkeley National Laboratory,
California 94720, United States Bowen Deng Department of Materials Science
and Engineering, University of California, Berkeley, California 94720, United
States Materials Sciences Division, Lawrence Berkeley National Laboratory,
California 94720, United States Tanjin He Department of Materials Science and
Engineering, University of California, Berkeley, California 94720, United
States Materials Sciences Division, Lawrence Berkeley National Laboratory,
California 94720, United States Zhengyan Lun Department of Materials Science
and Engineering, University of California, Berkeley, California 94720, United
States School of Chemical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China Gerbrand Ceder<EMAIL_ADDRESS>Department
of Materials Science and Engineering, University of California, Berkeley,
California 94720, United States Materials Sciences Division, Lawrence
Berkeley National Laboratory, California 94720, United States
###### Abstract
Artificial intelligence (AI) has emerged as a tool for discovering and
optimizing novel battery materials. However, the adoption of AI in battery
cathode representation and discovery is still limited due to the complexity of
optimizing multiple performance properties and the scarcity of high-fidelity
data. In this study, we present a machine-learning model (DRXNet) for battery
informatics and demonstrate the application in the discovery and optimization
of disordered rocksalt (DRX) cathode materials. We have compiled the
electrochemistry data of DRX cathodes over the past five years, resulting in a
dataset of more than 19,000 discharge voltage profiles on diverse chemistries
spanning 14 different metal species. Learning from this extensive dataset, our
DRXNet model can automatically capture critical features in the cycling curves
of DRX cathodes under various conditions. Illustratively, the model gives
rational predictions of the discharge capacity for diverse compositions in the
Li–Mn–O–F chemical space as well as for high-entropy systems. As a universal
model trained on diverse chemistries, our approach offers a data-driven
solution to facilitate the rapid identification of novel cathode materials,
accelerating the development of next-generation batteries for carbon
neutralization.
###### pacs:
## I Introduction
Figure 1: Discharge voltage profiles from experiments. (A) The discharge
voltage profile illustrates the relationship between capacity $Q$ and voltage
$V$, which is conditioned on the composition of the cathode material. (B) The
derivative quantity $dQ/dV$ is used to quantify the redox potentials of the
TM. The experimental discharge voltage profiles of Li1.2Mn0.2Cr0.2Ti0.4O2.0
DRX with (C) multi-rate tests from 20 – 1000 mA/g and (D) multi-cycle tests
from the first to the 30th cycles. (E) The parsed discharge profile is stored
in a voltage array $\\{V_{i}^{N}\\}$, and a capacity array $\\{Q_{i}^{N}\\}$,
where the subscript $i$ represents a point (state) on a discharge profile, and
the superscript $N$ represents the cycle number.
The pursuit of carbon neutrality has become a global imperative in the face of
climate change, driving the transition to renewable energy sources and the
widespread adoption of electric vehicles [1, 2, 3]. High-performance battery
cathode materials with large energy density, high-rate performance, and long
cycle life are central to these advancements. The development of new cathode
materials is essential to meeting the increasing demand for energy storage and
advancing the electrification of transportation systems [4].
Artificial intelligence (AI) has emerged as a potential tool in the discovery
and optimization of novel battery materials [5, 6]. By leveraging vast amounts
of experimental and computational data, AI-assisted techniques can accelerate
the design and synthesis of battery materials by identifying promising
candidates within large chemical spaces [7, 8, 9], uncovering hidden
structure-property relationships via machine-learned atomistic modeling [10],
predicting the remaining lifespan of batteries [11, 12, 13, 14, 15], and
optimizing protocols for fast charge/discharge protocol [16]. These efforts
significantly reduce the time and cost required for conventional trial-and-
error approaches. Most recently, a battery data genome initiative has been
proposed to use AI assistance to accelerate the discovery and optimization of
battery materials [17].
Despite these advancements, current machine-learning efforts in battery
research primarily focus on predicting the lifespan for a simple chemistry or
within a limited chemical space, such as NMC (Ni–Mn–Co) or LFP (LiFePO4). The
development of exploratory machine learning for representing comprehensive
compositional effects in a multi-dimensional chemical space remains
underdeveloped due to the challenges associated with simultaneously optimizing
multiple electrochemical properties (e.g., rate capability, cyclability, and
various test voltage windows) [18]. Moreover, the scarcity of high-fidelity
data further hinders the progress of AI in the battery field.
Disordered rocksalt (DRX) materials have emerged as promising cathode
materials that make use of earth-abundant precursors to enable scaling of Li-
ion energy storage to several TWh/year production [19]. Owing to the nearly
unlimited compositional design space and considerably more complex structure-
property relationship of DRX cathodes compared with conventional layered
cathodes (Figure 1A), their rational design requires the extensive involvement
of advanced characterization techniques (e.g., pair-distribution function
analysis [20], spherical-aberration-corrected transmission electron microscopy
[21], solid-state nuclear magnetic resonance spectroscopy [22]) as well as
sophisticated computational tools (e.g., high-dimensional cluster expansion
and Monte Carlo simulation [23, 24]). Data-driven methods offer alternative
means of compositional design and optimization of materials without having to
fully construct their structure-property relationships.
In light of these challenges, we developed DRXNet, an exploratory machine-
learning model for the discovery and optimization of battery cathode
materials. DRXNet uses composition, test current density, working voltage
window, and cycle number as inputs to predict entire discharge voltage
profiles. By training and testing over 19,000 experimental discharge voltage
profiles of DRX materials comprising various metal species, we show that the
model accurately captures the cathode electrochemistry under different test
conditions. Notably, DRXNet captures accessible discharge capacity in diverse
Li–Mn–O–F compositions and makes rational predictions for several high-entropy
systems. As a universal model trained on diverse chemistries, DRXNet offers a
data-driven solution to facilitate the rapid identification of novel cathode
materials with improved energy-storage capabilities.
Figure 2: Description of the collected experimental dataset and model design:
(A) The elemental distribution of collected experimental electrochemistry
data. The dataset contains 7,898 discharge profiles collected from DRX oxides
and 11,604 discharge profiles from oxyfluorides. The color-coding of the boxes
indicates the number of discharge profiles (cycles) on compounds that contain
that specific element. The number within each elemental box represents the
number of compounds with that element on which experiments were conducted. (B)
A histogram of the number of cycles ($N_{\text{cycle}}$) and current density
(rate) for all the individual electrochemical tests. (C) An end-to-end
pipeline that maps $Q_{i}=\mathcal{F}(V_{i}|\mathcal{O})$, which consists of
the electrochemical condition network $\mathcal{O}$ (left) and the state
prediction network $\mathcal{F}$ (right). The electrochemical condition
network encodes the DRX composition, current density rate, and cycle
information. The three encoded vectors are synthesized through gated-MLPs with
soft attention to obtain the condition vector $\vec{X}_{\mathcal{O}}$ [25].
The state prediction is approximated as a forward deep neural network that
takes the voltage state $V_{i}$ and cycling voltage window
$V_{\text{low}},V_{\text{high}}$ as inputs. The encoded condition vector
$\vec{X}_{\mathcal{O}}$ is element-wise added in the hidden layer of
$\mathcal{F}$. The circled symbols are all element-wise operations. The
message-passing graph neural network (GNN) is used for compositional encoding
of DRX, adapted from the Roost model [26].
## II Results
### II.1 Parsing discharge profiles
Unlike conventional NMC-based layered cathodes [27, 28], DRX materials exhibit
more diverse electrochemical behavior due to the significantly larger chemical
space over which they can exist and their more subtle structure involving
various forms of cation short-range order [29]. A prototype DRX cathode
(Li1+xM’aM”bO2-yFy) is composed of three primary compositional parameters: (1)
the redox-active species M’; (2) the inert high-valent transition metal M”,
which charge-compensates for the Li excess and stabilizes disordered
structures [30]; (3) fluorine, which enhances the cyclability and accommodates
more Li excess without losing TM redox by reducing the anion valence [31]. In
addition, other compositional modifications are often made to enhance
capacity, rate, or cyclability. For instance, Mg doping in Mn-based
oxyfluoride DRX can increase the discharge capacity while retaining a similar
voltage-profile shape [32]; Cr doping in Li1.2Mn0.4Ti0.4O2.0 results in
comparable low-rate capacity but significantly improves the high-rate
performance near the top of charge [33]. These non-linear effects arising from
compositional changes make both material design and machine-learning modeling
challenging, thereby necessitating a comprehensive, high-fidelity dataset to
address such issues.
Figure 1A introduces the typical discharge-voltage profile in battery tests.
The profile shape is tied to various factors, such as the DRX composition,
applied current density rate, and degradation that may have occurred in prior
cycles. Figure 1C and D show the multi-rate tests (the first cycle) and multi-
cycle tests (of 20 mA/g and 1000 mA/g) of Li1.2Mn0.2Cr0.2Ti0.4O2.0 cathode as
an example. The capacity $Q$ is measured in experiments by determining the
cumulative charge transferred in a galvanostatic test. Taking the derivative
of $Q$ with respect to $V$, the $dQ/dV$ value can be evaluated for a given
voltage profile, which is a crucial physical quantity for analyzing
characteristic redox potentials from different TMs [34].
### II.2 DRX Battery Test Dataset
We have compiled the electrochemical test data related to DRX compounds by
mining electronic experimental notebooks in our research group over the past
five years to construct the DRX Test Dataset (DRX-TD). The dataset contains
not only results on successful materials published in several papers [35, 36,
29, 37, 38, 39, 40, 33, 32] but also data on less well-performing DRX
compounds. This endeavor yielded a comprehensive dataset containing 19,000
discharge profiles across 16 different elements (14 metal species + O and F)
from lab experiments and published literature (see Methods). An individual
electrochemical test is defined as a group of $N_{\text{cycle}}$ discharge
profiles with a fixed current density rate, where $N_{\text{cycle}}$ is the
number of cycles conducted in such a test, corresponding to the results
obtained from one coin-cell. The distribution of elements in the DRX-TD is
shown in Figure 2A, where the number in each element’s box represents the
number of compounds with that element present for which an electrochemical
test is present. The box’s color indicates the total number of discharge
profiles for compounds containing that element. Comprising 7,898 discharge
profiles of DRX oxides and 11,604 discharge profiles of oxyfluorides, the
dataset offers extensive coverage of major redox-active TMs. Figure 2B
displays histograms for the number of cycles, $N_{\text{cycle}}$, and the
current rates at which experiments were performed. As is typical for
exploratory research programs in a research laboratory, most of the
electrochemical tests were conducted at a low current rate (20 mA/g) and for
10-100 cycles.
For each discharge profile, 100 points were uniformly sampled from the values
of $V$ and $Q$, resulting in a voltage series
$\boldsymbol{V}=\left[V_{1},V_{2},...,V_{i},...\right]$ and a capacity series
$\boldsymbol{Q}=\left[Q_{1},Q_{2},...,Q_{i},...\right]$. The $dQ/dV$ curve was
then calculated by differentiating $\boldsymbol{Q}$ with $\boldsymbol{V}$. As
$dQ/dV$ is a more intrinsic property for battery materials, including this
value in the modeling allows for a more representative analysis of the
electrochemical performance of DRX compounds under various conditions (see
Methods).
### II.3 DRXNet architecture
DRXNet aims to draw a connection between chemistry and cathode performance by
establishing a mapping between $\boldsymbol{V}$ and $\boldsymbol{Q}$ for
arbitrary cathode compositions under various test conditions. This idea can be
conceptualized as identifying a function $\mathcal{F}$ that maps cathode
parameters and the voltage state $V_{i}$ to produce the capacity state $Q_{i}$
as an output. The function $\mathcal{F}$ is conditionally defined by the
parameters $\mathcal{O}$, which consider the electrode composition, current
rate, and cycle number
$Q_{i}=\mathcal{F}(V_{i}|\mathcal{O}).$ (1)
We designed DRXNet with two main components, as shown in Figure 2C: (1) An
electrochemical condition network that generates a feature vector
$\vec{X}_{\mathcal{O}}$ based on the compound’s composition and
electrochemical test information; (2) A state prediction network to
approximate the discharge state of the cathode as a function of the voltage
state, $Q_{i}=\mathcal{F}(V_{i}|\mathcal{O})$, given the electrochemical
conditional encoding of $\mathcal{O}$. For instance, Algorithm 1 demonstrates
how DRXNet predicts the first-cycle discharge profile of
$\text{Li}_{1.2}\text{Mn}_{0.2}\text{Cr}_{0.2}\text{Ti}_{0.4}\text{O}_{2}$ at
a current rate of 20 mA/g between 1.5 and 4.8 V.
Condition Inputs:
$\mathcal{O}=\begin{cases}&\textbf{composition}=\text{Li}_{1.2}\text{Mn}_{0.2}\text{Cr}_{0.2}\text{Ti}_{0.4}\text{O}_{2}\\\
&\textbf{rate}=20~{}\text{mA/g},\\\ &\textbf{cycle}=1\end{cases}$
Condition Outputs:
$\vec{X}_{\mathcal{O}_{1}}=\vec{X}_{\text{comp}}+\sigma_{f_{1}}(\vec{X}_{\text{comp}}||\vec{X}_{\text{rate}})\cdot
f_{1}(\vec{X}_{\text{comp}}||\vec{X}_{\text{rate}})$
$\displaystyle\vec{X}_{\mathcal{O}_{N}}=\vec{X}_{\mathcal{O}_{1}}+\sigma_{f_{2}}(\vec{X}_{\mathcal{O}_{1}}||\vec{X}_{\text{cycle}})$
$\displaystyle\cdot f_{2}(\vec{X}_{\mathcal{O}_{1}}||\vec{X}_{\text{cycle}})$
$\displaystyle\cdot\boldsymbol{W}_{n}(N-1)$
Inputs: $\boldsymbol{V}=\left[1.5,...,V_{i},...,4.8\right]\rightarrow N$
series
for _$i=1$ to $N$_ do
Compute $Q_{i}=\mathcal{F}(V_{i}|\vec{X}_{\mathcal{O}_{N}})$
end for
Outputs: $\boldsymbol{Q}=\left[Q_{1},...,Q_{i},...,Q_{N}\right]$
Algorithm 1 The workflow of DRXNet with an example of
$\text{Li}_{1.2}\text{Mn}_{0.2}\text{Cr}_{0.2}\text{Ti}_{0.4}\text{O}_{2}$
Initially, three condition inputs (composition, rate, cycle) are encoded to
represent $\mathcal{O}$. We use Roost, a graph neural network model proposed
by Goodall and Lee [26], for compositional encoding. Roost takes elements as
graph nodes and updates the correlation between elements through weighted
message passing based on each element’s fractional concentration. The nodes
are initialized with elemental embedded vectors $\vec{h}_{s}$ ($s$: species)
from mat2vec to capture as much prior chemical information as possible through
text mining of previously published literature [41]. Moreover, we consider
only the cation species as independent nodes in Roost, treating the anion-
species information (fluorine) as a mean-field background, i.e.,
$\vec{h}^{\prime}_{\text{Li}}=\vec{h}_{\text{Li}}+c_{\text{F}}\cdot\vec{h}_{\text{F}}$,
where $c_{\text{F}}$ is the fractional concentration of fluorine and
$\vec{h}_{\text{Li/F}}$ is the embedded vector of Li/F. Rate and cycle
information is encoded using multi-layer perceptrons (MLPs).
Figure 3: Error and model variance analysis of DRXNet in compositional space:
The prediction error of discharge capacity between 2.0 and 4.4 V ($y$-axis)
vs. cycle number ($x$-axis). The model variance is represented by
$\sigma_{Q}$, a standard deviation of the ensemble of the models’ prediction,
which is plotted as scaled colored dots. (A)–(B): Predictions on 3TM/HE using
models trained on the 2TM dataset. (C)–(D): Predictions on 3TM/HE using models
trained on both the 2TM dataset and the first cycles of the 3TM/HE dataset.
(E)–(F): Predictions on 3TM/HE using models trained on the 2+3TM dataset.
Because the rate and cycle properties are intrinsically affected by the
composition, we used gated MLPs with soft attention for electrochemical
condition encoding via a hierarchical network structure [25]. The
$\vec{X}_{\mathcal{O}_{1}}=\vec{X}_{\text{comp}}+\sigma_{f_{1}}(\vec{X}_{\text{comp}}||\vec{X}_{\text{rate}})\cdot
f_{1}(\vec{X}_{\text{comp}}||\vec{X}_{\text{rate}})$ is a rate-informed
feature vector, where $\sigma_{f}$ and $f$ represent MLPs with different
activation functions and $||$ denotes the concatenation operation. In
addition, the cycle-informed vector
$\vec{X}_{\mathcal{O}_{N}}=\vec{X}_{\mathcal{O}_{1}}+\sigma_{f_{2}}(\vec{X}_{\mathcal{O}_{1}}||\vec{X}_{\text{cycle}})\cdot
f_{2}(\vec{X}_{\mathcal{O}_{1}}||\vec{X}_{\text{cycle}})\cdot\boldsymbol{W}_{n}(N-1)$
is linearly dependent on the cycle number with a trainable weight
$\boldsymbol{W}_{n}$. As such, the feature vector $\vec{X}_{\mathcal{O}_{1}}$
is used to represent the 1st cycle and $\vec{X}_{\mathcal{O}_{N}}$ is used to
represent the $N$-th cycle, respectively.
Lastly, we used several MLPs to construct the state prediction network
$\mathcal{F}$, as shown in Figure 2C. $\mathcal{F}$ takes the voltage state
$V_{i}$ and working window $V_{\text{low}},V_{\text{high}}$ as inputs, and the
$\vec{X}_{\mathcal{O}}$ is element-wise added to the hidden layer of
$\mathcal{F}$ to inform $\mathcal{F}$ of conditions $\mathcal{O}$ (see
Methods). As such, the state prediction network $\mathcal{F}$ is constructed
as a simple function mapping from the voltage state $V_{i}$ to the capacity
$Q_{i}$. In addition, $(dQ/dV)_{i}$ is obtained by auto-differentiation of
$\mathcal{F}$.
### II.4 Applicability domain
We explore the scope of DRXNet’s applicability in the realm of composition
space. Determination of the applicability domain in battery machine-learning
models can be challenging due to the unavailability of sufficient test data,
as generating new data necessitates the synthesis of new materials or
conducting battery cycling tests for weeks to months [18, 42]. Simply
separating the sequence of voltage and capacity signals $\\{V_{i},Q_{i}\\}$
into training and test sets can result in data leakage and a failure to
represent the expected error in real applications. To evaluate the
expressibility and generalization of DRXNet, we designed several experiments
by partitioning the dataset based on compositions. The electrochemical tests
with no more than two metal species (2TM, excluding Li) were designated as the
training set, whereas the tests with three metal species (3TM) and higher
numbers of TM components (high-entropy, HE) were assigned as test sets. For
each test, an ensemble of five independent models was trained to enhance the
overall prediction accuracy and robustness and to quantify the model variance.
The average value is used for the prediction, and the standard deviation of
the prediction from the ensemble of five DRXNet models ($\sigma_{Q}$) is used
to represent the model variance as an approximation of how uncertain the
predictions are.
A rational design of battery cathodes typically focuses on the capacity that
can be delivered within a certain voltage window. Therefore, we used DRXNet to
compute the voltage profiles with electrochemical test parameters in the test
set and compared the delivered capacity between 2.0 – 4.4 V of experiments and
predictions within 50 cycles (see Figure 3). The voltage range of 2.0 – 4.4 V
(vs. Li+/Li) is reasonable for current electrolytes, and most commercialized
cathodes such as LiFePO4, LiCoO2, and NMC operate within this voltage range.
Our choice of this voltage range for testing model performance is aligned with
these industry norms. The average voltages ($\bar{V}=\sum_{i}V_{i}\Delta
Q_{i}/\sum_{i}\Delta Q_{i}$) between 2.0 – 4.4 V were subsequently computed.
As a baseline, the mean absolute deviation (MAD) of average voltage is
0.16/0.21 V for 3TM/HE, and the MAD of discharge capacity is 36.59/38.54 mAh/g
for 3TM/HE. Figure 3A and B demonstrate the performance of the DRXNet models
trained on the 2TM dataset and tested on the 3TM and HE datasets. Mean
absolute errors (MAEs) of 0.1/0.13 V for the average voltage and 23.38/29.97
mAh/g for the capacity were obtained for the 3TM/HE test datasets,
respectively. It is found that large prediction errors already occur for the
first cycle and propagate into the subsequent cycles. Notably, a systematic
underestimation of capacity is observed for the HE compounds (Figure 3B),
which can be rationalized by the fact that 2TM compounds cannot capture the
improved performance arising from the novel high-entropy physics [40, 43].
For practical applications, new data points can be continuously collected as
experiments progress, enabling on-the-fly training with incoming data to
improve predictive performance. To evaluate possible improvement with
additional information specific to the system being tested, we evaluated the
improvement when DRXNet is trained on a dataset containing all 2TM data and is
provided with the first cycle data from 3TM/HE materials. The knowledge of
just the first cycle data results in a reduction of the mean capacity error
from 23.38/29.97 mAh/g to 14.84/17.58 mAh/g for 3TM/HE (Figure 3C and D). The
enhanced performance achieved by explicitly training with the first cycle
indicates that the model can better generalize cycling performance, even when
experiments for a specific composition are not extensively sampled. This
capability has the potential to significantly reduce the month-long timeframe
typically required for electrochemical testing to identify whether a new
cathode material has a desired cyclability or rate capability. Training the
model with first-cycle data led to a substantial decrease in both the
prediction error and model variance for the initial few cycles, although the
model variance increased subsequently with the cycle number for untrained
domains (Figures 3C and D).
To examine how data augmentation could improve the performance of DRXNet, we
further trained models on the 2+3TM dataset where chemical information in
addition to 2TM interactions is included. Figure 3E and F display the
predictions on the 3TM (MAE: 6.0 mAh/g) and HE (MAE: 19.63 mAh/g) datasets. It
is important to note that the models trained on 2+3TM data show an error
reduction of around 10 mAh/g for the HE capacity prediction compared to the
results obtained when training the 2TM model (Figures 3B and F), along with a
significant reduction on the model variance. This finding suggests that the
2TM dataset is inadequate for extracting relevant information and generalizing
it to other compositions. The scaling to electrode material with a high number
of components necessitates capturing more than 2TM correlations or
interactions in training the graph neural network. Failure to do so may lead
to systematic prediction errors, as demonstrated in Figure 3B. When the model
is able to acquire sufficient chemical domain knowledge (e.g., 2+3TM-model),
it becomes feasible to extrapolate the electrochemical properties of high-
component electrodes, which is evidenced in Figure 3F with reduced prediction
error as well as model variance, and only a few outlier experiments exhibit
large errors.
Figure 4: Illustration of predictions of discharge capacity in Li–Mn–O–F DRX
systems: (A) Compositional design principle includes the optimization of Li-
excess content, TM-redox and Li-F short-range order (SRO) [39].
(B, C) Prediction of discharge capacity in the Li–Mn–O–F chemical space for
the 1st (B) and 30th (C) cycle between 1.5 – 4.8 V at a current density rate
of 20 mA/g. The blue stars indicate the compositions included in the training
set.
### II.5 Recover design principles in Li–Mn–O–F chemical space
We present several examples to illustrate how DRXNet learns the underlying
cathode chemistry and assists in designing new materials, where the models
used for these applications are pretrained on all discharge profiles. As an
attractive earth-abundant, non-precious TM, Mn is of considerable interest for
designing next-generation cathode materials [35]. Lun _et al._ [39] proposed
three primary design degrees of freedom for Mn-based DRX (Figure 4A): (1) the
Li-excess content, which controls the presence of a percolating network
facilitating Li diffusion; (2) the Mn content, as achieving high capacity with
a low amount of Mn requires a large amount of oxygen redox, leading to poor
cyclability; and (3) the fluorine content, which lowers the total cation
valence and provides greater freedom to optimize the Li and Mn content.
Fluorine modifies cation short-range order (SRO) through the strong Li–F
attraction and lowers the initial capacity [44, 32], but can increase
stability at high voltage charging [21]. These principles are highly
correlated and exert non-linear effects on performance.
We used DRXNet to predict the discharge capacity of DRX compounds between 1.5
and 4.8 V at a current rate of 20 mA/g for the 1st and 30th cycles. The
results, as a function of Li and F content, are shown in Figures 4B and C. The
Mn content and valence follow directly from the Li and F content. The effect
of fluorine on performance, extensively characterized experimentally, is well
captured by our model: A higher F content ($y$ in O2-yFy) results in a lower
discharge capacity for the 1st cycle but a higher capacity for the 30th cycle,
consistent with its documented role in promoting surface stability [21]. In
particular, cation-disordered Li1.333Mn0.667O2 (bottom right corner of Figure
4C) is predicted to have the highest capacity ($>320$ mAh/g) for the first
cycle but the lowest capacity for the 30th cycle. In this compound, the
valence of Mn is 4+, and all capacity originates from oxygen. Such a large
amount of O-redox leads to rapid capacity fade consistent with the
experimental observations on disordered Li2MnO3 reported in Ref. [45].
To provide some context for the extrapolation capability of DRXNet, we have
illustrated the compositions in the training dataset with blue stars in
Figures 4B and C. From this, it can be observed that even with a limited
distribution of training points on the composition map, DRXNet offers
reasonably consistent predictions that seem to be in line with the
experimental observations beyond the training points. As DRXNet is trained on
various compositions beyond the Li–Mn–O–F chemical space, the ability to
extrapolate to other domains can be attributed to the transfer learning from
other F- and non-F-containing compounds. The example in this section
demonstrates how practitioners can generalize the design principles from a
data-driven perspective purely starting from the data mined from experiments.
### II.6 Exploratory search for high-entropy cathodes
Figure 5: Predicted discharge voltage profiles of two high-entropy DRX
materials. (A) Li1.2Mn0.1Mg0.1Cr0.3Ti0.2Nb0.1O1.8F0.2 (HE-1) and (B)
Li1.2Mn0.1Mg0.1Cr0.15V0.15Ti0.2Nb0.1O1.8F0.2 (HE-2) with various current
densities (from 20 mA/g to 1000 mA/g) between voltage window of 1.5 –4.8 V.
The inset displays the cycled discharge capacity at a current density of 20
mA/g. HE-2 with various current densities (from 10 mA/g to 10 A/g) between
voltage windows of (C) 2.0 –4.4 V and (D) 2.0 – 4.0 V.
High-entropy DRXs are composed of many species and present a vast chemical
space to explore for battery materials discovery. We used DRXNet for virtual
high-throughput screening considering redox-compatible species from the
bivalent (Mn2+, Fe2+, Ni2+, Mg2+) and trivalent group (Mn3+, Cr3+, V3+, Fe3+).
Two case studies of predicted high-entropy DRXs are presented:
Li1.2Mn0.1Mg0.1Cr0.3Ti0.2Nb0.1O1.8F0.2 (HE-1) and
Li1.2Mn0.1Mg0.1Cr0.15V0.15Ti0.2Nb0.1O1.8F0.2 (HE-2). The discharge profiles
predicted with DRXNet under various current densities are shown in Figures 5A
and B. A more comprehensive collection of predictions for other compositions
is included in Figure S6.
For HE-1, DRXNet predicts a discharge capacity of 276 mAh/g at a current rate
of 20 mA/g. The compound delivers its largest discharge capacity near 3V and
transitions to a higher voltage slope below 3V, a phenomenon that has been
widely observed in Mn redox and/or Cr redox-based DRXs [46, 33, 40]. HE-1 is
predicted to have an unusually high rate capability for a DRX compound when
discharging. A capacity of 196 mAh/g is estimated at 1000 mA/g, which is
$71\%$ of the capacity at 20 mA/g. Previous work has demonstrated that multi-
elemental substitution (i.e., high-entropy strategy) frustrates the
unfavorable short-range order that leads to poor Li kinetics. In addition, the
incorporation of Cr and its migration as Cr6+ at high voltage creates a more
extended 0-TM network for Li transport. Both of these features improve the Li
diffusion kinetics [33, 40]. DRXNet clearly learns those benefits and
extrapolates rationally into electrochemistry prediction of the high-entropy
compositions.
As a comparison to HE-1, we formulated HE-2 with partial V3+ to Cr3+
substitution. The change in the shape of the voltage profile due to the low
potential of V5+/V3+ reduction is well captured by DRXNet as shown in Figure
5B and $dQ/dV$ curves in SI. It is clearly demonstrated that with V3+
incorporation, a nearly constant slope can be observed down to the low-voltage
region, which is characteristic for reported V-based DRX cathodes [47, 48].
Nevertheless, similar to Cr6+, V5+ can migrate into the tetrahedral sites to
enhance Li transport, which benefits the rate capability [47]. Consistently,
with this concept, HE-2 is predicted to retrain 171 mAh/g capacity at 1000
mA/g ($64\%$ of the 266 mAh/g capacity at 20 mA/g), which is superior to the
majority of the DRX cathodes reported to date.
The inset plots in Figures 5A and B show the predicted discharge capacity of
HE-1 and HE-2 for 20 cycles. The capacity drop in the first five cycles is
predicted to slow down upon further cycling. This result is in full agreement
with experimental findings, which indicate some of the irreversibility in the
initial cycles, such as cathode–electrolyte interface formation [49]. These
examples illustrate how practitioners can effectively use DRXNet to navigate
the extensive chemical space of high-entropy DRXs and identify promising
candidates for cathode design and optimization.
### II.7 Electrochemical conditions
We further tested the depth and transferability of DRXNet’s predictive
capabilities by varying the HE-2 discharge voltage window and cycling rate,
which are typical parameters varied in the investigation of a new cathode
material. Figure 5C displays the discharge profiles between 2.0 – 4.4 V, with
two additional rates tested (10 mA/g for a low rate and 104 mA/g for an
extremely high rate). These conditions are infrequently incorporated into our
training data. The 10 mA/g exhibits a discharge profile very similar to that
obtained at 20 mA/g, which is entirely consistent with typical experimental
findings, as the discharge process at such a low rate exhibits a reduced
overpotential and is closer to the equilibrium. The 10 A/g rate discharge
profile demonstrates a sharp drop in voltage, reasonably indicating poor
performance at this extremely high rate. Some unphysical predictions start to
appear when the model is tested to predict the discharge profiles between 2.0
– 4.0 V. As Figure 5D shows a small non-zero offset $\sim 6$ mAh/g for the 20
mA/g rate profile appears at the onset of discharge (4.0 V). Since the start
of the capacity curve at the upper level of the voltage cutoff is not formally
enforced to zero by the model, but emerges from the linear embeddings of the
voltage state $V_{i}$ with the voltage window
$[V_{\text{low}},V_{\text{high}}]$, an offset can be created when there is not
enough data for that specific voltage window.
Based on the tests, our primary conclusion is that DRXNet exhibits a
reasonable ability to learn the cathode material’s chemical information in the
latent space and generalize to test conditions that are included in the
dataset. However, for test conditions that the model has not or rarely
encountered (e.g., experiments with $V_{\text{high}}<4.0$ V), discrepancies or
unphysical profiles may still arise. This highlights the data scarcity issue,
which is typical for human-generated experimental conditions, which are biased
toward what is needed to demonstrate performance rather than what is optimal
for model training [50].
## III Discussion
Most machine learning approaches predicting battery performance have been
focused on predictions for a specific chemistry or limited chemical space of
commercialized cathodes, typically, the remaining useful life forecasted from
the initial cycles [11, 51, 52]. However, the nature of battery cathode
material discovery and optimization lies in a broad domain of chemistries,
which is more challenging for AI as it needs to capture the direct (e.g.,
voltage) and indirect effect (e.g., cycle life) of chemical changes [18].
Recent studies have demonstrated the feasibility of building universal models
for atomistic modeling by harnessing more than ten years of ab-initio
calculations spanning the periodic table [53, 54, 10, 55, 56]. It becomes a
logical extension to envision universal models for the experimental discovery
of battery materials by leveraging the wealth of both ab-initio calculation
and experimental data generated on cathode materials worldwide [17, 57]. In
this work, we propose an end-to-end training pipeline to encode and learn the
(electro)chemical information of cathode materials from voltage profiles.
Focused on DRX cathodes, we data-mined years of lab-generated experimental
discharge voltage profiles and trained a universal machine-learning model
(DRXNet) to make predictions across diverse compositions. This was achieved
through a novel model design consisting of an electrochemical condition
network $\mathcal{O}$ and a state prediction network $\mathcal{F}$.
The design of the two networks promotes modularity in the architecture,
streamlining the optimization and interpretation of each network individually
and their learned features. For instance, composition is an intrinsic property
of the synthesized cathode materials, and the encoding of such features is
independent of other factors such as current density and cycle status,
rationalizing our approach to first extract the composition-only feature
$\vec{X}_{\text{comp}}$ via a GNN. Although it remains a challenge that the
composition may change as a function of current density and cycle status due
to TM dissolution and the irreversible reaction of lithium outside the
cathode, DRXNet encompasses these factors into the rate- and cycle-informed
feature vector representations. By leveraging a ResNet-inspired architecture
using skip connections [58], we achieve a more effective synthesis of the
feature vector within the latent space. This design allows for a direct
connection between the rate-informed feature, $\vec{X}_{\mathcal{O}_{1}}$, and
the prediction of the first cycle capacity. Such architecture has been proven
to boost model training and alleviate the well-known gradient vanishing
issues.
Given the inherent sequential nature of battery testing data – where
possessing information from the $N$-th cycle implies the availability of data
from the first cycle – it becomes crucial to design features that reflect this
causality. This insight leads to the formulation of the cycle-informed
feature, $\vec{X}_{\mathcal{O}_{N}}$. This feature accentuates the difference
between the first and the $N$-th cycles, guiding the prediction for the $N$-th
cycle capacity, as detailed in Eq.(6). Consequently, our loss function is
constructed for multi-task learning with both terms for the first and $N$-th
cycle capacities, ensuring the causal relationships in cycle-dependent
capacity predictions (refer to Eq.(12)). Through an ablation study on whether
to include the first cycle term, $\ell(Q^{1})$, in the loss function or not,
we found that the model without $\ell(Q^{1})$ tends to be underfitted (more
details in Figure S8). Our incorporation of loss terms for both the first and
$N$-th cycle capacities enhances the model expressibility, which is a crucial
factor in the optimization of battery materials.
In addition, the modular design of the electrochemical condition network
$(\mathcal{O})$ provides flexibility for the feature representation when
expanding the model to include other information. The training dataset, being
derived from our own experimental results, does not encompass testing
parameters such as particle size, electrolyte type, synthesis variations, etc.
Since the battery electrodes were fabricated in our laboratory using
standardized recipes and methodologies, these factors have been coarsely
integrated into the compositional model and are treated as constants across
our dataset. Currently, the model does not include features to capture
structural information (crystal structure, short-range order, etc.). In DRX
compounds, short-range order is known to influence performance and to the
extent that this is not a direct consequence of composition, but modified by
synthesis parameters its effects are not accounted for [59]. In principle,
researchers can choose to include such factors to design the electrochemical
feature vector, depending on the specific problem they are addressing. Given
the vast amount and complexity of these properties, a synthetic data
collection approach is necessary. Data-mining techniques, such as text mining
and figure mining, can automatically retrieve valuable experimental
information from decades of published literature [60, 61], though a challenge
with aggregating diverse data from literature is the numerous hidden and
unspecified variables relevant to materials synthesis and electrochemical
testing. Looking forward, automated labs can address both data scarcity and
transparency issues by enabling more extensive exploration of the experimental
space and even better collect data from ”failed” experiments [62, 63, 64, 65].
In conclusion, DRXNet represents a step forward in developing machine-learning
models for battery materials research. By continuously refining the model and
incorporating additional data and parameters, we anticipate that such a
machine-learning framework will play an increasingly critical role in
discovering and optimizing next-generation battery materials.
## IV Methods
### IV.1 Data collection
We collected coin-cell electrochemical test data from our lab starting in 2016
and converted them into a digital format (.json). Each .json file contains
information on one individual electrochemical test, including the electrode
composition, electrode mass (g), active mass (g), test current rate (mA/g),
low and high voltage value of the working window (V), and charge/discharge
profiles of $N_{\text{cycle}}$ collected cycles. The compositions used for the
model training were taken as the targeted composition in experiments. For the
fraction of our data set which was previously published, the composition
values were typically confirmed by Inductively Coupled Plasma (ICP) analysis.
For these compounds, the feature vectors of the targeted compositions and ICP-
analyzed composition exhibit $\geq 99.7\%$ in cosine similarity as shown in
Supplementary Information, which supports using the targeted composition for
the general prediction purpose. Nonetheless, minor variations between the
actual composition and the target composition can be a source of noise in the
data.
For the in-house battery tests, the CR2032 coin cells were assembled using
commercial 1 M LiPF6 in an ethylene carbonate and dimethyl carbonate solution
(volume ratio 1:1) as the electrolyte, glass microfiber filters (Whatman) as
separators, and Li-metal foil (FMC) as the anode. The coin cells were tested
on an Arbin battery cycler at room temperature. The cathode consisted of a
mixture of active material (DRX), Super C65 carbon black, and
polytetrafluoroethylene (PTFE). The capacity signal, collected in units of Ah
from the Arbin battery cycler, was normalized to mAh/g using the mass of the
active material (active mass). The data from the failed tests (e.g., Arbin
cycler breakdown, electrolyte failure, strong signal fluctuations, etc.) were
removed from the dataset (see Figure S1 for examples).
To enhance the generalization and expressibility of DRXNet, we expanded the
dataset by figure mining published voltage profiles in related systems not
covered by our lab tests (see Table S1), which was accomplished using the
WebPlotDigitizer [66]. We used the UnivariateSpline method to denoise and
resample the experimental profiles and compute the $dQ/dV$ curves. One hundred
points were uniformly sampled to form the voltage series
$\boldsymbol{V}=\left[V_{0},V_{1},...,V_{i},...\right]$ for each discharge
profile, and the capacity series and $dQ/dV$ series were calculated
accordingly from $\boldsymbol{V}$.
### IV.2 Model design
#### IV.2.1 Preliminaries
A linear layer with trainable weight $\boldsymbol{W}$ and bias
$\boldsymbol{b}$ is defined as
$L(\vec{X})=\vec{X}\boldsymbol{W}+\boldsymbol{b}.$ (2)
For simplicity of notion, each $L$ represents different trainable weights in
the following equations.
#### IV.2.2 Compositional encoding
For elemental information, each element is first embedded into a
200-dimensional vector using mat2vec [41]. The Roost (Representation Learning
from Stoichiometry) model is used for compositional encoding [26], which is a
graph neural network (GNN) with message passings as follows:
$\displaystyle\vec{h}_{i}^{t+1}$
$\displaystyle=\vec{h}_{i}^{t}+\sum_{j,m}a_{i,j}^{t,m}\cdot\sigma_{g}\circ
L_{c}\left(\vec{h}_{i}^{t}||\vec{h}_{j}^{t}\right),$ (3) $\displaystyle
a_{i,j}^{t,m}$
$\displaystyle=\frac{w_{j}\exp(e_{i,j}^{t,m})}{\sum_{k}w_{k}\exp(e_{i,k}^{t,m})},~{}e_{i,k}^{t,m}=\sigma_{g}\circ
L_{a}\left(\vec{h}_{i}^{t}||\vec{h}_{j}^{t}\right).$
In these equations, $\vec{h}_{i}^{t}$ represents the $t$-th hidden layer for
the $i$-th element; $||$ denotes the concatenation operation; and the soft-
attention coefficient $a_{i,j}^{t,m}$ describes the interaction between
elements $i$ and $j$, with $m$ as the index of multi-head attention. $L_{c}$
and $L_{a}$ denote the linear layer for the core and attention layer,
respectively. The fractional concentration $w_{j}$ of element $j$ depends on
the specific compound (e.g., $w_{j}=0.6/0.1/0.1/0.2$ for Li/Mn/Cr/Ti in
Li1.2Mn0.2Cr0.2Ti0.4O2.0). $\sigma_{g}$ is the SiLu activation function. After
$n$ graph convolution layers, the encoded composition vector
$\vec{X}_{\text{comp}}$ is obtained by average pooling over the elements with
weighted attention
$\vec{X}_{\text{comp}}=\text{Pooling}\left[\frac{w_{i}\exp\left(\sigma_{g}\circ
L_{a}(\vec{h}_{i}^{n})\right)}{\sum_{k}\exp\left(\sigma_{g}\circ
L_{a}(\vec{h}_{i}^{n})\right)}\cdot\left(\sigma_{g}\circ
L_{c}(\vec{h}_{i}^{n})\right)\right]$ (4)
#### IV.2.3 Electrochemical condition encoding
The electrochemical test primarily involves two pieces of information: the
current density rate and cycle number. We use MLPs to encode the rate and
cycle number:
$\vec{X}_{\text{rate}}=\sigma_{g}\circ
L(\text{rate}),~{}\vec{X}_{\text{cycle}}=\sigma_{g}\circ L(\text{cycle}).$ (5)
As the actual rate and cycle performance are strongly correlated with cathode
materials, the relationship between the composition, rate, and cycle is
synthesized using gated-MLPs with soft attention[25]:
$\displaystyle\vec{X}_{\mathcal{O}_{1}}$
$\displaystyle=\vec{X}_{\text{comp}}+\sigma_{f_{1}}(\vec{X}_{\text{comp}}||\vec{X}_{\text{rate}})\cdot
f_{1}(\vec{X}_{\text{comp}}||\vec{X}_{\text{rate}})$ (6)
$\displaystyle\vec{X}_{\mathcal{O}_{N}}$
$\displaystyle=\vec{X}_{\mathcal{O}_{1}}+\sigma_{f_{2}}(\vec{X}_{\mathcal{O}_{1}}||\vec{X}_{\text{cycle}})\cdot
f_{2}(\vec{X}_{\mathcal{O}_{1}}||\vec{X}_{\text{cycle}})$
$\displaystyle\quad\cdot\boldsymbol{W}_{n}(N-1)$
where $\sigma_{f}=\sigma_{s}\circ B\circ L$ is an MLP, $\sigma_{s}$ is the
Sigmoid activation function, and $f=\sigma_{g}\circ B\circ L$ is an MLP with
SiLu activation function $\sigma_{g}$. The BatchNormalization layer $B$ is
added before the activation function. In this equation,
$\vec{X}_{\mathcal{O}_{1}}$ is a feature vector jointly determined by the
composition and rate information, which is used to predict the first cycle
property. $\vec{X}_{\mathcal{O}_{N}}$ is a feature vector jointly determined
by the composition, rate, and cycle information, which is used to predict the
$N$-th cycle property. The difference between $\vec{X}_{\mathcal{O}_{1}}$ and
$\vec{X}_{\mathcal{O}_{N}}$ is linearly dependent on the number of cycles with
a trainable weight $\boldsymbol{W}_{n}$, allowing the model to learn cycle
performance contrastively.
#### IV.2.4 State prediction network
The state prediction network ($\mathcal{F}$) takes the inputs of voltage state
($V_{i}$) and outputs the discharge-capacity state ($Q_{i}$)
$Q_{i}=\mathcal{F}\left(V_{i}|\mathcal{O}\right).$ (7)
In practice, the voltage profile is measured within the applied voltage window
[$V_{\text{low}},V_{\text{high}}$]. To accommodate the voltage window in the
discharge state prediction, the first layer in $\mathcal{F}$ is encoded via an
MLP:
$\displaystyle\vec{Z}_{i}^{0}$
$\displaystyle=L\circ\sigma_{\mathcal{F}}\circ\left[L(V_{\text{low}},V_{\text{high}})+L(V_{i})\right],$
(8)
where $\sigma_{\mathcal{F}}(\cdot)$ is the $\mathtt{Softplus}$ activation
function. The test-condition information is element-wise added to the state
prediction network [58]
$\vec{Z}_{i}^{N}=\sigma_{\mathcal{F}}\circ
L\left(\vec{Z}_{i}^{0}+\vec{X}_{\mathcal{O}_{N}}\right)$ (9)
The state of capacity is obtained by
$Q^{N}_{i}=\sigma_{\mathcal{F}}\circ L\circ\sigma_{\mathcal{F}}\circ
L(\vec{Z}_{i}^{N})$ (10)
where $Q^{N}_{i}$ is the capacity for the $N$-th cycle (including the first
cycle). Because the discharge capacity is always positive,
$\sigma_{\mathcal{F}}$ is added to constrain the predicted capacity to be
positive and accelerate the training process. $dQ/dV$ for the redox potential
can be obtained via PyTorch auto-differentiation [67]
$\left.\frac{dQ}{dV}\right|_{i}=\text{AutoDiff}(Q_{i},V_{i}).$ (11)
### IV.3 Model training
The model is trained to minimize the sum of multi-task losses for the capacity
of the first cycle, the $n$-th cycle, and $dQ/dV$:
$\mathcal{L}=w_{Q}\ell(Q_{i}^{N})+w_{dQ}\ell(\frac{dQ^{N}}{dV_{i}})+w_{Q_{1}}\ell(Q_{i}^{1})+\mathcal{R}.$
(12)
A MSE loss function is used for $\ell(Q_{i}^{N})$ and
$\ell(\frac{dQ^{N}}{dV_{i}})$, whereas a MAE loss function is employed for the
first cycle as a contrastive term $\ell(Q_{i}^{1})$. The weights for
$Q_{i}^{N}$, $dQ/dV$, and $Q_{i}^{1}$ are set to $w_{Q}$ = 1, $w_{dQ}$ = 1,
and $w_{Q_{1}}$ = 5. The term $\mathcal{R}$ represents regularization, which
consists of two parts: (1) an $\ell_{2}$-norm regularization of the network’s
parameters $||\boldsymbol{\theta}||_{2}$ and (2) a smoothing term
$||dQ/d\textbf{c}||_{2}$ to avoid large, unphysical performance fluctuations
(c denotes the fractional concentration of elements). The weight of
regularization is $10^{-4}$.
To make predictions, an ensemble of five independent models was trained to
make predictions. Each model was trained with a batch size of 1024 within 30
epochs. The Adam optimizer was used with $10^{-3}$ as the initial learning
rate. The ExponentialLR scheduler was used to adjust the learning rate with a
decay of 0.9 per epoch.
## V Acknowledgments
This work was primarily supported by the U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences, Materials Sciences and Engineering
Division under Contract No. DE-AC0205CH11231 (Materials Project program
KC23MP). The data collection in this work was supported by the Assistant
Secretary for Energy Efficiency and Renewable Energy, Vehicle Technologies
Office, under the Advanced Battery Materials Research (BMR) Program of the US
Department of Energy (DOE) under contract No. DE-AC0205CH11231. The
computational modeling in this work was supported by the computational
resources provided by the Extreme Science and Engineering Discovery
Environment (XSEDE), supported by National Science Foundation grant number
ACI1053575; the National Energy Research Scientific Computing Center (NERSC);
and the Lawrencium computational cluster resource provided by the IT Division
at the Lawrence Berkeley National Laboratory. The authors thank Huiwen Ji,
Jianping Huang, and Zijian Cai for their help in experimental data collection,
and Yifan Chen for valuable discussions.
## VI Availability
The codes of DRXNet are open-sourced at https://github.com/zhongpc/DRXNet and
https://doi.org/10.5281/zenodo.10719829. The open-source dataset is available
at https://doi.org/10.6084/m9.figshare.25328578.v1 for public access, which
contains 12,688 experimental discharge voltage profiles excluding the Mn-rich
and Ti-based DRX. The open-source dataset is not identical to, but rather a
part of, the DRX-TD that was used for the pretrained models in the paper.
## VII Supplementary Information
Supplemental information can be found online at
https://doi.org/10.1016/j.joule.2024.03.010.
## VIII Author Contributions
P.Z. and G.C. conceived the initial idea. P.Z. collected the dataset and
developed the code base with help from B.D. and T.H.. Z.L. and G.C. offered
insight into the project. P.Z. and G.C. prepared the manuscript. All authors
contributed to discussions and approved the manuscript.
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|
# PAIRING SYMMETRY AND PAIRING STATE IN FERROPNICTIDES: THEORETICAL OVERVIEW
I.I. Mazina and J. Schmalianb Code 6391, Naval Research Laboratory,
Washington, DC 20375 Iowa State University and Ames Laboratory, Ames, IA,
50011
(February 17, 2009)
###### Abstract
We review the main ingredients for an unconventional pairing state in the
ferropnictides, with particular emphasis on interband pairing due to magnetic
fluctuations. Summarizing the key experimental prerequisites for such pairing,
the electronic structure and nature of magnetic excitations, we discuss the
properties of the $s^{\pm}$ state that emerges as a likely candidate pairing
state for these materials and survey experimental evidence in favor of and
against this novel state of matter.
One fist of iron, the other of steel
If the right one don’t get you, then the left one will
Merle Travis, 16 tons
## 1 Introduction
The discovery of cuprate superconductors has changed our mentality in many
ways. In particular, the question that would have sounded moot to most before
1988, what is the symmetry of the superconducting state, is now the first
question to be asked when a new superconductor has been discovered. The pool
of potential candidates, before considered at best a mental Tetris for
theorists, had acquired a practical meaning. It has been demonstrated that
superconductivity in cuprates is $d$-wave, while in MgB2 it is multi-gap
$s$-wave with a large gap disparity. There is considerable evidence that
Sr2RuO4 is a $p$-wave material. Other complex order parameters are routinely
discussed for heavy fermion systems or organic charge transfer salts. It is
likely that the newly discovered ferropnictides represent another
superconducting state, not encountered in experiment before.
Besides the general appreciation that pairing states may be rather nontrivial,
it has also been recognized that unconventional pairing is likely due, at
least to some extent, to electronic (Coulomb or magnetic) mechanisms and,
conversely, electronic mechanisms are much more likely to produce
unconventional pairing symmetries than the standard uniform-gap $s$-wave. It
has been appreciated that the actual symmetry is very sensitive to the
momentum dependence of the pairing interaction, as well as to the underlying
electronic structure (mostly, fermiology).
Therefore we have structured this overview so that it starts with a layout of
prerequisites for a meaningful discussion of the pairing symmetry. First of
all, we shall describe the gross features of the fermiology according to
density-functional (DFT) calculations, as well as briefly assess verification
of such calculations via ARPES and quantum oscillations experiments. Again,
detailed discussion of these can be found elsewhere in this volume. We will
also point out where one may expect caveats in using the DFT band structure:
it is in our view misleading to assume that these compounds are uncorrelated.
While not necessarily of the same nature as in cuprates, considerable
electron-electron interaction effects cannot be excluded and are even
expected.
We will then proceed to discuss the role of magnetic fluctuations as well as
other excitations due to electron-electron interactions. We discuss the
special role the antiferromagnetic (AFM) ordering vector plays for the pairing
symmetry and address the on-site Coulomb (Hubbard correlations), to the extent
of their possible effect on the pairing symmetry, and possible overscreeining
(Ginzburg-Little) interactions. We also discuss puzzling issues that are
related to the magnetoelastic interaction in these systems. As for a
discussion of the electron-phonon interaction we refer to the article by Boeri
et al in this volume. The final part of this review consists of a summary of
theoretical aspects of the pairing state, along with a discussion of its
experimental manifestations.
## 2 Prerequisites for addressing the Cooper pairing
### 2.1 Electronic structure and fermiology
#### 2.1.1 Density functional calculations
The two families of the Fe-based superconductors are $1111$ systems ROFeAs
with rare earth ions R[1, 2] and the $122$ systems AFe2As2 with alkaline earth
element A[3]. Both families have been studied in much detail by first
principles DFT calculations. Here and below, unless specifically indicated, we
use a 2D unit cell with two Fe per cell, and the corresponding reciprocal
lattice cell; the $x$ and $y$ directions are along the next-nearest-neighbor
Fe-Fe bond. It appears that all materials share the same common motif: two or
more hole-like Fermi surfaces near the $\Gamma$ point [$\mathbf{k=}(0,0)$],
and two electron-like surfaces near the M point [$\mathbf{k=}(\pi,\pi)$] (Fig.
1-5). This is true, however, in strictly non-magnetic calculations only, when
the magnetic moment on each Fe is restricted to zero. As discussed below, this
is not necessarily a correct picture.
Figure 1: (color online) The Fermi surface of the non-magnetic LaAsFeO for 10%
e-doping [4] Figure 2: (color online) The Fermi surface of the non-magnetic
BaFe2As2 for 10% e-doping (Co doping, virtual crysatl approximation)[4]
If, however, we neglect this potential caveat, and concentrate on the two best
studied systems, 1111 and 122, the following relevant characteristics can be
pointed out: First, the density of states (DOS) for holes and electrons is
comparable for undoped materials; with doping, respectively one or the other
becomes dominant. For instance, for Ba0.6K0.4Fe2As2 the calculated DOS (in the
experimental structure) for the three hole bands varies between $1.1$
st/eV/f.u. and $1.3$ st/eV/f.u., the inner cylinder having, naturally, the
smallest DOS and the outer the largest. For the electron bands the total DOS
is $1.2$ st/eV/f.u., that is, two to three times smaller than the total for
the hole bands[4]. We shall see later that this is important. Another
interesting effect is that in the 122 family doping in either direction
strongly reduces the dimensionality compared to undoped compounds (in the 1111
family this effect exists, but is much less pronounced), see Fig. 4. This
suggests that the reason that doping destroys the long-range magnetic order
(it is believed by many that such a destruction is prerequisite for
superconductivity in ferropnictides) is not primarily due to the change in the
2D electronic structure, as it was initially anticipated[5], but rather due to
the destruction of magnetic coupling between the layers. Indeed the most
striking difference between the undoped 1111 and undoped 122 electronic
structure is quasi two-dimensionality of the former and a more 3D character of
the latter (the difference is clear already in the paramagnetic calculations,
but is particularly drastic in the antiferromagnetic state), while at the same
time the observed magnetism in the 122 family is at least three times stronger
than in LaFeAsO (in the mean-field DFT calculation the difference is quite
small).
Figure 3: (color online) The Fermi surface of the non-magnetic BaFe2As2 for
10% h-doping (20% Cs doping, virtual crysatl approximation.[4]
The fact that the nesting is very imperfect is crucial from the point of view
of an SDW instability, making the material stable against infinitesimally
small magnetic perturbation. For superconductivity, however, it is less
important, as discussed later in the paper.
Figure 4: (color online) The Fermi surface of BaFe2As2 for 20% h-doping
(corresponding to Ba1.6K0.4Fe2As2, calculated as 40% Cs doping in the virtual
crystal approximation) [4]. Figure 5: (color online) The Fermi surface of
undoped nonmagnetic FeTe. [4]
#### 2.1.2 Experimental evidence
Experimental evidence regarding the band structure and fermiology of these
materials comes, basically, from two sources: Angular resolved photoemission
spectroscopy (ARPES) and quantum oscillations measurements. The former has an
additional advantage of being capable of probing the electronic structure in
the superconducting state, assessing the amplitude and angular variation of
the superconducting gap. A potential disadvantage is that it is a surface
probe, and pnictides, especially the 122 family, are much more three-
dimensional than cuprates. This means that, first, the in-plane bands as
measured by ARPES, strongly depend on the normal momentum, $k_{\perp},$ and,
second, there is a bigger danger of surface effects in the electronic
structure than in the cuprates. There are indications that the at least in
1111 compounds the surface is charged, that is to say, the doping level in the
bulk is different from that on the surface. Additionally, LDA calculations
suggest that in the magnetic prototypes, the band structure depends
substantially on interlayer magnetic ordering, again, not surprisingly, mostly
in the 122 compounds, as Fig.6 illustrates. Of course, there is no guarantee
that the last two layers order in the same way as the bulk (or even with the
same moment).
Figure 6: (color online) Band structure of the orthorhombic antiferromagnetic
BaFe2As2 calculated for two different interlayer ordering pattern: the
experimental antiferromagnetic one (space group #66, broken green) and the
hypothetical ferromagnetic (still antiferromagnetic in plane, space group #67,
solid red). In both cases the magnetic moment on Fe was artifically suppressed
to 1 $\mu_{B}$ by aplying a fictitious negative Hubbard U [4]. The point N is
above the point Y.
These caveats notwithstanding, ARPES has already provided invaluable
information. ARPES measurements have been performed for both 1111[6, 7] and
122 materials[8, 9, 10, 11]. These measurements demonstrated the existence of
a well-defined Fermi surface that consists of hole and electron pockets, in
qualitative agreement with the predictions of electronic structure
calculations. Thus, one can say that the topology of the Fermi surface,
including the location and the relative size of the individual Fermi surface
sheets agrees with the LDA expectation — which is most important for the
pairing models. Similarly, it is rather clear that the ARPES bandwidth is
reduced from the LDA one by a factor of 2–2.5, similar to materials with
strong itinerant magnetic fluctuations (cf., for instance, Sr2RuO4 near a
magnetic quantum critical point[12]). These findings are also consistent with
the deduced normal state linear specific heat coefficient in 1111 materials
(e.g., $4-6$ mJ/mol K2 in Ref. [13]) corresponding to a factor 1–2 compared to
the bare LDA value[14]. However, in the 122 compound a specific heat
coefficient 63 mJ/mol K2 was reported[13], to be compared with roughly
11.5mJ/mol K2 from the LDA calculations[4]. While a renormalization of 5.5 is
not consistent with either ARPES or quantum oscillations, consistency among
different experimental publications for the 122 systems is lacking as well
[15, 13].
Another experimental probe of the electronic structure is based on quantum
oscillations that measure extremal cross-section areas of the FS (ideally, for
different directions of the applied field) and the effective masses. Such
measurements are very sensitive to the sample quality, therefore so far only a
handful of results are available. However, data on the P-based 1111 compound
agree reasonably well with band structure calculations[16], and indicate the
same mass renormalization as ARPES[17]
Importantly, quantum oscillations measurements on AFM 122 compounds[18, 19]
indicate that even the undoped pnictides are well defined Fermi liquids, even
though a significant portion of the Fermi surface disappears due to the
opening of a magnetic gap. The frequencies of the magneto-oscillations then
suggest that the ordered magnetic state has small Fermi surface pockets
consistent with the formation of a spin-density wave. Thus, the electronic
structure of the pnictides is consistent with a metallic state with well
defined Fermi surfaces.
Besides determining the overall shape of the Fermi surface sheets, ARPES is
able to yield crucial information about the momentum dependence of the
superconducting gap. Several groups performed high quality ARPES measurements
of this effect[7, 8, 9, 10]. In some cases significant differences in the size
of the gap amplitude for different Fermi surface sheets have been observed.
However, there seems to be a consensus between all ARPES groups that the gap
amplitude on an individual Fermi surface sheet depends weakly on the
direction. While this seems to favor a pairing state without nodes, one has to
keep in mind that all measurements so far have been done for fixed values of
the momentum $k_{\perp}$, perpendicular to the planes. While it might be
premature to place too much emphesis on the relative magnitude of the gaps
observed in different bands in ARPES experiments, it is worth noting that most
experimentalists agree that in the hole-doped 122 material the inner hole
barrel and the electron barrel have comparable (and large) superconducting
gaps, while the outer hole barrel has about twice smaller gap. On the other
hand, there are first data[20] indicating that in the electron doped
BaFe1.85Co0.15As2 the hole and the electron bands have about the same gap
despite the hole pockets shrinking, and electron pocket extending. Even more
interesting, the most natural interpretation of the measured fermiology is
that the hole FS in BaFe1.85Co0.15As2 actually corresponds to the outer
($xz/yz)$ barrel in Ba0.6K0.4Fe2As2 that has a small gap in that compound.
#### 2.1.3 Role of spin fluctuations in electronic structure
As is clear from the above discussion, strong spin fluctuations have a
substantial effect upon the band structure. First of all, they dress one-
electron excitations providing mass renormalization, offering an explanation
for the factor 2–2.5. This is in fact a relatively modest renormalization: it
is believed that, for instance, in He3 or in Sr2RuO4 itinerant spin
fluctuations provide renormalization of a factor of 4 or larger. However, it
is likely that the effect goes beyond simple mass renormalziation. As will be
discussed in detail below, there is overwhelming evidence of large local
moments on Fe, mostly from the fact that the Fe-As bond length corresponds to
a fully magnetic (large) Fe ion. There is also evidence that the in-plane
moments are rather well correlated in the planes, and the apparent loss of the
long-range ordering above $T_{N}$ is mainly due to a loss of 3D coherency
between the planes[21]. It is only natural to expect a similar situation to be
true when magnetism is suppressed by doping.
If that is the case, the electronic structure in the paramagnetic parts of the
phase diagram, at least in the vicinity of the transition, should not be
viewed as dressed nonmagnetic band, but rather as an average between the bands
corresponding to various magnetic 3D stackings (cf. Fig. 6). Fig. 6,
corresponding to the $T=0$ magnetic moment of 1 $\mu_{B},$ is probably
exaggerating this effect, but it is still likely that in a considerable range
of temperatures and doping near the observed magnetic phase boundary a
nonmagnetic band structure is not a good starting point, and a theory based on
magnetic precursors is needed. More experiments, particularly using diffuse
scattering, and more theoretical work are needed to clarify the issue. A
discussion to this effect may be found in Ref. [22]. See also Section 2.3
below.
### 2.2 Magnetic excitations
#### 2.2.1 Experimental evidence
Compared to cuprates and other similar compounds, two peculiarities strike the
eye. First, the parent compounds of the pnictide superconductors assume an
antiferromagnetic structure, where neighboring Fe moments are parallel along
one direction withinin the FeAs plane and antiparallel along the other.
Neutron scattering data yield ordered moments per Fe of $0.35\mu_{B}$ for
LaFeAsO[23], $0.25\mu_{B}$ for NdFeAsO[24], $0.8\mu_{B}$ for CeFeAsO[25], and
$0.9$ $\mu_{B}$ for BaFe2As2[26]. Intriguingly, in NdFeAsO the ordered moment
at very low temperatures increases by a factor of 3 to 4 at the temperature
corresponding to the ordering of Nd-spins[27]. Note that the correct magnetic
structure has been theoretically predicted by DFT calculations[5, 28], which,
moreover, consistently overestimated the tendency to magnetism (as opposed to
the cuprates). Second, the magnetically ordered state remains metallic. As
opposed to cuprates or other transition metal oxides, the undoped systems
exhibit a small but well established Drude conductivity[29], display magneto-
oscillations[18] and have Fermi surface sheets of a partially gapped metallic
antiferromagnetic state[30]. Above the magnetic ordering temperature a sizable
Drude weight, not untypical for an almost semimetal has been observed.
Further, the ordered Fe magnetic moment in the 1111 systems depends
sensitively on the rare earth ion, very different from YBa2Cu3O6 where yttrium
can be substituted by various rare earth elements with hardly any effect on
the Cu moment. Note that the rare earth sites project onto the centers of the
Fe plaquettes and thus do not exchange-couple with the latter by symmetry.
Finally, the magnetic susceptibility of BaFe2As2 single crystals[31] above the
magnetic transition shows no sign for an uncoupled local moment behavior.
#### 2.2.2 Itinerant versus local magnetism
The vicinity of superconductivity to a magnetically ordered state is the key
motivation to consider pairing mechanisms in the doped systems that are linked
to magnetic degrees of freedom. Similar to cuprate superconductors, proposals
for magnetic pairing range from quantum spin fluctuations of localized
magnetic moments to fluctuations of paramagnons as expected in itinerant
electron systems. To judge whether the magnetism of the parent compounds is
localized or itinerant (or located in the crossover regime between these two
extremes) is therefore crucial for the development of the correct description
of magnetic excitations and possibly the pairing interactions in the doped
systems.
In our view the case at hand is different from such extreme cases as undoped
cuprate on one end and weak itinerant magnets like ZrZn2 on the other. While
being metals with partially gapped Fermi surface, there is evidence that Fe
ions are in a strongly magnetic states with strong Hund rule coupling for Fe.
This results in a large magnetic moment — but only for some particular
ordering patterns (for comparison, in FeO and similar materials LDA produce
large magnetic moment regardless of the imposed long range order). While it is
obvious that ferropnictides are not Mott insulators with localized spins,
interacting solely with near neighbors, a noninteracting electron system may
be not a perfect starting approximation either. To make progress we have to
decide what is the lesser of two evils and use it, even realizing the problems
with the selected approach. Given the above mentioned experimental facts, our
preference is that these systems are still on the itinerant side.
A feature that has attracted much interest is the quasi-nesting between the
electron and the hole pockets. The word “quasi” is instrumental here: even the
arguably most nested undoped LaFeAsO is very far from the ideal nesting and
even worse in the (more magnetic) BaFe2As${}_{2}.$ Indeed, it has been
observed that in the LDA calculations the nonmagnetic structure in either
compound is stable with respect to an infinitesimally small AFM perturbations,
but strongly unstable with respect to finite amplitude perturbations. This can
be understood from the point of view of the Stoner theory, applied to a finite
wave vector Q: the renormalized static spin susceptibility (in the DFT the RPA
approximation is formally exact) can be written as
$\chi_{LDA}(\mathbf{Q)}=\frac{\chi_{0}(\mathbf{Q})}{1-I\chi_{0}(\mathbf{Q})},$
(1)
where $I$ is the Stoner factor of iron, measuring the intra-atomic Hund
interaction (in the DFT, it is defined by the second variation of the
exchange-correlation functional with respect to the spin density). While the
denominator in Eq. 1 provides a strong enhancement of $\chi$, albeit not
exactly at $\mathbf{Q=}(\pi,\pi)$, but at a range of the wave vectors near
$\mathbf{Q}$), it does not by itself generate an instability. One can say that
an infinitesimally weak magnetization can only open a gap over a very small
fraction of the Fermi surface. However, a large-amplitude spin density wave
opens a gap of the order of the exchange splitting, $IM$, where $M$ is the
magnetic moment on iron, and, obviously, affects most of the conducting
electrons. In other words, the magnetism itself is generated by the strong
Hund rule coupling on Fe (just as in the metal iron), but the topology of the
Fermi surface helps select the right ordering pattern. Formation of the
magnetic moments is local; arranging them into a particular pattern is
itinerant.
There are several corollaries of this fact that are important for pairing and
superconductivity. First, despite the fact that the overall physics of these
materials is more on the itinerant side than on the localized side (see a
discussion to this effect later in the paper), it is more appropriate to
consider magnetic moments on Fe as local rather than itinerant (as for
instance in the classical spin-Peierls theory). Note that the same is true for
the metal iron as well. Second, the interaction among these moments is not
local, as for instance in superexchange systems (it appears impossible to map
the energetics of the DFT calculations onto a two nearest neighbor Heisenberg
model[32]). The AFM vector is not determined by local interactions in real
space (as for instance in the $J_{1}+J_{2}$ models, see below), but by the
underlying electronic structure in reciprocal space. Third, since the energy
gain due to formation of the SDW mainly occurs at finite (and large, $IM$ is
on the order of eV) energies, looking solely at the FS may be misleading.
Indeed, FeTe is one compound where the Fe moments apparently do not order into
a $\mathbf{Q=}(\pi,\pi)$ SDW, but in a more complex structure corresponding to
a different ordering vector[33], despite the fact that the FS shows about the
same degree of nesting (Fig.5) as LaFeAsO and a noticeably better nesting than
BaFe2As${}_{2}.$ DFT calculations correctly identify the ground state in all
these cases, and the origin can be traced down again to the opening of a
partial gap: in both 1111 and 122 compounds the $\mathbf{Q=}(\pi,\pi)$ is
about the only pattern that opens such a gap around the Fermi level, while in
FeTe comparable pseudogaps open in both magnetic structures (and the
calculated energies are very close, the actual experimental structure being
slightly lower[34]).
#### 2.2.3 Perturbative itinerant approach
Even if one accepts the point of view that the magnetism in the Fe-pnictides
is predominantly itinerant, the development of an adequate theory for the
magnetic fluctuation spectrum is still highly nontrivial. As pointed out
above, there are strong arguments that the driving force for magnetism is not
Fermi surface nesting but rather a significant local Hund’s and exchange
coupling. This can be quantitatively described in terms of a multiband Hubbard
type interaction of the Fe-$3d$ states
$\displaystyle H_{int}$
$\displaystyle=U\sum_{i,a}n_{ia\uparrow}n_{ia\downarrow}+U^{\prime}\sum_{i,a>b}n_{ia}n_{ib}$
$\displaystyle-
J_{H}\sum_{i,a>b}\left(2\mathbf{s}_{ia}\cdot\mathbf{s}_{ib}+\frac{1}{2}n_{ia}n_{ib}\right)$
$\displaystyle+J\sum_{i,a>b,\sigma}d_{ia\sigma}^{\dagger}d_{ia\overline{\sigma}}^{\dagger}d_{ib\overline{\sigma}}d_{ib\sigma},$
(2)
with intra- and inter-orbital Coulomb interaction $U$ and $U^{\prime}$, Hund’s
coupling $J_{H}$ and exchange coupling $J$, respectively. Here $a$, $b$ refer
to the orbitals in a Wannier type orbital at site $i$. $X$-ray absorption
spectroscopy measurements support large values for the Hund’s couplings that
lead to a preferred high spin configuration,[35] leading to larger values of
$J_{H}$. The importance of the Hund coupling for the normal state behavior of
the pnictides was recently stressed in Ref.[36].
Weak coupling expansions in these interaction parameters may not capture
quantitative aspects of the magnetism in the pnictides. Nevertheless, it is
instructive to summarize the main finding of the result of weak coupling
expansions, in particular as they demonstrate the very interesting and
nontrivial aspects that results from interband interactions with almost nested
hole and electron Fermi-surfaces[37, 38, 39]. For an ideal semimetal (two
identical hole and electron bands with the Fermi energies $E_{h}$ and $E_{e})$
all susceptibilities at the nesting vector Q diverge as $\log|E_{h}/E_{e}-1|$.
Depending on the details of electron-electron interaction this signals an
instability, at $E_{h}=E_{e},$ to a spin density wave state or to a
superconducting state for infinitesimal interaction. The corresponding
interference between particle-hole and particle-particle scattering events can
be analyzed by using a renormalization group approach. For $J_{H}=J=0$, the
authors of Ref.[38] find that at low energies the interactions are dominated
by Cooper pair-hopping between the two bands, favoring an
$s^{\pm}$-superconducting state that is fully gapped on each Fermi surface
sheet, but with opposite sign on the two sheets. It is worth pointing out that
this pairing mechanism is due to very generic interband scattering, not
necessarily due to _spin-fluctuations_ , as all particle-hole and particle-
particle scattering events enter in essentially the same matter. An
$s^{\pm}$-state was also obtained using a functional renormalization group
approach[37], where the authors argue that the pairing mechanism is due to
collective spin fluctuations that generate a pairing interaction at low
energies. The appeal of these calculations is clearly that controlled and thus
robust conclusions can be drawn. On the other hand, as discussed below, the
Fermi surface nesting is less crucial as is implied by these calculations.
Attempts to include sizable electron-electron interactions within an itinerant
electron theory are based on the partial summation of ladder and bubble
diagrams, in the spirit of Eq.1. This leads to the RPA type theory of Ref.[40,
41, 42, 43] and the fluctuation exchange approximation of multiband
systems[44, 45]. RPA calculations yield a magnetic susceptibility that is
peaked at or near $\mathbf{Q=}\left(\pi,\pi\right)$. For parameters where the
Fermi surface around $\Gamma$ is present, the dominant pairing channel is
again the $s^{\pm}$-state, while $d$-wave pairing occurs as one artificially
eliminates this sheet of the Fermi surface. The exchange of paramagnons
between Fermi surface sheets is shown to be an efficient mechanism for spin
fluctuation induced pairing. The fluctuation exchange (FLEX) approach is to
some extent a self consistent version of the RPA theory[46]. While the method
is not very reliable to address high energy features, the description of the
low energy dynamics spin response, the low energy electronic band
renormalization and, the nature of the pairing instabilityare rather reliable.
The fact that several orbitals matter in the FeAs systems is also of help as
FLEX type approaches can be formulated as theories that become exact in the
limit of large fermion flavor[47]. Refs.[44, 45] performed FLEX calculations
for the FeAs systems and find once again that the dominant pairing state is an
$s^{\pm}$-state, even though Ref.[44] also find a $d$-wave state in a regime
where the magnetic fluctuation spectrum is peaked at vectors away from
$\mathbf{Q=}\left(\pi,\pi\right)$. These authors find a solution that is
numerically close to a compact form
$\Delta\left(\mathbf{k}\right)=\Delta_{0}\cos(ak_{x})\cos(ak_{y}),$ (3)
but this form is neither required by symmetry nor can be consistently deduced
from any low-energy theory (where pairing occurs at or near the Fermi
surface). We will come back to this issue later in this review.
To summarize, numerous calculations that start from an itinerant description
of the magnetic interactions yield an $s^{\pm}$ pairing state caused by the
exchange of collective interband scattering or paramagnons.
#### 2.2.4 J1-J2 model
The initially assumed (although later refuted by the experiment[49]) absence
of the Drude weight in undoped ferropnictides has been taken as evidence for
the fact that they are in the vicinity of a Mott transition and should be
considered as bad metals with significant incoherent excitations[48]. If
correct, it is clearly appropriate to start from a theory of localized spins,
analogous to what is believed to be correct in the cuprate superconductors[50,
51] (it is worth noting that proximity to a Mott transition is a sufficient,
but not necessary condition for existence of local moments). If the dominant
magnetic interactions are between nearest and next nearest neighbor Fe-spins,
the following model describes the localized spins:
$H=J_{1}\sum_{\left\langle
i,j\right\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+J_{2}\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}$ (4)
Here, $J_{1}$ and $J_{2}$ are the superexchange interactions between two
nearest-neighbor and next-nearest-neighbor Fe sites, respectively. A
geometrical argument can be made[52, 48] that indeed the two superexchange
paths $via$ As have comparable strength (however, this argument fails to
recognize that the direct overlap between Fe orbitals in pnictides is very
large[53], thus leading to a strong enhancement of the nearest neighbor
antiferromagnetic exchange in the localized picture[54], and that in metals
superexchange is not the only and usually not the most important magnetic
interaction). When $J_{1}>2J_{2}$ the conventional Neel state has the lowest
energy, when $J_{1}<2J_{2}$ the stripe order emerging in the experiment is the
lowest magnetic state. The system is frustrated if $J_{1}=2J_{2}.$
Upon doping the poor metal (strictly the insulator) described by Eq. 4 with
charge carriers can be investigated for superconductivity, with pairing
stabilized by strong quantum spin fluctuations. In Ref.[55] a single band of
carriers was investigated leading to either $d_{x^{2}-y^{2}}+id_{xy}$ or
$d_{xy}$-pairing, depending on the carrier concentration and the precise ratio
of $J_{1}$ and $J_{2}$. A more realistic theory for the pairing in the
$J_{1}$-$J_{2}$ model in the pnictides must of course include at least two
bands and was developed in Ref.[56]. For sufficiently large $J_{2}$, the
$s^{\pm}$-state is once again the dominating pairing state. It may seem
strange that this strong coupling theory based upon the (unlikely, from the
experimental point of view) proximity to a Mott transition has essentially the
same pairing solutions ($d$-wave for one Fermi surface sheet and
$s^{\pm}$-wave for two Fermi surface sheets separated by $\mathbf{Q}$), as the
RPA calculation of [40]. In Section 3 we will explain that this is not
surprising at all and that even a totally unphysical theory may lead to
perfectly sensible results for superconductivity, as long as it has the same
structure of magnetic excitations in the reciprocal space.
### 2.3 Magneto-elastic coupling
The parent compounds exhibit a structural and a magnetic transition, strongly
suggesting that magnetoelastic coupling plays a role in the physics of
pnictides in general and in superconductivity in particular. Electronic
structure calculations for a non-magnetic state indicate that the electron-
phonon interaction in the pnictides is rather modest and definitely not
sufficient to explain superconducting transition temperatures of $50$ K[57,
5]. However, as these calculations were based on the nonmagnetic electronic
structure, effects of local magnetism on iron were entirely neglected. Indeed,
the equilibrium position of As calculated under this assumption are quite
incorrect and the force constant for the Fe-As bond is $30\%$ higher than it
should be. On the other hand, fully magnetic AFM calculations, while
overestimating the ordered moment, produce highly accurate equilibrium
structures and the force constant in agreement with experiment[22]. It was
pointed out that including soft magnetism in the calculation, i.e. magnetism
with directional and amplitude fluctuations, may substantially enhance the
electron-phonon coupling[58]. The emphasis is on “soft” : additional reduction
of the force constants of the Fe-As bonds does not come from the fact that the
moment exists, but from the fact that the amplitude of the moment depends on
the bond length. Intriguingly, in the 1111 systems the AFM transition occurs
somewhat below a structural phase transition. Both transitions seem to be of
the second order, or of very weakly first order[59]. In 122 compounds the
structural and magnetic orders emerge simultaneously through a strong first
order transition[60, 61].
In the ordered state, Fe spins are parallel along one direction and
antiparallel along the other. Since we expect the bond length for parallel and
antiparallel Fe-spin polarization to be distinct, magnetism couples strongly
to the shear strain
$\varepsilon_{\mathrm{shear}}=\varepsilon_{xy}-\varepsilon_{yx}$. Thus,
$\varepsilon_{\mathrm{shear}}\neq 0$ should invariably occur below the Neel
temperature. Experiment finds that the ferromagnetic bonds are shorter than
antiferromagnetic bonds. From the point of view of superexchange interaction
it seems somewhat surprising that ferromagnetic bonds shorten and the
superexchange-satisfied bonds expand. Yet this behavior is exactly the same as
the DFT calculations had predicted[52], and it can be traced down to one-
electron energy (the observed sign of the orthorhombic distortion simply
lowers the one-electron DOS at the Fermi level)[101].
What remains puzzling is however why in the 1111 family the structural
transition occurs above $T_{N}.$ Naively, this fact could be taken as evidence
for a hypothesis that elastic degrees of freedom are the driving force and
that magnetism is secondary. There are strong quantitative and qualitative
arguments against this view. First, numerous DFT calculations[62, 63, 22]
converge to the correct orthorhombic structure (with correct sign and
magnitude of the distortion), if performed with AFM magnetic ordering, and to
a tetragonal solution if done without magnetism. On the other hand, the
antiferromagnetism is obtained even without allowing for a structural
distortion. In other words, magnetism is essential for the distortion, but the
distortion is not needed for the magnetism.
There exists also a very general argument that demonstrates that the magnetism
is indeed primary and the structural distortion secondary. Historically the
relevant physics was first encountered in the 2D $J_{1}$-$J_{2}$ model[64],
and applied to ferropnictides in Refs.[50, 51]. Below we will reformulate this
argument form a general point of view. We begin with a unit cell that contains
two Fe sites (just as the actual cristallographic unit cell for the FeAs
trilayer). The most natural choice of the origin is in the middle between
these two Fe cites (Fig. 7a ). The coordinates of the atoms are
$\mathbf{r}_{ij}^{+}=\mathbf{R}_{ij}+\mathbf{d}$,
$\mathbf{r}_{ij}^{-}=\mathbf{R}_{ij}-\mathbf{d,}$
$\mathbf{d}=(\frac{1}{4},\frac{1}{4}),$ where $\mathbf{R}_{ij}$ ($i,j$
integer) are the coordinates of the centers of the unit cells. This naturally
implies partitioning the entire lattice into two sublattives, shown as open
and solid dots in Fig. 7a.
Both ferro- and antiferromagnetic checkerboard orderings correspond to a
$\mathbf{Q}=(0,0)$ perturbation of the uniform state, since in both cases all
unit cells remain identical. The Fourier transform of either patter contains
only momenta corresponding to the reciprocal lattice vectors. Conversely, a
spin density wave with the quasi-momentum $\mathbf{Q}=(\pi,\pi)$ corresponds
to flipping all spins in every other unit cell, as illustrated in Fig. 7b,c by
shading colors (blue cells have the magnetization density opposite to that of
the pink cells). It is evident from Fig. 7b and c that this imposes no
requirement upon the mutual orientation of the two sublattices. Again, one can
say that the susceptibility as a function of quasimomentum $\mathbf{q}$ inside
the first Brillouin zone does not describe fluctuations of the magnetic moment
of two ions in the same unit cell with respect to each other, for that purpose
one needs to know the linear response at all momenta $\mathbf{q+G}$, where
$\mathbf{G}$ is an arbitrary reciprocal lattice vector.
Figure 7: (color online) (a)Fe2 lattice with the fully symmetric unit cells
shown. The full circles denote one sublattice, the hollow ones the other.
Shading shows ordering corersponding to the vector
$\mathbf{Q=}\left(\pi,\pi\right)$ in the Fe2 lattice; for each ssublattice,
spins in the pink unit cells are opposite to the spins in the blue cells, but
relative orientation of the two sublattices is arbitrary. (b) Ordered state
with $\mathbf{Q=}\left(\pi,\pi\right)$ and with parallel orientation of the
spins in the unit cell ($\sigma=1$). (c) Same ordering vector
$\mathbf{Q=}\left(\pi,\pi\right)$, but with antiparallel orientation of the
spins in the unit cell ($\sigma=-1$).
Let us assume that the most stable mean field phase corresponds to Néel order
in each of the two sublattices. In the $J_{1}$-$J_{2}$ language that
corresponds to $J_{2}>J_{1}/2,$ in the itinerant language to an instability in
$\chi$ at $\mathbf{Q}=(\pi,\pi)$. Moreover, it is obvious from Fig. 7b,c that
in the classical ground state one sublattice does not exchange-couple at all
to the other, so the classical ground state is infinitely degenerate. this is
however not important for the following discussion, what matters is that the
two extreme cases are always degenerate, the one where two spin in the same
cell are parallel (Fig. 7b) or antiparallel (Fig. 7c). In the $J_{1}+J_{2}$
model the infinite degeneracy is reduced by quantum fluctuations, but the
double degeneracy remains, while in the LDA it is only double degenerate
already on the mean-field level[65].
It is instructive [64] to introduce two order parameters corresponding to the
Neel (checkerboard) ordering for each sublattice,
$\mathbf{m}_{\pm}=\sum_{ij}(-1)^{i+j}\mathbf{M}_{ij}^{\pm},$where
$\mathbf{M}_{ij}^{\pm}$ are the magnetic moments of the two Fe’s in the unit
cell $ij.$ Following Ref. [64] one can introduce the third (scalar) order
parameter,
$\sigma=\sum_{ij}\sigma_{ij}=\sum_{ij}\mathbf{M}_{ij}^{+}\cdot\mathbf{M}_{ij}^{-}$.
Now $\sigma>0$ corresponds to parallel orientation of the magnetization inside
the unit cell (Fig. 7b) while $\sigma<0$ refers to antiparallel orientation
(Fig. 7c). In the former case $\sigma>0$, neighboring Fe spins are parallel
along the diagonal and antiparallel along the counter-diagonal. The situation
is reversed for $\sigma<0$. These two configurations are degenerate and
correspond to the frequently discussed ’stripe’ magnetic order. In two
dimensions, according to the Mermin-Wagner theorem, $\sigma$ is the only order
parameter that can be finite at finite temperature. Therefore the presumably
largest energy scale of the system, the mean field transition temperature of
each sublattice, $T^{\ast}$ ($\sim J_{2}$ in the local model, and the energy
difference $E_{FM}-E_{AFM}$ in the itinerant picture), does not generate any
phase transition, but rather starts a crossover regime where the correlation
length $\xi_{m}$ for the $\mathbf{m}_{\pm}$ order parameter becomes much
longer that the lattice parameter.
In this regime, one can investigate a possibility of a phase transition
corresponding to the $\sigma$ order parameter. It is important to realize that
$\sigma$ does not have to change sign along a domain wall of the
magnetization. This ensures that $\sigma$ can order even though the sublattice
magnetization vanishes. $\sigma$ does couple to the (long-range) fluctuations
of $\mathbf{m;}$ integrating these fluctuations out one will obtain an
effective Hamiltonian coupling $\sigma_{ij}$ and
$\sigma_{i^{\prime}j^{\prime}}$ as far as $\xi_{m},$ meaning that even very
small coupling between $\mathbf{m}_{+}$ and $\mathbf{m}_{-}$ will produce a
phase transition to a finite $\sigma$ at a temperature $T_{s}\sim
J_{1}\xi_{m}^{2}(T_{s})\sim J_{1}\exp(J_{2}/T_{s})$. Solving this for $T_{s}$,
one gets $T_{S}\sim J_{2}/\log(J_{2}/J_{1})$. Note that here again $J_{1}$ and
$J_{2}\sim T^{\ast}$ just characterize the relevant energy scales and by no
means require the validity of the $J_{1}+J_{2}$ model.
As mentioned above $\sigma$ is positive (negative) for ferromagnetic
(antiferromagnetic) bonds, see Fig.8. Thus $\sigma$ couples bilinearly to the
order parameter of the orthorhombic structural transition
$F_{c}=\gamma\varepsilon_{\mathrm{shear}}\sigma.$ (5)
When the expectation value of $\sigma$ is nonzero below a transition
temperature $T_{s}$, the tetragonal symmetry is spontaneously broken leading
to $\varepsilon_{\mathrm{shear}}\neq 0$. We see that $T_{s}$ is suppressed
from $T^{\ast}$ rather weakly (logarithmically) and that even a weak coupling
between the two sublattices would produce a structural phase transition.
Figure 8: (color online) Magnetoelastic coupling: The two atoms per unit cell
are denoted by filled and open circles. A ferromagnetic bond leads to a
shortening of the nearest neighbor lattice constant (bold dashed lines), while
an antiferromagnetic bond leads to a longer lattice constanti (thin dashed
lines). Depending on the relative orientation of the two sublattices (i.e. the
sign of $\sigma$), two distortions with opposite sign of
$\varepsilon_{\mathrm{shear}}$ are possible.
The third energy scale existing in the problem is set by the interlayer
magnetic coupling, $J_{\perp}.$ In the DFT we found $J_{\perp}\lesssim 1$ meV
in LaFeAsO and $J_{\perp}\sim 16$ meV in BaFe2As2[4]. This huge difference
defines the different behavior of these two compounds. In the former the Neel
transition temperature for a sublattice ordering is on the order of
$T^{\ast}/\log(T^{\ast}/J_{\perp}),$ logarithmically smaller than $T_{s},$
while in the latter one expects a much larger $T_{N}$, and likely larger than
the $T_{s}$ for an individual FeAs plane.
The phase between $T_{N}$ and $T_{s},$ if $T_{s}>T_{N},$ was dubbed “nematic”
in Refs. [50, 51], as the order parameter $\left\langle\sigma\right\rangle\neq
0$ even though $\left\langle\mathbf{M}_{ij}\right\rangle=0$, as expected for
an axial, as opposed to vectorial order parameter. The first order nature of
the transition in the 122 systems is then likely a consequence of the coupling
to soft elastic degrees of freedom, and/or of nonlinear interactions. A more
rigorous treatment of the described physics will be published elsewhere[66].
There is another interesting experimental evidence for the unconventional
nature of the magneto-elastic coupling in these systems. In the 122 systems
the structural distortion $\propto\varepsilon_{\mathrm{shear}}$ and the
sublattice magnetization seem to be proportional to each other.[67] At a
second order transition, symmetry arguments imply however that the former
should be proportional to the square of the sublattice magnetization. At a
first order transition, no such strict connection can be established, however
one expects that the generic behavior is recovered as the strength of the
first order transition gets smaller, realizable via alcaline earth
substitution. Experiments show that the mentioned linear behavior is similar
for Ca, Ba or Sr[68]. In our view this behavior is evidence for the fact that
the first order transition in the 122 systems is never close to being weak.
Arguments that the first order character of the magneto-elastic phase
transition originates from the lattice instabilities near the onset of spin-
density wave order were recently given in Ref.[69]. However, further
discussion clearly goes beyond the limit of this review.
The fact that at the structural transition (and even above), magnetic
correlations in plane are already well established, with large correlation
lengths, explains many otherwise mysterious observations. A more detailed
discussion can be found in Ref. [22].
This picture is not without ramifications for superconductivity. First and
foremost, it implies that at superconducting composition ferropnictides,
especially the 1111 family, are not really paramagnetic, bat rather systems
with a large in-plane magnetic correlation length, much larger than the
lattice parameter and likely much larger than the superconducting correlation
length. Second, the excitation structure in such a system is unusual and
cannot be entirely described in terms of $\chi(\mathbf{Q),}$ where
$\mathbf{Q}=(\pi,\pi),$ since such a description loses the physics associated
with the parameter $\sigma.$ Finally, it implies that the lattice and spin
degrees of freedom do not fluctuate independently and are naturally connected
to each other. Therefore a detailed quantitative theory for the pairing state
will have to include lattice vibrations. Conversely, experiments that find
evidence for a lattice contribution to the pairing mechanism should not be
considered as evidence against magnetic pairing.
### 2.4 Other excitations
While everybody’s attention is attracted to magnetic pairing mechanisms and
spin fluctuations, it would be premature and preposterous to exclude any other
excitations from consideration. First of all, it might be still too early to
discard the venerable phonons. While there is no question that the
calculations performed so far [57, 5] were accurate and the linear response
technique used had proved very reliable before (MgB${}_{2},$ CaC6 $etc.),$
these calculation by definition do not take into account any effects of the
magnetism. As discussed above, it is very likely that the ground state even in
the so-called nonmagnetic region of the phase diagram is characterized by an
AFM correlation length long enough compared to the inverse Fermi vector. In
this case, the amplitude of the magnetic moment of Fe (even though its
direction fluctuates in time) is nonzero and the electronic structure is
sensitive to it. Calculations suggest that a phonon stretching the Fe-As bond
will strongly modulate this magnetic moment and thus affect the electronic
structure at the Fermi level more than for a nonmagnetic compound (or, for
that matter, a magnetic compound with a hard magnetic moment). Softness of the
Fe moments, variationally, provides an additional route for electron-phonon
coupling and should therefore always enhance the overall coupling constant.
Whether this is a weak or a strong effect, and whether the resulting coupling
is stronger in the intraband channel (enhancing the $s_{\pm}$
superconductivity) or in the interband channel (with the opposite effect), is
an open question. Only preliminary results are available[58].
Besides the phonons and the spin fluctuation, charge (polarization)
fluctuations can also, in principle, be pairing agents. To the great surprise
of the current authors, nobody has yet suggested an acoustic plasmon mechanism
for ferropnictides, a mechanism that was unsuccessfully proposed for cuprates,
for MgB2 and for CaC${}_{6}.$ Presumably the apparent lack of strong transport
anisotropy in 122 and the absence of carriers with largely disparate mass
prevented these usual suspects from being discussed.
It is not only the harsh condition on the very existence of acoustic plasmons,
but a very general malady (better known in the case of acoustic plasmons, but
generally existing for any sort of exciton pairing) that prevents plasmonic
superconductivity in most realistic cases: lattice stability. Basically,
efficient pairing of electrons via charge excitations of electronic origin
requires overscreening of electrostatic repulsion — which by itself does not
constitute a problem. But since the ion-ion interaction is screened by the
same polarization operator as electron-electron interaction, there is an
imminent danger that the former is overscreened as well. This is an
oversimplified picture (electron-electron susceptibility differs from the
response to an external field on the level of vertex corrections), but it
captures the essential physics.
This danger was appreciated by the early proponents of the excitonic
superconductivity, W. Little[70] and V. Ginzburg[71], therefore they proposed
space separation between a highly polarizable insulating media, providing
excitons, and a metallic layer or string where the superconducting electrons
live. The sandwich structure of the As-Fe-As trilayer reminds us of the
Ginzburg’s “sandwich” (“Ginzburger” ) and tempts to revisit his old proposal.
This was done recently by Sawatzky and collaborators[72] who pointed out that
As is a large ion (Pauling radius for As4- is 2.2 Å) and ionic polarizability
grows with the radius cube. Since the conducting electrons are predominantly
of Fe origin, they suggested pairing of Fe d electrons $via$ polarization of
As ions. So far, this proposal was received with a skepticism that can be
summarized as follows. (1) Analyzing the muffin-tin projected character of the
valence bands, as it was done in Ref. [72] is generally considered to be an
unreliable way to estimate the hybridization between different ions; indeed
the largest part of the electronic wave function refers to the interstitial
space, which is naturally identified as mostly As-like. (2) Removal of the As
orbitals from the basis leads to a strong reduction of the valence band width,
indicating that hybridization between Fe and As is about as strong as direct
Fe-Fe hopping. (3) When Bloch functions are projected upon the Fe-only Wannier
functions, the latter come out very diffuse and extend way beyond the Fe ionic
radius. That is to say, negligible hybridization between Fe and As, that is
prerequisite for the scenario promoted in Ref. [72], appears to be a rather
questionable proposition. Besides, above-mentioned calculations of the phonon
spectra and electron-phonon coupling implicitly account for the large
susceptibility of the As-4 ions (which comes mostly from the outer, valence
shell) yet they find no manifestation of strong As polarization: neither
particular phonon softening nor strong coupling with any phonon.
## 3 Pairing symmetry: general considerations
### 3.1 Geometrical consideration: excitation vectors and Fermi surface
Given such disparate views that different researchers hold about the origin of
magnetism in ferropnictides and of the character of spin fluctuations there,
it may seem strange that a great majority of model calculations predict the
same pairing symmetry, $s_{\pm},$ with full gaps in both electron and hole
bands, but with the opposite signs of the order parameters between the two. In
fact, this is not surprising at all. To begin with, let us point out that the
sign of the interaction mediated by boson exchange is always positive
(attraction) for charge excitations (phonons, plasmons, polarization
excitons), since the components of a Cooper pair have the same charge, but can
be either positive (for triplet pairing, where the electrons in the pair have
the same spin) or negative (repulsion) for singlet pairing, for spin
excitations. That is to say, exchange of spin fluctuations mediates repulsion.
A quick glance at the anisotropic BCS equation reveals that repulsive
interactions can be pairing when, and only when the wave vector of such a
fluctuation spans parts of the Fermi surface(s) with opposite signs of the
order parameter (equivalently, one can say that an interaction that is
repulsive everywhere in the momentul space, can be partially attractive in the
real space, for instance, for electrons located an nearest lattice sites).
This can be illustrated on a popular model of high-$T_{c}$ cuprates, which
considers a simplified cylindrical Fermi surface nearly touching the edge of
the Brillouin zone and superexchange-driven spin fluctuations with the wave
vector $(\pi,0)$. As Fig. 9a illustrates, such an interaction is pairing in
the $d_{x^{2}-y^{2}}$ symmetry, because it spans nearly perfectly the lobes of
the order parameter with the opposite signs.
Figure 9: (color online) (a) A cartoon illustrating how a repulsive
interaction corresponding to superexchange spin fluctuations $Q=(\pi,\pi$) may
generate $d$-wave pairing in cuprates. (b) The same, for an $s_{\pm}$ state
and spin fluctuations with $Q=(\pi,0)$ (in a Brillouin zone corresponding to
one Fe per cell). (c) If the central hole pocket is absent, the superexchange
interaction favors a nodeiless $d$ state.
Most models used for ferropnictides assume a simplified fermiology with one or
more hole FSs and one or more electron FSs displaced by the SDW vector
($\pi,0$) (in this Section, we use the notations corresponding to the
Brillouin zone with one Fe per cell). Any spin-fluctuation induced interaction
with this wave vector, no matter what the origin of these fluctuations (FS
nesting, frustrated superexchange, or anything else) unavoidably leads to a
superconducting state with the opposite signs of the order parameter for the
electrons and for the holes. Depending on the details of the model the ground
state maybe isotropic or anisotropic and the gap magnitudes on the different
sheets may be the same or may be different, but the general extended $s$
symmetry with the sign-reversal of the order parameter (an $s_{\pm}$ state) is
predetermined by the fermiology and the spin fluctuation wave vector (Fig.
9b).
It is worth noting that while most (but not all) models consider spin
fluctuations corresponding to the observed instability to be the leading
pairing agent, some include spin fluctuations of different nature [for
instance, nearest neighbor superexchange or nesting between the “X” and “Y”
electron pockets, both corresponding to the same wave vector, ($\pi,\pi)$ in
the unfolded zone and $(0,0)$ in the conventional zone], or phonons, or direct
Coulomb repulsion; these additional interactions may modify the gap ratios and
anisotropies (in extreme cases, creating nodes on some surfaces), but, for a
realistic choice of parameters, unlikely to change the symmetry.
Moreover, if the radius of the largest FS pocket is larger than the magnetic
vector, spin fluctuations start to generate an intraband pair-breaking
interaction, which by itself will lead to an angular anisotropy and possible
gap nodes.
The above reasoning, however, is heavily relying upon an assumption that the
topology predicted by the DFT is correct. So far, as discussed above, the
evidence from ARPES and from quantum oscillations has been favorable. It is
still of interest to imagine, for instance, electron-doped compounds not
having hole pockets at all or having them so small that the pairing energy for
them is negligible. It was pointed out[40, 73] that in this case spin
fluctuations with different momentum vectors dominate and create a nodeless
$d$-wave state in the electron pockets, as Fig. 9c illustrates.
### 3.2 General properties of the $s_{\pm}$ state
Since the $s_{\pm}$ states constitute the most popular candidate for the
superconducting symmetry of pnictides, it is worth recapitulating the physics
of this state. Let us start with the simplest possible case: two bands (two
Fermi surfaces) and interband repulsive interaction between the two. Let the
interaction strength be $-V,$ and the DOSs $N_{1}\neq N_{2}.$ To be specific,
let $N_{2}=\alpha N_{1},$ $\alpha\geq 1.$ Then in the weak coupling limit the
BCS equations read
$\displaystyle\Delta_{1}$ $\displaystyle=-\int
d\epsilon\frac{N_{2}V\Delta_{2}\tanh(E_{2}/2k_{B}T)}{2E_{2}}$
$\displaystyle\Delta_{2}$ $\displaystyle=-\int
d\epsilon\frac{N_{1}V\Delta_{1}\tanh(E_{1}/2k_{B}T)}{2E_{1}}$ (6)
where $E_{i}$ is the usual quasiparticle energy in band $i$ given by
$\sqrt{(\epsilon-\mu)^{2}+\Delta_{i}^{2}}.$ Near $T_{c}$ linearization gives
$\displaystyle\Delta_{1}$
$\displaystyle=\Delta_{2}\lambda_{12}\log(1.136\omega_{c}/T_{c})$
$\displaystyle\Delta_{2}$
$\displaystyle=\Delta_{1}\lambda_{21}\log(1.136\omega_{c}/T_{c}),$ (7)
where $\lambda_{12}=N_{2}V$, the dimensionless coupling constant, with a
similar expression for $\lambda_{21}.$ These equations readily yield
$\lambda_{eff}=\sqrt{\lambda_{12}\lambda_{21}}$ and
$-\Delta_{1}/\Delta_{2}=\sqrt{N_{2}/N_{1}}\equiv\sqrt{\alpha}.$ Note that the
Fermi surface with the larger DOS has a smaller gap. It can also be shown that
the gap ratio at zero temperature in the weak coupling limit is also given by
$\sqrt{N_{2}/N_{1}},$ and strong coupling effects tend to reduce the disparity
between the gaps.
The situation becomes more interesting for more than two orbitals with
distinct gaps. Let us consider a model for the hole-doped 122 compound. The
calculated FS (Fig.4) shows three sets of sheets: Two e-pockets at the corner
of the zone, two outer h-pockets, formed by the $xz$ and $yz$ orbitals
(degenerate at $\Gamma$ without the spin-orbit), and the inner pocket formed
by $x^{2}-y^{2}.$ In the DFT calculations all three hole cylinders are
accidentally close to each other, however, ARPES shows two distinct sets, the
inner barrel, one of which presumably corresponding to $x^{2}-y^{2}$ band, and
the outer one, presumably $xz/yz.$ The pairing interaction between the
e-pockets and the two different types of the h-pockets need not be the same
(by virtue of the the matrix elements). Using the same partial DOS as listed
above for Ba1.6K0.6Fe2As2 (both total and individual DOS depend weakly on the
position of the Fermi level, reflecting the 2D character of the band structure
at this doping), roughly 1.2 st/eV for each hole band and the same for the two
e-band together, we get the coupling matrix
$\left(\begin{array}[]{ccc}0&0&-\lambda_{1}\nu_{1}\\\
0&0&-\lambda_{2}\nu_{2}\\\ -\lambda_{1}&-\lambda_{2}&0\end{array}\right),$ (8)
where $\nu_{1,2}$ is the ratio of DOS of the first ($xz/yz)$ and the second
($x^{2}-y^{2})$ hole bands to that of the electron bands. Note that
$\nu_{1}\sim 2$ and $\nu_{2}\sim 1.$ Diagonalizing this matrix we find the gap
ratios to be
$\Delta_{1}:\Delta_{2}:\Delta_{e}=\lambda_{1}:\lambda_{2}:\sqrt{\lambda_{1}^{2}\nu_{1}+\lambda_{2}^{2}\nu_{2}.}$
The latest ARPES measurements[11] imply that $\Delta_{i}:\Delta_{o}\approx
2:1,$ where $i$ and $o$ stand for the inner and outer sets of hole Fermi
surfaces. This would mean that the two coupling constants are twice larger
that the other (although we do not know which), which is fairly possible.
However, that implies that the electron FS has a gap that is larger than that
of the largest hole band by at least a factor of $\sqrt{1.5}=1.22$ (assuming
that the outer FSs in the calculations, are formed by the $xz/yz$ bands; the
opposite assumptions leads to an even larger electron-band gap). This is in
some disagreement with the ARPES data that suggest that $\Delta_{e}$ is on the
order of $\Delta_{i}$ or slightly smaller. However, this is a small
discrepancy, which can be easily corrected by introducing small intraband
electron-phonon coupling for the hole bands, and/or taking into account
possible gap suppression by impurities in the electron band. It is also worth
noting that the spread of the measured values, depending on the sample and on
the location on the FS, is on the order of 10%.
### 3.3 Coulomb avoidance
It was realized quite some time ago that a $d$-wave pairing has an additional
advantage compared to an $s$-wave, namely that the electrons in a Cooper pair
avoid each other (the pair wave function has zero amplitude at
$\mathbf{r-r}^{\prime}=0$), strongly reducing their local Coulomb repulsion.
The leading contribution to the pairing interaction in the single band Hubbard
model $U\sum_{\mathbf{k}}\left\langle
c_{\mathbf{k\uparrow}}c_{-\mathbf{k\downarrow}}\right\rangle$ is repulsive,
but vanishes as $\sum_{\mathbf{k}}\Delta_{\mathbf{k}}=0$ due to the symmetry
of the $d$-wave state. Thus, a contact Coulomb repulsion does not affect
$d$-wave superconductivity.
The simplest possible $s^{\pm}$-wave function is given by Eq.3. In this case,
the sum over the Brillouin zone vanishes again due to nodes at $\pm ak_{x}\pm
ak_{y}=\pi/2$. This description is however somewhat misleading because it may
produce a false impression that there is a symmetry reason for the vanishing
of the Coulomb repulsion in the $s^{\pm}$state, or that this particular
functional form is essential for avoiding the Coulomb repulsion. To illustrate
that this is not the case, it is instructive to consider a toy problem in
reciprocal space. In the weak coupling regime, the effective coupling matrix
$\Lambda_{\mathbf{kk}^{\prime}}$ (note that the band index is uniquely defined
by the wave vector) is
$\Lambda_{\mathbf{kk}^{\prime}}=\lambda_{\mathbf{kk}^{\prime}}-\mu_{\mathbf{kk}^{\prime}}^{\ast},$
(9)
where $\lambda$ is the original coupling matrix in orbital space and
$\mu_{\mathbf{kk}^{\prime}}^{\ast}$ is the renormalized Coulomb
pseudopotential. The critical temperature is determined by the largest
eigenvalue of the matrix $\Lambda,$ and the $\mathbf{k}$ dependence of the
order parameter $\Delta_{\mathbf{k}}$ is given by the corresponding
eigenvector. If $\mu^{\ast}$ is a constant and
$\sum_{\mathbf{k}}\Delta_{\mathbf{k}}=0$ (as in the $d$-wave case), any
eigenvector of the matrix $\lambda$ is also an eigenvector of $\Lambda,$ with
the same eigenvalue. This proves that Coulomb avoidance takes place for any
superconductor where the order parameter averages to zero over the entire FS,
and not only for the $d$-wave symmetry.
Let us now consider a specific $s^{\pm}$ superconductor. For simplicity, let
us take two bands with the same DOS, $N_{1}=N_{2}=N$ and with an interband
coupling only:
$\lambda_{ij}=\left(\begin{array}[]{cc}0&-VN\\\ -VN&0\end{array}\right).$ (10)
We shall also assume that the Coulomb repulsion $U$ is a contact interaction,
so that $\mu_{ij}^{\ast}=UN$ is the same for all matrix elements. The maximal
eigenvalue of $\Lambda$, which corresponds to the effective coupling constant
$\lambda_{\mathrm{eff}}$, is indeed simply $VN$ and _independent_ of $U$. The
corresponding eigenvector is $\Delta_{1}=-\Delta_{2}$, i.e. the $s^{\pm}$
state. The Coulomb interaction is irrelevant, just like in case of $d$-wave
pairing. The effect is however a consequence of the assumed symmetry of the
two bands. In general, unlike the d-wave, no symmetry requires that
$\sum_{\mathbf{k}}\Delta_{\mathbf{k}}=0$. This can already be seen if one
considers a model with distinct densities of states: $N_{2}=\alpha
N_{1}=\alpha N$. We have
$\lambda_{ij}=\left(\begin{array}[]{cc}0&-\alpha VN\\\
-VN&0\end{array}\right).$ (11)
and the weak-coupling gap ratio near $T_{c}$ is $\sqrt{\alpha}$. Now the
effect of the Coulomb repulsion is not nullified, but is still strongly
suppressed. The eigenvalues are easily determined. The key result is that the
maximal eigenvalue remains positive for all finite $\alpha$. Even the extreme
limit $\lambda_{\mathrm{eff}}^{\pm}(U\rightarrow\infty)=2VN\alpha/(1+\alpha)$
is for realistic $\alpha$ only somewhat reduced compared to
$\lambda_{\mathrm{eff}}^{\pm}(U=0)=\sqrt{\alpha}VN$. This is qualitatively
different from the regular ($s_{++})$ interband-only pairing with an
attractive interband interaction of the same strength. In this case,
$\lambda_{\mathrm{eff}}^{++}(U>V/2)<0$, and the Coulomb interaction dominates
over the attractive interband pairing interaction. In the linear in $UN$
regime, the suppression rate of $\lambda_{eff}(U)$ is $(\sqrt{\alpha}-1)/2$
for $s^{\pm}$ and $(\sqrt{\alpha}+1)/2$ for $s^{++}$ pairing. For example, for
the DOSs ratio of $4$ (the gap ratio is then $2$) $\mu^{\ast}\approx
0.25\lambda_{eff}\left(U=0\right)$ will suppress an $s^{++}$ superconductivity
entirely, while in the $s^{\pm}$ case the effective coupling will be reduced
only by 8%.
The efficiency of the Coulomb avoidance is neither limited to the assumption
of a uniform Coulomb interaction among and within the bands, nor is a result
of the weak coupling approach. Strong coupling FLEX type calculations also
find pairing states with very small repulsive contribution due to Coulomb
interaction[44, 45].
## 4 Pairing symmetry: experimental manifestations
### 4.1 Parity
Since we want to review the experimental situation regarding the pairing
symmetry, the first question to ask is, whether superconductivity is singlet
or triplet? Fortunately, this question can be answered relatively confidently.
Measurements of the Knight shift on single crystals of the Co-doped BaFe2As2
superconductor[74] clearly indicate full suppression of spin susceptibility in
the superconducting state in all directions, incompatible with a triplet
pairing in a tetragonal crystal. For other compounds only polycrystalline,
direction-averaged data exist, but they fully agree with the above result,
virtually excluding triplet superconductivity. This leaves, of all possible
scenarios, essentially three: conventional $s$ (presumably multigap),
$s_{\pm}$ and $d$.
### 4.2 Gap amplitude
All experiments that distinguish between different pairing states can be,
roughly speaking, grouped into two classes: those probing the gap amplitude
and those probing the gap symmetry. The advantage of the former is that they
are comparatively easier to perform. The temperature dependence of any
observable sensitive to the excitation gap is sensitive to the presence of
nodes or multiple gaps. The disadvantage is that only a measurement of the
relative phase of the wave function will unambiguously determine the pairing
state, including its symmetry.
Important and very transparent probes of the gap amplitude are thermodynamic
measurements. The early reports of the specific heat leaned towards power-law
behavior characteristic of nodal superconductivity. The latest data [13, 15]
suggest a fully gapped superconductivity, or a dominant fully gapped component
with possible small admixture of a nodal state. While the experimental
situation is still far from consensus, especially regarding the 1111 family, a
few observations may be in place: (i) The specific heat jump in the h-doped
BaFe2As2 is strong and sharp, and in 1111 compounds is weak and poorly
expressed. This cannot be ascribed to a difference in calculated band
structures. This is either due to sample quality issues or possibly to the
more isotropic character of superconducting and magnetic properties in 122
systems. (ii) In no case can specific heat temperature dependence be fitted
with one gap. Multiple gap fits, having more parameters, are of course less
reliable. (iii) Another, usually more reliable signature of nodal
superconductivity is a square-root dependence of the specific heat coefficient
on the magnetic field. Existing reports[13] however show a clear linear
dependence, characteristic of a fully gapped superconductor.
Another popular probe is temperature dependence of the NMR relaxation rate.
Extensive studies have been done in this aspect (see other articles in this
volume). In all studied systems, the relaxation rate is non-exponential. The
initial impression was that the relaxation rate is cubic in temperature,
$1/T_{1}\propto T^{3},$ consistent with nodal lines[75, 76]. Later it was
argued that the data cannot be described by a single power law as in the
cuprates[77, 78]. These results were obtained for the 1111 systems. The
situation with the 122 family is even less clear. Published data[79, 74] do
not show exponential decay either, but the results are equally far from any
single power law behavior. Even more puzzling, the only paper reporting on the
low-$T_{c}$ LaFePO superconductor claims that the relaxation rate does not
decrease below $T_{c}$ at all[80].
The third relevant experiment is measuring the London penetration depth.
Reports are again contradictory. For instance, in Pr-based 1111 compound the
penetration depth was found[81] to barely change between $\approx 0.05T_{c}$
and $T^{\ast}\approx 0.35T_{c},$ and than increase roughly as
$(T-T^{\ast})^{2}$ between $T^{\ast}$ and $\approx 0.65T_{c},$ a picture
roughly consistent with a multi-gap nodeless superconductor. Malone $et$
$al$[82] measured Sm-based 1111 and were able to fit their data very well in
the entire interval from $T_{c}/30$ and $T_{c}$ using two full gaps. In Nd-
based 1111 the penetration depth was measured at $T>0.1T_{c}$ and fitted with
a single anisotropic gap for $0.1T_{c}<T<T_{c}/3$,[83] however, the latest
result from the same authors, taken at lower temperature, can be better fitted
with a quadratic law[84]. Similar quadratic behavior has been clearly seen in
the 122 compounds[85]. At the same time, the low-$T_{c}$ LaFePO is again odd:
it shows a linear behavior[86].
To summarize, the thermodynamic data on average lean towards a nodal
superconductivity. However, some data are not consistent with the gap nodes,
and there is no clear correlation with the sample quality either way.
Moreover, while some data suggest line nodes, others are consistent only with
point nodes, in the clean limit. One can say with a reasonable degree of
confidence that the entire corpus of the data cannot be described by any one
scenario in the clean limit. On the other hand, essentially any temperature
dependence of thermodynamic characteristics can be fitted if a particular
distribution of impurity scattering is assumed in an intermediate regime
between the Born and the unitary scattering, and a particular relation between
the intra- and interband scattering (there have been a number of paper doing
exactly that for the NMR relaxation rate, for instance, Ref. [87], or for the
penetration depth, for instance, Ref. [88]). However, the fact that all these
papers rely upon specific combinations of parameters, while the phenomena they
seek to describe are rather universal, calls for caution. Besides, except in
the pure unitary regime, scattering is accompanied by a $T_{c}$ suppression
and most papers do not find any correlation between thermodynamic probes and
$T_{c}$ among different samples. Another possibility is that required
scattering is provided not by impurities, but by intrinsic defects that are
thermodynamically or kinetically necessarily present in all samples (for
example, dynamic domain walls introduced in Ref. [22]). More measurements at
the lower temperature and on clean samples will probably clarify the matter.
At the moment one cannot consider this problem solved.
Close to the thermodynamic measurements are tunneling type experiments. As of
now, these have been nearly exclusively point-contact Andreev reflection
probes. Here, again, the experimental reports are quite inconsistent,
moreover, the situation is in some sense worse than in thermodynamic probes,
since uncontrollable surface properties enter the picture. Interpretation
generally includes fitting one curve with a large number of parameters, and
the procedure is not always well defined. Generally speaking, three types of
results have been reported: $d$-wave like, single full gap-like, and multigap.
Interpretation is particularly difficult because within the $s^{\pm}$ picture
formation of subgap Andreev bound states was predicted (e.g., Refs. [89, 90])
that can be easily mistaken for multiple gaps.
### 4.3 Phase-sensitive probes
In view of all that, experiments directly probing the gap symmetry are highly
desirable. The paramagnetic Meissner effect, also known as Wohlleben effect,
occurs in a polycrystalline sample when inter-grain weak links have random
order parameter phase shifts, $0$ or $\pi.$ It has been routinely observed in
cuprates and is considered a key signature of $d$-wave superconductivity. The
effect does not exist in conventional, even anisotropic and multi-gap
superconductors, even though sometimes it can be emulated by impurity effects
in the junctions. For $d-$wave superconductors without pronounced
crystallographic texture the Wohlleben effect is expected, and its absence can
be taken as evidence against $d$-wave. Finally, in the $s^{\pm}$ scenario the
phase is the same by symmetry for $(100)$ and $(010)$ grain boundaries, and
there are good reasons to expect the same phase for $(110)$ boundaries as
well. There may or may not be a $\pi$ phase shift for phase boundaries at some
specific orientation, likely for a narrow range of angles[91], but probably
not enough to produce a measurable Wohlleben effect. The absence of the effect
in experiment[92] is a significant argument against $d$-wave, but hardly helps
to distinguish $s$ from $s^{\pm}.$
Similarly, the $c$-axis tunneling provides evidence against the $d$-wave,
where the Josephson current strictly parallel to the crystallographic $c$
direction vanishes by symmetry. Experimentally a sizable current was
found[93].
Recalling the cuprates again, the ultimate argument in favor of the $d$-wave
was provided by the corner Josephson junction experiments that probe directly
the phase shift between two separate junctions; in cuprates, with their
$d_{x^{2}-y^{2}}$ symmetry, these junction were to be along the $(100)$ and
$(010)$ directions. Similarly, a potential $d_{xy}$ state could be detected by
the combination of $(110)$ and $(\bar{1}10)$ directions. On the other hand, a
conventional $s$ state would not produce a phase shift for any combination of
contacts. Again, the case of $s_{\pm}$ superconductivity is nontrivial. While
symmetry does not mandate a $\pi$ shift for any direction, it can be shown
that, depending on the electronic structure parameters and properties of the
interface, there may exist intermediate angles (between $0$ and $45^{o})$
where a $\pi$ shift is possible[91]. It also may be possible if the two
junctions have different tunneling properties, so that one of them filters
through only hole-pocket electrons, and the other only electron-pockets. It is
not as bizarre as it may seem, and some possibilities were discussed in Ref.
[91]. Probably the most promising design involves “sandwiches” of various
geometries. The first proposal of that kind was by Tsoi et al[90], who
suggested an $s/s^{\pm}/s^{\prime}$ trilayer, where $s$ is a conventional
quai-2D superconductor with a large Fermi surface that has no overlap with the
hole FS of the $s^{\pm}$ layer (equivalently, a superconductor with small
Fermi surfaces centered around the M points), and $s^{\prime}$ is a
conventional superconductor with a small FS centered around $\Gamma.$ This was
followed by another proposal of a bilayer of hole-doped and electron-doped 122
materials[91]. In both cases the idea is that the current through the top of
the sandwich will be dominated by the electron FS, and through the bottom by
the hole one. Both proposals require momentum conservation in the interfacial
plane, that is, basically, epitaxial or very high quality interface. The
former proposal has an additional disadvantage of requiring two high-quality
interfaces with very special conventional superconductors, particularly the
one that should filter through the electron FS is rather difficult to find. As
of now, no experiments have been reported pursuing any of the above
suggestions, but with better single crystals and thin films it should become
increasingly doable. It should be stressed, however, that in this case, unlike
the cuprates, an absence of the $\pi$ shifts in any of the proposed geometries
does not disprove the $s^{\pm}$ scenario, since the effect here is
quantitative rather than qualitative, but the presence of the sought effect
would be a very strong argument in favor of it. On the other hand, standard
90o corner junction experiments similar to cuprates are also important, as
they could prove unambiguously that the symmetry is not $d$-wave (even though
they cannot distinguish between $s$ and $s^{\pm}).$
Further properties of interfaces between an $s^{\pm}$ superconductor and
normal metal or conventional superconductor are now actively being studied
theoretically, encouraging further experimental research. Probably we will see
first results within the next year.
### 4.4 Coherence factor effects
Other signatures of the $s^{\pm}$ state are based on the fact, previously
pointed out by many in connection with the cuprates, that the coherence
factors are “reversed” for electronic transitions involving order parameters
of the opposite sign. In the conventional BCS theory, as is well known,
coherence factors of two kinds appear. The first kind, sometimes called “Type
I” or “minus” coherence factor, is given by the expression
$(1-\Delta_{\mathbf{k}}\Delta_{\mathbf{k}^{\prime}}/E_{\mathbf{k}}E_{\mathbf{k}^{\prime}}),$
where
$E_{\mathbf{k}}=\sqrt{\Delta_{\mathbf{k}}^{2}+\varepsilon_{\mathbf{k}}^{2}},$
and $\varepsilon_{\mathbf{k}}$ in the normal state excitation. The other kind,
Type II or the “plus” coherence factor has the opposite sign in front of the
fraction. If both order parameters entering this formula have the same sign,
the Type I factor is destructive, in the sense that it goes to zero when
$\varepsilon\rightarrow 0,$ and cancels out the peak in the superconducting
DOS. Type I factors appear, for instance, in the polarization operator, and as
a result there are no coherence peaks in phonon renormalization (as measured
by ultrasound attenuation, for instance) and in spin susceptibility (including
the Knight shift). Type II factors appear, for instance, in the NMR relaxation
rate, and they are constructive, resulting in the famous Hebel-Slichter peak
below $T_{c}.$
Obviously, if $\Delta_{\mathbf{k}}$ and $\Delta_{\mathbf{k}^{\prime}}$ have
opposite signs, the meaning of the coherence factors is reversed; the Type I
factors are now constructive and the Type II destructive. There are several
straightforward ramifications of that. For instance, as it was pointed out
already in the first paper proposing the $s^{\pm}$ scenario[5], the spin
susceptibility at the SDW wave vector should show resonance enhancement just
below $T_{c}$. For explicit calculations of this effect see for example
Refs.[94, 95]. There are indeed some reports of this effect, as measured by
neutron scattering[96]. In principle, one can expect a similar effect in the
phonon line-width, for the phonons with the same wave vector, just below
$T_{c},$ but this is really hard to observe.
Less straightforward are cases of the quantities that involve averaging over
the entire Brillouin zone, in which case the answer, essentially, depends on
which processes play a more dominant role in the measured quantity, those
involving intra-, or interband scattering. The answer usually depends on
additional assumptions about the matrix elements involved, which can rarely be
calculated easily from first principles. An example is electronic Raman
scattering; a possibility of a resonant enhancement in some symmetries has
been discussed recently[97].
## 5 Role of impurities
Impurity and defect scattering is believed to play an important role in
pnictide superconductors. Proximity to a magnetic instability implies that
ordinary defects may induce static magnetic moments on the neighboring Fe
sites and thus trigger magnetic scattering. If, as is nearly universally
believed, an order parameter with both signs is present, nonmagnetic
impurities are also pair-breaking. Thus the anticipation is that in regular
samples, and maybe in samples of much higher quality, impurity-induced pair
breaking will play a role.
Our intuition regarding the impurity effects in superconductors is largely
based upon the Abrikosov-Gorkov theory of Born-scattering impurities in BCS
superconductors. There was an observation at that time that folklore ascribes
to Mark Azbel: Soviet theorists do what can be done as good as it should be
done, and American ones do what shall be done as good as it could be done. For
many years the approach to the impurity effects in superconductors was largely
Soviet: most researchers refine the Abrikosov-Gorkov theory, applying it to
anisotropic gaps and to unconventional superconductors, and relatively little
has been done beyond the Born limit — despite multiple indications that most
interesting superconductors, from cuprates to MgB2 to pnictides are in the
unitary limit or in an intermediate regime.
The physics of the nonmagnetic scattering in the two different limits is quite
different. In the Born limit, averaging over all scattering events yields a
spatially uniform superconducting state and tries to reduce the variation of
the order parameter over the FS. Ultimately, for sufficiently strong
scattering, the order parameter becomes a constant, corresponding to the DOS-
weighted average over the FS. Note that unless this average is zero by
symmetry (like in d-wave) the suppression of $T_{c},$ while linear at small
concentrations, is never complete. As pointed out by Mishra $et$ $al.$ [98],
this effect should manifest itself most clearly in an extended s-wave pairing
with accidental nodes in the order parameter. Indeed, while in $d$-wave
superconductors impurities broadens nodes into finite gapless spots, in an
extended $s$ case it is likely that the order parameter of one particular sign
dominates a given FS pocket, in which case Born impurities will first make the
parts of the FS with the “wrong” order parameter gapless, and then lead to a
fully gapped superconductivity. Of course, this only holds for nonmagnetic
impurities. Isotropic magnetic impurities will be just pair-breaking as they
are in conventional superconductors, with the only interesting new physics
being that magnetic impurities cease being pair-breakers if they scatter a
pair such that the sign of the order parameter is flipping. The rule of thumb
is that a scattering path for which magnetic scattering is pair-breaking (no
change of sign of the order parameter), nonmagnetic scattering will not be
pair-breaking, and $vice$ $versa.$
The physics of the unitary limit is quite different. In that limit, the
concentration of impurities is relatively low, but the scattering potential of
an individual impurity is strong, $N(0)v_{imp}\gg 1.$ In that case rather than
suppressing superconductivity uniformly each impurity creates a bound state at
the chemical potential, thus creating a zero energy peak in the density of
states, without substantial suppression of the bulk superconductivity.
Increasing the impurity concentration broadens the peak, while increasing its
strength barely has any effect at all [99]. In an intermediate case between
the Born limit and the unitary limit, the bound state is formed inside the gap
at a finite energy and is the broader the closer it is to the gap (that is,
closer to the Born limit).
The principal difference from the point of view of the experiment is that the
unitary or intermediate scattering can create subgap density of states at
arbitrary low energy at any temperature, without a drastic suppression of
$T_{c}.$ It was shown in Ref. [87] that any standard code for solving the
Eliashberg equations in the Born limit can be easily modified, with minor
changes, to treat the unitary limit, as well as any intermediate regime.
Therefore we anticipate an imminent shift in the community from the “Soviet”
approach to the “Western” approach, with more quantitative understanding of
the effect beyond the Born approximation.
## 6 Conclusions
In this article we presented a brief overview of some proposals that have been
made for the pairing state in the Fe-pnictide superconductors. In particular,
we summarized arguments that support the view that the vicinity of
superconductivity and magnetism in these systems is not accidental. The
obvious appeal of this, and essentially any other electronic pairing mechanism
is, of course, that the involved energy scales, and thus $T_{c}$, can in
principle be larger if compared to pairing due to electron-phonon interaction.
Electronic mechanisms also promise a new level of versatility in the design of
new superconductors.
At this early stage in the research on the iron pnictide family, experiments
have not conclusively determined the pairing symmetry, the detailed pairing
state or the microscopic pairing mechanism. Still, in our view a plausible
picture emerges where superconductivity is caused by magnetic fluctuations.
Only two ingredients are vital to arrive at a rather robust conclusion for the
pairing state. First, pnictides need to have Fermi surface sheets of two
kinds, one near the center of the Brillouin zone, and the other near the
corner. Second, the typical momentum for the magnetic fluctuations should be
close to the ordering vectors $\mathbf{Q=}\left(\pi,\pi\right)$ of the parent
compounds. Then, magnetic interactions lead quite naturally to an efficient
inter-band coupling that yields an $s^{\pm}$ pairing state. This result is
general in the sense that it is obtained regardless of whether one develops a
theory based on localized quantum magnetism or itinerant paramagnons. There is
evidence that the two needed ingredients are present in the pnictides. Fermi
surface sheets at the appropriate locations have been predicted in non-
magnetic LDA calculations and seen in ARPES experiments. The magnetic ordering
vector has been determined via neutron scattering, even though we have to
stress that a clear identification of magnetic fluctuations for
superconducting systems without long range magnetic order is still lacking.
The resulting $s^{\pm}$ pairing state has a number of interesting properties.
As far as the a group theoretic classification is concerned, its symmetry is
the same as that for a conventional $s$-wave pairing, where the gap-function
has same sign on all sheets of the Fermi surface. However, there are
significant differences between the two states. The sign change in the gap
affects the coherence factors, leading to the resonance peak in the dynamic
spin susceptibility and the absence of a Hebel-Slichter peak in NMR.
Nonmagnetic impurities affect the $s^{\pm}$-state just like magnetic
impurities do in an ordinary $s$-wave state, i.e. here a behavior more akin to
$d$-wave superconductors. Another implication of the sign change in the
$s^{\pm}$-state leads to rather efficient Coulomb avoidance.
The presence of nodes in the superconducting gap in still an open issue. In
$d$-wave or $p$-wave pairing states, nodal lines or points are fixed by
symmetry. This is different for the $s^{\pm}$-state. In its most elementary
version, the sign change of the gap corresponds to a node located between two
Fermi surface sheets. This is the case for the $\Delta\left(\mathbf{k}\right)$
given in Eq.3. Energetic arguments favor such a gapless state as long as the
momentum transfer $\mathbf{Q}$ couples efficiently to large parts of distinct
Fermi surface sheets and Coulomb avoidance is efficient. However, as there is
no symmetry constraint for the location of the nodes, it is in principle
possible that there are nodes on some Fermi surface sheets.
Next to the nature of the pairing state, the microscopic understanding of the
magnetism of the Fe-pnictides is one of the most interesting aspects of these
materials. Are these systems made up of localized spins that interact via
short ranged, nearest neighbor exchange interactions or, are they better
described in terms of itinerant magnetism? While we emphasized that many
aspects of the pairing state emerge regardless of which of these points of
view is correct, this is really only true for the most elementary aspects of
the theory. As our understanding of these materials deepens, dynamical aspects
of the pairing state will become more and more important, and the details of
the magnetic degrees of freedom will matter. In our view, the most sensible
description starts from itinerant electrons, however with significant
electron-electron interaction. In detail, we find numerous arguments that
emphasize the role of magneto-elastic couplings and that favor a sizable Hund
coupling, i.e. the multi orbital character and the corresponding local multi-
orbital interactions are important to understand the magnetism and
superconductivity alike. Regardless of whether this specific point of view is
correct or not, it is already evident that the ferropnictides make up a whole
new class of materials that stubbornly refuse to behave according to one of
the simple minded categories of condensed matter theory.
## 7 Acknowledgements
This research was supported by the Ames Laboratory, operated for the U.S.
Department of Energy by Iowa State University under Contract No. DE-
AC02-07CH11358 (J.S.), and by the Office of Naval Research (I.I.M.). The
authors wish to thank all their friends and collaborators, without whom this
works could not be accomplished, and their numerous colleagues who read the
manuscript and sent us many useful and insightful comments.
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|
# Irreversibility of Structure Tensors of Modules
Maciej Wojtala
###### Abstract
Determining the matrix multiplication exponent $\omega$ is one of the greatest
open problems in theoretical computer science. We show that it is impossible
to prove $\omega=2$ by starting with structure tensors of modules of fixed
degree and using arbitrary restrictions. It implies that the same is
impossible by starting with $1_{A}$-generic non-diagonal tensors of fixed size
with minimal border rank. This generalizes the work of Bläser and Lysikov [3].
Our methods come from both commutative algebra and complexity theory.
Keywords: matrix multiplication complexity, minimal border rank tensors,
structure tensors for modules.
## 1 Introduction
Determining the matrix multiplication exponent $\omega$ is one of the most
important problems in theoretical computer science. Trivial bounds are
$2\leq\omega\leq 3$. In the classical paper V. Strassen proved a non-trivial
bound $\omega\leq\log_{2}7<2.81$ [15]. D. Coppersmith and S. Winograd proved
the bound $\omega<2.376$ [9]. This result was recently slightly improved [10,
16, 13] resulting with the best known upper bound $\omega<2.373$ with rounding
to the third decimal place.
No non-trivial lower bound is known and the conjecture states that $\omega=2$.
However since Coppersmith-Winograd there was very little progress in inventing
more efficient algorithms and obtaining better upper bounds. Recent papers
give an explanation for the phenomenon - many currently used approaches cannot
result with an algorithm giving $\omega=2$. For instance the laser method used
for big Coppersmith-Winograd tensors cannot show $\omega=2$, in fact it cannot
even show $\omega\leq 2.30$ [1, 2]. Also the framework proposed by Umans and
Cohn using reducing matrix multiplication to group algebra multiplication
cannot show $\omega=2$ for abelian groups and certain non-abelian groups [4,
5]. M. Christiandl, P. Vrana and J. Zuiddam introduced a quantity called
irreversibility and proved that it is impossible to show $\omega=2$ using
arbitrary restrictions starting with irreversible tensors (i.e. with
irreversibility greater than one) [8].
M. Bläser and V. Lysikov showed in their paper [3], that one cannot prove
$\omega=2$ over $\mathbb{C}$ using arbitrary restrictions and starting with
powers of structure tensors of non-semisimple algebras with bounded dimension.
This result is quite general, since the class of tensors that are structural
tensors of some algebra is quite large, larger than previously considered
classes, and using arbitrary restrictions is not a restrictive assumption. For
instance, Coppersmith-Winograd tensors are in this class. Let us name the
three coordinates of our tensors by $V_{1},\>V_{2},\>V_{3}$, i.e. our tensors
belong to the space $V_{1}\otimes V_{2}\otimes V_{3}$ where
$V_{i}\simeq\mathbb{C}^{n}$. Bläser and Lysikov use this result to conclude
that it is impossible to show $\omega=2$ using arbitrary restrictions and
starting with tensors that are both $1_{V_{1}}$\- and $1_{V_{2}}$-generic (so-
called binding tensors) with minimal border rank and being non-diagonal. This
result is really interesting since small border rank is believed to be
desirable in obtaining fast matrix multiplication. An important note is that
one could still try to prove $\omega=2$ by taking sequence of dimensions going
to infinity (since in the result there is an assumption that dimensions are
bounded).
This paper is an extension of results obtained by Bläser and Lysikov. We
consider an arbitrary algebraically closed field $\mathbb{K}$ (of arbitrary
characteristic) and tensors from
$\mathbb{K}^{n}\otimes\mathbb{K}^{n}\otimes\mathbb{K}^{n}$. From the
perspective of commutative algebra a natural generalisation is to consider
modules instead of algebras. In this paper we show such a generalisation. To
achieve that we define a structure tensor of a module. The generalisation
theorem is the main result of this paper:
###### Theorem 1.
For bounded $n$ it is impossible to prove $\omega=2$ over $\mathbb{K}$ using
arbitrary restrictions and starting with powers of tensors of size $n$ that
are isomorphic to some structure tensor of a non-semisimple module.
This allows us also to extend the corollary obtained by Bläser and Lysikov:
###### Corollary 2.
For bounded $n$ it is impossible to prove $\omega=2$ over $\mathbb{K}$ using
arbitrary restrictions and starting with powers of tensors of size $n$ that
are $1_{V_{1}}$-generic, have minimal border rank and have rank larger than
$n$.
It means that we only need to assume $1_{V_{1}}$-genericity - assuming
$1_{V_{2}}$-genericity is not necessary. There are plenty of tensors that are
$1_{V_{1}}$-generic but not $1_{V_{2}}$\- or $1_{V_{3}}$-generic; our
generalization applies to them as well.
The approach in the proof is based on the one in Bläser and Lysikov paper [3],
but there are several issues. The main part focuses on showing that a
structure tensor of a non-semisimple module is $0$-subtight-unstable. To
achieve this we use the quotient ring $A$ obtained by dividing the polynomial
ring by the annihilator of the considered module. We show that ring $A$ is
Artinian. It allows to create a filtration of the module induced by powers of
a nilradical of $A$. We also induce the filtration of the polynomial ring (by
taking the preimage of the filtration of $A$) and of its subset consisting of
forms of degree at most one (by restricting the filtration of the polynomial
ring). Then we will show that it suffices to prove that the last filtration is
non-trivial. To prove that this filtration is indeed non-trivial we will
analyze the spectrum of the quotient ring and by Hilbert’s Nullstellensatz use
it to analyze affine functions which correspond to forms of degree at most
one.
It is unclear if the assumption of $1_{V_{1}}$-genericity can be replaced with
conciseness; the following question remains open:
###### Question 3.
Do there exist concise minimal border rank (with rank greater than border
rank) tensors that are stable?
### Acknowledgements
During preparation of this publication the author was part of the Szkoła Orłów
programme. The publication was created under the supervision of Joachim
Jelisiejew, whose help and support were invaluable. The author also wants to
thank Markus Bläser, Vladimir Lysikov, Joseph Landsberg and the anonymous
referee for helpful comments.
## 2 Definitions
Let $\mathbb{K}$ be an algebraically closed field and let
$S=\mathbb{K}[x_{1},\>x_{2},\>\ldots,\>x_{n-1}]$.
We denote by $S_{\leq 1}$ the $\mathbb{K}$-linear subspace of polynomials of
degree at most one.
Let us consider an $S$-module $M$. Observe that structure of multiplication in
this module is uniquely determined by multiplication by elements of $S_{\leq
1}$. Furthermore, multiplication by $S_{\leq 1}$ is uniquely determined by the
multiplication by a basis of $S_{\leq 1}$. We define the structure tensor of
$M$ as $\sigma_{M}\in S_{\leq 1}^{*}\otimes M^{*}\otimes M$ or equivalently
$\sigma_{M}\colon S_{\leq 1}\otimes M\rightarrow M$. For a choice of
coordinates on the first, second and third factors
($\\{A_{i}\\},\>\\{B_{j}\\},\\{C_{k}\\}$ respectively), $\sigma_{M}$ simply
encodes the result of multiplying $A_{i}$ by $B_{j}$ in $\\{C_{k}\\}$ basis.
Note that using the polynomial algebra is a quite general assumption. Indeed,
let $A$ be a commutative unital algebra with $\dim_{\mathbb{K}}A=n$, let $M$
be a module over $A$ of dimension $n$ and $t$ be the tensor corresponding to
the bilinear map $A\otimes M\to M$. Let $(1,\>g_{1},\>g_{2},\ldots,\>g_{n-1})$
span the algebra as a $\mathbb{K}$-linear space. Consider the unique
surjection $S\to A$ that sends $x_{i}$ to $g_{i}$. Then $M$ becomes an
$S$-module and $t$ identifies with the structure tensor of the $S$-module $M$.
In particular, after putting $M=A$ we deduce that the structure tensor of any
commutative unital algebra with $\dim_{\mathbb{K}}A=n$ is isomorphic to the
structure tensor of the obtained $S$-module. Note that using the unity is
needed to assure $1_{V_{1}}$-genericity. In comparison to [3] we assume that
the algebra $A$ is commutative, however due to [12, Lemma 2.6] it is satisfied
when its structure tensor is of minimal border rank (so Corollary 2
strengthens analogous result by Bläser and Lysikov [3, Corollary 25]).
###### Example 4.
We consider $\mathbb{K}^{4}$, where we treat elements as column vectors. Let
us also consider matrices:
$A_{1}=\begin{bmatrix}0&0&1&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{bmatrix},\qquad A_{2}=\begin{bmatrix}0&0&0&1\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\ \end{bmatrix},\qquad A_{3}=\begin{bmatrix}0&0&0&0\\\ 0&0&1&0\\\
0&0&0&0\\\ 0&0&0&0\\\ \end{bmatrix}.$
The matrices clearly pairwise commute, so we can introduce a structure of a
$\mathbb{K}[x_{1},\>x_{2},\>x_{3}]$-module on $\mathbb{K}^{4}$ where
multiplication of a vector $v$ by $x_{i}$ is simply the left multiplication of
$v$ by $A_{i}$ (so we identify $x_{i}$ with $A_{i}$ and take as a
multiplication in the module the left multiplication of a vector by a matrix).
Let us take the standard basis $(e_{1},\>e_{2},\>e_{3},\>e_{4})$ of
$\mathbb{K}^{4}$. We also have the basis of
$\mathbb{K}[x_{1},\>x_{2},\>x_{3}]_{\leq 1}$: $(A_{0}\coloneqq
Id,\>A_{1},\>A_{2},\>A_{3})$ (since we identified $x_{i}$ with $A_{i}$ for
$i=1,\>2,\>3$). To obtain the structure tensor we need to verify the results
of pairwise multiplications.
We have
$A_{0}\cdot e_{i}=e_{i},\qquad A_{1}\cdot e_{i}=\begin{bmatrix}\delta_{i=3}\\\
0\\\ 0\\\ 0\\\ \end{bmatrix},\qquad A_{2}\cdot
e_{i}=\begin{bmatrix}\delta_{i=4}\\\ 0\\\ 0\\\ 0\\\ \end{bmatrix},\qquad
A_{3}\cdot e_{i}=\begin{bmatrix}0\\\ \delta_{i=3}\\\ 0\\\ 0\\\ \end{bmatrix}.$
So the structure tensor of this module is
$A_{0}^{*}\otimes e_{1}^{*}\otimes e_{1}+A_{0}^{*}\otimes e_{2}^{*}\otimes
e_{2}+A_{0}^{*}\otimes e_{3}^{*}\otimes e_{3}+A_{0}^{*}\otimes
e_{4}^{*}\otimes e_{4}+A_{1}^{*}\otimes e_{3}^{*}\otimes
e_{1}+A_{2}^{*}\otimes e_{4}^{*}\otimes e_{1}+A_{3}^{*}\otimes
e_{3}^{*}\otimes e_{2}.$
Above we explained that every structure tensor of a commutative unital algebra
is the structure tensor of a module. Here we show that for an algebra
$A=\nicefrac{{S}}{{I}}$ the structure tensor of $A$ as an algebra and as an
$S$-module may differ. The difference comes from the fact that a given algebra
can have many $S$-module structures: the structure coming from a span
$(1,\>g_{1},\>g_{2},\ldots,\>g_{n-1})$ is in general different from the
structure coming from $\nicefrac{{S}}{{I}}$.
###### Example 5.
Let us consider the algebra
$\nicefrac{{\mathbb{K}[x_{1},\>x_{2}]}}{{(x_{1}^{3},\>x_{2})}}$. We have
standard basis of $\mathbb{K}[x_{1},\>x_{2}]_{\leq 1}$: $(A_{0}\coloneqq
1,\>A_{1}\coloneqq x_{1},\>A_{2}\coloneqq x_{2})$ and standard basis of the
algebra: $(A_{0},\>A_{1},\>A_{3}\coloneqq x_{1}^{2})$. To obtain the structure
tensors we need to verify results of pairwise multiplications, however for
structure tensor of module we multiply elements from the basis of
$\mathbb{K}[x_{1},\>x_{2}]_{\leq 1}$ by the basis of the algebra and for the
structure tensor of algebra - elements from the basis of the algebra by
themselves.
So the structure tensor of the module and the algebra are respectively
$\displaystyle A_{0}^{*}\otimes A_{0}^{*}\otimes A_{0}+A_{0}^{*}\otimes
A_{1}^{*}\otimes A_{1}+A_{0}^{*}\otimes A_{3}^{*}\otimes
A_{3}+A_{1}^{*}\otimes A_{0}^{*}\otimes A_{1}+A_{1}^{*}\otimes
A_{1}^{*}\otimes A_{3},$ $\displaystyle A_{0}^{*}\otimes A_{0}^{*}\otimes
A_{0}+A_{0}^{*}\otimes A_{1}^{*}\otimes A_{1}+A_{0}^{*}\otimes
A_{3}^{*}\otimes A_{3}+A_{1}^{*}\otimes A_{0}^{*}\otimes
A_{1}+A_{1}^{*}\otimes A_{1}^{*}\otimes A_{3}+A_{3}^{*}\otimes
A_{0}^{*}\otimes A_{3}.$
Note that both structure tensors come from the
$\mathbb{K}[x_{1},\>x_{2}]$-module structure on $A$: the structure tensor of
the module encodes the results of multiplication by $1,\>x_{1},\>x_{2}$, while
the structure tensor of the algebra encodes the results of multiplication by
$1,\>x_{1},\>x_{1}^{2}$.
A module is simple if it is non-zero and has no non-zero proper submodules. A
module is semi-simple if it is a direct sum of simple modules.
An arbitrary restriction of a tensor $t\in V_{1}\otimes V_{2}\otimes V_{3}$ is
$\phi(t)$, where $\phi\colon V_{1}\otimes V_{2}\otimes V_{3}\to
V_{1}^{\prime}\otimes V_{2}^{\prime}\otimes V_{3}^{\prime}$ is a linear map
induced by a triple of linear maps $\phi_{i}\colon V_{i}\to V_{i}^{\prime}$.
For a tensor $t\in V_{1}\otimes V_{2}\otimes V_{3}$ and a linear form $x\in
V_{1}^{*}$, the contraction $t\cdot x$ is defined as $(v_{1}\otimes
v_{2}\otimes v_{3})\cdot x=x(v_{1})(v_{2}\otimes v_{3})$ for rank one tensors
and extended to arbitrary tensors by linearity. Thus, a tensor $t\in
V_{1}\otimes V_{2}\otimes V_{3}$ defines a map $V_{1}^{*}\rightarrow
V_{2}\otimes V_{3}$ sending $x$ to $t\cdot x$. The two other maps
$V_{2}^{*}\rightarrow V_{1}\otimes V_{3}$ and $V_{3}^{*}\rightarrow
V_{1}\otimes V_{2}$ can be defined similarly. These maps are called
flattenings of the tensor $t$. A tensor is called concise if all its
flattenings are injective. Such a tensor does not lie in any non-trivial
subspace $V_{1}^{\prime}\otimes V_{2}^{\prime}\otimes V_{3}^{\prime}$ with
$V_{k}^{\prime}\subset V_{k}$. We denote the maximum of the three ranks of the
flattenings by $N(t)$. For a concise tensor, the ranks of the flattenings are
the dimensions of $V_{k}$, and
$N(t)=\max\left\\{\dim_{\mathbb{K}}V_{1},\>\dim_{\mathbb{K}}V_{2},\>\dim_{\mathbb{K}}V_{3}\right\\}$.
A tensor $t\in V_{1}\otimes V_{2}\otimes V_{3}$ is called $1_{V_{1}}$-generic
if $\dim_{\mathbb{K}}V_{2}=\dim_{\mathbb{K}}V_{3}$ and there exists $x\in
V_{1}^{*}$ such that the matrix $t\cdot x\in V_{2}\otimes V_{3}$ has full
rank. The notions of $1_{V_{2}}$-genericity and $1_{V_{3}}$-genericity are
defined analogously [3].
A block tensor is a tensor $t\in V_{1}\otimes V_{2}\otimes V_{3}$ with a
triple of direct sum decompositions $V_{1}=\bigoplus_{i\in
I_{1}}V_{1,\>i},\>V_{2}=\bigoplus_{i\in
I_{2}}V_{2,\>i},\>V_{3}=\bigoplus_{i\in I_{3}}V_{3,\>i}$. The decompositions
of $V_{k}$ induce the decomposition of the tensor space $V_{1}\otimes
V_{2}\otimes V_{3}=\bigoplus_{(i_{1},\>i_{2},\>i_{3})\in I_{1}\times
I_{2}\times I_{3}}V_{1,\>i_{1}}\otimes V_{2,\>i_{2}}\otimes V_{3,\>i_{3}}$.
For a block tensor $t$, we denote by $t_{i_{1}i_{2}i_{3}}$ its projection onto
$V_{1,\>i_{1}}\otimes V_{2,\>i_{2}}\otimes V_{3,\>i_{3}}$ [3, Definition 8].
The support of a block tensor $t$ is defined as
$supp\;t=\\{(i_{1},\>i_{2},\>i_{3})\in I_{1}\times I_{2}\times I_{3}\mid
t_{i_{1}i_{2}i_{3}}\neq 0\\}$ [3, Definition 9].
The block format of a block tensor is a triple $(n_{1},\>n_{2},\>n_{3})$ of
maps $n_{k}\colon I_{k}\rightarrow\mathbb{N}$ defined as
$n_{k}(i)=\dim_{\mathbb{K}}V_{k,\>i},\>k=1,\>2,\>3$. The relative block format
is a triple $(f_{1},\>f_{2},\>f_{3})$ defined as
$f_{k}(i)=\frac{n_{k}(i)}{N_{k}}$ where $N_{k}=\dim_{\mathbb{K}}V_{k}$. [3,
Definition 10]. A subset $S\subset I_{1}\times I_{2}\times I_{3}$ is called
s-subtight with numbering given by three maps $a_{k}\colon
I_{k}\rightarrow\mathbb{Z}$ if for each $(i_{1},\>i_{2},\>i_{3})\in S$ we have
$a_{1}(i_{1})+a_{2}(i_{2})+a_{3}(i_{3})\leq s$ [3, Definition 13].
A tensor $t\in V_{1}\otimes V_{2}\otimes V_{3}$ is unstable, if $0$ is
contained in the Zariski closure of $SL(V_{1})\times SL(V_{2})\times
SL(V_{3})$ orbit of $t$.
A set $S\in I_{1}\times I_{2}\times I_{3}$ is a combinatorially unstable
support in block format $(n_{1},\>n_{2},\>n_{3})$ if there exist exponents
$u_{k}\colon I_{k}\rightarrow\mathbb{Q}$ such that $\sum_{i\in
I_{k}}n_{k}(i)u_{k}(i)=0$ for each $k$ and
$u_{1}(i_{1})+u_{2}(i_{2})+u_{3}(i_{3})>0$ for each
$(i_{1},\>i_{2},\>i_{3})\in S$ [3, Definition 19]. A tensor is combinatorially
unstable if its support is a combinatorially unstable support.
It turns out that combinatorially unstable tensors are unstable [3,
Proposition 20].
Let a tensor $t$ have $s$-subtight support with numbering
$(a_{1},\>a_{2},\>a_{3})$. Let $(f_{1},\>f_{2},\>f_{3})$ be a relative block
format of $t$ and $\overline{a}_{k}=\sum_{i\in I_{k}}f_{k}(i)a_{k}(i)$ for
$k=1,\>2,\>3$. If the inequality
$\overline{a}_{1}+\overline{a}_{2}+\overline{a}_{3}>s$ holds, then we say then
that the tensor $t$ is $s$-subtight-unstable. It turns out that $s$-subtight-
instability implies combinatorial instability [3, Theorem 21].
By $SR(t)$ we denote slice rank of tensor $t$. By $\widetilde{SR}(t)$ we
denote the asymptotic slice rank of tensor $t$, i.e.
$\widetilde{SR}(t)=\limsup_{m\in\mathbb{N}}{SR(t^{\otimes m})^{\frac{1}{m}}}$.
For a ring $R$ we denote by $rad(R)$ the nilradical of $R$, i. e. the ideal
consisting of elements which raised to some power give zero.
## 3 Irreversibility of Structure Tensors of Modules
Let $\mathbb{K}$ be an algebraically closed field. Let
$S=\mathbb{K}[x_{1},\>x_{2},\>\ldots,\>x_{n-1}]$. For $S$-module $M$ of rank
$n=\dim_{\mathbb{K}}S_{\leq 1}$ let $\sigma_{M}\colon S_{\leq 1}\otimes
M\rightarrow M$ be its structure tensor, so $\sigma_{M}\in S_{\leq
1}^{*}\otimes M^{*}\otimes
M\simeq\mathbb{K}^{n}\otimes\mathbb{K}^{n}\otimes\mathbb{K}^{n}$.
Let $F_{n}$ be the set of tensors
$t\in\mathbb{K}^{n}\otimes\mathbb{K}^{n}\otimes\mathbb{K}^{n}$ such that there
exist a non-semisimple $S$-module $M$ such that $t\simeq\sigma_{M}$ (so
$F_{n}$ is the set of tensors considered in Theorem 1).
Let us distinguish the coordinates by taking
$\mathbb{K}^{n}\otimes\mathbb{K}^{n}\otimes\mathbb{K}^{n}=A\otimes B\otimes
C$.
Before proving this corollary we introduce a useful lemma which classifies
simple modules. It is quite known but we present it for self-containment.
###### Lemma 6.
An $S$-module $M$ is simple $\iff$ there exists a maximal ideal $\mathfrak{m}$
of $S$ such that $M=\nicefrac{{S}}{{\mathfrak{m}}}$.
###### Proof.
"$\impliedby$": $\nicefrac{{S}}{{\mathfrak{m}}}$ is a field which is clearly
simple.
"$\implies$": Let $N$ be a simple $S$-module. Let $n_{0}$ be a non-zero
element of $N$. Then $Sn_{0}$ is a non-trivial submodule and thus $Sn_{0}=N$.
Let us define $\pi_{N}\colon S\rightarrow N$, $\pi_{N}(s)=sn_{0}$. By
$Sn_{0}=N$, we have that $\pi_{N}$ is surjective and thus
$N\simeq\nicefrac{{S}}{{\ker\pi_{N}}}$. It now suffices to show that
$\ker\pi_{N}$ is a maximal ideal of $S$.
Since $N\neq 0$, $\ker\pi_{N}$ is a subideal of some maximal ideal
$\mathfrak{m}$ of $S$. Let us suppose that
$\ker\pi_{N}\subsetneq\mathfrak{m}$. Then
$\nicefrac{{\mathfrak{m}}}{{\ker\pi_{N}}}\neq 0$ and clearly
$\nicefrac{{\mathfrak{m}}}{{\ker\pi_{N}}}\subseteq\nicefrac{{S}}{{\ker\pi_{N}}}=N$.
However $N$ is simple, so it must hold
$\nicefrac{{\mathfrak{m}}}{{\ker\pi_{N}}}=\nicefrac{{S}}{{\ker\pi_{N}}}$.
However it implies that
$\nicefrac{{S}}{{\mathfrak{m}}}=\nicefrac{{(\nicefrac{{S}}{{\ker\pi_{N}}})}}{{\nicefrac{{(\mathfrak{m}}}{{\ker\pi_{N}}})}}=0$,
which is a contradiction. ∎
###### Proof of Corollary 2.
Let $T_{n}$ be the set of tensors
$t\in\mathbb{K}^{n}\otimes\mathbb{K}^{n}\otimes\mathbb{K}^{n}$ such that $t$
is $1_{A}$-generic, the rank of $t$ is larger than $n$ and the border rank of
$t$ is $n$. Let $t\in T_{n}$. We need to show that $t\in F_{n}$.
By the assumption that $t$ is $1_{A}$-generic we know that there exist
$\alpha$ such that $t(\alpha):C\rightarrow B^{*}$ has maximal rank. Thus
$a\otimes b\otimes c\mapsto a\otimes b\otimes t(\alpha)(c)$ is a tensor
isomorphism between $t$ and a tensor $\widetilde{t}\in A\otimes B\otimes
B^{*}$. Let us denote $V=\widetilde{t}(A^{*})\subseteq B\otimes B^{*}=End(B)$.
By [12, Lemma 2.6] the linear subspace $V$ consists of commutative matrices.
Since $\widetilde{t}(\alpha)$ is an identity matrix, we have $Id\in V$. We
also clearly have $\dim_{\mathbb{K}}V\leq n$.
We can now choose a linear span $\\{1,\>v_{1},\>v_{2},\>\ldots,\>v_{n-1}\\}$
of $V$ (here $1$ is $Id$). We define a
$\mathbb{K}[x_{1},\>x_{2},\>\ldots,\>x_{n-1}]$-module structure on $V$ by
$x_{i}\cdot b=v_{i}(b)$ for $i=1,\>2,\>\ldots,\>n-1$ and for all $b\in B$.
By commutativity it is indeed a module structure, since $x_{i}\cdot(x_{j}\cdot
b)=v_{i}(v_{j}(b))=v_{j}(v_{i}(b))=x_{j}\cdot(x_{i}\cdot b)$. One can read
more about such a construction in [11]. Let us denote the module as
$M_{\widetilde{t}}$. By construction its structure tensor
$\sigma_{M_{\widetilde{t}}}$ is isomorphic to $\widetilde{t}$. Since
$\widetilde{t}\simeq t$, we obtain that $t\simeq\sigma_{M_{\widetilde{t}}}$,
so $t$ is isomorphic to $S_{n}$-module structure tensor.
We now need to argue that $M_{\widetilde{t}}$ is non-semisimple. However, if
the module $M_{\widetilde{t}}$ were semisimple, then by Lemma 6 it would be a
direct sum of modules of rank $1$, so its structure tensor would be diagonal
in some basis and thus its rank would be $n$. Since $t$ is isomorphic to this
structure tensor, it would imply that $rank(t)=n$, which is contradiction with
the assumption $rank(t)>n$. ∎
Let us now fix a degree $n$ and a module $M$.
To prove 1, we will use an approach that directly generalizes the approach
from [3] \- we will show that if $M$ is not semisimple, then $\sigma_{M}$ is
$0$-subtight-unstable.
###### Theorem 7.
The tensor $\sigma_{M}$ is $0$-subtight-unstable or $M$ is semisimple.
The proof is given later, after handful lemmas.
By $Ann(M)$ we denote the annihilator of module $M$ over $S$. Let
$A=\nicefrac{{S}}{{Ann(M)}}$ and let $\pi$ be a surjection from $S$ to $A$.
We use the quotient ring $A$ because it has much more convenient structure: in
fact, as we show in Lemma 9, it is an Artinian ring, which will let us
conclude a lot about its spectrum. By Hilbert’s Nullstellensatz it will allow
us to analyze affine functions over $\mathbb{K}$, which correspond to $S_{\leq
1}$. Artinian structure will also allow us to use the nilradical construction
to obtain filtrations induced by powers of nilradical. However, we will have
to struggle with one fundamental issue - we want the preimage of $rad(A)$ in
$S$ to be non-trivial. In fact, we want more - preimage of $rad(A)$ has to
have a non-trivial intersection with $S_{\leq 1}$.
The next three lemmas are quite classical commutative algebra arguments, but
we present them to assure self-containment of the paper.
###### Lemma 8.
There exist a natural number $r$ such that $rad(A)^{r}=0$.
###### Proof.
By Hilbert’s basis theorem $S$ is Noetherian ring and since $A$ is its
quotient, it is Noetherian too. Hence $rad(A)$ is finitely generated. Let us
denote its generators as $g_{1},\>g_{2}\>,\ldots,\>g_{k}$. Let $p_{i}$ be such
positive integers that $g_{i}^{p_{i}}=0$ for $i=1,\>2,\>\ldots\>k$. Let
$r=k\max(p_{i})$. Then $rad(A)^{r}=0$.
Indeed, if $a\in rad(A)$, then $a=\sum_{i=1}^{k}{c_{i}g_{i}}$ for some
$c_{i}\in A$ and
$a^{r}=(\sum_{i=1}^{k}{c_{i}g_{i}})^{r}=\sum_{a_{1}+a_{2}+\ldots+a_{k}=r}{b_{(a_{1},\>a_{2},\>\ldots,\>a_{k})}g_{1}^{a_{1}}g_{2}^{a_{2}}\ldots
g_{k}^{a_{k}}}$. For every component of the sum by the pigeonhole principle
for some $i$ we have $a_{i}\geq\max(p_{i})$, thus $g_{i}^{a_{i}}=0$ and so
$b_{(a_{1},\>a_{2},\>\ldots,\>a_{k})}g_{1}^{a_{1}}g_{2}^{a_{2}}\ldots
g_{k}^{a_{k}}=0$. ∎
###### Lemma 9.
$A$ is an Artinian ring.
###### Proof.
Since $A$ is an $\mathbb{K}$-algebra, it suffices to show that $A$ has finite
dimension over $\mathbb{K}$ (every descending sequence of ideals is a
descending sequence of $\mathbb{K}$-linear subspaces so if $A$ has finite
dimension over $\mathbb{K}$ then such a sequence clearly stabilizes).
Let us define $\psi\colon A\rightarrow Hom_{\mathbb{K}}(M,\>M)$ by
$\psi(s+Ann(M))=(m\rightarrow sm)$. By definition of $Ann(M)$ the map $\psi$
is well-defined. Clearly $\psi$ is $\mathbb{K}$-linear.
Let us now observe that $\psi$ is injective. Indeed, if $\psi(s+Ann(M))=0$,
then for all $m\in M$ $sm=0$ and thus $s\in Ann(M)$. Thus $A$ is isomorphic as
a $\mathbb{K}$-linear subspace with $\operatorname{im}(\psi)$, which has
finite dimension over $\mathbb{K}$ as a linear subspace of
$Hom_{\mathbb{K}}(M,\>M)$ which has finite dimension. ∎
###### Lemma 10.
The spectrum of an Artinian ring is finite and equals its maximal spectrum.
This lemma is well-known, but we add a proof for completeness.
###### Proof.
Let $R$ be an Artinian ring and let $p$ be its prime ideal. We will first
argue that $p$ is maximal. Let $\pi_{p}$ be a projection from $R$ to
$\nicefrac{{R}}{{p}}$. Let us observe that $\nicefrac{{R}}{{p}}$ is also
Artinian - for every descending sequence of ideals $I_{1},\>I_{2},\>\ldots$ in
$\nicefrac{{R}}{{p}}$ the sequence of preimages
$\pi_{p}^{-1}(I_{1}),\>\pi_{p}^{-1}(I_{2}),\>\ldots$ in $R$ stabilizes and
thus the sequence $I_{1},\>I_{2},\>\ldots$ also stabilizes. The ring
$\nicefrac{{R}}{{p}}$ is a domain and it suffices to show that it is a field.
Let $x\in\nicefrac{{R}}{{p}},\>x\neq 0$. Let us consider the descending
sequence $(x)\supseteq(x^{2})\supseteq\ldots$ Since it stabilizes, for some
$k$ holds equality $(x^{k})=(x^{k+1})$ with $k\geq 1$. Thus there exist
$a\in\nicefrac{{R}}{{p}}$ such that $x^{k}=x^{k+1}a$. So $x^{k}(1-xa)=0$ and
since $\nicefrac{{R}}{{p}}$ is a domain and $x\neq 0$ we obtain $xa=1$, so $x$
is invertible. So $\nicefrac{{R}}{{p}}$ is a field and thus $p$ is a maximal
ideal of $R$.
We will now show that maximal spectrum of $R$ is finite. Let us suppose
otherwise. Then there exists an infinite sequence of maximal ideals
$\mathfrak{m_{i}}$. Let us consider the descending sequence
$\mathfrak{m_{1}},\>\mathfrak{m_{1}}\cap\mathfrak{m_{2}},\>\mathfrak{m_{1}}\cap\mathfrak{m_{2}}\cap\mathfrak{m_{3}},\>\ldots$
Since it stabilizes there exists such $l$ that
$\mathfrak{m_{1}}\cap\mathfrak{m_{2}}\cap\ldots\cap\mathfrak{m_{l}}=\mathfrak{m_{1}}\cap\mathfrak{m_{2}}\cap\ldots\cap\mathfrak{m_{l}}\cap\mathfrak{m_{l+1}}$.
Clearly $\mathfrak{m_{i}}\not\subseteq\mathfrak{m_{j}}$ for $i\neq j$. Let
$a_{i}\in\mathfrak{m_{i}}\setminus\mathfrak{m_{l+1}}$ for
$i=1,\>2,\>\ldots,\>l$. Then $a_{1}a_{2}\ldots
a_{l}\in\mathfrak{m_{1}}\cap\mathfrak{m_{2}}\cap\ldots\cap\mathfrak{m_{l}}$,
so by our assumption $a_{1}a_{2}\ldots
a_{l}\in\mathfrak{m_{1}}\cap\mathfrak{m_{2}}\cap\ldots\cap\mathfrak{m_{l}}\cap\mathfrak{m_{l+1}}$,
so $a_{1}a_{2}\ldots a_{l}\in\mathfrak{m_{l+1}}$. Thus by primeness of
$\mathfrak{m_{l+1}}$ there exists $c$ such that $a_{c}\in\mathfrak{m_{l+1}}$.
It is a contradiction with the definition with $a_{i}$. ∎
We now will be using definitions referring to block tensors, which are
introduced in Section 2.
###### Proposition 11.
If $\pi^{-1}(rad(A))\cap S_{\leq 1}\neq 0$, then $\sigma_{M}$ is $0$-subtight-
unstable.
###### Proof.
We will use the same approach as in [3, Example 14].
By $r$ we denote the minimal natural number such that $rad(A)^{r}=0$; by Lemma
8 such a natural number exists. Let us take the sequence
$M\supseteq rad(A)M\supseteq rad(A)^{2}M\supseteq\ldots\supseteq
rad(A)^{r}M=0$
. Let us take the sequence $(M_{k})$ of linear subspaces satisfying:
* •
$rad(A)^{k-1}M=M_{k-1}\oplus rad(A)^{k}M$ for $r>k>1$,
* •
$M_{r}=0$.
By definition $(M_{i})_{i=0}^{r}$ is a decomposition of $M$.
Let $(R_{i})_{i=0}^{r}$ be a decomposition of $S_{\leq 1}$ induced by
$\pi^{-1}({rad(A)}^{k})\cap S_{\leq 1}$, i.e. $\pi^{-1}({rad(A)}^{k})\cap
S_{\leq 1}=R_{k}\oplus\pi^{-1}({rad(A)}^{k+1})\cap S_{\leq 1}$, $R_{r}=0$.
We have $R_{1}\oplus R_{2}\oplus\ldots\oplus R_{r}=\pi^{-1}(rad(A))\cap
S_{\leq 1}$, so the assumption $R_{1}\oplus R_{2}\oplus\ldots\oplus R_{r}\neq
0$.
For all $i,\>j\geq 0$ we have
$R_{i}\>M_{j}\subseteq(rad(A)^{i}+Ann(M))\>rad(A)^{j}M=rad(A)^{i+j}M=\bigoplus_{k\geq
i+j}M_{k}$.
Let us now consider the numbering $a_{1}(i)=a_{2}(i)=i.\>a_{3}(i)=-i$. The
structure tensor $\sigma_{M}$ is a block tensor with decompositions obtained
from decompositions of $R_{i}$ and $M_{i}$. Let $I_{1},\>I_{2},\>I_{3}$ be its
indexing sets. As we observed above, for all $(i_{1},\>i_{2},\>i_{3})$ such
that $a_{1}(i_{1})+a_{2}(i_{2})+a_{3}(i_{3})>0$ it holds that
$\sigma_{M}(R_{i_{1}},\>M_{i_{2}},\>M_{i_{3}})=0$. It means that $\sigma_{M}$
has $0$-subtight support. Let $(f_{1},\>f_{2},\>f_{3})$ be a relative block
format of $\sigma_{M}$. Let $\overline{a}_{k}=\sum_{i\in
I_{k}}f_{k}(i)a_{k}(i)$ for $k=1,\>2,\>3$. We need to to show that
$\overline{a}_{1}+\overline{a}_{2}+\overline{a}_{3}>0$. Since decompositions
on the second and the third coordinate are dual, we have
$I_{2}=I_{3},\>f_{2}=f_{3}$ and thus
$\overline{a}_{2}+\overline{a}_{3}=\sum_{i\in
I_{2}}f_{2}(i)\>a_{2}(i)+\sum_{i\in I_{3}}f_{3}(i)\>a_{3}(i)=\sum_{i\in
I_{2}}f_{2}(i)\>i+\sum_{i\in I_{3}}f_{3}(i)\>(-i)=0$. So we just need to show
$\overline{a}_{1}>0$. We have $\overline{a}_{1}=\sum_{i\in
I_{1}}f_{1}(i)\>a_{1}(i)=\sum_{i\in I_{1}}f_{1}(i)\>i$. In our case
$I_{1}=\\{0,\>1,\>\ldots,\>r\\}$ and as we observed before $R_{1}\oplus
R_{2}\oplus\ldots\oplus R_{r}\neq 0$, so there exist
$s\in\\{1,\>2,\>\ldots,\>r\\}$ such that $f_{1}(s)>0$. Thus we have
$\sum_{i\in I_{1}}f_{1}(i)\>i=\sum_{i=0}^{r}f_{1}(i)\>i\geq f_{1}(s)\>s>0$,
which ends the proof. ∎
Now we will argue that $\pi^{-1}(rad(A))\cap S_{\leq 1}$ is zero only for
semisimple modules. To prove it, we will analyze the spectrum of $A$ and the
support of $M$.
Let us define $supp\>M$ as a set of such maximal ideals $\mathfrak{m}$ of $S$
that $\mathfrak{m}M\neq M$. Let us also denote by $V(I)$ the Zariski closure
of ideal $I$, i.e. the set of ideals which include $I$.
###### Lemma 12.
It holds that $supp\>M=V(Ann(M))$.
###### Proof.
First we argue that $V(Ann(M))$ contains only maximal ideals. Since by Lemma 9
we have that $A=\nicefrac{{S}}{{Ann(M)}}$ is Artinian, by Lemma 10 all its
prime ideals are maximal. Taking preimage induces bijection between
$\operatorname{Spec}(\nicefrac{{S}}{{Ann(M)}})$ and $V(Ann(M))$ and this
bijection preserves maximality of an ideal. Thus all elements of $V(Ann(M))$
are maximal ideals.
Let now $\mathfrak{m}$ be a maximal ideal of $S$. We need to show that
$Ann(M)\not\subseteq\mathfrak{m}\iff\mathfrak{m}M=M$.
On the one hand, if $Ann(M)\not\subseteq\mathfrak{m}$, then
$\mathfrak{m}+Ann(M)=(1)$ and thus
$\mathfrak{m}M=\mathfrak{m}M+Ann(M)M=(\mathfrak{m}+Ann(M))M=(1)M=M.$
On the other hand, if $\mathfrak{m}M=M$, then by Nakayama’s lemma there exist
an element $m_{0}\in\mathfrak{m}$ such that $(1-m_{0})M=0$ and thus
$1-m_{0}\in Ann(M)$. Since $1-m_{0}\not\in\mathfrak{m}$, it holds that
$Ann(M)\not\subseteq\mathfrak{m}$. ∎
###### Lemma 13.
We have the inequality $|supp\>M|\leq\deg M$ and if the equality holds, then
$M$ is semisimple.
###### Proof.
By Lemma 12 the set $supp\>M$ can be treated as a subset of $SpecMax(A)$. By
Lemma 9 we have that $A$ is Artinian and thus by Lemma 10 we have that
$SpecMax(A)$ is finite, so $supp\>M$ is also finite.
Let
$supp\>M=\\{\mathfrak{m}_{1},\>\mathfrak{m}_{2},\>\ldots,\>\mathfrak{m}_{k}\\}$.
We have $\deg M=dim_{\mathbb{K}}M\geq
dim_{\mathbb{K}}\nicefrac{{M}}{{\mathfrak{m}_{1}M\cap\mathfrak{m}_{2}M\cap\ldots\cap\mathfrak{m}_{k}M}}$.
By the Chinese Remainder Theorem we have
$\nicefrac{{M}}{{\mathfrak{m}_{1}M\cap\mathfrak{m}_{2}M\cap\ldots\cap\mathfrak{m}_{k}M}}\simeq\nicefrac{{M}}{{\mathfrak{m}_{1}M}}\times\nicefrac{{M}}{{\mathfrak{m}_{2}M}}\times\ldots\times\nicefrac{{M}}{{\mathfrak{m}_{k}M}}.$
By definition of $\mathfrak{m}_{i}$ we have
$\nicefrac{{M}}{{\mathfrak{m}_{i}M}}\neq 0$ and thus
$dim_{\mathbb{K}}(\nicefrac{{M}}{{\mathfrak{m}_{i}M}})\geq 1$ and so
$dim_{\mathbb{K}}(\nicefrac{{M}}{{\mathfrak{m}_{1}M}}\times\nicefrac{{M}}{{\mathfrak{m}_{2}M}}\times\ldots\times\nicefrac{{M}}{{\mathfrak{m}_{k}M}})\geq
k.$
Thus $\deg
M\geq\deg\nicefrac{{M}}{{\mathfrak{m}_{1}M\cap\mathfrak{m}_{2}M\cap\ldots\cap\mathfrak{m}_{k}M}}\geq
k=|supp\>M|$, which proves the first part of the statement.
Let us now suppose that equality holds, i.e. $\deg M=k$. Thus all inequalities
from the first part must be equalities and hence
$\mathfrak{m}_{1}M\cap\mathfrak{m}_{2}M\cap\ldots\cap\mathfrak{m}_{k}M=0$ and
for all $i$ we have $dim_{\mathbb{K}}\nicefrac{{M}}{{\mathfrak{m}_{i}M}}=1$.
First equality and Chinese Remainder Theorem give us
$M\simeq\nicefrac{{M}}{{\mathfrak{m}_{1}M}}\times\nicefrac{{M}}{{\mathfrak{m}_{2}M}}\times\ldots\times\nicefrac{{M}}{{\mathfrak{m}_{k}M}}.$
Now by Lemma 6 it suffices to show that for all $i$ we have
$\nicefrac{{M}}{{\mathfrak{m}_{i}M}}\simeq\nicefrac{{S}}{{\mathfrak{m}_{i}}}$
as $S$-modules.
Since $\nicefrac{{M}}{{\mathfrak{m}_{i}M}}$ is a linear space of dimension
one, there exist $m\in\nicefrac{{M}}{{\mathfrak{m}_{i}M}}$ such that
$\mathbb{K}m=\nicefrac{{M}}{{\mathfrak{m}_{i}M}}$.
Let us define $S$-module homomorphism $\varphi\colon
S\rightarrow\nicefrac{{M}}{{\mathfrak{m}_{i}M}}$ by $\varphi(1)=m$,
$\varphi(s)=sm$. Since $\mathbb{K}m=\nicefrac{{M}}{{\mathfrak{m}_{i}M}}$, the
map $\varphi$ is surjective. Thus
$\nicefrac{{M}}{{\mathfrak{m}_{i}M}}\simeq\nicefrac{{S}}{{\ker(\varphi)}}$.
Clearly $\mathfrak{m}_{i}\subseteq\ker(\varphi)$ and thus
$\nicefrac{{S}}{{\ker(\varphi)}}\subseteq\nicefrac{{S}}{{\mathfrak{m}_{i}}}$.
However
$dim_{\mathbb{K}}\nicefrac{{M}}{{\mathfrak{m}_{i}M}}=1=dim_{\mathbb{K}}\nicefrac{{S}}{{\mathfrak{m}_{i}}}$
and so the inclusion must be an equality and thus indeed
$\nicefrac{{M}}{{\mathfrak{m}_{i}M}}\simeq\nicefrac{{S}}{{\mathfrak{m}_{i}}}$
as $S$-modules. ∎
###### Proposition 14.
If it holds that $\pi^{-1}(rad(A))\cap S_{\leq 1}=0$, then we have an equality
$|supp\>M|=\deg M$.
###### Proof.
By definition $\pi^{-1}(rad(A))=\sqrt{Ann(M)}$. The ideal $rad(A)$ is a
nilradical of $A$ so it is an intersection of its all prime ideals. Since by
Lemma 9 we have that $A$ is Artinian, by Lemma 10 all its prime ideals are
maximal and the number of maximal ideals is finite. Thus $rad(A)$ is an
intersection of finite number of maximal ideals and so $\pi^{-1}(rad(A))$ is
also intersection of finite number of maximal ideals.
By Lemma 13 we have that $|supp\>M|\leq\deg M$. Let us suppose by
contradiction that $|supp\>M|<\deg M$. By Lemma 12 we have that
$|supp\>M|=|V(Ann(M))|$. Let
$V(Ann(M))=\\{\mathfrak{m}_{1},\>\mathfrak{m}_{2},\ldots,\>\mathfrak{m}_{r}\\}$,
let us remark that by Lemma 12 ideals $\mathfrak{m}_{i}$ are maximal for all
$i$. Moreover
$\pi^{-1}(rad(A))=\sqrt{Ann(M)}=\bigcap_{p_{i}\in
V(Ann(M))}{p_{i}}=\mathfrak{m}_{1}\cap\mathfrak{m}_{2}\cap\ldots\cap\mathfrak{m}_{r}.$
By Hilbert’s Nullstellensatz maximal ideals in polynomial ring over
algebraically closed field can be treated as points, we will be using this
equivalence. In it, the elements of $\pi^{-1}(rad(A))\cap S_{\leq 1}$ are
exactly affine functions $f$ such that $f(\mathfrak{m}_{i})=0$ for all $i$.
We have $r=|V(Ann(M))|=|supp\>M|<\deg M=\deg S_{\leq 1}$. So the space of
affine functions has the dimension at least $r+1$. Thus the dimension of a
subspace of affine functions such that $f(\mathfrak{m}_{i})=0$ for all $i$ is
at least one. Thus there exist a non-zero function $f$ satisfying these
conditions. Since $f$ is an element of $\pi^{-1}(rad(A))\cap S_{\leq 1}$, we
obtain $\pi^{-1}(rad(A))\cap S_{\leq 1}\neq 0$, which gives us expected
contradiction. ∎
###### Proof of Theorem 7.
Let us suppose that $M$ is not semisimple. We need to prove that $\sigma_{M}$
is $0$-subtight-unstable.
By Lemma 13 we have that $|supp\>M|<\deg M$. Thus by Proposition 14 we have
$\pi^{-1}(rad(A))\cap S_{\leq 1}\neq 0$. Proposition 11 implies that
$\sigma_{M}$ is $0$-subtight-unstable. ∎
###### Theorem 15.
Let $t\in\mathbb{K}^{n}\otimes\mathbb{K}^{n}\otimes\mathbb{K}^{n}$ be a non-
zero $s$-subtight-unstable tensor with numbering $(a_{1},\>a_{2},\>a_{3})$ and
relative block format $(f_{1},\>f_{2},\>f_{3})$. Let also
$\overline{a}_{k}=\sum_{i\in I_{k}}f_{k}(i)a_{k}(i)$ for $k=1,\>2,\>3$. Then
it holds that
$\widetilde{SR}(t)\leq
n\exp\left(-\frac{(\overline{a}_{1}+\overline{a}_{2}+\overline{a}_{3}-s)^{2}}{6\sum_{k=1}^{3}\sum_{i_{k}\in
I_{k}}a_{k}^{2}}\right).$
###### Proof.
The proof is given in the proof of [3, Theorem 22]. Although in a statement
there is an assumption that tensors are over $\mathbb{C}$, there proof does
not use this assumption (it operates on supports of the tensors), which is
also noted in [3, Remark 26].
Note that the fraction is well defined, since if it held that $a_{k}\equiv 0$
for $k=1,\>2,\>3$, then $\overline{a}_{k}=0$. We have that
$\overline{a}_{1}+\overline{a}_{2}+\overline{a}_{3}>s$, so $s<0$. However
since $t$ is non-zero, its support is non-empty, and by definition for some
$(i_{1},\>i_{2},\>i_{3})$ it holds that
$0=a_{1}(i_{1})+a_{2}(i_{2})+a_{3}(i_{3})\leq s$. So we would have $s\geq
0>s$, which is a contradiction. ∎
Having Theorem 7 and Theorem 15 proved, we are almost ready to prove the main
theorem, using similar tools as Bläser and Lysikov. However we have to
slightly change the assumptions, so we will need to prove one more lemma.
Let us remind that for tensor $t\in V_{1}\otimes V_{2}\otimes V_{3}$ we denote
by $N(t)$ the maximal rank of flattening of $t$ (see Section 2).
###### Lemma 16.
We have an equality $N(\sigma_{M})=n$.
###### Proof.
Clearly $N(\sigma_{M})\leq n$, since $n=\dim_{\mathbb{K}}S_{\leq
1}=\dim_{\mathbb{K}}M$ and $\sigma_{M}\in S_{\leq 1}^{*}\otimes M^{*}\otimes
M$. Since $1\in S_{\leq 1}$, the matrix of $\sigma_{M}(1,-)$ is an identity
matrix and thus the ranks of flattenings of $\sigma_{M}$ over the second and
the third coordinate are at least $\dim_{\mathbb{K}}M=n$, so
$N(\sigma_{M})\geq n$, and thus the equality must hold. ∎
We now present more general result which implies Theorem 1. In particular, to
prove Theorem 1 we will use $0$-subtight-instability.
###### Proposition 17.
In $\mathbb{K}^{n\times n\times n}$ with $n$ fixed, it is impossible to prove
$\omega=2$ using arbitrary restrictions from powers of elements of tensor
family $G_{n}$ such that for every $t\in G_{n}$ tensor $t$ is $s$-subtight-
unstable and satisfies $N(t)=n$.
###### Proof.
We use the approach as in [3, Theorem 16], but we have slightly weaker
assumptions - we do not require conciseness.
Note that zero tensor does not belong to $G_{n}$, since $N(\mathbf{0})=0$.
If we let
$B(n)\coloneqq\inf\left\\{\frac{\log N(t)}{\log\widetilde{SR}(t)}\mid t\in
G_{n}\right\\},$
then by [3, Proposition 15] the irreversibility of any tensor with $N(t)\leq
n$ is bounded from below by $B(n)$, and by [8, Theorem 9] [3, Theorem 7] the
best bound on $\omega$ we can get is at least $2B(n)$.
By the assumption we have for all $t\in G_{n}$ that $N(t)=n$. Also by the
assumption all tensors from family $G_{n}$ are $s$-subtight-unstable, so by
Theorem 15 we have for all $t\in G_{n}$ the bound
$\widetilde{SR}(t)\leq
n\exp\left(-\frac{(\overline{a}_{1}(t)+\overline{a}_{2}(t)+\overline{a}_{3}(t)-s)^{2}}{6\sum_{k=1}^{3}\sum_{i_{k}\in
I_{k}}a_{k}(t)^{2}}\right).$
Let us denote
$\overline{B}(n)(t)=\frac{\log n}{\log
n-\frac{(\overline{a}_{1}(t)+\overline{a}_{2}(t)+\overline{a}_{3}(t)-s)^{2}}{6\sum_{k=1}^{3}\sum_{i_{k}\in
I_{k}}a_{k}(t)^{2}}}$
and
$\overline{B}(n)=\inf\left\\{\overline{B}(n)(t)\mid t\in G_{n}\right\\}.$
We can bound $B(n)$ from below by $\overline{B}(n)$. So it is sufficient to
show that $\overline{B}(n)>1$.
Clearly the term
$\frac{(\overline{a}_{1}(t)+\overline{a}_{2}(t)+\overline{a}_{3}(t)-s)^{2}}{6\sum_{k=1}^{3}\sum_{i_{k}\in
I_{k}}a_{k}(t)^{2}}$ is non-negative. Moreover, all tensors from the family
$G_{n}$ are $s$-subtight-unstable, so by definition the term
$\overline{a}_{1}(t)+\overline{a}_{2}(t)+\overline{a}_{3}(t)-s$ is strictly
positive. It implies that for $t\in G_{n}$ the value of $\overline{B}(n)(t)$
is strictly greater than one. Moreover, the value of $\overline{B}(n)(t)$
depends only on the support of the tensor $t$. Since for fixed $n$ the number
of possible supports of tensors is finite, the set of possible values of
$\overline{B}(n)(t)$ is also finite. Thus we have
$\overline{B}(n)=\inf\left\\{\overline{B}(n)(t)\mid t\in
G_{n}\right\\}=\min\left\\{\overline{B}(n)(t)\mid t\in G_{n}\right\\}>1,$
which ends the proof. ∎
###### Proof of Theorem 1.
By Theorem 7 all tensors $t\in F_{n}$ are $0$-subtight-unstable. By Lemma 16
it also holds that $N(t)=n$. Applying result of Proposition 17 for
$G_{n}=F_{n}$, we obtain the claim. ∎
Now we present the exact bound of irreversibility of the structure tensor of a
non-semisimple module. Note that we have already proved that this
irreversibility is strictly greater than one.
###### Corollary 18.
If the module $M$ is non-semisimple, then the irreversibility of its structure
tensor $\sigma_{M}$ is at least
$\left(1-\left(\frac{\left(\sum_{i=0}^{r-1}is_{i}\right)^{2}}{6n^{2}\log
n\left(2\sum_{i=0}^{r-1}m_{i}^{2}+\sum_{i=0}^{r-1}s_{i}\right)}\right)\right)^{-1},$
where $r$ is minimal such that $rad(A)^{r}=0$ ($r$ exists by Lemma 8),
$t_{i}=\dim_{\mathbb{K}}\pi^{-1}({rad(A)}^{i})\cap S_{\leq 1}$
$s_{i}=t_{i}-t_{i+1}$,
$m_{i}=\dim_{\mathbb{K}}\left({rad(A)}^{i}M/{rad(A)}^{i+1}M\right)$.
###### Proof.
As we observed, if $M$ is non-semisimple, then its structure tensor is
$0$-subtight unstable and satisfies $N(\sigma_{M})=n$, so it satisfies the
assumptions from Proposition 17, so we can use the bound
$i(\sigma_{M})\geq\frac{\log n}{\log
n-\frac{(\overline{a}_{1}(t)+\overline{a}_{2}(t)+\overline{a}_{3}(t))^{2}}{6\sum_{k=1}^{3}\sum_{i_{k}\in
I_{k}}a_{k}(t)^{2}}}.$
Now it is sufficient to use definitions of $a_{i}$ and $\overline{a}_{i}$ from
the proof of Proposition 11 and we obtain the desired bound. Analogical bound
for the structure tensors of non-semisimple algebras is given in [3, Corollary
23]. ∎
## 4 Characteristic zero
We will now show that in Proposition 17 we can weaken the assumption of
$s$-subtight-instability for fields of characteristic zero.
###### Proposition 19.
In $\mathbb{C}^{n\times n\times n}$ with $n$ fixed, it is impossible to prove
$\omega=2$ using arbitrary restrictions from powers of elements of tensor
family $G_{n}$ such that for every $t\in G_{n}$ tensor $t$ is unstable and
satisfies $N(t)=n$.
###### Proof.
From [3, Theorem 5] (originally in [6]) and [3, Theorem 4] (originally in
[14]) it follows that the set of possible values of $\widetilde{SR}(t)$ for
tensors in $\mathbb{C}^{n\times n\times n}$ is finite. Therefore, the set of
possible ratios $\frac{\log N(t)}{\log\widetilde{SR}(t)}$ is also finite. So
we can let
$B(n)\coloneqq\min\left\\{\frac{\log N(t)}{\log\widetilde{SR}(t)}\mid t\in
G_{n}\right\\},$
and by [3, Proposition 15] the irreversibility of any tensor with $N(t)\leq n$
is bounded from below by $B(n)$, and also by [8, Theorem 9] [3, Theorem 7] the
best bound on $\omega$ we can get is at least $2B(n)$.
Since all tensors form $G_{n}$ are unstable, we have by [3, Theorem 3] that
for all $t\in G_{n}$ we have $\widetilde{SR}(t)<n$. Since we assumed that for
all $t\in G_{n}$ we have $N(t)=n$, it holds that $2B(n)>2$, so it is
impossible to show $\omega=2$. ∎
###### Proposition 20.
In $\mathbb{K}^{n\times n\times n}$ with $n$ fixed with characteristic of
$\mathbb{K}$ equal to zero, it is impossible to prove $\omega=2$ using
arbitrary restrictions from powers of elements of tensor family $G_{n}$ such
that for every $t\in G_{n}$ tensor $t$ is combinatorially unstable and
satisfies $N(t)=n$.
###### Proof.
Let us suppose that it is possible to prove $\omega=2$ over $\mathbb{K}$ using
arbitrary restrictions. We will prove that then it is possible to prove
$\omega=2$ over $\mathbb{C}$ using arbitrary restrictions and obtain
contradiction with Proposition 19.
Possibility to prove $\omega=2$ over $\mathbb{K}$ using arbitrary restrictions
means that there exists a sequence of pairs $(t_{i},\>N_{i})$ such that for
all $i$ it holds that $t_{i}\in G_{n}$ and using arbitrary restrictions (which
are basically some linear equations) we can obtain from $t_{i}^{\otimes
N_{i}}$ large matrix multiplication, i.e. such that for a limit at infinity we
obtain $\omega=2$.
For a tensor $t_{i}$ let $(x_{i_{j}})$ be a (finite) tuple of vectors
corresponding to the flatenning of $t_{i}$ with maximal rank (each of three
flattenings is a linear map, we choose basis of tensors $x_{i_{j}}$ for which
contractions with $t_{i}$ give linear space of maximal dimension). Let also
$Z_{i}$ be a set of coefficients of arbitrary restrictions (which are
basically linear equations) used with $t_{i}$. All sets $Z_{i}$ are finite.
Now we want to transfer this sequence to $\mathbb{C}$. Firstly let us observe
that we only need to consider powers of tensors from sequence $t_{i}$, which
means that if $\mathbb{K}$ is large we can drop some part of it. More
precisely, let $\mathbb{\widetilde{L}}$ be a smallest subfield of $\mathbb{K}$
such that $\mathbb{\widetilde{L}}$ contains all elements of sets $Z_{i}$ and
tensors $t_{i}$ and $x_{i_{j}}$ (which means that we treat tensors as tuples
and want $\mathbb{\widetilde{L}}$ to contain all elements of corresponding
tuples). Let the field $\mathbb{L}$ be an algebraic closure of the field
$\mathbb{\widetilde{L}}$. Since $\mathbb{K}$ is algebraically closed,
$\mathbb{L}$ is a subfield of $\mathbb{K}$. The field $\mathbb{\widetilde{L}}$
is a countable field and thus $\mathbb{L}$ is also a countable field as an
algebraic closure of a countable field, so $\mathbb{L}$ is a countable
extension of $\mathbb{Q}$. By construction of $\mathbb{L}$ there exists an
injection from $\mathbb{L}$ to $\mathbb{K}$. This injection is a bijection on
tensors $t_{i}$, so it preserves the whole construction of obtaining large
matrix multiplications using arbitrary restrictions. Since the field
$\mathbb{L}$ is a countable extension of $\mathbb{Q}$, there exists an
injection from $\mathbb{L}$ to $\mathbb{C}$. Let us denote by $t_{i}^{\prime}$
the preimages of $t_{i}$ in injection from $\mathbb{L}$ to $\mathbb{K}$ and by
$\widetilde{t_{i}}$ the images of $t_{i}^{\prime}$ in injection from
$\mathbb{L}$ to $\mathbb{C}$. Now we can induce a construction of large matrix
multiplications in $\mathbb{C}$ using tensors $\widetilde{t_{i}}$ and
arbitrary restrictions.
We now want to obtain a contradiction with Proposition 19. To do so, we need
to prove that tensors $\widetilde{t_{i}}$ are unstable and
$N(\widetilde{t_{i}})=n$. By [3, Theorem 20] combinatorial instability implies
instability, so it is sufficient to show that tensors $\widetilde{t_{i}}$ are
unstable and $N(\widetilde{t_{i}})=n$. First we show that $t_{i}^{\prime}$ are
combinatorially unstable and $N(t_{i}^{\prime})=n$. The latter statement is
quite clear, since we have that $N(t_{i})=n$, which is obtained by
contractions with tensors $x_{i_{j}}$ (by definition) and because tensors
$x_{i_{j}}$ can be injected to $\mathbb{L}$, we have also
$N(t_{i}^{\prime})=n$. Being combinatorially unstable is also quite clear -
because we added to $\mathbb{L}$ tensors $t_{i}$, supports of these tensors
over $\mathbb{K}$ and $\mathbb{L}$ are the same sets. Since being
combinatorially unstable is a property of tensor support, and supports over
both fields are equal, being combinatorially unstable over $\mathbb{K}$
implies being combinatorially unstable over $\mathbb{L}$. Because $\mathbb{L}$
can be injected to $\mathbb{C}$, we can inject $x_{i_{j}}$ to $\mathbb{C}$, so
analogically $\widetilde{t_{i}}$ are combinatorially unstable. Also $t_{i}$
can be injected to $\mathbb{C}$, so the support of $\widetilde{t_{i}}$ is
equal to supports of $t_{i}$ and $t_{i}^{\prime}$, so we also have
$N(\widetilde{t_{i}})=n$. ∎
###### Remark 21.
It is well-known (a result due to Schönhage) that $\omega$ depends only on the
characteristic of the field, not on the field itself [7, Corollary 15.18].
However, in Proposition 19 and Proposition 20 we do not show that $\omega$ in
characteristic zero has some specific value, but that it cannot be proved that
$\omega=2$ using arbitrary restrictions from the class of starting tensors.
Thus, the generalisation from the case of $\mathbb{C}$ to the case of any
field of characteristic zero needed justification.
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|
# From Hyperbolic Geometry Back to Word Embeddings
Sultan Nurmukhamedov
Yandex School of Data Analysis
<EMAIL_ADDRESS>
&Thomas Mach
University of Potsdam
<EMAIL_ADDRESS>Arsen Sheverdin
University of Amsterdam
<EMAIL_ADDRESS>
&Zhenisbek Assylbekov
Nazarbayev University
<EMAIL_ADDRESS>
###### Abstract
We choose random points in the hyperbolic disc and claim that these points are
already word representations. However, it is yet to be uncovered which point
corresponds to which word of the human language of interest. This
correspondence can be approximately established using a pointwise mutual
information between words and recent alignment techniques.
From Hyperbolic Geometry Back to Word Embeddings
Sultan Nurmukhamedov Yandex School of Data Analysis<EMAIL_ADDRESS>Thomas
Mach University of Potsdam<EMAIL_ADDRESS>
Arsen Sheverdin University of Amsterdam<EMAIL_ADDRESS>Zhenisbek Assylbekov Nazarbayev University<EMAIL_ADDRESS>
## 1 Introduction
Vector representations of words are ubiquitous in modern natural language
processing (NLP). There are currently two large classes of word embedding
models: they build (1) static and (2) contextualized word vectors
correspondingly.
Static embeddings map each word type into a vector of real numbers, regardless
of the context in which the word type is used. The most prominent
representatives of this class of models are word2vec Mikolov et al. (2013b, a)
and GloVe Pennington et al. (2014). The obvious problem with this approach is
the representation of polysemous words, such as bank—it becomes unclear
whether we are talking about a financial institution, or we are talking about
the river bank.
Contextualized word embeddings, such as ELMo Peters et al. (2018) and BERT
Devlin et al. (2019), solve this problem by mapping each word token into a
vector space depending on the context in which the given word token is used,
i.e. the same word will have different vector representations when used in
different contexts. The second approach can nowadays be considered mainstream,
despite relatively few papers offering theoretical justifications for
contextualized word embeddings.
For static embeddings, on the contrary, there is a number of theoretical
works, each of which offers its own version of what is happening when word
vectors are trained. An incomplete list of such works includes those of Levy
and Goldberg (2014), Arora et al. (2016), Hashimoto et al. (2016), Gittens et
al. (2017), Tian et al. (2017), Ethayarajh et al. (2019), Allen et al. (2019),
Allen and Hospedales (2019), Assylbekov and Takhanov (2019), Zobnin and
Elistratova (2019). Other advantages of static embeddings over contextualized
ones include faster training (few hours instead of few days) and lower
computing requirements (1 consumer-level GPU instead of 8–16 non-consumer
GPUs). Morevoer, static embeddings are still an integral part of deep neural
network models that produce contextualized word vectors, because embedding
lookup matrices are used at the input and output (softmax) layers of such
models. Therefore, we consider it necessary to further study static
embeddings.
Several recent works Nickel and Kiela (2017); Tifrea et al. (2019) argue that
static word embeddings should be better trained in hyperbolic spaces than in
Euclidean spaces, and provide empirical evidence that word embeddings trained
in hyperbolic spaces need less dimensions to achieve the same quality as
state-of-the-art Euclidean vectors.111The quality of word vectors is usually
measured by the performance of downstream tasks, such as similarity,
analogies, part-of-speech tagging, etc. Usually such works motivate the
hyperbolicity of word embeddings by the fact that hyperbolic spaces are better
suited for embedding hierarchical structures. Words themselves often denote
concepts with an underlying hierarchy. An example of such a hierarchy is the
WordNet database, an excerpt of which is shown in Fig. 1.
carnivorefelinebig catliontigercatcaninedogwolffox Figure 1: An excerpt from
the WordNet database.
In the present paper we will investigate where the hyperbolicity originates
from. If we take the state-of-the-art Euclidean embeddings, is it possible to
establish a direct connection between them and their counterparts from a
hyperbolic word embedding? This was answered positively by Assylbekov and
Jangeldin (2020) who established a chain of connections: from word embeddings
to co-occurrence matrices, then to complex networks, and, finally, to
hyperbolic spaces. In this paper, to provide an additional justification for
the constructed chain, we propose a way to move from the final point,
hyperbolic spaces, to the initial one, word embeddings. We show that drawing
random points from the hyperbolic plane results in a set of points that
reasonably well resembles word embeddings. In fact, we can match these points
to word embeddings. Contrary, the same trick does not work with points drawn
at random in the Euclidean space. Thus, one can argue that the hyperbolic
space provides the underlying structure for word embeddings, while in the
Euclidean space this structure has to be superimposed.
### Notation
We denote with $\mathbb{R}$ the real numbers. Bold-faced lowercase letters
($\mathbf{x}$) denote vectors, plain-faced lowercase letters ($x$) denote
scalars, bold-faced uppercase letters ($\mathbf{A}$) denote matrices,
$\langle\mathbf{x},\mathbf{y}\rangle$ is the Euclidean inner product. We use
$\mathbf{A}_{a:b,c:d}$ to denote a submatrix located at the intersection of
rows $a,a+1,\ldots,b$ and columns $c,c+1,\ldots,d$ of $\mathbf{A}$. ‘i.i.d.’
stands for ‘independent and identically distributed’, ‘p.d.f’ stands for
‘probability distribution function’. We use the sign $\propto$ to abbreviate
‘proportional to’, and the sign $\sim$ to abbreviate ‘distributed as’.
Assuming that words have already been converted into indices, let
$\mathcal{W}:=\\{1,\ldots,n\\}$ be a finite vocabulary of words. Following the
setup of the widely used word2vec model Mikolov et al. (2013a, b), we use two
vectors per each word $i$: (1) $\mathbf{w}_{i}\in\mathbb{R}^{d}$ when
$i\in\mathcal{W}$ is a center word, (2) $\mathbf{c}_{i}\in\mathbb{R}^{d}$ when
$i\in\mathcal{W}$ is a context word; and we assume that $d\ll n$.
In what follows we assume that our dataset consists of co-occurence pairs
$(i,j)$. We say that “the words $i$ and $j$ co-occur” when they co-occur in a
fixed-size window of words. Let $\\#(i,j)$ be the number of times the words
$i$ and $j$ co-occur.
Figure 2: Random hyperbolic graph.
Figure 3: Distribution of PMI values (top) and of $R-X$.
## 2 Background: From Word Embeddings to Hyperbolic Space
Our departure point is the skip-gram with negative sampling (SGNS) word
embedding model of Mikolov et al. (2013b) that maximizes the following
objective function
$\sum_{i\in\mathcal{W}}\sum_{j\in\mathcal{W}}\\#(i,j)\log\sigma(\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle)\\\
+k\cdot\mathbb{E}_{j^{\prime}\sim
p}[\log\sigma(-\langle\mathbf{w}_{i},\mathbf{c}_{j^{\prime}}\rangle)],$ (1)
where $\sigma(x)=\frac{1}{1+e^{-x}}$ is the logistic sigmoid function, $p$ is
a smoothed unigram probability distribution for words,222The authors of SGNS
suggest $p(i)\propto\\#(i)^{3/4}$. and $k$ is the number of negative samples
to be drawn. Interestingly, training SGNS is approximately equivalent to
finding a low-rank approximation of a shifted pointwise mutual information
(PMI) matrix Levy and Goldberg (2014) in the form
$\log\frac{p(i,j)}{p(i)p(j)}-\log
k\approx\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle,$ (2)
where the left-hand side is the shifted PMI between $i$ and $j$, and the
right-hand side is an $ij$-th element of a matrix with rank $\leq d$ since
$\mathbf{w}_{i},\mathbf{c}_{j}\in\mathbb{R}^{d}$. This approximation was later
re-derived by Arora et al. (2016), Zobnin and Elistratova (2019), Assylbekov
and Takhanov (2019), and Allen et al. (2019) under different sets of
assuptions. In a recent paper, Assylbekov and Jangeldin (2020) showed that the
removal of the sigmoid transformation in the SGNS objective (1) gives word
embeddings comparable in quality with the original SGNS embeddings. A
maximization of such modified objective results in a low-rank approximation of
a squashed shifted PMI ($\sigma$SPMI) matrix, defined as
$\mathbf{A}_{ij}:=\sigma\left(\log\frac{p(i,j)}{p(i)p(j)}-\log k\right).$ (3)
Moreover, treating the $\sigma$SPMI matrix as a connection probabilities
matrix of a random graph, the authors show that such graph is a complex
network, that is it has strong clustering and scale-free degree distribution,
and according to Krioukov et al. (2010), such graph possesses an effective
hyperbolic geometry underneath. The following chain summarizes this argument:
$\boxed{\text{Word
Embeddings}}\quad\longrightarrow\quad\boxed{\sigma\text{SPMI}}\quad\longrightarrow\\\
\boxed{\text{Complex Network}}\quad\longrightarrow\quad\boxed{\text{Hyperbolic
Space}}$ (4)
In our work, we go from the final point (hyperbolic space) to the starting one
(word embeddings), and the next section provides the details of our method.
Method | Word Similarity | POS Tagging
---|---|---
WS353 | Men | M. Turk | CoNLL-2000 | Brown
SGNS | .678 | .656 | .690 | 90.77 | 92.60
PMI + SVD | .669 | .674 | .666 | 92.25 | 93.76
$\sigma$SPMI + SVD | .648 | .622 | .666 | 92.76 | 93.78
RHG + SVD + Align | .406 | .399 | .509 | 92.23 | 93.19
Random + Align | .165 | .117 | .111 | 81.89 | 89.39
Table 1: Evaluation of word embeddings on the similarity and POS tagging
tasks. For the similarity tasks the evaluation metric is the Spearman’s
correlation with human ratings, for the POS tagging tasks it is accuracy.
Random stands for random vectors that were obtained as i.i.d. draws from
$\mathcal{N}(\mathbf{0},\mathbf{I})$.
## 3 Method: From Hyperbolic Geometry to Word Embeddings
It is difficult to visualize hyperbolic spaces because they cannot be
isometrically embedded into any Euclidean space.333This means that we cannot
map points of a hyperbolic space into points of a Euclidean space in such way
that the distances between points are preserved. However, there exist models
of hyperbolic spaces: each model emphasizes different aspects of hyperbolic
geometry, but no model simultaneously represents all of its properties. We
will consider here the so-called native model Krioukov et al. (2010), in which
the hyperbolic plane $\mathbb{H}^{2}$ is represented by a disk of radius $R$,
and we use polar coordinates $(r,\theta)$ to specify the position of any point
$v\in\mathbb{H}^{2}$, where the radial coordinate $r$ equals the hyperbolic
distance of $v$ from the origin. Given this notation, the distance $x$ between
two points with coordinates $(r,\theta)$ and $(r^{\prime},\theta^{\prime})$
satisfies the hyperbolic law of cosines
$\textstyle\cosh x=\cosh r\cosh r^{\prime}\\\ -\sinh r\sinh
r^{\prime}\cos(\theta-\theta^{\prime}),$ (5)
for the hyperbolic space of constant curvature $-1$.444Defining constant
curvature is beyond the scope of our paper. We just mention here that there
are only three types of isotropic spaces: Euclidean (zero curvature),
spherical (positively curved), and hyperbolic (negatively curved). A key
property of hyperbolic spaces is that they expand faster than Euclidean
spaces. E.g., a circle with radius $r$ has in the Euclidean plane a length of
$2\pi r=\Theta(r)$ and an area of $\pi r^{2}=\Theta(r^{2})$, while its length
and area in the hyperbolic plane are $2\pi\sinh(r)=\Theta(e^{r})$ and
$2\pi(\cosh r-1)=\Theta(e^{r})$ correspondingly. It is noteworthy that in a
balanced tree with branching factor $b$, the number of nodes that are $r$
edges from the root grows as $\Theta(b^{r})$, i.e. exponentially with $r$,
leading to the suggestion that hierarchical complex networks with tree-like
structures might be easily embeddable in hyperbolic space.
Based on the above facts, we construct a random hyperbolic (RHG) graph as in
the work of Krioukov et al. (2010): we place randomly $n$ points (nodes) into
a hyperbolic disk of radius $R$, and each pair of nodes $(i,j)$ is connected
with probability $\sigma(R-x_{ij})$, where $x_{ij}$ is the hyperbolic distance
(5) between points $i$ and $j$. Angular coordinates of the nodes are sampled
from the uniform distribution: $\theta\sim\mathcal{U}[0,2\pi]$, while the
radial coordinates are sampled from the exponential p.d.f.
$\rho(r)=\frac{\alpha\sinh\alpha r}{\cosh\alpha R-1}=\Theta(e^{\alpha r}).$
The hyperparameters $R$ and $\alpha$ are chosen based on the total number of
nodes $n$, the desired average degree $\bar{k}$ and the power-law exponent
$\gamma$ according to the equations (22) and (29) of Krioukov et al. (2010).
An example of such RHG is shown in Figure 3. Notice, that the connection
probabilities matrix of our graph is
$\mathbf{B}_{ij}:=\sigma(R-x_{ij}),$
Comparing this to (3), we see that if $\mathbf{A}$ and $\mathbf{B}$ induce
structurally similar graphs then the distribution of the PMI values
$\log\frac{p(i,j)}{p(i)p(j)}$ should be similar to the distribution of
$R-x_{ij}$ values (up to a constant shift). To test this empirically, we
compute a PMI matrix of a well-known corpus, text8, and compare the
distribution of the PMI values with the p.d.f. of $R-X$, where $X$ is a
distance between two random points of a hyperbolic disk (the exact form of
this p.d.f. is given in Proposition A.1). The results are shown in Figure 3.
As we can see, the two distributions are similar in the sense that both are
unimodal and right-skewed. The main difference is in the shift—distribution of
$R-X$ is shifted to the left compared to the distribution of the PMI values.
We hypothesize that the nodes of the RHG treated as points of the hyperbolic
space are already reasonable word embeddings for the words of our vocabulary
$\mathcal{W}$. The only thing that we do not know is the correspondence
between words $i\in\mathcal{W}$ and nodes of the RHG. Instead of aligning
words with nodes, we can align their vector representations. For this, we take
singular value decompositions (SVD) of $\mathbf{A}$ and $\mathbf{B}$:
$\mathbf{A}=\mathbf{U}_{A}\boldsymbol{\Sigma}_{A}\mathbf{V}_{A}^{\top},\quad\mathbf{B}=\mathbf{U}_{B}\boldsymbol{\Sigma}_{B}\mathbf{V}_{B}^{\top},$
and then obtain embedding matrices by
$\displaystyle\mathbf{W}_{A}$
$\displaystyle:=\mathbf{U}_{A,1:n,1:d}\boldsymbol{\Sigma}^{1/2}_{A,1:d,1:d}\in\mathbb{R}^{n\times
d}$ $\displaystyle\mathbf{W}_{B}$
$\displaystyle:=\mathbf{U}_{B,1:n,1:d}\boldsymbol{\Sigma}^{1/2}_{B,1:d,1:d}\in\mathbb{R}^{n\times
d}$
as in the work of Levy and Goldberg (2014). An $i^{\text{th}}$ row in
$\mathbf{W}_{A}$ is an embedding of the word $i\in\mathcal{W}$, while an
$i^{\text{th}}$ row in $\mathbf{W}_{B}$ is an embedding of the RHG’s node $i$.
To align these two sets of embeddings we apply a recent stochastic
optimization method of Grave et al. (2019) that solves
$\min_{\mathbf{Q}\in\mathcal{O}_{d}}\min_{\mathbf{P}\in\mathcal{P}_{n}}\|\mathbf{W}_{A}\mathbf{Q}-\mathbf{P}\mathbf{W}_{B}\|^{2}_{2},$
where $\mathcal{O}_{d}$ is the set of $d\times d$ orthogonal matrices and
$\mathcal{P}_{d}$ is the set of $n\times n$ permutation matrices. As one can
see, this method assumes that alignment between two sets of embeddings is not
only a permutation from one set to the other, but also an orthogonal
transformation between the two. Once the alignment is done, we treat
$\mathbf{PW}_{B}$ as an embedding matrix for the words in $\mathcal{W}$.
## 4 Evaluation
In this section we evaluate the quality of word vectors resulting from a
RHG555Our code is available at https://github.com/soltustik/RHG against those
from the SGNS, PMI, and $\sigma$SPMI. We use the text8 corpus mentioned in the
previous section. We were ignoring words that appeared less than 5 times
(resulting in a vocabulary of 71,290 tokens). We set window size to 2,
subsampling threshold to $10^{-5}$, and dimensionality of word vectors to 200.
The SGNS embeddings were trained using our custom
implementation.666https://github.com/zh3nis/SGNS The PMI and BPMI matrices
were extracted using the hyperwords tool of Levy et al. (2015) and SVD was
performed using the PyTorch library of Paszke et al. (2019).
The embeddings were evaluated on word similarity and POS tagging tasks. For
word similarity we used WordSim (Finkelstein et al., 2002), MEN (Bruni et al.,
2012), and M.Turk (Radinsky et al., 2011) datasets. For POS tagging we trained
a simple classifier777feedforward neural network with one hidden layer and
softmax output layer by feeding in the embedding of a current word and its
nearby context to predict its part-of-speech (POS) tag:
$\widehat{\mathrm{POS}}_{t}=\operatorname{softmax}(\sigma(\mathbf{A}[\mathbf{w}_{t-2};\ldots;\mathbf{w}_{t+2}]+\mathbf{b}))$
where $[\mathbf{x};\mathbf{y}]$ is concatenation of $\mathbf{x}$ and
$\mathbf{y}$. The classifier was trained on CoNLL-2000 Tjong Kim Sang and
Buchholz (2000) and Brown Kucera et al. (1967) datasets.
The results of evaluation are provided in Table 1. As we see, vector
representations of words generated from a RHG lag behind in word similarity
tasks from word vectors obtained by other standard methods. Note, however,
that the similarity task was designed with Euclidean geometry in mind. Even
though our RHG-based vectors are also ultimately placed in the Euclidean space
(otherwise the alignment step would not have been possible), their nature is
inherently non-Euclidean. Therefore, the similarity scores for them may not be
indicative. So, for example, when RHG vectors are fed into a nonlinear model
for POS tagging, they are comparable with other types of vectors.
We notice that random vectors—generated as i.i.d. draws from
$\mathcal{N}(\mathbf{0},\mathbf{I})$ and then aligned to the embeddings from
$\sigma$SPMI—show poor results in the similarity tasks and underperform all
other word embedding methods in the POS tagging tasks. This calls into
question whether multivariate Gaussian is a reasonable (prior) distribution
for word vectors as was suggested by Arora et al. (2016), Assylbekov and
Takhanov (2019).
## 5 Conclusion and Future Work
In this work we show that word vectors can be obtained from hyperbolic
geometry without explicit training. We obtain the embeddings by randomly
drawing points in the hyperbolic plane and by finding correspondence between
these points and the words of the human language. This correspondence is
determined by the relation (hyperbolic distance) to other words. This method
avoids the, often expensive, training of word vectors in hyperbolic spaces as
in Tifrea et al. (2019). A direct comparison is not what this paper
attempts—our method is cheaper but produces word vectors of lower quality. Our
method simply shows that word vectors do fit better into hyperbolic space than
into Euclidean space.
Finally, we want to sketch a possible direction for future work. The
hyperbolic space is a special case of a Riemannian manifold. Are Riemannian
manifolds better suited for word vectors? In particular which manifolds should
one use? At the moment, there is only limited empirical knowledge to address
these questions. For instance, Gu et al. (2019) obtained word vectors of
better quality, according to the similarity score, in the product of
hyperbolic spaces, which is still a Riemannian manifold but not a hyperbolic
space anymore. We are hopeful that future work may provide an explanation for
this empirical fact.
## Acknowledgements
Zhenisbek Assylbekov was supported by the Program of Targeted Funding “Economy
of the Future” #0054/ ПЦФ-НС-19. The work of Sultan Nurmukhamedov was
supported by the Nazarbayev University Faculty-Development Competitive
Research Grants Program, grant number 240919FD3921. The authors would like to
thank anonymous reviewers for their feedback.
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## Appendix A Auxiliary Results
###### Proposition A.1.
Let $X$ be a distance between two points that were randomly uniformly placed
in the hyperbolic disk of radius $R$. The probability distribution function of
$X$ is given by
$f_{X}(x)=\int_{0}^{R}\int_{0}^{R}\\\
\frac{\sinh(x)\rho(r_{1})\rho(r_{2})dr_{1}dr_{2}}{\pi\sqrt{1-A(r_{1},r_{2},x)}\sinh(r_{1})\sinh(r_{2})},$
(6)
where
$A(r_{1},r_{2},x)=\frac{\cosh(r_{1})\cosh(r_{2})-\cosh(x)}{\sinh(r_{1})\sinh(r_{2})}$,
and $\rho(r)=\frac{\alpha\sinh\alpha r}{\cosh\alpha R-1}$.
###### Proof.
Let us throw randomly and uniformly two points $(r_{1},\theta_{1})$ and
$(r_{2},\theta_{2})$ into the hyperbolic disk of radius $R$, i.e.
$r_{1},r_{2}\,\,{\stackrel{{\scriptstyle\text{i.i.d.}}}{{\sim}}}\,\,\rho(r)$,
$\theta_{1},\theta_{2}\,\,{\stackrel{{\scriptstyle\text{i.i.d.}}}{{\sim}}}\,\,\text{Uniform}[0,2\pi)$.
Let $X$ be the distance between these points ($X$ is a random variable). Let
$\gamma$ be the angle between these points, then
$\gamma:=\pi-|\pi-|\theta_{1}-\theta_{2}||\sim\text{Uniform}[0,\pi)$ and thus
$f_{\cos\gamma}(t)=\frac{1}{\pi\sqrt{1-t^{2}}},\quad t\in[-1,1].$
Since the distance in our model of hyperbolic plane is given by
$X=\cosh^{-1}[\cosh r_{1}\cosh r_{2}-\sinh r_{1}\sinh r_{2}\cos\gamma]$
we have
$\displaystyle\Pr(X\leq x)$
$\displaystyle=\Pr\left(\cos\gamma\geq\underbrace{\frac{\cosh r_{1}\cosh
r_{2}-\cosh x}{\sinh r_{1}\sinh r_{2}}}_{A(r_{1},r_{2},x)}\right)$
$\displaystyle=\Pr(\cos\gamma\geq A(r_{1},r_{2},x))$
$\displaystyle=\int_{A(r_{1},r_{2},x)}^{+\infty}\frac{1}{\pi\sqrt{1-t^{2}}}$
$\displaystyle=\frac{1}{2}-\frac{\sin^{-1}A(r_{1},r_{2},x)}{\pi},$
and therefore
$f_{X\mid
r_{1},r_{2}}(x)=\frac{d}{dx}\left[\frac{1}{2}-\frac{\sin^{-1}A(r_{1},r_{2},x)}{\pi}\right]\\\
=\frac{\sinh x}{\pi\sqrt{1-A(r_{1},r_{2},x)}\sinh(r_{1})\sinh r_{2}}$
for $x\in(|r_{1}-r_{2}|,r_{1}+r_{2})$. Integrating $f_{X\mid
r_{1},r_{2}}(x)\rho(r_{1})\rho(r_{2})$ with respect to $r_{1}$ and $r_{2}$ we
get (6). ∎
|
# Unravelling the structure of magnetised molecular clouds with SILCC-Zoom:
sheets, filaments and fragmentation
Shashwata Ganguly,1 S. Walch,1,2 D. Seifried,1,2 S. D. Clarke1,3 and M. Weis1
1I. Physikalisches Insitut, Universität zu Köln, Zülpicher Str. 77, 50937
Köln, Germany
2Cologne Centre for Data and Simulation Science, University of Cologne,
Cologne, Germany
3Academia Sinica, Institute of Astronomy and Astrophysics, Taipei, Taiwan
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
To what extent magnetic fields affect how molecular clouds (MCs) fragment and
create dense structures is an open question. We present a numerical study of
cloud fragmentation using the SILCC-Zoom simulations. These simulations follow
the self-consistent formation of MCs in a few hundred parsec sized region of a
stratified galactic disc; and include magnetic fields, self-gravity,
supernova-driven turbulence, as well as a non-equilibrium chemical network. To
discern the role of magnetic fields in the evolution of MCs, we study seven
simulated clouds, five with magnetic fields, and two without, with a maximum
resolution of 0.1 parsec. Using a dendrogram we identify hierarchical
structures which form within the clouds. Overall, the magnetised clouds have
more mass in a diffuse envelope with a number density between 1-100 cm-3. We
find that six out of seven clouds are sheet-like on the largest scales, as
also found in recent observations, and with filamentary structures embedded
within, consistent with the bubble-driven MC formation mechanism. Hydrodynamic
simulations tend to produce more sheet-like structures also on smaller scales,
while the presence of magnetic fields promotes filament formation. Analysing
cloud energetics, we find that magnetic fields are dynamically important for
less dense, mostly but not exclusively atomic structures (typically up to
$\sim 100-1000$ cm-3), while the denser, potentially star-forming structures
are energetically dominated by self-gravity and turbulence. In addition, we
compute the magnetic surface term and demonstrate that it is generally
confining, and some atomic structures are even magnetically held together. In
general, magnetic fields delay the cloud evolution and fragmentation by $\sim$
1 Myr.
###### keywords:
MHD – methods: numerical – stars: formation – ISM: clouds – ISM: kinematics
and dynamics
††pubyear: 2022††pagerange: Unravelling the structure of magnetised molecular
clouds with SILCC-Zoom: sheets, filaments and fragmentation–F
## 1 Introduction
Magnetic fields are ubiquitous in the interstellar medium (ISM, Crutcher et
al., 2003; Heiles & Troland, 2005; Fletcher et al., 2011; Beck, 2015). Since
the discovery of interstellar magnetic fields by Hiltner (1951) and Hall
(1951), they have been known to be integral to the dynamical evolution of the
ISM. Magnetic fields, however, are also notoriously difficult to measure
accurately and model theoretically. Decades of dedicated observations have
resulted in a good understanding of the morphology and strength of the
magnetic field in different ISM phases (Crutcher, 1999; Bourke et al., 2001;
Heiles & Crutcher, 2005; Troland & Crutcher, 2008; Crutcher, 2012; Beck, 2015;
Planck Collaboration et al., 2020; Lopez-Rodriguez et al., 2023).
However, the exact nature of how magnetic fields affect molecular cloud (MC)
formation and evolution is an open question and subject of intense scrutiny
(see e.g. reviews by Crutcher, 2012; Hennebelle & Inutsuka, 2019; Girichidis
et al., 2020; Pattle et al., 2022). Various numerical studies have performed
detailed analysis on the interplay of magnetic fields with other physical
processes (e.g. turbulence, thermal pressure) in order to determine how MCs
are shaped, formed, and how they evolve (e.g. Heitsch et al., 2001; Federrath
& Klessen, 2012; Walch et al., 2015; Körtgen & Banerjee, 2015; Girichidis et
al., 2016b; Körtgen et al., 2018; Seifried et al., 2019; Ibáñez-Mejía et al.,
2022).
On galactic scales, ordered magnetic fields have been observed, with a
correlation between the direction of the spiral arms and the magnetic field
(Beck, 2009; Fletcher et al., 2011; Li & Henning, 2011). In the diffuse ISM,
the magnetic field strength, $B$, does not show any correlation with the
density for number densities of up to roughly 300 cm-3 (Crutcher et al.,
2010). Above these densities, Crutcher et al. (2010) find
$B\propto\rho^{\kappa}$, with $\kappa\approx 2/3$, consistent with sub-
dominant magnetic field strengths, although there remains considerable scatter
in the observations.
The lack of correlation between the strength of the magnetic field and the
density of the ambient medium implies that in the diffuse ISM, magnetic fields
can channelise gas flows along the field lines and therefore influence the
environment in which MCs form. Pardi et al. (2017) show that magnetic fields
are more likely to cause a smoother gas distribution, while Molina et al.
(2012) find that they are more likely to affect the dynamics of lower-density
gas. Magnetic fields can add to the thermal pressure exerted by the gas and
slow down the formation of dense gas (Hill et al., 2012), as well as molecular
gas (Girichidis et al., 2018; Seifried et al., 2020a). A sufficiently strong
magnetic field can prevent the collapse of a MC altogether (Mouschovias, 1991;
Spitzer, 1978) or slow down cloud evolution (Heitsch et al., 2001; Padoan &
Nordlund, 2011; Federrath & Klessen, 2012; Ibáñez-Mejía et al., 2022).
In terms of morphology, they can facilitate the formation of elongated
filamentary structures (Hacar et al., 2022; Pineda et al., 2022) and are
essential in understanding the filamentary nature of the ISM (see e.g. Bally
et al., 1987; André et al., 2014). The direction of such elongation relative
to the direction of the magnetic field is a matter of great active research
(e.g. Soler & Hennebelle, 2017; Seifried et al., 2020b). In the lower density
range, for sub-Alfvénic gas, anisotropic turbulence can lead to structures
elongated parallel to field lines. In contrast, at higher densities, magnetic
fields can channelise flows along field lines and therefore facilitate
structures perpendicular to the field direction.
Magnetic fields are likely to also affect the fragmentation of clouds and
cloud cores. Commerçon et al. (2011) find that fragments in magnetized cloud
cores are more massive compared to those formed without magnetic fields.
Although the probability density function (PDF) of lower density gas is found
to be different in the presence of magnetic fields (Molina et al., 2012), the
high density, potentially star-forming part does not seem to significantly
affected (Klessen & Burkert, 2001; Slyz et al., 2005; Girichidis et al., 2014;
Schneider et al., 2015).
In this work, we perform a numerical investigation of the role that magnetic
fields play in the formation and shaping of density structures within MCs. We
do a detailed analysis of realistic MC simulations based on the SILCC-Zoom
simulations (Seifried et al., 2017) by comparing the morphological, dynamical,
and fragmentation properties in seven simulated clouds, five with magnetic
fields (magnetohydrodynamic or MHD clouds) and two without (hydrodynamic or HD
clouds).
The paper is structured as follows: In Section 2, we outline the numerical
setup of the simulation. Section 3 discusses the procedure for identifying and
classifying structures (Ganguly et al., 2022). We highlight the differences
density PDFs between HD and MHD clouds in Section 4. The morphological
properties of the obtained structures are presented in Section 5. We find all
the MCs to be sheet-like on the largest scales (tens of parsecs). On smaller
scales, we see that the presence of magnetic fields enhances the formation of
filamentary over sheet-like sub-structures. In Section 6, we analyse the
dynamics and energetic balance of magnetized structures and relate them to the
fragmentation of cloud sub-structures. We find that the presence of magnetic
fields slows down cloud evolution and, in particular, leads to more massive
fragments at low to intermediate densities (<100 cm-3). We attempt to make an
order of magnitude estimate of this slow-down effect in Section 6.5. Finally,
we present the summary of our findings in Section 7.
## 2 Numerical methods and simulation
We present here results based on the SILCC-Zoom simulations (Seifried et al.,
2017; Seifried et al., 2019). The SILCC-Zoom simulations are MCs with
realistic boundary conditions, generated by embedding the clouds within the
SILCC simulations of multi-phase interstellar gas, thus having realistic
initial conditions (Walch et al., 2015; Girichidis et al., 2016a). In this
section, we highlight some key features of the simulations. More details on
the simulations can be found in Seifried et al. (2017) and Seifried et al.
(2019).
All simulations were executed using the adaptive mesh refinement code FLASH,
version 4 (Fryxell et al., 2000; Dubey et al., 2008), which solves the ideal
MHD equations for an ideal fluid. If we consider a fluid parcel of density
$\rho$, velocity $\mathbf{v}$, total energy $e_{\mathrm{tot}}$, and magnetic
field vector $\mathbf{B}$ (zero if pure hydrodynamics), these are given as
follows:
$\displaystyle\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{v})=0,$
(1) $\displaystyle\frac{\partial\rho\mathbf{v}}{\partial
t}+\nabla\cdot\left[\rho\mathbf{v}\otimes\mathbf{v}+\left(P+\frac{B^{2}}{8\pi}\right)\mathbf{I}-\frac{\mathbf{B}\otimes\mathbf{B}}{4\pi}\right]=\rho\mathbf{g},$
(2) $\displaystyle\frac{\partial e_{\mathrm{tot}}}{\partial
t}+\nabla\cdot\left[\left(e_{\mathrm{tot}}+P\right)\mathbf{v}-\frac{(\mathbf{B}\cdot\mathbf{v})\mathbf{B}}{4\pi}\right]=\rho\mathbf{v}\cdot\mathbf{g}+\dot{u}_{\mathrm{heat}},$
(3) $\displaystyle\frac{\partial\mathbf{B}}{\partial
t}-\nabla\times(\mathbf{v}\times\mathbf{B})=0.$ (4)
Here, Eqs. 1 to 4 represent the conservation of mass, momentum, energy, and
magnetic flux, respectively. $P$ represents the thermal pressure, $\mathbf{g}$
is the local gravitational acceleration obtained from solving the Poisson
equation, $u$ is the internal energy, and $\dot{u}_{\mathrm{heat}}$ is the
internal energy input rate due to the combination of heating and cooling
processes. The $\otimes$ is the outer product (i.e.
$(\mathbf{a}\otimes\mathbf{b})_{ij}=a_{i}b_{j}$).
The total energy and the pressure are computed as follows:
$\displaystyle e_{\mathrm{tot}}=u+\frac{1}{2}\rho v^{2}+\frac{1}{8\pi}B^{2},$
(5) $\displaystyle P=(\gamma-1)u,$ (6)
with $\gamma$ being the adiabatic index.
Here, we present results from runs with and without magnetic fields. The MHD
simulations shown are performed using an entropy-stable solver that guarantees
minimum possible dissipation (Derigs et al., 2016; Derigs et al., 2018). The
hydrodynamic simulations have been performed using the MHD Bouchut 5-wave
solver (Bouchut et al., 2007; Waagan, 2009) that guarantees positive entropy
and density. The magnetic field strength has been set to zero for these runs.
All simulations include self-gravity as well as an external galactic potential
due to the presence of old stars. This external potential is calculated
assuming a stellar population density of $\Sigma_{\rm
star}=30~{}\mathrm{M}_{\odot}~{}\mathrm{pc}^{-2}$, a sech2 vertical profile
and a scale height of 100 pc, according to Spitzer (1942). The self-gravity of
the gas is calculated using a tree-based algorithm (Wünsch et al., 2018).
The entire simulation domain consists of a box of 500 pc $\times$ 500 pc
$\times$ $\pm$ 5 kpc size, with the long axis representing the vertical
$z-$direction of a galactic disc. The box is set with periodic boundary
conditions in the $x-$ and $y-$ direction, and outflow boundary condition in
the $z-$direction. The initial gas surface density is set to $\Sigma_{\rm
gas}=10\ \mathrm{M}_{\odot}~{}\mathrm{pc}^{-2}$ which corresponds to solar
neighbourhood conditions. The vertical distribution of the gas is modelled as
a Gaussian, i.e. $\rho=\rho_{0}\ \mathrm{exp}(-z^{2}/2h_{z}^{2})$, where
$h_{z}$=30 pc is the scale height and $\rho_{0}=9\times 10^{-24}$ g cm-3. The
initial gas temperature is set to 4500 K. For runs with magnetic fields, the
magnetic field is initialized along the $x-$direction, i.e.
$\mathbf{B}=(B_{x},0,0)$ with $B_{x}=B_{x,0}\sqrt{\rho(z)/\rho_{0}}$ and the
magnetic field strength at the midplane $B_{x,0}=3\ \mu$G. The field strength
is chosen to be in accordance with recent observations (e.g. Beck &
Wielebinski, 2013).
The turbulence in the simulations is generated by supernova explosions. The
explosion rate is set to 15 SNe Myr-1, which is consistent with the Kennicutt-
Schmidt relation, which observationally determines the star formation rate
surface density for a given gas surface density (Schmidt, 1959; Kennicutt,
1998). 50% of the supernovae are placed following a Gaussian random
distribution along the $z-$direction up to a height of 50 pc, while the other
50% are placed at density peaks of the gas. This prescription of supernova
driving creates a multi-phase turbulent ISM which can be used as initial
conditions for the zoom-in simulations (Walch et al., 2015; Girichidis et al.,
2016a).
Apart from the dynamics of the gas, we also model its chemical evolution using
a simplified non-equilibrium chemical network based on hydrogen and carbon
chemistry (Nelson & Langer, 1997; Glover & Mac Low, 2007; Glover et al.,
2010). For this purpose, we follow the abundance of H+, H, H2, CO, C+, e-, and
O. At the beginning of the simulation, all hydrogen in the disc midplane is
neutral and carbon is in its ionized form (i.e. H and C+, respectively).
To correctly model the chemistry of the gas, we include an interstellar
radiation field (ISRF) of strength $G_{0}=1.7$ in Habing units (Habing, 1968;
Draine, 1978). The attenuation of this radiation field is taken into
consideration by computing the true optical depth inside any given point in
the simulation domain. This is computed as follows:
$\mathrm{A_{V,3D}=-\frac{1}{2.5}ln\left[\frac{1}{{\it
N}_{PIX}}\sum_{i=1}^{{\it N}_{PIX}}exp\left(-2.5\frac{N_{H,tot,i}}{1.87\times
10^{21}\ \mathrm{cm}^{-2}}\right)\right]},$ (7)
where the sum is carried over each Healpix pixel, with $N_{\rm PIX}$ being the
total number of such pixels (here 48), and $N_{\mathrm{H,tot},i}$ is the
column density computed for the $i-$th pixel. In essence, for any given point,
we compute the column density along various lines of sight and use that for an
effective $\mathrm{A_{V,3D}}$. The averaging is performed in an exponential
manner because the intensity of radiation decreases in an exponential manner
due to extinction caused by the gas column density along the line of sight.
The calculation for this is performed by the TreeRay Optical Depth module
developed by Wünsch et al. (2018).
To study the formation of MCs, all supernova explosions are stopped at a
certain time $t_{0}$. Up to this point, the maximum grid resolution is 3.9 pc.
At time $t_{0}$, different regions are identified for the zoom-in process,
primarily by determining which regions form molecular gas when the simulations
are run further at the original SILCC resolution of 3.9 pc. The time $t=t_{0}$
refers to the start of the evolution of the different clouds and is set as an
evolutionary time $t_{\rm evol}=0$. The total simulation time $t$ is related
to the evolution time as
$t=t_{0}+t_{\rm evol}.$ (8)
From $t_{\rm evol}=0$ on, in the selected regions, the AMR grid is allowed to
be refined to a higher resolution to capture structures that form as MCs.
These regions are called zoom-in regions and are of primary importance to us
as sites of MCs. Each SILCC simulation we run contains two such "zoom-in"
boxes simultaneously. All runs present here have a maximum resolution of 0.125
pc. For details of how the zoom-in process is achieved, see Seifried et al.
(2017).
## 3 Classification of structures
For the analysis presented in this work, we look at eight different cubic
boxes of 62.5 pc in size, each from a different SILCC zoom-in region. These
boxes are chosen by visual inspection, in order to capture the most
interesting features contained in each zoom-in region. For the purpose of this
work, we will refer to these cubic regions as MCs. They are named MC1-HD and
MC2-HD for the two hydrodynamic clouds, and MC$x$-MHD for the MHD clouds,
where $x$ is between one and six. We present some basic details of the
different MCs in Table 1. A projected view of all the different MCs is added
in the Appendix A. For more information on the presented clouds, we refer the
reader to Seifried et al. (2017) for the HD clouds and Seifried et al. (2019)
for the MHD clouds.
Run name | MHD | $t_{0}$ | Total mass | H2 mass | $\langle\mathrm{B}\rangle$
---|---|---|---|---|---
| | [Myr] | $[10^{4}\,{\rm M}_{\odot}]$ | $[10^{4}\,{\rm M}_{\odot}]$ | [$\mu$G]
MC1-HD | no | 12 | 7.3 | 2.1 | 0
MC2-HD | no | 12 | 5.4 | 1.6 | 0
MC1-MHD | yes | 16 | 7.8 | 1.3 | 4.8
MC2-MHD | yes | 16 | 6.2 | 0.86 | 3.9
(MC3-MHDa | yes | 16 | 2.0 | 0.19 | 2.0)
MC4-MHD | yes | 11.5 | 6.8 | 1.2 | 6.4
MC5-MHD | yes | 11.5 | 10.1 | 1.6 | 6.8
MC6-MHD | yes | 16 | 6.6 | 1.4 | 4.3
Table 1: Basic information on the eight analysed simulations. From left to
right we list the run name, whether magnetic fields are present or not, the
time when the AMR "zoom-in" starts, as well as the total mass, molecular
hydrogen mass and the average magnetic field strength at $t_{\rm evol}=2$ Myr.
aWe discard MC3-MHD from our further analysis because of its low molecular gas
content and lack of interesting density features (see also Fig. 14).
We perform a detailed analysis of the different clouds, following their
evolution from $t_{\rm evol}=2$ Myr to $t_{\rm evol}=3.5$ Myr, primarily
focusing on the latter time. The beginning and the end time are chosen to look
at relatively early stages of structure formation in the MCs. We do not look
at times earlier than 2 Myr primarily because the clouds undergo the
refinement process and are not fully resolved until $t_{\rm evol}\sim 1.5$
Myr.
### 3.1 Structure identification
To identify structures in our MCs, we use a dendrogram algorithm (Rosolowsky
et al., 2008). Dendrogram is a model-independent method to determine
hierarchical structures in two and three dimensions. Since we are interested
in 3-dimensional structures, we perform the dendrogram analysis on
3-dimensional density cubes. We do not use the 3D AMR grid structure inherent
in the data, but rather convert it into a uniform mesh at 0.125 and 0.25 pc
resolution (see also Table 2).
Given an initial density field, $\rho$, the dendrogram essentially depends on
three free parameters: the initial starting threshold, $\rho_{0}$, the density
jump, $\Delta\rho$, and the minimum number of cells that need to be included
in any structure, $N_{\rm cells}$. Due to high density contrasts, we build the
dendrogram tree on the logarithmic density profile of the gas, and therefore
have used density bins of $\Delta\mathrm{log}_{10}\ \rho$, rather than
$\Delta\rho$. In addition to the three parameters mentioned, we can choose a
pruning peak, $\rho_{\rm prune}$, to allow the dendrogram to create new
structures only when such a structure will have peak density $\rho_{\rm
peak}>\rho_{\rm prune}$, although this has not been used in the present work.
Using these parameters, the dendrogram algorithm allows us to define volumes
of gas as structures in a hierarchical tree, primarily defined by their
threshold density $\rho_{\rm thr}$, which is the minimum density value inside
a given structure. This can be thought of as equivalent to contour values for
two dimensional maps. The hierarchy is characterised by different dendrogram
branches, where a branch is a given dendrogram structure and all its parent
structures, up to the largest and most diffuse ancestor in the dendrogram
tree.
For probing both the higher and lower density ends of the data, we perform two
dendrgram analyses on the same regions: a higher density dendrogram analysis
performed at a resolution of 0.125 pc for probing gas above densities of 10-22
g cm-3 (referred to as high-den), and a lower density analysis performed at
0.25 pc for gas between the densities of 10-24 and 10-22 g cm-3 (referred to
as low-den). The low-den values are computed as volume averaged values from
the higher resolution grid. We present the dendrogram parameters used for both
analyses in Table 2.
dendrogram | Resolution | $\rho_{0}$ | $\Delta$ log${}_{10}\ \rho$ | $N_{\rm cells}$ | $\rho_{\rm prune}$ | additional
---|---|---|---|---|---|---
type | [pc] | [g cm-3] | | | [g cm-3] | criteria
high-den | 0.125 | $10^{-22}$ | 0.1 | 100 | None | None
low-den | 0.25 | $10^{-24}$ | 0.2 | 100 | None | $\rho_{\rm thr}<10^{-22}$ g cm-3
Table 2: Information on the parameters used for the two different kinds of
dendrogram analyses. From left to right are: the type of dendrogram, the grid
resolution at which it is performed, the starting density, the logarithmic
density jump, the minimum number of cells in structures, the density of the
pruning peak used, and if any additional criteria were used to select
structures.
In addition to the difference in the basic parameters between the two
dendrogram analyses, we remove all structures with $\rho_{\rm thr}>10^{-22}$ g
cm-3 for the low-den analysis. This is done in order to avoid double counting
of structures.
The parameter values mentioned in Table 2 have been chosen from a mixture of
practical considerations, such as CPU memory, computation time, and through
trial and error. We note that in principle the same analysis could be
performed by a single dendrogram analysis at $\rho_{\rm thr}=10^{-24}$ g cm-3
at the highest resolution of 0.125 pc. However, the computation cost of such
an analysis was prohibitive in our case. Combining the high-den and low-den
dendrogram analyses allows us to probe a much higher density range than would
be otherwise possible.
In terms of the parameters used, we have seen no unexpected change in the
results by changing the free parameters within a reasonable range. We refer
the reader to our companion paper (Ganguly et al., 2022) for a more thorough
discussion of the effect of altering the parameter values on the analysis.
Overall, we find that changing the parameters, while resulting in a varying
number of obtained structures, leaves the statistical properties of the
structures virtually unaffected.
An example of the leaf density structures (structures that contain no further
sub-structures) from the dendrogram analysis can be seen in Fig. 1 for MC1-MHD
at $t_{\rm evol}=3.5$ Myr, as contours over column density maps. The three
panels show, from left to right, the cloud projected along the $x-$, $y-$ and
$z-$direction. The contours are drawn as projections of the 3D dendrogram
structure outlines in the projected direction. We distinguish between
structures depending on their molecular H2 content, by plotting structures
with over 50% of their total hydrogen mass in molecular form (referred to as
molecular structures) in solid lines and otherwise in dashed lines (referred
to as atomic structures).
Figure 1: Left to right: Projections of MC1-MHD at $t_{\rm evol}=$3.5 Myr
along the $x$-, $y$-, and $z$-axis, respectively. The contours show the
projections of the leaf dendrogram structures along the same axis. Molecular
structures ($>50\%\;\mathrm{H}_{2}$ mass fraction) are plotted with solid, and
atomic structures ($<50\%\;\mathrm{H}_{2}$ mass fraction) are plotted with
dashed lines. The molecular structures nicely trace the dense spine of the two
main filaments, while the atomic structures mostly represent the envelope.
Due to the nature of the dendrogram algorithm, there are some structures which
touch the edge of the box. This can lead to structures whose morphology is
determined by their proximity to the edge. To avoid this, we do not classify
the morphology of any structures which have more than 5% of their surface
cells touching any edge. This is relevant especially for the large-scale
structures from the low-den dendrogram analysis. However, they can still be of
interest while considering cloud dynamics, and in such a case we add them as
an additional category of "unclassified". While in a different context, Alves
et al. (2017) have shown the importance of having closed contours while
studying 2D maps. We have attempted to follow the same principle here as much
as possible.
### 3.2 Structure classification
Once we obtain the tree of dendrogram density sub-structures, we aim to
classify their morphology. For each structure, we compute an equivalent
ellipsoid that has the same mass and the same moments of inertia (MOI) as the
original structure. We then use the axes lengths of this equivalent ellipsoid
to classify the shape of the different structures.
Let us consider a uniform density ellipsoid of mass $M$ and semi-axes lengths
$a,\ b,\ c$ with $a\geq b\geq c$. The moments of inertia along the three
principal axes will be given as follows:
$\begin{split}I_{a}&=\frac{1}{5}M(b^{2}+c^{2}),\\\
I_{b}&=\frac{1}{5}M(c^{2}+a^{2}),\\\
I_{c}&=\frac{1}{5}M(a^{2}+b^{2}),\end{split}$ (9)
where $I_{c}\geq I_{b}\geq I_{a}$. If we now compute the principal moments of
inertia of our given dendrogram structure to be $A$, $B$ and $C$,
respectively, then the ellipsoid has an equivalent moment of inertia if
$A=I_{a},\ B=I_{b},\ C=I_{c}.$ (10)
This leads to the following equation for computing the axis lengths of the
equivalent ellipsoids:
$\begin{split}a&=\sqrt{\frac{5}{2M}(B+C-A)}\ ,\\\
b&=\sqrt{\frac{5}{2M}(C+A-B)}\ ,\\\ c&=\sqrt{\frac{5}{2M}(A+B-C)}\
.\end{split}$ (11)
We then use the aspect ratio of the semi-axes of the corresponding ellipsoid
and the position of the center of mass (COM) of the structure relative to its
boundary (i.e. whether the COM is contained by the structure itself) to
categorise the different structures into four categories - sheets, curved
sheets (referred to as sheet_c in this paper), filaments, and spheroids:
$\begin{split}&\textbf{sheet: }\frac{a}{b}\leq f_{\rm asp},\frac{a}{c}>f_{\rm
asp}\\\ &\textbf{filament: }\frac{a}{b}>f_{\rm asp}\\\ &\textbf{spheroidal:
}\frac{a}{c}\leq f_{\rm asp},\text{ contains its own COM}\\\
&\textbf{sheet\\_c: }\frac{a}{c}\leq f_{\rm asp},\text{ does not contain its
own COM}\end{split}$ (12)
where we set the aspect ratio factor $f_{\rm asp}=3$.
The inclusion of the COM criterion in addition to the ratio of the ellipsoid
axes help us deal with especially the larger-scale structures which can be
highly curved. A highly curved sheet could have comparable MOI eigenvalues
along the different eigen-directions, but would not contain its own COM. We
highlight some visual examples of such highly curved sheet-like structures
when we discuss the large scale morphology of our clouds in Section 5. In
contrast to curved sheets, a spheroidal structure would contain its own COM.
Apart from using the normal moment of inertia, we also perform the
classification by computing a volume-weighted moment of inertia, where we
compute the moment of inertia of the structures (the quantities $A$, $B$ and
$C$) by assuming the structure is of the same mass but with uniform density,
but find statistically little to no difference in the resulting morphologies.
The discussion above highlights some possible caveats of our method. If we
have a situation of multiple crossing filaments (hub-like structure), or
parallel filaments joined by a more diffuse intermediate medium - the method
will identify it as a sheet-like structure splitting into filaments in the
dendrogram tree hierarchy. We must therefore emphasise that our definition of
a sheet in this context is more general and contains also situations where
multiple filamentary structures are connected by a more diffuse medium.
Further, for highly curved structures, it is possible that the simple fit
ellipsoid method may not result in a good description of the ellipsoid axis
lengths.
## 4 Density distribution and magnetic fields
We first consider the bulk properties of the different MCs to quantify the
differences between the hydrodynamic and MHD clouds. From Table 1, we see that
the volume-weighted root-mean-square average magnetic field strength for all
MHD clouds is comparable and varies between 3.9-6.8 $\mu$G. These values are
slightly higher than the initial magnetic field strength $B_{x,0}=3\,\mu{\rm
G}$. The cloud masses and their H2 masses are also within a factor of roughly
2 to each other (with the exception of MC3-MHD, see below). For a view of the
time evolution of the total and H2 masses, as well as the H2 mass fraction, we
refer the reader to Appendix A.
MC3-MHD stands out as it has a much lower H2 mass and H2 mass fraction
compared to the other clouds (Table 1). Visual inspection of this cloud shows
that its structures are still diffuse and not as prominent, suggesting that it
perhaps needs much longer to collapse, or may not collapse at all (see Fig.
14, bottom row left). Its molecular content remains at a roughly constant
level of 10% throughout. Since we are interested primarily in the problem of
density structures that eventually form stars, we exclude MC3-MHD from further
analysis considering its unevolved state and low molecular content.
It is of interest to examine whether the mass distribution in different clouds
is affected by the presence of magnetic fields. This can be seen in Fig. 2,
which shows the volume-weighted density PDF of all different clouds at $t_{\rm
evol}=2$ Myr (top) and $t_{\rm evol}=3.5$ Myr (bottom) in the density range
probed by the dendrogram analysis ($>10^{-24}$ g cm-3). The respective density
PDFs for the full density range can be found in Appendix B. The two
hydrodynamic clouds are plotted using reddish lines (red and salmon), while
the magnetised clouds are shown using darker colours. For all clouds, the
shown density range contains more than 99% of their total mass.
Figure 2: Volume-weighted density PDF for different HD and MHD clouds $t_{\rm
evol}=2$ Myr (top) and 3.5 Myr (bottom). The density range shown is used for a
dendrogram analysis, and contains more than 99% of the total mass of the
clouds. The two hydrodynamic clouds are plotted in reddish lines. The vertical
line demarcate the boundaries of the high-den ($>10^{-22}$ g cm-3) and the
low-den (between $10^{-24}-10^{-22}$ g cm-3) dendrogram analyses (see also
Table 2). The MHD clouds have more fraction of gas in the density range
between roughly $10^{-24}$ and $10^{-22}$ g cm-3, or between approximately 1
and 100 cm-3.
From Fig. 2, we see that between $10^{-24}$ and $10^{-22}$ $\mathrm{g\
cm^{-3}}$, corresponding to the rough number densities between 1 and 100
$\mathrm{cm^{-3}}$, the MHD clouds contain much more gas. This is more
prominent at $t_{\rm evol}=2$ Myr, but remains also clearly visible at $t_{\rm
evol}=3.5$ Myr. This effect can also be visually seen in the column density
plots of Fig. 14, where the denser parts of the hydrodynamic clouds seem to be
embedded in a more rarefied medium compared to their MHD counterparts. We
calculate the mass percentage at 2 Myr in different density regimes in Table
3, which shows that, at this time, the MHD clouds contain almost 50% of their
mass between $10^{-24}$ and $10^{-22}$ $\mathrm{g\ cm^{-3}}$, in contrast to
only around 26% for the hydrodynamic MCs.
Cloud | Mass percentage at 2 Myr
---|---
sample | $\frac{\rho}{{\rm g}\,{\rm cm}^{-3}}<10^{-24}$ | $10^{-24}\leq\frac{\rho}{{\rm g}\,{\rm cm}^{-3}}<10^{-22}$ | $\frac{\rho}{{\rm g}\,{\rm cm}^{-3}}\geq 10^{-22}$
HD | 0.9 | 25.9 | 73.2
MHD | 0.5 | 50.0 | 49.5
Table 3: The average mass percentage in different density regimes for the HD
and MHD clouds at $t_{\rm evol}=2$ Myr. The MHD clouds have twice the amount
of mass in the intermediate density range between $10^{-24}$ g cm-3 and
$10^{-22}$ g cm-3 compared to their HD counterparts.
Magnetic fields in our simulations therefore play an important role in shaping
the environment inside which denser, molecular, and potentially star-forming
structures live. This is consistent with the picture that magnetic fields have
a noticeable effect on the dynamics of low (here, $\lesssim 10^{-22}$ g cm-3)
density gas (Molina et al., 2012). Similar conclusions have been reached by
Seifried et al. (2020b) using the technique of relative orientation of
magnetic fields with respect to filaments who find that this change in the
relative impact of magnetic fields occurs around $\sim 100$ cm-3. We explore
the effects of magnetic fields in more detail by looking at the clouds’
fragmentation properties in Section 6.4.
For $\rho>10^{-22}$ g cm-3 we see no clear trend in the slope of the density
PDF between the hydrodynamic and MHD clouds. This is consistent with
simulations and observations showing that column density PDFs are not
sensitive to the presence of a magnetic fields in the high column density
regime (Klessen & Burkert, 2001; Slyz et al., 2005; Girichidis et al., 2014;
Schneider et al., 2015).
However, at $t_{\rm evol}=2$ Myr, the two hydrodynamic clouds seem to have a
bit more dense gas mass (see also Table 3), although the effect is visually
far from clear. If there were a "delay" in the formation of denser gas when
magnetic fields are present, this would be extremely relevant for the
formation of well-shielded, molecular gas. In Table 4, we show the mass above
an $\mathrm{A_{V,3D}}$ (Eq. 7) of 1 and 10 for one magnetised and one non-
magnetised cloud of comparable mass (MC1-MHD and MC1-HD, see Table 1).
Additionally, the mass-weighted PDF of $\mathrm{A_{V,3D}}$ for these two
clouds is shown in Appendix C, Fig. 18. From Table 4, as well as from Fig. 18,
we find that the amount of gas above $\mathrm{A_{V,3D}}>1$ and
$\mathrm{A_{V,3D}}>10$ in MC1-MHD is consistently lower compared to MC1-HD. In
Section 6.5, we attempt to quantify such a delay timescale due to magnetic
fields. For a more detailed analysis on the connection between magnetic fields
and $\mathrm{A_{V,3D}}$, we refer the reader to Seifried et al. (2020a).
cloud, time | mass above | mass above
---|---|---
| $\mathrm{A_{V,3D}}>1$ [%] | $\mathrm{A_{V,3D}}>10$ [%]
MC1-HD, 2 Myr | 41.1 | 2.3
MC1-HD, 3.5 Myr | 44.0 | 9.8
MC1-MHD, 2 Myr | 26.6 | 0
MC1-MHD, 3.5 Myr | 31.7 | 0.8
Table 4: The percentage of mass above values of $\mathrm{A_{V,3D}}=1,\ 10$ for
two similar mass clouds, MC1-HD and MC1-MHD.
## 5 Morphology
We perform a morphological classification of all simulated cloud structures
using the method described in Section 3.2. As an intuitive visual aid, we
first present 3D surfaces of three large-scale cloud dendrogram
structures111We show the largest structures from the high-den dendrogram
analysis ($\rho>10^{-22}$ g cm-3) as they are on the maximum resolution and
therefore capture the finer complexities of the cloud better. The large scale
structures for the low-den dendrogram analysis follow the same trend. (from
top to bottom: MC1-MHD, MC5-MHD, and MC6-MHD) seen from three different
viewing angles (different columns) in Fig. 3. The lighter blue colour shows
the large-scale structure (identified at $\rho_{\rm thr}\approx 10^{-22}$ g
cm-3) and in red, we show one of the primary embedded filamentary structures
(identified using values of $\rho_{\rm thr}$ between $10^{-20}-10^{-21}$ g
cm-3). Visual inspection seems to suggest that the large-scale, lighter blue
structures are rather thin and sheet-like, and indeed all three clouds shown
in Fig. 3 are identified as sheets or curved sheets according to the
classification algorithm of Section 3.2. This is even clearer in a video view,
which can be found here222https://hera.ph1.uni-koeln.de/~ganguly/silcc_zoom/.
The visual suggestion of the clouds being sheet-like on the largest scales is
also confirmed for all clouds in a quantitative analysis, presented below.
Figure 3: 3D surface rendering of example large-scale dendrogram structures
from the high-den dendrogram analysis for MC1-MHD (top row), MC5-MHD (middle
row), and MC6-MHD (bottom row), from different viewing angles (left to right).
The blue structures represent the large-scale sheets or curved sheets at
$\rho_{\rm thr}\approx 10^{-22}$ g cm-3, while the embedded red structures
show one of the more prominent embedded filaments ($\rho_{\rm thr}$ between
$10^{-20}-10^{-21}$ g cm-3). The units in the axes are in parsec. A video link
for the various structures can be found in https://hera.ph1.uni-
koeln.de/~ganguly/silcc_zoom/morphology_3d/.
We estimate the size of the structures simply from the volume $V$ as:
$R=V^{1/3}.$ (13)
We define $N_{\rm tot}$ as the total number of morphologically classified
structures, i.e. $N_{\rm tot}$ is
$N_{\rm tot}=N_{\rm sheet}+N_{\rm sheet\\_c}+N_{\rm filament}+N_{\rm
spheroid},$ (14)
with $N_{x}$ being the total number of structures (i.e. both parents and
leaves) of morphological class $x$ (where $x\in$ [sheet, sheet_c, filament,
spheroid]). We express the number of structures of type $x$ at a given size
$R$ by $N_{x}(R)$.
In Fig. 4 we plot the cumulative fraction (i.e. $N_{x}(R)/N_{\rm tot}$) of
sheets, curved sheets, filaments, and spheroidal structures against $R$ for
all structures (i.e. both parents and leaves) in the two hydrodynamic clouds
(left panel) and the five MHD clouds (right panel) at $t_{\rm evol}=3.5$ Myr.
The numerical values of the overall fractions across all scales, $\int
N_{x}(R)/N_{\rm tot}\ \mathrm{d}R$, for both HD and MHD clouds at two
different times can be found in Table 5.
Figure 4: Cumulative histogram of different morphologies (sheets, curved sheets, filaments, or spheroids) for all HD (left) and MHD (right) clouds at $t_{\rm evol}=$ 3.5 Myr. 6 out of the 7 analysed clouds are sheet-like on large scales, with filamentary networks embedded inside. Spheroidal structures are rarer in the presence of magnetic fields. Both HD and MHD clouds produce more sheets than filaments, but the MHD runs tend to have a relative increase in the fraction of filaments. cloud, time | $\frac{N_{\rm sheet}}{N_{\rm tot}}$ | $\frac{N_{\rm sheet\\_c}}{N_{\rm tot}}$ | $\frac{N_{\rm filament}}{N_{\rm tot}}$ | $\frac{N_{\rm spheroid}}{N_{\rm tot}}$ | $N_{\rm tot}$
---|---|---|---|---|---
HD, 2 Myr | 0.58 | 0.12 | 0.22 | 0.08 | 910
HD, 3.5 Myr | 0.63 | 0.07 | 0.19 | 0.11 | 1167
MHD, 2 Myr | 0.57 | 0.03 | 0.31 | 0.09 | 487
MHD, 3.5 Myr | 0.56 | 0.04 | 0.33 | 0.07 | 2087
Table 5: Fraction of sheets, curved sheets, filaments, and spheroids among all
morphologically classified structures, for both HD and MHD clouds at $t_{\rm
evol}=2,\ 3.5$ Myr. While all clouds are dominated by sheet-like structures,
the MHD clouds have a higher fraction of filaments compared to their
hydrodynamic counterparts.
We find that spheroidal structures, shown in green, are generally less
numerous compared to sheet-like or filamentary structures ($\sim$10% of
$N_{\rm tot}$ are spheroidal, Table 5). Sheets (including curved sheets)
appear to be the most abundant structures within all clouds (summing up to
$\sim$70% for the HD case and $\sim 60$% for the MHD case). However, filaments
are considerably more abundant in the MHD clouds compared to their HD
counterparts ($>30$% for MHD as opposed to $\sim$20% for HD clouds). In terms
of size, we find that at the largest $R$ values, indeed almost all clouds (six
out of seven) are either sheets or curved sheets, confirming the visual trend
we found in Fig. 3.
We highlight the morphological trends as a function of the molecular fraction
in Fig. 5. Similar to Fig. 4, we plot here the cumulative fraction of (curved)
sheets, filaments, and spheroids, but this time as a function of the molecular
mass fraction $f_{\rm H_{2}}$, which is the H2 mass in a structure divided by
the total hydrogen mass in the structure. Note that structures with high
$f_{\rm H_{2}}$ are usually small (located mostly at small $R$ in Fig. 4). We
see that around $f_{\rm H_{2}}>0.7$, there are more filaments than sheet-like
structures in the MHD case (right panel). This trend is absent for the HD
clouds (left panel). This implies that magnetic fields particularly enhance
the formation of filaments on the small scales, shaping the morphology of the
denser, well-shielded, molecular gas. This is in line with the fact that
magnetic fields can, in general, aid the formation of filamentary sub-
structures (Hacar et al., 2022; Pineda et al., 2022).
Figure 5: Cumulative histogram of different morphologies (sheets, curved
sheets, filaments, or spheroids) against H2 mass fraction for all HD (left)
and MHD (right) clouds at $t_{\rm evol}=$ 3.5 Myr. The most molecular
structures are more filamentary in presence of magnetic fields.
Gravitational collapse naturally proceeds anisotropically and tends to create
elongated structures (e.g. Burkert & Hartmann, 2004). However, we show in
Ganguly et al. (2022) that most of our cloud structures are unbound or only
marginally bound. This being the case, gravity cannot be the principal
contributor to forming elongated structures, and we must therefore identify
other possible sources of the lack of spheroidal structures. Shock compression
and turbulence are two such methods for producing elongated structures (see,
e.g., Inoue & Inutsuka 2016 for shock compression; Federrath 2016 for
turbulence; and Hacar et al. 2022 for a general overview). Sheets and
filaments are both elongated structures. However, it is interesting that for
the hydrodynamic clouds, sheets are by far the most numerous, whereas for the
MHD clouds filaments and sheets are more comparable in total number. This is
consistent with the results of Hennebelle (2013), who investigate setups of
both decaying supersonic turbulence and colliding flows, and find that their
simulations tend to produce more sheet-like structures for hydrodynamical
simulations, and more filamentary structures for MHD simulations.
Overall, we see primarily sheet-like MCs with an abundance of elongated
structures (filamentary or sheet-like), irrespective of whether the simulation
contains magnetic fields or not. Sheets are generally more numerous, probably
representing the fact that we trace a large number of structures belonging to
the sheet-like atomic envelope of the MCs. This is supported by the fact that,
in Fig. 5, both HD and MHD clouds show an abundance of sheet-like structures
below $f_{\rm H_{2}}\approx 0.5$. The presence of magnetic fields, however,
tends to somewhat increase the fraction of filamentary over sheet-like
structures.
The sheet-like nature of our clouds is consistent with a number of recent
observations. Kalberla et al. (2016) have argued that the cold, neutral
hydrogen in the ISM is organised in sheet-like structures. Investigating the
L1495 region of the Taurus molecular cloud, Arzoumanian et al. (2018) report
evidence of extended sheet-like structures too. Using the recent GAIA data,
Rezaei Kh. & Kainulainen (2022) have concluded that the California molecular
cloud is sheet-like in nature. Tritsis et al. (2022) have reached a similar
conclusion regarding the Musca molecular cloud using 3D dust extinction maps.
Based on a Herschel study of the giant molecular filament G214.5, Clarke et
al. (2023) have also posited that the filament is a result of the HI shell of
an expanding superbubble interacting with the local medium. Our findings here
are thus perfectly in line with these observations.
The morphology of MCs at larger (tens of parsecs) scales is of paramount
importance in relation to how the MCs themselves form. Our analysis shows that
the clouds are preferentially sheet-like, with and without magnetic fields.
The ISM in the SILCC simulations (and therefore also in the SILCC-Zoom
simulations) has a multi-phase structure (Walch et al., 2015; Girichidis et
al., 2016a). The MCs in these simulations form primarily at the shells or
intersections of expanding supernova bubbles. The large-scale sheets we see,
can therefore be interpreted as tracing these supernova-driven shells, with a
complex network of different morphological sub-structure contained within.
This picture is consistent with the bubble-driven structure formation scenario
(Koyama & Inutsuka, 2000; Inoue & Inutsuka, 2009; Inutsuka et al., 2015;
Pineda et al., 2022).
## 6 Dynamics and Fragmentation
### 6.1 The magnetic field - density scaling
The impact of magnetic fields on the MCs is naturally correlated to the field
strength. The initial 3 $\mu$G seed field in the original simulations is
expected to be enhanced when we look at denser structures inside the MCs. The
scaling behaviour of the magnetic field $B$ with $\rho$ is integral to
understanding the importance of magnetic fields at different scales.
If contraction of gas occurs exclusively along the magnetic field lines, this
should lead to no dependence of the magnetic field strength on the density,
i.e. $B\propto\rho^{0}$. If magnetic field lines do contract with the
enhancement of gas density, then one expects a scaling similar to
$B\propto\rho^{\kappa}$, with $\kappa=0.5,0.67$ for the strong and weak field
limits, respectively (see e.g. the review by Hennebelle & Inutsuka, 2019).
In the ISM, indeed the $\kappa=0$ relation is observed up to number densities
of $\sim$300 cm-3 (Troland & Heiles, 1986; Crutcher et al., 2010). This
corresponds to densities of roughly 1.1$\times 10^{-21}$ g cm-3, using a mean
molecular weight of 2.35. Crutcher et al. (2010) find that above these
densities, the data is consistent with $\kappa=2/3$, with considerable
scatter. The transition in power law is usually associated with the magnetic
fields becoming dynamically sub-dominant (Seifried et al., 2020b; Pattle et
al., 2022) and roughly matches with our observation that below $\sim 100$ cm-3
the mass in the MHD clouds is enhanced.
We can attempt to capture whether this transition in the importance of the
magnetic field is seen in the Alfvénic Mach number, $\mathcal{M}_{\rm A}$. For
a given sub-structure, we can compute $\mathcal{M}_{\rm A}$ as
$\mathcal{M}_{\rm A}=\sigma_{\rm 1D}/v_{\rm A}.$ (15)
Here $\sigma_{\rm 1D}$ is the one-dimensional velocity dispersion and $v_{\rm
A}$ is an estimate of the average Alfvén wave group velocity. For a structure
of mass $M$, we compute $\sigma_{\rm 1D}$ from
$\sigma_{\rm
1D}^{2}=\frac{1}{3M}\int_{V}\rho(\mathbf{v}-\mathbf{v}_{0})^{2}\mathrm{d}^{3}r,$
(16)
with $\mathbf{v}_{0}$ being the centre of mass velocity computed as
$\mathbf{v}_{0}=\frac{1}{M}\int_{V}\rho\mathbf{v}\mathrm{d}^{3}r.$ (17)
The integration is performed over the entire volume $V$ of the given
structure.
The Alfvén velocity can be computed as
$v_{\rm A}=\sqrt{\frac{\langle|\mathbf{B}|^{2}\rangle}{4\pi\rho_{\rm avg}}}.$
(18)
The density $\rho_{\rm avg}$ here is the volume-averaged density, i.e.
$\rho_{\rm avg}=M/V,$ (19)
and $\langle|\mathbf{B}|^{2}\rangle$ is the volume-averaged square of the
magnetic field $\mathbf{B}$,
$\langle|\mathbf{B}|^{2}\rangle=\frac{1}{V}\int_{V}|\mathbf{B}|^{2}\mathrm{d}^{3}r.$
(20)
The behaviour of the magnetic field strength with density for the MHD clouds
can be seen in Fig. 6, where we plot the root-mean-square magnetic field
strength against the threshold (minimum) density $\rho_{\rm thr}$ for all
dendrogram structures at $t_{\rm evol}=3.5$ Myr. The different dendrogram
structures are marked with filled/empty symbols depending on whether their H2
mass fraction (with respect to their total hydrogen mass) is greater/less than
50%. The colour bar shows $\mathcal{M}_{\rm A}$, as computed from Eq. 15. The
reddish points represent super-Alfvénic ($\mathcal{M}_{\rm A}>1$) structures,
while the blueish points are sub-Alfvénic ($\mathcal{M}_{\rm A}<1$). In the
sub-Alfvénic case, the fluid speed is smaller than the magnetic wave speed,
meaning that the magnetic field is dynamically important and guides the flow.
The vertical dotted line at $10^{-22}$ g cm-3 represents the boundary between
the points obtained from the low-den (left half) and high-den (right half)
dendrograms, respectively. The dash-dotted black line and the dotted power-law
represent the Crutcher et al. (2010) relation discussed previously and
$B\propto\rho^{0.5}$, respectively.
Figure 6: Relation between the root-mean-square magnetic field and $\rho_{\rm
thr}$ for all MHD clouds at $t_{\rm evol}$=3.5 Myr. The colour bar shows the
Alfvénic Mach number $\mathcal{M}_{\rm A}$. The dash-dotted line represents
the B$-\rho$ relation from Crutcher et al. (2010), while the dotted line
represents a $B\propto\rho^{0.5}$ power law. The cyan dashed line represents
the best fit power law for all points with $\rho_{\rm thr}>1.1\times 10^{-21}$
g cm-3.
The cyan dashed line represents the linear least-squares best fit performed on
the logarithm of the points for high densities ($\rho_{\rm thr}>1.1\times
10^{-21}$ g cm-3). The best fit of $\kappa=0.47\pm 0.03$ is consistent with
the strong-field limit of $B\propto\rho^{0.5}$. We have already shown in the
previous section (Section 5) that our structures are on average highly
elongated, and magnetic fields clearly help to deform the shape of the forming
structures. It is therefore not unexpected that we find a shallower scaling
compared to the weak field limit ($\kappa=0.67$).
We see that, while there is no clear transition from the sub- to the super-
Alfvénic regime, there is clearly a trend that higher Alfvénic Mach numbers
are preferentially obtained at the higher density end. This is confirmed by a
Kolmogorov-Smirnov (KS) two-sample test, which compares if two distributions
belong to the same population. In this case, we compare the $\rho_{\rm
thr}$-distributions of structures with $\mathcal{M}_{A}>1$ and
$\mathcal{M}_{A}\leq 1$. We find the $p$-values333If the $p$-value is larger
than a certain value (typically 0.05), this means that we cannot reject the
null hypothesis that the sub-Alfvénic and super-Alfvénic structures have the
same underlying density distribution. to be very low: $6\times 10^{-4}$ at 2
Myr and $5.2\times 10^{-15}$ at 3.5 Myr (see Table 6).
Crutcher et al. (2010) found that the observed magnetic field distribution is
rather flat at low density, in agreement with the idea that denser clouds are
swept up along the magnetic field lines on large scales, while at higher
density there is a power-law increase of the magnetic field strength. If
spherical clouds start to collapse and the magnetic field is not strong enough
to stop the collapse, one expects a power-law slope of $\kappa=0.5-0.67$ (see
above).
In the case of our clouds, we find that the high-density end is well
consistent with $\kappa=0.5$, and the lower-density end clearly shows a much
shallower slope. Nonetheless, there does not seem to be a clear single density
at which there is a sharp change in slope. Simulations by Li et al. (2015),
Mocz et al. (2017), Girichidis et al. (2018), Zhang et al. (2019) find
similarly the lack of a sharp transition density. Auddy et al. (2022) predict
that the transition density depends on the fourth power of $\mathcal{M}_{A}$.
While of potential interest, this is unfortunately not demonstrable from the
present analysis.
variable 1 | variable 2 | time [Myr] | p-value
---|---|---|---
$\rho_{\rm thr}(\mathcal{M}_{A}>1)$ | $\rho_{\rm thr}(\mathcal{M}_{A}\leq 1)$ | 2 | $6\times 10^{-4}$
| | 3.5 | $5.2\times 10^{-15}$
Table 6: The $p$-values of the 2-sample KS test for the density distribution
of sub-Alfvenic and super-Alfvénic structures. We can see that the $p$-value
is low for both 2 and 3.5 Myr, suggesting that sub-Alfvénic and super-Alfvénic
structures (corresponding to bluish and reddish points in Fig. 6,
respectively) have statistically significant differences in their density
distributions.
### 6.2 Impact of magnetic fields on the energetics of sub-structures
We are also interested in assessing the energetic relevance of magnetic fields
over different length scales in the MCs, especially with respect to
potentially star-forming structures. For this purpose, we compute the volume
term of the magnetic energy and compare it with the kinetic and potential
energies. Similar work for the same simulations has been performed by Ganguly
et al. (2022), who assess the virial balance of the cloud sub-structures.
Here, we extend the range of our analysis to include the dynamics of lower-
density gas (between 10-24 and 10-22 g cm-3; low-den dendrogram analysis, see
Table 2).
The magnetic energy of a given structure is computed as
$E_{\rm B}=\int_{V}\frac{1}{8\pi}|\mathbf{B}|^{2}\mathrm{d}^{3}r,$ (21)
where the integration is computed over the entire volume $V$ of the structure.
The kinetic energy is computed using the following relation:
$E_{\mathrm{KE}}=\frac{1}{2}\int_{V}\rho(\mathbf{v}-\mathbf{v}_{0})^{2}\mathrm{d}^{3}r.$
(22)
Here, $\mathbf{v}_{0}$ is the centre of mass velocity computed from Eq. 17.
The self-gravitating potential energy of a given structure is obtained using
the following relation:
$E_{\rm
PE}=-\frac{1}{2}G\int_{V}\int_{V}\frac{\rho(\mathbf{r})\rho(\mathbf{r^{\prime}})}{|\mathbf{r}-\mathbf{r^{\prime}}|}\mathrm{d}^{3}r\mathrm{d}^{3}r^{\prime},$
(23)
where $G$ is the gravitational constant. We compute the self-gravity of each
dendrogram structure using a KD-tree algorithm (Bentley, 1975) instead of an
$\mathcal{O}(N^{2})$ direct computation.
We show the relative importance of magnetic fields with respect to potential
and kinetic energy in the left and right panel of Fig. 7, respectively, for
all MHD cloud structures at $t_{\rm evol}=3.5$ Myr. For both plots, the
$x$-axis represents the density threshold $\rho_{\rm thr}$, and the $y$-axis
represents $E_{\rm B}/|E_{\rm PE}|$ (left) and $E_{\rm B}/|E_{\rm KE}|$
(right), respectively. The colours of the points represent their morphologies.
Here, for the purpose of understanding the dynamics of low-density gas, we
also include the "unclassified" structures (i.e. structures with >5% of their
surface cells touching the edge of the analysis box, see Section 3). The side
panels to the right and top of each plot show the marginal distributions of
$N_{x}/N_{\rm tot}$ for each morphology. Note that, since the definition of
$N_{\rm tot}$ (Eq. 14) does not contain unclassified structures, the fractions
in the two side panels add up to greater than unity. The filled symbols are
molecular structures, while the open symbols are atomic.
Figure 7: Ratio of magnetic energy to self-gravitating potential energy (left)
and to kinetic energy (right), respectively, plotted against the density
threshold for all dendrogram structures of all MHD clouds at time $t_{\rm
evol}=3.5$ Myr. The colours represent different morphologies. The dash-dotted
lines indicate a $\rho^{-1/2}$ relation. The top and the right panels show the
marginalised distributions (separated by morphology) over the density and the
corresponding energy ratio.
Typically, for low-density structures, which mostly consist of atomic gas, the
magnetic energy is either comparable to or much larger than the potential
energy (left panel of Fig. 7). The magnetic energy is also comparable to or
larger than the kinetic energy (right panel), but the spread in this energy
ratio is much smaller compared to the $E_{\rm B}/E_{\rm PE}$ ratio. For some
branches (a dendrogram branch is defined as a given structure and all its
parent structures, see Section 3.1), the energy ratio seems to roughly follow
a $\rho^{-1/2}$ power law. These branches represent the evolution from
diffuse, large-scale structures to denser, embedded structures. Camacho et al.
(2022) also find a tight power-law scaling between the potential and magnetic
energies. While not exactly the same, both scaling behaviours seem to imply
that magnetic fields become less important as we go deeper into the MCs
themselves. This is also in accordance with the findings of Seifried et al.
(2020b), Ibáñez-Mejía et al. (2022), as well as Ganguly et al. (2022), as
discussed previously.
From the marginal distributions, we find a weak trend that the high-density
end is dominated by filaments. Curved sheets and unclassified structures only
appear at lower densities because they are usually larger-scale structures.
There is no obvious correlation between the morphology of the structures and
the energy ratios. This suggests that the different morphological
configurations are created by the same formation mechanism, most likely
turbulent compression.
There also seems to be a difference in the energy ratios between atomic and
molecular structures. This can be clearly seen in the average behaviour of
these ratios over time. Fig. 8 plots the time evolution of the average value
of $E_{\rm B}/|E_{\rm PE}|$ (left) and $E_{\rm B}/E_{\rm KE}$ (right) for all
atomic (red), molecular (blue), and dense molecular (yellow) structures from
the MHD clouds, where we define dense molecular structures to be structures
that are both molecular and have $\rho_{\rm thr}>10^{-20}$ g cm-3. The error
bars here represent the standard error on the mean. From Fig. 8, we see that
magnetic energy dominates over potential and kinetic energies for atomic
structures, while it plays a subordinate role in molecular structures. There
is no clear trend indicating that this behaviour changes as a function of
time.
Figure 8: Time evolution of the average ratio of magnetic to potential energy
(left) and kinetic energy (right). The different colours represent atomic,
molecular, and dense molecular (molecular and $\rho_{\rm thr}>10^{-20}$ g
cm-3) structures in red, blue, and yellow, respectively. The errors bars are
the standard errors on the mean. For denser and molecular structures, magnetic
energy is less important compared to potential or kinetic energies. The atomic
structures, representing more the envelope of the molecular gas, have high
magnetic energies, especially compared to self-gravity.
The subservient role of magnetic energy for dense structures compared to
potential or kinetic energy suggests that while magnetic fields help to shape
the cloud structures across different scales, the dynamics of the denser, and
potentially star-forming structures, is determined by the interaction between
gravity and turbulence 444We explore the interplay between turbulence and
gravity in much greater detail in our companion paper by means of a virial
analysis (Ganguly et al., 2022). This explains why there is no discernible
difference in the power-law tail of the density PDFs between hydrodynamic and
MHD clouds (Fig. 2), confirming that the star-forming gas (see e.g. Klessen &
Burkert, 2001; Girichidis et al., 2014; Schneider et al., 2015) is virtually
unaffected by the presence of magnetic fields. However, magnetic fields change
the gas properties of the environment from which denser structures form,
accrete, and sit in (i.e. by making the surrounding envelope "fluffier"),
thereby also influencing the shape of these structures.
### 6.3 Magnetic surface energy
In the previous section, we have discussed the magnetic pressure term in
comparison to self-gravity and kinetic energy. The magnetic pressure relates
to the stretching and compression of magnetic field lines, and does not take
into account the effect of curvature in the field.
The magnetic surface term can be computed as an integral over the surface of a
given structure, $S$, as follows:
$E_{\rm B}^{\rm
surface}=\oint_{S}(\mathbf{r}-\mathbf{r}_{0})\mathbf{T}\hat{\mathbf{n}}\
\mathrm{d}S.$ (24)
Here $\mathbf{r}_{0}$ is the centre of mass, $\hat{\mathbf{n}}$ is the surface
normal vector that points outwards, and $\mathbf{T}$ is the Maxwell stress
tensor, which can be written as follows for ideal MHD:
$\mathbf{T}=\frac{1}{4\pi}\left(\mathbf{B}\otimes\mathbf{B}-\frac{1}{2}|\mathbf{B}|^{2}\hat{\mathbf{I}}\right).$
(25)
$\hat{\mathbf{I}}$ is here an identity matrix of rank two.
We evaluate Eq. 24 as a volume integral using the Gauss’ divergence theorem
for convenience. This gives us the following relation:
$E_{\rm B}^{\rm surface}=-E_{\rm
B}+\int_{V}(\mathbf{r}-\mathbf{r}_{0})\cdot\nabla\mathbf{T}\ \mathrm{d}V$ (26)
From Eq. 26, we can see that $E_{\rm B}^{\rm surface}$ can be both positive or
negative. When it is is positive, it adds to the magnetic pressure term and
acts as a dispersive term. In contrast, when $E_{\rm B}^{\rm surface}<0$, it
acts as a confining term.
The importance of $E_{\rm B}^{\rm surface}$ with respect to the volume term,
$E_{\rm B}$, can be seen in Fig. 9, left panel, which plots the magnitude of
the ratio of $E_{\rm B}^{\rm surface}/E_{\rm B}$ to the density threshold of
the cloud sub-structures for all MHD clouds at $t_{\rm evol}=3.5$ Myr.
Structures where $E_{\rm B}^{\rm surface}$ helps to disperse them ($E_{\rm
B}^{\rm surface}>0$) are marked in red, while structures where $E_{\rm B}^{\rm
surface}$ acts as a confining term ($E_{\rm B}^{\rm surface}<0$) are marked in
cyan. The vertical dotted line marks the difference between the results of the
low-den and high-den dendrogram runs at $\rho=10^{-22}$ g cm-3, as in the
previous plots. The horizontal dotted line represents a value of one, where
the volume and surface terms are equally important magnitude-wise. The top and
side panels show the marginal distributions.
Figure 9: Left: Ratio of the absolute value of the magnetic surface to volume
energy, plotted against the density threshold. The different colours represent
whether the magnetic surface term is positive and resists collapse or negative
and promotes collapse. The magnetic surface energy seems to be as relevant as
the volume energy, and for more than half of the structures acts as a
confining term. Right: The ratio of the total magnetic energy (surface plus
volume) to the self-gravitating potential energy, plotted against the magnetic
volume energy over the self-gravitating potential energy. The dashed line
represents a 1:1 ratio, and the shaded region represents a factor of 2. For
many small-scale atomic structures, the magnetic surface term seems to be
important as a confining force.
From the marginal distributions, we see that $E_{\rm B}^{\rm surface}$ acts as
a confining term for somewhat more number of structures compared to where the
surface term is dispersive. The magnetic surface term seems to be comparable
to and in some cases, even exceeding the volume term $E_{\rm B}$. This implies
that for diffuse and mostly atomic structures, where magnetic energy is
comparable or dominant, the surface term is important. This is especially
relevant when $E_{\rm B}^{\rm surface}$ acts as a confining term. However, for
dense structures, where $E_{\rm B}$ is one to two orders of magnitude smaller
than the potential and kinetic energies, the surface term is unlikely to
significantly affect the dynamics.
In the right panel of Fig. 9 we plot the magnitude of $(E_{\rm B}^{\rm
surface}+E_{\rm B})/E_{\rm PE}$ against $E_{\rm B}/E_{\rm PE}$ for all MHD
cloud sub-structures at 3.5 Myr. The colour bar here represents the size of
the structures. The horizontal and vertical dotted lines both represent a
value of unity along the $y-$ and $x-$ axes, respectively. The dashed line
represents a 1:1 line, and the shaded region around it represents a factor of
2 in each direction. The magnetic surface energy is not significant compared
to the volume energy for structures on or close to the 1:1 line. Structures
with strong dispersive $E_{\rm B}^{\rm surface}$ terms lie above the 1:1 line,
while points that lie below the 1:1 line represent structures where $E_{\rm
B}^{\rm surface}$ is confining in nature. Most interesting here are the points
that lie in the bottom right quadrant of the plot. They represent structures
where the magnetic pressure $E_{\rm B}$ is higher compared to the self-
gravity, and would be completely unbound in a traditional virial analysis.
However, the confining $E_{\rm B}^{\rm surface}$ term is strong enough that
the overall magnetic contribution becomes far less, thus allowing for a sort
of "magnetic confinement". These structures are mostly atomic and typically
seem to be $\lessapprox 1$ pc.
Two examples of structures belonging to MC2-MHD that exhibit such magnetic
confinement are plotted in Fig. 10 as black contour lines over a density slice
in the $y-z$ plane. The background colour here represents the density, while
the planar magnetic field is shown using the line integral convolution (LIC)
technique555The package used can be found in
https://github.com/alexus37/licplot.. For both structures, we mention the
magnitude of the $E_{\rm B}/E_{\rm PE}$ and $(E_{\rm B}^{\rm surface}+E_{\rm
B})/E_{\rm PE}$ ratios in the figure title. As can be clearly seen, the
magnetic surface term reduces the $|(E_{\rm B}^{\rm surface}+E_{\rm B})/E_{\rm
PE}|$ ratio to less than one. However, this naturally does not take into
account other energy terms, i.e. kinetic and thermal energy, and hence it is
not fully clear whether these structures are overall confined. Interestingly,
the structures for which the magnetic surface energy is important and of
confining nature (see right panel of Fig. 9) are usually located at the
"kinks" of magnetic field lines.
Figure 10: Two examples of structures confined by $E_{\rm B}^{\rm surface}$
from MC2-MHD, plotted as black contours over density slices in the $y-z$
plane, at $t_{\rm evol}=2$ Myr. The colour map is the logarithmic density, and
the direction of the planar magnetic field is plotted as line integral
convolution. The relevant energy ratios of the indicated structures are
denoted in the title. Structures for which the magnetic surface energy is
important and of confining nature, are usually located at the "kinks" of
magnetic field lines.
### 6.4 Fragmentation
In this section, we attempt to quantify to what extent magnetic fields affect
the fragmentation properties of molecular clouds. For this purpose, we study
the numbers and masses of different fragments, represented by leaf structures
(i.e. structures containing no further sub-structures) found in our dendrogram
analysis, and in addition perform a magnetic Jeans analysis on these
fragments.
#### 6.4.1 Number and mass distribution of fragments
Representing fragments by the leaves in the dendrogram analysis suffers from
the caveat of depending on the dendrogram parameters. Increasing the minimum
number of cells required in a dendrogram structure, for example, would
naturally reduce the number of fragments and increase their masses. The
absolute values of the masses and numbers we find, therefore, are sensitive to
the parameter values we have used. However, since we used the exact same
parameters for each HD and MHD run, and because all molecular clouds have
similar masses and identical environmental parameters (solar neighbourhood
parameters), the relative difference between the average behaviour of the HD
and MHD clouds is meaningful. With this caveat in mind, let us look at the
fragmentation properties of our dendrogram structures.
Figure 11: Top row: Cumulative distribution of the average number of leaf
structures against $\rho_{\rm thr}$ for HD and MHD clouds at $t_{\rm evol}=$
2, 2.5, 3.5 Myr, respectively (from left to right). The hydrodynamic clouds
have on average more new structures forming at earlier times, but this
distinction slowly disappears later on. Bottom row: Distribution of average
mass of leaf structures for both HD and MHD clouds at $t_{\rm evol}=$ 2, 2.5,
3.5 Myr, respectively (from left to right). The leaf structures, representing
fragments, are more massive for MHD clouds at earlier times, while this
distinction mostly disappears later on as gravity takes over.
We study the numbers and masses of leaf fragments in Fig. 11. The top row
plots the cumulative distribution of the average number of leaf structures,
$\langle N_{\rm structure}^{\rm leaf}\rangle$, as a function of $\rho_{\rm
thr}$ for both HD (blue) and MHD (red) clouds. The average here simply means
that we divide the total number of obtained structures by the number of
clouds, i.e. 5 for MHD and 2 for HD. The three panels (left to right) show
three different times, $t_{\rm evol}=2,\ 2.5$ and 3.5 Myr, respectively. The
vertical line at $10^{-22}$ g cm-3 marks the difference between the low-den
and the high-den dendrogram analysis. We see that at $t_{\rm evol}=2$ and 2.5
Myr, up to densities between $10^{-23}-10^{-22}$ g cm-3, the HD and MHD clouds
form roughly similar numbers of leaf fragments. However, at higher densities,
$\langle N_{\rm structure}^{\rm leaf}\rangle$ is much higher for the HD
clouds. This difference largely disappears at 3.5 Myr. This suggests that the
formation of structures is somewhat slowed down in the presence of magnetic
fields in the beginning, but at later stages, as gravity becomes dynamically
more and more important, this difference diminishes.
In the bottom row of Fig. 11 we plot the average mass of the leaf structures,
$\langle M_{\rm structure}^{\rm leaf}\rangle$, as a function $\rho_{\rm thr}$
for HD and MHD structures for the three different times. The shaded regions
represent the standard deviation of the average mass at a given $\rho_{\rm
thr}$. We see that at $t_{\rm evol}=2$ Myr, the MHD fragments are slightly
more massive compared to their hydrodynamic counterparts, in particular for
$\rho_{\rm thr}\lesssim 10^{-21}$ g cm-3. This difference disappears later.
For the densest structures, we do not seem to see a systematic difference in
$\langle M_{\rm structure}^{\rm leaf}\rangle$. This is in line with Fig. 2,
which shows that the difference in the density PDFs between the HD and MHD
clouds in the density range that corresponds primarily to the cloud envelope
(i.e. $\lessapprox 10^{-21}$ g cm-3) is most striking at $t_{\rm evol}=2$ Myr,
and less so later on.
Overall, the results shown in Fig. 11 indicate that the MHD clouds fragment
more slowly than the HD clouds but therefore have slightly more massive
fragments at early times. This is consistent with the result that magnetic
fields affect the dynamics of lower density gas more (Molina et al., 2012;
Seifried et al., 2020a, b; Ibáñez-Mejía et al., 2022). We also see that the
number and mass of the leaf structures are comparable at later times. This
suggests that the magnetic fields "slow down" the evolution of the cloud but
are less relevant once the cloud is more evolved, and gravity becomes
energetically more and more important, as shown in the previous energetic
analysis. This effect could be related to the overall strength of the magnetic
field. We investigate this further in the next Section.
#### 6.4.2 Magnetic Jeans analysis
The classic thermal Jeans analysis (Jeans, 1902) is a useful tool to
investigate the stability of MCs and their substructures (clumps and cores)
under thermal perturbations. Here, we perform its magnetic equivalent. The
thermal Jeans length, $\lambda_{\rm T}$, defines the largest length-scale
stable to thermal perturbations. For a given structure, this is defined as
$\lambda_{\rm T}=c_{s}\sqrt{\frac{\pi}{G\rho_{\rm avg}}},$ (27)
where $c_{s}$ is the average sound speed given by
$\displaystyle c_{s}=\frac{1}{V}\int_{V}\sqrt{\frac{P}{\rho}}\mathrm{d}^{3}r.$
(28)
Here, $P$ is the thermal pressure and the sound speed is calculated assuming
an isothermal equation of state due to the densities under consideration. We
remind the reader that $\rho_{\rm avg}$ is the volume-averaged density
computed in Eq. 19. From the Jeans length, a maximum mass stable under thermal
perturbations can be calculated. This mass is referred to as the thermal Jeans
mass, $M_{\rm T}$, and is given by
$M_{\rm T}=\frac{4}{3}\pi\rho_{\rm avg}\left(\frac{\lambda_{\rm
T}}{2}\right)^{3}.$ (29)
Similar to the thermal analysis, we can perform a magnetic Jeans analysis and
a Jeans analysis combining both magnetic and thermal support. For the magnetic
Jeans analysis, the relevant length ($\lambda_{\rm B}$) and mass ($M_{\rm B}$)
scales are given by,
$\displaystyle\lambda_{\rm B}=c_{\rm B}\sqrt{\frac{\pi}{G\rho_{\rm avg}}},$
(30) $\displaystyle M_{\rm B}=\frac{4}{3}\pi\rho_{\rm
avg}\left(\frac{\lambda_{\rm B}}{2}\right)^{3}.$ (31)
For a combination of thermal and magnetic effects, the relevant magneto-
thermal Jeans length ($\lambda_{\rm B,T}$) and Jeans mass ($M_{\rm B,T}$) are
$\displaystyle\lambda_{\rm B,T}=c_{\rm B,T}\sqrt{\frac{\pi}{G\rho_{\rm
avg}}},$ (32) $\displaystyle M_{\rm B,T}=\frac{4}{3}\pi\rho_{\rm
avg}\left(\frac{\lambda_{\rm B,T}}{2}\right)^{3}.$ (33)
The characteristic speeds are given by,
$\displaystyle c_{\rm B}=v_{\rm A},$ (34) $\displaystyle c_{\rm
B,T}=\sqrt{c_{s}^{2}+v_{\rm A}^{2}},$ (35)
where $v_{\rm A}$ is the Alfvén speed (Eq. 18).
In Fig. 12, we show the ratio of a structure’s mass to its magneto-thermal
Jeans mass, $M/M_{\rm B,T}$, as a function of $\rho_{\rm thr}$ for all MHD
cloud branch structures (top) and leaves (bottom) at $t_{\rm evol}=3.5$ Myr.
We remind the reader that branch structures contain sub-structures and leaves
do not. The Jeans mass can only be properly used when the corresponding length
is resolved. This is shown in Appendix E, Fig. 20, which depicts that some
structures with $\rho_{\rm thr}\gtrapprox 10^{-20}$ g cm-3 seem to be not
properly Jeans resolved. These are marked with black outlines in Fig. 12. Note
that the structures are (un-)resolved in the context of our dendrogram
analysis, which requires a minimum number of 100 cells per structure, and
therefore at least 200 cells to resolve fragmentation (as in the case of
fragmentation, each fragment would need to contain at least 100 cells). The
colour-bar denotes the ratio of $c_{s}$ to $v_{\rm A}$. Most of the structures
have $v_{\rm A}>c_{s}$ (blue points), suggesting support by magnetic fields
rather than by thermal pressure. This is confirmed by our purely magnetic
Jeans analysis (Fig. 21), which shows an almost identical distribution to the
magneto-thermal Jeans analysis of Fig. 12 (as well as Fig. 20). From Fig. 12,
top panel, we find that roughly below $10^{-22}$ g cm-3, all structures are
Jeans stable ($M/M_{\rm B,T}<1$). At higher densities, we have both, Jeans
stable and unstable structures. Some prominent branches clearly have $M/M_{\rm
B,T}>1$ above $10^{-22}$ g cm-3, indicating the growing importance of gravity
for fragmentation at higher densities. For leaves, this transition density
seems to occur at higher densities.
Figure 12: The ratio of the mass of a given structure to its magneto-thermal
Jeans mass ($M_{\rm B,T}$, Eq. 33) as a function of $\rho_{\rm thr}$ for all
MHD branch (top) and leaf (bottom) sub-structures at $t_{\rm evol}=3.5$ Myr. A
branch sub-structure has further sub-structures, while a leaf does not. The
horizontal dotted line represents a ratio of unity. The vertical line
separates the points obtained from the high-den and the low-den dendrogram
analysis. The colour-bar shows the ratio of the sound speed to the Alfvén wave
speed. For blue points, $v_{\rm A}>c_{s}$. A power-law is plotted in each
panel for rough guidance. Structures whose fragmentation is not well-resolved
(see Fig. 20) are marked with an additional black outline and mostly populate
the right-hand top corner of the plot. The magneto-thermal forces seem unable
to keep all the structures Jeans stable beyond $\sim 10^{-22}$ g cm-3,
suggesting the growing importance of gravity.
Interestingly, the leaves seem to have an overall sharper scaling behaviour
compared to the branches. However, this separation cannot be seen in the Jeans
length, where all structures show a consistent scaling of roughly
$\lambda_{\rm B,T}\propto\rho^{-2/3}$ (Fig. 20). This can be understood as
follows:
The mass of a structure is dependent on the density and size, i.e.
$M\propto\rho R^{3}.$ (36)
Combining Eq. 36 with Eq. 33, we obtain
$\frac{M}{M_{\rm B,T}}\propto R^{3}\lambda_{\rm B,T}^{-3}.$ (37)
As the size of the leaf structures is more determined by the choice of $N_{\rm
cells}$, they typically show very weak or no scaling between density and size,
and we can therefore approximate $R\propto\rho^{0}$. For the leaves, this
leads to $M/M_{\rm B,T}\propto\lambda_{\rm B,T}^{-3}$. As $\lambda_{\rm
B,T}\propto\rho^{-2/3}$ approximately (Fig. 20), this leads to $M/M_{\rm
B,T}\propto\rho^{2}$. For the branches, we find a shallower slope. In Ganguly
et al. (2022), we find many branches to follow $M\propto R$, which would lead
to $M/M_{\rm B,T}\propto\rho^{1/2}$, roughly consistent with the trend seen
for the branches here. The relation of the scaling between $\lambda_{\rm B,T}$
and $\rho$ is in itself interesting and we discuss it in Appendix F.
Overall, the Jeans analysis seems to show the emergence of potentially Jeans-
unstable structures at slightly lower densities ($\sim 30$ cm-3) compared to
that found in the previous energetic analysis. This could reflect the fact
that the Jeans analysis performed here does not include the kinetic energy,
which is often larger compared to $E_{\rm B}$ and the thermal energy (see
Section 4 in Ganguly et al., 2022). Turbulent motions can act as an effective
kinetic pressure term. Although the kinetic energy is often treated as an
effective pressure in the literature (see e.g. Chandrasekhar, 1951; Bonazzola
et al., 1987; Federrath & Klessen, 2012), we show in Ganguly et al. (2022)
that the volume and surface terms of the kinetic energy combine in a highly
non-trivial manner, with structures often being confined or even compressed
under ram pressure. This suggests that including a kinetic pressure in the
Jeans analysis would be too simplistic and not meaningful.
Overall, most leaf fragments in the Jeans analysis have $M/M_{\rm B,T}<1$,
suggesting that their fragmentation is unlikely to be primarily Jeans-like.
However, above 10-20 g cm-3, we begin to obtain Jeans unstable fragments which
are mostly unresolved and will likely undergo further fragmentation, possibly
ending up as the precursors of protostars.
### 6.5 Delay introduced by magnetic fields
The fragmentation analysis performed in 6.4 seems to suggest that magnetic
fields at least delay fragmentation in many cases. To estimate how much the
evolution of the cloud is slowed down by the effect of magnetic fields, we
define a delay timescale $\Delta t_{\rm B}$. Consider a structure of size $S$
which is compressed by an external flow with velocity $v$. In the absence of
either gravity or magnetic fields, as well as neglecting internal thermal and
kinetic pressure, the structure would be compressed on a crossing time,
$t_{\rm v}=S/v.$ (38)
For simplicity, we estimate the size of a structure using the shortest axis of
the equivalent ellipsoid, i.e. $S=2c$ (Section 3.2). We approximate the sweep-
up velocity $v$ to be equal to the bulk velocity of the structure, i.e.
$v=|\mathbf{v}_{0}|$, where $\mathbf{v}_{0}$ is obtained from Eq. 17. So
overall we have
$t_{\rm v}=2c/|\mathbf{v}_{0}|.$ (39)
Next, we consider an additional gravitational acceleration $a_{\rm g}$
assisting the sweeping up, where
$a_{\rm
g}=-\frac{1}{V}\int_{V}\mathbf{g}\cdot\frac{\mathbf{r}-\mathbf{r}_{0}}{|\mathbf{r}-\mathbf{r}_{0}|}d^{3}r$
(40)
is the average acceleration towards the centre of mass, $\mathbf{r}_{0}$. We
can then estimate (to first order) the gravitationally assisted sweep-up
timescale, $t_{\rm v,\ g}$, from
$S=vt_{\rm v,\ g}+\frac{1}{2}a_{\rm g}t_{\rm v,\ g}^{2}.$ (41)
For non-gravitating structures, this reduces to $t_{\rm v}$. For non-zero
gravitational field, taking the real root, we get
$t_{\rm v,g}=\frac{(-v+\sqrt{v^{2}+2Sa_{\rm g}})}{a_{\rm g}}.$ (42)
In the presence of magnetic fields, we can represent the combined acceleration
by gravity and magnetic fields as $a_{\rm g,B}$, where
$a_{\rm
g,B}=-\frac{1}{V}\int_{V}\left(\mathbf{g}-\frac{\nabla|\mathbf{B}|^{2}}{8\pi\rho}\right)\cdot\frac{\mathbf{r}-\mathbf{r}_{0}}{|\mathbf{r}-\mathbf{r}_{0}|}d^{3}r.$
(43)
We can then rewrite Eq. 42 to estimate a combined timescale
$t_{\rm v,g,B}=\frac{(-v+\sqrt{v^{2}+2Sa_{\rm g,B}})}{a_{\rm g,B}}.$ (44)
The time delay due to the presence of magnetic fields, $\Delta t_{\rm B}$, can
then be estimated as
$\Delta t_{\rm B}=t_{\rm v,g,B}-t_{\rm v,g}.$ (45)
Figure 13: The estimated delay timescale, $\Delta t_{\rm B}$ (Eq. 45), for
various MHD cloud structures from the high-den analysis at $t_{\rm evol}=2$
Myr. A power-law proportional to $R^{3}$ is plotted to show the rough scaling.
In Fig. 13, we plot $\Delta t_{\rm B}$ for various structures from the high-
den dendrogram analysis. We see that at the largest cloud scales at $t_{\rm
evol}=2$ Myr, $\Delta t_{\rm B}$ is of the order of $\sim 1$ Myr, and then
steadily decreases as a power-law roughly consistent with $\Delta t_{\rm
B}\propto R^{3}$. This timescale of $\sim 1$ Myr seems to be consistent with
the results of the fragmentation analysis in Section 6.4, where we found that
the significant differences in the cloud fragmentation properties at $t_{\rm
evol}=2$ Myr seem to have completely disappeared at $t_{\rm evol}=3.5$ Myr.
The general power-law trend We emphasise, however, that the calculation of
$\Delta t_{\rm B}$ should only be considered a first-order approximation. Note
that $\Delta t_{\rm B}$ does not directly depend on the magnetic field
strength but rather on its gradient. Hence, it is difficult to predict how
$\Delta t_{\rm B}$ would scale with different strengths of the background
field. This means that molecular clouds that form in a more magnetised medium
do not necessarily form structures more slowly.
### 6.6 Densities at which magnetic fields become dynamically sub-dominant
From the results presented in the previous sections, we can attempt to answer
the question of at what densities magnetic fields become dynamically sub-
dominant. From the density PDF of different clouds (Fig. 2), we find that the
density distribution is significantly different in the presence of magnetic
fields only below $\sim$ 100 cm-3. This is in accordance with previous
simulations and observations (Klessen & Burkert, 2001; Slyz et al., 2005;
Girichidis et al., 2014; Schneider et al., 2015), as well as the conclusions
drawn by Seifried et al. (2020a) using distributions of the three-dimensional
true optical depth, $A_{\rm V,3D}$. From the energetic analysis (Fig. 7), we
find that, magnitude-wise, gravity and kinetic energy supersede magnetic
fields above a few $\sim 100$ cm-3, consistent with the results of Ibáñez-
Mejía et al. (2022). Moreover, this density range is also in accordance with
the results of Seifried et al. (2020b), who find that relative orientation of
magnetic fields with respect to elongated filamentary structures changes at a
few $\sim 100$ cm-3 due to the occurrence of gravity-driven converging flows
(Soler & Hennebelle, 2017), suggesting energetic sub-dominance of magnetic
fields at higher densities. Lastly, also the fragmentation analysis presented
in this work (Fig. 11) shows differences in fragmentation patterns below a
similar density regime of $\sim 100$ cm-3. A Jeans fragmentation analysis
yields roughly consistent limits as well.
In summary, for clouds born from an ISM with typical magnetic field strengths
as in our Milky Way (Beck & Wielebinski, 2013), the density PDFs, the
energetic analysis, the histogram of relative orientation technique applied by
Seifried et al. (2020b), and the fragmentation analysis in this work - all
seem to point to the fact that the magnetic field becomes sub-dominant above
densities of around $100-1000$ cm-3. This overall trend is also fully
consistent with the $B-\rho$ relation obtained by Crutcher et al. (2010), who
conclude a transition density of $\sim 300$ cm-3.
## 7 Conclusions
We investigate the role magnetic fields play in determining the morphology,
energetics, and fragmentation properties of young molecular clouds by
analysing seven different simulated clouds (five with magnetic fields and two
without) from the SILCC-Zoom simulations. These simulations are geared to
study the evolution of the multi-phase interstellar medium in a supernova-
driven, turbulent, stratified galactic disc environment. To identify forming
structures, we use a dendrogram algorithm, and trace the statistical
properties of the identified structures. We include a simple chemical network
which allows us to follow the formation of H2 as the cloud assembles and
thereby distinguish between mostly atomic (H2 mass fraction < 50%) and mostly
molecular (H2 mass fraction > 50%) structures.
* •
We observe that the MHD clouds are fluffier, meaning that they have more
intermediate density gas between the number densities of roughly $1-100$ cm-3,
compared to their hydrodynamic counterparts. In the hydrodynamic clouds, the
lack of magnetic fields results in the denser structures being surrounded by a
comparatively more rarefied envelope.
* •
In terms of morphology, we find that almost all clouds are sheet-like, which
is consistent with recent observations of sheet-like envelopes around denser
filamentary cloud structures (Kalberla et al., 2016; Arzoumanian et al., 2018;
Rezaei Kh. & Kainulainen, 2022; Tritsis et al., 2022; Pineda et al., 2022;
Clarke et al., 2023). In our case, the MCs form due to compressions caused by
expanding supernova shells, consistent with the bubble-driven MC formation
scenario (Koyama & Inutsuka, 2000; Inoue & Inutsuka, 2009; Inutsuka et al.,
2015).
* •
We find that spheroidal structures within the clouds are rare on all spatial
scales, with $\sim 90$% of the structures being elongated. We further see that
the runs with magnetic fields have a roughly comparable fraction of filaments
and sheets, whereas the hydrodynamic runs overall produce more sheet-like
structures compared to filaments.
* •
Energetically, magnetic fields in our simulations are important for less dense
(up to $\sim$1000 cm-3) and mostly, but not exclusively, atomic structures.
The dynamics for denser and potentially star-forming structures is dominated
by the interplay of turbulence and gravity. This density threshold, above
which the magnetic fields seems to become sub-dominant, is supported by the
previous works of Seifried et al. (2020b), Ibáñez-Mejía et al. (2022) and is
consistent with the observed transition in the $B-\rho$ relation (Crutcher et
al., 2010).
* •
By investigating the magnetic surface energy term, we find that for most
structures it acts in a confining manner, and, for some low-density
structures, it even leads to overall magnetic confinement.
* •
By studying the numbers and masses of cloud fragments that form, we find that
at densities below roughly $\sim 100$ cm-3, the presence of magnetic fields
helps to create more massive fragments, but generally does not result in an
increase in the number of such structures. A stability analysis suggests that
in the resolved range, leaf fragments are mostly Jeans stable and the
fragmentation is not primarily governed by magnetic Jeans instabilities.
Instead of significantly altering the nature of fragmentation, magnetic fields
seem to rather slow down the fragmentation process. Using a simple order-of-
magnitude estimate, we find that this delay timescale is $\sim 1$ Myr.
Overall, using density PDFs, and an energetic as well as a fragmentation
analysis, we find a scenario where magnetic fields significantly affect the
flows and fragmentation in the lower density gas (below $\sim 100$ cm-3),
channelling flows and thereby affecting both, the morphology of the forming
structures as well as the formation timescale of the dense gas. Once the dense
structures (typically above $\sim 1000$ cm-3) form, however, the further
evolution and fragmentation of the dense gas seems to be mostly unaffected by
the magnetic field.
## Acknowledgements
We would like to thank the referee, Prof. Dr. Robi Banerjee, for their helpful
comments, suggestions, and overall discussion, which have increased the
quality of the paper. SG, SW, DS and MW would like to acknowledge the support
of Bonn-Cologne Graduate School (BCGS), which is funded through the German
Excellence Initiative, as well as the DFG for funding through SFB 956
’Conditions and Impact of Star Formation’ (subprojects C5 and C6). SDC is
supported by the Ministry of Science and Technology (MoST) in Taiwan through
grant MoST 108-2112-M-001-004-MY2. This research made use of astrodendro, a
Python package to compute dendrograms of Astronomical data
(http://www.dendrograms.org/); as well as yt, an open-source, permissively-
licensed python package for analyzing and visualizing volumetric data
(https://yt-project.org/). The 3D renderings in Fig. 3 were computed using
paraview. The FLASH code used in this work was partly developed by the Flash
Center for Computational Science at the University of Rochester.
## Data Availability
The data underlying this article can be shared for selected scientific
purposes after request to the corresponding author.
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## Appendix A Basic information of clouds
We present here some basic properties of the different analysed molecular
clouds. Fig. 14 plots the column density projections of all different clouds,
both HD and MHD. Fig. 15 plots the total and H2 mass of the different MCs in
the left panel, and the H2 mass fraction in the right panel. We see that there
is no difference in the overall mass of the clouds depending on the magnetic
field, but that the H2 mass fraction in the HD clouds is higher. The cloud
MC3-MHD (cyan line), which has been excluded from this analysis, has the
lowest total H2 mass, as well as the lowest H2 mass fraction.
Figure 14: Column density projection along the x axis for different molecular
clouds at t${}_{\rm evol}=3.5$ Myr. The MHD clouds have typically more diffuse
emission. Note that we have excluded MC3-MHD from further analysis due to its
low molecular content (see Fig. 15).
Figure 15: Left: Time evolution of total mass and total H2 mass in the
different molecular clouds, both HD and MHD, from $t_{\rm evol}=2$ to 3.5 Myr.
The solid lines represent the total mass, and the dashed lines represent the
H2 mass. Middle: H2 mass fraction for the same clouds, both HD and MHD. The
two HD clouds are plotted in reddish lines. Apart from MC3-MHD, which we
discard due to its low molecular gas mass, the other MHD and HD and clouds
have comparable masses. The two HD clouds, however, have a much higher H2 mass
fraction. Right: The mass in dendrogram for each cloud. Note that the
discarded MC3-MHD is missing as we did not perform a dendrogram on it. The
dendrogram mass has similar trends to the total mass.
## Appendix B Alternative PDF views
We present additional views of density PDFs (both with a linear scale and
mass-weighted) in Figs. 16 and 17 as a complementary addition to Fig. 2.
Figure 16: Density PDF with a linear y-axis for all HD and MHD clouds at
tevol=3.5 Myr. Figure 17: Mass-weighted PDF for all clouds, both HD and MHD,
at t${}_{\rm evol}=3.5$ Myr. The average mass percentage in the different
regimes (shown by the vertical dotted bars), for both HD and MHD clouds, is
shown as text. The mass contained at $\rho<10^{-24}$ g cm-3 is $<1\%$ for all
clouds. The mass difference in the intermediate regime, even at 3.5 Myr, is
clearly seen.
## Appendix C Distribution of visual extinction for magnetized and non-
magnetized clouds
The gas mass distribution at different $\mathrm{A_{V,3D}}$ values (Eq. 7) for
one example HD and MHD cloud of comparable mass is presented in Fig. 18, for 2
Myr (top) and 3.5 Myr (bottom). The vertical dashed line represents
$\mathrm{A_{V,3D}}=1$. The HD cloud has consistently higher mass at high
$\mathrm{A_{V,3D}}$ values.
Figure 18: Mass weighted $\mathrm{A_{v,3D}}$ PDF for different HD and MHD
clouds at $t_{\rm evol}$=3.5 Myr. MC2-MHD stands out as having much less
shielded gas mass compared to the other clouds. The other HD and MHD clouds
have similar behaviour.
## Appendix D Alternative view of the magnetic field - density relation
As a companion view to Fig.6, we show here the same relation between the
magnetic field and density, but this time using the average density,
$\rho_{\rm avg}$ instead of $\rho_{\rm thr}$. Since for any given structure,
$\rho_{\rm avg}\geq\rho_{\rm thr}$, this results in a shallower fit at the
high density end using $\rho_{\rm avg}$.
Figure 19: Similar to Fig. 6, but using $\rho_{\rm avg}$ instead of $\rho_{\rm
thr}$. This creates a shallower slope at particularly the high density end, as
$\rho_{\rm avg}\geq\rho_{\rm thr}$. However, the overall statistical trend is
similar.
## Appendix E Supplement to the magnetic Jeans analysis
The Jeans mass analysis is only conclusive provided the Jeans length is
resolved. In Fig. 20, we plot the ratio of the magneto-thermal Jeans length,
$\lambda_{\rm B,T}$, to the maximum resolution, $\Delta x$ ($\sim 0.125$ pc
for $\rho_{\rm thr}>10^{-22}$ g cm-3 and $\sim 0.25$ pc for $\rho_{\rm
thr}<10^{-22}$ g cm-3, see also Table 2), as a function of $\rho_{\rm thr}$
for all sub-structures at 3.5 Myr. The colour-bar, similar to Fig. 12, denotes
$c_{s}/v_{\rm A}$. The horizontal dotted line denotes $(2N_{\rm
cells})^{1/3}$. As $N_{\rm cells}$ denotes the minimum number of cells
required in the dendrogram analysis for any structure, $2N_{\rm cells}$ is the
minimum number of cells a structure must have in order to fragment. Therefore,
$(2N_{\rm cells})^{1/3}$ represents the minimum number of cells required in
one direction by which the Jeans length should be resolved. We find that this
seems to not be the case only for some structures with $\rho_{\rm thr}\gtrsim
10^{-20}$ g cm-3. When we fit $\lambda_{\rm B,T}$ against $R$ using a linear
least-squares fit on the logarithm of the data, we obtain an exponent of
$-0.70\pm 0.01$, roughly consistent with an exponent of $-2/3$.
Figure 20: The ratio of the magneto-thermal Jeans length, $\lambda_{\rm B,T}$
to the maximum resolution $\Delta x$ ($\sim 0.125$ pc for $\rho_{\rm
thr}>10^{-22}$ g cm-3 and $\sim 0.25$ pc for $\rho_{\rm thr}<10^{-22}$ g cm-3,
see also Table 2), as a function of $\rho_{\rm thr}$ for all MHD sub-
structures at 3.5 Myr. The horizontal dotted line denotes the resolution limit
for the present dendrogram analysis ($2N_{\rm cells}^{1/3}$, with $N_{\rm
cells}=100$). The red dashed line denotes the best-fit exponent for a linear
least-squares fit on the logarithm of the data. Structures above $\rho_{\rm
thr}\approx 10^{-20}$ g cm-3 seem to be not well resolved enough to be
conclusive regarding the fragmentation analysis.
The analysis performed in Fig. 12 considers the combined contribution of
magnetic and thermal perturbations. It might be interesting to note their
relative contributions. For this purpose, we show a purely magnetic Jeans
analysis in Fig. 21. Comparing $M/M_{\rm B}$ to $M/M_{\rm B,T}$ (Fig. 12), we
find little to no difference, suggesting that the magnetic contribution is in
this density range more important than the thermal contribution. This can also
be seen in the fact that most of the points have larger $v_{\rm A}$ compared
to $c_{s}$ (bluish in the colour-bar). For completeness, we explicitly include
the thermal Jeans mass and length plot in Fig. 22.
Figure 21: Top: same as the combined panels of Fig. 12, but for a purely
magnetic Jeans mass. Bottom: same as Fig. 20, but for a purely magnetic Jeans
length.
Figure 22: Top: same as Fig. 21, but for a purely thermal Jeans mass. Bottom:
same as Fig. 20, but for a purely thermal Jeans length. Note that the
$y$-range in the top panel is different to the previous similar plots.
## Appendix F The scaling-relation between Jeans length and density
The Jeans length, $\lambda$, depends on the characteristic wave speed, $c$
($c_{s}$, $v_{\rm A}$, or a combination of the two), and the density, i.e.
$\lambda\propto c\rho^{-1/2}.$ (46)
In our case, we are dominated by the magnetic over kinetic pressure, i.e.
$\lambda_{\rm B,T}\approx\lambda_{\rm B}$. The Alfvén wave speed scales as
$v_{\rm A}\propto\frac{B}{\rho^{1/2}}.$ (47)
This leads to
$\lambda_{\rm B,T}\propto\frac{B}{\rho}.$ (48)
For a scaling of $B\propto\rho^{1/2}$, this leads to $\lambda_{\rm
B,T}\propto\rho^{-1}$. A scaling of $B\propto\rho^{1/3}$ leads to
$\lambda_{\rm B,T}\propto\rho^{-2/3}$. The fitted value seems to be somewhere
in-between, closer to $\rho^{-2/3}$, and is also roughly consistent with the
overall $B-\rho$ scaling in Fig. 6 and Fig. 19.
|
11institutetext: Université Paris-Saclay, Université Paris Cité, CEA, CNRS,
AIM, F-91191, Gif-sur-Yvette, France 22institutetext: INAF - Osservatorio di
Astrofisica e Scienza dello Spazio di Bologna, via Piero Gobetti 93/3, I-40129
Bologna, Italy 33institutetext: INFN - Sezione di Bologna, viale Berti Pichat
6/2, I-40127 Bologna, Italy
33email<EMAIL_ADDRESS>
# Fast multi-scale galaxy cluster detection with weak lensing:
Towards a mass-selected sample
G. Leroy 11 S. Pires 11 G.W. Pratt 11 C. Giocoli 2233
(Received XXXX ; accepted XXXX)
The sensitivity and wide area reached by ongoing and future wide-field optical
surveys allows for the detection of an increasing number of galaxy clusters
uniquely through their weak lensing (WL) signal. This motivates the
development of new methods to analyse the unprecedented volume of data faster
and more efficiently. Here we introduce a new multi-scale WL detection method
based on application of wavelet filters to the convergence maps. We compare
our results to those obtained from four commonly-used single scale approaches
based on the application of aperture mass filters to the shear in real and
Fourier space. The method is validated on Euclid-like mocks from the
DUSTGRAIN-pathfinder simulations. We introduce a new matching procedure that
takes into account the theoretical signal-to-noise of detection by WL and the
filter size. We perform an analysis of the filters, and a comparison of the
purity and the completeness of the resulting detected catalogues. We show that
equivalent results are obtained when the detection is undertaken in real and
Fourier space, and when the algorithms are applied to the shear and the
convergence. We show that the multiscale method applied to the convergence is
faster and more efficient at detecting clusters than single scale methods
applied to the shear. We obtained an increase of 25$\%$ in the number of
detections while maintaining the same purity compared to the most up-to-date
aperture mass filter. We analyse the detected catalogues and quantify the
efficiency of the matching procedure, showing that less than $5\%$ of the
detections from the multiscale method can be ascribed to line-of-sight
alignments. The method is well-adapted to the more sensitive, wider-area,
optical surveys that will be available in the future, and paves the way to
cluster samples that are as near as possible to being selected by total matter
content.
###### Key Words.:
Gravitational lensing : weak ; \- Galaxies: clusters : general ; \- methods :
data analysis ; \- Cosmology : dark matter, large-scale structure of Universe
\-
## 1 Introduction
Modern cosmological models show that gravitational collapse drives cosmic
structure formation through a hierarchical assembly process in which objects
merge into larger and larger structures. Clusters of galaxies sit at the
endpoint of this process, so the formation and evolution of the cluster
population directly traces the growth of cosmic structure over time. This
information can be used to constrain cosmological models. For example, the
number of clusters as a function of mass and redshift is highly sensitive to
the underlying cosmological parameters (e.g. White & Rees, 1978; Perrenod,
1980; Voit, 2005; Allen et al., 2011).
The matter content of clusters – composed of dark matter (DM; 85%), ionised
hot gas in the intracluster medium (12%), and stars ($\sim 3$%) – reflects
that of the Universe. Clusters are typically detected through their baryonic
components. Studies based on samples from X-ray and Sunyaev-Zeldovich (SZ)
surveys have been very successful at providing cosmological constraints (e.g.
Pacaud et al., 2006; Vikhlinin et al., 2009; Hasselfield et al., 2013; Bocquet
et al., 2019; Salvati et al., 2021), and increasingly-wide-field optical
surveys have become competitive in recent years (e.g. Rozo et al., 2010;
Hamana et al., 2015; Lesci et al., 2022). However, leveraging the data from
these surveys requires linking the baryonic observables (optical richness,
X-ray luminosity, and SZ flux) to the underlying mass. Today, survey sample
sizes are sufficiently large that the dominant uncertainties in cosmological
parameter estimation with clusters lie in systematic effects in the mass
estimates and the selection function (e.g. Pratt et al., 2019, and references
therein).
The vast amount of matter contained in clusters bends the light of background
galaxies. This coherent deflection of the path of light from background
sources by an intervening mass is termed gravitational lensing (e.g. Schneider
et al., 1992; Pyne & Birkinshaw, 1993). At large scales, the distortion in the
shapes of the background galaxies is detectable only statistically, and is
consequently termed weak gravitational lensing. The image distortions due to
the weak lensing effect can be characterised by the shear (a warping of the
background source images) and the convergence (a magnification effect on the
same). Both shear and convergence can provide insights into the statistical
properties of the weak lensing field, and has been shown that they contain
precisely the same information (e.g. Schneider et al., 2002; Pires et al.,
2020). The weak lensing effect is directly sensitive to the total projected
mass along the line of sight (LOS). As such, it is an attractive method for
cluster detection, potentially paving the way towards true mass-selected
samples. However, the sensitivity of cluster detection through the weak
lensing signal depends critically on the number of background sources that are
available to be lensed by the intervening matter. Only recently have deep,
wide-field surveys yielded the background source densities needed to detect
substantial numbers of objects.
A number of methods have been developed to detect galaxy clusters through
their weak lensing shear or convergence signal in optical imaging data.
Application of a simple Gaussian filter to the convergence, combined with
thresholding, was widely used in early optical surveys (e.g. White et al.,
2002; Miyazaki et al., 2002; Hamana et al., 2004; Tang & Fan, 2005; Gavazzi &
Soucail, 2006; Miyazaki et al., 2007; Fan et al., 2010; Shan et al., 2012,
2018). Such methods demonstrated that it was possible to detect clusters
through their weak lensing signal, while motivating the development of new,
more efficient, approaches. In the subsequent development of optimal filtering
techniques, the filter kernel was adapted to include information on the shape
of the expected halo profile, while excluding the shape noise and the
contribution of large-scale structure (LSS) to the noise budget (e.g. Hennawi
& Spergel, 2005; Maturi et al., 2005; Wittman et al., 2006).
The widely used aperture mass (AM) technique, introduced by Schneider (1996),
consists of convolving the lensing signal with a filter function of a specific
scale. A number of new filters have since been developed to maximise the
effectiveness of the AM method; they have been tested on simulations and
applied to various optical surveys (e.g. Schneider, 1996; Schneider et al.,
1998; Jarvis et al., 2004; Schirmer et al., 2004; Hetterscheidt et al., 2005;
Hennawi & Spergel, 2005; Maturi et al., 2005, 2007; Pace et al., 2007;
Schirmer et al., 2007; Dietrich & Hartlap, 2010; Hamana et al., 2012; Lin et
al., 2016; Miyazaki et al., 2018; Hamana et al., 2020; Oguri et al., 2021).
The filter function is a key component of the AM method and must be designed
to obtain the optimal signal-to-noise ratio (S/N) at a given scale. A defining
characteristic of the above approaches is that they all operate on a single
scale. As such, their detection efficiency is highly dependent on the filter
design and on its relation to the size of the structures we want to detect.
While some studies have proposed using multi-scale wavelet filters to de-noise
the convergence map (Starck et al., 2006; Lanusse et al., 2016), they were not
optimised for cluster detection because they do not take the LSS contribution
into account. Nevertheless, such multi-scale de-noising techniques have been
shown to be a promising approach for cluster detection (Leonard et al., 2015).
Motivated by the potential of current and upcoming deep wide-field optical
surveys with sufficient background source densities, such as the Hyper
Supreme-Cam (HSC) survey (Aihara et al. 2018), the Legacy Survey of Space and
Time (LSST111https://www.lsst.org/; Closson Ferguson et al. 2009; Ivezić et
al. 2019), Euclid222https://www.euclid-ec.org/ (Laureijs et al. 2011), and the
Roman Space Telescope333https://roman.gsfc.nasa.gov/ (formerly WFIRST; Spergel
et al. 2015), we revisit here the question of cluster detection through the
weak lensing effect. We perform a quantitative comparison of existing single-
scale detection methods, including a complete analysis of the filters. We
introduce a new multi-scale detection approach based on the wavelet transform
applied to the convergence. We chose to focus on the convergence because this
quantity explicitly traces the total matter distribution integrated along the
LOS and is computationally less expensive to analyse, making it ideal for
application to upcoming large-scale survey data. We quantify the performance
of our new multi-scale approach by applying it to the DUSTGRAIN-pathfinder
simulations detailed in Giocoli et al. (2018), which feature source densities
similar to those expected from the Euclid survey. We find that the new multi-
scale method operating on the convergence is faster and more efficient at
detecting clusters than currently used single-scale methods operating on the
shear.
This paper is organised as follows. In Sect. 2 we summarise the key aspects of
gravitational lensing. The mock dataset DUSTGRAIN-pathfinder simulations are
described in Sect. 3. Section 4 introduces the AM formalism and several
commonly used AM filters. A description of the wavelet formalism is also
provided. Section 5 details the detection procedure that we use to compare the
different filters. An analysis of the different options in the implementation
of the detection algorithm is provided in Sect. 6. In Sect. 7 we provide the
details of the matching procedure that we developed to allow for a fair
comparison of the methods. Finally, the performance of the detection methods
is evaluated in Sect. 8, and we conclude in Sect. 9.
Figure 1: Simulated shear maps covering a field of $5^{\circ}\times 5^{\circ}$
(left and middle panels) and the corresponding E-mode convergence map (right).
The E-mode convergence map directly traces the projected matter distribution,
the overdensities appearing as bright structures.
## 2 Weak gravitational lensing theory
### 2.1 Weak lensing theory
The gravitational field of massive objects affects the path of light in their
vicinity. Thus, the light from background galaxies is deflected as it travels
towards us and their images appear distorted. These distortions, or (reduced)
shear, are a direct observable and are an imprint of the intervening large-
scale matter distribution. Different structures, such as clusters of galaxies,
filaments, or even individual galaxies, can act as lenses and create this
warping effect.
We summarise here the gravitational lens theory that is sufficient for the
treatment of lensing by galaxy clusters (see e.g. Bartelmann & Schneider,
2001). We consider a lens at angular position $\boldsymbol{\theta}$ and at
distance $D_{\rm l}$ from the observer. Its surface mass density,
$\Sigma(\theta)$, is integrated from its 3D mass density,
$\rho(\boldsymbol{\theta},z),$ along the LOS:
$\Sigma(\boldsymbol{\theta})=\int_{0}^{+\infty}\rho(\boldsymbol{\theta},z)dz.$
(1)
From the surface mass density, we can define the lensing potential,
$\psi(\boldsymbol{\theta}),$ according to the position of the background
galaxy sources with distance $D_{\rm s}$ from the observer and distance
$D_{\rm ls}$ from the lens:
$\psi(\boldsymbol{\theta})=\frac{4G}{c^{2}}\frac{D_{\rm l}D_{\rm ls}}{D_{\rm
s}}\int_{\rm\mathbb{R}^{2}}\Sigma(\boldsymbol{\theta}^{\prime})\ln{(\lvert\boldsymbol{\theta}^{\prime}-\boldsymbol{\theta}\rvert)}\,d^{2}\boldsymbol{\theta}^{\prime}.$
(2)
When light rays travel close to a lens, they are bent, leading to image
distortions and, potentially, to multiple images. These images appear with a
deflection angle, $\alpha(\boldsymbol{\theta}),$ induced by the lensing
potential, $\psi(\boldsymbol{\theta})$:
$\alpha(\boldsymbol{\theta})=\nabla\psi(\boldsymbol{\theta}).$ (3)
Here $\alpha(\boldsymbol{\theta})$ is the difference between the angular
position, $\boldsymbol{\beta}$, where the images would be without the lens and
the observed position, $\boldsymbol{\theta}$. This is summarised by the lens
equation:
$\alpha(\boldsymbol{\theta})=\boldsymbol{\theta}-\boldsymbol{\beta}.$ (4)
From Eqs. 2, 3, and 4, the deviation, $\boldsymbol{\beta}$, of the observed
image with respect to the undistorted image can be derived:
$A\equiv\frac{\partial\boldsymbol{\beta}}{\partial\boldsymbol{\theta}}=\delta_{ij}-\frac{\partial^{2}\psi(\boldsymbol{\theta})}{\partial\theta_{i}\partial\theta_{j}}=\pmatrix{1}-\kappa-\gamma_{1}&-\gamma_{2}\\\
-\gamma_{2}1-\kappa+\gamma_{1}.$ (5)
Here
$\displaystyle\gamma_{1}=\frac{1}{2}(\partial_{1}^{2}-\partial_{2}^{2})\psi\
\rm{and}$ (6) $\displaystyle\gamma_{2}=\partial_{1}\partial_{2}\psi$ (7)
correspond to the two components of the shear $\gamma$, and
$\kappa=\frac{1}{2}(\partial_{1}^{2}+\partial_{2}^{2})\psi$ (8)
corresponds to the convergence that is the shape contraction or dilation of
the image.
From the above, we can express the convergence as a function of the critical
value of the surface mass density, $\Sigma_{\rm crit}$:
$\kappa(\boldsymbol{\theta})=\frac{\Sigma(\boldsymbol{\theta})}{\Sigma_{\rm
crit}},$ (9)
with
$\Sigma_{\rm crit}=\frac{c^{2}}{4\pi G}\frac{D_{\rm s}}{D_{\rm l}D_{\rm ls}}.$
(10)
Thus, the convergence is a tracer of the total matter distribution integrated
along the LOS, weighted by the redshift-dependent factor $1/\Sigma_{\rm
crit}$.
The direct observable is the reduced shear, $g$, that is derived from the
ellipticities of the observed background galaxies. The reduced shear, $g$, is
defined as follows:
$g=\frac{\gamma}{1-\kappa}.$ (11)
In the weak lensing regime ($\kappa\ll 1$), it approximates the shear,
$\gamma$.
### 2.2 Application
In practice, we observe a distorted image, from which we measure the galaxy
ellipticities and estimate the shear components, $\gamma_{1}$ and
$\gamma_{2}$. Kaiser & Squires (1993) introduced the mass inversion technique,
which involves the computation of the convergence map $\kappa$ from the
measured shear field.
We can consider a complex notation to represent the shear field,
$\gamma=\gamma_{1}+\rm{i}\gamma_{2}$, and the convergence field,
$\kappa=\kappa_{E}+\rm{i}\kappa_{B}$, with $\kappa_{E}$ and $\kappa_{B}$,
called E- and B-modes. Then, taking the Fourier transform of Eqs. 6, 7, and 8,
we obtain
$\displaystyle\hat{\gamma}=\hat{P}\,\hat{\kappa},$ (12) $\displaystyle{{\rm
with:}\,\,}\hat{P}=\hat{P}_{1}+i\,\hat{P}_{2}\rm{\,\,and\,\,},$ (13)
$\displaystyle\hat{P}_{1}=\frac{k_{1}^{2}-k_{2}^{2}}{k_{1}^{2}+k_{2}^{2}},$
(14)
$\displaystyle\hat{P}_{2}=\frac{2\,k_{1}^{2}k_{2}^{2}}{k_{1}^{2}+k_{2}^{2}},$
(15)
where the hat symbol refers to the Fourier transform, and $k_{i}$ is the wave
number at the angular position, $\theta_{i}$.
Considering the conjugate $\hat{P}*=\hat{P}_{1}-i\hat{P}_{2}$, we can
reconstruct $\kappa_{E}$ and $\kappa_{B}$ from the complex shear $\gamma$,
obtaining
$\displaystyle\hat{\kappa}_{E}=\hat{P}_{1}\hat{\gamma}_{1}+\hat{P}_{2}\hat{\gamma}_{2}$
(16)
$\displaystyle\hat{\kappa}_{B}=-\hat{P}_{2}\hat{\gamma}_{1}+\hat{P}_{1}\hat{\gamma}_{2}.$
(17)
Given that $\psi$ is a scalar potential, it can be shown that weak lensing
does not in principle produce B-modes. Thus, B-modes can be used to estimate
the level of the noise in the data.
Figure 1 shows an example of simulated shear and convergence maps without
shape noise derived from the DUSTGRAIN-pathfinder simulation (described in
Sect. 3) covering a field of $5^{\circ}\times 5^{\circ}$. The left and middle
panels correspond to the two components of the shear, $\gamma_{1}$ and
$\gamma_{2}$ and the right panel to the corresponding E-mode convergence map,
$\kappa_{E}$. Although both the shear and the convergence can be used for
cluster detection, the convergence is easier and computationally less
expensive to analyse because it is a scalar field that is proportional to the
projected matter distribution.
## 3 Weak lensing simulations
### 3.1 The DUSTGRAIN N-body simulations
We make use of a suite of cosmological N-body simulations called The
DUSTGRAIN-pathfinder (see Giocoli et al., 2018, for a detailed description).
This DM-only simulation traces the collisionless evolution of $783^{3}$ DM
particles with a mass $m_{\rm CDM}=8.1\times 10^{10}$ $h^{-1}$ M⊙ , contained
within a periodic cosmological box of side 750 $h^{-1}$ Mpc. In the present
work, we use a subset of the full DUSTGRAIN-pathfinder runs consisting of 256
realisations sharing the same standard cosmological parameters in agreement
with Planck Collaboration XIII 2016: $\Omega_{M}=\Omega_{\rm CDM}+\Omega_{\rm
b}+\Omega_{\nu}=0.31345$, $\Omega_{\rm b}=0.0481$, $\Omega_{\Lambda}=0.68655$,
$h=0.6731,$ and $\sigma_{8}=0.847$. The different LOS realisations were
obtained by randomising the stacked comoving cosmological boxes through
combinations of the following procedures: (i) changing the sign of the
Cartesian coordinates, (ii) redefining the position of the observer, and (iii)
modifying the order of the axes in the coordinate system. By construction,
these variations preserve the clustering properties of the particle
distribution at the scale of the comoving simulation snapshot.
### 3.2 Mock dark matter halo catalogues
The mock DM halo catalogues were generated by identifying the DM haloes in the
DUSTGRAIN simulation through a friends-of-friends algorithm (Davis et al.
1985) with linking distance $\lambda=0.16\times d$, where $d$ is the mean
separation distance between particles. The `SUBFIND` algorithm (Springel et
al. 2001) was then used to evaluate the standard parameters of each friends-
of-friends-identified halo: the redshift, $z$, and the virial mass, $\rm
M_{200c}$, and radius $\rm R_{200c}$, corresponding to the mass and radius of
a spherical region around the fiducial centre of each halo enclosing 200 times
the critical density of the Universe. Thus, for each of the 256 realisations,
the catalogue is composed of the positions (i.e. the right ascension and the
declination) of the identified DM haloes, their estimated redshift, $z_{l}$,
and their virial mass, $\rm M_{200c}$, and radius, $\rm R_{200c}$. This DM
halo catalogue is used in our study for the matching procedure to quantify the
purity and completeness of each selected sample.
### 3.3 Mock galaxy catalogues
The particles stored in 21 different snapshots were used to construct
continuous past light cones from $z=0$ to $z=4$ using the `MAPsim` pipeline
(Giocoli et al., 2015, 2017). The routine extracts the positions of each
particle to recreate the past light cone and at the same time, the particles
are binned to build $27$ different lens planes to recompose the projected
matter density distribution. Then, the shear of each galaxy is computed by
projection, using the Born approximation, which assumes unperturbed light
paths to integrate the lensing distortions (Bartelmann & Schneider, 2001). The
redshift distribution of the galaxies, $n(z)$, was built to follow a realistic
distribution approaching the one expected for the Euclid survey (e.g. Cropper
et al. 2013). Each realisation contains around 3 million galaxies in a field
of view of $5^{\circ}\times 5^{\circ}$, with a galaxy distribution extending
up to $z=4$. This leads to a galaxy density of about $n_{\rm{g}}=30$ gal.
arcmin-2.
Noise was added to the shear to mimic realistic surveys. Uncertainties in the
shear (referred to as ‘shape noise’) arise from a combination of the
unavoidable intrinsic shape of the galaxies (referred to as the ‘intrinsic
shape noise’), and measurement errors that include, among other factors,
uncertainties in the galaxy shape measurement and the point spread function
correction. The intrinsic shape of the galaxies being the dominant component
and the galaxies being randomly distributed, the shape noise can be modelled
as an additive noise. This can be well approximated by a Gaussian distribution
with a mean of $\mu=0$ and a standard deviation of
$\sigma_{\rm{\epsilon}}=0.26$ (e.g. Leauthaud et al., 2007; Schrabback et al.,
2015, 2018). For each galaxy, the shape noise was therefore included in the
catalogue by adding Gaussian noise to the two components of the shear. For
each of the 256 realisations, the galaxy catalogue is composed of: (i) the
positions (i.e. the right ascension and the declination) of the galaxies, (ii)
the two components of the shear, and (iii) the redshift. These noisy galaxy
catalogues are then used as the inputs to evaluate the detection algorithms
described below.
## 4 Filters
In this section we present several different filters that have been used for
weak lensing cluster detection in previous studies, and we compare them to the
wavelet filters that we use in our new multi-scale detection approach. A
complete analysis and comparison of the different filters both in real and
Fourier space is also provided.
Although used since the very first weak lensing detection algorithms (e.g.
White et al., 2002; Miyazaki et al., 2002; Hamana et al., 2004; Tang & Fan,
2005; Gavazzi & Soucail, 2006; Miyazaki et al., 2007; Fan et al., 2010; Shan
et al., 2012, 2018), we excluded the Gaussian filter from this analysis
because it behaves as a low-pass filter and is thus not optimal for cluster
detection. Although low-pass filters appear to be natural tools for reducing
the shape noise that dominates at high-frequencies, they are not adapted to
diminish the low-frequency signal coming from LSS, which causes many spurious
peaks (see e.g. Maturi et al., 2005). Since the use of such low-pass filters
is sub-optimal for the current application, we focus our analysis on pass-band
filters that are better able to target the signal around a given scale.
### 4.1 The aperture mass formalism
The AM method (Schneider, 1996) is commonly used to reconstruct maps of the
projected matter distribution at a given scale. The AM map can be evaluated at
a position $\boldsymbol{\theta_{\rm o}}$ by convolving the convergence field,
$\kappa$, with a filter, $U$, such that
$M_{\rm ap}(\boldsymbol{\theta_{\rm
o}})=\int_{\mathbb{R}^{2}}\kappa(\boldsymbol{\theta})\,U(\lvert\,\boldsymbol{\theta}\,-\,\boldsymbol{\theta_{\rm
o}}\,\rvert)\,d^{2}\boldsymbol{\theta},$ (18)
where $U$ must be compensated; that is,
$\int_{0}^{\rm\theta_{\rm
ap}}\rm\theta^{\prime}\,U(\rm\theta^{\prime})\,d\theta^{\prime}=0$ (19)
must be fulfilled within the aperture of radius $\theta_{\rm ap}$.
Alternatively, the AM map can also be computed from the tangential component
of the shear, $\gamma_{\rm t}$, which is a quantity that can directly be
computed from $\gamma_{1}$ and $\gamma_{2}$:
$\gamma_{\rm t}=\gamma_{1}\cos(\rm\phi)+\gamma_{2}\sin(\rm\phi),$ (20)
where $\phi$ is the polar angle
$\phi(\boldsymbol{\theta},\boldsymbol{\theta_{0})}$ relative to the centre of
the aperture, $\boldsymbol{\theta_{0}}$. The AM map at position
$\boldsymbol{\theta}_{\rm o}$ can then be obtained by convolving the
tangential shear with a filter, $Q$, such that
$M_{\rm
ap}(\boldsymbol{\theta_{o}})=\int_{\mathbb{R}^{2}}\gamma_{t}(\boldsymbol{\rm\theta})\,Q(\lvert\,\boldsymbol{\rm\theta}\,-\,\boldsymbol{\rm\theta_{\rm
o}}\,\rvert)\,d^{2}\boldsymbol{\rm\theta},$ (21)
the filters $Q$ and $U$ being linked by the relations (Schneider & Bartelmann,
1997)
$Q(\rm\theta)=\frac{2}{\rm\theta^{2}}\int_{0}^{\rm\theta}\rm\theta^{\prime}U(\rm\theta^{\prime})\,d\theta^{\prime}-U(\rm\theta)$
(22)
and
$U(\rm\theta)=2\int_{\theta}^{+\infty}\rm\frac{Q(\rm\theta^{\prime})}{\theta^{\prime}}\,d\theta^{\prime}-Q(\rm\theta).$
(23)
The performance of the AM technique at finding clusters on a map depends on
the exact choice of filter, and on its capacity to isolate the lensing signal
from the shape noise and the LSS component. The optimal filters $U$ or $Q$
should be designed to fulfil two properties. In real-space, the filters should
be local, that is, decrease smoothly to zero within a finite radius. Ideally,
they should fit the shape of the structures of interest. Moreover, they should
be local in Fourier space, focusing on a particular angular scale. Ideally,
their size should match as closely as possible the size of the structures of
interest.
A number of functions or families of functions have been proposed (e.g.
Schneider, 1996; Schirmer, 2004; Schirmer et al., 2007; Jarvis et al., 2004;
Hamana et al., 2012; Miyazaki et al., 2018). In the present study, we consider
the following four well-known AM filters.
The first is the original AM family of filters defined in the paper by
Schneider (1996, hereafter $\rm S96$):
$U_{\rm S96}(\theta)=\left\\{\begin{array}[]{ll}1&\mbox{x $\in$ [0,
$\nu_{1}R$],}\\\
\frac{1}{1-c}\big{[}\frac{\nu_{1}R}{\sqrt{(x-\nu_{1}R)^{2}+(\nu_{1}R)^{2}}}\big{]}&\mbox{x
$\in$ [$\nu_{1}R$, $\nu_{2}R$],}\\\ \frac{b}{R^{3}}(R-x)^{2}\,(x-\alpha
R)&\mbox{x $\in$ [$\nu_{2}R$, $R$],}\\\ 0&\mbox{x > $R$},\end{array}\right.$
(24)
where parameters $\alpha$, $b,$ and $c$ ensure that the filter is compensated
and continuous in real space. The parameters $\nu_{1}$, $\nu_{2}$, and $R$
define the angular extent of the filter and must be adapted to the size and
shape of the clusters we want to detect. We implemented the different
parameterisations presented in Schneider (1996) and we use $\nu_{1}=0.1$,
$\nu_{2}=0.9$, $R=9\arcmin$, $\alpha=0.8531$, $b=-329.8,$ and $c=0.2415$,
which we found to maximise the purity and completeness. The corresponding
$Q_{\rm S96}$ can be calculated analytically with Eq. 22.
The second is the filter developed by Schirmer (2004, hereafter $\rm TANH$):
$Q_{{\rm TANH}}(\theta)=\frac{1}{1+{\rm e}^{6-150\,\frac{\theta}{R}}+{\rm
e}^{-47+50\,\frac{\theta}{R}}}\frac{\tanh\left(\frac{\theta}{x_{c}R}\right)}{\frac{\theta}{x_{c}R}},$
(25)
where $R$ is the truncation radius and $x_{c}$ (dimensionless) defines the
width of the filter. This filter was directly derived from the shape of the
tangential shear of the Navarro-Frenk-White (NFW) profile (Navarro et al.,
1996). Following Schirmer et al. (2007), we used $x_{c}=0.1$. We fixed the
truncation radius $R=7\arcmin$ as the optimal choice to maximise the number of
detections, following the procedure described by Hetterscheidt et al. (2005).
The corresponding $U_{\rm TANH}$ can be computed numerically using Eq. 23.
The third is the filter proposed by Jarvis et al. (2004, hereafter J04):
$U_{\rm J04}(\rm\theta)=\frac{1}{\rm
2\pi\sigma^{2}}\,\left(1-\frac{\rm\theta^{2}}{\rm
2\sigma^{2}}\right)\,\exp{\left(-\frac{\rm\theta^{2}}{2\sigma^{2}}\right)},$
(26) $Q_{\rm J04}(\rm\theta)=\frac{1}{\rm
2\pi\sigma^{2}}\frac{\rm\theta^{2}}{\rm
2\sigma^{2}}\exp{\left(-\frac{\rm\theta^{2}}{\rm 2\sigma^{2}}\right)},$ (27)
where $U_{\rm J04}$ was defined following Van Waerbeke (1998) as the second
derivative of a Gaussian function with standard deviation $\sigma$. The
$U_{\rm J04}$ is also known as the Mexican hat wavelet filter. In the
following mass aperture computation, we use $\sigma=4\arcmin$, corresponding
to an apparent angular radius of about $4^{\prime}$, and a truncation radius
$R=20\arcmin$, as suggested by Leonard et al. (2012) to minimise oscillations
and high frequency mode contamination.
The fourth is the filter introduced by Hamana et al. (2012) and used by
Miyazaki et al. (2018, hereafter M18), which is defined as a truncated
Gaussian filter:
$U_{\rm
M18}(\rm\theta)=\frac{1}{\rm\pi\,\theta_{s}^{2}}\,\exp{\left(\frac{\rm\theta^{2}}{\rm\theta_{s}^{2}}\right)}\,-\,U_{\rm
o},$ (28)
where $U_{0}$ is a parameter to ensure that the filter $U$ is compensated and
$Q_{\rm
M18}(\theta)=\frac{1}{\rm\pi\,\theta^{2}}\bigg{[}1\,+\,\left(1-\frac{\rm\theta}{\rm\theta_{s}}\right)^{2}\bigg{]}\,\exp{\left(\frac{\rm\theta^{2}}{\rm\theta_{s}^{2}}\right)},$
(29)
with $\theta_{s}$ being defined as the angular scale of the aperture. In
practice, the choice of $\theta_{s}$ influences the angular radius of the
cluster the filter targets, but the relation is not direct. Following Miyazaki
et al. (2018), we use $\theta_{s}=1.5\arcmin$, and $U_{\rm M18}(\theta)=0$ if
$\theta>R$, with a truncation radius $R=15\arcmin$. We fixed this value of $R$
to reduce the impact of the truncation on the filter behaviour as it derives
from a Gaussian filter. Indeed, the filter $U_{\rm M18}$ reaches negative
values at $U_{\rm M18}(R)$ and is subsequently forced to $0$ by the
truncation. This step in the function results in contamination in the signal.
As the choice of truncation radius strongly impacts the computation time,
there is a clear trade off between the computation time and the filter
behaviour.
### 4.2 The wavelet formalism
The wavelet formalism can also be used to reconstruct maps of the projected
matter distribution at a given scale (see e.g. Leonard et al., 2012; Pires et
al., 2012). In Leonard et al. (2012), the authors showed that wavelet filter
functions at a given scale are formally identical to AM filter functions at
that scale. Similar to the AM functions, many different wavelet functions
exist, including starlet, Mexican hat, Morlet, and biorthogonal (see e.g.
Starck et al., 1998, 2006, for reviews). Many wavelet functions have been
specifically designed to fulfil the properties of localisation in real and
Fourier spaces, which is a distinct advantage compared to the AM approach.
A wavelet map of the convergence $\kappa$ at position $\theta$ and scale $a$
can be computed as
$W_{a}(\theta)=\frac{1}{\sqrt{a}}\int\kappa(t)\,\Psi_{a}(\,t-\theta\,)\,dt,$
(30)
where $\Psi_{a}$ is the wavelet filter of scale $a$. All wavelet filters are
defined such that they respect the condition of zero mean. Therefore, they are
compensated by definition.
Figure 2: Comparison of different filter functions, $U$ (left) and $Q$
(right), for AM map reconstruction. The figure shows in blue (dotted lines)
the AM filter functions defined by Schneider (1996, S96) with $R=9\arcmin$; in
red (dash-dotted lines) the AM filter functions defined in Schirmer (2004,
TANH) with $R=7\arcmin$; in purple (dash-dotted lines) the AM filter functions
defined in Jarvis et al. (2004, J04) with $\sigma=4\arcmin$; and in green
(dashed lines) the AM filter functions defined by Miyazaki et al. (2018, M18).
The filter $U_{\rm M18}$ is also a compensated filter thanks to the $U_{0}$
term, but due to the high truncation radius we used, this is not distinctly
visible. Finally, the AM filter functions corresponding to the wavelet filter
functions at scale $i=4$ are displayed in orange (solid lines). All the
filters are normalised to have a maximum amplitude of 1.
Another advantage of the wavelet formalism is that it can decompose a given
image into several complementary scale components. This is possible using the
wavelet transform, for which fast algorithms exist. For our study, we use the
isotropic un-decimated wavelet transform, also called the starlet transform
(e.g. Starck et al., 1998, 2006), which is able to simultaneously compute
several wavelet maps on dyadic scales. The starlet functions are isotropic,
which makes them well suited to extracting galaxy clusters, which appear
roughly circular in the convergence maps. As such functions decompose an image
into several complementary scales (see Sect. 4.3), they are ideal for
undertaking a multi-scale analysis and for targeting structures of different
angular sizes. The starlet transform decomposes the convergence as follows:
$\kappa(\theta)=C_{J}(\theta)+\sum_{i=1}^{J}W_{i}(\theta),$ (31)
where $J$ is the number of scales of the decomposition, $C_{J}$ is the
corresponding smoothed version of the convergence $\kappa$, and $W_{i}$ are
the wavelet maps targeting clusters with an apparent angular radius of
$2^{i-1}$ times the size of the pixel. As a consequence, the wavelet scales
are fixed by the pixel size, and, by definition, increase as a power of two.
In the starlet transform, the wavelet functions are defined as the difference
between two $B^{3}$-spline functions at different resolutions. Their
application to the convergence map was shown to be equivalent to applying the
following AM filter (Leonard et al., 2012):
$U_{W_{i}}(u)=\frac{1}{9}\,\bigg{[}\,93\,\left\lvert
u\right\rvert^{3}\,-\,64\,\left(\,\left\lvert\frac{1}{2}\,-\,u\,\right\rvert^{3}\,+\,\left(\,\frac{1}{2}\,+\,u\,\right)^{3}\right)\\\
+18\,\left(\,\left\lvert\,1\,-\,u\,\right\rvert^{3}\,+\,\left\lvert\,1+\,u\,\right\rvert^{3}\,\right)-\frac{1}{2}\,\left(\,\left\lvert\,2-\,u\,\right\rvert^{3}+\left\lvert\,2+\,u\,\right\rvert^{3}\right)\bigg{]},$
(32)
where
$u=\frac{\theta}{2^{i}x},$ (33)
and $x$ is the pixel size. The new multi-scale detection method that we
introduce in Sect. 5.3 takes advantage of this multi-scale decomposition, the
wavelet transform being able rapidly to decompose the convergence into
different scales.
### 4.3 Filter analysis
The performance of each detection method is closely linked to the capacity of
the filter functions to reduce the shape noise and the contribution from LSS,
while simultaneously minimising the signal loss. In the following, we analyse
the properties of the filters described above both in real and Fourier spaces
to help in the interpretation of the results presented in Sect. 8.
Figure 3: Fourier space comparison of several AM filters and wavelet filters,
expressed as a function of wave number $k$ in arcmin-1 and filter radius in
arcmin. The dotted blue line (upper panel) represents the S96 filter, the
dash-dotted red and purple lines (middle panels) represent the TANH and J04
filters, and the dashed green line (lower panel) corresponds to the M18
filter. The solid lines show the different wavelet filters: the teal, green,
orange, and red light lines correspond to the W2, W3, W4, and W5 wavelet
filters with radii of $1\aas@@fstack{\prime}17$, $2\aas@@fstack{\prime}32$,
$4\aas@@fstack{\prime}68$, and $9\aas@@fstack{\prime}37,$ respectively. For
comparison purposes, the J04 and M18 filters are forced to the same amplitude
as the W4 function to illustrate their behaviour in terms of filter radius.
In Fig. 2 we compare the different filter functions $Q$ and $U$, defined in
Sects. 4.1 and 4.2, in real space. This comparison underlines the differences
in filter shape. In particular, we focus our attention on their compensated
and local properties. The dotted blue curves correspond to the AM functions
defined in Schneider (1996). The filter $U_{\rm S96}$ is compensated and local
by design. The dash-dotted red curves correspond to the AM functions defined
in Schirmer (2004). The filter $U_{\rm TANH}$ drops rapidly to zero, but it is
not local and is only compensated at infinity. The localisation is imposed by
a truncation radius at $R=7^{\prime}$. The dash-dotted purple curves
correspond to the AM functions defined in Jarvis et al. (2004). This filter
$U_{\rm J04}$ is also not local, and is also only compensated at infinity
since it tends to zero when $\theta$ tends to $+\infty$. In practice, the
filter is truncated at R=$20\arcmin$ to have a local support. The dashed green
curves correspond to the AM functions defined in Miyazaki et al. (2018). In
the definition of $U_{\rm M18}$, $U_{0}$ ensures that the filter is
compensated within an aperture of radius $R$. However, the $Q_{\rm M18}$
function is non-local, which can introduce errors when truncating the
aperture. The solid orange curves correspond to the wavelet filter function
$U_{W_{4}}$ defined in Eq. 32 for $i=4$ (hereafter W4). The wavelet filter is
shown to be compensated within an aperture radius equal to two times the
filter radius.
It is also instructive to study the representation of the filter functions in
Fourier space as this allows us to fully understand how the filters are
designed to locate structures. In Fourier space, an optimal band-pass filter
should be local, in order to select only a specific range of frequencies
around the scale of interest, and thus reduce the noise and the contribution
from LSS. The design of AM filters can be delicate, and some such filters
suffer from oscillations in Fourier space owing to truncations applied in real
space that deteriorate the band-pass ability of the filter (see e.g. Leonard
et al., 2012, for more details). In contrast, many wavelet functions have been
specifically designed to fulfil the properties of localisation in real and
Fourier space.
To analyse the filters precisely in Fourier space, we studied their impulse
response. For this purpose, we simulated a null convergence map with a single
peak at the centre. Then we passed this convergence map through the four AM
filters and each of the wavelet functions, and computed the corresponding
power spectra in Fourier space. The resulting power spectra are compared in
Fig. 3. This representation highlights each characteristic filter radius,
corresponding to the characteristic size of the structures the filter targets.
The 3′, radius of the S96 filter falls between that of the W3 and W4 wavelet
filters. The TANH filter, with a radius of about 2′, matches that of the W3
wavelet filter. The J04 filter, which has a radius of about 4′, matches that
of the W4 wavelet filter. Finally, the M18 filter can be compared to a
combination of the W3 and W4 wavelet filters. In the Fourier domain, we can
clearly see the band-pass behaviour of the different functions. In particular,
the S96 and TANH filters show oscillatory behaviour, which can be explained by
their design.
Figure 4: Wavelet decomposition of an E-mode convergence map (including shape
noise), covering a simulated field of $5^{\circ}\times 5^{\circ}$. The upper-
left panel corresponds to the E-mode convergence map to be decomposed. The
other panels correspond to the wavelet maps at scale W1 =
$1\aas@@fstack{\prime}17$ (upper-middle panel), scale W2 =
$2\aas@@fstack{\prime}32$ (upper-right panel), scale W3 =
$4\aas@@fstack{\prime}68$ (lower-left panel), scale W4 =
$9\aas@@fstack{\prime}37$ (lower-middle panel), and scale W5 =
$18\aas@@fstack{\prime}75$ (lower-right panel). The wavelet maps are sensitive
to structures of different apparent angular sizes. The W1 map (upper-middle
panel) appears to be dominated by the shape noise. The wavelet decomposition
also includes the smoothed version, which we excluded from this
representation.
This Fourier space representation further shows that the selected scales of
the wavelet filter appear to be well defined. In the wavelet formalism, the
wavelet filter functions have complementary dyadic scales, defined by the
pixel resolution as explained in Sect. 4.2. Their comparison with the four AM
filters highlights that a wavelet filter at a given scale can directly be
compared to an AM filter. However, unlike AM filters, which focus essentially
on one scale, the wavelet formalism allows us to extract signal at more than
one complementary scale. Generally, the standard AM methods target clusters
located at relatively low redshift with a large apparent angular size. Thus,
typical AM filter radii are taken between $1\arcmin$ and $5\arcmin$ (e.g.
Hamana et al., 2012) to maximise the S/N for clusters with a similar angular
extent. In contrast, the lensing signal from clusters with different apparent
angular sizes can easily be targeted using the wavelet formalism.
## 5 Detection algorithms
In this section we discuss the implementation of the detection approach we
used to compare the filters. The detection algorithms are all based on the
same basic principles. The lensing signal is first convolved with a filter
function of a specific scale (described in the previous section) to
reconstruct E- and B-mode AM maps. Thresholding is then applied to the E-mode
AM map to locate the overdensity peaks corresponding to clusters.
### 5.1 Binning and filtering
In the following, we describe the steps used to produce the E- and B-mode AM
maps on which the detection is performed. As explained in Sect. 3.3, the noisy
galaxy catalogues are the basic inputs for the detection algorithms. These
contain the shear at the discrete positions of each of the galaxies.
At this stage the convolution by the filter function can be undertaken in real
space, on the shear directly at the position of the galaxies, or in Fourier
space, by binning the shear on a regular grid to build a shear map. A third
possibility is to reconstruct the convergence from the shear maps and then to
perform the filter convolution on the resulting convergence map. A comparison
of these approaches is provided in Sects. 6.1 and 6.2, where we show that we
obtain statistically similar results irrespective of approach. For the
following, we use the third approach because it is computationally faster, in
particular for wavelet filters, with the use of the wavelet transform.
We first binned the observed galaxy ellipticities on a regular grid to create
what we refer to as the noisy shear maps. In our study, the simulated field
being $5^{\circ}\times 5^{\circ}$, we decided to bin the galaxies in a grid of
$512\times 512$ pixels, yielding a pixel size of $0\aas@@fstack{\prime}58$. On
average, with a galaxy density of 30 galaxies per arcmin2, about 10 galaxies
fall into each pixel. The standard method for binning the shear on a regular
grid consists of simply calculating the average shear per pixel (see e.g.
Kaiser & Squires, 1993; van Waerbeke, 2000; Pires et al., 2020). In this
paper, we introduce a new binning strategy to be closer to the real space
approach. This consists of summing the shear of all the galaxies that fall
into each pixel. Thus, the shear in each pixel is effectively weighted by the
number of galaxies in that pixel. This binning strategy is more adapted to
weak lensing detection because additional information on the galaxy density is
included in each pixel. The resulting shear map can then be normalised by the
mean number of galaxies per pixel to facilitate the comparison with other
approaches. However, this global normalisation has no impact on the detection
because it affects the E- and B-modes in a similar way. The impact of these
two different binning approaches is discussed in more detail in Sect. 6.1. The
disadvantage of this approach is that the weak lensing signal depends
significantly on the number of galaxies per pixel, complicating its
interpretation in terms of mass. However, such calculations can be undertaken
in a second step, after detection.
We then applied the Kaiser & Squires inversion described in Sect. 2.2 to the
noisy shear maps $\gamma_{1}$ and $\gamma_{2}$, to reconstruct the E- and
B-mode convergence maps. Convolution by the different filters was then
performed both on the E- and B-mode convergence maps, to produce the
corresponding AM maps.
In the following, we refer to maps of the projected matter distribution at a
given scale as AM maps. In practice, these can be obtained by applying AM
filters or wavelet filters to the shear or to the convergence. The procedure
to detect the haloes on these E- and B-mode AM maps using a single scale is
identical, regardless of the method that has been used to produce them.
### 5.2 Single-scale detection procedure
Once the E- and B-mode AM maps at a specific scale are produced, the detection
procedure starts by applying a thresholding step. Since the weak lensing
effect produces only E-modes, the B-modes are simply due to the noise and can
therefore be used to evaluate the level of the noise in the E-mode AM map. As
proposed by Miyazaki et al. (2018), we used the maximum pixel value in the
B-modes to define the threshold that is then applied to the E-mode AM map. We
then only keep the pixels in the E-mode AM map that are greater than this
threshold, setting the other pixels to zero.
In their study of the HSC data, Miyazaki et al. (2018) find that this
threshold corresponds to $4.7$ times the standard deviation of the noise. The
standard deviation depends on the filter under consideration and is computed
from the B-mode AM map. For the DUSTGRAIN-pathfinder simulation we use here,
the equivalent average thresholds for the 256 realisations correspond to
$4.09$, $4.37$, $4.01$, $4.35$, $4.57$, $4.35,$ and $4.04$ times the standard
deviation for the S96, TANH, J04, M18, W2, W3, and W4 filters, respectively.
The final detection step is to identify the peaks in each of the thresholded
E-mode AM maps. A peak is defined as a pixel whose value is greater than its
eight nearest neighbours. The peaks at the border of the map over a width
equal to the filter radius are discarded to avoid contamination due to
boundary effects introduced by the convolution step. Following this procedure,
we obtain a list of detections and associated peak coordinates. While this
procedure is sufficient for single-scale methods, it needs to be further
developed in the multi-scale case to deal with multiple detections, as we
describe below.
### 5.3 Multi-scale detection procedure
In the detection method based on the wavelet transform, we chose to decompose
the lensing signal into $J=5$ scales (see Eq. 31), as this will encompass a
maximum of possible galaxy cluster apparent angular sizes. Figure 4 shows the
result of such a decomposition applied to the noisy convergence map of a
$5^{\circ}\times 5^{\circ}$ simulated field. The top left panel shows a noisy
version of the E-mode convergence map displayed on the right panel of Fig. 1,
and the other panels show the wavelet maps corresponding to scales $i=1$ to 5
(W1-W5). Each wavelet map gives details of the original convergence map at
different scales. The first scale (W1) is mostly dominated by the noise. In
contrast, the scales from W2 to W5 shed light on different signals within the
input E-mode convergence map. In our analysis, we decide to only keep the
scales W2, W3 and W4. We remove scale W1 because it is dominated by noise.
Scales W5 and above have also been removed because these filter sizes greatly
exceed the expected angular size of the clusters we want to detect.
On the remaining scales W2, W3, and W4, we apply the single-scale detection
procedure described above to extract the local maxima at each individual scale
(hereafter W234). We now have to deal with an issue that is specific to the
multi-scale approach. A signal from a given object can be detected at several
scales, and with a position that varies slightly from one scale to another. We
refer to these henceforth as multiple detections. Once we have obtained a peak
candidate list for each scale, it is important to identify such multiple
detections and to recombine them. This is achieved by computing the separation
distance between all the pairs of detections on consecutive scales. If two
peaks on different scales have a separation distance smaller than the larger
of the two filter radii, we consider them to be a multiple detection. These
are then recombined, and the position of the recombined peak is set by the
position of the detection at the finest scale. If there are more than two
peaks, we repeat this procedure for all the detection peaks and all the scales
consecutively. We keep track of the index $i$ of the finest scale where the
recombined detection appears, as this will be considered as the scale at which
the signal was detected for the matching procedure described below in Sect. 7.
We also saved the detections at each individual wavelet scale to study their
complementarity before combination and to allow a comparison of their
performance with respect to the corresponding AM filters.
Figure 5: Comparison of the real and Fourier space implementations. Left:
Power spectra of a $5^{\circ}\times 5^{\circ}$ AM map with $512\times 512$
pixels obtained using the real space implementation (blue) and using the
Fourier space implementations (orange and green lines for summed and averaged
pixel binning, respectively). Compared to the real space implementation, there
is a $1.2\%$ and $1.5\%$ loss of power for the Fourier space implementation
with summed and averaged pixel binning, respectively. Right: Detections
obtained on the same field when the computation is undertaken in real space
(blue circles) or in Fourier space using either the summed pixel binning
(golden circles) or the averaged pixel binning (green circles). The Fourier
space implementation with summed pixel binning is very close to the real space
implementation in terms of detections. When the binning is performed by
averaging the shear values, there are fewer detections, and about 30% of these
detections are different from those obtained with the real space
implementation.
Figure 6: Comparison of the shear and convergence approaches. Left: Power
spectra of a $5^{\circ}\times 5^{\circ}$ AM map with $512\times 512$ pixels
obtained by applying the AM J04 filter to the shear maps (blue line) and to
the E-mode convergence map (orange dots) computed from the same data as in
Fig. 6. The power spectra agree perfectly. Right: Detections obtained on the
same field when the convolution by the J04 AM filter is performed on the shear
map (blue circles) and on the convergence map (golden circles). There is a
one-to-one correspondence between the detections.
## 6 Analysis of implementation options for the detection algorithms
In practice, the detection is undertaken on a catalogue of galaxies, for each
of which a noisy shear measurement is available. As described above, this is
typically undertaken through the convolution of the shear with a compensated
filter. This convolution can be performed in real space, directly at the
position of the galaxies (e.g. Maturi et al., 2005; Wittman et al., 2006;
Miyazaki et al., 2018; Hamana et al., 2020), or in Fourier space, binning the
shear on a regular grid (e.g. Hennawi & Spergel, 2005; Leonard et al., 2015;
Oguri et al., 2021). Moreover, some cluster weak lensing detection algorithms
are based on the convolution of the convergence (derived from the shear and
also defined on a regular grid) by a compensated filter (e.g. White et al.,
2002; Hamana et al., 2004; Tang & Fan, 2005; Miyazaki et al., 2007). In this
section we explore how binning impacts the signal, and whether the choice to
use the shear or the convergence has any effect on the detection efficiency.
### 6.1 Detection in real space versus Fourier space
In the limit of a perfectly homogeneous distribution of background sources,
the detection of convergence peaks, whether in real space or in Fourier space,
depends only on the number density of these sources. However, as the
background sources are not homogeneously distributed, an additional
uncertainty is introduced into the method by the binning. This can impact the
position and the S/N of the detections. This point has not been fully studied
here because the distribution of the background sources is homogeneous in the
simulations. However, as the number of background sources per pixel varies
slightly for statistical reasons, our study already provides some insights on
the impact of the background source distribution.
#### 6.1.1 Binning strategy
The computation time for AM-based detection algorithms is considerable for the
real space approach. Computation time is significantly reduced for AM-based
detection algorithms when using the Fourier approach. However, there is a loss
of resolution compared to the real space approach because the galaxies must be
binned on a shear map before performing the Fourier transform, instead of
using their exact position. This loss of resolution introduces a loss of power
for the high S/N pixels in the shear map, affecting the efficiency of the
detection method. An additional disadvantage of the Fourier approach is that
there are artefacts at the border of the field due to the periodic assumption
of the discrete Fourier transform. Such effects can be significantly mitigated
by removing the image boundaries or by dealing correctly with borders during
the mass inversion, as proposed in Pires et al. (2020).
The impact of the loss of resolution when working in Fourier space depends on
the map resolution (i.e. the number of pixels of the map), and can be
mitigated by using an appropriate value. Indeed, there is a trade-off to be
made on the map resolution. Decreasing the resolution too much leads to power
leakage, which can affect the detection method. On the other hand, finer
binning can oversample the underlying data, and can introduce missing data
into the map.
Both real space and Fourier space approaches can be found in the literature
(e.g. Hennawi & Spergel, 2005; Gavazzi & Soucail, 2006; Miyazaki et al., 2018;
Hamana et al., 2020; Oguri et al., 2021). In this paper we have introduced an
additional Fourier space approach with a different binning strategy as
explained in Sect. 5.1.
#### 6.1.2 Impact on the signal
We first assessed the impact of the choice of a real or Fourier space approach
on the signal conservation, when using an appropriate map resolution. We
considered three different implementations: (i) application of an AM filter to
the shear at the positions of the galaxies, directly in real space; (ii)
application of an AM filter in Fourier space to the binned shear obtained by
averaging the signal from the galaxies that fall in each pixel (averaged pixel
binning); and (iii) application of an AM filter in Fourier space to the binned
shear obtained by summing the signal from the galaxies that fall in each pixel
(summed pixel binning).
First, we studied the differences between the three implementations through
their spectra in the Fourier domain. For this test, we used the filter
proposed by Jarvis et al. (2004) because it can be easily applied to the
shear, either at the position of the galaxies or binned on a regular grid.
From the shear catalogue of a $5^{\circ}\times 5^{\circ}$ field without noise,
we computed the E-mode AM maps for the three different implementations and
derived the corresponding power spectrum.
The results are presented in the left-hand panel of Fig. 6, with the power
spectra shown for implementations 1, 2, and 3. The power spectra are very
similar, showing that on average the signal is equally well preserved by all
three approaches. To quantitatively compare the implementations, we computed
the integral of the power spectra. Compared to the real space approach, we
find that the power spectrum integral of the Fourier space implementation
suffers from a $1.2\%$ loss for summed pixel binning, and a $1.5\%$ loss for
averaged pixel binning.
#### 6.1.3 Impact on the detections
We now analyse the impact of the three implementation choices discussed above
on the detection efficiency. For each of the three implementations, we built
the E- and B-mode AM map and applied the detection procedure described in
Sect. 5.2 using the Jarvis et al. (2004) AM filter. This was undertaken on all
256 realisations. The right-hand panel of Fig. 6 compares the detections
obtained with the three different implementations for one of the realisations.
When averaging these results over the 256 realisations, we found that the mean
number of detections using the real space approach ($49.2\pm 0.6$) is very
close to that from the Fourier space approach with summed pixel binning
($47.9\pm 0.5$). In contrast, the Fourier space approach with averaged pixel
binning gives a smaller mean number of detections ($43.4\pm 0.5$). Moreover, a
closer look on the detections made by the different approaches reveals
noticeable differences. We found that $28.2\%\pm 2.1\%$ of the detections
obtained from the averaged pixel binning approach differ from those made by
the real space approach. In contrast, with the summed pixel binning approach
only $7.6\%\pm 1.3\%$ of the detections are different.
These results show that the binning strategy has a non-negligible impact on
the detection efficiency, both in terms of number and distribution of the
detections. In light of these results, we conclude that the Fourier space
approach with summed pixel binning strategy agrees very well with the real
space approach, producing results that are statistically similar. In the
following, we decided to perform all tests and comparisons using the third
implementation (i.e. the Fourier space approach using a summed pixel binning
strategy) because it is computationally less demanding. It has the additional
advantage of providing a substantial speed increase with respect to detection
in real space: for our applications, we find a speed increase of two orders of
magnitude, in agreement with the study by Leonard et al. (2012).
### 6.2 Detection using shear map versus convergence map
Both shear and convergence can give insights into the statistical properties
of the weak lensing field, and indeed, it can be shown that they contain
precisely the same information (e.g. Schneider et al., 2002; Pires et al.,
2020). The (reduced) shear is a direct observable and is usually preferred for
reasons of simplicity. However, the convergence has the key advantage that
$\kappa_{E}$ encapsulates all the lensing signal, while it is inevitably
shared between $\gamma_{1}$ and $\gamma_{2}$ in the shear. In this connection,
the convergence is more adapted for galaxy cluster detection because it
explicitly traces the total matter distribution integrated along the LOS. A
further advantage of using the convergence is that it is computationally less
expensive to analyse.
Figure 7: Application of a theoretical S/N pre-selection to the halo
catalogue. Left: Weak-lensing selection function for clusters of galaxies, in
the redshift–mass plane.The selection function has been computed assuming a
Euclid-like redshift distribution with $\rm n_{\rm g}=30\,gal.{arcmin}^{-2}$
and $\sigma_{\epsilon}=0.26$. Contours denote the theoretical S/N for an NFW
profile. Right: Example of halo catalogue pre-selection using the theoretical
S/N, for one $5^{\circ}\times 5^{\circ}$ field. The light blue dots correspond
to all the haloes in the field, and the dark blue dots are the selected haloes
with S/N$>2$.
Although the reconstruction of the convergence field from survey data is a
difficult task, Pires et al. (2020) show that the lensing signal is preserved
in the convergence maps provided the mass inversion is performed without noise
regularisation, and systematic effects such as irregular sampling and complex
survey geometries are well controlled.
The AM formalism can be expressed either on the shear or on the convergence
maps. We now assess the difference between these two approaches. We first
compared them in Fourier space. For each realisation of the noisy shear
catalogue, we produced the E- and B-mode AM maps using the J04 AM filter
applied to either the shear or the convergence maps. Then, we computed and
compared the power spectra. The result of this comparison for one realisation
is shown in the left-hand panel of Fig. 6, where we see that the power spectra
obtained from the shear aperture map and the convergence aperture map are in
good agreement. Averaging over all 256 realisations, we measure a discrepancy
of $0.03\%$ in the power spectrum integral between the shear and the
convergence approaches. This difference can be explained by residual border
effects that are not totally suppressed when removing the image boundaries.
We then compared the two approaches in terms of detections, applying the
detection procedure described in Sect. 5.2. The results, shown in the right-
hand panel of Fig. 6, highlight the perfect agreement between the approaches.
This result remains valid for wavelet filters that are formally identical to
AM filters (see Leonard et al. 2012 for more details). However, while the AM
formalism can be applied to either the shear or the convergence, the wavelet
transform is not designed to process spin-2 fields (i.e. with two independent
components) such as the shear field. So, the shear and convergence approaches
being equivalent, we decided to perform all the tests and comparisons applying
the AM and Wavelet filters to the convergence maps, to be able to use the
wavelet transform.
## 7 Matching
Once the peaks are detected in the E-mode AM maps, they must be cross-matched
with the position of the haloes in the DM halo catalogue. This matching
procedure is essential to characterise the performance of the different
methods, and the results can be biased if not undertaken correctly. This is
particularly true when the filter characteristics differ significantly between
methods. We first identify two main points in the matching procedure that can
significantly increase the number of false detections.
The first concerns the characteristics of the halo catalogue used for
matching. The DM halo catalogue contains objects down to a mass of $\rm
M_{200c}=10^{12}\,h^{-1}\,M_{\odot}$ and up to a redshift of $\rm z=3.6$. For
the cosmology under consideration, a halo catalogue with these characteristics
contains about $15,000$ haloes in a field of $5^{\circ}\times 5^{\circ}$.
Obviously, only a small fraction of these haloes will be detectable through
the weak lensing effect, which is only sensitive to the most massive clusters
with sufficient galaxy sources behind them to trace their mass distribution.
Changing the characteristics of the halo catalogue, such as imposing limits in
mass and/or redshift, changes the performance of the methods with respect to a
number of key points (e.g. matching, false associations).
The second is the association distance. Typically, the matching is performed
within a fixed physical, comoving, or angular radius centred on the candidate
peak, the association distance of which is usually optimised for a given
filter (e.g. Hennawi & Spergel, 2005; Gavazzi & Soucail, 2006; Miyazaki et
al., 2018; Hamana et al., 2020; Oguri et al., 2021). However, when comparing
different filters, this association distance must be adapted for each
detection method to allow a fair comparison.
In the following, we address the first issue by applying a pre-selection to
the halo catalogue based on the theoretical S/N of the detection of an NFW
halo profile by weak lensing. Regarding the second issue, we developed a
method that allows us to adapt the matching distance to the filter
characteristics used in the detection method.
### 7.1 Halo catalogue pre-selection by S/N
Imposing a pre-selection on the DM halo catalogue, by removing those haloes
that are unlikely to be detected through weak lensing, reduces significantly
the number of false associations. We defined our pre-selection by deriving a
measure of the halo detectability in the form of the theoretical S/N of the
detection by weak lensing (e.g. Hamana et al., 2004; Hetterscheidt et al.,
2005; Bergé et al., 2010; Andreon & Bergé, 2012). We followed the approach
proposed by Bergé et al. (2010). Assuming the filter perfectly represents the
signal $\kappa$, the S/N of an halo of mass $\rm M_{200c}$ and redshift
$z_{l}$ can be expressed as
$\nu=\frac{\sqrt{\rm n_{\rm
g}}}{\sigma_{\epsilon}}\sqrt{\iint\kappa(\boldsymbol{\theta})^{2}\,\rm
d\boldsymbol{\theta}^{2}},$ (34)
where $\sigma_{\epsilon}$ is the mean intrinsic shape noise of the source
galaxies and $\rm n_{\rm g}$ the mean density of background sources. Assuming
the mass of the cluster follows an NFW distribution (Navarro et al., 1996) its
density profile is given by
$\rho(r)=\frac{\rho_{s}}{(r/r_{s})\,(1+r/r_{s})^{2}},$ (35)
where $\rho_{s}$ is the characteristic density and $r_{s}=R_{200c}/c$ is the
scale radius. The surface mass density $\Sigma$, projected along the LOS, can
be written as (see Bartelmann & Schneider, 2001)
$\Sigma(x)=2\,r_{\rm s}\,\rho_{\rm s}\,g(x),$ (36)
where $x=r/r_{\rm s}$ is a dimensionless radius. The function $g$ is defined
with respect to the distance to the halo centre and the concentration, $c$. If
$c>1$, it can be written as (Wright & Brainerd, 2000)
$g(x)=\frac{1}{1+c}\left\\{\begin{array}[]{ll}-\frac{\sqrt{c^{2}-x^{2}}}{(1-x^{2})(1+c)}+\frac{1}{(1-x^{2})^{3/2}}\rm
arccosh\left(\frac{x^{2}+c}{x(1+c)}\right)&\mbox{(x\,<\,1),}\\\
\frac{\sqrt{c^{2}+1}}{3}\left(1+\frac{1}{1+c}\right)&\mbox{ (x\,=\,1),}\\\
-\frac{\sqrt{c^{2}-x^{2}}}{(1-x^{2})(1+c)}-\frac{1}{(x^{2}-1)^{3/2}}\rm
arccos\left(\frac{x^{2}+c}{x(1+c)}\right)&\mbox{(1<x<c),}\\\
0&\mbox{(x>c).}\end{array}\right.$ (37)
The concentration parameter $c$ is derived using the semi-analytical model
introduced by Diemer & Joyce (2019), as implemented in COLOSSUS (Diemer,
2018). The S/N of the halo can then be written as (Bergé et al., 2010)
$\nu=\,<Z>\,2\sqrt{2\,\pi}\,\frac{\sqrt{\rm n_{\rm
g}}}{\sigma_{\epsilon}}\frac{\rm\,r_{\rm s}^{2}\,\rho_{\rm s}}{D_{\rm
d}\Sigma_{\infty}}\sqrt{\int_{0}^{\,c}xg(x)^{2}dx}.$ (38)
The weight parameter $Z$ is designed to take into account the impact of the
distribution of the sources (e.g. Seitz & Schneider, 1997). the angular-
diameter distance to the lens is described by $D_{\rm d}$, and
$\Sigma_{\infty}$ is the value of $\Sigma_{\rm crit}$ for a source at a
redshift of infinity. For each halo in the catalogue, we computed its
theoretical S/N ($\nu$) assuming a Euclid-like redshift source distribution
$\rm p_{\rm z_{s}}$ (derived from the simulation), a galaxy density of $\rm
n_{\rm g}=30\,gal.{arcmin}^{-2}$ and a shape noise of
$\sigma_{\rm\epsilon}=0.26$.
The left-hand panel of Fig. 7 shows the resulting ideal weak lensing detection
selection function in the mass-redshift plane for the characteristics of the
simulation. The figure is obtained by generating a grid in the mass-redshift
plane and computing the S/N value for each point of the grid using Eq. 38. The
colour map highlights the increase in S/N towards the upper-left corner,
corresponding to the detection of massive clusters at low redshift. The solid
white lines correspond to S/N contours of level S/N
=[0,1,2,3,4,5,6,7,10,12,15,20,25,30,35], which have been interpolated from the
$(z,M)$ grid points.
However, the above only yields an average value for the S/N of detection
assuming a mean source distribution and an optimal filter. In practice, the
filters are never perfectly adapted to the cluster shape as assumed in the
theoretical S/N estimation. Furthermore, the detection S/N depends on the
redshift distribution of the sources that are behind the cluster. The galaxies
behind the cluster being few in number, small variations in their redshift can
artificially boost or decrease the weak lensing signal.
Figure 8: Dispersion of the association distance (without maximum association
distance) for the W3 wavelet filter, considering all 256 realisations. In all
the panels, the blue dots correspond to an association between a peak
detection and its closest halo. Upper left: Absolute association distance,
$D$, as a function of its normalised expression with respect to the
characteristic halo radius, $\theta_{200c}$. Lower right: Zoomed-in view of
the central $20\times 20$ arcminute distribution of absolute association
distances expressed in RA and Dec, to highlight their dispersion. The two
remaining panels show the projection of the absolute association distance
distribution onto the RA and Dec axes: the projection onto the Dec axis
(lower-left panel) and the projection onto the RA axis (upper-right panel).
The RA and Dec axes are zoomed around the central area. On these two panels,
the dashed orange line shows the result of a fit by a Gaussian function whose
parameters ($\mu$, $\sigma$) are labelled ‘Mean’ and ‘Stddev’.
We applied a conservative threshold to the 256 DM halo catalogues by keeping
all the haloes with S/N$>2$. Although a halo with S/N = 2 is unlikely to be
detected, we applied this value to take into account the variance in the
detection due to the distribution of the sources, and the use of an ideal
filter. Below this value we do not expect to detect many clusters. We discuss
this choice of pre-selection threshold further in Sect 8.3.2. With this pre-
selection, in practice, from a typical $5^{\circ}\times 5^{\circ}$ field, only
2-3% of the $15,000$ haloes in the catalogue are kept. The remaining $300-400$
haloes are then the basis for the matching procedure. The right-hand panel of
Fig. 7 shows the selected haloes in the mass-redshift plane for one
$5^{\circ}\times 5^{\circ}$ field.
Figure 9: Matched detections for the W3 filter. The two panels are similar to
Fig. 8. The blue dots correspond to an association between a peak detection
and its closest halo. The orange dots correspond to the detections with an
absolute association distance lower than the defined MMD.
### 7.2 Matching procedure
The matching procedure between the catalogue haloes and the peak candidates is
a complex task. The typical approach consists of evaluating the distance
between the candidate peak positions and the closest halo, and to consider
these matched if the distance is below a given maximum matching distance
(MMD).
In practice, most matching methods simply draw a circle of a given apparent
angular radius around the candidate peak and match the closest halo that falls
in the circle (e.g. Hamana et al., 2004; Gavazzi & Soucail, 2006; Hamana et
al., 2015, 2020). However, in some procedures the matching is more refined,
drawing a circle of a given comoving distance (in Mpc) around the halo and
then matching the detection with highest S/N within this radius (e.g. Miyazaki
et al., 2018; Oguri et al., 2021). The latter is more precise, but requires
access to the redshift of the haloes in the catalogue.
Our study is further complicated by the characteristics of the different
filter functions, and the inherent multi-scale nature of the wavelet approach.
We therefore used a different method, drawing on SZ cluster survey matching
procedures (see e.g. Planck Collaboration et al., 2014). In our study, the
matching is undertaken by computing the apparent angular distance $D$
(hereafter the association distance) between each candidate peak and each halo
in the DM halo catalogue, after S/N pre-selection. This association distance
is then normalised by the apparent halo radius $\theta_{200c}$. This
normalised distance $D/\theta_{200c}$, together with the MMD defined below,
are our matching criteria. Together, they take into account the filter size
and have the effect of favouring the closest and most massive clusters in the
matching.
The matching procedure is implemented iteratively. For $N$ detections and $M$
haloes, we obtain an array of $N\times M$ measured distances. We start the
iterative process for a given candidate peak by selecting the minimum
normalised distance $D/\theta_{200c}$ among all the measured normalised
distances. The detection and halo corresponding to this minimum normalised
distance are considered as matched and the corresponding row and column are
removed from the array. The procedure is repeated until we reach the defined
MMD. This procedure is perfectly suited to cross-match detections with
simulated halo catalogues, yielding excellent results compared to typical
methods. However, for application to cross-matching with real cluster
catalogues (e.g. optical, X-ray, etc.) access to an estimate of the $\rm
R_{200c}$ (or $\rm M_{200c}$) of the haloes is required.
### 7.3 Maximum matching distance
The choice of the MMD is critical, and must take the positional precision of
the detection method into account, as larger filter sizes will lead to a loss
of precision on the coordinates of the peak centre. At the same time, if the
chosen MMD is too large, false associations may be introduced even after pre-
selecting the halo catalogue, while associations may be missed if the MMD is
too small. Therefore, the MMD must be optimised and adapted to the detection
method in question.
For this study, we developed an empirical method that is able to adapt the MMD
to the detection method. Our approach derives the MMD from the uncertainty in
the position of the detection introduced by the size of the filter. In
practice, this is undertaken by performing an association procedure on all the
256 fields, which corresponds to applying the matching procedure without any
limit in the matching distance.
Figure 8 shows an example of the distribution of the corresponding association
distances for all the 256 realisations (about $75,000$ haloes after catalogue
pre-selection) for the filter W3, for which the filter radius equals
$2\aas@@fstack{\prime}$32\. Each blue dot corresponds to a pairing between a
peak detection and its closest halo. The upper-left panel shows the absolute
association distance $D$, expressed in arcminutes, as a function of the
association distance $D$ normalised by the angular halo radius
$\theta_{200c}$. We clearly distinguish two distinct groups of associations.
In the upper group, the detections are outside the characteristic halo radius.
Therefore, we can consider this group to be false associations. The lower
group is composed of associations for which the peak position is within the
$\theta_{200c}$ radius of the halo and can thus be considered as true
associations. To isolate the correct associations and define a correct MMD, we
use a different visualisation of the results. In the lower-right panel of Fig.
8, the blue dots represent a unique peak detection-halo pair whose association
distance $D$ has been decomposed in terms of RA and Dec (we show only the
central $20\times 20$ arcminutes). The high-density central region corresponds
to peak detections that have been correctly matched to an underlying halo. The
dispersion in this high-density central region in the RA-Dec plane highlights
the uncertainties in the position of the peak detection introduced by the size
of the filter.
To highlight the properties of the dispersion, we projected the distribution
on both RA and Dec axes, displayed for the W3 filter in the upper-right and
lower-left panels of Fig. 8, respectively. The positional uncertainty is well
approximated by a Gaussian distribution. In practice, we obtain the same
parameters if we fit a 2D Gaussian distribution to the RA-Dec plane or a 1D
Gaussian distribution separately to the RA and Dec components. From the fit we
extracted the standard deviation $\sigma$, which corresponds to the dispersion
of the positional uncertainty of the detection with respect to the true halo
position. The Gaussian fit shows a y-offset that can be explained by randomly
matched haloes. The Gaussian fit also shows an x-offset corresponding, to the
mean of the Gaussian distribution $\mu$, which is due to the pixelisation.
From these parameters, we defined the MMD to be $5\sigma\,+\,\lvert\mu\rvert$.
We then applied this procedure to all the filters, and the resulting MMDs for
each detection method are shown in Table 1. Comparing these values with the
values of the filter radii shown in Fig. 3, we see that the MMD is directly
proportional to the filter radius, as expected.
Table 1: Estimated maximum matching distance (MMD) for different filter functions, evaluated using the procedure described in Sect. 7.3. Filter function | $\sigma$ | $\lvert\mu\rvert$ | MMD [arcmin]
---|---|---|---
S96 | 0.83 | 0.26 | 4.4
TANH | 0.47 | 0.28 | 2.6
J04 | 0.87 | 0.31 | 4.7
M18 | 0.60 | 0.28 | 3.3
W2 | 0.23 | 0.27 | 1.7
W3 | 0.45 | 0.30 | 2.5
W4 | 0.89 | 0.29 | 4.7
Table 2: Completeness of the different detection methods for different S/N
halo pre-selections, in percent.
Filter function | Completeness (S/N¿2) | Completeness (S/N¿5) | Completeness (S/N¿7) | Completeness (S/N¿10)
---|---|---|---|---
S96 | $13.3\pm 0.3$ | $65.6\pm 0.1$ | $89.3\pm 0.1$ | $98.1\pm 0.1$
TANH | $11.9\pm 0.2$ | $62.3\pm 0.1$ | $85.3\pm 0.1$ | $95.8\pm 0.1$
J04 | $12.9\pm 0.2$ | $59.7\pm 0.2$ | $87.7\pm 0.1$ | $97.4\pm 0.1$
M18 | $17.1\pm 0.2$ | $71.5\pm 0.1$ | $93.9\pm 0.1$ | $98.1\pm 0.1$
W4 | $12.7\pm 0.2$ | $60.0\pm 0.2$ | $89.1\pm 0.1$ | $98.2\pm 0.1$
W234 | $21.0\pm 0.2$ | $75.3\pm 0.1$ | $95.7\pm 0.1$ | $99.2\pm 0.1$
444The completeness is computed on each field of 5∘ $\times$ 5∘ using
different S/N halo pre-selection in the matching procedure, with S/N ¿ 2, 5, 7
and 10. The results give the mean and associated uncertainty in the
completeness (in percent), estimated from the 256 realisations.
Once the MMDs are defined, we can apply the matching procedure defined in
Sect. 7.2. Figure 9 shows the results of the matching for the W3 filter. The
blue dots still refer to the associations from all the detections and their
closest halo. The orange dots correspond to the detections that have been
successfully matched. Considering only the pairs for which the association
distance is lower than the MMD allows us to separate the two groups of
associations distinctly. This further allows us to build two catalogues: one
of matched detections, and another of unmatched detections.
### 7.4 Multi-scale matching procedure
In the multi-scale approach, a same cluster can be detected on several
successive scales, complicating the matching analysis. To assess the
performance of the multi-scale approach, it is important to recombine the
multiple detections and to associate them with one single scale (the finest
scale), as explained in Sect. 5.3.
In practice, the multi-scale matching procedure works as follows. Starting
with the finest scale, the matching procedure is performed using the
appropriate matching distance defined in Table 1. Associations at that scale
are considered to be unique, and these haloes are removed from the catalogue
that is used on subsequent scales. This procedure is repeated for each
subsequent scale.
The final result is a catalogue of associated and non-associated detections
for each wavelet scale. These are important to measure the individual
performance of each wavelet scale, and to compare their overall contribution
in terms of detections and associations. The final catalogues of associated
and non-associated detections are then obtained by concatenation of the
individual single-scale catalogues.
## 8 Results and performance
We now assess and compare the performance of the detection methods. We start
by quantifying the completeness and purity of each sample, as commonly done by
other studies (e.g. Miyazaki et al., 2018; Euclid Collaboration et al., 2019;
Hamana et al., 2020). Then we analyse in more detail our detections by
performing an analysis of the distribution of the matched detections in terms
of redshift, mass and scale. We then quantify the number of false detections
due to the shape noise for each method. Finally, we undertake a
characterisation and a classification of the matched and unmatched detections.
For the wavelet filters, we undertake the above both on one single scale (W4),
and using the full multi-scale approach (W234).
### 8.1 Detection method performance
#### 8.1.1 Completeness
To quantify the performance of the different methods and to compare them, we
used the completeness, C, defined as follows:
$\rm
C=\frac{Number\,\,of\,\,matched\,\,detections}{Number\,\,of\,\,clusters\,\,in\,\,the\,\,halo\,\,catalogue}.$
(39)
We note that the completeness depends on the characteristics of the halo
catalogue used for matching. Changing the pre-selection S/N in the halo
catalogue will change the values of the completeness for each method. However,
this does not affect the comparison between the methods.
To further develop our analysis, we computed the completeness for each
detection method using different S/N thresholds in the halo catalogue. For
each method, we computed the completeness of the 256 realisations considering
only the haloes in the catalogue whose theoretical S/N is above a given value.
We repeated this operation for different S/N limits. The results, averaged
over the 256 realisations, are shown in Table 2. We see that considering the
haloes with S/N$>2$ results in a completeness below 25% for all the detection
methods. As expected, the completeness increases with the halo pre-selection
S/N threshold, and almost all of the highest S/N clusters are detected by the
different detection methods. Even if the overall behaviour is comparable, this
quantification shows that the detection methods based on the W234 and M18
filters outperform those based on the S96, TANH, and J04 filters in terms of
completeness.
#### 8.1.2 Purity
Although the completeness for a given S/N halo selection is a good indicator
of the performance of the different detection methods, it does not provide any
information on the proportion of unmatched detections. That is why a measure
of the purity of the sample is also needed to refine the comparison. The
purity, P, is defined as
$\rm
P=\frac{Number\,\,of\,\,matched\,\,detections}{Total\,\,number\,\,of\,\,detections}.$
(40)
This quantity measures the proportion of matched detections in the sample
(e.g. Euclid Collaboration et al., 2019) and can be used to evaluate the false
detection rate. We computed the purity of the samples obtained from each
detection method for a fixed catalogue pre-selection threshold of S/N$>2$.
This threshold was chosen so as to not decrease the purity artificially as
explained in Sect. 7.1. The results for the different detection methods,
averaged over the 256 realisations, are summarised in Table 3. The estimated
purity for all detection methods is comparable, at around 85%. This is
somewhat lower than the expected number from the false detections estimated in
Sect. 8.2.1. This led us to perform a case-by-case analysis of the unmatched
detections, as detailed in Sect. 8.3.2. In particular, we observed that for
some detections the matching procedure has failed or is too conservative.
Although this obviously has an impact on the estimated purity, it does not
affect the comparison between the methods because it affects all of them to
the same extent.
#### 8.1.3 Purity versus completeness
Table 3: Detection method performance.
Filter function | Number of detections | Purity P [%] | Completeness C [%]
---|---|---|---
S96 | $57.6\pm 1.0$ | $83.7\pm 0.3$ | $13.3\pm 0.3$
TANH | $50.9\pm 0.9$ | $84.5\pm 0.4$ | $11.9\pm 0.2$
J04 | $47.9\pm 0.5$ | $88.3\pm 0.3$ | $12.9\pm 0.2$
M18 | $66.6\pm 0.7$ | $83.1\pm 0.3$ | $17.1\pm 0.2$
W4 | $48.0\pm 0.6$ | $88.3\pm 0.3$ | $12.7\pm 0.2$
W234 | $83.2\pm 0.9$ | $82.7\pm 0.3$ | $21.0\pm 0.2$
555 For each detection method, the detections are obtained applying the
detection procedure described in Sect. 5. The purity and the completeness are
measured using the matching described in Sect. 7 using a S/N$>2$ pre-selection
in the halo catalogue. The results correspond to the mean and its
uncertainties (in percent) estimated from the 256 realisations.
Comparing the results in Table 3, we see that the detection methods based on
the S96, TANH, J04, and W4 filters lead to higher purity and lower
completeness. Although the completeness is not directly linked to the number
of detections, but rather to the number of matched detections, the lower
completeness for these filters is consistent with their lower mean number of
detections compared to the two other methods, and also with their Fourier
space representation, whose band-pass width is narrower. We note that the
wavelet filter W4 alone attains similar performance to the single-scale
filters S96, TANH and J04. Conversely, the completeness is higher and the
purity is lower for the M18 and W234 filters, which use larger band-pass
filters. However, the multi-scale approach spans a wider range of possible
cluster scales, and therefore allows us to reach a higher completeness. As a
consequence, the multi-scale approach results in a higher mean number of
detections for the same purity. Compared to the M18 AM filter, for instance,
the multi-scale approach yields 25% more detections at a purity of $\sim
85\%$.
Table 3 also shows that some detection methods are more efficient in terms of
completeness, while others are more efficient in terms of purity. Since the
purity and completeness cannot be compared separately, a fair comparison
between the detection methods requires an analysis of the evolution of the
completeness as a function of the purity. We therefore computed this for each
detection method using the pre-selected S/N$>2$ catalogue for the matching. In
practice, we varied the threshold in the detection procedure described in
Sect. 5.2 in steps of $5\%$ and obtained new values of the completeness and
purity for each new detection threshold value.
Figure 10: Completeness as a function of purity for the detection methods
based on the S96 (blue), TANH (red), J04 (purple), and M18 (green) AM filters,
the W4 wavelet filter (light blue), and the W234 wavelet filter (orange). The
cross markers represent precisely where the methods are evaluated in terms of
purity and completeness. The purity and the completeness are computed using a
S/N$>2$ halo pre-selection and averaging over the 256 realisations.
The results, averaged over the 256 realisations, are shown in Fig. 10. We see
that the multi-scale approach outperforms the other detection methods at all
values of purity. Compared to the detection method based on the TANH filter,
there is an increase in the completeness of up to 65% at low purity. The
increase in completeness is less for the detection method based on the M18
filter, but it remains significant in the range of purities studied. We also
note that the performance of the wavelet filter W4 alone and the single-scale
filter J04 are comparable.
### 8.2 Analysis of the detections
Here we estimate the number of false detections due to the shape noise, and
characterise the matched detections in terms of mass, redshift and scale.
Figure 11: Distribution of the matched clusters in the $z-M$ plane for one
$5^{\circ}\times 5^{\circ}$ patch, using the S96 AM filter in the upper-left
panel, the TANH AM filter in the upper-right panel, the J04 AM filter in the
middle-left panel, the M18 filter in the middle-right panel, the W4 wavelet
filter in the bottom-left panel, and the W234 wavelet filters in the bottom-
right panel. The light blue dots correspond to all the haloes within the halo
catalogue, and the dark blue dots to the haloes we have pre-selected to have
S/N$>2$. The overplotted orange crosses correspond to the haloes that have
been matched with a detection, and in parentheses we note the number of
matches for each method on one patch. This patch was chosen to be
representative of the performance of the different methods in terms of
detection number.
#### 8.2.1 False detections
In the following we define false detections as those detections not related to
any lensing signal (i.e. overdensity), and therefore due only to the noise.
Then, we quantify the number of false detections due to the shape noise for
each detection method. For this purpose, we randomised the orientation of the
sources in the noisy galaxy catalogues to suppress the lensing signal and to
generate pure noise galaxy catalogues. We then derived the associated
convergence maps as described in Sect. 5 and applied the various detection
methods. For each method, we then computed the mean false detection rate
averaged over the 256 realisations. This was obtained from the ratio of the
number of detections due purely to the shape noise to the number of detections
obtained when the signal is not suppressed. The resulting false detection rate
due to the shape noise is given, for each detection method, in the second
column of Table 4.
The false detection rate due to shape noise is $1.1\%$ for the J04 detection
method, $1.8\%$ for the detection methods based on the S96, TANH and W234
filters, and 3% for the M18 detection method. The impact of the shape noise
depends on the filter. The noise level is lower at large scales compared to
smaller scales, which explains why the J$04$ detection method is less
sensitive to the noise compared to the other methods. In contrast, the M$18$
detection method is the most sensitive because it includes small scales.
Similarly, the oscillation behaviour of the S96 and TANH filters increases
their sensitivity to the noise, by integrating smaller scales. The W$234$
detection method also includes small scales, but the impact of the shape noise
is reduced owing to its multi-scale detection procedure. Despite these
differences, the number of false detections due to the shape noise is small,
irrespective of detection method.
#### 8.2.2 Characterisation of the matched detections
Once the matching is completed, we characterise the matched detections by
plotting their distribution in the redshift-mass plane. Figure 11 shows the
distribution of the clusters detected by the five different detection methods
in the redshift-mass plane. The figure is obtained from one single
realisation, but is representative of the full 256 realisation sample. The
matching is performed using only the haloes with S/N$>2$.
As expected, all the detection methods are most sensitive to the signal from
high mass clusters lying at low redshift. Comparison shows that the multi-
scale detection method W234 has more matched detections than the others, in
agreement with the completeness results presented in Table 2. In addition,
almost all the clusters detected by the S96, TANH, J04, and M18 detection
methods are also detected by the multi-scale detection procedure. Compared to
the four other detection methods, the multi-scale approach generally detects
more clusters at lower masses, and up to higher redshifts. In Appendix A we
further analyse the multi-scale approach by investigating the contribution of
each wavelet scale to the total number of matched detections.
A striking aspect of Fig. 11 is that some haloes with a very high predicted
S/N are not detected, regardless of the method used. We performed a case-by-
case analysis of these haloes and we found several explanations for this. In
some cases, the detection is missed because of the shape noise. In other
cases, the detection is missed because the lensing signal is decreased by the
redshift distribution of the sources behind the cluster (through Eqs. 9 and
10). In some other cases, the cluster is detected but the matching procedure
fails because the normalised distance is not always the best criterion.
### 8.3 Classification of the detections
We now undertake a further analysis of the matched and unmatched detections by
classifying them into different categories. In particular, Fig. 12 summarises
these categories and the results for the W234 method, after detection and
matching of all 256 realisations.
#### 8.3.1 Classification of the matched detections
The matched detections can be classified into two categories.
Table 4: Classification of the detections.
Filter function | Matched [%] | False detections [%] | Boosted objects ($\rm p_{z}$) [%] | Blended haloes [%] | Others [%]
---|---|---|---|---|---
S96 | $83.7\pm 0.3$ (48) | $2.0\pm 0.2$ (2) | $3.5\pm 0.2$ (2) | $5.8\pm 0.3$ (3) | $\sim 5.0(3)$
TANH | $84.5\pm 0.3$ (43) | $1.7\pm 0.3$ (1) | $4.3\pm 0.3$ (2) | $6.2\pm 0.3$ (3) | $\sim 3.3(2)$
J04 | $88.3\pm 0.3$ (42) | $1.1\pm 0.2$ (1) | $3.6\pm 0.2$ (2) | $5.9\pm 0.4$ (2) | $\sim 1.1$ (1)
M18 | $83.1\pm 0.3$ (56) | $3.0\pm 0.3$ (2) | $4.0\pm 0.1$ (3) | $5.6\pm 0.3$ (3) | $\sim 4.3$ (3)
W234 | $82.7\pm 0.3$ (67) | $1.8\pm 0.3$ (1) | $6.4\pm 0.2$ (6) | $5.5\pm 0.2$ (5) | $\sim 3.6$ (4)
666The detections have been classified in the categories described in Sect.
8.3.1 and 8.3.2, for each detection method and averaged on each field of size
5∘ $\times$ 5∘. The first column gives the proportion of matched detection
using a S/N¿2 pre-selected halo catalogue in the matching procedure described
in Sect. 7. Then, the unmatched detections have been classified in the
categories described in Sect. 8.3.2. The false detections due to shape noise
are in the second column. The unmatched detections due to haloes whose lensing
signal has been boosted are in the third column. The unmatched detections due
to blended lensing signal from LOS alignments or the merging of two haloes of
S/N¿1.5 are in the fourth column. The last column gives the proportion of
unmatched detections that remain unclassified. These results correspond to the
mean and associated uncertainty of the detections that fall into these
categories. The numbers in parentheses correspond to the average number of
detections in each category in a 5∘ $\times$ 5∘ realisation (rounded up to an
integer number).
True detections correctly matched. Almost all the matched detections
correspond to correct associations to a single halo of the catalogue (‘Single
matched’ in Fig. 12). However, it is inevitable that some true detections can
be correctly matched but should in fact be associated with two (or possibly
more) haloes whose lensing signal is merged. This may occur when two haloes
are aligned in the LOS or when their apparent angular distance is small
compared to the size of the filter. In this case, the matching procedure does
not differentiate and the association is undertaken simply with the closest
halo. We estimate that about 5% of the matched detections fall in this
category by searching for multiple haloes in the vicinity of the matched
detections (’Multiple Matched (LOS)’ in Fig. 12). These imperfect associations
can slightly reduce the estimated completeness, but all the methods are
impacted in the same way by this effect.
False detections incorrectly matched. Among the matched detections, a small
fraction are in fact randomly-matched false detections. These are detections
due to the shape noise that are matched owing to being close to a halo in the
catalogue. Their number depends both on the number of false detections (see
Sect. 8.2.1) and on the false association rate (given in Table 5). In this
study, we intentionally keep this number very small for all the detection
methods (less than 0.3 %). However, one must be careful because both the
detection and matching procedures can make this number become large. Thus, it
is essential to always compute this number to fully assess the performance of
the detection methods.
#### 8.3.2 Classification of the unmatched detections
Here we focus on the detections that are not matched with a halo, to fully
complete the analysis of the different detection methods. These unmatched
detections can be classified into several categories.
False detections unmatched correctly. The most intuitive class for of
unmatched detections is that composed of those due to the shape noise (see
Sect. 8.2.1.) Beyond the small fraction that is incorrectly matched owing to
being located close to a halo in the catalogue (see above), almost all of
these false detections are not matched. Their number thus corresponds
approximately to the false detection rate, and is given in the second column
of Table 4. This number is very small compared to the estimated purity in
Sect. 8.1, and other explanations need to be found to explain the remaining
$\sim 15\%$ of the unmatched detections.
Table 5: False association rate for different filter functions.
Filter function | False association rate [%] |
---|---|---
S96 | $12.8\pm 0.3$ |
TANH | $9.0\pm 0.3$ |
J04 | $14.2\pm 0.4$ |
M18 | $10.6\pm 0.4$ |
W234 | $10.8\pm 0.4$ |
W2 | $2.8\pm 0.3$ |
W3 | $8.7\pm 0.4$ |
W4 | $13.6\pm 0.4$ |
777The false association rate is computed for each detection method including
each individual wavelet filter W2, W3 and W4. The fraction of false
associations is estimated by matching random detection positions with the halo
catalogues with a S/N$>2$ halo pre-selection. The errors are computed using
the 256 realisations.
Boosted lensing signal unmatched incorrectly. One possible explanation for the
remaining unmatched detections is that some of these detections should be
associated with haloes that have been removed from the halo catalogue after
the S/N pre-selection. This could happen because the S/N estimated for each
halo in the catalogue to perform the selection is theoretical. It is computed
assuming an average theoretical source density and a theoretical distribution,
$\rm p_{\rm z_{s}}$, for the redshift of the sources. In practice the sources
behind a cluster are few in number. As a consequence their number and their
redshift distribution can deviate significantly from the theoretical values,
and can in some cases boost the lensing signal of a cluster, regardless of its
mass and redshift. At the same time, selecting haloes with a much lower S/N in
the halo catalogue serves to increase the false association rate, and so the
number of false detections matched incorrectly (see Sect 8.3.1). Table 5 gives
the false association rate when using a S/N$>2$ pre-selection in the halo
catalogue. This remains below 15% for all detection methods. Thus, the choice
of imposing a S/N$>2$ pre-selection in the halo catalogue is a trade-off
between the false association rate and the missed association rate.
We estimated the fraction of unmatched detections due to a boosted lensing
signal by undertaking the matching procedure a second time for all 256
realisations, but with a halo pre-selection of S/N$>1.5$. The results for each
method is given in the third column of Table 4.
Blended lensing signal unmatched incorrectly. A further explanation for the
remaining unmatched detections is the case of two haloes contributing to the
same detection:
1. 1.
Blended LOS: The blended lensing signal can be caused by LOS alignments (e.g.
filamentary structures). This means that the detection is coming from two
haloes with S/N$<2$ that are aligned along the LOS and whose lensing signals
cannot be separated without considering the redshift distribution of the
background galaxies in the detection step.
2. 2.
Blended haloes: The blended lensing signal can also be linked to the size of
the filter. Indeed, some unmatched detections can come from two haloes whose
apparent distance is small compared to the filter radius. The lensing signal
of these two haloes being merged within the AM map, there is a single
detection associated with the two haloes, the position of which depends on the
mass and redshift of the two haloes. In this case, the detection is unmatched
when the theoretical S/N of the two haloes is less than 2, or when the
position of the detection is too far away from the centre of the two haloes.
We estimated the proportion of blended lensing signal falling in the two
categories by searching for multiple haloes with S/N $>1.5$ in the vicinity of
the unmatched detections. The proportion of unmatched detections due to the
blending of the signal coming from two (or possibly more) haloes is given in
the fourth column of Table 4. In this class of unmatched detections, we found
that about a third of them are due to intrinsic LOS alignments.
A small fraction of unmatched detections do not fall into the above three
categories. The fraction for each method is listed as ‘others’ in the last
column of Table 4. Most of these remaining detections can be matched with
haloes with S/N$<1.5$. But further decreasing the halo pre-selection threshold
increases the false association rate, making this classification very
uncertain.
Figure 12: Characterisation of the detections from the W234 method. The
matched detections (blue segments) are categorised as correctly matched with
single or multiple haloes, or incorrectly matched as defined in Sect. 8.3.1.
The unmatched detections are sub-categorised as blended signal, boosted signal
from the source distribution, false detections, or others, as defined in Sect.
8.3.2. The results are averaged over the 256 realisations.
## 9 Summary and conclusions
The sensitivity of wide-field optical surveys now allows for the blind
detection of galaxy clusters through their weak lensing signal. This situation
will vastly improve with the launch of Euclid and the Roman Space Telescope.
However, the construction of weak-lensing-selected galaxy cluster catalogues
requires the improvement or development of new galaxy cluster detection
algorithms able to cope with the increasingly large volume of data.
In this paper we have introduced a new, fast, multi-scale approach, based on
wavelet filters operating on complementary scales, for detecting galaxy
clusters through their weak lensing signal. This new method, W234, was
compared to four commonly used approaches (S96, TANH, J04, and M18) based on
AM filters. The comparison was performed on the same set of Euclid-like mocks
obtained from the DUSTGRAIN-pathfinder simulations, composed of 256
realisations of a field of 5∘ $\times$ 5∘, corresponding to a total area of 6
400 deg2.
We first undertook a thorough examination of the filter characteristics in
real and Fourier space. This allowed us to demonstrate the equivalence between
some of the commonly used AM filters (S96, TANH, J04, and M18) and wavelet
filters at individual scales; moreover, we show that the M18 AM filter is
comparable to a combination of two wavelet filters. We then investigated
different options for implementing the detection methods. Applying the
detection methods to the 256 realisations, we find:
* •
Similar results are obtained when applying the detection algorithms to the
shear directly at the galaxy positions in real space or to the binned shear in
Fourier space. Binning a 5∘ $\times$ 5∘ field into $512\times 512$ bins, the
power spectrum integral of the AM map suffers a maximum 1.5% loss when passing
from real to Fourier space.
* •
The binning scheme that is adopted can have an impact on the detection
efficiency in terms of the number and distribution of detections. In
particular, binning by averaging the shear in each pixel yields a smaller
number of detections than binning by summing the shear, and a significant
fraction of these detections ($\sim 28\%$) are different from those detected
in the real space approach. In contrast, the summed pixel binning yields very
similar results, in terms of both number and distribution, to application to
the shear in real space.
* •
Application of the detection methods to the shear or the convergence yields
similar results. In particular, the power spectrum loss on the AM map is
negligible when passing from the shear to the convergence, and the number and
distribution of detections agree exactly.
To quantitatively compare the results obtained from the different filters,
each having different characteristic radii, we developed a new adaptive
matching procedure to match the detections to the haloes in the simulations.
We find:
* •
The characteristics of the halo catalogue used for the matching can have a
dramatic effect on the number of false associations. We thus applied a pre-
selection to the halo catalogue based on the theoretical weak lensing S/N
expected for an NFW halo, and kept only haloes with S/N$>2$ for the subsequent
matching step.
* •
The association distance is a critical parameter when dealing with filters
that have different characteristic radii. It must take both the spatial
resolution of the filter and the angular size of the haloes into account. We
obtained an optimal matching distance for the detections from each filter by
considering the absolute association distance, $D$, of all haloes as a
function of the association distance normalised by the characteristic radius
of each halo, $D/\theta_{200c}$. Then, decomposing the distribution of the
absolute association distance in terms of RA and Dec and fitting with a
Gaussian, we obtained an optimal MMD for each filter.
* •
For the multi-scale wavelet detections, a halo might be detected on more than
one successive scale. We thus carried out a special matching procedure for the
detections from the wavelet filters, building a cumulative catalogue from the
finest to the coarsest scales.
We then compared the performance of the detection methods, after running them
on the 256 simulations and using the matching procedure developed in this
paper. We find:
* •
The completeness is a strong function of the halo catalogue pre-selection
threshold. However, irrespective of the exact S/N pre-selection threshold, the
S96, TANH, and J04 filters exhibit a lower completeness than the M18 and W234
filters. In contrast, the J04 filter exhibits slightly higher purity at the
expense of completeness.
* •
The evolution of the completeness as a function of purity shows that the
multi-scale approach outperforms the methods based on AM filters at all values
of purity.
* •
The multi-scale method detects practically all the detections from the
individual AM approaches, with the addition of supplemental detections,
particularly at lower masses and higher redshifts.
Finally, we performed an exhaustive analysis of the matched and unmatched
detections for each method, using a S/N$>2$ pre-selection on the halo
catalogue. We found similar results for all methods. For the W234 method, we
find:
* •
More than $82\%$ of the detections are correctly matched, and of these, the
overwhelming majority (95%) are due to a single halo. For our matching
procedure, correctly matched detections due to multiple haloes along the LOS
account for less than 5% of the total. Furthermore, because our detection and
matching methods aim at high purity, the number of incorrectly matched haloes
is negligible ($<1\%$).
* •
About $17\%$ of the detections are unmatched. Only a small fraction of these
($10\%$) are due to false detections. The remainder can be explained mostly by
haloes below the S/N threshold whose signal has been boosted because of a
fortuitous redshift distribution of background sources ($37\%$), or a blended
lensing signal from LOS alignments, or angular separation distances smaller
than the filter size ($32\%$).The final 20% of the unmatched detections can
likely be associated with haloes with S/N$<1.5$ that do not appear in the
catalogue due to the pre-selection.
In this work we have shown that a multi-scale approach is faster and gives
better results, in terms of purity and completeness, than the currently used
AM methods. We have also introduced a new adaptative matching procedure that
allows a fair comparison between detection methods that operate on different
scales. A significant advantage of our method is its computational speed.
Compared to methods that apply single-scale AM filters to the shear directly
at the galaxy positions, our multi-scale method applied to the convergence
yields a gain of two to three orders of magnitude in speed.
Based on these results, we aim to apply our multi-scale approach to current
and future large-area survey data, which will allow the construction of large
catalogues of clusters selected through their weak lensing signal. This next
step also involves new challenges. As detailed in Hamana et al. (2020), the
presence of foreground galaxies that are not part of the cluster can dilute
the lensing signal. This dilution effect contaminates the weak lensing dataset
and analyses. One way to tackle this issue is to use photometric redshifts to
select sources and minimise contamination. By taking the redshift distribution
within the dataset into account, the tomographic approach reduces the impact
of the dilution effect. This solution has already shown promising results for
the detection of galaxy clusters in the HSC survey (e.g. Hamana et al., 2020;
Oguri et al., 2021). The tomographic approach may also yield coarse
information on the redshift of the detected sources. We are confident that a
multi-scale approach combined with tomography will improve on the results
shown here. Application to upcoming deep, large-area surveys will allow us to
develop our method further.
###### Acknowledgements.
The authors would like to thank Joel Bergé for providing his code to estimate
the theoretical S/N of detection of a halo by weak lensing, Jean-Baptiste
Melin for his advices and Peter Schneider for his helpful comments on the
filters. GL acknowledges funding from CNES, the French space agency. GC
acknowledges support from INAF theory Grant 2022: Illuminating Dark Matter
using Weak Lensing by Cluster Satellites.
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## Appendix A Contribution of each wavelet scale
In this section we study the contribution of each wavelet scale to the matched
detections made by the multi-scale approach.
Figure 13 shows the distribution of the matched clusters in the $z-M$ plane
for the W2 wavelet filter (upper-left panel), the W3 wavelet filter (upper-
right panel) and the W4 wavelet filter (lower-left panel). The lower-right
panel shows the distribution of the matched clusters in the $z-M$ plane for
the multi-scale detection method. The colours allow us to see the contribution
of each scale after recombination of the multiple detections. The different
panels also highlight the differences in terms of targeted clusters by the
different wavelet scales. In particular, we see the complementarity of the
different wavelet filters.
If we look at the number of matched detections for each wavelet scale after
recombination, we see that the proportion of detections made at each scale
does not vary much from one realisation to another. On average, the W2 wavelet
filter provides up to $29\%\pm 0.39$, W3 around $33\%\pm 0.57,$ and W4 around
$38\%\pm 0.42$ of the matched detections. Hence, the three wavelet scales are
useful and contribute similarly to the total number of matched detections.
Figure 13: Distribution of the matched clusters in the z-M plane for one
$5^{\circ}\times 5^{\circ}$ patch, obtained using the W2 wavelet filter
(upper-left panel), W3 wavelet filter (upper-right panel), W4 wavelet filter
(lower-left panel), and the multi-scale approach (lower-right panel).
|
# Non-imaging metasurface design for collimated beam shaping
Kirstine E. S. Nielsen Department of Physics, Technical University of
Denmark, Fysikvej, DK-2800 Kongens Lyngby, Denmark<EMAIL_ADDRESS>Mads A. Carlsen Department of Physics, Technical University of Denmark,
Fysikvej, DK-2800 Kongens Lyngby, Denmark Søren Raza Department of Physics,
Technical University of Denmark, Fysikvej, DK-2800 Kongens Lyngby, Denmark
<EMAIL_ADDRESS>
## Abstract
Metasurfaces provide a versatile platform for realizing ultrathin flat optics
for use in a wide variety of optical applications. The design process involves
defining or calculating the phase profile of the metasurface that will yield
the desired optical output. Here, we present an inverse design method for
determining the phase profile for shaping the intensity profile of a
collimated incident beam. The model is based on the concept of optimal
transport from non-imaging optics and enables a collimated beam with an
arbitrary intensity profile to be redistributed to a desired output intensity
profile. We derive the model from the generalized law of refraction and
numerically solve the resulting differential equation using a finite-
difference scheme. Through a variety of examples, we show that our approach
accommodates a range of different input and output intensity profiles, and
discuss its feasibility as a design platform for non-imaging optics.
## Introduction
Metasurfaces are gaining a lot of attention in recent years due to their
highly engineerable properties, excellent optical performance and resulting
applications as ultrathin flat optical devices [1, 2, 3, 4, 5, 6, 7, 8, 9,
10]. Their functionality is based on imparting a spatially-dependent phase
delay using judiciously designed nanoantennas [11, 12] to shape the
transmitted [13, 14, 15, 16] or reflected wavefront [17]. The freedom offered
by metasurfaces to implement any phase profile enables the design of flat
free-form optics [18, 19] with new opportunities for non-imaging applications
[20]. Non-imaging free-form optics focuses on the optimal transfer of light
energy from a light source onto a target without requiring the formation of an
image of the source. These optics have important applications in light
concentration and illumination [21].
While determining the phase profiles for imaging optics is well established,
either by analytical expressions [22] or numerical calculations using
commercial ray-tracing software [23] and Fourier methods [24], these methods
are generally not applicable for non-imaging optics. Non-imaging free-form
lenses are formulated as an inverse problem with the aim to redistribute a
light source with a known intensity profile into a desired output intensity
[21]. A variety of different strategies have been implemented to tackle this
inverse problem [25, 26, 27, 28, 29, 21], but a particularly elegant and
general way to do it is through an optimal transport formulation [30, 31].
This method makes no a priori symmetry assumptions on the lens shape and has
been demonstrated to work for a variety of incident light distributions and
output illumination patterns [32].
Here, we demonstrate a metasurface formulation of the optimal transport
approach to determine the phase profile for arbitrary one-dimensional
intensity shaping of a collimated beam. Using the generalized law of
refraction [33], we derive a nonlinear differential equation for the
metasurface phase profile, which depends on the intensity profile of the
incident beam and the desired output intensity profile. The differential
equation is transformed into a system of nonlinear algebraic equations using a
finite difference scheme and numerically solved with a root-finding algorithm.
The results from four different combinations of incident and output
illuminations are shown to demonstrate the versatility of the method in the
design of non-imaging metasurfaces.
## Theoretical framework
Figure 1: Schematic representation of the functionality of the non-imaging
metasurface. A normally-incident collimated beam $\boldsymbol{\mathrm{I}}$
with intensity profile $I(x)$ is incident on a metasurface with phase profile
$\phi(x)$. The metasurface, positioned at $z=0$, refracts the beam onto the
target plane positioned a distance $L$ from the metasurface. The refracted ray
$\boldsymbol{\mathrm{O}}$ hits the target plane at a specific coordinate
$t_{x}$ to produce the desired intensity profile $E(t_{x})$. The boundary rays
of the beam are marked with blue.
We consider a normally-incident collimated beam of light with a one-
dimensional intensity profile $I(x)$, which is redistributed by a refracting
metasurface with phase profile $\phi(x)$ to a desired target illumination with
intensity profile $E(t_{x})$ (Fig. 1). The coordinate $t_{x}$ at the target is
located a distance $L$ away from the metasurface and is connected to the
$x$-coordinate at the metasurface through the refraction caused by the
metasurface phase profile. Our aim is to determine the metasurface phase
profile that maps the incident intensity profile to the desired target
intensity profile. This constitutes an inverse problem, which we formulate by
drawing inspiration from the concept of optimal transport in non-imaging free-
form optics [30].
We assume that the metasurface is a refractor, which transmits all incident
light and is lossless. Metasurfaces with near unity transmission and $2\pi$
phase coverage can be realized with high-refractive-index dielectrics using
Huygens’-type [14, 34] and nanopost metaatoms [15]. Energy conservation then
dictates that the power incident on the metasurface is equal to the power
received at the target [35]
$\displaystyle\int_{S_{1}}I(x)\textrm{d}x=\int_{S_{2}}E(t_{x})\textrm{d}t_{x}=\int_{S_{1}}E(t_{x})\left|\frac{\partial
t_{x}}{\partial x}\right|\textrm{d}x,$ (1)
where $S_{1}$ and $S_{2}$ denote the metasurface and target widths,
respectively, and $\left|\frac{\partial t_{x}}{\partial x}\right|$ is the
Jacobian describing the change of variable from the target coordinate $t_{x}$
to the metasurface coordinate $x$. Physically, the Jacobian describes the ray
expansion or contraction due to the refraction by the metasurface. Equation
(1) requires that the power is locally conserved
$\displaystyle E(t_{x})\left|\frac{\partial t_{x}}{\partial x}\right|=I(x).$
(2)
The aim is to establish a coordinate relationship between the metasurface and
target planes. From geometrical considerations (Fig. 1), we obtain
$t_{x}=x+L\frac{O_{x}}{O_{z}},$ (3)
where $\boldsymbol{\mathrm{O}}=(O_{x},O_{z})$ is the unit vector describing
the direction of the refracted ray. The direction of the refracted ray is
determined by its wave vector $\boldsymbol{\mathrm{k_{r}}}$, which, in turn,
is found from the generalized law of refraction [33, 36]. Since the incident
beam impinges normally and the metasurface is flat, the direction of the
refracted ray is solely determined by the metasurface phase gradient as
$\boldsymbol{\mathrm{O}}=\frac{\boldsymbol{\mathrm{k_{r}}}}{|\boldsymbol{\mathrm{k_{r}}}|}=\frac{1}{k}(k_{x},k_{z})=\left(\frac{\partial\tilde{\phi}}{\partial
x},\sqrt{1-\left(\frac{\partial\tilde{\phi}}{\partial x}\right)^{2}}\right).$
(4)
Here, we have introduced a scaled wavelength-independent phase
$\tilde{\phi}=\phi/(n_{0}k_{0})$, where $k_{0}$ is the vacuum wave number and
$n_{0}$ is the refractive index of the medium between the metasurface and the
target plane. Combining Eqs. (3-4), we evaluate the Jacobian in Eq. (2) to
arrive at
$\displaystyle E(t_{x})\left|1+L\frac{\partial^{2}\tilde{\phi}}{\partial
x^{2}}\left[1-\left(\frac{\partial\tilde{\phi}}{\partial
x}\right)^{2}\right]^{-\frac{3}{2}}\right|-I(x)=0.$ (5)
Equation (5) is our main result and is a nonlinear differential equation for
the metasurface phase profile, given a known input intensity profile $I(x)$
and a desired target intensity profile $E(t_{x})$. We note that only the phase
gradient $\partial\tilde{\phi}/\partial x$ impacts the beam shaping, as the
phase itself does not explicitly appear in Eq. (5). We exploit this in the
numerical implementation described in Sec. Numerical implementation.
As the final step we treat the boundaries of the system. The boundary
condition is defined by the edge-ray principle, which is the basis for most
non-imaging design problems. The edge-ray principle states that a mapping of
all rays from the edge of the source distribution should be directed to the
edge of the target illumination distribution, in order to ensure that all rays
fall within the target [37]. In our case this condition is represented by the
coordinate mapping in Eq. (3), which at $x=x_{\text{min}}$ (Fig. 1) requires
the phase gradient to satisfy the boundary condition
$\displaystyle\left.\frac{\partial\tilde{\phi}}{\partial
x}\right|_{x=x_{\text{min}}}$
$\displaystyle=A_{\text{min}}\left(1+A_{\text{min}}^{2}\right)^{-\frac{1}{2}},\,\,\,\,\,\,\,\,A_{\text{min}}=\frac{t_{x,\text{min}}-x_{\text{min}}}{L},$
(6)
and similarly at $x=x_{\text{max}}$ for $A_{\text{max}}$ and
$t_{x,\text{max}}$.
The theoretical framework given by Eqs. (5-6) describes the metasurface phase
profile needed to transport the incident intensity to the target intensity.
The nonlinear nature of Eq. (5) renders a general analytical solution
unfeasible. Instead, we transform the nonlinear differential equation into a
system of nonlinear algebraic equations using a finite difference scheme, and
solve the resulting system of equations numerically.
## Numerical implementation
We noted that the phase does not explicitly appear in Eqs. (5-6), and thus the
problem can be simplified by solving for the phase gradient directly. We
discretize the metasurface into $N$ grid points in the interval
$\left[-\tfrac{X}{2};\tfrac{X}{2}\right]$ with a spacing $h=X/(N-1)$, where
$X$ is the width of the metasurface. Denoting the scaled phase gradient as
$y\equiv\partial\tilde{\phi}/\partial x$ and employing a centered finite
difference scheme [38], Eq. (5) turns into the following system of algebraic
equations for the interior grid points
$\displaystyle F_{i}^{\text{int}}$
$\displaystyle=E(t_{x}(y_{i}))\left|1+L\frac{y_{i+1}-y_{i-1}}{2h(1-y_{i}^{2})^{\frac{3}{2}}}\right|-I(x_{i})=0,$
(7)
where $t_{x}(y_{i})=x_{i}+Ly_{i}/\sqrt{1-y_{i}^{2}}$. The boundary points
$i=1,N$ described by Eq. (6) are given by
$\displaystyle F_{i}^{\text{BC}}$
$\displaystyle=y_{i}-A_{i}\left(1+A_{i}^{2}\right)^{-\frac{1}{2}}=0.$ (8)
These discretized equations describe a system of $N$ equations with $N$
variables, which we solve numerically using a root-finding algorithm
implemented in the commercial software MATLAB. The implementation is based on
the Levenberg–Marquardt algorithm, which employs the Jacobian matrix for fast
and robust convergence. The elements $J_{i,j}=\partial F_{i}/\partial y_{j}$
in the Jacobian matrix for our system of equations [Eqs. (7-8)] are only non-
zero for $j=i+1,\,i,\,i-1$ for the interior points and for $j=i$ on the
boundary, making the Jacobian matrix sparse. For the interior points we find
the Jacobian elements
$\displaystyle J_{i,i}$
$\displaystyle=E^{\prime}(t_{x}(y_{i}))\frac{L|K_{i}|}{\left(1-y_{i}^{2}\right)^{\frac{3}{2}}}+E(t_{x}(y_{i}))\frac{3L(y_{i+1}-y_{i-1})y_{i}}{2h\left(1-y_{i}^{2}\right)^{\frac{5}{2}}}\frac{K_{i}}{|K_{i}|},$
(9) $\displaystyle J_{i,i+1}$
$\displaystyle=-J_{i,i-1}=\frac{E(t_{x}(y_{i}))L}{2h\left(1-y_{i}^{2}\right)^{\frac{3}{2}}}\frac{K_{i}}{|K_{i}|},$
(10)
where we have defined
$K_{i}=1+L\frac{y_{i+1}-y_{i-1}}{2h(1-y_{i}^{2})^{\frac{3}{2}}}$. In Eq. (9),
prime denotes the differentiation with respect to the argument. For the
boundary points described by Eq. (8), the Jacobian elements are given by
$J_{i,i}=1$.
The iterative root-finding algorithm based on Eqs. (7-10) requires an initial
guess for the phase gradient. For complex beam shaping, the phase gradient may
vary in a manner that is a priori difficult to guess. To alleviate this
problem, we use numerical continuation where we iteratively increase the
complexity of the beam shaping, using in each iteration the obtained phase
gradient from the previous iteration as an initial guess. For example, in the
Gaussian beam shaping presented in Sec. Design examples, our starting point is
a broad Gaussian profile (close to constant) which we gradually sharpen by
decreasing the linewidth to obtain the desired beam shaping funcitonality. We
find that this remedies the dependence on providing an accurate initial guess
and ensures robust convergence. With this numerical implementation we obtain
the phase gradient to which we fit a function using linear interpolation and
perform integration to extract the phase profile.
Figure 2: (a) Scaled phase profiles and phase gradients, and (b) resulting ray
plots for a focusing non-imaging metasurface. (c-d) Similar to (a-b) for a
defocusing metasurface. In (a,c) the dashed black lines indicate the
analytical phase profile [Eq. 11], while in (b,d) they indicate the boundary
rays. The ray plots (b,d) show that the calculated phase profile indeed
achieves the desired focus (b) and defocus (d) effect.
## Design examples
We verify the theoretical model by testing four beam shaping functionalities.
In the following examples, the metasurface width is $X=1$ mm and the distance
between the metasurface and the target is $L=10$ cm. First we demonstrate
focusing and defocusing functionalities which can be verified analytically
(Fig. 2). For focusing, the target width $T$ is set to 1000 times smaller than
the metasurface width [Fig. 2(a-b)], while defocusing is modelled by setting
the target width to $T=15X$, i.e., 15 times larger than the metasurface width
[Fig. 2(c-d)]. The incident beam and target illumination both have constant
intensity profiles, and we ensure energy conservation by scaling the target
intensity profile using Eq. (1). In these cases the scaled phase profile of
the metasurface is given by [22]
$\tilde{\phi}(x)=\pm\left(\sqrt{x^{2}+f^{2}}-f\right).$ (11)
The $-$ ($+$) solution in Eq. (11) gives a focusing (defocusing) phase profile
with the focal length given by $f=L$ [$f=LX/(T-X)$]. Consequently, we expect
convex and concave phase distributions for the metasurface in the focus and
defocus cases, respectively. For the focusing case [Fig. 2(a)], we indeed
obtain a convex phase profile in excellent agreement with Eq. (11). The ray
plot in Fig. 2(b) shows to which point $t_{x}$ on the target plane a ray
incident on the metasurface in point $x$ will be refracted. The target points
$t_{x}$ are evaluated using Eq. (3) with the numerically obtained phase
gradients as input. The ray plots visualize the redistribution of the incident
beam onto the target plane and show that the boundary rays are indeed mapped
to the boundary of the target in accordance with the edge-ray principle. For
the defocus case [Fig. 2(c-d)], the numerically obtained concave phase profile
is again in agreement with the analytical expression. These intuitive cases
demonstrate that our model produces results in agreement with existing theory.
Figure 3: (a) Incident Gaussian intensity profile $I(x)$ and constant target
intensity $E(t_{x})$. (b) Numerically calculated scaled phase profile and
scaled phase gradient. (c) Comparison between the intended target intensity
$E(t_{x})$ (black dashed line) and the resulting target intensity profile
based on the calculated scaled phase gradient (green). (d) Ray distribution
calculated from the obtained phase gradient, which illustrates the refraction
of the incident beam by the metasurface phase profile to obtain the desired
constant intensity.
From here we move to more complex beam shaping, where the input has a Gaussian
intensity profile which is mapped to a constant target intensity profile [Fig.
3(a)]. The metasurface and target widths are kept the same. We choose a
configuration where the input intensity is not centered with respect to the
metasurface, in order to stress the model. In Fig. 3b we show the numerically
obtained phase and phase gradient profiles. We find a convex-type phase
profile, which is reasonable as the beam shaping taking place is similar to
the defocus case [Fig. 2(a)]. The rays are directed towards the perimeter of
the target plane, which is consistent with the flattening of the Gaussian beam
shape [Fig. 3(d)]. We also check that the calculated phase gradient produces
the expected target intensity profile by using Eq. (2). The calculated target
intensity profile is in excellent agreement with the desired target intensity
[Fig. 3(c)].
Finally, we investigate the inverse case where the target is a Gaussian
intensity profile $E(t_{x})=E_{0}e^{-t_{x}^{2}/2b^{2}}$ with $b=0.2$ mm, while
the input intensity profile remains constant [Fig. 4(a)]. In Fig. 4(b), we see
that the phase profile has a concave shape similar to the focus case [Fig.
2(a)], which is reasonable since the constant incident beam is focused to a
Gaussian target illumination [Fig. 4(c-d)]. However, the phase gradient
differs from the focusing case as the edge rays at the metasurface must be
mapped to the edge of the target. This forces the phase gradient to go to zero
at the edges of the metasurface, since the target and metasurface widths are
identical. This example serves to show that rather complex beam shaping can be
handled by our theoretical framework.
Figure 4: (a) Incident constant intensity $I(x)$ and Gaussian target intensity
profile $E(t_{x})$. (b) Numerically calculated scaled phase profile and scaled
phase gradient. (c) The intended target intensity $E(t_{x})$ (black dashed
line) and the resulting target intensity profile based on the calculated phase
gradient (green). (d) Ray distribution calculated from the obtained phase
gradient.
## Metasurface design and efficiency
The theoretical model and its numerical implementation determines a
wavelength-independent scaled phase and phase gradient profile for the desired
beam shaping. We can retrieve the actual phase profile of the metasurface at
any desired operating wavelength $\lambda$ by the operation
$\phi(x)=2\pi/(n_{0}\lambda)\tilde{\phi}(x)$. Fig. 5 shows half of the
symmetric phase profile for the defocus case [Fig. 2(c)] calculated for
$\lambda=660$ nm and $\lambda=1550$ nm and wrapped on the interval $[0,2\pi]$.
Here, we have assumed that the rays travel through air ($n_{0}=1$). Since a
metasurface is made up of a periodic array of nanostructures, which each
provide a specific phase shift to incident light, we need to discretize and
wrap the phase profile in modulo of $2\pi$. This gets increasingly tougher for
shorter wavelengths, as they yield larger phase gradients (Fig. 5). It is
advantageous to have as many phase levels in the discretization as possible in
order to best approximate the continuous phase profile [16, 39]. However, for
simple metalenses 4 phase levels have proven to give sufficient resolution and
efficiency [40, 41]. For more advanced lens designs as many as 8 phase levels
are required [42] , and for production of detailed holograms 8-32 phase levels
have been proven necessary to achieve the required resolution [43, 44, 45]. In
our case the steepest phase gradient is found for $\lambda=660$ nm at the very
edge of the metasurface where the largest refraction occurs (Fig. 5). Here we
need to cover the full $2\pi$ phase range over 9 µm. The metasurface period
needs to be subwavelength in order to limit diffraction, meaning that 16 phase
levels are reasonable to achieve with a theoretical efficiency of 99% [16].
For more complex beam shaping with larger phase gradients the phase
discretization impacts the achievable efficiency of the metasurface.
Figure 5: Phase profiles for the defocus lens described in Fig. 2 at the
wavelengths $\lambda=660$ nm and $\lambda=1550$ nm. Since the phase profile is
symmetric, only one half is plotted. The phase is wrapped on the interval
$[0,2\pi]$ to illustrate the phase changes. The shortest distance over which
the phase goes from $0$ to $2\pi$ is 9 µm (23 µm) for $\lambda=660$ nm
($\lambda=1550$ nm).
## Conclusion
In summary we have demonstrated a theoretical optimal transport framework for
obtaining the phase profile of a metasurface for arbitrary one-dimensional
beam shaping with collimated incident light. We show how the resulting
nonlinear differential equation can be solved numerically using a finite
difference scheme and a root-finding algorithm. Four specific cases are
investigated to demonstrate the versatility of the method within the design of
non-imaging metasurfaces. The resulting phase profiles are smooth and our
results are visualized through ray tracing. We find that the phase profiles
can be realized with a reasonable phase resolution.
### Funding
S. R. acknowledges support by the Independent Research Funding Denmark
(7026-00117B).
### Acknowledgments
We thank Hugh Simons and Gor Nahapetyan for assistance with the numerical
implementation.
### Disclosures
The authors declare no conflicts of interest.
### Data availability
Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.
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|
11institutetext: State University of Campinas (UNICAMP) 22institutetext:
Maritaca AI 33institutetext: NeuralMind AI
# BLUEX: A benchmark based on Brazilian Leading Universities Entrance eXams
Thales Sales Almeida 1122 Thiago Laitz 1133 Giovana K. Bonás 11 Rodrigo
Nogueira 1122
###### Abstract
One common trend in recent studies of language models (LMs) is the use of
standardized tests for evaluation. However, despite being the fifth most
spoken language worldwide, few such evaluations have been conducted in
Portuguese. This is mainly due to the lack of high-quality datasets available
to the community for carrying out evaluations in Portuguese. To address this
gap, we introduce the Brazilian Leading Universities Entrance eXams (BLUEX), a
dataset of entrance exams from the two leading universities in Brazil: UNICAMP
and USP. The dataset includes annotated metadata for evaluating the
performance of NLP models on a variety of subjects. Furthermore, BLUEX
includes a collection of recently administered exams that are unlikely to be
included in the training data of many popular LMs as of 2023. The dataset is
also annotated to indicate the position of images in each question, providing
a valuable resource for advancing the state-of-the-art in multimodal language
understanding and reasoning. We describe the creation and characteristics of
BLUEX and establish a benchmark through experiments with state-of-the-art LMs,
demonstrating its potential for advancing the state-of-the-art in natural
language understanding and reasoning in Portuguese. The data and relevant code
can be found at https://github.com/Portuguese-Benchmark-Datasets/BLUEX
## 1 Introduction
Recent advances in Language Models (LMs) have generated significant interest
due to their demonstrated capabilities on a wide range of language tasks,
including text classification, language translation, and text generation [3,
7]. LM performance has been particularly impressive on standardized tests,
which present challenging questions requiring high levels of domain-specific
knowledge and reasoning. For instance, recent benchmarks on GPT-4 [16] showed
that it can achieve human-level performance on a variety of graduate-level
benchmarks.
Despite the impressive performance of LMs on standardized tests, few
evaluations have been performed in Portuguese [15], partially due to the lack
of available datasets in the language. This lack of high-quality, standardized
datasets presents a significant challenge for researchers interested in
developing and evaluating LMs in Portuguese. To address this gap for Brazilian
Portuguese, we introduce BLUEX, a dataset consisting of entrance exams for the
two leading universities in Brazil. Our dataset offers a rich source of high-
quality high school-level questions annotated with their respective subjects,
as well as flags indicating the required capabilities necessary to respond
accurately to the questions, such as knowledge of Brazilian culture and the
application of mathematical reasoning. These annotations can be used to
evaluate the performance of LMs on a variety of subjects and capabilities such
as domain-specific knowledge and reasoning. Additionally, BLUEX includes a
collection of recently administered entrance exams that are unlikely to be
included in the training data of many currently popular LMs.
In anticipation of the emergence of multimodal models that combine text and
image understanding, we have annotated BLUEX to indicate the position of
images in each question. Additionally, we have included all necessary images
with the dataset to facilitate research on multimodal language tasks. We
believe that this resource will be essential in evaluating the performance of
models that reason with both text and image inputs to solve complex problems.
In this paper, we describe the creation and characteristics of BLUEX and
establish a benchmark through experiments with state-of-the-art LMs. Our
findings suggest that BLUEX provides a valuable resource for benchmarking and
advancing the state-of-the-art in natural language understanding and reasoning
in Portuguese. This is particularly relevant since even the current state-of-
the-art models, such as GPT-4, still have considerable room for improvement
and do not achieve the highest cutoff grades for both universities.
## 2 Related Work
In the realm of Portuguese Natural Language Processing (NLP) datasets, there
appears to be a limited availability.
For question-answering tasks, Faquad [21] is available, which exhibits an
extractive style akin to SQuAD [18]. It features questions concerning
Brazilian higher education institutions, with documents sourced from a federal
university and supplemented by Wikipedia articles. Another option is the
Multilingual Knowledge Questions and Answers (MKQA) dataset, which covers 26
languages [12]. This dataset was generated by selecting 10,000 queries from
the Natural Questions dataset [10] and acquiring new passage-independent
answers for each question. Subsequently, human translators translated the
questions and answers into 25 non-English, typologically diverse languages,
including Portuguese.
Regarding sentence entailment tasks, ASSIN 1 and 2 [5, 19] are available.
These datasets encompass Recognizing Textual Entailment (RTE), also referred
to as Natural Language Inference (NLI), and Semantic Textual Similarity (STS)
tasks. The former involves predicting if a given text (premise) implies
another text (hypothesis), while the latter quantifies the semantic
equivalence between two sentences.
The Portuguese Language Understanding Evaluation (PLUE) benchmark [6] provides
Portuguese translations of the GLUE [26], SNLI [1], and SciTAIL [8] datasets.
These translations have been generated using automatic translation tools
including Google Translate and OpusMT [24].
The Winograd Schema Challenge (WSC) dataset [9] contains pairs of sentences
with minimal differences, featuring an ambiguous pronoun that is resolved
divergently between the two sentences. Melo et al. [13] manually translated
and adapted this dataset to Portuguese.
For sentiment analysis tasks, the TweetsentBr dataset [2] consists of 15,000
tweets related to the TV show domain, collected between January and July 2017.
The tweets were manually annotated by seven annotators into three classes:
positive, neutral, and negative.
The Multilingual Amazon Slu resource package (SLURP) for Slot-filling, Intent
classification, and Virtual assistant Evaluation (MASSIVE) [4] is a 1M-example
dataset containing realistic virtual utterances in 51 languages, including
Portuguese. Professional translators translated the dataset from English, and
it is annotated for slot (55 classes) and intent (60 classes) prediction
tasks.
A dataset more closely related to BLUEX is the ENEM-challenge dataset [22],
which includes the editions of the Brazilian national exam, Exame Nacional do
Ensino Medio (ENEM), from 2009 to 2017. Additionally, Nunes et al. [15]
introduced a dataset containing the ENEM exam of 2022, the same paper
evaluated the performance of LMs such as GPT-3.5-Turbo and GPT-4 on both the
ENEM-challenge and the ENEM 2022 datasets.
## 3 The BLUEX Dataset
### 3.1 Dataset Creation
BLUEX is a dataset comprising more than 1,000 multiple choice questions from
the entrance exams of the two leading universities in Brazil, Unicamp and USP,
administered between 2018 and 2023. The dataset was created by automatically
extracting each question text, alternatives, and related images using scripts,
and subsequently each example was manually annotated to correct extraction
errors and provide additional metadata such as image positioning.
### 3.2 Annotated Question Metadata
The annotated metadata is described below.
* •
Prior Knowledge (PRK) \- Indicates whether the question requires knowledge
from outside of what has been provided in the question, such as familiarity
with a particular author’s work or a specific mathematical formula.
* •
Text Understanding (TU) \- Indicates whether the question requires
understanding of a particular text.
* •
Image Understanding (IU) \- Indicates whether the question requires
understanding of an image. It should be noted that not all questions with
images require their understanding to answer the question.
* •
Mathematical Reasoning (MR) \- Indicates whether the question requires
mathematical reasoning, such as the ability to perform calculations and
symbolic manipulations.
* •
Multilingual (ML) \- Indicates whether the question requires knowledge of two
or more languages, such as questions designed to test English skills of
Portuguese speakers.
* •
Brazilian Knowledge (BK) \- Indicates whether the question involves knowledge
specific to Brazil, such as Brazilian history, literature, geography, or
culture.
* •
Subjects \- A list of subjects related to the question, such as geography,
physics, etc.
* •
Related Images \- A list of all the related images for the question.
* •
Alternative Type \- Indicates whether the answer choices are presented as text
or as images. This is important because some questions may use images as
answer choices, which requires different processing techniques than questions
with only textual answers.
By providing such annotations along with the questions we aim to facilitate
research into language understanding and reasoning in Portuguese for both pure
language models and multimodal models. We believe that BLUEX will be a
valuable resource for researchers to evaluate and improve the performance of
future language models in the context of Portuguese-language standardized
tests.
### 3.3 Image Positioning
Many of the questions in the exams require a contextual or informational
understanding of images. Despite active research in the field of multimodal
models, models that can adeptly process both text and image data and yield
satisfactory results remain scarce in the public domain. We believe that BLUEX
can serve as an essential evaluation tool for such models. Anticipating the
use of models that will process images and text in an interleaved manner, we
also provide precise information regarding the placement of images within the
question, as illustrated in Figure 1.
Figure 1: Example of image annotation in BLUEX.
### 3.4 Dataset Distribution
The BLUEX dataset covers a wide range of high school subjects, including
Mathematics, Physics, Chemistry, Biology, History, Geography, English,
Philosophy and Portuguese, as well as multidisciplinary questions that involve
two or more subjects. The distribution of questions is shown in Table 1, where
we also provide the distribution for the subset of questions without images,
which accounts for approximately 58% of the total dataset.
Furthermore, Table 2 shows the distribution of the dataset across annotated
categories, as explained in Section 3.2. We observe that the majority of
questions require specific knowledge and the ability to comprehend text, two
expected capabilities in students taking these exams. Note that any given
question can be part of multiple categories.
biology chemistry english geography history mathematics philosophy physics
portuguese multidisciplinary Total UNICAMP 60 45 57 51 60 89 1 61 86 46 556
USP 50 63 41 55 63 69 4 63 88 43 539 BLUEX 110 108 98 106 123 158 5 124 174 89
1095 No images UNICAMP 35 15 20 22 49 64 1 36 65 31 338 USP 23 15 25 16 52 36
4 26 80 23 300 BLUEX 58 30 45 38 101 100 5 62 145 54 638
Table 1: Distribution over subjects.
DS TU IU MR ML BK UNICAMP 431 440 160 209 60 69 USP 446 442 203 174 43 63
BLUEX 877 882 363 383 103 132 No Images UNICAMP 273 282 0 118 23 46 USP 237
269 0 70 25 43 BLUEX 510 551 0 188 48 89
Table 2: Distribution over categories.
## 4 Results
Model BLUEX UNICAMP USP MR BK Highest Cutoff Score 0.863 0.855 0.872 - -
Average Human Score 0.521 0.530 0.511 - - Random 0.220 0.250 0.200 0.223 0.228
GPT-4 [16] 0.748 0.749 0.747 0.447 0.854 Sabiá 65B [17] 0.632 0.615 0.650
0.239 0.775 GPT-3.5-Turbo 0.582 0.580 0.583 0.277 0.764 LLaMA 65B [25] 0.542
0.530 0.557 0.271 0.652 OPT 66B [30] 0.223 0.246 0.197 0.186 0.258 Sabiá 7B
[17] 0.466 0.494 0.433 0.25 0.551 Alpaca 7B [23] 0.284 0.308 0.257 0.261 0.258
BloomZ 7B [14] 0.284 0.275 0.293 0.17 0.326 LLaMA 7B [25] 0.255 0.275 0.233
0.255 0.247 Bertin 6B [20] 0.241 0.293 0.183 0.261 0.315 Bloom 7B [29] 0.238
0.302 0.167 0.255 0.281 XGLM 7.5B [11] 0.205 0.219 0.19 0.213 0.202 OPT 6.7B
[30] 0.205 0.240 0.167 0.207 0.281 GPT-J 6B [27] 0.197 0.222 0.17 0.186 0.236
Table 3: Accuracy in the BLUEX dataset.
To enable future comparisons, we evaluated our dataset using several language
models, ranging from 6B to 66B parameters, including OpenAI’s GPT-4 and
GPT-3.5-Turbo models. Our experiments were conducted using large language
models with no specific training for this task. Each model was provided with
one example in the input and then asked to answer a question from the test
set. The example was randomly selected from an exam of the same university as
the current question, but from a different year. For example, if the current
question is from UNICAMP 2019, the example provided in the prompt would be a
question from a UNICAMP exam, but not from 2019. We excluded all questions
containing images from our experiments since the language models we used can
only process text. This resulted in a total of 638 questions being used, which
corresponds to approximately 60% of the dataset
Table 3 summarizes our experimental findings, including the mean score
achieved by exam-taking students, as well as the mean cutoff score of the most
competitive major, which is medicine in both universities.111The average and
cutoff scores are reported by the entities responsible for administering the
exams. The results presented in Table 3 are the average of all the exams
contained in the BLUEX dataset. The BLUEX column shows the accuracy of the
whole subset used in the evaluation, while the UNICAMP and USP columns account
for only the questions from the respective universities. The MR and BK columns
account only for questions that include those categories.
Among the language models tested in the 7B-parameter range, Sabiá [17], a
model further pre-trained in Portuguese, consistently outperformed all other
models, coming close to matching the average human score. Among the open-
source models in the 60B-parameter range, LLaMA 65B [25] significantly
outperformed OPT 66B [30] and achieved similar performance to GPT-3.5-Turbo.
Sabiá 65B achieved better performance than GPT-3.5-Turbo but still lagged
behind GPT-4 by ten points. GPT-4 was by far the best model in our evaluations
but did not achieve an average score high enough to pass in medicine, the most
competitive major. It is worth noting that the average and cutoff scores
provided in Table 3 are computed taking into account the whole exam, including
questions with images, while the scores obtained by the language models
utilize only the subset of questions with no images.
We also conducted a more detailed analysis of the models’ performance by
examining their ability to handle specific question types. Table 3 presents
the findings for questions that required Mathematical Reasoning (MR) and
Brazilian Knowledge (BK). We observe that, with the exception of GPT-4, all
models struggled to perform significantly better than random chance in
questions that required Mathematical Reasoning. Even GPT-4 only achieved an
accuracy of 44% in MR questions. On the other hand, when considering questions
that require brazilian knowledge, Sabiá greatly outperformed all the other
models in the 7B-parameter range, indicating that the extra pretraining in
Portuguese provided the model with additional regional knowledge. In the
60B-parameter range, Sabiá also showed improvement over LLaMA, increasing the
accuracy in these questions by 10 points and slightly outperforming
GPT-3.5-Turbo. Nevertheless, it could not match the remarkable performance of
GPT-4.
Figure 2: Accuracy of the best models over the years of the exams.
Moreover, Figure 2 displays the performance of the top four models on the
exams conducted each year. It can be observed that the models have a small
variance between the years, which is expected as the difficulty of each exam
and the number of questions in the subset vary across years. A surprising
result, however, is the increased performance that all models seem to exhibit
in 2023. The average and highest cutoff scores also increased slightly over
the years, indicating that the exams became slightly easier in recent years.
Since the 2023 exams were very recently administered, it is unlikely that they
are part of any of the studied models’ training data. Therefore, since the
models’ performance in the most recent years is comparable to that in older
exams, it is reasonable to assume that the models are not merely memorizing
the answers for the questions in the dataset.
## 5 Conclusion
This work introduced BLUEX, a new dataset that consists of 13 college entrance
exams applied between 2018 and 2023 from two of the leading Brazilian
universities, UNICAMP and USP. Each question of these exams was extensively
annotated to help measure different abilities across multiple subjects in
Portuguese. Beyond that, by providing images and their corresponding positions
within the text, BLUEX is one of the few Portuguese datasets that are ready to
evaluate multimodal models. We provide results from multiple LMs as baselines
and reference scores based on students performance to facilitate future
comparisons. We believe that BLUEX will be a important benchmark in the
evaluation of the Portuguese capabilities of future models.
## 6 Future Work
The models used in this study employed a single in-context example. However,
there’s room for further investigation, such as determining whether increasing
the number of few-shot examples could boost the performance of each model, as
well as assessing their zero-shot performance. Furthermore, Nunes et al. [15]
showed that GPT-4’s performance on ENEM questions was significantly boosted
when chain-of-thought prompts [28] were used. Adopting a similar approach here
could potentially lead to performance improvement.
Finally, regarding multimodal models, their performance can be assessed
utilizing the BLUEX dataset. This provides an opportunity for researchers to
investigate the models’ capabilities in integrating visual and textual
information to address high school level questions.
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open-access multilingual language model (2023)
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## 7 Appendix
### 7.1 Prompt for evaluation
The prompt used for all the experiments in this paper is shown in the Figure
3.
| Select the correct alternative
---|---
… Few-shot examples go here, separated with ###
Nth example:
Question: | Times change, desires change, Beings change, trust changes: The whole world is composed of change, Always taking on new qualities. Continually we see novelties, Different in everything from hope: From evil, only the sorrows remain in memory, And from good (if any existed), the longing. Time covers the ground with a green cloak, Which was once covered in cold snow, And in me, it turns the sweet song into tears. And besides this daily change, Another change causes even more astonishment, That no longer changes as it used to. (Luís Vaz de Camões). (Luís de Camões, 20 sonnets. Campinas: Unicamp Publisher, p.91.) Indicate the statement that applies to the sonnet written by Camões.
Alternatives: | A. The poem takes up the Renaissance theme of the change of things, which the poet feels as a reason for hope and faith in life.
B. The idea of transformation refers to worldly things, but it does not affect
the poet’s state of mind due to his love belief.
C. Everything is always renewed, unlike the poet’s hopes, which harbor his
sorrows and longings.
D. Not only does the poet’s state of mind change, but also his experience of
change itself.
Answer: | D.
Figure 3: Example of prompt used in the experiments, the question was
translated into English for the convenience of readers. The text in red is the
expected output.
### 7.2 Benchmark per Subject
Table 4 provides a detailed report of each model achieved accuracy by subject.
Questions that were associated with more than one subject contributed to the
accuracy of both scores. For example, a question related to mathematics and
English will be taken into account when calculating the accuracy of both
mathematics and English subjects.
Model Biology Chemistry English Geography History Mathematics Philosophy
Physics Portuguese GPT-4 0.871 0.675 0.918 0.935 0.930 0.389 1.000 0.557 0.805
Sabiá 65B 0.771 0.350 0.837 0.774 0.883 0.278 1.000 0.257 0.755 GPT-3.5-Turbo
0.700 0.350 0.714 0.806 0.805 0.259 0.714 0.329 0.629 LLaMA 65B 0.657 0.350
0.816 0.677 0.719 0.306 0.429 0.286 0.572 OPT 66B 0.229 0.275 0.286 0.161
0.273 0.176 0.286 0.200 0.189 Sabiá 7B 0.514 0.350 0.592 0.565 0.672 0.241
0.571 0.271 0.509 Alpaca 7B 0.286 0.225 0.347 0.306 0.320 0.269 0.143 0.229
0.264 BloomZ 7B 0.243 0.075 0.551 0.371 0.336 0.185 0.143 0.171 0.308 LLaMA 7B
0.229 0.325 0.286 0.210 0.266 0.231 0.000 0.314 0.245 Bertin 6B 0.186 0.225
0.347 0.226 0.234 0.259 0.286 0.243 0.245 Bloom 7B 0.243 0.225 0.327 0.210
0.219 0.259 0.143 0.214 0.239 XGLM 7.5B 0.143 0.300 0.245 0.161 0.164 0.204
0.000 0.171 0.264 OPT 6.7B 0.186 0.250 0.143 0.145 0.234 0.185 0.000 0.257
0.214 GPTJ 6B 0.214 0.200 0.204 0.113 0.227 0.194 0.000 0.200 0.195
Table 4: Results for each model by subject in BLUEX.
|
problemgen Input: Output: problemdec Input: Question:
# On the enumeration of signatures of XOR-CNF’s
Nadia Creignou Aix-Marseille Université, CNRS, LIS, Marseille, France. Oscar
Defrain Aix-Marseille Université, CNRS, LIS, Marseille, France. Frédéric
Olive Aix-Marseille Université, CNRS, LIS, Marseille, France. Simon Vilmin
Aix-Marseille Université, CNRS, LIS, Marseille, France.
(February 28, 2024)
###### Abstract
Given a CNF formula $\varphi$ with clauses $C_{1},\dots,C_{m}$ over a set of
variables $V$, a truth assignment $\mathbf{a}:V\to\\{0,1\\}$ generates a
binary sequence
$\sigma_{\varphi}(\mathbf{a})=(C_{1}(\mathbf{a}),\ldots,C_{m}(\mathbf{a}))$,
called a signature of $\varphi$, where $C_{i}(\mathbf{a})=1$ if clause $C_{i}$
evaluates to 1 under assignment $\mathbf{a}$, and $C_{i}(\mathbf{a})=0$
otherwise. Signatures and their associated generation problems have given rise
to new yet promising research questions in algorithmic enumeration. In a
recent paper, Bérczi et al. interestingly proved that generating signatures of
a CNF is tractable despite the fact that verifying a solution is hard. They
also showed the hardness of finding maximal signatures of an arbitrary CNF due
to the intractability of satisfiability in general. Their contribution leaves
open the problem of efficiently generating maximal signatures for tractable
classes of CNFs, i.e., those for which satisfiability can be solved in
polynomial time. Stepping into that direction, we completely characterize the
complexity of generating all, minimal, and maximal signatures for XOR-CNFs.
Keywords: algorithmic enumeration, XOR-CNF, signatures, maximal bipartite
subgraphs enumeration, extension, proximity search.
## 1 Introduction
Propositional formulas are ubiquitous in computer science. The complexity of
the satisfiability problem has been extensively studied from the point of view
of several algorithmic tasks such as decidability, counting, or enumeration to
mention but a few. Given a CNF formula $\varphi$ with clauses
$C_{1},\dots,C_{m}$ over a set of variables $V=\\{v_{1},\dots,v_{n}\\}$, a
truth assignment $\mathbf{a}:V\to\\{0,1\\}$ leads to a binary sequence
$\sigma_{\varphi}(\mathbf{a})$, called a _signature_ of $\varphi$, defined by
$\sigma_{\varphi}(\mathbf{a})=(C_{1}(\mathbf{a}),\ldots,C_{m}(\mathbf{a}))$
where $C_{i}(\mathbf{a})=1$ if clause $C_{i}$ evaluates to 1 under assignment
$\mathbf{a}$, and $C_{i}(\mathbf{a})=0$ otherwise. Deciding the satisfiability
of $\varphi$ boils down to deciding whether the all-one sequence is a
signature of $\varphi$.
In this paper, we investigate the problems of listing all, minimal, and
maximal signatures of a CNF, where minimal and maximal are meant bitwise. The
task of finding all signatures of a CNF originates from well-design pattern
trees and has first been posed during the Dagstuhl seminar on enumeration in
data management [BKPS19]. However, enumerating the signatures of a given
CNF—may it be all, minimal, or maximal ones—will generally produce an
exponential number of solutions. Therefore, input sensitive polynomial time
complexity is not a suitable yardstick of efficiency when analyzing algorithms
performance. Instead, _output-sensitive complexity_ estimates the running time
of an enumeration algorithm using both input and output size. More precisely,
an enumeration algorithm runs in _total-polynomial time_ (or _output-
polynomial_ time) if its execution time is bounded by a polynomial in the
combined size of the input and the output. Still, the regularity of
enumeration algorithms is more relevant than their total running time. For
this reason, _polynomial delay_ and _incremental-polynomial time_ are
customarily regarded as better notions of tractability for enumeration
complexity. On the one hand, polynomial delay means that the delay between
consecutive outputs is polynomial in the input size. Incremental-polynomial
time, on the other hand, means that the time to produce the $i^{\text{th}}$
solution is bounded by a polynomial in the input size plus $i$ (see e.g.,
[JYP88, Str19]).
Generating all the signatures of a CNF turns out to be of great fundamental
interest, for it is an example of enumeration problem which can be solved in
total-polynomial time even though solutions cannot be recognized in polynomial
time.111These problems thus lie outside of the class EnumP which is commonly
studied in algorithmic enumeration; see e.g. [Str19]. In a recent contribution
[BBČ+21], Bérczi et al. address this problem. They give an incremental-
polynomial time algorithm for listing all the signatures of a CNF, and improve
their result to polynomial delay for tractable CNF’s. On the other hand, they
show that generating maximal signatures is hard, while minimal signatures can
be listed with polynomial delay, regardless of the CNF under consideration.
Their positive result relies on the equivalence with maximal independent sets
enumeration in graphs, known to be tractable [TIAS77]. As of their hardness
result, it relies on the fact that the existence of at least two solutions
would imply the non-satisfiability of the CNF at hand. Thus, the difficulty of
the enumeration heavily relies on the intractability of the satisfiability of
CNF’s in general. This observation naturally leads to the following open
question, where by tractable CNF’s we mean families of CNF’s for which the
satisfiability of a formula $\varphi$ in the family (as well as its sub-
formulas) can be decided in polynomial time.
###### Question 1.
Can the maximal signatures of tractable CNF’s be enumerated in total-
polynomial time?
Among tractable formulas, Horn and 2-CNF’s are two natural candidates for
approaching Question 1. These two cases however proved to be very challenging,
and the question of their tractability was explicitly stated as an open
problem in the 2022’s edition of the Workshop on Enumeration Problems and
Applications (WEPA’22).
Another natural step toward answering Question 1 is to consider XOR-CNF
formulas, being conjunctions of clauses using the “exclusive-or” connective
instead of the usual “or” connective. On top of being tractable for
satisfiability, XOR-CNF’s appear in many complexity classifications and enjoy
polynomial time algorithms for a broad variety of problems including counting
and enumerating their models (see e.g., [CH96, CH97, COS11]). Note that these
formulas do not fall in the framework of CNF’s though. Indeed, rewritting a
XOR-CNF as a CNF may result in a blowup on the number of clauses and
signatures, as discussed in Section 2. Hence, we remark that the positive
results from [BBČ+21] do not directly apply to XOR-CNF. Yet, we show that
signatures enumeration remains tractable using flashlight search as it is done
in [BBČ+21]. Namely, we prove the following.
###### Theorem 1.1.
The set of signatures of a XOR-CNF formula can be enumerated with polynomial
delay and polynomial space.
Concerning the enumeration of minimal signatures, we show that the problem is
in fact equivalent to the case of listing maximal signatures.
###### Proposition 1.2.
There is polynomial-delay and polynomial space algorithm listing all minimal
signatures of a XOR-CNF formula if and only if there is one listing all
maximal signatures of a XOR-CNF formula.
This leave us with the study of maximal signatures enumeration for XOR-CNF’s.
Quite surprisingly, we prove this problem to relate well with other problems
from graph theory and matroid theory. Specifically, we show the case of 2-XOR-
CNF to generalize maximal bipartite subgraphs enumeration, a problem which was
recently shown to admit a polynomial-delay algorithm [CGM+22]. As of the
general case of XOR-CNF’s, it can be seen as listing all maximal satisfiable
subsystems of a linear system of equations, which is related to the
enumeration of circuits passing through a given element in a matroid [BEGK03];
see also [KBE+05, KBB+09]. Relying on these results, we derive that maximal
signatures enumeration for XOR-CNF’s is tractable. Namely, we obtain the
following.
###### Theorem 1.3.
There is an incremental-polynomial time algorithm generating the maximal (or
minimal) signatures of a XOR-CNF.
Concerning the case of 2-XOR-CNF, we show it to be even more tractable, that
is, we show that it can be solved with polynomial delay using proximity
search.
###### Theorem 1.4.
There is a polynomial-delay algorithm generating the maximal (or minimal)
signatures of a 2-XOR-CNF.
To prove Theorem 1.4, we represent the input 2-XOR-CNF as an edge-bicolored
multigraph where each color codes the parity of the XOR clause. Then, the
enumeration amounts to list maximal bipartite subgraphs with additional
constraints on colored edges.
A caveat of Theorems 1.3 and 1.4 is that a queue of solutions has to be
maintained in the generation. More precisely, in [BEGK03] the queue is needed
to generate a new solution as the algorithm uses the saturation technique
where new solutions are derived from obtained ones. In [CGM+22], the queue has
the other purpose of preventing repetitions. In both cases, getting rid of a
potentially exponential space use is a natural enhancement one may seek. It
should however be noted that improving the algorithm from [BEGK03] to get
polynomial space is open for two decades now. As of improving Theorem 1.4 to
use polynomial space, it would imply such a result for maximal bipartite
subgraphs enumeration, another open question [CGM+22]. Toward such directions,
we prove that the folklore technique of flashlight search used to obtain
Theorem 1.1 may probably not be of great use as deciding whether a subsequence
of clauses can be extended into a signature is NP-complete.
### Organization of the paper.
In the next section we introduce the notions that will be used in this paper,
basic properties, and prove Theorem 1.1. The equivalence between maximal and
minimal signatures enumeration for XOR-CNF is detailed in Section 3. Theorems
1.3 and 1.4 are proved in Section 4. Future directions and the limitations of
flashlight search toward an improvement of Theorem 1.4 are finally discussed
in Section 5.
## 2 Preliminaries
All the objects considered in this paper are finite. If $V$ is a set,
$\mathbf{2}^{V}$ is its powerset. Let $m\in\mathbb{N}$ and let
$\sigma\in\\{0,1\\}^{m}$ be a binary sequence. We write $\sigma[i]$ to denote
the value of the $i^{\text{th}}$ element of $\sigma$. It will be furthermore
convenient to note $\mathbf{1}(\sigma)$ the set of indices
$I\subseteq\\{1,\dots,m\\}$ whose corresponding values in $\sigma$ is 1;
$\mathbf{0}(\sigma)$ is defined analogously. The _complementary_ of $\sigma$
is the binary sequence $\bar{\sigma}$ obtained from $\sigma$ by flipping all
the bits, i.e., $\bar{\sigma}[i]=1+\sigma[i]$ (mod $2$) for $1\leq i\leq m$.
If $\sigma,\tau\in\\{0,1\\}^{m}$, we write $\sigma\leq\tau$ if
$\sigma[i]\leq\tau[i]$ for all each $1\leq i\leq m$. In other words, $\leq$
corresponds to the bitwise order.
### Enumeration complexity.
An enumeration algorithm aims at enumerating a set of solutions of some
problem, one after the other, with no repetition. We already have defined
relevant time complexities in the introduction. However, space is also an
important consideration in the analysis of enumeration algorithms. We assume
that the solutions we produce are not stored in memory but rather flashed and
discarded. Thus, when measuring space we only consider the space needed by the
algorithm in order to conduct the enumeration, in terms of the input size. We
refer to [JYP88, Str19] for more details on the complexity of enumeration
algorithms.
### Graphs.
Given an undirected graph $G$, we write $V(G)$ its set of vertices and $E(G)$
its set of edges. For convenience, we will note $uv$ instead of $\\{u,v\\}$
for edges of $G$. A graph $G$ is bipartite if there exists a bipartition
$V_{1},V_{2}$ of $V(G)$ such that each edge of $E(G)$ has one endpoint in
$V_{1}$ and the other in $V_{2}$. Equivalently, a graph $G$ is bipartite if
and only if it does not contain any odd cycles. In this paper we will consider
_multigraphs_ as well, meaning graphs with potentially several edges sharing
two same endpoints, i.e., with $E(G)=\\{e_{1},\dots,e_{m}\\}$ a multiset. In
this case and to avoid any ambiguity we will refer to the edges of $G$ by
their label rather than their endpoints.
### XOR-formulas.
We assume basic familiarity with propositional logic. Let
$V=\\{x_{1},\dots,x_{n}\\}$ be a set of Boolean variables. A _literal_ is a
variable $x_{i}$ (positive literal) or its negation $\neg x_{i}$ (negative
literal). A _clause_ $C$ is a disjunction of literals. The size $|C|$ of $C$
is its number of literals. We say that $C$ is a $k$-clause if $|C|\leq k$. A
formula in conjunctive normal form (or a CNF for short) $\varphi$ is a
conjunction of clauses, i.e., $\varphi=C_{1}\land\dots\land C_{m}$ where
$C_{i}$, for $1\leq i\leq m$. It will be convenient to denote by $V(\varphi)$
the variables of $\varphi$. The (total) _size_ of $\varphi$ is defined as
$\sum_{1\leq i\leq m}|C_{i}|$. An assignment of the variables in $V$ is a
mapping $\mathbf{a}:V\to\\{0,1\\}$. Given an assignment $\mathbf{a}$, we write
$C_{i}(\mathbf{a})=1$ (resp. $C_{i}(\mathbf{a})=0$) if $C_{i}$ evaluates to
$1$ (resp. $0$) under assignment $\mathbf{a}$. The notations
$\varphi(\mathbf{a})=1$ and $\varphi(\mathbf{a})=0$ are defined analogously.
The formula $\varphi$ is _satisfiable_ if there exists an assignment
$\mathbf{a}$ of $V(\varphi)$ such that $\varphi(\mathbf{a})=1$; it is
unsatisfiable otherwise.
A _XOR-clause_ is a clause in which the usual connective “or” is replaced by
the “exclusive-or” connective. It is well-known that any XOR-clause can be
represented as a linear equation $x_{1}+\dots+x_{k}=\varepsilon$,
$\varepsilon\in\\{0,1\\}$ over the two-elements field $\mathbf{F}_{2}$. In
particular, the $2$-XOR-clause $x_{1}+x_{2}=0$ (resp. $x_{1}+x_{2}=1$) is
satisfied if and only if $x_{1}=x_{2}$ (resp. $x_{1}\neq x_{2}$). A XOR-clause
is _odd_ (resp. _even_) if $\varepsilon=1$ (resp. $0$). Given a XOR-clause
$C=(x_{1}+\dots+x_{k}=\varepsilon)$, with $\varepsilon\in\\{0,1\\}$, we put
$\bar{C}=(x_{1}+\dots+x_{k}=1-\varepsilon)$ and call $\smash{\bar{C}}$ the
_negation_ of $C$. In other words, the bar operator _changes the parity_ of
$C$. A XOR-CNF $\varphi$ is a conjunction of XOR-clauses. Equivalently,
$\varphi$ can be seen as a system of linear equations over $\mathbf{F}_{2}$.
For a XOR-CNF $\varphi=C_{1}\land\dots\land C_{m}$, we denote by
$\bar{\varphi}$ the XOR-CNF $\bar{C}_{1}\land\dots\land\bar{C}_{m}$ and call
it the _inverse_ of $\varphi$. We recall that the satisfiability of a XOR-CNF
can be tested in polynomial time; see e.g., [CH11, Theorem 2.18].
### Signatures.
The next notations and terminology are borrowed from [BBČ+21]. Let
$\varphi=C_{1}\land\dots\land C_{m}$ be a CNF. We refer to the introduction
for the definition of a signature, and shall say that an assignment
$\mathbf{a}$ of $\varphi$ _produces_ the signature
$\sigma_{\varphi}(\mathbf{a})=(C_{i}(\mathbf{a}),\ldots,C_{i}(\mathbf{a}))$.
We will further drop the subscript $\varphi$ from this notation when it is
clear from the context. Note that in particular, $\varphi$ is satisfiable if
and only if $(1,\dots,1)$ is a signature of $\varphi$. A signature $\sigma$ of
$\varphi$ is _minimal_ if it is minimal with respect to $\leq$ among all
signatures of $\varphi$. Similarly, we call _maximal_ a signature which is
maximal for $\leq$ among signatures of $\varphi$.
In this paper we are interested in the following problems.
Observe that in the above problems, and by the previous discussions, we may
indifferently consider $\varphi$ as a set of XOR clauses or as a set of binary
equations over $\mathbf{F}_{2}$. A subset of simultaneously satisfiable
clauses on one side corresponds to a feasible subset of equations on the other
side. In the rest of the paper, we will assume that $\varphi$ is in this
latter form. In particular, when $\varphi$ is seen as a system of linear
equations over $\mathbf{F}_{2}$, there is a one-to-one correspondence between
maximal signatures of $\varphi$ and maximal feasible subsets of equations.
Another remark is that any XOR-CNF may be rewritten as a CNF, i.e., the
$\oplus$ operator may be rewritten using $\land$ and $\lor$. For example, the
XOR-clause $x_{1}\neq x_{2}$ can be rewritten as $(x_{1}\vee
x_{2})\land(\bar{x_{1}}\vee\bar{x_{2}})$. We however note that we do not have
a bijection between the sets of signatures of the two formulas. Typically,
generalizing the above example to disjoint clauses will produce a XOR-CNF with
one minimal signature while the equivalent CNF has exponentially many such
signatures. Though this does not rule out the existence of a reduction between
(maximal) signatures of XOR-CNF’s to (maximal) signatures of CNF’s it should
be noted that the status of maximal signatures enumeration is still open for
2-CNF’s, a point that is discuss in Section 5. As of the case of all
signatures, it admits a simple and direct proof using flashlight search, as
explained in Section 3.
We finally argue that duplicate clauses in $\varphi$ may be ignored as far as
the enumeration of signatures is concerned. Indeed, having one of these
clauses to 1 implies that all other copies are to 1, and vice versa. Hence, we
will assume without loss of generality that all formulas are without
duplicated clauses in the rest of the paper, i.e., that they are defined as
pairwise distinct XOR-clauses.
## 3 Signatures of XOR-CNF’s and their properties
We prove that testing whether a binary sequence is a signature (resp. minimal
signature, maximal signature) of a XOR-CNF can be done in polynomial time in
the size of the CNF at hand, and use it to show that listing all signatures of
a XOR-CNF is tractable. For the rest of the section, let us fix some XOR-CNF
$\varphi=C_{1}\land\dots\land C_{m}$ on variable set
$V=\\{x_{1},\dots,x_{n}\\}$. A preliminary step is the following.
###### Proposition 3.1.
A sequence $\sigma\in\\{0,1\\}^{m}$ is a signature of $\varphi$ if and only if
$\overline{\sigma}$ is a signature of $\bar{\varphi}$. In particular, $\sigma$
is a minimal (resp. maximal) signature of $\varphi$ if and only if
$\overline{\sigma}$ is a maximal (resp. minimal) signature of $\bar{\varphi}$.
###### Proof.
Let $\mathbf{a}$ be an assignment producing the signature $\sigma$ of
$\varphi$. By definition we have $C_{i}(\mathbf{a})=1$ and
$C_{j}(\mathbf{a})=0$ for each $i,j\in\\{1,\dots,m\\}$ such that $\sigma[i]=1$
and $\sigma[j]=0$. This can be equivalently written as
$\smash{\bar{C}_{i}(\mathbf{a})=0}$ if $\overline{\sigma}[i]=0$ and
$\smash{\bar{C}_{i}(\mathbf{a})=1}$ when $\overline{\sigma}[i]=1$. This is in
turn equivalent to $\mathbf{a}$ producing the signature $\overline{\sigma}$ of
$\bar{\varphi}$. The second part of the statement follows from the fact that
for every sequences $\sigma,\tau\in\\{0,1\\}^{m}$ we have $\sigma\leq\tau$ if
and only if $\overline{\tau}\leq\overline{\sigma}$. ∎
From Proposition 3.1 we already derive Proposition 1.2.
Given two disjoint subsets $A,B\subseteq\\{1,\dots,m\\}$, we define
$\varphi(A,B):=\left(\bigwedge_{i\in A}C_{i}\right)\land\left(\bigwedge_{j\in
B}\bar{C}_{j}\right)$
to be the formula obtained from $\varphi$ by taking the clauses with index in
$A$ and the negation of clauses with index in $B$. We furthermore set this
formula to be defined on variable set $V(\varphi)$ even though some variables
in $V(\varphi)$ may not appear in any of the clauses of $\varphi(A,B)$—we want
this to ensure that assignments of $\varphi(A,B)$ are well-defined for
$\varphi$. Remark that the size of $\varphi(A,B)$ is at most that of
$\varphi$, and that it is defined on precisely $m$ clauses whenever $A,B$ is a
bipartition of $\\{1,\dots,m\\}$. In the next proposition, we show that
testing whether a binary sequence is a signature of a XOR-CNF can be reduced
to satisfiability testing.
###### Proposition 3.2.
A sequence $\sigma\in\\{0,1\\}^{m}$ is a signature of $\varphi$ if and only if
the formula $\varphi(\mathbf{1}(\sigma),\mathbf{0}(\sigma))$ is satisfiable.
###### Proof.
We start with the only if part. Assume that $\sigma$ is a signature of
$\varphi$. Then there is an assignment $\mathbf{a}$ that produces $\sigma$. By
definition $\mathbf{a}$ satisfies the clauses $C_{i}$ such that $\sigma[i]=1$,
and it does not satisfy the clauses $C_{j}$ such that $\sigma[j]=0$. This
latter case is equivalent to $\mathbf{a}$ satisfying the clauses
$\smash{\bar{C}_{j}}$. Hence $\mathbf{a}$ is an assignment which satisfies the
formula $\varphi(\mathbf{1}(\sigma),\mathbf{0}(\sigma))$ as desired.
We move to the if part. Assume that
$\varphi(\mathbf{1}(\sigma),\mathbf{0}(\sigma))$ has a satisfying assignment
$\mathbf{a}$ defined over $V$. Now since $\mathbf{a}$ satisfies the clauses
$C_{i}$ of $\varphi$ such that $\sigma[i]=1$, and not the clauses $C_{j}$ of
$\varphi$ such that $\sigma[j]=0$, we conclude that $\sigma$ is indeed a
signature of $\varphi$ as required. ∎
###### Proposition 3.3.
Deciding whether $\sigma$ is a minimal (resp. maximal) signature of $\varphi$
can be done in polynomial time in the size of $\varphi$.
###### Proof.
To check that $\sigma$ is a signature, we rely on Proposition 3.2: we build
the XOR-CNF $\varphi(\mathbf{1}(\sigma),\mathbf{0}(\sigma))$ and test its
satisfiability, all of which in polynomial time.
We prove the if part. Suppose that $\psi(j)$ is satisfiable for some
$j\in\mathbf{0}(\sigma)$, and consider an arbitrary assignment $\mathbf{a}$
over $V$ satisfying $\psi(j)$. Let $\tau$ be the signature of $\varphi$
produced by assignment $\mathbf{a}$. Since $\mathbf{a}$ satisfies $\psi(j)$,
every $1$ in $\sigma$ is a $1$ in $\tau$, so that $\sigma\leq\tau$. Moreover,
$\sigma[j]<\tau[j]$ holds, and $\sigma<\tau$ thus follows. Hence $\sigma$ is
not a maximal signature concluding the if implication. As for the only if
part, the existence of a signature $\tau>\sigma$ implies by definition that
$\psi(j)$ is satisfiable for some $j\in\mathbf{0}(\sigma)$ such that
$\tau[j]>\sigma[j]$.
Henceforth, deciding whether $\sigma$ is a maximal signature of $\varphi$ can
be done by $m$ calls to the satisfiability of a XOR-CNF whose size is no
greater than $\varphi$. We conclude that it can be done in polynomial time.
For minimality, we conclude using Proposition 3.1. ∎
To conclude this section, we argue that enumerating all the signatures of a
XOR-CNF is tractable using the above properties and flashlight search. We
recall that the main idea of this technique is to conduct binary search over
the space of solutions. In order for flashlight search to run with polynomial
delay and polynomial space, it is sufficient to solve the so-called “extension
problem” in polynomial time: given two disjoint subsets $A,B$ of the
groundset, decide whether there exists a solution including $A$ and avoiding
$B$. We refer to e.g. [BEG04, MS19] for a more detailed description of this
folklore technique.
In the case of XOR-CNF signatures, $A$ will consist of indices of clauses to
be satisfied in the signature—the 1’s in the sequence—while $B$ will consist
of indices of clauses not to be satisfied—the 0’s in the sequence. Then, it
follows from the discussion above that $\varphi$ admits a signature $\sigma$
satisfying $\sigma[i]=1$ and $\sigma[j]=0$ for all $i\in A$ and $j\in B$ if
and only if $\varphi(A,B)$ is satisfiable. Indeed, a satisfying assignment
$\mathbf{a}$ of $\varphi(A,B)$ satisfies the clauses of $\varphi$ with index
in $A$, and not those with index in $B$. Thus the signature of $\varphi$
produced by $\mathbf{a}$ meets the requirements. As for the converse
direction, it follows directly from the definition of a signature. We conclude
by Proposition 3.3 that the extension problem can be solved in polynomial time
for XOR-CNF signatures, hence proving Theorem 1.1 that we recall here.
See 1.1
We point that Theorem 1.1 is not a direct consequence of [BBČ+21, Theorem 1].
Indeed, in the later theorem the authors consider families of tractable CNF’s
which do not contain XOR-CNF’s in general. The arguments, however, follow the
same line.
## 4 Maximal signatures of XOR-CNF’s
We investigate the complexity of enumerating the maximal signatures of a given
XOR-CNF. More precisely, we prove the general case to be solvable in
incremental-polynomial time by a result of Boros et al. [BEGK03], and the
2-XOR-CNF case to be solvable with polynomial delay using proximity search.
In Section 2 we have mentioned that the clauses of a given XOR-CNF $\varphi$
may be seen as linear equations $x_{1}+\dots+x_{k}=\varepsilon$,
$\varepsilon\in\\{0,1\\}$ over the two-elements field $\mathbf{F}_{2}$. Hence
$\varphi$ can be seen as a system of linear equations whose maximal feasible
subsystems correspond to its maximal signatures. In [BEGK03], the authors
present an incremental polynomial-time algorithm for enumerating all circuits
of a matroid containing an element, and more generally, one for enumerating
all maximal subsets not spanning an element. Using Farkas’ lemma (more
precisely the Fredholm alternative) it can be derived an algorithm which,
given an infeasible system of linear equations, enumerates all its maximal
feasible (or minimal infeasible) subsystems within the same time; see also
[KBE+05, KBB+09] on the use of this characterization of infeasible systems.
This immediately yields Theorem 1.3 that we restate here.
See 1.3
We now focus on 2-XOR-CNF’s. Recall from Section 2 that we can consider these
formulas as a set of 2-clauses of the form $x=y$ or $x\neq y$ for positive
literals $x,y$ only, together with a set of 1-clauses $(z)$ with $z$ a
positive or negative literal. Furthermore we may assume that no clause is
repeated in the formula, as far as signatures enumeration is concerned. We
shall call _graphic_ a 2-XOR-CNF with clauses of the first type only—that is,
without 1-clauses—and will assume for now on that our XOR-CNF’s are graphic.
Later, we will show how to deal with 1-clauses with small modifications on the
instance.
We start by defining an underlying multigraph structure of graphic 2-XOR-CNF
that will be convenient for applying the framework of proximity search. Given
such a XOR-CNF $\varphi$ we define $G(\varphi)$ (or simply $G$ when it is
clear from the context) as the edge-bicolored multigraph defined on vertex set
$V(\varphi)$ and where there is a blue edge $xy$ whenever $x\neq y$ is a
clause of $\varphi$, and a red edge $xy$ whenever $x=y$ is a clause of
$\varphi$. A same pair of variables $x,y$ can produce both a red and a blue
edge: this corresponds to having clauses $x=y$ and $x\neq y$ in $\varphi$
which typically makes it unsatisfiable. However as no clause is repeated in
$\varphi$, no two edges of a same color share the same endpoints. In other
words, our multigraph has edge multiplicity at most two, and two edges between
a same pair of endpoints means that one is blue and the other is red; see
Figure 1 for an illustration of such multigraphs. In the following, we will
label edges of $G$ as $e_{1},\dots,e_{m}$ so that the edge $e_{j}$ corresponds
to the clause $C_{j}$ in $\varphi$. We note $R(G)$ the set of red edges of
$G$, and $B(G)$ the set of its blue edges. By definition, we have $R(G)\cap
B(G)=\emptyset$ and $R(G)\cup B(G)=E(G)$. A naive but crucial property we get
from that partition is that the set of red edges characterizes the set of blue
edge, and vice versa. By subgraph $H$ of an edge-bicolored multigraph $G$ we
mean the graph on same vertex set $V(H)=V(G)$ and $E(H)\subseteq E(G)$, that
is, we consider subgraphs as sets of edges. Finally, given two disjoint
subsets $X,Y\subseteq V(\varphi)$, by $\delta(X,Y)$ we mean the set of edges
having an endpoint in $X$ and the other in $Y$.
Figure 1: An edge-bicolored multigraph $G$ on nine vertices, with red edges
denoted by thin red lines and blue edges denoted by bold blue lines.
Note that an arbitrary (simple, uncolored) graph $H$ is bipartite whenever it
admits a bipartition $(X,Y)$ of its vertex set with $E(H)=\delta(X,Y)$. In the
next lemma, we relate the satisfiability of a graphic 2-XOR-CNF to a property
of its underlying edge-bicolored multigraph which generalizes bipartiteness.
We stress the fact that this property may only be stated on blue edges, which
is still reasonable as blue and red edges partition the edges of the graph.
###### Lemma 4.1.
Let $\varphi$ be a graphic 2-XOR-CNF. Then $\varphi$ is satisfiable if and
only if there is a bipartition $(X,Y)$ of the vertex set of $G(\varphi)$ such
that $B(G)=\delta(X,Y)$. Moreover, every such bipartition corresponds to a
model $\mathbf{a}$ of $\varphi$ with $X=\mathbf{a}^{-1}(1)$ and
$Y=\mathbf{a}^{-1}(0)$.
###### Proof.
We prove the first implication. If $\varphi$ is satisfiable then it has a
satisfying assignment $\mathbf{a}$ defining a bipartition $(X,Y)$ of
$V(\varphi)$ where $X=\mathbf{a}^{-1}(1)$ and $Y=\mathbf{a}^{-1}(0)$. Clearly
$B(G)\subseteq\delta(X,Y)$ as otherwise we have two variables $x,y$ with
$x\neq y$ yet $\mathbf{a}(x)=\mathbf{a}(y)$. On the converse, if there is a
red edge in $\delta(X,Y)$ we deduce the existence of two variables $x,y$ with
$x=y$ yet $\mathbf{a}(x)\neq\mathbf{a}(y)$. Hence, $\delta(X,Y)\subseteq
E(G)\setminus R(G)=B(G)$ and $B(G)=\delta(X,Y)$ follows. This concludes the
first implication.
We prove the other implication. Let $(X,Y)$ be a bipartition of $V(G)$ such
that $B(G)=\delta(X,Y)$. Consider the assignment $\mathbf{a}$ defined by
$\mathbf{a}(x)=1$ and $\mathbf{a}(y)=0$ for all $x\in X$ and $y\in Y$. By
construction to each clause of $\varphi$ of the form $x\neq y$ corresponds a
blue edge lying in $\delta(X,Y)$ and hence such that
$\mathbf{a}(x)\neq\mathbf{a}(y)$. As of the clauses of the form $x=y$ they
correspond to red edges not in $\delta(X,Y)$ and hence such that
$\mathbf{a}(x)=\mathbf{a}(y)$. All these clauses are satisfied by $\mathbf{a}$
and we conclude that $\varphi$ is indeed satisfiable. ∎
###### Lemma 4.2.
Let $\varphi$ be a graphic 2-XOR-CNF. Then the maximal signatures of $\varphi$
are exactly the maximal subgraphs $H$ of $G(\varphi)$ admitting a bipartition
$(X,Y)$ of their vertex set such that $\delta(X,Y)=B(H)$.
###### Proof.
Let $\sigma$ be a maximal signature of $\varphi$ and fix $H:=G(\psi)$ for
$\psi:=\varphi(\mathbf{1}(\sigma),\emptyset)$—recall that $\psi$ is defined on
the full variable set $V(\varphi)$ while some of its variables may not appear
in any clause. Since $\psi$ is satisfiable by definition of a signature, and
using Lemma 4.1, we derive that $B(H)=\delta(X,Y)$ for a bipartition $(X,Y)$
of $V(G)$ corresponding to a model $\mathbf{a}$ of $\psi$. Suppose that $H$ is
not maximal with this property. Then there exist an edge $e_{j}\in
E(G)\setminus E(H)$, $1\leq j\leq m$ and a bipartition
$(X^{\prime},Y^{\prime})$ with $H^{\prime}:=H+e_{j}$ satisfying
$B(H^{\prime})=\delta(X^{\prime},Y^{\prime})$. Then by Lemma 4.1 we derive
that $\psi:=\varphi(\mathbf{1}(\sigma)\cup\\{j\\},\emptyset)$ is satisfiable.
Let $\mathbf{a}$ be an assignment satisfying $\psi$. Since $j$ belongs to
$\mathbf{0}(\sigma)$, we obtain that $\mathbf{a}$ produces a signature $\tau$
with $\tau>\sigma$, a contradiction.
Let $H$ be a maximal subgraph of $G$ admitting a bipartition $(X,Y)$ of its
vertex set such that $B(G)=\delta(X,Y)$. By Lemma 4.1 the formula
$\psi:=\varphi(\\{j:e_{j}\in E(H)\\},\emptyset)$ corresponding to the edges of
$H$ is satisfiable. Let $\mathbf{a}$ be a satisfying assignment of $\psi$ and
consider the signature $\sigma$ it produces for $\varphi$. We argue that it is
a maximal signature of $\varphi$. Suppose that this is not the case and let
$\tau$ be a signature of $\varphi$ such that $\tau>\sigma$. Then
$\psi^{\prime}:=\varphi(\mathbf{1}(\sigma)\cup\\{j\\},\emptyset)$ is
satisfiable for some $1\leq j\leq m$ with $\sigma[j]<\tau[j]$. Hence by Lemma
4.1, the graph $H^{\prime}:=G(\psi^{\prime})$ satisfies
$B(H^{\prime})=\delta(X^{\prime},Y^{\prime})$. Moreover $E(H)\subset
E(H^{\prime})$ which contradicts the fact that $H$ is chosen maximal. We
conclude that $\sigma$ is a maximal signature. ∎
In the following, given an edge-bicolored multigraph, we call _red-blue
bipartite_ any of its subgraph with the property described in Lemma 4.2. In
Figure 2, we give an example of a maximal red-blue bipartite subgraph $H$ of
the multigraph $G$ of Figure 1. Note that red-blue bipartite subgraphs do not
contain both a red edge and a blue edge between a same pair of vertices, i.e.,
they define edge-bicolored graphs.
Figure 2: A maximal (connected) red-blue bipartite subgraph $H$ of the
multigraph $G$ defined in Figure 1. The bipartition $(H_{0},H_{1})$ of $H$ is
highlighted. The edges of $G$ not in $H$ are dotted.
###### Theorem 4.3.
There is a polynomial-delay and polynomial-space algorithm enumerating the
maximal signatures of a graphic 2-XOR-CNF $\varphi$ if and only if there is
one listing the maximal red-blue bipartite subgraphs of its underlying
multigraph $G(\varphi)$.
###### Proof.
This is a consequence of Lemma 4.2 observing that constructing either the
formula or the edge-bicolored multigraph can be done in polynomial time in the
size of the structure at hand. As there is a bijection between the two
solutions sets this reduction preserves polynomial delay and polynomial space.
∎
A consequence of Lemma 4.2 is that the maximal signatures of a 2-XOR-CNF only
consisting of disequalities $x\neq y$ is equivalent to the enumeration of
maximal (edge) bipartite subgraphs of a given graph. Indeed in that case,
$R(G)$ is empty—hence it is a graph—and we only require $E(G)=\delta(X,Y)$. We
derive the following using a recent result by Conte and Uno [CGM+22] on
maximal bipartite subgraphs enumeration using proximity search.
###### Corollary 4.4.
There is a polynomial-delay algorithm listing the maximal signatures of a
2-XOR-CNF consisting of disequalities $x\neq y$ only.
We end the section showing that the result from [CGM+22] can be extended into
our context using proximity search, and arguing how to capture clauses of size
one.
In the following, we say that an edge-bicolored multigraph $G$ is _connected_
if its underlying graph (where colors are ignored and multi-edges are
considered as a single edge) is connected. Note that we can assume without
loss of generality that $G(\varphi)$ is connected, as far as signatures
enumeration is concerned. Indeed, if it was not the case we could enumerate
the signature of each subformula of $\varphi$ corresponding to connected
components of $G(\varphi)$ and combine the results by doing their Cartesian
product. Hence, we may further assume that $G(\varphi)$ is connected, and
shall call _connected_ a formula such that its underlying multigraph is. We
first show that if $G$ is connected then its maximal red-blue bipartite
subgraphs are as well, yielding a canonical bipartition.
###### Lemma 4.5.
Let $G$ be a connected edge-bicolored multigraph. Then any of its maximal red-
blue bipartite subgraph is connected.
###### Proof.
Let $H$ be an arbitrary maximal red-blue bipartite subgraph of $G$ and suppose
toward a contradiction that it is not connected. Since $G$ is connected we can
find a shortest path $P$ connecting two distinct and connected components
$C_{1},C_{2}$ of $H$. For convenience, we consider two such components
minimizing the length of $P$ so that adding $P$ only connects $C_{1},C_{2}$.
We obtain a red-blue bipartite supergraph of $H$ by the following procedure:
we start from the endpoint of $P$ in $C_{1}$ and for each of the edges
$e_{1},\dots,e_{k}$ of $P$ in order, we either keep its other endpoint in the
same part if $e_{i}$ is red, or put its endpoint in the other part if $e_{i}$
is blue. When reaching $e_{k}$, we may need to swap the original bipartition
of $C_{2}$. No contradiction can be found during this process and the obtained
graph is indeed a supergraph of $H$. This contradicts the fact that $H$ is
maximal and we conclude that $H$ is connected as desired. ∎
###### Lemma 4.6.
If $G$ is a connected edge-bicolored bipartite graph then it admits a unique
bipartition $(X,Y)$ of its vertex set with $B(G)=\delta(X,Y)$ and $X$
containing the vertex of smallest label in $G$.
###### Proof.
We prove the statement using induction on the number of vertices of $G$. If
$G$ has 0, 1, or 2 vertices, the result is clear. Now, assume that the
statement holds for every graph with at most $k$ vertices. Let $G$ be a
connected edge-bicolored bipartite graph with vertices
$V(G)=\\{v_{1},\dots,v_{k+1}\\}$. We consider the edge-colored bipartite
induced subgraph $H:=G[\\{v_{1},\dots,v_{k}\\}]$ with $v_{k+1}$ chosen so that
$H$ is connected. Note that this can be done by considering a BFS of $G$
launched at its vertex $v_{1}$ of smallest label, and choosing $v_{k+1}$ as
the vertex obtained last in the traversal. By inductive hypothesis, there
exists a unique bipartition $(X^{\prime},Y^{\prime})$ of $V(H)$ such that
$\delta(X^{\prime},Y^{\prime})=B(H)$ and $v_{1}\in X^{\prime}$. Since $G$ is
bipartite, there exists at least one bipartition $(X,Y)$ of $V(G)$ such that
$\delta(X,Y)=B(G)$. Moreover, every such bipartition must comply with
$(X^{\prime},Y^{\prime})$, that is, without loss of generality,
$X^{\prime}\subseteq X$ and $Y^{\prime}\subseteq Y$. Henceforth, $G$ has at
most two possible valid bipartitions:
$(X^{\prime},Y^{\prime}\cup\\{v_{k+1}\\})$ and
$(X^{\prime}\cup\\{v_{k+1}\\},Y^{\prime})$. Now, $G$ is connected, so there
exists a vertex $v_{i}$ in $V(G)\setminus\\{v_{k+1}\\}$, say in $X^{\prime}$,
such that $v_{i}v_{k+1}$ is an edge of $G$. We have two disjoint cases:
* •
$v_{i}v_{k+1}$ is red in which case $(X^{\prime}\cup\\{v_{k+1}\\},Y^{\prime})$
is the unique correct bipartition; and
* •
$v_{i}v_{k+1}$ is blue in which case
$(X^{\prime},Y^{\prime}\cup\\{v_{k+1}\\})$ is the unique correct bipartition.
We deduce using induction that there exists exactly one bipartition $(X,Y)$ of
$G$ satisfying $\delta(X,Y)=B(G)$ with $X$ containing the vertex of smallest
label in $G$. ∎
Proximity search has been introduced in [CU19] as a special case of solution
graph traversal in which the proof that the solution graph is strongly
connected relies on a notion of proximity between solutions that is more
involved than their intersection. More precisely, the proximity between two
solutions $S,S^{*}$ is defined as the length of the longest prefix of $S^{*}$
(according to a carefully chosen ordering of its elements) that is subset of
$S$, and need not to be symmetric. We refer to [CGM+22] for more details on
the technique. In [CGM+22], the authors extract the necessary conditions that
make an enumeration problem “proximity searchable” and thus to be solvable
with polynomial delay. We restate these conditions here in the context of XOR-
CNF for self-containment and better readability.
In the reformulation below, let $G$ be an edge-bicolored multigraph and
$\mathcal{H}(G)$ denote the set of maximal (connected) red-blue bipartite
subgraphs of $G$ we shall call _maximal solutions_. We further call
_solutions_ the connected red-blue bipartite subgraphs of $G$.
###### Theorem 4.7 (Reformulation of [CGM+22, Definition 4.2 and Theorem
4.3]).
For arbitrary $H,H^{*}\in\mathcal{H}(G)$, let $\mu(H^{*})$ denote an ordering
of the edges in $H^{*}$, and $H\,\tilde{\cap}\,H^{*}$ denote the elements in
the longest prefix of $\mu(H^{*})$ that is a subset of $H$. Then, there is a
polynomial-delay algorithm enumerating $\mathcal{H}(G)$ whenever:
1. 1.
For $H\in\mathcal{H}(G)$ any prefix of $\mu(H)$ is a solution;
2. 2.
There is a polynomial-time computable function $\operatorname{{COMP}}$, which
given a solution $H^{\prime}$ produces $H=\operatorname{{COMP}}(H^{\prime})$ a
maximal solution such that $H\supseteq H^{\prime}$;
3. 3.
Given $H\in\mathcal{H}(G)$ and $e\in E(G)\setminus E(H)$ there is a family
$\mathcal{K}\subseteq 2^{E(H)}$ of sets called removables such that:
* •
the family $\mathcal{K}$ can be computed in polynomial time;
* •
for any $K\in\mathcal{K}$ the set $H\setminus K\cup\\{e\\}$ is a solution; and
* •
for any $H^{*}\in\mathcal{H}(G)$ and minimal element $e$ in $\mu(H^{*})$ that
is not in $H$, there exists $K\in\mathcal{K}$ with
$(H\,\tilde{\cap}\,H^{*})\cap K=\emptyset$.
The rest of the section is dedicated to proving that the conditions of Theorem
4.7 are fulfilled for well-chosen $\mu$ and set $\mathcal{K}$. The fact that
$\operatorname{{COMP}}$ can be computed in polynomial time follows from the
fact that the red-blue bipartiteness is a hereditary property which can be
tested in polynomial time, hence that we can obtain a maximal solution by a
greedy procedure.
We start with the definition of $\mu$. Given a maximal red-blue bipartite
graph $H$ of $G$ we denote by $(H_{0},H_{1})$ the unique bipartition given by
Lemma 4.6 where $H_{0}$ is the side containing the vertex $v_{0}$ of smallest
index in $G$. Consider a BFS ordering of the vertices of $G$ starting at
$v_{0}$: $v_{0}$ is first in the order and at each step, given the last vertex
in the ordering, we add its neighborhood in ascending order of their label.
Clearly, each prefix of such an ordering defines a connected induced subgraph.
We define $\mu(H)$ as the increasing ordering of the edges of $H$ with respect
to their endpoint occurring later in the BFS ordering launched at $v_{0}$, and
break the tie by increasing order of their earlier endpoint in the BFS; see
Figure 3 for an example of such an edge-ordering. This choice of an edge-
ordering of a maximal solution yields the following property which will be
crucial in the rest of the proof.
Figure 3: A BFS on the maximal red-blue bipartite subgraph $H$ of $G$ (Figure
2) will give the following order on the vertices:
$v_{0},v_{1},v_{3},v_{4},v_{2},v_{5},v_{8},v_{7},v_{6}$. The order of the
edges of $H$ it produces is
$\mu(H)=v_{0}v_{1},v_{0}v_{3},v_{0}v_{4},v_{3}v_{4},v_{1}v_{2},v_{1}v_{5},v_{2}v_{5},v_{1}v_{8},v_{3}v_{7},v_{8}v_{6},v_{7}v_{6}$.
###### Observation 4.8.
If $H\in\mathcal{H}(G)$ then any prefix of $\mu(H)$ is a connected red-blue
bipartite subgraph of $G$ and the first edge in this ordering is incident to
$v_{0}$.
Let us now consider arbitrary $H,H^{*}\in\mathcal{H}(G)$ and put
$H^{\prime}:=H\,\tilde{\cap}\,H^{*}$ as defined in Theorem 4.7. Note that if
$H^{\prime}$ is empty then the trivial removable $K=E(G)$ satisfies the above
conditions. Hence we may assume for convenience (in the rest of the analysis)
that $|H\,\tilde{\cap}\,H^{*}|\neq 0$, while it will appear clear later that
this assumption is in fact not needed. By Observation 4.8 since the first edge
in $\mu(H^{*})$ belongs to both $H^{\prime}$, $H$ and $H^{*}$ we derive that
$v_{0}$ belongs to all three graphs. Again by Observation 4.8, $H^{\prime}$ is
a connected red-blue bipartite subgraph of $G$. Thus by Lemma 4.6 it admits a
unique bipartition $(H^{\prime}_{0},H^{\prime}_{1})$ with $v_{0}\in
H^{\prime}_{0}$. Now since $H$ and $H^{*}$ contain $H^{\prime}$ they must
agree with that bipartition, meaning that $H^{\prime}_{0}\subseteq H_{0}\cap
H^{*}_{0}$ and $H^{\prime}_{1}\subseteq H_{1}\cap H^{*}_{1}$, which is a
crucial point. In the following, for each edge $e=ab$ in $E(G)\setminus E(H)$,
we define $K_{1}:=\\{av:av\in E(G)\\}$ and $K_{2}:=\\{bv:bv\in E(G)\\}$ to be
the two removables of $(H,e)$ and argue that $\mathcal{K}:=\\{K_{1},K_{2}\\}$
meets the requirements of the theorem.
Consider the first edge $e=ab$ in $\mu(H^{*})$ that is not an edge of $H$. We
distinguish two symmetric cases on whether $e$ is red or blue, and only detail
the blue case here. This situation is depicted in Figure 4.
We assume without loss of generality that the endpoint $a$ belongs to
$H^{*}_{0}$ and that the other endpoint $b$ lies in $H^{*}_{1}$. Recall that
$e\not\in H$. Hence since $e$ is blue, either the two endpoints $a,b$ belong
to $H_{0}$, or they both belong to $H_{1}$. Two symmetric cases arise. Let us
assume $\\{a,b\\}\subseteq H_{0}$. Note that in $H^{*}$ we have no blue edge
between $b$ and $H^{\prime}_{1}$, and no red edge between $b$ and
$H^{\prime}_{0}$. In $H$ this is the opposite: we have no blue edge between
$b$ and $H^{\prime}_{0}$, and no red edge between $b$ and $H^{\prime}_{1}$.
Now as $E(H^{\prime})\subseteq E(H^{*})\cap E(H)$ we conclude that $b$ is not
incident to blue or red edges in $H^{\prime}$, that is, $E(H^{\prime})\cap
K_{2}=\emptyset$. This concludes the case and the other situation of
$\\{a,b\\}\subseteq H_{1}$ yields the symmetric situation of
$E(H^{\prime})\cap K_{1}=\emptyset$ by the same arguments. The other situation
of $e$ being red follows the exact same line of arguments, swapping red for
blue.
Figure 4: The situation occurring when considering two solutions $H,H^{*}$ in
Theorem 4.7 with $H^{\prime}=H\,\tilde{\cap}\,H^{*}$: $H^{\prime}_{0}\subseteq
H_{0}\cap H^{*}_{0}$, $H^{\prime}_{1}\subseteq H_{1}\cap H^{*}_{1}$ and the
blue edge $ab$ of $H^{*}$ satisfies $E(H^{\prime})\cap K_{2}=\emptyset$ where
$K_{2}:=\\{bv:bv\in E(G)\\}$.
We finish the proof observing that the graphic conditions of 2-XOR-CNF’s can
be relaxed. As a preliminary step, note that 1-clauses containing variables
that do not appear in 2-clauses can be removed from the instance as they will
be set to true in any maximal signature. Thus we may assume that $\varphi$
does not contain such isolated 1-clauses. We code every other 1-clauses $(x)$
and $(\overline{y})$—which can also be seen as clauses $x=1$ and $y=0$—by a
blue edge $xu$ and a red edge $yu$ in the edge-bicolored multigraph $G$ we
have defined above, for some special vertex $u$ that will be connecting all
such 1-clauses. Note that $G$ stays connected by the above assumption that
$\varphi$ has no isolated 1-clause. Then we set $u$ to be the vertex of
smallest label in $G$, followed by $v_{0}$. Then by the definition of $\mu(H)$
the vertex $u$ will be placed in $H_{0}$ which can be seen as forcing the
gadget to false, satisfying the clause $y=0$ if the red edge $uy$ is selected,
and satisfying the clause $x=1$ if the blue edge $ux$ is selected. We conclude
with Theorem 1.4 that we restate here as a consequence of Theorem 4.7 and the
above discussion.
See 1.4
## 5 Discussions
In this work we have showed that enumerating all (resp. minimal, maximal)
signatures of a XOR-CNF formula is tractable, namely, Theorems 1.1, 1.3 and
1.4. Observe that the algorithm of Theorem 1.3 runs in incremental-polynomial
time while the other two run with polynomial delay. Hence it is natural to ask
whether Theorem 1.3 can be improved to polynomial delay.
###### Question 2.
Can the maximal signatures of a XOR-CNF be generated with polynomial delay?
The algorithm by Boros et al. [BEGK03] underlying Theorem 1.3 relies on the
enumeration of the circuits of a matroid, for which the same question seems
open since two decades. Hence, answering Question 2 would either require new
techniques, or would need to answer the open question from [BEGK03].
Furthermore, except for Theorems 1.1 and 1.3 that run with polynomial space,
the algorithm of Theorem 1.4 requires a space that is potentially exponential
as solutions must be stored into a queue. A natural question is the following.
###### Question 3.
Can the maximal signatures of a 2-XOR-CNF be generated with polynomial delay
and polynomial space?
We however stress the fact that a positive answer to Question 3 would improve
the algorithm by Conte et al. for maximal bipartite subgraphs enumeration for
which the questions of achieving polynomial delay and space is open [CGM+22].
Toward this direction, we now argue that flashlight search may not be adapted
in order to give a positive answer to Question 3, as the extension problem is
hard in that case. Moreover, this result even holds when restricted to 2-XOR-
CNF’s consisting of disequalities, that is, in the context of maximal
bipartite subgraph enumeration, which may be of independent interest.
###### Theorem 5.1.
The problem of deciding, given a graph $G$ and two disjoint edge sets
$A,B\subseteq E(G)$, whether there exists a maximal bipartite (edge) subgraph
$H$ of $G$ such that $A\subseteq E(H)$ and $B\cap E(H)=\emptyset$ is ${\sf
NP}$-complete.
###### Proof.
We first argue that the problem belongs to NP. Note that bipartiteness is
hereditary, meaning that every subgraph of a bipartite graph is bipartite.
Moreover since bipartiteness can be decided in polynomial time, we can check
maximality of $H$ by trying to add every edge in $E(G)\setminus E(H)$ and
checking whether the resulting graph is bipartite. Now since the two extra
conditions $A\subseteq E(H)$ and $B\cap E(H)=\emptyset$ can be checked in
polynomial time, we conclude that $H$ itself is a polynomial certificate.
We now prove NP-hardness by reduction from 3-SAT. Let
$\varphi=C_{1}\land\dots\land C_{m}$ be an instance of 3-SAT on variable set
$V(\varphi)=\\{v_{1},\dots,v_{n}\\}$. We construct $G$ by creating two
vertices $x_{i},y_{i}$ representing the literals of variable $v_{i}$ for every
$1\leq i\leq n$, a special vertex $u$, and a vertex $c_{j}$ representing the
clause $C_{j}$ for every $1\leq j\leq m$. We add the edges $x_{i}y_{i}$ for
every $1\leq i\leq n$, the edge $x_{i}c_{j}$ if $v_{i}$ appears positively in
$C_{j}$, and the edge $y_{i}c_{j}$ if it appears negated. Finally, we connect
$u$ to every other vertex in $G$ making it a universal vertex. This completes
the construction of the graph. Note that it is not bipartite as it contains
triangles $x_{i}y_{i}u$ for $1\leq i\leq n$. The two sets $A,B\subseteq E(G)$
are defined as $A:=\\{x_{i}y_{i}:1\leq i\leq n\\}$ and $B:=\\{uc_{j}:1\leq
j\leq m\\}$. We illustrate the reduction in Figure 5.
Figure 5: The reduction of Theorem 5.1 with the CNF $\varphi=(v_{1}\lor
v_{2}\lor v_{3})\land(\bar{v_{1}}\lor v_{2}\lor v_{3})\land(\bar{v_{1}}\lor
v_{2}\lor\bar{v_{3}})$. We have $A=\\{x_{1}y_{1},x_{2}y_{2},x_{3}y_{3}\\}$ and
$B=\\{c_{1}u,c_{2}u,c_{3}u\\}$. A maximal bipartite subgraph including $A$ and
avoiding $B$ is highlighted in grey. It corresponds to the model of $\varphi$
which assigns $1$ to $v_{1}$ and $v_{2}$, and $0$ to $v_{3}$.
Let us first show that there exists a maximal bipartite subgraph of $G$
containing $A$ and avoiding $B$ whenever $\varphi$ is satisfiable. Consider a
model $\mathbf{a}$ of $\varphi$ and consider the subgraph $H$ of $G$ defined
as
$\displaystyle E(H):=A$
$\displaystyle\cup\\{ux_{i}:\mathbf{a}(v_{i})=1\\}\cup\\{uy_{i}:\mathbf{a}(v_{i})=0\\}$
$\displaystyle\cup\\{c_{j}x_{i}:\mathbf{a}(v_{i})=1\ \text{and}\ v_{i}\in
C_{j}\\}$ $\displaystyle\cup\\{c_{j}y_{i}\hskip 0.85355pt:\mathbf{a}(v_{i})=0\
\text{and}\ \overline{v}_{i}\in C_{j}\\}.$
Clearly $H$ extends subset $A$. We first argue that it is bipartite. Consider
the bipartition $(V_{1},V_{2})$ where
$V_{1}=\\{x_{i}:\mathbf{a}(v_{i})=1\\}\cup\\{y_{i}:\mathbf{a}(v_{i})=0\\}$ and
$V_{2}=V\setminus V_{1}$. By construction $V_{1}$ is edgeless. The set $V_{2}$
is edgeless too as by the definition of $H$ vertices adjacent to $u$ or to
$c_{j}$, $1\leq j\leq m$ in $H$ are precisely those in $V_{1}$. Thus $H$ is
bipartite. We greedily extend $H$ into a maximal bipartite subgraph and note
$H^{\prime}$ the result. Suppose toward a contradiction that $H^{\prime}$ does
not satisfy the required properties, i.e., that it contains an edge $uc_{j}$
for some $1\leq j\leq m$. Then by definition of $V_{1}$ and the fact that
$\mathbf{a}$ is a model of $\varphi$, we get that $uc_{j}w$ would induce a
triangle for $w$ the neighbor of $c_{j}$ in $V_{1}$. This contradicts the fact
that $H^{\prime}$ is bipartite. Consequently $H^{\prime}$ extends $A$ and
avoids $B$ as required.
Suppose now that there exists a maximal bipartite subgraph $H$ of $G$ with
$A\subseteq E(H)$ and $B\cap E(H)=\emptyset$. Let $(V_{1},V_{2})$ be a
bipartition of $H$. Note that since $x_{1}y_{1},\dots,x_{n}y_{n}$ belong to
$A$ (hence to $H$), the sets $V_{1}$ and $V_{2}$ contain exactly one and
distinct elements from $\\{x_{i},y_{i}\\}$ for each $1\leq i\leq n$. Let us
assume without loss of generality that $u\in V_{2}$. Then since every edge
$uc_{j}$, $1\leq j\leq m$ belongs to $B$, it must be that every such $c_{j}$
lies in $V_{2}$ as well. Thus, except for the the choice of which of
$\\{x_{i},y_{i}\\}$ belongs to $V_{1}$, $1\leq i\leq n$, the bipartition
$(V_{1},V_{2})$ is now completely characterized. Consider a triangle
$ux_{i}y_{i}$, $1\leq i\leq n$. Then one of $x_{i},y_{i}$ belongs to $V_{1}$,
call it $\ell_{1}$, and the other belongs to $V_{2}$, call it $\ell_{2}$.
Clearly $u\ell_{2}$ is not an edge of $H$ as both $u$ and $\ell_{2}$ lie in
the same side of the bipartition. We argue that $u\ell_{1}$ is an edge of $H$.
Suppose that it is not the case. Then by maximality of $H$, adding $u\ell_{1}$
creates an odd cycle. Consequently there exists an even path $P$ going from
$u$ to $\ell_{1}$ in $H$, where by even we mean that $P$ contains an even
number of edges. Since $\ell_{1}\ell_{2}$ belongs to $H$ we deduce that the
path $P\ell_{2}$ obtained by adding $\ell_{2}$ to $P$ is odd, which
contradicts the fact that $u$ and $\ell_{2}$ both lie in $V_{2}$. A
consequence of these observations is that the set of edges $u\ell$ in $H$ for
$\ell\in V_{1}$ defines an assignment $\mathbf{a}$ of $\varphi$, where
$\mathbf{a}(v_{i})=1$ if $\ell=x_{i}$, and $\mathbf{a}(v_{i})=0$ if
$\ell=y_{i}$. Hence the edges of $H$ having some $c_{j}$ for endpoint are
precisely those connecting $c_{j}$ to $V_{1}$. We claim that every such
$c_{j}$ has at least one such incident edge, hence that $\mathbf{a}$ is a
satisfying truth assignement of $\varphi$. Indeed, if it was not the case then
$c_{j}$ would be disconnected from the rest of the graph. But in that case, we
could have added $uc_{j}$ to the graph still maintaining the bipartiteness
(changing $c_{j}$ to the other side of the bipartition) contradicting the
maximality of $H$. This concludes the proof. ∎
Concerning prospective research directions, let us restate here the implicit
question from [BBČ+21] that motivated this work, which additionally was posed
at the WEPA’19 open problem session, and that is still open to date. Recall
that a family of CNF’s is tractable if there exists a polynomial-time
algorithm to decide the satisfiability of any formula (and its subformulas) in
the family.
See 1
Natural examples of tractable CNF’s include 2-CNF’s and Horn CNF’s for which,
to the best of our knowledge, no progress has been made. Toward this
direction, the case of Horn 2-CNF also seems open.
Another question left open by the work of Bérczi et al. deals with the
_dimension_ of the formula, defined as the maximum size of a clause it
contains. Namely, the algorithm of [BBČ+21, Theorem 4] only performs in
incremental-polynomial time for fixed dimension, and it is open whether the
same result can be obtained for formulas of arbitrary dimension.
Finally, another promising direction concerns the parameterized study of
signatures enumeration. More particularly, it can be seen that the algorithm
of [BBČ+21, Theorem 4] is XP-incremental parameterized by the dimension $k$,
that is, it generates the $i^{\text{th}}$ solution in time
$(\|\varphi\|+i)^{f(k)}$ for some computable function $f$. It would be
interesting to know whether this algorithm can be improved to run with FPT-
delay, that is, to run with $f(k)\cdot\|\varphi\|^{O(1)}$ delay; see e.g.
[CMM+17, GKKL22] for more details on the parameterized aspects of enumeration
problems. A preliminary step in that direction would be to get such running
times for a double parameterization. Among parameters, the maximum number
$\omega$ of occurrence of a variable in the formula is natural. This parameter
is relevant as in [BBČ+21, Theorem 3] the authors give a simpler algorithm for
signatures enumeration when both $k$ and $\omega$ are bounded. Improving this
XP-incremental algorithm into one that is FPT-delay parameterized by $k$ plus
$\omega$ seems open.
### Acknowledgements.
The authors would like to thank Kazuhiro Kurita, Kazuhisa Makino, Kunihiro
Wasa, and Yasuaki Kobayashi for preliminary discussions on the topic of this
paper.
## References
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* [BEG04] Endre Boros, Khaled Elbassioni, and Vladimir Gurvich. Algorithms for generating minimal blockers of perfect matchings in bipartite graphs and related problems. In European Symposium on Algorithms, pages 122–133. Springer, 2004.
* [BEGK03] Endre Boros, Khaled Elbassioni, Vladimir Gurvich, and Leonid Khachiyan. Algorithms for enumerating circuits in matroids. In International Symposium on Algorithms and Computation, pages 485–494. Springer, 2003.
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* [CGM+22] Alessio Conte, Roberto Grossi, Andrea Marino, Takeaki Uno, and Luca Versari. Proximity search for maximal subgraph enumeration. SIAM Journal on Computing, 51(5):1580–1625, 2022.
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* [CMM+17] Nadia Creignou, Arne Meier, Julian-Steffen Müller, Johannes Schmidt, and Heribert Vollmer. Paradigms for parameterized enumeration. Theory of Computing Systems, 60:737–758, 2017.
* [COS11] Nadia Creignou, Frédéric Olive, and Johannes Schmidt. Enumerating all solutions of a boolean CSP by non-decreasing weight. In Karem A. Sakallah and Laurent Simon, editors, Theory and Applications of Satisfiability Testing - SAT 2011 - 14th International Conference, SAT 2011, Ann Arbor, MI, USA, June 19-22, 2011. Proceedings, volume 6695 of Lecture Notes in Computer Science, pages 120–133. Springer, 2011.
* [CU19] Alessio Conte and Takeaki Uno. New polynomial delay bounds for maximal subgraph enumeration by proximity search. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 1179–1190, New York, NY, USA, 2019. Association for Computing Machinery.
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# Freeze-in bino dark matter in high scale supersymmetry
Chengcheng Han Peiwen Wu Jin Min Yang Mengchao Zhang
###### Abstract
We explore a scenario of high scale supersymmetry where all supersymmetric
particles except gauginos stay at a high energy scale $M_{\rm SUSY}$ which is
much larger than the reheating temperature $T_{\text{RH}}$. The dark matter is
dominated by bino component with mass around the electroweak scale and the
observed relic abundance is mainly generated by the freeze-in process during
the early universe. Considering the various constraints, we identify two
available scenarios in which the supersymmetric sector at an energy scale
below $T_{\text{RH}}$ consists of: a) bino; b) bino and wino. Typically, for a
bino mass around 0.1-1 TeV and a wino mass around 2 TeV, we find that $M_{\rm
SUSY}$ should be around $10^{12-14}$ GeV with $T_{\text{RH}}$ around
$10^{4-6}$ GeV.
## 1 Introduction
Supersymmetry (SUSY) [1, 2, 3, 4, 5, 6] is a significant theoretical framework
aiming at extending the Standard Model (SM), drawing inspiration from the
pursuit of a quantum gravity theory, particularly within the context of
superstring theory. In the field of phenomenology, SUSY not only provides a
viable candidate for dark matter (DM) which plays a crucial role in the
formation of large-scale structures in the universe, but also contributes to
the renormalization group running of gauge couplings through the inclusion of
additional particles near the electroweak scale. This property of SUSY
facilitates the potential unification of the three fundamental forces at high
energy scales. It has long been postulated that SUSY DM takes the form of
Weakly Interacting Massive Particles (WIMPs) that can be probed through
diverse experiments [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. However, the
absence of confirmed DM signals poses significant challenges to the standing
of SUSY DM. The current LHC search results indicate that SUSY particles seem
to be heavier than the electroweak (EW) scale [18, 19], thus challenging the
WIMP paradigm of SUSY (for recent reviews on SUSY in light of current
experiments, see, e.g., [20, 21, 22]).
Given the current situation, in this study we consider an alternative scenario
of SUSY DM in which gauginos are located at a low energy scale while all other
SUSY partners exist at a significantly higher scale $M_{\rm SUSY}$. This
scenario is a special case of the Split SUSY [23, 24, 25, 26] where higgsinos
are also taken to be a similar scale as sfermions. One should note that the
Higgs sector in this scenario is fine-tuned [27, 28, 29, 30, 31] and it might
be a consequence of the anthropic principle. However, in this work we will
assume that SUSY still provides a candidate of DM and we will specifically
consider the Minimal Supersymmetric Standard Model (MSSM). Since the
measurement of gamma-ray from the MAGIC [32] has strongly constrained the
possibility of wino DM111There is still viable parameter space for wino dark
matter assuming core profile of the DM., the only viable DM candidate in the
MSSM is bino. However, it is widely known that pure bino DM is typically
overabundant from the freeze-out mechanism [33] due to its weak coupling with
the visible sector [34, 35]. Alternatively, a bino particle with a rather weak
coupling may serve as a suitable candidate for Feebly Interacting Massive
Particle (FIMP) DM with a correct relic abundance generated via the freeze-in
mechanism [36], with assumptions that the reheating process solely occurs in
the Standard Model (SM) sector and the reheating temperature $T_{\text{RH}}$
is lower than the SUSY scale $M_{\text{SUSY}}$ .
In this work we study the possibility that the bino DM in MSSM is generated
via the freeze-in process during the early universe. We assume that all MSSM
particles except gauginos share similar mass $M_{\rm SUSY}$ which is much
higher than the reheating temperature $T_{\text{RH}}$ of the universe. To
generate enough relic abundance of bino dark matter, we always require the
bino mass lower than the reheating temperature. While for the mass of wino or
gluino, they could be either higher or lower than the reheating temperate
$T_{\text{RH}}$ depending on the different scenarios we consider.
The paper is organized as follows. In Section 2 we present the model set up.
In Section 3 we first overview the physics related to dark matter and then
study the dominate channels for bino freeze-in production. In Section 4 we
give the numerical results and discuss the experimental limits on the model
parameter space relevant for our scenarios. We draw the conclusions in Section
5 and leave the calculation details in Appendices.
## 2 Model of heavy supersymemtry
Since we are considering a scenario of high scale supersymmetry in which only
gauginos are at low energy scale, the relevant Lagrangian terms are
$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle-\sum_{f=q,l}M^{2}_{\tilde{f}}\tilde{f}^{\ast}\tilde{f}+\bigg{[}\bigg{(}\sum_{A=1,2,3}-\frac{1}{2}M_{A}\tilde{V}^{A,a}\tilde{V}^{A,a}\bigg{)}-\mu\tilde{H}_{u}\cdot\tilde{H}_{d}+b\mu
H_{u}\cdot H_{d}+h.c.\bigg{]}$ (2.1)
$\displaystyle-\sum_{A=1,2}\sqrt{2}g_{A}\bigg{[}{H}^{\ast}_{u}\bigg{(}T^{A,a}\tilde{V}^{A,a}\bigg{)}\tilde{H}_{u}+{H}^{\ast}_{d}\bigg{(}T^{A,a}\tilde{V}^{A,a}\bigg{)}\tilde{H}_{d}+h.c.\bigg{]}$
$\displaystyle-\sum_{A=1,2,3}\sqrt{2}g_{A}\bigg{[}\sum_{f=q,l}\tilde{f}^{\ast}\bigg{(}T^{A,a}\tilde{V}^{A,a}\bigg{)}f+h.c.\bigg{]}$
$\displaystyle-(M^{2}_{H_{u}}+|\mu|^{2})H^{\ast}_{u}H_{u}-(M^{2}_{H_{d}}+|\mu|^{2})H^{\ast}_{d}H_{d}~{},$
where $A=1,2,3$ correspond to the SM gauge group $\rm
U(1)_{Y},SU(2)_{L},SU(3)_{C}$, respectively, and $a$ denotes the corresponding
indices in adjoint representation of group $A$. Fields
$\tilde{V}^{A,a},\tilde{H}_{u},\tilde{H}_{d},\tilde{f}$ are the superpartners
of the SM vector gauge bosons $V^{A,a}=B,W^{1\sim 3},G^{1\sim 8}$, scalar
doublets $H_{u},H_{d}$ and fermions $f$. The fields $H_{u}$, $H_{d}$,
$\tilde{H}_{u}$, $\tilde{H}_{d}$ are defined as
$\displaystyle H_{u}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}H_{u}^{+}\\\
H_{u}^{0}\end{array}\right),\quad\tilde{H}_{u}=\left(\begin{array}[]{c}\tilde{H}_{u}^{+}\\\
\tilde{H}_{u}^{0}\end{array}\right),\quad
H_{d}=\left(\begin{array}[]{c}H_{d}^{0}\\\
H_{d}^{-}\end{array}\right),\quad\tilde{H}_{d}=\left(\begin{array}[]{c}\tilde{H}_{d}^{0}\\\
\tilde{H}_{d}^{-}\end{array}\right).$ (2.10)
For the Higgs sector, we need a SM-like Higgs boson $H_{\text{SM}}$ near the
electroweak scale [37, 38]. This is obtained from the mixing between the two
Higgs doublets $H_{u}$ and $H_{d}$ in the MSSM:
$\displaystyle H_{u}=\left(\begin{array}[]{c}H_{u}^{+}\\\
H_{u}^{0}\end{array}\right)$ $\displaystyle=$ $\displaystyle\sin\beta\,H_{\rm
SM}+\cos\beta\,H_{\rm NP}=\sin\beta\left(\begin{array}[]{c}G_{\rm SM}^{+}\\\
H_{\rm SM}^{0}\end{array}\right)+\cos\beta\left(\begin{array}[]{c}H_{\rm
NP}^{+}\\\ H_{\rm NP}^{0}\end{array}\right)~{},$ (2.17)
$\displaystyle(-i\sigma^{2})H^{\ast}_{d}=\left(\begin{array}[]{c}-(H_{d}^{-})^{*}\\\
(H_{d}^{0})^{*}\end{array}\right)$ $\displaystyle=$
$\displaystyle\cos\beta\,H_{\rm SM}-\sin\beta\,H_{\rm
NP}=\cos\beta\left(\begin{array}[]{c}G_{\rm SM}^{+}\\\ H_{\rm
SM}^{0}\end{array}\right)-\sin\beta\left(\begin{array}[]{c}H_{\rm NP}^{+}\\\
H_{\rm NP}^{0}\end{array}\right)~{},$ (2.24)
where $\sigma^{2}$ is the second Pauli matrix, and $\tan\beta=\langle
H^{0}_{u}\rangle/\langle H^{0}_{d}\rangle$ with $\langle H^{0}_{u}\rangle$ and
$\langle H^{0}_{d}\rangle$ being the vacuum expectation values (VEVs). Such
mixings can be realized by properly choosing Higgs mass parameters
$\mu,M_{H_{u}},M_{H_{d}}$, and $b$. The subscription "NP" in $H_{\text{NP}}$
denotes the new physics (NP) Higgs doublet in the MSSM accompanying the SM
one222Note that in order not to increase the complexity of notation, we don’t
further perform the expansion of the complex but electrically neutral scalars
$H_{\rm SM}^{0},H_{\rm NP}^{0}$ into real and imaginary parts. However, one
needs to beware that $G_{\rm SM}^{\pm},H_{\rm SM}^{0}$ contain the Goldstone
boson modes to be absorbed into vector gauge bosons $W^{\pm},Z^{0}$ after the
electroweak symmetry breaking (EWSB).. Since the mass parameters $M_{H_{u}}$,
$M_{H_{d}}$, $b$, $\mu$ are all much larger than the electroweak scale, a
tuning of these parameters are needed to get a light Higgs at electroweak
scale [27, 28, 29, 30, 31]. We need also match the Higgs self-coupling to be
the SUSY value at the scale of $M_{\rm SUSY}$,
$\displaystyle\lambda(M_{\rm
SUSY})=\frac{{g^{\prime}_{1}}^{2}+g_{2}^{2}}{4}\cos^{2}2\beta~{}.$ (2.25)
Note that the Higgs self-coupling $\lambda$ becomes very small at high energy
scale due to the RGE running, and thus the $\beta$ value should get close to
$\pi/4$ and $\tan\beta\approx 1$. We will fix $\tan\beta=1$ as the benchmark
parameter throughout this work for simplicity.
Generally, when considering physical processes at temperature $T\ll
M_{\text{SUSY}}$, we can integrate out the heavy mediators with mass
$\mu,M_{\tilde{f}}\sim M_{\text{SUSY}}\gg T_{\text{RH}}$ and get the following
effective operators at the level of dimension 5 and 6, respectively,
$\displaystyle{\text{dimension-5:}}\quad\propto\quad\frac{1}{\mu}\,|H_{\rm
SM}|^{2}(\tilde{B}\tilde{B},\tilde{B}\tilde{W})~{},$ (2.26)
$\displaystyle{\text{dimension-6:}}\quad\propto\quad\frac{1}{M^{2}_{\tilde{f}}}(f^{\dagger}\tilde{B}^{\dagger})(f\tilde{B},f\tilde{W},f\tilde{G})~{}.$
(2.27)
Since we assume the mass parameters of higgsinos $\mu$ and sfermions
$M_{\tilde{f}}$ around $M_{\text{SUSY}}$, the dominant process would be from
the dimension-5 (dim-5) operators. Nevertheless we also present the processes
related to dim-6 operators for completeness.
We acknowledge that a majority of the significant processes are evaluated at
energy scales considerably beneath $M_{\text{SUSY}}$. The recommended approach
entails initiating the integration procedure for the massive particle to
derive the effective operators of dimension 5 and 6, along with their
corresponding Wilson coefficients, within the realm of $M_{\text{SUSY}}$.
Subsequently, the computation of these Wilson coefficients at the pertinent
scale is achieved by employing the Renormalization Group Equations to track
the evolution of the operators. Notably, there exists a potential correction
to the primary outcome, potentially on the order of $O(1)$, yet the
fundamental framework remains robust. We leave the investigation of this
effect for future study.
## 3 Freeze-in bino dark matter in MSSM
### 3.1 Particle spectrum
Despite the existence of new Higgs bosons and many supersymmetric partners of
the SM particles, the MSSM particle spectrum we consider in this work consist
of two sectors distinguished by their characteristic mass scales. Although not
making significant difference for the mass spectrum structure before and after
EWSB, we take the pre-EWSB case as an illustration.
* •
Heavy sector, inactive after cosmological reheating
Mass: $M\sim M_{\text{SUSY}}\gg T_{RH}$
* $\star$
Higgs bosons not in SM: $H^{0}_{\text{NP}}$, $A$, $H^{\pm}_{\text{NP}}$
* $\star$
Sfermions $\tilde{f}$
* $\star$
Higgsinos $\tilde{H}_{u},\tilde{H}_{d}$
* •
Light sector, active after cosmological reheating
Mass: $M\sim\mathcal{O}(1)\,\text{TeV}\ll M_{\text{SUSY}}$
* $\star$
SM particles
* $\star$
Bino $\tilde{B}$, consisting cosmological DM with mass $M_{1}<T_{RH}$
* $\star$
Winos $\tilde{W}$, with mass $M_{2}$
* $\star$
Gluinos $\tilde{G}$, with mass $M_{3}$
In the above we utilized gauge eigenstates for description, since
$\tilde{B},\tilde{W}$ do not mix with higgsinos $\tilde{H}_{u},\tilde{H}_{d}$
before EWSB when the SM Higgs $H_{\text{SM}}$ has not acquired the VEV.
### 3.2 Bino production from freeze-in mechanism
In the early stage of universe before EWSB when the gaugino states
$\tilde{B},\tilde{W}$ do not mix with higgsinos $\tilde{H}_{u},\tilde{H}_{d}$,
pure $\tilde{B}$ acting as DM can only interact with SM via mediators with
heavy mass near the scale $M_{\text{SUSY}}$, as shown in Fig. 1. Due to the
suppressed interacting strength, the cosmological production of bino DM in our
scenario proceed via the freeze-in mechanism. In the follows we consider the
contributions to bino DM production from several typical processes333After
electroweak phase transition occurs and $H_{\text{SM}}$ acquires VEV, the top
and bottom vertex in the left panel of Fig.1 imply the mixing between
$\tilde{B},\tilde{W}$ and $\tilde{H}_{u},\tilde{H}_{d}$, resulting in the mass
eigenstates of electrically neutral neutralinos $\tilde{\chi}^{0}_{1,2,3,4}$
and charged $\tilde{\chi}^{\pm}_{1,2}$ (see discussions in Section 4.1)..
Figure 1: Schematic plots for interactions of DM composed of pure $\tilde{B}$
with SM after cosmological reheating considered in this work, which would
induce dimension-5 (left) and dimension-6 (right) effective operators. The SM
Higgs $H_{\text{SM}}$ originates from the mixing between MSSM Higgs doublets
$H_{u},H_{d}$. Colored lines indicate the direction of freeze-in production
when applicable. Additional Hermitian conjugated processes also exist when the
amplitudes are complex. See more discussions in the main texts.
### 3.3 Case I: bino freeze-in from $HH^{\ast}\to\tilde{B}\tilde{B}$
This case corresponds to the left panel of Fig. 1 but without winos
$\tilde{W}$. After integrating out the heavy higgsinos, the relevant dim-5
effective interaction is given by (the details are given in Appendix A)
$\displaystyle\mathcal{L}^{\text{eff}}_{HH^{\ast}\to\tilde{B}\tilde{B}}=\frac{2g_{1}^{2}\,Y_{H}^{2}}{\mu}\sin\beta\cos\beta(|H_{\rm
SM}|^{2})(\tilde{B}\tilde{B}+\tilde{B}^{\dagger}\tilde{B}^{\dagger})~{},$
(3.1)
where $|H_{\rm SM}|^{2}=G_{\rm SM}^{+}(G_{\rm SM}^{+})^{\ast}+(H_{\rm
SM}^{0})(H_{\rm SM}^{0})^{*}$. In the subscription
$HH^{\ast}\to\tilde{B}\tilde{B}$ on the left side (and hereafter when not
causing any confusion), we denote $H_{\text{SM}}$ as $H$ to simplify the
notation, and all fields in the initial and final states of the process should
be understood in the sense of physical particles444Discussion on the naming
convention of particles, states and filed can be found in, e.g. [39].. With
more details given in Appendix B, Eq.(3.1) would induce the Boltzmann equation
of the bino number density:
$\displaystyle\frac{d}{dt}n_{\tilde{B}}+3\mathcal{H}n_{\tilde{B}}=\textbf{C}_{HH^{\ast}\to\tilde{B}\tilde{B}}\approx\frac{g_{1}^{4}}{4}\frac{1}{\pi^{5}}\frac{\,\sin^{2}\beta\cos^{2}\beta}{\mu^{2}}T^{6}~{}.$
(3.2)
The above equation can be modified to a differential equation about bino yield
$Y_{\tilde{B}}=n_{\tilde{B}}/S$ ($S$ is the entropy density) and temperature
$T$:
$\displaystyle\frac{dY_{HH^{\ast}\to\tilde{B}\tilde{B}}(T)}{dT}=-\frac{\textbf{C}_{HH^{\ast}\to\tilde{B}\tilde{B}}}{ST\mathcal{H}}$
$\displaystyle\approx-(1.25\times 10^{-3})\times
M_{\text{Pl}}\frac{\textbf{C}_{HH^{\ast}\to\tilde{B}\tilde{B}}}{T^{6}}\approx-(1\times
10^{-6})\times M_{\text{Pl}}\
g_{1}^{4}\frac{\,\sin^{2}\beta\cos^{2}\beta}{\mu^{2}}~{},$
where $M_{\text{Pl}}\approx 1.22\times 10^{19}$ GeV is the Planck mass,
$S=2\pi^{2}g_{\ast}T^{3}/45$, and Hubble expansion rate $\mathcal{H}\approx
1.66\sqrt{g_{\ast}}T^{2}/M_{\text{Pl}}$ with $g_{\ast}=106.75$ before EWSB.
Performing a simple integral from reheating temperature, it can be found that
the final yield of $\tilde{B}$ depends on the reheating temperature
$T_{\text{RH}}$ which corresponds to the Ultraviolet (UV) freeze-in scenario
[36, 40]:
$\displaystyle Y_{HH^{\ast}\to\tilde{B}\tilde{B}}(\infty)\approx(1\times
10^{-6})\times M_{\text{Pl}}\
g_{1}^{4}\frac{\,\sin^{2}\beta\cos^{2}\beta}{\mu^{2}}\ T_{\text{RH}}~{},$
(3.3)
and the corresponding current relic abundance is given by
$\displaystyle\left(\Omega_{\tilde{B}}h^{2}\right)_{HH^{\ast}\to\tilde{B}\tilde{B}}=M_{1}\
\frac{Y_{HH^{\ast}\to\tilde{B}\tilde{B}}(\infty)S_{0}}{\rho_{cr}}\approx
Y_{HH^{\ast}\to\tilde{B}\tilde{B}}(\infty)\left(\frac{M_{1}}{\text{TeV}}\right)\times(2.72\times
10^{11})~{}.$
### 3.4 Case II: fermion scattering process $f\bar{f}\to\tilde{B}\tilde{B}$
After integrating out sfermions with heavy mass $M_{\tilde{q},\tilde{l}}\sim
M_{\text{SUSY}}$ in the right panel of Fig.1, the effective interactions
between SM fermion pair and $\tilde{B}$ pair have the following form at
dimention 6 (for more details, see Appendix C):
$\displaystyle\mathcal{L}^{\text{eff}}_{f\bar{f}\to\tilde{B}\tilde{B}}=\sum_{f=q,l}\frac{(\sqrt{2}g_{1}Y_{f})(\sqrt{2}g_{1}Y_{f})}{M^{2}_{\tilde{f}}}(f^{\dagger}\tilde{B}^{\dagger})(f\tilde{B})~{},$
(3.5)
where for simplicity, we consider an universal mass for all the fermions, i.e.
$M_{\tilde{f}}\equiv M_{\tilde{q}}=M_{\tilde{l}}$.
Thus the Boltzmann equation is
$\displaystyle\frac{dY_{f\bar{f}\to\tilde{B}\tilde{B}}(T)}{dT}=-\frac{\textbf{C}_{f\bar{f}\to\tilde{B}\tilde{B}}}{ST\mathcal{H}}$
$\displaystyle\approx-(1.25\times 10^{-3})\times
M_{\text{Pl}}\frac{\textbf{C}_{f\bar{f}\to\tilde{B}\tilde{B}}}{T^{6}}\approx-(8.6\times
10^{-5})\times M_{\text{Pl}}\frac{g_{1}^{4}}{M^{4}_{\tilde{f}}}T^{2}~{},$
(3.6)
and correspondingly,
$\displaystyle Y_{f\bar{f}\to\tilde{B}\tilde{B}}(\infty)$
$\displaystyle\approx$ $\displaystyle(4.7\times
10^{-7})\times\frac{M_{\text{Pl}}}{M^{4}_{\tilde{f}}}\ T^{3}_{\text{RH}}~{},$
(3.7)
$\displaystyle\left(\Omega_{\tilde{B}}h^{2}\right)_{f\bar{f}\to\tilde{B}\tilde{B}}$
$\displaystyle=$ $\displaystyle M_{1}\
\frac{Y_{f\bar{f}\to\tilde{B}\tilde{B}}(\infty)S_{0}}{\rho_{cr}}\approx
Y_{f\bar{f}\to\tilde{B}\tilde{B}}(\infty)\left(\frac{M_{1}}{\text{TeV}}\right)\times(2.72\times
10^{11})~{}.$ (3.8)
### 3.5 Case III: gluino/wino scattering or decay processes
As indicated by blue colored arrows in Fig. 1, the $2\to 2$ scattering
processes consist of two ways of generating bino DM when combining $\rm
U(1)_{Y}$ with $\rm SU(2)_{L}$ or $\rm SU(3)_{C}$ interactions, related by the
cross symmetry. Moreover, we can also have the red colored arrow indicating
$1\to 3$ ($1\to 2$) decay processes generating binos before (after) EWSB when
the cosmological temperature drops below the scale of $M_{2}$ or $M_{3}$
(equivalently, when the age of the universe reach the lifetime of $\tilde{W}$
and $\tilde{G}$).
Similar to the previous two cases, integrating out heavy higgsino and
sfermions would generate the following dim-5 and dim-6 effective operators:
$\displaystyle\mathcal{L}^{\text{eff}}_{\text{case-III}}=$
$\displaystyle\bigg{\\{}-\sum_{b=1}^{3}\frac{(\sqrt{2}g_{1}Y_{H})(\sqrt{2}g_{2})}{\mu}\sin\beta\cos\beta(H^{\ast}\frac{1}{2}\sigma^{b}H)(\tilde{B}\tilde{W}^{b})$
(3.9)
$\displaystyle+\sum_{f=u_{L},d_{L},e_{L},\nu}\quad\sum_{b=1}^{3}\frac{(\sqrt{2}g_{1}Y_{f})(\sqrt{2}g_{2})}{M^{2}_{\tilde{f}}}(f^{\dagger}\tilde{B}^{\dagger})(\frac{1}{2}\sigma^{b}f\tilde{W}^{b})$
$\displaystyle+\sum_{f=u_{L},d_{L},{u^{\dagger}_{R}},{d^{\dagger}_{R}}}\quad\sum_{a=1}^{8}\frac{(\sqrt{2}g_{1}Y_{f})(\sqrt{2}g_{3})}{M^{2}_{\tilde{f}}}(f^{\dagger}\tilde{B}^{\dagger})(\frac{1}{2}\lambda^{a}f\tilde{G}^{a})\bigg{\\}}+h.c.~{}.$
Note that the index $f$ in the second line includes only $\rm SU(2)_{L}$
doublets, while the index $f$ in the third line includes only quarks. To
highlight the difference, we use index $a$ and $b$ to denote generators of
$\rm SU(3)_{C}$ and $\rm SU(2)_{L}$ interactions, respectively.
Correspondingly, $\lambda^{a}$ and $\sigma^{b}$ are Gell-Mann and Pauli
matries, respectively.
In the following, we consider the contributions to the bino DM production from
$2\to 2$ scattering and $1\to 3$ decay separately, while leaving the effects
of $1\to 2$ decay appearing after EWSB in Section 4.1.
#### 3.5.1 Case III A: $2\to 2$ scattering involving gluino/wino
With more detailed given in Appendix D, the collision terms in the Boltzmann
equation for dim-5 and dim-6 operators are approximated as (ignoring the
masses of all external particles)
$\displaystyle\textbf{C}_{\rm dim-5}$ $\displaystyle=$
$\displaystyle\frac{T}{2048\pi^{6}}\int^{\infty}_{4M_{1}^{2}}ds\
(s-4M_{1}^{2})^{1/2}K_{1}({\sqrt{s}}/{T})\sum_{\rm
internal\,d.o.f}\int\,d\Omega$ (3.10)
$\displaystyle\times\bigg{(}|\mathcal{M}|^{2}_{HH^{\ast}\to\tilde{B}\tilde{B}}+|\mathcal{M}|^{2}_{HH^{\ast}\to\tilde{B}\tilde{W}}+N_{\text{conj}}|\mathcal{M}|^{2}_{\tilde{W}H\to\tilde{B}H}\bigg{)}$
$\displaystyle=\bigg{(}\frac{1}{4}g_{1}^{4}+\frac{3}{2}g_{1}^{2}g_{2}^{2}\bigg{)}\frac{1}{\pi^{5}}\frac{\,\sin^{2}\beta\cos^{2}\beta}{\mu^{2}}T^{6}~{},$
$\displaystyle\textbf{C}_{\rm dim-6}$ $\displaystyle=$
$\displaystyle\frac{T}{2048\pi^{6}}\int^{\infty}_{4M_{1}^{2}}ds\
(s-4M_{1}^{2})^{1/2}K_{1}({\sqrt{s}}/{T})\sum_{\rm
internal\,d.o.f}\int\,d\Omega$ (3.11)
$\displaystyle\times\bigg{(}|\mathcal{M}|^{2}_{f\bar{f}\to\tilde{B}\tilde{B}}+|\mathcal{M}|^{2}_{f\bar{f}\to\tilde{B}\tilde{W}}+N_{\text{conj}}|\mathcal{M}|^{2}_{\tilde{W}f\to\tilde{B}f}+|\mathcal{M}|^{2}_{f\bar{f}\to\tilde{B}\tilde{G}}+N_{\text{conj}}|\mathcal{M}|^{2}_{\tilde{G}f\to\tilde{B}f}\bigg{)}$
$\displaystyle=$
$\displaystyle(\frac{190}{9}g^{4}_{1}+30g_{1}^{2}g_{2}^{2}+\frac{440}{3}g_{1}^{2}g_{3}^{2})\frac{1}{\pi^{5}}\frac{1}{M^{4}_{\tilde{f}}}T^{8}~{},$
where $N_{\text{conj}}=2$ denotes the effects of conjugated process.
#### 3.5.2 Case III B: decay of gluino/wino
Following the method in [36] with $f_{\tilde{G}}$ and $f_{\tilde{W}}$
approximated by $e^{-E_{\tilde{G}}/T}$ and $e^{-E_{\tilde{W}}/T}$, the
Boltzmann equation of freeze-in production for the $1\to 3$ decay processes is
$\displaystyle\frac{d}{dt}n_{\tilde{B}}+3\mathcal{H}n_{\tilde{B}}=\textbf{C}$
$\displaystyle\approx\frac{g_{\tilde{G}}M_{3}^{2}}{2\pi^{2}}TK_{1}(\frac{M_{3}}{T})\Gamma_{\tilde{G}\to
f\bar{f}\tilde{B}}+\frac{g_{\tilde{W}}M_{2}^{2}}{2\pi^{2}}TK_{1}(\frac{M_{2}}{T})(\Gamma_{\tilde{W}\to
f\bar{f}\tilde{B}}+\Gamma_{\tilde{W}\to HH^{\ast}\tilde{B}})~{},$ (3.12)
where $g_{\tilde{G}}=16$ and $g_{\tilde{W}}=6$ are the internal d.o.f. of
$\tilde{G}$ and $\tilde{W}$, respectively. The expressions of decay width
involved in the above results are listed in Appendix E. Changing variables to
yield $Y_{\tilde{B}}$ and temperature $T$, we then integrate over temperature
evolution to obtain the final yield. If reheating temperature $T_{\text{RH}}$
is much larger than $M_{2}$ and $M_{3}$, then the final yield from $1\to 3$
decay can be approximated by
$\displaystyle Y^{\rm 1\to
3}_{\tilde{B}}(\infty)\approx\int^{T_{\text{RH}}}_{T_{min}}\frac{\textbf{C}}{ST\mathcal{H}}dT$
$\displaystyle\approx(3\times 10^{-4})\times
M_{\text{Pl}}\left(\frac{1}{M^{2}_{3}}g_{\tilde{G}}\Gamma_{\tilde{G}\to
f\bar{f}\tilde{B}}+\frac{1}{M^{2}_{2}}g_{\tilde{W}}\Gamma_{\tilde{W}\to
f\bar{f}\tilde{B}}+\frac{1}{M^{2}_{2}}g_{\tilde{W}}\Gamma_{\tilde{W}\to
HH^{\ast}\tilde{B}}\right)~{}.$ (3.13)
It is worth pointing out that the above result is not sensitive to
$T_{\text{RH}}$. Taking a low reheating temperature $T_{\text{RH}}=1.1\,M_{3}$
as an example, increasing the value of $T_{\text{RH}}$ does no modify the
result significantly.
In addition to the $1\to 3$ decay, we should also note that wino $\tilde{W}$
with mass $M_{2}<T_{\text{RH}}$ keeps staying in the thermal bath until
reaching its freeze-out moment yielding a relic wino number density, which
would later convert to the equal amount of bino number density $n_{\tilde{B}}$
via $1\to 2$ decay $\tilde{W}\to\tilde{B}+h$ after EWSB occurs. Depending on
the bino mass $M_{1}$, this freeze-out component would also contribute to the
total bino DM abundance in today’s epoch. We checked that with wino mass
$M_{2}=2$ TeV, the $1\to 2$ decay contribution of $Y^{1\to 2}_{\tilde{B}}$ to
final bino yield is around $25\%$ ($1\%$) on the percentage level for $M_{1}=$
1(0.1) TeV [41], thus not affecting the freeze-in domination scenario of this
work. We properly include the wino freeze-out contribution in our results.
There is also contribution from gluino late time decay. However, to avoid the
constraints from BBN, we have to set the gluino mass higher than the
$T_{\text{RH}}$, thus we do not include its contribution here.
## 4 Numerical results and discussion
In Fig. 2 we show the required scales of $\mu$ ($M_{\tilde{f}}$) for dim-5(6)
operators with various $T_{\text{RH}}$ to produce the observed bino DM relic
abundance. The upper (lower) two lines correspond to dim-5 (6) operators. We
can see that due to the more suppression of dim-6 operators, the needed
$M_{\tilde{f}}$ are generally $\mathcal{O}(10^{-4})$ smaller than $\mu$ in the
dim-5 case. If we assume $\mathcal{O}(\mu)\approx\mathcal{O}(M_{\tilde{f}})$,
in order not to overclose the Universe, the dim-6 contributions would be
completely negligible.
From Fig. 2, we can see that for the case $M_{\tilde{B}}<T_{RH}\ll
M_{\tilde{W}}$, the dominant production of bino dark matter is from the
process $HH^{*}\rightarrow\tilde{B}\tilde{B}$ from the dim-5 operator.
Generally, $M_{\rm SUSY}$ should be around $10^{13-14}$ GeV for
$T_{RH}<10^{6}$ GeV. Since the final relic abundance is proportional to
${T_{RH}}/{\mu^{2}}$, the $M_{\rm SUSY}$ could continue increasing if the
reheating temperature $T_{\text{RH}}$ becomes higher. Note that this is
similar to the model of Higgs portal to fermion dark matter which are studied
in [42], with which we find our result are consistent. We emphasize that our
model is motivated by a more complete framework and [42] falls into one of
cases we consider. Moreover, For the case
$M_{\tilde{B}},M_{\tilde{W}}<T_{RH}$, we find the wino-included process can
largely enhance the annihilate rate and a higher scale is needed to satisfy
the relic abundance. In this case, $M_{\rm SUSY}$ should be around
$10^{14-15}$ GeV for $T_{RH}<10^{6}$ GeV.
Notice that if the gluino is in the thermal equilibrium with SM in the early
universe and the sfermions mediating the gluino decay are heavier than
$10^{9}$ GeV, gluino’s lifetime could be longer than the age of Universe when
the big bang nucleosynthesis (BBN) happens, leading to energy injection into
the cosmic plasma and altering the BBN profile. In all cases considered in
this work we find $M_{\rm SUSY}$ is much larger than $10^{9}$ GeV, therefore
we always need $M_{\tilde{G}}\gg T_{RH}$ to avoid the limit from BBN [43].
More discussions on BBN limits are given in 4.1.
Figure 2: Values of $\mu$ and $M_{\tilde{f}}$ to produce the observed DM
abundance via the UV freeze-in processes. See more discussions in the main
texts.
In Fig. 3 we show the comparison of final contributions and intermediate
profile of UV and IR freeze-in processes to the bino DM relic abundance. It
can be clearly seen that the IR freeze-in final yields from wino 3-body decays
are negligible compared to that of UV freeze-in processes generated by $2\to
2$ annihilation. Moreover, the critical production moment determining the
final yield of UV freeze-in locates in a much smaller $x$ (and thus much
higher temperature) than the IR freeze-in case.
Figure 3: Comparison between UV freeze-in and IR freeze-in. Note the
difference between temperatures indicated by $x=M_{2}/T$ producing the correct
relic density of bino DM.
### 4.1 Limits from BBN
After EWSB, the SM-like Higgs doublet needs to be replaced by:
$\displaystyle H=\left(\begin{array}[]{c}G^{+}\\\
\frac{1}{\sqrt{2}}(v+h+iG^{0})\end{array}\right)~{},$ (4.3)
where $v=246$ GeV is the VEV of SM Higgs 555If wino decays much later than
electroweak phase transition, then $v=246$ GeV is a good approximation. and
$h$ is the observed SM-like Higgs scalar. $G^{\pm}$ ($G^{-}=(G^{+})^{\ast}$)
and $G^{0}$ are Goldstone bosons that form the longitudinal modes of SM gauge
bosons $W^{\pm}$ and $Z$. As mentioned earlier, the SM-like Higgs VEV will
generate mixings among the gauge states
$\tilde{B},\tilde{W},\tilde{H}_{u},\tilde{H}_{d}$ and form mass eigenstates of
charge-neutral neutralinos $\tilde{\chi}_{1,2,3,4}$ and charged charginos
$\tilde{\chi}^{\pm}_{1,2}$ (with ascending mass order inside sectors of
neutralinos and charginos, respectively). For the scenario considered in this
work, the component of neutralino $\tilde{\chi}^{0}_{1}$
($\tilde{\chi}^{0}_{2}$) is dominated by bino $\tilde{B}$ (wino
$\tilde{W}^{3}$), and component of chargino $\tilde{\chi}^{\pm}_{1}$ is
dominated by winos $\frac{1}{\sqrt{2}}(\tilde{W}^{1}\mp i\tilde{W}^{2})$. More
details of the approximated masses and couplings can be found in [44, 45, 46].
In the following, we would utilize the language of gauge states (bino
$\tilde{B}$, wino $\tilde{W}$, higgsinos $\tilde{H}_{u},\tilde{H}_{d}$ ) and
mass eigenstates (neutralino $\tilde{\chi}^{0}$, chargino
$\tilde{\chi}^{\pm}$) interchangeably before and after EWSB.
Now we study the limit of BBN on our scenario from lifetimes of neutralinos,
charginos. In our scenario, only neutralino
$\tilde{\chi}^{0}_{1}\approx\tilde{B},\
\tilde{\chi}^{0}_{2}\approx\tilde{W}^{3}$ and chargino
$\tilde{\chi}^{\pm}_{1}\approx\frac{1}{\sqrt{2}}(\tilde{W}^{1}\mp
i\tilde{W}^{2})$ existed in the primordial thermal bath. Due to the loop
induced mass-splitting between $\tilde{\chi}^{\pm}_{1}$ and
$\tilde{\chi}^{0}_{2}$, chargino $\tilde{\chi}^{\pm}_{1}$ can have the 2-body
decay $\tilde{\chi}^{\pm}_{1}\to\tilde{\chi}^{0}_{2}\pi^{\pm}$ [47, 48, 49,
50]. It makes the lifetime of $\tilde{\chi}^{\pm}_{1}$ much shorter than 1
sec, and thus not affecting the BBN profile. However, we need to scrutinize
the lifetime of $\tilde{\chi}^{0}_{2}$ more carefully. If
$\tilde{\chi}^{0}_{2}$ decays after the onset of BBN, then the highly
energetic decay products will cause the photodissociation or hadrodissociation
and thus change the final abundances of light elements. So a bound from BBN
can be put on the model parameters, especially on the SUSY scale
$M_{\text{SUSY}}$ [51, 52].
It is easy to see that Fig. 1 implies the 2-body decay mode of
$\tilde{\chi}^{0}_{2}\to\tilde{\chi}^{0}_{1}h$ at the level of dim-5 after
EWSB, in which case we will have:
$\displaystyle\mathcal{L}_{\text{eff}}$ $\displaystyle=$
$\displaystyle-\sum_{b=1}^{3}\frac{(\sqrt{2}g_{1}Y_{H})(\sqrt{2}g_{2})}{\mu}\sin\beta\cos\beta(H^{\ast}\frac{1}{2}\sigma^{b}H)(\tilde{B}\tilde{W}^{b})+h.c.$
(4.4) $\displaystyle=$
$\displaystyle-\frac{g_{1}g_{2}v}{2\mu}\sin\beta\cos\beta(G^{\mp}\tilde{W}^{\pm}\tilde{B}-h\tilde{W}^{0}\tilde{B}+h.c.)$
$\displaystyle\approx$
$\displaystyle-\frac{g_{1}g_{2}v}{2\mu}\sin\beta\cos\beta(G^{\mp}\tilde{\chi}^{\pm}_{1}\tilde{\chi}^{0}_{1}-h\tilde{\chi}^{0}_{2}\tilde{\chi}^{0}_{1}+h.c.)~{},$
where the first term containing Goldstone boson $G^{\mp}$ can be understood in
the context of Goldstone equivalence theorem (GET) for
$\tilde{\chi}^{\pm}_{1}\to\tilde{\chi}^{0}_{1}W^{\pm}$. It should be noticed
that Eq.(4.4) does not contain the three-particle coupling
$G^{0}\tilde{W}\tilde{B}$ and thus would not provide a way of inferring the
2-body decay mode $\tilde{\chi}^{0}_{2}\to\tilde{\chi}^{0}_{1}Z$ via the GET.
In fact, $\tilde{\chi}^{0}_{2}\to\tilde{\chi}^{0}_{1}\ Z$ comes from the gauge
covariant kinetic terms of gauginos and higgsinos combined with gaugino
mixings after EWSB. However, the decay width of
$\tilde{\chi}^{0}_{2}\to\tilde{\chi}^{0}_{1}Z$ suffers from an extra
suppression of $\frac{1}{\mu^{2}}$ embedded in the mass mixings compared to
$\tilde{\chi}^{0}_{2}\to\tilde{\chi}^{0}_{1}h$ and thus can be ignored [53] .
Therefore, we have the following dominant 2-body decay (see Appendix F for
more details):
$\displaystyle\Gamma_{\tilde{\chi}^{0}_{2}\to\tilde{\chi}^{0}_{1}h}\approx
M_{2}\frac{1}{16\pi}\left(\frac{v}{\mu}g_{1}g_{2}\sin\beta\cos\beta\right)^{2}\left(1-\frac{M_{1}^{2}}{M_{2}^{2}}\right)\left(1+\frac{M_{1}}{M_{2}}\right)^{2}~{}.$
(4.5)
Using the GET we would obtain the same results for
$\Gamma_{\tilde{\chi}^{\pm}_{1}\to\tilde{\chi}^{0}_{1}W^{\pm}}$ when
neglecting the gauge boson masses.
In this work, we apply the limit of BBN to the requirement that lifetime of
$\tilde{\chi}^{0}_{2}$ must be less than 0.3 second [36]. In Fig. 4 , we show
the interplay between BBN constraints and freeze-in production, where regions
below black lines are allowed while region above blue lines are allowed. We
can see that for bino mass around 0.1-1 TeV, an upper bound of
$M_{\text{SUSY}}\sim 10^{14}$ TeV is needed to satisfy both phenomenological
requirements.
Figure 4: Interplay between BBN constraints and freeze-in production, where
regions below black lines are allowed while region above blue lines are
allowed.
### 4.2 Limits from direct/indirect detection
Our scenario can easily escape from the current limits from the direct and
indirect detection. In the case of direct detection, Eq.(3.1) after EWSB would
generate the $t$-channel scattering of $\tilde{\chi}^{0}_{1}$ with quarks and
gluons in SM neucleons mediated by SM Higgs, of which the event rate is
suppressed by $1/\mu^{2}$ and thus negligibly small. In the case of indirect
detection, which is basically the inversed process of the freeze-in DM
production, would generate cosmic rays via DM pair annihilations
$\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}\to h^{\ast}\to\text{SM}$ and
$\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}\to hh\to\text{SM}$, of which the
flux is again suppressed by $1/\mu^{2}$ and thus not violating the current
experimental bounds.
### 4.3 Limits from the LHC
The collider signals of our scenario mainly come from
$pp\to\tilde{\chi}^{\pm}_{1}\tilde{\chi}^{\mp}_{1},\tilde{\chi}^{\pm}_{1}\tilde{\chi}^{0}_{2}$
followed by $\tilde{\chi}^{\pm}_{1}\to\tilde{\chi}^{0}_{2}\pi^{\pm}$ and
$\tilde{\chi}^{0}_{2}\to\tilde{\chi}^{0}_{1}h$ which both generate the long-
lived particle (LLP) signals. The LLP signatures manifest as disappearing
track for $\tilde{\chi}^{\pm}_{1}\to\tilde{\chi}^{0}_{2}\pi^{\pm}$ and
displaced vertices for $\tilde{\chi}^{0}_{2}\to\tilde{\chi}^{0}_{1}h$,
respectively. However,
$\tau_{\tilde{\chi}^{0}_{2}\to\tilde{\chi}^{0}_{1}h}>\mathcal{O}(10^{-2})\,\text{s}$
would make $\tilde{\chi}^{0}_{2}$ traverse through the whole detector before
decaying without leaving any energy deposit in the calorimeters, thus can
easily evade the current ATLAS [54] and CMS [55] searches for displaced vertex
signals at $\sqrt{s}=13\,\text{TeV}$. As for the disappearing track signature
of $\tilde{\chi}^{\pm}_{1}\to\tilde{\chi}^{0}_{2}\pi^{\pm}$, ATLAS [56] and
CMS [57] also performed dedicated searches using dataset at
$\sqrt{s}=13\,\text{TeV}$ and imply that
$\tilde{\chi}^{\pm}_{1},\tilde{\chi}^{0}_{2}$ should be heavier than 500-600
GeV, therefore our benchmark points with $M_{2}=2\,\text{TeV}$ are still
available.
## 5 Conclusion
We studied a scenario of dark matter generated from UV freeze-in mechanism,
realized in the framework of high scale MSSM. The bino is the dark matter
candidate and its relic abundance is generated by the freeze-in processes via
the dim-5 or dim-6 operators. We found that the SUSY scale $M_{\text{SUSY}}$
should be around $10^{13-15}~{}\text{GeV}$ for reheating temperature in the
range of $10^{4-6}~{}\text{GeV}$. We also illustrated the interplay between
BBN constraints from neutral wino decay and the experimentally observed dark
matter relic abundance, implying an upper bound of $M_{\text{SUSY}}$ around
$10^{14}~{}\text{GeV}$ for wino mass around 2 TeV and bino mass of $0.1\sim 1$
TeV.
## Acknowledgments
This work was supported by the Natural Science Foundation of China (NSFC)
under grant numbers 12105118, 11947118, 12075300, 11821505 and 12335005, the
Peng-Huan-Wu Theoretical Physics Innovation Center (12047503), the CAS Center
for Excellence in Particle Physics (CCEPP), and the Key Research Program of
the Chinese Academy of Sciences under grant No. XDPB15. CH acknowledges
support from the Sun Yat-Sen University Science Foundation and the Fundamental
Research Funds for the Central Universities, Sun Yat-sen University under
Grant No. 23qnpy58. PW acknowledges support from Natural Science Foundation of
Jiangsu Province (Grant No. BK20210201), Fundamental Research Funds for the
Central Universities, Excellent Scholar Project of Southeast University (Class
A), and the Big Data Computing Center of Southeast University. PW also
acknowledges his wife for being tolerant and supportive (freezing in or
squeezing in time to give birth to a son the same day the work was finished).
## Appendix A Notation conventions and dim-5 operator in Case I
In Eq.(2.1), the dot product means
$\tilde{H}_{u}\cdot\tilde{H}_{d}=\tilde{H}_{u,i}(i\sigma^{2})^{ij}\tilde{H}_{d,j}=\tilde{H}^{+}_{u}\tilde{H}^{-}_{d}-\tilde{H}^{0}_{u}\tilde{H}^{0}_{d}$
to realize the isospin symmetry ${\rm SU(2)_{L}}$ where $\sigma^{2}$ is the
second Pauli matrix. The Kronecker delta function $\delta_{i}^{\,\,\,j}$
manifests the ${\rm SU(2)_{L}}$-blindness of the ${\rm U(1)_{Y}}$ interactions
under consideration for binos production and $Y_{H_{u}}=+1/2,Y_{H_{d}}=-1/2$
are the hypercharges of doublets $H_{u},H_{d}$, respectively. We follow the
convention of [39] and impose the left-chiral two-component spinor formalism
for higgisnos
$\tilde{H}^{+}_{u},\tilde{H}^{0}_{u},\tilde{H}^{0}_{d},\tilde{H}^{-}_{d}$ and
bino $\tilde{B}$ (as well as winos $\tilde{W}$ and gluinos $\tilde{g}$ in
later discussion). For the Case I in Section 3.2, the relevant Lagrangian
terms are
$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle-\frac{1}{2}M_{1}\tilde{B}\tilde{B}-\mu\left(\tilde{H}^{+}_{u}\tilde{H}^{-}_{d}-\tilde{H}^{0}_{u}\tilde{H}^{0}_{d}\right)+h.c.$
(A.1)
$\displaystyle-\frac{g_{1}}{\sqrt{2}}({H}^{+}_{u})^{\ast}\tilde{H}^{+}_{u}\tilde{B}-\frac{g_{1}}{\sqrt{2}}({H}^{0}_{u})^{\ast}\tilde{H}^{0}_{u}\tilde{B}+\frac{g_{1}}{\sqrt{2}}({H}^{-}_{d})^{\ast}\tilde{H}^{-}_{d}\tilde{B}+\frac{g_{1}}{\sqrt{2}}({H}^{0}_{d})^{\ast}\tilde{H}^{0}_{d}\tilde{B}+h.c.~{},$
After integrating out higgsinos with mass $\mu$, we obtain dim-5 operator
between SM Higgs $H_{\text{SM}}$ and $\tilde{B}$ DM:
$\displaystyle\mathcal{L}^{\text{eff}}_{HH^{\ast}\to\tilde{B}\tilde{B}}$
$\displaystyle=$
$\displaystyle-\frac{(\sqrt{2}g_{1}Y_{H})(\sqrt{2}g_{1}Y_{H})}{\mu}(H_{u}^{\ast}\cdot
H_{d}^{\ast})\tilde{B}\tilde{B}+h.c.$ (A.2) $\displaystyle=$
$\displaystyle-\frac{2g_{1}^{2}\,Y_{H}^{2}}{\mu}\sin\beta\cos\beta(H^{\ast}_{\rm
SM}\cdot i\sigma^{2}H_{\rm
SM})(\tilde{B}\tilde{B}+\tilde{B}^{\dagger}\tilde{B}^{\dagger})$
$\displaystyle=$
$\displaystyle\frac{2g_{1}^{2}\,Y_{H}^{2}}{\mu}\sin\beta\cos\beta(|H_{\rm
SM}|^{2})(\tilde{B}\tilde{B}+\tilde{B}^{\dagger}\tilde{B}^{\dagger})~{},$
where $Y_{H}=|Y_{H_{u}}|=|Y_{H_{d}}|=1/2$ and the dot products are
$H_{u}^{\ast}\cdot
H_{d}^{\ast}=({H}^{+}_{u})^{\ast}({H}^{-}_{d})^{\ast}-({H}^{0}_{u})^{\ast}({H}^{0}_{d})^{\ast}$.
## Appendix B Boltzmann equation and calculation details of freeze-in DM in
Case I
In the homogeneous and isotropic universe, the production of bino is described
by following Boltzmann equation [33]:
$\displaystyle\frac{d}{dt}n_{\tilde{B}}+3\mathcal{H}n_{\tilde{B}}=\textbf{C}~{},$
(B.1)
with $n_{\tilde{B}}$ denoting the number density of bino particle, and
$\mathcal{H}$ is the Hubble expansion rate. Taking
$HH^{\ast}\to{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{B}\tilde{B}}$
($\tilde{B}$ means the physical bino particle) in Case I of Section 3.2 as an
example, we have [58]
$\displaystyle\textbf{C}_{ij\to kl}$ $\displaystyle=$ $\displaystyle
N\times\frac{1}{S}\times\bigg{\\{}\int\frac{d^{3}p_{i}}{(2\pi)^{3}2E_{i}}\frac{d^{3}p_{j}}{(2\pi)^{3}2E_{j}}\frac{d^{3}p_{k}}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{l}}{(2\pi)^{3}2E_{l}}$
(B.2) $\displaystyle\times(2\pi)^{4}\delta^{4}(p_{i}+p_{j}-p_{k}-p_{l})\
\bigg{[}f_{i}f_{j}(1-f_{k})(1-f_{l})-f_{k}f_{l}(1+f_{i})(1+f_{j})\bigg{]}$
$\displaystyle\times\sum_{\rm internal\,d.o.f}|\mathcal{M}|^{2}_{ij\to kl}\
\bigg{\\}}~{},$
where $f_{i,j,k,l}$ are the phase space distribution functions. The number
density, taking $f_{i}$ as example, is defined as
$\displaystyle n_{i}\equiv g_{i}\int\frac{d^{3}p}{(2\pi)^{3}}f_{i}(p)~{},$
(B.3)
in which $g_{i}$ is the internal degree of freedom (d.o.f.) of particle $i$.
The factor $N$ denotes the number of particles under consideration produced in
the final state and the factor $1/S$ originates from the phase space
suppression due to the identical particles in the initial and final states.
For
$HH^{\ast}\to{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{B}\tilde{B}}$
we have $N=2$ and $1/S=1/(N!)=1/2$. After some manipulations and neglecting
the negligible backward process, we have [58]
$\displaystyle\textbf{C}_{ij\to kl}$ $\displaystyle\approx$
$\displaystyle\frac{T}{32\pi^{4}}\int^{\infty}_{(m_{k}+m_{l})^{2}}ds\,p_{ij}\,W_{ij\to
kl}\,K_{1}({\sqrt{s}}/{T})$ (B.4) $\displaystyle W_{ij\to kl}$
$\displaystyle=$ $\displaystyle\frac{p_{kl}}{16\pi^{2}\sqrt{s}}\,\sum_{\rm
internal\,d.o.f}\int\,d\Omega\,|\mathcal{M}|^{2}_{ij\to kl}$ (B.5)
$\displaystyle p_{ij}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{s-(m_{i}+m_{j})^{2}}\sqrt{s-(m_{i}-m_{j})^{2}}}{2\sqrt{s}}~{},$
(B.6)
where $p_{kl}$ is similar to $p_{ij}$. After summing over all bino spin states
$s_{1},s_{2}$ and isospin states of the SM-like Higgs , we have the amplitude
square ($s$ is the square of the central energy):
$\displaystyle\sum_{\rm internal\,d.o.f}\int d\Omega\
|\mathcal{M}|^{2}_{HH^{\ast}\to\tilde{B}\tilde{B}}~{},$
$\displaystyle\approx(2\pi)\times\bigg{[}\sum_{i,j=1}^{2}(\delta_{i}^{\,\,\,j})^{2}\bigg{]}\bigg{[}Y^{4}_{H}\bigg{]}\bigg{(}\frac{g_{1}g_{2}\sin\beta\cos\beta}{\mu}\bigg{)}^{2}\bigg{[}64\
s\bigg{(}1-\frac{4M_{1}^{2}}{s}\bigg{)}^{\frac{3}{2}}\bigg{]}$
$\displaystyle\approx(16\pi)\times\frac{g_{1}^{4}}{\mu^{2}}\sin^{2}\beta\cos^{2}\beta\
s~{}.$ (B.7)
We modify the MSSM model file available in FeynRules [59, 60] to highlight the
gauge state interactions and then export to FeynArts [61] augmented with
FeynCalc [62] to perform the calculation.
Since we are considering freeze-in production of $\tilde{B}$, $f_{1,2}$ in Eq.
(B.2) can be ignored. We can further approximate $f_{3,4}$ by Maxwell-
Boltzmann distribution, i.e. $f_{3,4}\approx e^{-E_{3,4}/T}$. Then the
collision term can be rewritten as [58, 63, 40]
$\displaystyle\textbf{C}_{HH^{\ast}\to\tilde{B}\tilde{B}}$
$\displaystyle\approx$
$\displaystyle\frac{T}{2048\pi^{6}}\int^{\infty}_{4M_{1}^{2}}ds\
(s-4M_{1}^{2})^{1/2}K_{1}({\sqrt{s}}/{T})\sum_{\rm
internal\,d.o.f}\int\,d\Omega\,|\mathcal{M}|^{2}_{HH^{\ast}\to\tilde{B}\tilde{B}}~{},$
(B.8) $\displaystyle\approx$
$\displaystyle\frac{T}{128\pi^{5}}\frac{g_{1}^{4}\,\sin^{2}\beta\cos^{2}\beta}{\mu^{2}}\int^{\infty}_{4M_{1}^{2}}ds\
s^{3/2}K_{1}({\sqrt{s}}/{T})~{}.$
Here $K_{1}$ is the Bessel function of the second kind, and we treat the SM-
like Higgs in the initial state as being massless. In the case where $M_{1}\ll
T$, the collision term can be approximated as (using
$\int^{\infty}_{0}dxx^{4}K_{1}(x)=16$)
$\displaystyle\int^{\infty}_{4M_{1}^{2}}ds\ s^{3/2}K_{1}({\sqrt{s}}/{T})$
$\displaystyle\approx$ $\displaystyle\int^{\infty}_{0}(dx\,T)\
(2xT)(xT)^{3}K_{1}(x)$ (B.9) $\displaystyle=$ $\displaystyle
2\,T^{5}\int^{\infty}_{0}dx\,x^{4}K_{1}(x)=32\ T^{5}~{}.$
## Appendix C The calculation details in Case II
We use $f=q,l$ with $q=u_{L},d_{L},u^{\dagger}_{R},d^{\dagger}_{R}$ and
$l=\nu,e_{L},e^{\dagger}_{R}$ to denote the left-handed two-component Weyl
spinor of SM quarks and leptons, where the bars are simply notations and do
not mean the Dirac conjugation. Hypercharges are given by
$\\{Y_{Q_{L}}=Y_{u_{L}}=Y_{d_{L}},\ Y_{{u^{\dagger}_{R}}},\
Y_{{d^{\dagger}_{R}}},\ Y_{L_{L}}=Y_{e_{L}}=Y_{\nu},\
Y_{{e^{\dagger}_{R}}}\\}=\\{1/6,\ -2/3,\ 1/3,\ -1/2,\ 1\\}$. After integrating
out sfermions with mass $M_{\tilde{f}}$ in the right panel of Fig.1, we obtain
dim-6 operators between SM fermion pair and $\tilde{B}$ pair:
$\displaystyle\mathcal{L}_{\text{eff}}=\sum_{f=q,l}\frac{(\sqrt{2}g_{1}Y_{f})(\sqrt{2}g_{1}Y_{f})}{M^{2}_{\tilde{f}}}(f^{\dagger}\tilde{B}^{\dagger})(f\tilde{B})~{},$
(C.1)
where for simplicity we consider an universal mass for all the fermions, i.e.
$M_{\tilde{f}}\equiv M_{\tilde{q}}=M_{\tilde{l}}$.
The amplitude squared terms in the collision term for
$f\bar{f}\to\tilde{B}\tilde{B}$ scattering process is given by666 Again,
fields in the initial and final states in the process should be understood in
the sense of physical particles, where $\bar{f}$ denotes the physical anti-
particle. Discussion on the naming convention of particles, states and filed
can be found in, e.g. [39].
$\displaystyle\sum_{\rm internal\,d.o.f}\int d\Omega\
|\mathcal{M}|^{2}_{f\bar{f}\to\tilde{B}\tilde{B}}$ $\displaystyle\approx$
$\displaystyle 2\pi N_{\rm flavor}\bigg{[}N_{\rm
color}\bigg{(}\sum_{i,j=1}^{2}(\delta_{i}^{\,\,\,j})^{2}Y^{4}_{Q_{L}}+Y_{{u^{\dagger}_{R}}}^{4}+Y_{{d^{\dagger}_{R}}}^{4}\bigg{)}$
(C.2)
$\displaystyle+\bigg{(}\sum_{i,j=1}^{2}(\delta_{i}^{\,\,\,j})^{2}Y^{4}_{L_{L}}+Y_{{e^{\dagger}_{R}}}^{4}\bigg{)}\bigg{]}\bigg{(}\frac{g_{1}^{2}}{M^{2}_{\tilde{f}}}\bigg{)}^{2}\bigg{[}\frac{16}{3}\
s^{2}\bigg{]}$ $\displaystyle=$
$\displaystyle\frac{1520\pi}{27}\frac{g_{1}^{4}}{M^{4}_{\tilde{f}}}\ s^{2}$
where $N_{\rm flavor}=N_{\rm color}=3$. As in Eq. (B.2), if we neglect bino
mass, then the collision term can be approximately given by (using
$\int^{\infty}_{0}dxx^{6}K_{1}(x)=384$)
$\displaystyle\textbf{C}_{f\bar{f}\to\tilde{B}\tilde{B}}$
$\displaystyle\approx$
$\displaystyle\frac{T}{2048\pi^{6}}\int^{\infty}_{4M_{1}^{2}}ds\
(s-4M_{1}^{2})^{1/2}K_{1}({\sqrt{s}}/{T})\sum_{\rm
internal\,d.o.f}\int\,d\Omega\,|\mathcal{M}|^{2}_{ff^{\dagger}\to\tilde{B}\tilde{B}^{\dagger}}$
(C.3) $\displaystyle\approx$
$\displaystyle\frac{T}{2048\pi^{6}}\bigg{(}\frac{1520\pi}{27}\frac{g_{1}^{4}}{M^{4}_{\tilde{f}}}\bigg{)}\int^{\infty}_{4M_{1}^{2}}ds\
s^{5/2}K_{1}({\sqrt{s}}/{T})$ $\displaystyle\approx$
$\displaystyle\frac{T}{2048\pi^{6}}\bigg{(}\frac{1520\pi}{27}\frac{g_{1}^{4}}{M^{4}_{\tilde{f}}}\bigg{)}\int^{\infty}_{0}(Tdx)(2Tx)\
(xT)^{5}K_{1}({x})$ $\displaystyle=$
$\displaystyle\frac{190}{9}g_{1}^{4}\frac{1}{\pi^{5}}\frac{1}{M^{4}_{\tilde{f}}}T^{8}~{}.$
## Appendix D The calculation details in Case III A
When neglecting all particle masses in the final state, we have
$\displaystyle\sum_{\rm internal\,d.o.f}\int d\Omega\
|\mathcal{M}|^{2}_{HH^{\ast}\to\tilde{B}\tilde{W}}$
$\displaystyle\approx(2\pi)\times\bigg{[}\sum_{b=1}^{3}\text{tr}\bigg{(}\frac{1}{2}\sigma^{b}\frac{1}{2}\sigma^{b}\bigg{)}\bigg{]}\bigg{[}Y^{2}_{H}\bigg{]}\bigg{(}\frac{g_{1}g_{2}\sin\beta\cos\beta}{\mu}\bigg{)}^{2}\bigg{[}64\
s\bigg{]}$
$\displaystyle=(48\pi)\times\frac{g_{1}^{2}g_{2}^{2}}{\mu^{2}}\sin^{2}\beta\cos^{2}\beta\
s~{},$ (D.1) $\displaystyle\sum_{\rm internal\,d.o.f}\int d\Omega\
|\mathcal{M}|^{2}_{\tilde{W}H\to\tilde{B}H}=\sum_{\rm internal\,d.o.f}\int
d\Omega\ |\mathcal{M}|^{2}_{\tilde{W}H^{\ast}\to\tilde{B}H^{\ast}}$
$\displaystyle\approx(2\pi)\times\bigg{[}\sum_{b=1}^{3}\text{tr}\bigg{(}\frac{1}{2}\sigma^{b}\frac{1}{2}\sigma^{b}\bigg{)}\bigg{]}\bigg{[}Y^{2}_{H}\bigg{]}\bigg{(}\frac{g_{1}g_{2}\sin\beta\cos\beta}{\mu}\bigg{)}^{2}\bigg{[}32\
s\bigg{]}$
$\displaystyle=(24\pi)\times\frac{g_{1}^{2}g_{2}^{2}}{\mu^{2}}\sin^{2}\beta\cos^{2}\beta\
s~{},$ (D.2) $\displaystyle\sum_{\rm internal\,d.o.f}\int d\Omega\
|\mathcal{M}|^{2}_{f\bar{f}\to\tilde{B}\tilde{W}}$ $\displaystyle\approx
2\pi\bigg{[}\sum_{b=1}^{3}\text{tr}\bigg{(}\frac{1}{2}\sigma^{b}\frac{1}{2}\sigma^{b}\bigg{)}\bigg{]}\bigg{[}N_{\rm
flavor}\bigg{(}Y^{2}_{L_{L}}+N_{\rm
color}Y^{2}_{Q_{L}}\bigg{)}\bigg{]}\bigg{(}\frac{g_{1}g_{2}}{M^{2}_{\tilde{f}}}\bigg{)}^{2}\bigg{[}\frac{16}{3}\
s^{2}\bigg{]}$
$\displaystyle=(16\pi)\times\frac{g_{1}^{2}g_{2}^{2}}{M^{4}_{\tilde{f}}}\
s^{2}~{},$ (D.3) $\displaystyle\sum_{\rm internal\,d.o.f}\int d\Omega\
|\mathcal{M}|^{2}_{\tilde{W}f\to\tilde{B}f}=\sum_{\rm internal\,d.o.f}\int
d\Omega\ |\mathcal{M}|^{2}_{\tilde{W}\bar{f}\to\tilde{B}\bar{f}}$
$\displaystyle\approx
2\pi\bigg{[}\sum_{b=1}^{3}\text{tr}\bigg{(}\frac{1}{2}\sigma^{b}\frac{1}{2}\sigma^{b}\bigg{)}\bigg{]}\bigg{[}N_{\rm
flavor}\bigg{(}Y^{2}_{L_{L}}+N_{\rm
color}Y^{2}_{Q_{L}}\bigg{)}\bigg{]}\bigg{(}\frac{g_{1}g_{2}}{M^{2}_{\tilde{f}}}\bigg{)}^{2}\bigg{[}\frac{32}{3}\
s^{2}\bigg{]}$
$\displaystyle=(32\pi)\times\frac{g_{1}^{2}g_{2}^{2}}{M^{4}_{\tilde{f}}}\
s^{2}~{},$ (D.4) $\displaystyle\sum_{\rm internal\,d.o.f}\int d\Omega\
|\mathcal{M}|^{2}_{f\bar{f}\to\tilde{B}\tilde{G}}$ $\displaystyle\approx
2\pi\bigg{[}\sum_{a=1}^{8}\text{tr}\bigg{(}\frac{1}{2}\lambda^{a}\frac{1}{2}\lambda^{a}\bigg{)}\bigg{]}\bigg{[}N_{\rm
flavor}\bigg{(}\sum_{i,j=1}^{2}(\delta_{i}^{\,\,\,j})^{2}Y^{2}_{Q_{L}}+Y_{{u^{\dagger}_{R}}}^{2}+Y_{{d^{\dagger}_{R}}}^{2}\bigg{)}\bigg{]}\bigg{(}\frac{g_{1}g_{3}}{M^{2}_{\tilde{f}}}\bigg{)}^{2}\bigg{[}\frac{16}{3}\
s^{2}\bigg{]}$
$\displaystyle=(\frac{704\pi}{9})\times\frac{g_{1}^{2}g_{3}^{2}}{M^{4}_{\tilde{f}}}\
s^{2}~{},$ (D.5) $\displaystyle\sum_{\rm internal\,d.o.f}\int d\Omega\
|\mathcal{M}|^{2}_{\tilde{G}f\to\tilde{B}f}=\sum_{\rm internal\,d.o.f}\int
d\Omega\ |\mathcal{M}|^{2}_{\tilde{G}\bar{f}\to\tilde{B}\bar{f}}$
$\displaystyle\approx
2\pi\bigg{[}\sum_{a=1}^{8}\text{tr}\bigg{(}\frac{1}{2}\lambda^{a}\frac{1}{2}\lambda^{a}\bigg{)}\bigg{]}\bigg{[}N_{\rm
flavor}\bigg{(}\sum_{i,j=1}^{2}(\delta_{i}^{\,\,\,j})^{2}Y^{2}_{Q_{L}}+Y_{{u^{\dagger}_{R}}}^{2}+Y_{{d^{\dagger}_{R}}}^{2}\bigg{)}\bigg{]}\bigg{(}\frac{g_{1}g_{3}}{M^{2}_{\tilde{f}}}\bigg{)}^{2}\bigg{[}\frac{32}{3}\
s^{2}\bigg{]}$
$\displaystyle=(\frac{1408\pi}{9})\times\frac{g_{1}^{2}g_{3}^{2}}{M^{4}_{\tilde{f}}}\
s^{2}~{}.$ (D.6)
## Appendix E The calculation details in Case III B
The $1\to 3$ decay processes are indicated by the red colored arrow in Fig.1.
When neglecting all particle masses in the final state, we have
$\displaystyle\Gamma_{\tilde{W}\to\tilde{B}HH^{\ast}}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{32M_{2}^{3}}\
\frac{1}{g_{\tilde{W}}}\sum_{\text{internal d.o.f.}}\int dm^{2}_{12}\
dm^{2}_{23}\ |\mathcal{M}|^{2}_{\tilde{W}\to\tilde{B}HH^{\ast}}$ (E.1)
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{32M_{2}^{3}}\
\frac{1}{\sum_{b=1}^{3}(2s_{\tilde{W}}+1)}\bigg{[}\sum_{b=1}^{3}\text{tr}\bigg{(}\frac{1}{2}\sigma^{b}\frac{1}{2}\sigma^{b}\bigg{)}\bigg{]}$
$\displaystyle\times\bigg{[}Y^{2}_{H}\bigg{]}\bigg{[}\frac{32}{3}M_{2}^{6}\left(\frac{g_{1}g_{2}\sin\beta\cos\beta}{\mu}\right)^{2}\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{1}{384\pi^{3}}\left(\frac{g_{1}g_{2}\sin\beta\cos\beta}{\mu}\right)^{2}M_{2}^{3}~{}.$
$\displaystyle\Gamma_{\tilde{W}\to\tilde{B}f\bar{f}}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{32M_{2}^{3}}\
\frac{1}{g_{\tilde{W}}}\sum_{\text{internal d.o.f.}}\int dm^{2}_{12}\
dm^{2}_{23}\ |\mathcal{M}|^{2}_{\tilde{W}\to\tilde{B}f\bar{f}}$ (E.2)
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{32M_{2}^{3}}\
\frac{1}{\sum_{b=1}^{3}(2s_{\tilde{W}}+1)}\bigg{[}\sum_{b=1}^{3}\text{tr}\bigg{(}\frac{1}{2}\sigma^{b}\frac{1}{2}\sigma^{b}\bigg{)}\bigg{]}$
$\displaystyle\times\bigg{[}N_{\rm flavor}\bigg{(}Y^{2}_{L_{L}}+N_{\rm
color}Y^{2}_{Q_{L}}\bigg{)}\bigg{]}\bigg{[}\frac{2}{3}M_{2}^{8}\left(\frac{g_{1}g_{2}}{M^{2}_{\tilde{f}}}\right)^{2}\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{1}{1536\pi^{3}}\left(\frac{g_{1}g_{2}}{M^{2}_{\tilde{f}}}\right)^{2}M_{2}^{5}~{},$
$\displaystyle\Gamma_{\tilde{G}\to\tilde{B}f\bar{f}}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{32M_{3}^{3}}\
\frac{1}{g_{\tilde{G}}}\sum_{\text{internal d.o.f.}}\int dm^{2}_{12}\
dm^{2}_{23}\ |\mathcal{M}|^{2}_{\tilde{G}\to\tilde{B}f\bar{f}}$ (E.3)
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{32M_{3}^{3}}\
\frac{1}{\sum_{a=1}^{8}(2s_{\tilde{G}}+1)}\bigg{[}\sum_{a=1}^{8}\text{tr}\bigg{(}\frac{1}{2}\lambda^{a}\frac{1}{2}\lambda^{a}\bigg{)}\bigg{]}$
$\displaystyle\times\bigg{[}N_{\rm
flavor}\bigg{(}N_{\text{iso},Q_{L}}Y^{2}_{Q_{L}}+Y_{{u^{\dagger}_{R}}}^{2}+Y_{{d^{\dagger}_{R}}}^{2}\bigg{)}\bigg{]}\bigg{[}\frac{2}{3}M_{2}^{8}\left(\frac{g_{1}g_{3}}{M^{2}_{\tilde{f}}}\right)^{2}\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{11}{9216\pi^{3}}\left(\frac{g_{1}g_{3}}{M^{2}_{\tilde{f}}}\right)^{2}M_{2}^{5}~{}.$
where $dm^{2}_{12},dm^{2}_{23}$ are defined in [64].
## Appendix F The calculation details of 2-body decay after EWSB
As discussed in Section 4.1, we have the following $1\to 2$ decay possibly
affecting the cosmological BBN:
$\displaystyle\Gamma_{\tilde{\chi}^{0}_{2}\to\tilde{\chi}^{0}_{1}h}$
$\displaystyle=$
$\displaystyle\frac{1}{2s_{\tilde{\chi}^{0}_{2}}+1}\frac{1}{2M_{2}}\sum_{\text{spin
d.o.f.}}\int d\Pi_{2}\ |M|^{2}_{\tilde{\chi}^{0}_{2}\to\tilde{\chi}^{0}_{1}h}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\frac{1}{2M_{2}}\ \int d\Pi_{2}\
\left(\frac{v}{\mu}g_{1}g_{2}\sin\beta\cos\beta\right)^{2}\bigg{[}4(p_{\tilde{\chi}^{0}_{2}}\cdot
p_{\tilde{\chi}^{0}_{1}}+M_{\tilde{\chi}^{0}_{2}}M_{\tilde{\chi}^{0}_{1}})\bigg{]}$
$\displaystyle\approx$ $\displaystyle\frac{1}{2}\frac{1}{2M_{2}}\ \bigg{[}\int
d\Omega\frac{1}{16\pi^{2}}\frac{|\vec{p}_{\tilde{\chi}^{0}_{1}}|}{M_{2}}\bigg{]}\
\left(\frac{v}{\mu}g_{1}g_{2}\sin\beta\cos\beta\right)^{2}\bigg{[}4(M_{2}E_{\tilde{\chi}^{0}_{1}}+M_{2}M_{1})\bigg{]}~{},$
where
$\displaystyle E_{\tilde{\chi}^{0}_{1}}$ $\displaystyle=$
$\displaystyle\frac{M_{2}^{2}+M_{1}^{2}-M_{h}^{2}}{2M_{2}}\approx\frac{M_{2}^{2}+M_{1}^{2}}{2M_{2}}~{},$
(F.2) $\displaystyle|\vec{p}_{\tilde{\chi}^{0}_{1}}|$ $\displaystyle=$
$\displaystyle\sqrt{E_{\tilde{\chi}^{0}_{1}}^{2}-M_{1}^{2}}=\frac{\left(M_{2}^{4}+M_{1}^{4}+M_{h}^{4}-2M_{2}^{2}M_{1}^{2}-2M_{2}^{2}M_{h}^{2}-2M_{1}^{2}M_{h}^{2}\right)^{\frac{1}{2}}}{2M_{2}}$
(F.3) $\displaystyle\approx$
$\displaystyle\frac{M_{2}^{2}-M_{1}^{2}}{2M_{2}}~{}.$
Finally, we have [44]
$\displaystyle\Gamma_{\tilde{\chi}^{0}_{2}\to\tilde{\chi}^{0}_{1}h}$
$\displaystyle\approx$
$\displaystyle\frac{1}{2}\frac{1}{2M_{2}}\Bigg{[}4\pi\frac{1}{16\pi^{2}}\frac{1}{2}\left(1-\frac{M_{1}^{2}}{M_{2}^{2}}\right)\bigg{]}\left(\frac{v}{\mu}g_{1}g_{2}\sin\beta\cos\beta\right)^{2}\bigg{[}4M_{2}\frac{(M_{2}+M_{1})^{2}}{2M_{2}}\bigg{]}$
(F.4) $\displaystyle\approx$ $\displaystyle
M_{2}\frac{1}{16\pi}\left(\frac{v}{\mu}g_{1}g_{2}\sin\beta\cos\beta\right)^{2}\left(1-\frac{M_{1}^{2}}{M_{2}^{2}}\right)\left(1+\frac{M_{1}}{M_{2}}\right)^{2}~{}.$
Using the GET we would obtain the same results in the high energy limit for
$\Gamma_{\tilde{\chi}^{\pm}_{1}\to\tilde{\chi}^{0}_{1}W^{\pm}}$.
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|
# A Framework using Contrastive Learning for Classification with Noisy Labels
Madalina Ciortan§
EURA NOVA BE
Mont-Saint-Guibert, Belgium
<EMAIL_ADDRESS>
&Romain Dupuis§
EURA NOVA BE
Mont-Saint-Guibert, Belgium
<EMAIL_ADDRESS>
&Thomas Peel
EURA NOVA BE
Mont-Saint-Guibert, Belgium
<EMAIL_ADDRESS>
###### Abstract
We propose a framework using contrastive learning as a pre-training task to
perform image classification in the presence of noisy labels. Recent
strategies such as pseudo-labelling, sample selection with Gaussian Mixture
models, weighted supervised contrastive learning have been combined into a
fine-tuning phase following the pre-training. This paper provides an extensive
empirical study showing that a preliminary contrastive learning step brings a
significant gain in performance when using different loss functions: non
robust, robust, and early-learning regularized. Our experiments performed on
standard benchmarks and real-world datasets demonstrate that: i) the
contrastive pre-training increases the robustness of any loss function to
noisy labels and ii) the additional fine-tuning phase can further improve
accuracy, but at the cost of additional complexity.
§§footnotetext: Equal contribution
## 1 Introduction
Collecting large and well-annotated datasets for image classification tasks
represents a challenge as human quality annotations are expensive and time-
consuming. Alternative methods exist, such as web crawlers [27]. Nevertheless,
these methods generate noisy labels decreasing the performance of deep neural
networks. They tend to overfit to noisy labels due to their high capacity
[44]. That is why developing efficient noisy-label learning (NLL) techniques
is of great importance.
Various strategies have been proposed to deal with NLL: i) Noise transition
matrix [33, 9, 41] estimates the noise probability and corrects the loss
function, ii) a small and clean subset can help to avoid overfitting [14],
iii) samples selection identifies true-labeled samples [15, 10, 22], and iv)
robust loss functions solve the classification problem only by adapting the
loss function to be less sensitive to noisy labels [47, 37, 26]. Methods also
combine other strategies (eg. ELR+ [22], DivideMix [25]): two networks, semi-
supervised learning, label correction, or mixup. They show the most promising
results but lead to a large number of hyperparameters. That is why we explore
improvement strategies for robust loss functions. They are simpler to
integrate and faster to train, but as illustrated in Figure 1, they tend to
overfit and have lower performance for high noise ratios.
Figure 1: Top-1 test accuracy for a ResNet18 trained on the CIFAR-100 dataset
with a symmetric noise of 80% for three losses: Cross Entropy (CE), Normalized
Focal Loss + Reverse Cross Entropy (NFL+RCE), and Early Learning
Regularization (ELR).
Meanwhile, new self-supervised learning algorithms for image representations
have been recently developed [5, 12]. Such algorithms extract representation
(or features) in unsupervised settings. These representations can then be used
for downstream tasks such as classification. Methods based on contrastive
learning compete with fully supervised learning while fine-tuning only on a
small fraction of all available labels. Therefore, using contrastive learning
for NLL appears as promising. In this work, contrastive learning aims to pre-
train the classifier to improve its robustness.
The key contributions of this work are:
* •
A framework increasing robustness of any loss function to noisy labels by
adding a contrastive pre-training task.
* •
The adaptation of the supervised contrastive loss to use sample weight values,
representing the probability of correctness for each sample in the training
set
* •
An extensive empirical study identifying and benchmarking additional state of
the art strategies to boost the performance of pre-trained models: pseudo-
labeling, sample selection with GMM, weighted supervised contrastive learning,
and mixup with bootstrapping.
## 2 Related works
Existing approaches dealing with NLL and contrastive learning in computer
vision are briefly reviewed. Extra details can be found in Song et al. [36],
Le-Khac et al. [21].
### 2.1 Noise tolerant classification
Sample Selection: This method identifies noisy and clean samples within the
training data. Several strategies leverage the interactions between multiple
networks to identify the probably correct labels [10, 15, 22]. Recent works
[1, 35] exploit the small loss trick to identify clean and noisy samples by
considering a certain number of small-loss training samples as true-labeled
samples. This approach can be justified by the memorization effect: deep
neural networks first fit the training data with clean labels during a so-
called early learning phase, before overfitting the noisy samples during the
memorization phase [2, 25].
Robust Loss Function: Commonly used loss functions, such as Cross Entropy (CE)
or Focal Loss, are not robust to noisy labels. Therefore, new loss functions
have been designed. Such robust loss functions can be easily incorporated into
existing pipelines to improve performance regarding noisy labels. The
symmetric cross entropy [37] has been proposed by adding a reverse CE loss to
the initial CE. This combination improves the accuracy of the model compared
to classical loss functions. Ma et al. [26] show theoretically that
normalization can convert classical loss functions into loss functions robust
to noise labels. The combination of two robust loss functions can also improve
robustness. However, the performance of normalized loss functions remains
quite low for high noise rates as illustrated in Figure 1.
Semi-supervised: Semi-supervised approaches deal with both labeled and
unlabeled data. Recent works [30, 22, 38] combine sample selection with semi-
supervised methods: the possibly noisy samples are treated as unlabeled and
the possibly clean samples are treated as labeled. Such approaches leverage
information contained in noisy data, for instance by using MixMatch [3]. Semi-
supervised approaches show competitive results. However, they use several
hyperparameters that can be sensitive to changes in data or noise type [36,
31].
Contrastive learning: recent developments in self-supervised and contrastive
learning [46, 31, 23] inspire new approaches in NLL. Li et al. [23] employed
features learned by contrastive learning to detect out-of-distribution
samples.
### 2.2 Contrastive learning for vision data
Contrastive learning extracts features by comparing each data sample with
different samples. The central idea is to bring different instances of the
same input image closer and spread instances from different images apart. The
inputs are usually divided into positive (similar inputs) and negative pairs
(dissimilar inputs). Frameworks have been recently developed, such as CPCv2
[13], SimCLR [5], Moco [12]. Once the self-supervised model is trained, the
extracted representations can be used for downstream tasks.In this work, the
representations are used for noisy label classification.
Chen et al. [5] demonstrate that large sets of negatives (and large batches)
are crucial in learning good representations. However, large batches are
limited by GPU memory. Maintaining a memory bank accumulating a large number
of negative representations is an elegant solution decoupling the batch size
from the number of negatives [28]. Nevertheless, the representations get
outdated in a few iterations. The Momentum Encoder [12] addresses the issues
by generating a dynamic memory queue of representations. Other strategies aim
at getting more meaningful negative samples to reduce the memory/batch size
[16].
## 3 Preliminaries
Let
$D=\\{(\bm{x_{i}},\overline{y_{i}})\\}_{i=1..n},\bm{x_{i}}\in\mathbb{R}^{d},\overline{y_{i}}\in\\{1,\cdots,K\\}$
denote a noisy input dataset with an unknown number of samples incorrectly
labelled. The associated true and unobservable labels are written $y_{i}$. The
images $\bm{x_{i}}$ are of size $d$ and the classification problem has $K$
classes. The goal is to train a deep neural network (DNN) $f$. Using a robust
loss function for training consists of minimizing the empirical risk defined
by robust loss functions in order to find the set of optimal parameters
$\theta$. The one-hot encoding of the label is denoted by the distribution
$q(k|\bm{x})$ for a sample $\bm{x}$ and a class $k$, such as
$q(y_{i}|\bm{x_{i}})=1$ and $q(k\neq y_{i}|\bm{x_{i}})=0,\>\forall
i\in\\{1,\cdots,n\\}$. The probability vector of $f$ is defined by the softmax
function $p(k|\bm{x})=\frac{e^{z_{k}}}{\sum_{j=1}^{K}e^{z_{j}}}$ where $z_{k}$
denotes the logits output with respect to class $k$.
### 3.1 Classification with robust loss functions
The method employs noise-robust losses to train the classifier in the presence
of noisy labels. Such losses improve the classification accuracy compared to
the commonly used Cross Entropy (CE), as illustrated in Figure 1. In this
section, the general empirical risk for a given mini-batch is defined by
$L=\sum_{i=1}^{N}\mathcal{L}(f(x_{i}),\overline{y_{i}})=\sum_{i=1}^{N}l_{i}$ .
The term $l_{i}$ is modified by each loss function.
The classical CE is used as a baseline loss function not robust to noisy
labels [8] and is defined as:
$l_{ce}=-\sum_{k=1}^{K}q(k|\bm{x_{i}})log(p(k|\bm{x_{i}})).$ (1)
As presented in section 2, Ma et al. [26] introduce robust loss functions
called Active Passive Losses that do not suffer from underfitting. We
investigate the combination between the Normalized Focal Loss (NFL) and the
Reversed Cross Entropy (RCE) called NFL+RCE. It shows promising results on
various benchmarks. The NFL is defined as:
$l_{nfl}=\frac{-\sum\limits_{k=1}^{K}q(k|\bm{x_{i}})(1-p(k|\bm{x_{i}}))^{\gamma}log(p(k|\bm{x_{i}}))}{-\sum\limits_{j=1}^{K}\sum\limits_{k=1}^{K}q(y=j|\bm{x_{i}})(1-p(k|\bm{x_{i}}))^{\gamma}log(p(k|\bm{x_{i}}))},$
(2)
where $\gamma\geq 0$ is an hyperparameter. The RCE loss is:
$l_{rce}=-\sum_{k=1}^{K}p(k|\bm{x_{i}})log\left(q(k|\bm{x_{i}})\right).$ (3)
The final combination following the framework simply gives a different
$\alpha$ and $\beta$ to each loss:
$l_{nfl+rce}=\alpha.l_{nfl}+\beta.l_{rce}.$ (4)
The two hyperparameters $\alpha$ and $\beta$ control the balancing between
more active learning and less passive learning. For simplicity, $\alpha$ and
$\beta$ are set to 1.0 without any tuning.
Liu et al. [25] propose another framework to deal with noisy annotations based
on the “early learning” phase. The loss, called Early Learning Regularization
(ELR), adds a regularization term to capitalize on early learning. ELR is not
strictly speaking a robust loss but belongs to robust penalization and label
correction methods. The penalization term corrects the CE based on estimated
soft labels identified with semi-supervised learning techniques. It prevents
memorization of false labels by steering the model towards these targets. The
regularization term maximizes the inner product between model outputs and
targets:
$l_{elr}=l_{ce}+\frac{\lambda_{elr}}{N}log\left(1-\sum_{k=1}^{K}p(k|\bm{x_{i}})t(k|\bm{x_{i}})\right).$
(5)
The target is not set equal to the model output but is estimated with a
temporal ensembling from semi-supervised methods. Let $t(k|\bm{x_{i}})^{(l)}$
denote the target for example $\bm{x_{i}}$ at iteration $l$ of training with a
momentum $\beta$:
$t(k|\bm{x_{i}})^{(l)}=\beta
t(k|\bm{x_{i}})^{(l-1)}+(1-\beta)p(k|\bm{x_{i}})^{(l)}.$ (6)
### 3.2 Contrastive learning
Contrastive learning methods learn representations by contrasting positive and
negative examples. A typical framework is composed of several blocks [7]:
* •
Data augmentation: Data augmentation is used to decouple the pretext tasks
from the network architecture. Chen et al. [5] study broadly the impact of
data augmentation. We follow their suggestion combining random crop (and
flip), color distortion, Gaussian blur, and gray-scaling.
* •
Encoding: The encoder extracts features (or representation) from augmented
data samples. A classical choice for the encoder is the ResNet model [11] for
image data. The final goal of the contrastive approach is to find correct
weights for the encoder.
* •
Loss function: The loss function usually combines positive and negative pairs.
The Noise Contrastive Estimation (NCE) and its variants are popular choices.
The general formulation for such loss function is defined for the i-th pair as
[40]:
$L_{i}=-log\frac{exp(\bm{z_{i}}^{T}\bm{z_{j(i)}}/\tau)}{\sum_{a\in
A(i)}exp(\bm{z_{i}}^{T}\bm{z_{a}}/\tau)},\;\text{with}\;i\in I,$ (7)
where $\bm{z}$ is a feature vector, $I$ is the set of indexes in the mini-
batch, $i$ is the index of the anchor, $j(i)$ is the index of an augmented
version of the anchor source image, $A(i)=I\setminus\\{i\\}$, and $\tau$ is a
temperature controlling the dot product. The denominator includes one positive
and $K$ negative pairs.
* •
Projection head: That step is not used in all frameworks. The projection head
maps the representation to a lower-dimensional space and acts as an
intermediate layer between the representation and the embedding pairs. Chen et
al. [5, 6] show that the projection head helps to improve the representation
quality.
## 4 A framework coupling contrastive learning and noisy labels
As illustrated in Figure 2, our method classifies noisy samples in a two
phased process. First, a classifier pre-trained with contrastive learning
produces train-set pseudo-labels (pre-training phase, in panel a), used during
the training of a subsequent fine-tuning phase (panel b). The underlying
intuition is that the predicted pseudo-lables are more accurate than the
original noisy labels. The contrastive learning performed in the first phase
(panel a1) improves the performance the classifier (panel a2), sensitive to
noisy labels; the resulting model can be also used in a standalone way with a
reduced number of hyperparameters, without the underlying fine-tuning phase.
The second phase leverages the pseudo-labels predicted by the pre-training in
all underlying steps (b1-b3). To mitigate the effect of potentially
incorrectly predicted pseudo-labels, a Gaussian Mixture Model (GMM, panel b1)
with 2 components follows the small loss-trick to predict for each sample the
probability of correctness. This value is used as a weight in a supervised
contrastive step (panel b2), performed to improve the learned representations
by taking advantage of the label information. A classification head is added
to the contrastive model in order to produce the final predictions (panel b3).
The fine-tuning phase can be seen as an adaptation of the pre-training phase
to handle pseudo-labels.
Figure 2: Overview of the framework consisting of two phases: pre-training
(panel a) and fine-tuning (panel b). After a contrastive learning phase (a1) a
classifier (a2) is trained to predict train-set pseudo-labels $\widehat{y}$.
The fine-tuning phase uses $\widehat{y}$ as a new ground truth. First, a GMM
model (b1) predicts the probability of correctness for each sample, used as a
corrective weight factor in a supervised contrastive training (panel b2). The
final predictions $\widehat{y}_{final}$ are produced by the (b3) classifier.
To maximize the impact of the contrastive learning on the underlying
classification, the supervised training is performed in 2 steps: a warm-up
step, updating only the classifier layer (while keeping the encoder frozen) is
followed by the full model training. We compared three different loss
functions for the supervised classification: the classical CE, the robust
NFL+RCE, and the ELR loss.
### 4.1 Sample selection and correction with pseudo-labels
Pseudo labels represent one hot encoded model’s predictions on the training
set. Pseudo-labels were initially used in semi-supervised learning to produce
annotations for unlabelled data; in the noisy label setting, various
techniques (e.g. DivideMix, etc) identify a subset with a high likelihood of
correctness and treat the remaining samples as the unlabeled counterpart in
semi-supervised learning. In this work, we elaborate on the observation that
the training set labels, predicted after training the model with a noise-
robust loss function (i.e. the pseudo labels), are more accurate than the
ground truth. This observation is supported by the results in Figure 3,
depicting the accuracy of pseudo labels predicted on CIFAR100, contaminated
with various levels of asymmetric (panel a) and symmetric (panel b) noise. The
pseudo labels are more accurate than the corrupted ground truth in both
settings and bring a higher gain in performance as the noise ratio increases.
Figure 3: Accuracy of pseudo labels on all simulated settings with asymmetric
(a) and symmetric (b) noise, evaluated on CIFAR100. The correctness of the
ground truth is represented on the x axis, while the accuracy of predicted
pseudo labels on the y axis. In all experiments, the pseudo labels have a
higher accuracy than the corrupted ground truth and this gain increases with
the noise ratio
As proposed by other approaches [1], the loss value on train samples can be
used to discriminate between clean and mislabeled samples. The sample
correctness probability is computed by fitting a 2 components GMM on the
distribution of losses [22]. The underlying probability is used as a sample
weight:
$w_{i}=p(k=0|l_{i}),$ (8)
where $l_{i}$ is the loss for sample $i$ and $k=0$ is the GMM component
associated to the clean samples (lowest loss). Figure 4 depicts the evolution
of the clean training set identified by GMM on an example: its accuracy grows
from 0.6 to 0.93 while the size stabilizes at 60% of the training set.
Figure 4: Accuracy of the entire training set (in blue) compared to the clean
train subset (in red); The clean subset’s percentual size is depicted in
green. The example is performed on CIFAR100, with 40% symmetric noise.
### 4.2 Weighted supervised contrastive learning
A modification to the contrastive loss defined in Equation 7 has been proposed
to leverage label information [18]:
$L_{i}=-log\frac{1}{|P(i)|}\sum_{p\in
P(i)}\frac{exp(\bm{z_{i}}^{T}\bm{z_{p}}/\tau)}{\sum_{a\in
A(i)}exp(\bm{z_{i}}^{T}\bm{z_{a}}/\tau)},$ (9)
where $P(i)=\\{j\in I\setminus\\{i\\},y_{j}=\widetilde{y_{i}}\\}$ with
$\widetilde{y_{i}}$ the prediction of the model for input $\bm{x_{i}}$.
As explained in the previous section, the loss value for the training set
samples is used to fit a GMM with 2 components, corresponding to correctly and
incorrectly labeled samples. We adapted the supervised representation loss to
employ $w$, a weighting factor representing the sample probability of
membership to the correctly labeled component. Thus, likely mislabeled samples
having large loss values would contribute only marginally to the supervised
representations:
$L_{i}=-log\frac{1}{|P(i)|}\sum_{p\in
P(i)}\frac{\widetilde{w_{p,i}}exp(\bm{z_{i}}^{T}\bm{z_{p}}/\tau)}{\sum_{a\in
A(i)}exp(\bm{z_{i}}^{T}\bm{z_{p}}/\tau)},$ (10)
where $\widetilde{w_{p,i}}$ is a modified version of $w_{p}$ such as
$\widetilde{w_{p,i}}=1$ if $p=j(i)$ else $\widetilde{w_{p,i}}=w_{i}$. If all
samples are considered as noisy, Equation 10 is simplified into the classical
unsupervised contrastive loss in Equation 7.
## 5 Experiments
The framework is assessed on three benchmarks and the contribution of each
block identified in Figure 2 is analyzed.
### 5.1 Datasets
CIFAR10 and CIFAR100 [20]. These experiments assess the accuracy of the method
against synthetic label noise. The two datasets are contaminated with
simulated symmetric or asymmetric label noise reproducing the heuristic in Ma
et al. [26]. The symmetric noise consists in corrupting an equal arbitrary
ratio of labels for each class. The noise level varies from $0.2$ to $0.8$.
For asymmetric noise [33, 25], sample labels have been flipped within a
specific set of classes, thus providing confusion between predetermined pairs
of labels. For CIFAR100, 20 groups of super-classes have been created, each
consisting of 5 sub-classes. The label flipping is performed only within each
super-class circularly. The asymmetric noise ratio is explored between $0.2$
and $0.4$.
Webvision [24]. This is a real-world dataset with noisy labels. It contains
2.4 million images crawled from the web (Google and Flickr) that share the
same 1,000 classes from the ImageNet dataset. The noise ratio varies from 0.5%
to 88%, depending on the class. In order to speed-up the training time, we
used mini Webvision [15], consisting of only top 50 classes in the Google
subset (66,000 images).
Clothing1M [42]. Clothing 1M is a large real-world dataset consisting of 1
million images on 14 classes of clothing articles. Being gathered from
e-commerce websites, Clothing1M embeds an unknown ratio of label noise.
Additional validation and test sets, consisting of 14k and 10k clean labeled
samples have been made available. In order to speed-up the training time, we
selected a subset of 56,000 images keeping the initial class distribution.
Both Webvision and Clothing1M images were resized to $128\times 128$.
Therefore, the reported results may differ from other papers cropping the
images to a $224\times 224$ resolution.
### 5.2 Settings
We use the contrastive SimCLR framework [5] with a ResNet18 [11] (without
ImageNet pre-training) as encoder. A projection head was added after the
encoder for the contrastive learning with the following architecture: a multi-
layer perceptron with one hidden layer and a ReLu non-linearity. The
classifier following the contrastive learning step has a simple multilayer
architecture: a single hidden layer with batch normalization and a ReLU
activation function. A comparison with a linear classifier is provided in the
supplementary materials.
For all supervised classification, we use SGD optimizer with momentum 0.9 and
cosine learning rate annealing. The NFL hyperparameter $\gamma$ is set to
$0.5$. Unlike the original paper, the ELR hyperparameters do no depend on the
noise type: the regularization coefficient $\lambda_{elr}$ and the momentum
$\beta$ are set to $3.0$ and $0.7$. Details on the experiment setting can be
found in the supplementary materials.
All codes are implemented in the PyTorch framework [32]. The experiments for
CIFAR are performed with a single Nvidia TITAN V-12GB and the experiments for
Webvision and Clothing1M are performed with a single Nvidia Tesla V100-32GB,
demonstrating the accessibility of the method. Our implementation has been
made available along with the supplementary materials.
## 6 Results
All experiments presented in this secion evaluate our method’s performance
with the top-1 accuracy score.
### 6.1 Impact of contrastive pre-training
To evaluate the impact of the contrastive pre-training on the classification
model, the proposed method (pre-training phase) is compared with a baseline
classifier, trained for 200 epochs without contrastive learning. For each
simulated dataset, we compare robust losses (e.g. NLF+RCE and ELR) and cross
entropy. Results for CIFAR10 and CIFAR100 are depicted in Table 1 for
different levels of symmetric and asymmetric noise. The pre-training improves
the accuracy of the three different baselines for both datasets with different
types and ratios of label noise. The largest differences are observed for the
noisiest case with $80\%$ noise. The pre-training outperforms the baselines by
large margins between $10$ and $75$ for CIFAR10 and between $5$ and $30$ for
CIFAR100.
Table 1: Results on both CIFAR10 and CIFAR100 using symmetric noise (0.2 -
0.8) and asymmetric noise (0.2 - 0.4). We compare training from scratch or
from pre-trained representation. Best scores are in bold for each noise
scenario and each loss.
| | | CIFAR10 | CIFAR100
---|---|---|---|---
Type | $\eta$ | Loss | Base | Pre-t. | Base | Pre-t.
Sym | 0.2 | ce | 77.2 | 87.7 | 55.6 | 56.5
elr | 90.3 | 93.0 | 64.1 | 67.4
nfl+rce | 91.0 | 92.7 | 66.6 | 68.8
0.4 | ce | 58.2 | 78.0 | 39.9 | 41.9
elr | 82.3 | 92.0 | 56.9 | 62.0
nfl+rce | 87.0 | 91.4 | 60.2 | 66.3
0.6 | ce | 35.2 | 59.2 | 21.8 | 26.8
elr | 64.2 | 90.4 | 40.6 | 55.7
nfl+rce | 80.2 | 88.1 | 47.0 | 61.8
0.8 | ce | 17.0 | 27.3 | 7.80 | 12.4
elr | 18.3 | 84.8 | 16.2 | 45.3
nfl+rce | 42.8 | 59.9 | 20.1 | 50.2
Asym | 0.2 | ce | 84.0 | 87.9 | 59.0 | 57.8
elr | 91.8 | 92.4 | 70.3 | 70.2
nfl+rce | 90.2 | 91.5 | 63.9 | 68.4
0.3 | ce | 79.2 | 83.9 | 50.6 | 50.4
elr | 89.6 | 91.7 | 69.8 | 69.3
nfl+rce | 86.7 | 89.9 | 53.5 | 63.5
0.4 | ce | 75.3 | 77.8 | 41.8 | 42.4
elr | 72.3 | 89.5 | 67.6 | 67.6
nfl+rce | 80.0 | 82.4 | 40.6 | 47.8
In addition to the comparisons with ELR and NFL+RCE, performed using our
implementations (column Base in Table 1), we present the results reported by
other recent competing methods. As shown in the introduction, numerous
contributions have been made to the field in the last years. Six recent
representative methods are selected for comparison: Taks [34], Co-teaching+
[43], ELR [25], DivideMix [22], SELF [30], and JoCoR [39]. The results are
presented in Table 2. The difference between the scores reported by ELR and
those obtained with our run (using the same implementation, but slightly
different hyper-parameters and a ResNet18 instead of a ResNet34) suggests that
the method is less stable on data contaminated with asymmetric noise and
sensitive to small changes hyperparameters. Moreover, ELR proposes
hyperparameters having different values depending on the type of dataset (i.e.
CIFAR10/CIFAR100) and underlying noise (i.e. symmetric/asymmetric), identified
after a hyperparameter search exercise. The best scores are reported by
DivideMix and they surpass all other techniques. One can note DivideMix uses a
PreAct ResNet18 while we use a classical ResNet18. Moreover, a recent study
[31] attempted to replicate these values and reported significantly lower
results on CIFAR100 (i.e. $49.5\%$ instead of $59.6\%$ on symmetric data and
$50.9\%$ instead of $72.1\%$ on asymmetric data). Our framework compares
favourably with the other competing methods, both on symmetric and asymmetric
noise.
Table 2: Accuracy scores compared with 6 methods (Taks, Co-teaching+, ELR,
DivideMix, SELF, and JoCoR) on CIFAR10 (C10) and CIFAR100 (C100). The cases
most affected by dropout are presented, with symmetric (S) and asymmetric (A)
noise. Top-2 scores are in bold
| | C10
---
80% S
| C10
---
40% A
| C100
---
80% S
| C100
---
40% A
| Ours
---
(ELR)
84.8 | 89.5 | 45.3 | 67.6
| ELR [25]
---
73.9 | 91.1 | 29.7 | 73.2
| Taks [34]
---
40.2 | 73.4 | 16.0 | 35.2
| Co-teach+ [43]
---
23.5 | 68.5 | 14.0 | 34.3
| DivideMix [22]
---
92.9 | 93.4 | 59.6 | 72.1
| SELF [30]
---
69.9 | 89.1 | 42.1 | 53.8
| JoCoR [39]
---
25.5 | 76.1 | 12.9 | 32.3
Webvision and Clothing1M results are presented in Table 3. The contrastive
framework outperforms the respective baselines for the three loss functions.
Because the images have a reduced size, and for Clothing1M, we use a smaller
training set, the direct comparison with competing methods is less relevant.
However, the observed gap in performance is significant and promising for
training images with higher resolution. Moreover, a ResNet50 model has been
trained with our framework on the Webvision dataset with a higher resolution
($224\times 224$). The accuracy reaches respectively $75.7\%$ and $76.2\%$ for
CE and ELR. These results are very close to the values reported with DivideMix
($77.3\%$) and ELR+ ($77.8\%$) using a larger model, Inception-ResNet-v2 (the
difference is more than $4\%$ on the ImageNet benchmark [4]).
Table 3: Top-1 accuracy for mini-Webvision and Clothing1M. Best scores are in
bold for each dataset and each loss. Pre-t represents the pre-training phase
while Fine-tune refers to the results after the fine-tuning step.
| Webvision | Clothing1M
---|---|---
Loss | Base. | Pre-t. | Fine-tune | Base. | Pre-t. | Fine-tune
ce | 51.8 | 57.1 | 58.4 | 54.8 | 59.1 | 61.5
elr | 53.0 | 58.1 | 59.0 | 57.4 | 60.8 | 60.4
nfl+rce | 49.9 | 54.8 | 58.2 | 57.4 | 59.4 | 60.1
Supported by this first set of experiments, the preliminary pre-training with
contrastive learning shows great performances. The accuracy of both
traditional and robust-loss classification models is significantly improved.
### 6.2 Sensitivity to the hyperparameters
Estimating the best hyperparameters is complex for datasets with noisy labels
as clean validation sets are not available. For instance, Ortego et al. [31]
show that two efficient methods (eg. ELR and DivideMix) could be sensitive to
specific hyperparameters. Therefore a hyperparameter sensitivity study has
been carried out to estimate the stability of the framework for the learning
rate. Figure 5 depicts the sensitivity on CIFAR100 with $80\%$ noise. CE and
NFL+RCE seem to have opposite behaviors. The CE reaches competitive results
with small learning rates but is prompt to overfitting for higher learning
rates. The NFL+RCE loss tends to underfitting for the lowest learning rates
but is quite robust for higher values. The ELR loss has the smallest
sensitivity to the learning for the investigated range but does not reach the
best values obtained with CE or NFL+RCE. We can assume that the regularization
term coupled with pre-training is very efficient. It prevents memorization of
the false labels as observed with CE. Results for other noise ratios have been
documented in the supplementary materials.
Figure 5: Learning rate sensitivity for CIFAR100 with 80% noise. The explored
learning rate values are $\\{0.001,0.01,0.1,1.0\\}$. The baseline (dashed
line) is compared with our framework (solid line).
This sensitivity analysis is limited to the learning rate. Investigating the
impact of other hyperparameters, such as the momentum $\beta$ or the
regularization factor $\lambda_{elr}$, could be interesting. In their original
papers, ELR and NFL+RCE reach respectively $25.2\%$ and $30.3\%$ with other
hyperparameters. These values are still far from the improvements brought by
the contrastive pre-training but it suggests that the results could be
improved with different hyperparameters.
Our empirical results indicate that the analyzed methods may be sensitive to
hyperparameters. Despite the promised robustness to label noise, the analyzed
robust losses are also affected by overfitting or underfitting. Our
experiments have been built upon the parameters recommended in each issuing
paper (e.g. ELR, SIMCLR) but, since the individual building blocks can be
affected by small variations in input parameters, the performance of our
method may also be impacted. Finding a relevant method to estimate proper
hyperparameters in NLL remains a challenge. In the absence of a clean
validation set, identifying when overfitting starts also remains an open
challenge. This is demonstrated by our studies on the behaviour of the (also
noise-corrupted) validation set and another two recently proposed methods,
analyzing the stability of the loss function on the train set and the changes
in the upstream layers. These experiments are detailed in Supplementary
Materials.
### 6.3 Impact of the fine-tuning phase
Experimental results on synthetic label noise, depicted in Figure 6, show that
continuing the presented pre-training block (Figure 2) with the fine-tuning
phase increases the accuracy in over 65% of cases on CIFAR10 and over 80% of
cases on CIFAR100. For both datasets, asymmetric noise data benefit more from
this approach than symmetric noise. All experiments only use the input
parameters proposed in the loss-issuing papers.
Figure 6: Accuracy gain when performing the fine-tuning phase after the pre-
training block (computed as the difference between fine-tuning accuracy and
pre-training accuracy). The plot gathers the results for all noise ratios on
CIFAR10 (panels a, b) and CIFAR100 (c, d) with symmetric (first column) and
asymmetric (second column) noise.
The sample selection has also got a positive impact on the two real-world
datasets, as shown in Table 3 by the ”Fine-tune” columns. The average accuracy
improvement is about $1.8\%$. Only the ELR loss function slightly decreases
the performance on Clothing1M.
Enriching pretrained models with sample weighting and selection, pseudo labels
instead of corrupted targets, and supervised contrastive pre-training can
improve the classification accuracy. However, such an approach raises the
question of a trade-off between complexity, accuracy improvement, and
computation time.
## 7 Discussion and limits of the framework
In addition to the presented fine-tuning phase, we evaluated the performance
of other promising techniques, such as the dynamic bootstrapping with mixup
[1]. This strategy has been developed to help convergence under extreme label
noise conditions. Details can be found in the supplementary materials. The
improvement that dynamic bootstraping can bring when used after pre-training
is depicted in Figure 7. In most of the cases, this technique improves the
accuracy, as indicated by the positive accuracy gain scores, measuring the
difference between the accuracy after dynamic bootstraping and the accuracy of
the pre-training phase. ELR and CE benefit most from this addition for
CIFAR100. The impact of the dynamic boostrapping should also be analyzed for
the fine-tuning phase and for larger datasets, such as Webvision or
Clothing1M.
Figure 7: Top-1 accuracy gain for the dynamic bootstrapping on CIFAR100 with
asymmetric (a) and symmetric noise (b). Dynamic bootstrapping is an
alternative to the proposed fine-tuning phase.
One of the major drawbacks of our method is the extra computational time
needed to learn representations with contrastive learning. A detailed study,
comparing the execution time of our framework with 6 other competing methods
has been provided in supplementary materials. The pre-training phase doubles
the execution time of a reference baseline, consisting of performing only a
single classification step, while the entire framework increases the execution
time 3 to 4 times the baseline value. However, the constrastive learning does
not increase the need for GPU memory if the batch size is limited for the
contrastive learning [29, 12]. The computational time could be reduced by
initializing the contrastive step with the pretrained weights from ImageNet.
Most state-of-the-art approaches also leverage computationally expensive
settings, consisting of larger models (e.g. ResNet50), dual model training, or
data augmentation such as mixup. In this work, we explored the limits of a
restricted computational setting, consisting of a single GPU and 8GB RAM. All
experiments use a ResNet18 model, batch sizes of $256$, and for real-world
datasets, the images have been rescaled (e.g. $128\times 128$ instead of
$224\times 224$). We also foresee that the constrastive learning step could be
improved by images with higher resolutions as smaller details could be
identified in the representation embedding.
There remain multiple open problems for future research, such as: i)
identifying the start of the memorization phase in the absence of a clean
dataset, ii) studying the impact of contrastive learning on other models for
noisy labels such as DivideMix, iii) comparing SimCLR approach in the context
of noisy labels with other contrastive frameworks (the impact of Moco is
studied in the supplementary materials) and other self-supervised approaches,
and iv) having a better theoretical understanding of the interaction between
the initial state precomputed with contrastive learning and the classifier in
presence of noisy labels. Moreover, the analysis carried out in this work
should be validated on larger settings, in particular on Clothing1M with a
ResNet50, higher resolutions, and the full dataset.
## 8 Conclusions
In this work, we presented a contrastive learning framework optimized with
several adaptations for noisy label classification. Supported by an extensive
range of experiments, we conclude that a preliminary representation pre-
training improves the performance of both traditional and robust-loss
classification models. Additionally, multiple techniques can be used to fine-
tune and further optimize these results; however, no approach provides a
significant improvement systematically on all types of datasets and label
noise. The cross-entropy penalized by Early-Learning Regularization (ELR)
shows the best overall results for synthetic noise but also real-world
datasets.
However, the training phases remain sensitive to input configuration.
Overfitting is the common weakness of all studied models. When trained with
tuned parameters, even traditional (cross-entropy) models provide competitive
results, while robust-losses are less sensitive. The typical noisy label
adaptations, such as sample selection or weighting, the usage of pseudo
labels, or supervised contrastive losses, improve the performance to a lesser
extent but increase the framework’s complexity. We hope that this work will
promote the use of contrastive learning to improve the robustness of the
classification process with noisy labels.
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## Supplementary Materials
## Appendix A Description of the datasets
Table 4 gives a detailed description of datasets, including size of the
training and test sets, the image resolution, and the number of classes.
Table 4: Description of the datasets used in the experiments. Data set | Train | Test | Size | # classes
---|---|---|---|---
CIFAR10 | 50K | 10K | 32x32 | 10
CIFAR100 | 50K | 10K | 32x32 | 100
Clothing1M | 56K | 5K | 128x128 | 14
Mini-Webvision | 66K | 2.5K | 128x128 | 50
## Appendix B Detailed settings of the experiments
All experiments use the ResNet18 as encoder. The classification steps are
combined with data augmentation: a random crop with a padding of $4$, an
horizontal flip with a probability of $50\%$, and a random rotation of
$20^{\circ}$. All other hyperparameters are resumed in Table 5.
Table 5: Training parameters. Symbols: l.r means learning rate, w.d means weight decay, opti. means optimizer, Repre. means representation step, Classi. means the supervised classification step. | | C10/C100 | Webvision | Clothing1M
---|---|---|---|---
Repre. | Batch | 512 | 512 | 512
Opti. | Adam | Adam | Adam
l.r. | $10^{-3}$ | $10^{-3}$ | $10^{-3}$
w.d. | $10^{-6}$ | $10^{-6}$ | $10^{-6}$
epochs | 500 | 500 | 500
Classi. | Batch | 256 | 256 | 256
Opti. | SGD | SGD | SGD
l.r. | 0.01/0.1 | 0.4 | 0.01
w.d. | $10^{-5}$ | $3.10^{-5}$ | $10^{-4}$
epochs | 200 | 200 | 200
## Appendix C Ablation study
### C.1 Contrastive learning with a momentum encoder
The momentum encoder from the Moco framework [12] maintains a dynamic memory
queue of representations. The current mini-batch is added to the memory queue
while the oldest mini-batch is dequeued. The offline momentum encoder is a
copy of the online encoder by taking an exponentially-weighted average of the
parameter of the online encoder. The main advantage of Moco is to be able to
reduce the batch size (and the GPU memory) while keeping a very large number
of negative pairs for the contrastive learning.
Table 6: Top-1 accuracy on CIFAR100 with $80\%$ noise. Two different contrastive learning frameworks are evaluated for the pre-training: SimCLR and Moco. The third column gives the accuracy for a classifier with a smaller learning rate. | SimCLR | Moco | | Moco - Fine
---
tune
CE | 12.4 | 12.0 | 49.0
ELR | 45.3 | 38.8 | 42.3
NFL+RCE | 50.2 | 26.3 | 47.0
The different representations computed by SimCLR and Moco are compared on
CIFAR100. Both approaches are trained for 500 epochs following the usual
hyperparameter parameters from the initial papers. As the two methods use
different strategies to compute the representations, their quality is assessed
by learning a linear classifier on top of the frozen encoder network. It can
be seen as a proxy for representation quality. The SimCLR framework reaches
$55.3\%$ of accuracy while Moco gets $55.0\%$ of accuracy. However, the two
encoders do not behave in a similar way with regard to noisy labels. The same
classifier (multi-layer, same learning rate and weight decay) is trained
starting from the representation computed by SimCLR and Moco. As depicted in
Table 6, the representations computed by Moco are more sensitive to the noisy
labels. However, reducing the learning rate of the optimizer by a factor $10$
(column Moco - Fine Tune) significantly increases the accuracy.
Even if pretraining the encoder increases the accuracy for both contrastive
methods, the two approaches do not have the same behavior. In particular, the
best parameters for the classifier optimizer seem to be different. This raises
several questions about the difference between the two representations and
what properties of these representations improve the robustness of the
classifier.
### C.2 Sensitivity to the learning rate
We perform an hyperparameter search on the CIFAR100 datasets. The learning
rate is chosen in $\\{10^{-3},10^{-2},10^{-1},10^{0}\\}$. Results are
presented in Figure 8. The configuration with $80\%$ noise is clearly the most
sensitive case, in particular for the NFL+RCE loss and the CE. The ELR method
is quiet robust over the investigated range.
(a) CIFAR100 with $80\%$ noise. (b) CIFAR100 with $60\%$ noise.
(c) CIFAR100 with $40\%$ noise. (d) CIFAR100 with $20\%$ noise.
Figure 8: Hyperparameter sensitivity for CIFAR100.
### C.3 Impact of the classifier architecture
The impact of the 2 classifier architectures is detailed in Table 7. The
multilayer architecture performs better on datasets contaminated with a
significant amount of asymmetric noise.
Table 7: Results on both CIFAR10 and CIFAR100 using symmetric noise (0.2 -
0.8) and asymmetric noise (0.2 - 0.4). We compare a single linear layer (L) to
multiple layers (M) final classification head, for three losses: CE, ELR, and
NFL+RCE.
| | | CIFAR10 | CIFAR100
---|---|---|---|---
Type | $\eta$ | Loss | L | M | L | M
Sym | 0.2 | ce | 91.7 | 87.7 | 58.6 | 56.5
elr | 92.9 | 93.0 | 66.4 | 67.4
nfl_rce | 93.2 | 92.7 | 69.7 | 68.8
0.4 | ce | 90.6 | 78.0 | 44.2 | 41.9
elr | 92.1 | 92.0 | 60.8 | 62.0
nfl_rce | 92.1 | 91.4 | 67.0 | 66.3
0.6 | ce | 88.1 | 59.2 | 28.9 | 26.8
elr | 89.7 | 90.4 | 54.0 | 55.7
nfl_rce | 90.2 | 88.1 | 63.7 | 61.8
0.8 | ce | 72.6 | 27.3 | 14.1 | 12.4
elr | 82.0 | 84.8 | 41.6 | 45.3
nfl_rce | 78.9 | 59.9 | 54.2 | 50.2
Asym | 0.2 | ce | 91.6 | 87.9 | 60.1 | 57.8
elr | 92.7 | 92.4 | 69.3 | 70.2
nfl_rce | 92.5 | 91.5 | 69.1 | 68.4
0.3 | ce | 90.2 | 83.9 | 52.3 | 50.4
elr | 90.6 | 91.7 | 68.5 | 69.3
nfl_rce | 91.2 | 89.9 | 68.0 | 63.5
0.4 | ce | 84.7 | 77.8 | 43.7 | 42.4
elr | 68.4 | 89.5 | 65.5 | 67.6
nfl_rce | 62.6 | 82.4 | 63.0 | 47.8
## Appendix D Dynamic bootstrapping with mixup
In addition to the presented fine-tuning phase, we also evaluated the
performance of other techniques recently proposed for noisy label
classification. The weights $w$ computed by the sample selection phase can
also be combined with a mixup data augmentation strategy [45]. A specific
strategy for noisy labels, called dynamic bootstrapping with mixup [1], has
been developed to help convergence under extreme label noise conditions. The
convex combinations of sample pairs $\bm{x_{p}}$ (loss $l_{p}$) and
$\bm{x_{q}}$ (loss $l_{q}$) is weighted by the probability $w$ to belong to
the clean dataset:
$\bm{x}=\frac{w_{p}}{w_{p}+w_{q}}\bm{x_{p}}+\frac{w_{q}}{w_{p}+w_{q}}\bm{x_{q}}.$
(11) $l=\frac{w_{p}}{w_{p}+w_{q}}l_{p}+\frac{w_{q}}{w_{p}+w_{q}}l_{q}.$ (12)
The associated CE is corrected according to the weights:
$l_{ce}=-\sum_{k=1}^{K}\left(w_{i}q(k|\bm{x_{i}})+(1-w_{i})z_{i}\right)log(p(k|\bm{x_{i}})),$
(13)
where $z(k|\bm{x_{i}})=1$ if $k=\operatorname*{\arg\\!\max}p(k|\bm{x_{i}})$ or
zero for all the other cases. If the GMM probability are well estimated,
combining one noisy sample with one clean sample leads to a large weight for
the clean sample and a small weight for the noisy sample. Clean-clean and
noisy-noisy cases remain similar to a classical mixup with weights around
$0.5$.
The dynamic bootstrapping for ELR is derived by replacing the CE term by the
corrected version:
$l_{elr}(\theta)=l_{ceb}(\theta)+\frac{\lambda_{elr}}{N}log\left(1-\sum_{k=1}^{K}p(k|\bm{x_{i}}).t(k|\bm{x_{i}})\right).$
(14)
Regarding the robust loss function NFL+RCE, the two losses have to be
modified:
$\displaystyle
l_{nfl}=w_{i}\frac{-\sum\limits_{k=1}^{K}q(k|\bm{x_{i}})(1-p(k|\bm{x_{i}}))^{\gamma}log(p(k|\bm{x_{i}}))}{-\sum\limits_{j=1}^{K}\sum\limits_{k=1}^{K}q(y=j|\bm{x_{i}})(1-p(k|\bm{x_{i}}))^{\gamma}log(p(k|\bm{x_{i}}))}$
(15)
$\displaystyle+(1-w_{i})\frac{-\sum\limits_{k=1}^{K}z(k|\bm{x_{i}})(1-p(k|\bm{x_{i}}))^{\gamma}log(p(k|\bm{x_{i}}))}{-\sum\limits_{j=1}^{K}\sum\limits_{k=1}^{K}z(y=j|\bm{x_{i}})(1-p(k|\bm{x_{i}}))^{\gamma}log(p(k|\bm{x_{i}}))}$
where $q$ is the one-hot encoding of the label (the zero value is fixed to a
low value to avoid $log(0)$).
$l_{rce}=-\sum_{k=1}^{K}p(k|\bm{x_{i}})log\left(w_{i}.q(k|\bm{x_{i}})+(1-w_{i})z_{i}\right)$
(16)
## Appendix E Classification warmup
This section compares the classification accuracy of models trained with and
without a warm-up phase after the representation learning. The warm-up phase
consists of freezing the entire model except for the classification head.
Figure 9 depicts the gain in performance brought by the warm-up phase. When
using the default values, its inclusion is beneficial only for significant
amounts of symmetric noise. Our experiments have been performed using only the
recommended classifier learning rates, detailed in the experimental setup.
Having different learning rates for the warm-up phase and the classification
optimizing all weights (encoder and classifier) could have a different impact
on the warmup phase.
Figure 9: Gain in performance when using a supplementary classifier warm-up
phase before training the entire model on CIFAR 100 with symmetric (panel a)
and asymmetric noise (panel b).
## Appendix F Execution time analysis
In order to estimate our method’s computational cost, we compared the
execution time of both approaches, consisting of performing only the pre-
training phase and the pre-training followed by fine-tuning with the execution
time of performing only one supervised classification phase (i.e. the
baseline). The number of times our methods were slower than the baseline has
been depicted in Table 8. We provided similar metrics for the methods making
available this informations (i.e. Taks, Co-teaching+, JoCoR). As expected, the
pre-training doubles the execution time of the baseline as, in addition to
training the classifier, a contrastive learning phase has to be performed
beforehand. The entire framework introduces a computational cost 3 to 4.5
times higher. However, all methods leveraging pre-trained models (using for
instance supervised pre-training) also hide a similar computational cost.
Table 8: Comparison of execution time results reported as a factor with respect to the training time of the baseline, representing the supervised training of the model with the CE loss. The abbreviations Ours (Pre-t) indicate the pre-training phase while Ours (Fine-tune) the pre-training phase followed by fine-tuning | | C10
---
80% S
| C10
---
40% A
| C100
---
80% S
| C100
---
40% A
| Ours
---
(Pre-t)
2.36 | 2.53 | 2.40 | 2.32
| Ours
---
(Fine-tune)
3.42 | 3.63 | 4.31 | 4.36
Taks | 0.53 | 1.04 | 0.52 | 0.98
Co-teach+ | 2.00 | 2.00 | 2.00 | 2.01
JoCoR | 1.73 | 1.74 | 1.72 | 1.74
## Appendix G An attempt to prevent overfitting with early stopping
Overfitting is the common weakness of all studied models. Several strategies
understanding and preventing overfitting have been explored: i) analysing the
model behaviour on a validation set, ii) identifying the start of the
memorization phase using Training Stop Point [17], and iii) characterizing
changes in the model using Centered Kernel Alignment [19]. A clean validation
set is generally used to find the best moment for early stopping and to
estimate the hyperparameter sets. However, we assume that clean validation
samples are not available. Therefore, the methods must be robust to
overfitting and to a wide range of hyperparameter values.
As typical noisy label settings lack a clean reference set, we contrasted the
behavior of the model on a corrupted validation set with that on a clean test
set, where overfitting can be easily identified. Train/validation sets have
been generated using 5 cross validation folds. In the figure below, panel (a)
depicts the evolution of accuracy scores on the corrupted train/validation
sets as well as on the test set. After the first 50 epochs, the model starts
overfitting as the test accuracy drops by 10% ( Figure 10 panel a). The
accuracy on the corrupted train continues to increase as the model memorizes
the input labels. However, on the corrupted validation set a plateau followed
by a loss of performance is indicative of the same phenomena, but without
being always aligned with the overfitting phase observed on the test-set. The
memorization phenomena of the train-set labels incapacitates the model to
generalize on the corrupted validation set and explains the significant
difference in scores between the train and validation accuracies.
A second perspective on the analysis of overfitting explores the stability of
the network’s predictions on the validation set. Panel (b) depicts the number
of samples predicted in different classes across consecutive epochs. As the
model starts overfitting, the prediction stability also increases. After 200
epochs, only 500 from 10000 samples on the validation set change class from
one epoch to another. As expected, the network stability is correlated with
model overfitting on severe label noise.
Figure 10: Evolution of accuracy across train/validation/test sets (a)
Prediction stability on the validation set computed as the number of samples
changing class across consecutive epochs. The rolling mean average of the
number of predictions has been depicted in black. The experiments have been
performed on CIFAR 100, with 80% symmetric noise during the first
classification phase and used NFL + RCE loss.
Figure 11: Evolution of accuracy across train/validation/test sets (a)
Prediction stability on the validation set computed as the number of samples
changing class across consecutive epochs. The rolling mean average of the
number of predictions has been depicted in black. The experiments have been
performed on CIFAR 100, with 40% asymmetric noise during the first
classification phase and used NFL + RCE loss.
Several recent contributions studied the overfitting phenomena of neural
networks in an attempt to identify an early stopping point corresponding to
the maximum obtainable test accuracy. Traditional approaches leverage a clean
test set which is often unavailable when confronted with noisy labelled data.
Kamabattula et al. [17] proposed to find a Training Stop Point (TSP), a
heuristic analyzing the rate of change in the training accuracy and correlated
its transition towards the memorization phase with a transition towards a
smoother (smaller variance) regime, as depicted below. Our experimental
results showed that the theoretical conditions to identify the early stopping
point are not always met as suggested by TSP. Figure 12 indicates that the
overfitting phase, starting after the first 5 epochs, does not change the
variance of the train loss.
Figure 12: Evolution of train loss and test accuracy on CIFAR, 60% symmetric
noise. The theoretical conditions of higher variance on the train loss,
associated with the start of the memorization phase, as suggested by TSP, are
not fulfilled.
Centered Kernel Alignment (CKA) [19] provides a similarity index comparing
representations between layers of different trained models. In particular, CKA
shows interesting properties as CKA can consistently identify correspondences
between layers trained from different initializations.
The objective is twofold: i) observing if a specific behavior can be
identified for the overfitting and ii) comparing the CKA values with and
without contrastive pre-training. The CKA index is computed at three different
locations in the network: the input layer, the middle of the network, and the
final layer. Figure 13 shows the CKA similarity computed between the
initialization/pre-trained model and the same layer at different epochs during
the training process. It is interesting to note that the first layer of the
pre-trained model remains very similar to the same layer computed by
contrastive learning. Such behavior was expected in order to improve the
robustness against noisy labels. Indeed, if contrastive learning can extract
good representations for semi-supervised or transfer learning, being close to
such representations can also help to avoid learning noisy labels. As
expected, all layers of the model trained from a random initialization vary
much more during the training.
The training phase of the pre-trained model reaches its maximum accuracy
around 50 epochs but the CKA values of the middle and last layers continue to
drop until 130 epochs. On the other hand, the CKA values of the initialized
model remain stable after $150$ epochs when the test accuracy reaches almost
its maximum value. At first glance, the CKA behavior cannot be related to
overfitting.
(a) CKA from a pre-trained encoder with contrastive learning. (b) CKA from a
random initialization.
Figure 13: CKA similarity for a model trained with NFL+RCE loss function on
CIFAR100 with $80\%$ noise.
None of the studied approaches provides a solution preventing overfitting
across all our experiments and this problem remains an open question.
|
order estimate in perturbation theory gives for the nucleon ground state with
$n=0$
$\displaystyle\Delta
M_{\pm}=\mp\frac{g_{X}}{4}\int\frac{dz}{z}X_{0}(z)\left(\left|\tilde{f}_{L}^{0}(z)\right|^{2}-\left|\tilde{f}_{R}^{0}(z)\right|^{2}\right)=\mp\frac{g_{X}\sigma}{8\tilde{\kappa}^{2}}$
(B.181)
with $X_{0}(z)$ given in (II.5.1). (B.180) through the expansion around the
vev, $X(x,z)\approx X_{0}(z)e^{i\Pi(x,z)}$, would also generate a contribution
to the pion-nucleon coupling and also the axial-charge of the direct and
transition axial form factors YEE . Since our central interest is neutrino DIS
scattering we can neglect this coupling and its effects on our results, as
most of our analysis involves the behavior near the UV boundary where the
effects of $\sigma$ is negligible both in the nucleonic wavefunctions and the
spectrum.
## Appendix C Details of the Reggeon exchange
The bulk gauge field $L^{0}_{\mu}(k,z)$ exchange contribution to the
diffractive Compton scattering amplitude in the t-channel is given by
$\displaystyle i{\cal A}^{L}_{Lp\rightarrow Lp}(s,t)=\sum_{n}i\tilde{{\cal
A}}^{L}_{Lp\rightarrow Lp}(m_{n},s,t)$ $\displaystyle i\tilde{{\cal
A}}^{L}_{Lp\rightarrow
Lp}(m_{n},s,t)=(-i)V_{LLL}^{\mu}(q,q^{\prime},k,m_{n})\times\tilde{G}_{\mu\nu}(m_{n},t)\times(-i)V_{L\bar{\Psi}\Psi}^{\nu}(p_{1},p_{2},k,m_{n})\,,$
with the bulk vertices ($k=p_{2}-p_{1}=q-q^{\prime}$)
$\displaystyle V_{LLL}^{\mu}(q,q^{\prime},k,m_{n})\equiv$
$\displaystyle\left(\frac{\delta
S_{LLL}^{k}}{\delta(\epsilon^{0}_{\mu}\partial_{z}L^{0}(k,z))}\right)\,J_{L}(m_{n},z)+\left(\frac{\delta
S_{LLL}^{k}}{\delta(\epsilon^{0}_{\mu}L^{0}(k,z))}\right)\,J_{L}(m_{n},z)$
$\displaystyle=$ $\displaystyle
g_{5}^{3}\kappa_{CS}B^{\mu}(q,q^{\prime},\epsilon^{\pm})\int
dz\,\Big{(}\mathcal{V}(Q,z)\mathcal{V}(Q^{\prime},z)\partial_{z}J_{L}(m_{n},z)-\partial_{z}\mathcal{V}(Q,z)\mathcal{V}(Q^{\prime},z)J_{L}(m_{n},z)\Big{)}\,,$
$\displaystyle V_{L\bar{\Psi}\Psi}^{\nu}(p_{1},p_{2},k,m_{n})\equiv$
$\displaystyle\left(\frac{\delta
S_{L\bar{\Psi}\Psi}^{k}}{\delta(\epsilon^{0}_{\nu}L^{0}(k,z))}\right)\,J_{L}(m_{n},z)=g_{5}\int
dz\sqrt{g}\,e^{-\phi}z\bar{\Psi}(p_{2},z)\gamma^{\nu}\Psi(p_{1},z)J_{L}(m_{n},z)\,,$
We have defined $p=({p_{1}+p_{2}})/{2}$, $t=-K^{2}$, $\mathcal{V}(Q,z)\equiv
L(q=\sqrt{-Q^{2}},z)$ as given in (A.1.155), and used the bulk-to-bulk gauge
field propagator (A.1.2) with the substitutions $q\rightarrow k$,
$Q\rightarrow K$, and $V\rightarrow L$. We have also used the vertices in
(LABEL:vertices33), and defined
$\displaystyle
B^{\mu}(q,q^{\prime},\epsilon^{\pm})\equiv(-i)\epsilon^{\rho\sigma\nu\mu}\epsilon_{\rho}^{+}(q)\epsilon_{\sigma}^{-}(q^{\prime})(q^{\prime}_{\nu}+q_{\nu})\,.$
For $z^{\prime}\rightarrow 0$, we can use (A.1.159) and simplify (C) as
$\displaystyle i{\cal A}^{L}_{Lp\rightarrow
Lp}(s,t)\approx(-i)\mathcal{V}^{\mu}_{LLL}(q_{1},q_{2},k_{z})\times\big{(}-i\eta_{\mu\nu}\big{)}\times(-i)\mathcal{V}^{\nu}_{L\bar{\Psi}\Psi}(p_{1},p_{2},k_{z})\,,$
with
$\displaystyle\mathcal{V}^{\mu}_{LLL}(q_{1},q_{2},k_{z})=g_{5}^{3}\kappa_{CS}B^{\mu}(q,q^{\prime},\epsilon^{\pm})\int
dz\,\Big{(}\mathcal{V}(Q,z)\mathcal{V}(Q^{\prime},z)z-\partial_{z}\mathcal{V}(Q,z)\mathcal{V}(Q^{\prime},z)\frac{z^{2}}{2}\Big{)}\,,$
$\displaystyle\mathcal{V}^{\nu}_{L\bar{\Psi}\Psi}(p_{1},p_{2},k_{z})=g_{5}\times\frac{3}{2}\times\int
dz\,\sqrt{g}\,e^{-\phi}z\,\bar{\Psi}(p_{2},z)\gamma^{\nu}\Psi(p_{1},z)\mathcal{V}(K,z)=g_{5}F_{1}^{(LN)}(K)\,,$
where $\mathcal{V}(K,z)\equiv L^{0}(k=\sqrt{-K^{2}},z)$, and $F_{1}^{(LN)}(K)$
is the form factor of the the nucleon due to $L_{\mu}^{0}$.
The Reggeization of the bulk spin-1 gauge field $L_{\mu}^{0}(k,z)$ exchange
can be obtained, in a similar way to the Reggezation of the spin-2 graviton
exchange, through the substitution
$\displaystyle J_{L}(m_{n}(j),z)\rightarrow
z^{-(j-1)}\phi_{n}(j,z)=z^{-(j-1)}\frac{\tilde{\phi}_{n}(j,z)}{z}$ (C.187)
followed by the summation over all spin-j meson exchanges using the
Sommerfeld-Watson formula
$\displaystyle\frac{1}{2}\sum_{j\geq
1}(s^{j-1}+(-s)^{j-1})\rightarrow-\frac{\pi}{2}\int_{\mathbb{C}}\frac{dj}{2\pi
i}\left(\frac{s^{j-1}+(-s)^{j-1}}{{\rm sin}\,\pi j}\right)$
The contour ${\mathbb{C}}$ is to the left of all odd poles $j=1,3,...$ (in
contrast to the Reggeized graviton where the contour is chosen to the left of
the even poles), and requires the analytical continuation of the exchanged
amplitudes to the complex j-plane.
The spin-j normalized meson wavefunctions $J_{L}(m_{n}(j),z)$ (C.187) are
expressed in terms of the wavefunctions of massive scalar fields
$\tilde{\phi}_{n}(j,z)$ which are given, for the soft wall model, in terms of
the generalized Laguerre polynomials as
$\displaystyle\tilde{\phi}_{n}(j,z)=c_{n}(j)\,z^{\Delta(j)}L_{n}^{\Delta(j)-2}(w)\,,$
(C.189)
with $w=\tilde{\kappa}^{2}z^{2}$. The normalization coefficients are
$\displaystyle
c_{n}(j)=\Big{(}\frac{2\tilde{\kappa}^{2(\Delta(j)-1)}\Gamma(n+1)}{\Gamma(n+\Delta(j)-1)}\Big{)}^{\frac{1}{2}}\,,$
(C.190)
and the dimension of the massive scalar fields (with an additional mass coming
from the massive open string states attached to the D9 or D7-branes)
$\Delta(j)$ is given by
$\displaystyle\Delta(j)$ $\displaystyle=$ $\displaystyle
2+\sqrt{4+m^{2}R^{2}+\frac{R^{2}}{\alpha^{\prime}}(j-1)}$ (C.191)
$\displaystyle=$ $\displaystyle 2+\sqrt{\sqrt{\lambda}(j-j_{0})}\,,$
where, in the last line, we have used the fact that $m^{2}R^{2}=-3$. The
spin-1 transverse bulk gauge field defined as $zL_{\mu}^{0}(m_{n},z)$ obeys
the same bulk equation of motion as a bulk massive scalar field
$\tilde{\phi}_{n}(j=1,z)$ with $m^{2}R^{2}=-3$ which is manifest in (E.265).
We have also used the open string quantized mass spectrum
$m_{j}^{2}R^{2}=(j-1)({R^{2}}/{\alpha^{\prime}})=\sqrt{\lambda}(j-1)$ for open
strings attached to the D9 or D7-branes in bulk, and we have defined
$j_{0}=1-{1}/{\sqrt{\lambda}}$.
We now recall that the non-normalized bulk-to-boundary propagators of massive
scalar fields are given in terms of Kummer’s (confluent hypergeometric)
function of the second kind, and their integral representations are (for
space-like momenta $k^{2}=-K^{2}$)
$\displaystyle\mathcal{\tilde{V}}(j,K,z)=$ $\displaystyle
z^{\Delta(j)}U\Big{(}a_{K}+\frac{\Delta(j)}{2},\Delta(j)-1;w\Big{)}=z^{\Delta(j)}w^{2-\Delta(j)}U\Big{(}\tilde{a}(j),\tilde{b}(j);w\Big{)}$
$\displaystyle=$ $\displaystyle
z^{\Delta(j)}w^{2-\Delta(j)}\frac{1}{\Gamma(\tilde{a}(j))}\int_{0}^{1}dx\,x^{\tilde{a}(j)-1}(1-x)^{-\tilde{b}(j)}{\rm
exp}\Big{(}-\frac{x}{1-x}w\Big{)}\,,$ (C.192)
with $w=\tilde{\kappa}^{2}z^{2}$
$\displaystyle
a_{K}=a=\frac{K^{2}}{4\tilde{\kappa}^{2}}\qquad\tilde{a}(j)=a_{K}+2-\frac{\Delta(j)}{2}\qquad\tilde{b}(j)=3-\Delta(j)$
(C.193)
after using the identity $U(m,n;y)=y^{1-n}U(1+m-n,2-n,y)$. Therefore, the
bulk-to-bulk propagator of spin-j mesons
$\displaystyle
J_{L}(m_{n}(j),z)\,G(j,z,z^{\prime})\,J_{L}(m_{n}(j),z^{\prime})=z^{-(j-1)}\,G(j,z,z^{\prime})\,z^{\prime-(j-1)}$
can be approximated at the boundary as (for space-like momenta $k^{2}=-K^{2}$)
$\displaystyle G(j,z\rightarrow 0,z^{\prime})\approx$
$\displaystyle-\bigg{[}\frac{\phi_{n}(j,z\rightarrow
0)}{-g_{5}\mathcal{F}_{n}(j)}\bigg{]}\times\sum_{n}\frac{-g_{5}\mathcal{F}_{n}(j)\phi_{n}(j,z^{\prime})}{K^{2}+m_{n}^{2}(j)}$
$\displaystyle=$
$\displaystyle(\tilde{\kappa}^{2})^{\Delta(j)-2}\frac{z^{\Delta(j)-1}}{\Delta(j)-1}\frac{\Gamma(\Delta(j)-2+a)}{\Gamma(\Delta(j)-2)}\,\mathcal{V}(j,K,z^{\prime})$
where $\phi_{n}(j,z\rightarrow 0)=\frac{1}{z}\tilde{\phi}_{n}(j,z\rightarrow
0)$. We have defined the non-normalized bulk-to-boundary propagator of spin-j
mesons
$\displaystyle\mathcal{V}(j,K,z^{\prime})=\sum_{n}\frac{-g_{5}\mathcal{F}_{n}(j)\phi_{n}(j,z^{\prime})}{K^{2}+m_{n}^{2}(j)}=\frac{1}{z}\mathcal{\tilde{V}}(j,K,z^{\prime})|_{a+\frac{\Delta(j)}{2}\rightarrow
a+1+(\Delta(j)-3)}\,,$
with the shift $a+\frac{\Delta(j)}{2}\rightarrow a+1+(\Delta(j)-3)$ defined in
such a way that the mass spectrum of massive scalar fields
$m_{n}^{2}=4\tilde{\kappa}^{2}(n+\frac{\Delta(j=1)}{2})$ and the mass spectrum
of spin-1 gauge fields $m_{n}^{2}=4\tilde{\kappa}^{2}(n+1)$ match, i.e., we
shift $n+\frac{\Delta(j=1)}{2}\rightarrow n+1+(\Delta(j=1)-3)$, giving the
mass spectrum of spin-j mesons
$\displaystyle m_{n}^{2}(j)=4\tilde{\kappa}^{2}(n+1+(\Delta(j)-3))\,.$ (C.197)
We have also used
$\displaystyle\mathcal{F}_{n}(j)=$
$\displaystyle\frac{\mathcal{C}(j,K,\epsilon)}{g_{5}}\bigg{(}-\sqrt{g}\,e^{-\phi}\,\big{(}g^{xx}\big{)}^{2}\,\partial_{z^{\prime}}\phi_{n}(j,z^{\prime})\bigg{)}_{z^{\prime}=\epsilon}\,,$
$\displaystyle\mathcal{C}(j,K,\epsilon)=$
$\displaystyle\mathcal{V}(j,K,\epsilon)$ (C.198)
and the substitution $\phi_{n}(j,z\rightarrow
0)=\frac{1}{z}\tilde{\phi}_{n}(j,z\rightarrow 0)\approx
c_{n}(j)\,z^{\Delta(j)-1}L_{n}^{\Delta(j)-2}(0)$ for the soft wall model.
After the Reggeization, the scattering amplitude for the spin-j meson exchange
becomes
$\displaystyle i{\cal A}^{L}_{Lp\rightarrow
Lp}(j,s,t)\approx(-i)\mathcal{V}^{\mu}_{LLL}(j,q_{1},q_{2},k_{z})\times\big{(}-i\eta_{\mu\nu}\big{)}\times(-i)\mathcal{V}^{\nu}_{L\bar{\Psi}\Psi}(j,p_{1},p_{2},k_{z})\,,$
with
$\displaystyle\mathcal{V}^{\mu}_{LLL}(j,q_{1},q_{2},k_{z})=$
$\displaystyle\frac{1}{g_{5}^{2}}\times
g_{5}^{3}\kappa_{CS}B^{\mu}(q,q^{\prime},\epsilon^{\pm})\int dz\,z^{2(j-1)}$
$\displaystyle\times\bigg{(}\mathcal{V}(Q,z)\mathcal{V}(Q^{\prime},z)\times\frac{(\Delta(j)-1-(j-1))z^{\Delta(j)-1-(j-1)-1}}{\Delta(j)-1}$
$\displaystyle-\partial_{z}\mathcal{V}(Q,z)\mathcal{V}(Q^{\prime},z)\times\frac{z^{\Delta(j)-1-(j-1)}}{\Delta(j)-1}\bigg{)}\times(\tilde{\kappa}^{2})^{\Delta(j)-2}\frac{\Gamma(\Delta(j)-2+a)}{\Gamma(\Delta(j)-2)}$
$\displaystyle=$ $\displaystyle\mathcal{V}_{LLL}(j,Q,Q^{\prime})\times
B^{\mu}(q,q^{\prime},\epsilon^{\pm})\,,$
$\displaystyle\mathcal{V}^{\nu}_{L\bar{\Psi}\Psi}(p_{1},p_{2},k_{z})=$
$\displaystyle g_{5}\times\frac{3}{2}\times\int
dz\,\sqrt{g}\,e^{-\phi}z^{1+2(j-1)}\,\bar{\Psi}(p_{2},z)\gamma^{\nu}\Psi(p_{1},z)z^{-(j-1)}\mathcal{V}(j,K,z)$
$\displaystyle=$ $\displaystyle
g_{5}F_{1}^{(LN)}(j,K)\times\bar{u}(p_{2})\gamma^{\nu}u(p_{1})\,,$
We have defined
$\displaystyle\mathcal{V}_{LLL}(j,Q,Q^{\prime})=$
$\displaystyle\frac{1}{g_{5}^{2}}\times g_{5}^{3}\kappa_{CS}\int
dz\,z^{2(j-1)}$
$\displaystyle\times\bigg{(}\mathcal{V}(Q,z)\mathcal{V}(Q^{\prime},z)\times\frac{(\Delta(j)-1-(j-1))z^{\Delta(j)-1-(j-1)-1}}{\Delta(j)-1}$
$\displaystyle-\partial_{z}\mathcal{V}(Q,z)\mathcal{V}(Q^{\prime},z)\times\frac{z^{\Delta(j)-1-(j-1)}}{\Delta(j)-1}\bigg{)}(\tilde{\kappa}^{2})^{\Delta(j)-2}\frac{\Gamma(\Delta(j)-2+a)}{\Gamma(\Delta(j)-2)}$
$\displaystyle=$ $\displaystyle\frac{1}{g_{5}^{2}}\times
g_{5}^{3}\kappa_{CS}\times Q^{2-j-\Delta(j)}\times
I_{\xi}(j,Q,Q^{\prime})\times\tilde{\kappa}^{2\Delta(j)-4}\times\frac{\Gamma(\Delta(j)-2+a)}{\Gamma(\Delta(j)-2)}\times\frac{1}{\Delta(j)-1}$
with
$\displaystyle I_{\xi}(j,Q,Q^{\prime})$ $\displaystyle=$ $\displaystyle\int
d\xi\,\xi^{j-2+\Delta(j)}\bigg{(}\mathcal{V}(\xi)\mathcal{V}(\xi
Q^{\prime}/Q)\times(\Delta(j)-1-(j-1))\xi^{-1}-\partial_{\xi}\mathcal{V}(\xi)\mathcal{V}(\xi
Q^{\prime}/Q)\bigg{)}$ (C.204) $\displaystyle\approx$
$\displaystyle(\Delta(j)-1-(j-1))\times
2^{\Delta(j)+j-3}\,\frac{Q^{\prime}}{Q}\,G_{2,2}^{2,2}\left(\frac{Q^{2}}{Q^{\prime
2}}\bigg{|}\begin{array}[]{c}\frac{1}{2},\frac{3}{2}\\\
\frac{1}{2}(j+\Delta(j)-1),\frac{1}{2}(j+\Delta(j)+1)\\\ \end{array}\right)$
$\displaystyle+$ $\displaystyle
2^{\Delta(j)+j-2}\,\frac{Q^{\prime}}{Q}\,G_{2,2}^{2,2}\left(\frac{Q^{2}}{Q^{\prime
2}}\bigg{|}\begin{array}[]{c}\frac{1}{2},\frac{3}{2}\\\
\frac{1}{2}(j+\Delta(j)+1),\frac{1}{2}(j+\Delta(j)+1)\\\
\end{array}\right)\,,$ (C.207)
where $G_{p,q}^{m,n}\left(z\bigg{|}\begin{array}[]{c}a_{1},...,a_{p}\\\
b_{1},...,b_{q}\\\ \end{array}\right)$ is the Meijer G-function. We have used
the identities
$\displaystyle\lim_{\frac{Q}{\tilde{\kappa}}\rightarrow\infty}\mathcal{V}(\xi)=\xi
K_{1}(\xi)\qquad{\rm
and}\qquad\partial_{\xi}(\xi^{\nu}K_{\nu}(\xi))=-\xi^{\nu}K_{\nu-1}(\xi)$
(C.209)
to evaluate the integrals with $\xi=Qz$. The function $F_{1}^{(LN)}(j,K)$ in
(C) admits the integral representation
$\displaystyle F_{1}^{(LN)}(j,K)$ $\displaystyle=$
$\displaystyle\frac{3}{2}\times\frac{1}{2}\frac{\tilde{\kappa}^{-(j-1)-\Delta(j)-1}}{\Gamma(a)}\int_{0}^{1}dx\,x^{a-1}(1-x)^{-\tilde{b}(j)}$
$\displaystyle\times\bigg{(}\bigg{(}\frac{\tilde{n}_{R}}{\tilde{\kappa}^{\tau-1}}\bigg{)}^{2}\times\Gamma(c(j))\bigg{(}\frac{1}{1-x}\bigg{)}^{-c(j)}+\bigg{(}\frac{\tilde{n}_{L}}{\tilde{\kappa}^{\tau}}\bigg{)}^{2}\times\Gamma(c(j)+1)\bigg{(}\frac{1}{1-x}\bigg{)}^{-(c(j)+1)}\bigg{)}\,,$
$\displaystyle=$
$\displaystyle\frac{3}{2}\times\frac{1}{2}\tilde{\kappa}^{-(j-1)-\Delta(j)-1}$
$\displaystyle\times\bigg{(}\bigg{(}\frac{\tilde{n}_{R}}{\tilde{\kappa}^{\tau-1}}\bigg{)}^{2}\times\frac{\Gamma(c(j))\Gamma(1-\tilde{b}(j)+c(j))}{\Gamma(1-\tilde{b}(j)+c(j)+a)}+\bigg{(}\frac{\tilde{n}_{L}}{\tilde{\kappa}^{\tau}}\bigg{)}^{2}\times\frac{\Gamma(c(j)+1)\Gamma(2-\tilde{b}(j)+c(j))}{\Gamma(2-\tilde{b}(j)+c(j)+a)}\bigg{)}$
where
$\displaystyle\tilde{b}(j)=3-\Delta(j)\,,\qquad\qquad
c(j)=(\tau+1)+\frac{j-1}{2}-\frac{\Delta(j)}{2}-\frac{1}{2}\,.$ (C.211)
After summing over all contributions from the spin-j mesons, the total
amplitude ${\cal A}^{tot}_{Lp\rightarrow Lp}(s,t)$ is given by
$\displaystyle{\cal A}^{tot}_{Lp\rightarrow Lp}(s,t)=$
$\displaystyle-\int_{\mathbb{C}}\frac{dj}{2\pi
i}\left(\frac{s^{j-1}+(-s)^{j-1}}{{\rm sin}\,\pi j}\right){\cal
A}^{L}_{Lp\rightarrow Lp}(j,s,t)$ $\displaystyle{\cal A}^{L}_{Lp\rightarrow
Lp}(j,s,t)=$ $\displaystyle\mathcal{V}_{LLL}(j,Q,Q^{\prime})\times
B^{\mu}(q,q^{\prime},\epsilon^{\pm})\times
g_{5}F_{1}^{(LN)}(j,K)\times\bar{u}(p_{2})\gamma_{\mu}u(p_{1})\,,$ (C.212)
The contour $\mathbb{C}$ is at the rightmost of the branch-point of
$F_{1}^{LN}(j,K)$ and the leftmost of $j=1,3,...$. From (C), we determine the
single Reggeon amplitude (total amplitude) in momentum space, after wrapping
the j-plane contour ${\mathbb{C}}$ to the left,
$\displaystyle{\cal A}^{tot}_{Lp\rightarrow
Lp}(s,t)=-s^{j_{0}-1}\int_{-\infty}^{j_{0}}\frac{dj}{\pi}\left(\frac{1+e^{-i\pi}}{{\rm
sin}\,\pi j}\right)s^{j-j_{0}}\,\text{Im}[{\cal A}^{L}_{Lp\rightarrow
Lp}(j,s,t)]$ (C.213)
The imaginary part follows from the discontinuity of the $\Gamma$-function
$\displaystyle\text{Im}[{\cal A}^{L}_{Lp\rightarrow Lp}(j,s,t)]$
$\displaystyle\approx$
$\displaystyle\bigg{(}\Gamma(\Delta(j)-2)\mathcal{V}_{LLL}(j,Q,Q^{\prime})\times
B^{\mu}(q,q^{\prime},\epsilon^{\pm})\times
g_{5}F_{1}^{(LN)}(j,K)\times\bar{u}(p_{2})\gamma_{\mu}u(p_{1})\bigg{)}\bigg{|}_{j\rightarrow
j_{0},\Delta(j)\rightarrow 2}$ $\displaystyle\times$
$\displaystyle\text{Im}\bigg{[}\frac{1}{\Gamma(\tilde{\Delta}(j))}\bigg{]}$
with the complex argument
$\displaystyle\tilde{\Delta}(j)=\Delta(j)-2=i\sqrt{\sqrt{\lambda}(j_{0}-j)}\equiv
iy$
and $j_{0}=1-{1}/{\sqrt{\lambda}}$. For $y\rightarrow 0$, we may approximate
$1/\Gamma(iy)\approx iy\,e^{i\gamma y}$, with the Euler-Mascheroni constant
$\gamma=0.55772...$. The single Reggeon amplitude (total amplitude) in
momentum space (C) can now be cast in block form
$\displaystyle{\cal A}^{tot}_{Lp\rightarrow Lp}(s,t)=I(j_{0},s)\times
G_{5}(j_{0},s,t)$ (C.216)
with
$\displaystyle
I(j_{0},s)=-\tilde{s}^{j_{0}}\int_{-\infty}^{j_{0}}\frac{dj}{\pi}\left(\frac{1+e^{-i\pi}}{{\rm
sin}\,\pi
j}\right)\tilde{s}^{j-j_{0}}\,\sin\left[\tilde{\xi}\sqrt{\sqrt{\lambda}(j_{0}-j)}\right]$
$\displaystyle
G_{5}(j_{0},s,t)=\frac{1}{\tilde{s}}\bigg{(}\tilde{\kappa}^{2(j-1)}\Gamma(\Delta(j)-2)\mathcal{V}_{LLL}(j,Q,Q^{\prime})\times
B^{\mu}(q,q^{\prime},\epsilon^{\pm})\times
g_{5}F_{1}^{(LN)}(j,K)\times\bar{u}(p_{2})\gamma_{\mu}u(p_{1})\bigg{)}\bigg{|}_{j\rightarrow
j_{0},\,\Delta(j)\rightarrow 2}$
We have set $\tilde{s}\equiv{s}/{\tilde{\kappa}^{2}}$, and
$\tilde{\xi}-\pi/2=\gamma=0.55772.....$ is Euler-Mascheroni constant. We note
that the apparent pole in the Gamma-function at the Reggeon intercept, cancels
out in the combination $\Gamma(\Delta(j_{0})-2){\cal
V}_{LLL}(j_{0},Q,Q^{\prime})$.
In the block form (C.216), the spin-j integral $I(j_{0},s)$ is similar to the
spin-j integral in POLX (see Eq. 4.19), with the identifications
$\mathcal{K}(s,b^{\perp},z,z^{\prime})\leftrightarrow{\cal
A}^{tot}_{Lp\rightarrow Lp}(s,t)$,
$(zz^{\prime}/R^{4})G_{3}(j_{0},v)\leftrightarrow G_{5}(j_{0},s,t)$,
$\xi(v)\leftrightarrow\tilde{\xi}$, and $\widehat{s}\leftrightarrow\tilde{s}$.
We then follow POLX to evaluate the spin-j integral by closing the j-contour
appropriately. In the high energy limit
$\sqrt{\lambda}/\tilde{\tau}\rightarrow 0$
($\tilde{\tau}\equiv\log\tilde{s}$), the single Reggeon contribution to the
amplitude is
$\displaystyle{\cal A}^{tot}_{Lp\rightarrow Lp}(s,t)\simeq
e^{j_{0}\tilde{\tau}}\left[(\sqrt{\lambda}/\pi)+i\right](\sqrt{\lambda}/2\pi)^{1/2}\;\tilde{\xi}\;\frac{e^{-\sqrt{\lambda}\tilde{\xi}^{2}/2\tilde{\tau}}}{\tilde{\tau}^{3/2}}\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\tilde{\tau}}\bigg{)}\right)\times
G_{5}(j_{0},s,t)\,.$ (C.218)
We can rewrite the amplitude (C.218) as
$\displaystyle{\cal A}^{tot}_{Lp\rightarrow Lp}(s,t)$ $\displaystyle\simeq$
$\displaystyle 4\times 4\times g_{5}\times\frac{1}{g_{5}^{2}}\times
g_{5}^{3}\kappa_{CS}\times\Big{(}\frac{Q}{\tilde{\kappa}}\Big{)}^{2-j_{0}-\Delta(j_{0})}\times\Big{(}\frac{s}{\tilde{\kappa}^{2}}\Big{)}^{j_{0}}\times\frac{e^{-\sqrt{\lambda}\tilde{\xi}^{2}/2\log[s/\tilde{\kappa}^{2}]}}{(\log[s/\tilde{\kappa}^{2}])^{3/2}}$
$\displaystyle\times$
$\displaystyle\left[(\sqrt{\lambda}/\pi)+i\right]\times(\sqrt{\lambda}/2\pi)^{1/2}\;\tilde{\xi}\;\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\log[s/\tilde{\kappa}^{2}]}\bigg{)}\right)\times\tilde{G}_{5}(j_{0},t,Q,Q^{\prime})$
where
$\displaystyle\tilde{G}_{5}(j_{0},s,t)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4}\times\frac{1}{4}\times\frac{1}{g_{5}}\times\frac{1}{\frac{1}{g_{5}^{2}}\times
g_{5}^{3}\kappa_{CS}}\times\frac{1}{\tilde{\kappa}^{2}}\times\frac{1}{Q^{2-j-\Delta(j)}}\times\frac{1}{\kappa^{2\Delta(j)-4}}\times\frac{1}{\tilde{\kappa}^{-(j-1)-\Delta(j)-1}}\times\frac{1}{\tilde{\kappa}^{2(j-1)}}\times
G_{5}(j_{0},s,t)$ (C.220) $\displaystyle=$ $\displaystyle
I_{\xi}(j_{0},Q,Q^{\prime})\times\mathcal{F}_{1}^{(LN)}(j_{0},K)\times
s^{-1}\times\frac{1}{4}\times
B^{\mu}(q,q^{\prime},\epsilon^{\pm})\times\frac{1}{4}\times\bar{u}(p_{2})\gamma_{\mu}u(p_{1})$
with
$\displaystyle\mathcal{F}_{1}^{(LN)}(j_{0},K)\equiv\frac{\Gamma(\Delta(j_{0})-2+a)}{\Delta(j_{0})-1}\times\frac{1}{\tilde{\kappa}^{-(j_{0}-1)-\Delta(j_{0})-1}}\times
F_{1}^{(LN)}(j_{0},K)\,.$ (C.221)
## Appendix D Details of the Pomeron exchange
The transverse and traceless part of the graviton
($\eta_{\mu\nu}\rightarrow\eta_{\mu\nu}+h_{\mu\nu}$) follows from the
quadratic part of the Einstein-Hilbert action in de-Donder gauge,
$\displaystyle S=$ $\displaystyle\int
d^{5}x\sqrt{g}\,e^{-2\phi}\,\mathcal{L}_{h}\,,$
$\displaystyle\mathcal{L}_{h}=$
$\displaystyle-\frac{1}{4\tilde{g}_{5}^{2}}\,g^{\mu\nu}\,\eta^{\lambda\rho}\eta^{\sigma\tau}\partial_{\mu}h_{\lambda\sigma}\partial_{\nu}h_{\rho\tau}\,,$
(D.222)
with Newton constant $16\pi
G_{N}={8\pi^{2}}/{N_{c}^{2}}=\tilde{g}_{5}^{2}=2\kappa^{2}$. The massive
glueball spectrum is determined by solving the equation of motion for
$h_{\mu\nu}$ following from (D), with for spin-2 glueballs
$\displaystyle
m_{n}^{2}=8\tilde{\kappa}^{2}_{N}(n+1)\qquad\tilde{g}_{5}f_{n}=2\tilde{\kappa}_{N}$
(D.223)
### D.1 Graviton coupling in bulk
For the graviton in the axial gauge $h_{\mu z}=h_{zz}=0$. Using
$\eta_{\mu\nu}\rightarrow\eta_{\mu\nu}+h_{\mu\nu}$ in the linearized bulk
action gives
$\displaystyle h\overline{\Psi}\Psi:\quad$
$\displaystyle-\frac{\sqrt{2\kappa^{2}}}{2}\int
d^{5}x\,\sqrt{g}\,h_{\mu\nu}T_{F}^{\mu\nu}$ $\displaystyle hLL:\quad$
$\displaystyle-\frac{\sqrt{2\kappa^{2}}}{2}\int
d^{5}x\,\sqrt{g}\,h_{\mu\nu}T_{L}^{\mu\nu}$
with the energy-momentum tensors for the fermions and left gauge fields
$\displaystyle T_{F}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
e^{-\phi}\frac{i}{2}\,z\,\overline{\Psi}\gamma^{\mu}\overset{\leftrightarrow}{\partial^{\nu}}\Psi-\eta^{\mu\nu}\mathcal{L}_{F}\,,$
$\displaystyle T_{L}^{\mu\nu}$ $\displaystyle=$
$\displaystyle-e^{-\phi}\Big{(}z^{4}\eta^{\rho\sigma}\eta^{\mu\beta}\eta^{\nu\gamma}\,F^{L}_{\beta\rho}F^{L}_{\gamma\sigma}-z^{4}\,\eta^{\mu\beta}\eta^{\nu\gamma}\,F^{L}_{\beta
z}F^{L}_{\gamma z}\Big{)}-\eta^{\mu\nu}\mathcal{L}_{L}\,.$ (D.225)
and the rescaling
$\displaystyle\Psi\rightarrow\sqrt{2g_{5}^{2}}\Psi\qquad L_{N}\rightarrow
g_{5}L_{N}\qquad h_{\mu\nu}\rightarrow\sqrt{2\kappa^{2}}\,h_{\mu\nu}\,.$
(D.226)
Evaluating the couplings or the vertices (LABEL:vertices1) on the solutions,
Fourier transforming the fields to momentum space, and integrating by part the
trace-full part for the fermions, we find for the couplings to the fermions
($h\overline{\Psi}\Psi$) to the left gauge fields ($hLL$)
$\displaystyle h\overline{\Psi}\Psi:\quad$
$\displaystyle\int\frac{d^{4}p_{2}d^{4}p_{1}d^{4}k}{(2\pi)^{12}}(2\pi)^{4}\delta^{4}(p_{2}-k-p_{1})\big{(}S^{k}_{h\bar{\Psi}\Psi}\big{)}$
$\displaystyle hLL:\quad$
$\displaystyle\int\frac{d^{4}q^{\prime}d^{4}qd^{4}k}{(2\pi)^{12}}(2\pi)^{4}\delta^{4}(q-k-q^{\prime})\big{(}S^{k}_{hLL}\big{)}$
with
$\displaystyle S^{k}_{h\bar{\Psi}\Psi}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2\kappa^{2}}}{2}\int
dz\sqrt{g}\,e^{-\phi}z\,\epsilon^{TT}_{\mu\nu}h(k,z)\bar{\Psi}(p_{2},z)\gamma^{\mu}p^{\nu}\Psi(p_{1},z)\,,$
$\displaystyle S^{k}_{hLL}$ $\displaystyle=$
$\displaystyle\sqrt{2\kappa^{2}}\int
dz\sqrt{g}\,e^{-\phi}z^{4}\,\epsilon^{TT}_{\mu\nu}h(k,z)K^{\mu\nu}(q,q^{\prime},\epsilon,\epsilon^{\prime},z)\,.$
We have set $h_{\mu\nu}=\epsilon^{TT}_{\mu\nu}h(k,z)$ (where
$\epsilon^{TT}_{\mu\nu}$ is transverse and traceless polarization tensor),
$q^{2}=-Q^{2}$, $q^{\prime 2}=-Q^{\prime 2}$ for space-like momenta, and
defined
$\displaystyle K^{\mu\nu}(q,q^{\prime},\epsilon,\epsilon^{\prime},z)\equiv
B_{1}^{\mu\nu}\mathcal{V}(Q,z)\mathcal{V}(Q^{\prime},z)-B_{0}^{\mu\nu}\partial_{z}\mathcal{V}(Q,z)\partial_{z}\mathcal{V}(Q^{\prime},z)\,,$
$\displaystyle
B_{0}^{\mu\nu}(\epsilon,\epsilon^{\prime})\equiv\epsilon^{\mu}\epsilon^{\prime\nu}\,,$
$\displaystyle
B_{1}^{\mu\nu}(q,q^{\prime},\epsilon,\epsilon^{\prime})\equiv\epsilon\cdot\epsilon^{\prime}\,q^{\mu}q^{\prime\nu}-q\cdot\epsilon^{\prime}\,\epsilon^{\mu}q^{\prime\nu}-q^{\prime}\cdot\epsilon\,q^{\mu}\epsilon^{\prime\nu}+q\cdot
q^{\prime}\,\epsilon^{\mu}\epsilon^{\prime\nu}\,.$ (D.229)
with $B_{1,0}=\eta_{\mu\nu}B_{1,0}^{\mu\nu}$, $K=\eta_{\mu\nu}K^{\mu\nu}$, and
the non-normalizable wave function for the virtual photon $\mathcal{V}(Q,z)$
given in (LABEL:Gauge).
### D.2 Scattering amplitude
The t-channel Compton exchange of a spin-2 glueball of mass $m_{n}$ in AdS
reads
$\displaystyle i{\cal A}^{h}_{Lp\rightarrow Lp}(s,t)=\sum_{n}i\tilde{{\cal
A}}^{h}_{Lp\rightarrow Lp}(m_{n},s,t)$ $\displaystyle i\tilde{{\cal
A}}^{h}_{Lp\rightarrow
Lp}(m_{n},s,t)=(-i)V_{hLL}^{\mu\nu(TT)}(q,q^{\prime},k,m_{n})\times\tilde{G}^{TT}_{\mu\nu\alpha\beta}(m_{n},t)\times(-i)V_{h\bar{\Psi}\Psi}^{\alpha\beta(TT)}(p_{1},p_{2},k,m_{n})\,,$
with the bulk vertices ($k=p_{2}-p_{1}=q-q^{\prime}$)
$\displaystyle V_{hLL}^{\mu\nu(TT)}(q,q^{\prime},k,m_{n})\equiv$
$\displaystyle\left(\frac{\delta
S_{hLL}^{k}}{\delta(\epsilon^{TT}_{\mu\nu}h(k,z))}\right)\,J_{h}(m_{n},z)=\sqrt{2\kappa^{2}}\times\frac{1}{2}\int
dz\sqrt{g}\,e^{-\phi}z^{4}K^{\mu\nu}(q,q^{\prime},\epsilon,\epsilon^{\prime},z)J_{h}(m_{n},z)\,,$
$\displaystyle
V_{h\bar{\Psi}\Psi}^{\alpha\beta(TT)}(p_{1},p_{2},k,m_{n})\equiv$
$\displaystyle\left(\frac{\delta
S_{h\bar{\Psi}\Psi}^{k}}{\delta(\epsilon^{TT}_{\alpha\beta}h(k,z))}\right)\,J_{h}(m_{n},z)=-\sqrt{2\kappa^{2}}\times\int
dz\sqrt{g}\,e^{-\phi}z\bar{\Psi}(p_{2},z)\gamma^{\alpha}p^{\beta}\Psi(p_{1},z)J_{h}(m_{n},z)\,,$
with $p=({p_{1}+p_{2}})/{2}$. The bulk-to-bulk transverse and traceless
graviton propagator $G_{\mu\nu\alpha\beta}=G_{\mu\nu\alpha\beta}^{TT}$ for the
$2^{++}$ glueball is Raju:2011mp ; DHoker:1999bve
$\displaystyle
G_{\mu\nu\alpha\beta}^{TT}(m_{n},t,z,z^{\prime})=J_{h}(m_{n},z)\tilde{G}_{\mu\nu\alpha\beta}^{TT}(m_{n},t)J_{h}(m_{n},z^{\prime})\,,$
$\displaystyle\tilde{G}_{\mu\nu\alpha\beta}^{TT}(m_{n},t)={1\over
2}\left({\cal T}_{\mu\alpha}{\cal T}_{\nu\beta}+{\cal T}_{\mu\beta}{\cal
T}_{\nu\alpha}-\frac{2}{3}{\cal T}_{\mu\nu}{\cal
T}_{\alpha\beta}\right)\frac{i}{t-m_{n}^{2}+i\epsilon}\,,$
with
$\displaystyle{\cal T}_{\mu\nu}=$
$\displaystyle-\eta_{\mu\nu}+k_{\mu}k_{\nu}/m_{n}^{2}\,,$ $\displaystyle
J_{h}(m_{n},z)\equiv$
$\displaystyle\psi_{n}(z)=c_{n}\,z^{4}L_{n}^{\Delta(j)-2}(2\xi)$
and
$\displaystyle
c_{n}=\Bigg{(}\frac{2^{4}\tilde{\kappa}_{N}^{6}\Gamma(n+1)}{\Gamma(n+3)}\Bigg{)}^{\frac{1}{2}}\,,$
(D.233)
normalized according to
$\displaystyle\int
dz\,\sqrt{g}e^{-\phi}\,{\left|g^{xx}\right|}\,\psi_{n}(z)\psi_{m}(z)=\delta_{nm}\,.$
(D.234)
For $z^{\prime}\rightarrow 0$, we can simplify (D.2) as ($t=-K^{2}$),
$\displaystyle i{\cal A}^{h}_{Lp\rightarrow
Lp}(s,t)\approx(-i)\mathcal{V}^{\mu\nu(TT)}_{hLL}(q_{1},q_{2},k_{z})\times\bigg{(}\frac{i}{2}\eta_{\mu\alpha}\eta_{\nu\beta}\bigg{)}\times(-i)\mathcal{V}^{\alpha\beta(TT)}_{h\bar{\Psi}\Psi}(p_{1},p_{2},k_{z})\,,$
with
$\displaystyle\mathcal{V}^{\mu\nu(TT)}_{hLL}(q_{1},q_{2},k_{z})=\sqrt{2\kappa^{2}}\times\frac{1}{2}\int
dz\sqrt{g}\,e^{-\phi}z^{4}K^{\mu\nu}(q,q^{\prime},\epsilon,\epsilon^{\prime},z)\frac{z^{4}}{4}\,,$
$\displaystyle\mathcal{V}^{\alpha\beta(TT)}_{h\bar{\Psi}\Psi}(p_{1},p_{2},k_{z})=-\sqrt{2\kappa^{2}}\times\int
dz\,\sqrt{g}\,e^{-\phi}z\,\bar{\Psi}(p_{2},z)\gamma^{\mu}p^{\nu}\,\Psi(p_{1},z)\mathcal{H}(K,z)\,,$
where
$\displaystyle\mathcal{H}(K,z)$ $\displaystyle=\sum_{n}\frac{\sqrt{2}\kappa
F_{n}\psi_{n}(z^{\prime})}{K^{2}+m_{n}^{2}}$ (D.237)
$\displaystyle=4z^{4}\Gamma(a_{K}+2)U\Big{(}a_{K}+2,3;2\xi\Big{)}=\Gamma(a_{K}+2)U\Big{(}a_{K},-1;2\xi\Big{)}$
$\displaystyle=\frac{\Gamma(a_{K}+2)}{\Gamma(a_{K})}\int_{0}^{1}dx\,x^{a_{K}-1}(1-x){\rm
exp}\Big{(}-\frac{x}{1-x}(2\xi)\Big{)}\,,$
with $a_{K}={a}/{2}={K^{2}}/{8\tilde{\kappa}_{N}^{2}}$,
$\displaystyle F_{n}=\frac{1}{\sqrt{2}\kappa}\bigg{(}-\frac{1}{z^{\prime
3}}\partial_{z^{\prime}}\psi_{n}(z^{\prime})\bigg{)}_{z^{\prime}=\epsilon}=-\frac{4}{\sqrt{2}\kappa}c_{n}L_{n}^{2}(0)\,,$
We have used the transformation $U(m,n;y)=y^{1-n}U(1+m-n,2-n,y)$ in the second
line of (D.237).
### D.3 High energy limit
In the high energy limit $\sqrt{\lambda}/\tilde{\tau}\rightarrow 0$ with
$\tilde{\tau}\equiv\log\tilde{s}=\log[s/\tilde{\kappa}_{N}^{2}]$, the single
Pomeron (or spin-j gluballs) contribution to the Compton Scattering amplitude
has been evaluated in Mamo:2019mka , with the result
$\displaystyle{\cal A}^{tot}_{Lp\rightarrow Lp}(s,t)\simeq
e^{j_{0}\tilde{\tau}}\left[(\sqrt{\lambda}/\pi)+i\right](\sqrt{\lambda}/2\pi)^{1/2}\;\tilde{\xi}\;\frac{e^{-\sqrt{\lambda}\tilde{\xi}^{2}/2\tilde{\tau}}}{\tilde{\tau}^{3/2}}\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\tilde{\tau}}\bigg{)}\right)\times
G_{5}(j_{0},s,t,Q)$ (D.239)
with $\tilde{\xi}-\pi/2=\gamma=0.55772.....$ is Euler-Mascheroni constant, and
$\displaystyle G_{5}(j_{0},s,t,Q)=$
$\displaystyle\Big{(}\frac{\tilde{\kappa}_{N}}{\tilde{\kappa}_{V}}\Big{)}^{4-\Delta(j)+j-2}$
(D.240)
$\displaystyle\times\frac{1}{s^{2}}\bigg{(}\frac{1}{2}\tilde{\kappa}_{V}^{4-\Delta(j)+j-2}\Gamma(\Delta(j)-2)\Big{(}\mathcal{V}^{T}_{hLL}(j,Q,Q^{\prime})\times
B_{1}^{\alpha\beta}-\mathcal{V}^{L}_{hLL}(j,Q,Q^{\prime})\times
B_{0}^{\alpha\beta}\Big{)}$
$\displaystyle\times\frac{\sqrt{2\kappa^{2}}}{g_{5}^{2}}\times\tilde{\kappa}_{N}^{j-2+\Delta(j)}A(j,K)\bar{u}(p_{2})\gamma_{\alpha}p_{\beta}u(p_{1})\bigg{)}\bigg{|}_{j\rightarrow
j_{0},\,\Delta(j)\rightarrow 2}$
with, $Qz=\xi$,
$\displaystyle\mathcal{V}^{T}_{hLL}(j,Q,Q^{\prime})=$
$\displaystyle\frac{\sqrt{2\kappa^{2}}}{2}\int_{0}^{\infty}dz\sqrt{g}e^{-z^{2}\tilde{\kappa}_{V}^{2}}\,z^{4+2(j-2)}\times\mathcal{V}(Q,z)\times\mathcal{V}(Q^{\prime},z)\times
C(j)\times z^{\Delta(j)-(j-2)}$ $\displaystyle=$ $\displaystyle
Q^{4-(j+\Delta(j))}\times\frac{\sqrt{2\kappa^{2}}}{2}\int_{0}^{\infty}d\xi\,e^{-\xi^{2}\frac{\tilde{\kappa}_{V}^{2}}{Q^{2}}}\,\xi^{4+2(j-2)}\times\mathcal{V}(\xi)\times\mathcal{V}(\xi
Q^{\prime}/Q)\times C(j)\times\xi^{\Delta(j)-(j-2)}\,,$
$\displaystyle\mathcal{V}^{L}_{hLL}(j,Q,Q^{\prime})=$
$\displaystyle\frac{\sqrt{2\kappa^{2}}}{2}\int_{0}^{\infty}dz\sqrt{g}e^{-z^{2}\tilde{\kappa}_{V}^{2}}\,z^{4+2(j-2)}\times\partial_{z}\mathcal{V}(Q,z)\times\partial_{z}\mathcal{V}(Q^{\prime},z)\times
C(j)\times z^{\Delta(j)-(j-2)}$ $\displaystyle=$ $\displaystyle
Q^{6-(j+\Delta(j))}\times\frac{\sqrt{2\kappa^{2}}}{2}\int_{0}^{\infty}d\xi\,e^{-\xi^{2}\frac{\tilde{\kappa}_{V}^{2}}{Q^{2}}}\,\xi^{4+2(j-2)}\times\partial_{\xi}\mathcal{V}(\xi)\times\partial_{\xi}\mathcal{V}(\xi
Q^{\prime}/Q)\times C(j)\times\xi^{\Delta(j)-(j-2)}\,,$
and
$\displaystyle
A(j,K)=\frac{\tilde{\kappa}_{N}^{-(j-2)-\Delta(j)}}{2}\,\frac{\Gamma(c)\Gamma(1-\tilde{b}+c)}{\Gamma(1-\tilde{b}+c+\tilde{a})}$
$\displaystyle\times\bigg{(}\bigg{(}\frac{\tilde{n}_{R}}{\tilde{\kappa}_{N}^{\tau-1}}\bigg{)}^{2}\,_{2}F_{1}(\tilde{a},c+1,1-\tilde{b}+c+\tilde{a},-1)+\bigg{(}\frac{\tilde{n}_{L}}{\tilde{\kappa}_{N}^{\tau}}\bigg{)}^{2}\frac{c(1-\tilde{b}+c)}{1-\tilde{b}+c+\tilde{a}}\,_{2}F_{1}(\tilde{a}+1,c+1,2-\tilde{b}+c+\tilde{a},-1)\bigg{)}\,.$
The parameters are fixed as
$\displaystyle 1-\tilde{b}+c=(\tau-1)+\frac{j-2}{2}+\frac{\Delta(j)}{2}$
$\displaystyle 1-\tilde{b}+c+\tilde{a}=(\tau+1)+\frac{j-2}{2}+a_{K}$
$\displaystyle c=(\tau+1)+\frac{j-2}{2}-\frac{\Delta(j)}{2}$
$\displaystyle\tilde{n}_{R}=\tilde{n}_{L}\tilde{\kappa}_{N}^{-1}\sqrt{\tau-1}\qquad\tilde{n}_{L}=\tilde{\kappa}_{N}^{\tau}\sqrt{{2}/{\Gamma(\tau)}}$
and
$\displaystyle
C(j)=\tilde{\kappa}_{V}^{2\Delta(j)-4}\times\frac{1}{\Delta(j)}\frac{2^{\Delta(j)-2}\Gamma(a_{K}+\frac{\Delta(j)}{2})}{\Gamma(\Delta(j)-2)}\qquad$
$\displaystyle\Delta(j)=2+\sqrt{2\sqrt{\lambda}(j-j_{0})}\qquad{\rm and}\qquad
a_{K}=\frac{a}{2}=\frac{K^{2}}{8{\tilde{\kappa}}^{2}}\qquad{\rm and}\qquad
j_{0}=2-\frac{2}{\sqrt{\lambda}}\,.$ (D.244)
We can rewrite $G_{5}(j_{0},s,t,Q,Q^{\prime})$ of (D.240) more compactly as
$\displaystyle
G_{5}(j_{0},s,t,Q,Q^{\prime})=\frac{2\kappa^{2}}{g_{5}^{2}}\times\Big{(}\frac{Q}{\tilde{\kappa}}\Big{)}^{2-(j+\Delta(j))}$
$\displaystyle\times\frac{1}{s^{2}}\mathcal{F}(j,K)\bigg{(}I_{\xi}^{T}(j,Q,Q^{\prime})\times
B_{1}^{\alpha\beta}p_{\alpha}p_{\beta}-I_{\xi}^{L}(j,Q,Q^{\prime})\times
B_{0}^{\alpha\beta}p_{\alpha}p_{\beta}Q^{2}\bigg{)}\bigg{|}_{j\rightarrow
j_{0},\,\Delta(j)\rightarrow 2}$
where we have set $\tilde{\kappa}_{V}=\tilde{\kappa}_{N}=\tilde{\kappa}$, and
defined the dimensionless functions
$\displaystyle\mathcal{F}(j,K)$ $\displaystyle\equiv$
$\displaystyle\tilde{\kappa}^{j+2-\Delta(j)}\times\Gamma(\Delta(j)-2)\times
C(j,K)\times A(j,K)\,,$ $\displaystyle I_{\xi}^{T}(j,Q,Q^{\prime})$
$\displaystyle\equiv$
$\displaystyle\frac{1}{2}\int_{0}^{\infty}d\xi\,e^{-\xi^{2}\frac{\tilde{\kappa}_{V}^{2}}{Q^{2}}}\,\xi^{\Delta(j)+j+2}\times\mathcal{V}(\xi)\times\mathcal{V}(\xi
Q^{\prime}/Q)\,,$ (D.248) $\displaystyle\approx$
$\displaystyle\frac{1}{2}\times
2^{\Delta(j)+j+2}\,\frac{Q^{\prime}}{Q}\,G_{2,2}^{2,2}\left(\frac{Q^{2}}{Q^{\prime
2}}\bigg{|}\begin{array}[]{c}\frac{1}{2},\frac{3}{2}\\\
\frac{1}{2}(j+\Delta(j)+4),\frac{1}{2}(j+\Delta(j)+6)\\\
\end{array}\right)\,,$ $\displaystyle I_{\xi}^{L}(j,Q,Q^{\prime})$
$\displaystyle\equiv$
$\displaystyle\frac{1}{2}\int_{0}^{\infty}d\xi\,e^{-\xi^{2}\frac{\tilde{\kappa}_{V}^{2}}{Q^{2}}}\,\xi^{\Delta(j)+j+2}\times\partial_{\xi}\mathcal{V}(\xi)\times\partial_{\xi}\mathcal{V}(\xi
Q^{\prime}/Q)$ (D.251) $\displaystyle\approx$ $\displaystyle\frac{1}{2}\times
2^{\Delta(j)+j+2}\,\frac{Q^{\prime
2}}{Q^{2}}\,G_{2,2}^{2,2}\left(\frac{Q^{2}}{Q^{\prime
2}}\bigg{|}\begin{array}[]{c}1,1\\\
\frac{1}{2}(j+\Delta(j)+5),\frac{1}{2}(j+\Delta(j)+5)\\\
\end{array}\right)\,,$
using the identities (C.209).
We can also rewrite the amplitude (D.239) as
$\displaystyle{\cal A}^{tot}_{Lp\rightarrow Lp}(s,t)$ $\displaystyle\simeq$
$\displaystyle\frac{2\kappa^{2}}{g_{5}^{2}}\times\Big{(}\frac{Q}{\tilde{\kappa}}\Big{)}^{2+2/\sqrt{\lambda}}\times\Big{(}\frac{s}{\tilde{\kappa}^{2}}\Big{)}^{-2/\sqrt{\lambda}}\times\frac{e^{-\sqrt{\lambda}\tilde{\xi}^{2}/2\log[s/\tilde{\kappa}^{2}]}}{(\log[s/\tilde{\kappa}^{2}])^{3/2}}$
$\displaystyle\times$
$\displaystyle\left[(\sqrt{\lambda}/\pi)+i\right]\times(\sqrt{\lambda}/2\pi)^{1/2}\;\tilde{\xi}\;\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\log[s/\tilde{\kappa}^{2}]}\bigg{)}\right)\times\tilde{G}_{5}(j_{0},t,Q,Q^{\prime})$
where we have explicitly used $\tilde{\tau}=\log[s/\tilde{\kappa}^{2}]$,
$j_{0}=2-\frac{2}{\sqrt{\lambda}}$, $\Delta(j_{0})=2$,
$\tilde{\xi}-\pi/2=\gamma=0.55772.....$ is Euler-Mascheroni constant, and
defined
$\displaystyle\tilde{G}_{5}(j_{0},s,t,Q,Q^{\prime})\equiv\frac{1}{Q^{4}}\times\frac{g_{5}^{2}}{2\kappa^{2}}\times\Big{(}\frac{\tilde{\kappa}}{Q}\Big{)}^{2-(j_{0}+\Delta(j_{0}))}\times
s^{2}\times G_{5}(j_{0},s,t,Q,Q^{\prime})$ (D.253)
For small-x, we have $s\simeq{Q^{2}}/{x}$, we can rewrite the amplitude (D.3)
in terms of $x$ as
$\displaystyle{\cal A}^{tot}_{Lp\rightarrow Lp}(x,Q,t)$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}\times\frac{2\kappa^{2}}{g_{5}^{2}}\times\Big{(}\frac{Q}{\tilde{\kappa}}\Big{)}^{2-2/\sqrt{\lambda}}\times\Big{(}\frac{1}{x}\Big{)}^{1-2/\sqrt{\lambda}}\times\frac{e^{-\sqrt{\lambda}\tilde{\xi}^{2}/2(\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x])}}{(\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x])^{3/2}}$
$\displaystyle\times$
$\displaystyle\left[(\sqrt{\lambda}/\pi)+i\right]\times(\sqrt{\lambda}/2\pi)^{1/2}\;\tilde{\xi}\;\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{)}\right)\times\tilde{\tilde{G}}_{5}(j_{0},x,t,Q,Q^{\prime})$
where we have defined $\tilde{\tilde{G}}_{5}(j_{0},x,t,Q,Q^{\prime})\equiv
2x\,\tilde{G}_{5}(j_{0},s,t,Q,Q^{\prime})$ with ($\epsilon\cdot q=0$)
$\displaystyle\tilde{\tilde{G}}_{5}(j_{0},x,t,Q,Q^{\prime})=\mathcal{F}(j,K)\bigg{(}I_{\xi}^{T}(j,Q,Q^{\prime})\times\Big{(}\frac{1}{2x}\epsilon^{2}-\frac{2x}{Q^{2}}(\epsilon\cdot
p)^{2}\Big{)}-I_{\xi}^{L}(j,Q,Q^{\prime})\times\frac{2x}{Q^{2}}(\epsilon\cdot
p)^{2}\bigg{)}\bigg{|}_{j\rightarrow j_{0},\,\Delta(j)\rightarrow 2}\,,$
## Appendix E Operator Product Expansion
The parton model emerges in QCD through a leading twist contribution to the
structure functions. The twist expansion follows from the operator product
expansion (OPE) of the JJ currents. We now illustrate this expansion to
leading order for the charged current contributions in T-ordered product in
(IV.92). Specifically, we have as $x\rightarrow 0$
$\displaystyle T^{*}\left(J^{W^{-}}_{\mu}(x)J^{W^{+}}_{\nu}(0)\right)\approx$
$\displaystyle
e_{W}^{2}\left(\overline{q}(x)T^{-}T^{+}\gamma_{\mu}\frac{1}{2}(1-\gamma_{5})\,S(x)\,\gamma_{\nu}\frac{1}{2}(1-\gamma_{5})q(0)+\overline{q}(0)T^{+}T^{-}\gamma_{\nu}\frac{1}{2}(1-\gamma_{5})\,S(-x)\,\gamma_{\mu}\frac{1}{2}(1-\gamma_{5})q(x)\right)$
with $S(x)=2i\gamma\cdot x/(2\pi x^{2})^{2}$. With the help of the identity
$\displaystyle\gamma_{\mu}\gamma\cdot
x\gamma_{\nu}=\left(S_{\mu\nu\alpha\beta}+i\epsilon_{\mu\nu\alpha\beta}\gamma_{5}\right)x^{\alpha}\gamma^{\beta}$
(E.257)
with the symmetric tensor
$\displaystyle
S_{\mu\nu\alpha\beta}=\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta{\nu\alpha}-\eta_{\mu\nu}\eta_{\alpha\beta}$
(E.258)
in (E), the short distance contribution to the T-ordered product in (IV.92) is
$\displaystyle T_{\mu\nu}^{-+}\approx
2e_{W}^{2}\frac{q^{\alpha}}{q^{2}}\left<P\right|\bigg{(}S_{\mu\nu\alpha\beta}\overline{q}(0)\tau^{3}\gamma^{\beta}(1-\gamma_{5})q(0)-i\epsilon_{\mu\nu\alpha\beta}\overline{q}(0)\gamma^{\beta}(1-\gamma_{5})q(0)\bigg{)}\left|P\right>$
(E.259)
For unpolarized scattering, a comparison of (E.259) to (IV.92) suggests that
the parity odd structure function $F_{3}$ can be identified with the
antisymmetric tensor contribution,
$\displaystyle F_{3}(x,Q^{2})P^{\beta}\approx(2e_{W})^{2}{\rm
Im}\bigg{(}\frac{1}{x}\left<P\right|\overline{q}\gamma^{\beta}(1-\gamma_{5})q\left|P\right>\bigg{)}$
(E.260)
(E.260) is suggestive of a $j=1$ exchange through the left singlet current,
since $s^{j=1}\leftrightarrow 1/x$.
Although the twist-2 contribution to the OPE is real, it provides the relevant
starting point for the Reggeization by summing over higher spin-j states in
holography HATTAX . For that, we first note that the holographic dual of the
singlet current form factor is
$\displaystyle\left<P_{X}\right|\overline{q}\gamma^{\beta}(1-\gamma_{5})q\left|P\right>=$
$\displaystyle\left[e_{N}\overline{u}_{N}(P_{X})\gamma^{\beta}(1-\gamma^{5})u_{N}(P)\right]$
(E.261) $\displaystyle\times\int dz\sqrt{g}e^{-{\phi}}\,{{\cal
V}(Q,z)}\,\frac{z}{R}\,\bigg{[}\frac{1}{2}\bigg{(}\frac{z^{2}}{R^{2}}\bigg{)}^{2}\,\tilde{f}_{L}^{n_{x}}(z)\tilde{f}_{L}^{0}(z)+\frac{1}{2}\bigg{(}\frac{z^{2}}{R^{2}}\bigg{)}^{2}\,\tilde{f}_{R}^{n_{x}}(z)\tilde{f}_{R}^{0}(z)\bigg{]}$
The bulk-to-boundary propagator ${\cal V}(Q,z^{\prime})$ relates to the bulk-
to-bulk propagator $G(Q,z,z^{\prime})$ through
$\displaystyle\lim_{z\to 0}\frac{2}{z^{\prime 2}}\,G_{1}(Q,z^{\prime},z)={\cal
V}(Q,z)$ (E.262)
Using (E.262) into (E.261) gives
$\displaystyle\left<P_{X}\right|\overline{q}\gamma^{\beta}(1-\gamma_{5})q\left|P\right>=$
$\displaystyle\left[e_{N}\overline{u}_{N}(P_{X})\gamma^{\beta}(1-\gamma^{5})u_{N}(P)\right]\,$
(E.263) $\displaystyle\times\lim_{z^{\prime}\to 0}\frac{2}{z^{\prime 2}}\int
dz\sqrt{g}e^{-{\phi}}\,G_{1}(Q,z^{\prime},z)\,$
$\displaystyle\times\frac{z}{R}\,\bigg{[}\frac{1}{2}\bigg{(}\frac{z^{2}}{R^{2}}\bigg{)}^{2}\,\tilde{f}_{L}^{n_{x}}(z)\tilde{f}_{L}^{0}(z)+\frac{1}{2}\bigg{(}\frac{z^{2}}{R^{2}}\bigg{)}^{2}\,\tilde{f}_{R}^{n_{x}}(z)\tilde{f}_{R}^{0}(z)\bigg{]}$
### E.1 Hard wall
For the hard wall model with $\phi=0$, the bulk-to-bulk propagator can be
readily constructed
$\displaystyle
G_{1}(Q,z^{\prime},z)=zz^{\prime}\bigg{(}I_{0}(Qz_{0})K_{1}(Qz_{>})+K_{0}(Qz_{0})I_{1}(Qz_{>})\bigg{)}\frac{I_{1}(Qz_{<})}{I_{0}(Qz_{0})}$
(E.264)
and (E.262) explicitly checked. However, for the Reggeization it is more
useful to recall that the bulk-to-bulk propagator for the $U_{L}(1)$ vector
field, obeys the Green′s equation in warped space ($Q^{2}=-q^{2}$)
$\displaystyle\bigg{(}\bigg{(}\Delta_{j=1}-z^{2}Q^{2}+m_{j=1}^{2}\bigg{)}=\bigg{(}-z^{2}(\partial_{z}^{2}+Q^{2})+z\partial_{z}\bigg{)}\bigg{)}G_{j=1}(Q,z^{\prime},z)=\frac{\delta(z-z^{\prime})}{\sqrt{g}}$
(E.265)
with $m^{2}_{j=1}=-3$. Using the open-string Regge trajectory
$\displaystyle j=1+\alpha^{\prime}(m^{2}_{j}-m_{1}^{2})\qquad{\rm
with}\qquad\alpha^{\prime}=l_{s}^{2}=1/\sqrt{\lambda}$ (E.266)
(E.265) generalizes to spin-j
$\displaystyle\bigg{(}\Delta_{j}-z^{2}Q^{2}+m_{j}^{2}\bigg{)}G_{j}(Q,z^{\prime},z)=\frac{\delta(z-z^{\prime})}{\sqrt{g}}$
(E.267)
with the recursive relation for the warped Laplacian-like
$\displaystyle\Delta_{j}=z^{1-j}\,\Delta_{1}\,z^{j-1}$ (E.268)
(E.267) can be formally inverted
$\displaystyle{\sqrt{g}}\,G_{j}(Q,z^{\prime},z)=\frac{1}{(\Delta_{j}-z^{2}Q^{2}+m_{j}^{2})}\,{\delta(z-z^{\prime})}=z^{2-j}\,\frac{1}{(\Delta_{2}-z^{2}Q^{2}+m_{j}^{2})}\,z^{j-2}\,{\delta(z-z^{\prime})}$
(E.269)
Changing to the conformal variable $z^{2}=e^{-\rho}$, noting that
$\Delta_{2}=-4\partial_{\rho}^{2}+4$ and using the plane-wave identity
$\displaystyle
z^{\prime}\delta(z-z^{\prime})=\bigg{(}\frac{z^{\prime}}{z}\bigg{)}^{j-2}\int\frac{d\nu}{\pi}\,e^{i\nu(\rho-\rho^{\prime})}$
(E.270)
we can recast (E.269) in the form
$\displaystyle{\sqrt{g}}\,G_{j}(0,z^{\prime},z)=\sqrt{g^{\prime}}(g^{\prime
xx})^{j}(zz^{\prime})^{2-j}\int\frac{d\nu}{\pi}\,\frac{1}{4\nu^{2}+4+m_{j}^{2}}\,e^{i\nu(\rho-\rho^{\prime})}$
(E.271)
for $Q=0$. The Reggeized form of the spin-j and twist-2 extension of (E.263)
is
$\displaystyle\sum_{j}\,\frac{1}{x^{j}}\,\left<P\right|\overline{q}\gamma^{\beta}\,\partial^{j-1}\,(1-\gamma_{5})q\left|P\right>=\int_{\mathbb{C}}\frac{dj}{4i}\frac{1-e^{-i\pi
j}}{{\rm sin}\pi
j}\frac{1}{x^{j}}\,\left[e_{N}\overline{u}_{N}(P)\gamma^{\beta}\,\partial^{j-1}\,(1-\gamma^{5})u_{N}(P)\right]$
$\displaystyle\times\lim_{z^{\prime}\to 0}\frac{2}{z^{\prime 2}}\int
dz\sqrt{g^{\prime}}(g^{\prime
xx})^{j}(zz^{\prime})^{2-j}\int\frac{d\nu}{\pi}\,\frac{1}{4\nu^{2}+1+\sqrt{\lambda}(j-1)}\,e^{i\nu(\rho-\rho^{\prime})}\,\frac{z}{R}\,\bigg{[}\frac{1}{2}\psi_{L}^{2}(z)+\frac{1}{2}\psi_{R}^{2}(z)\bigg{]}\,$
using the open string Regge trajectory (E.266) in the forward limit ($Q=0$).
Here $\psi_{L}(z)$ is the lowest left-chirality bulk fermionic wavefunction
for the hard wall. The contour $\mathbb{C}$ is to the left-most of the poles
$j=1,3,...$ and to the right of the pole $j_{0}=1-1/\sqrt{\lambda}$. Undoing
the contour integration-${\mathbb{C}}$ by closing to the left and picking the
single pole $j_{0}$, and then performing the $\nu$-integration yield
$\displaystyle{\rm Im}\,\sum^{\rm
odd}_{j}\,\frac{1}{x^{j}}\,\left<P\right|\overline{q}\gamma^{\beta}\,\partial^{j-1}\,(1-\gamma_{5})q\left|P\right>=\frac{1}{x^{j_{0}}}\,\left[e_{N}\overline{u}_{N}(P)\gamma^{\beta}\,\partial^{j_{0}-1}\,(1-\gamma^{5})u_{N}(P)\right]$
$\displaystyle\times\frac{\pi}{2\sqrt{\lambda}}\lim_{z^{\prime}\to
0}\frac{2}{z^{\prime 2}}\int dz\sqrt{g^{\prime}}(g^{\prime
xx})^{j_{L}}(zz^{\prime})^{2-j_{0}}\,\frac{e^{-(\rho-\rho^{\prime})/4D\chi}}{\sqrt{\pi
D\chi}}\,\frac{z}{R}\,\bigg{[}\frac{1}{2}\psi_{L}^{2}(z)+\frac{1}{2}\psi_{R}^{2}(z)\bigg{]}\,$
The Gribov time is $\chi={\rm ln}(1/x)$ and the diffusion constant of the
Reggeon is $D=4/\sqrt{\lambda}$. (E.1) fixes the odd structure function in
(E.260) in the forward direction using this semi-quantitative OPE argument,
$\displaystyle
F_{3}(0,x)\approx\frac{1}{x^{j_{0}}}\approx\frac{1}{x^{1-1/\sqrt{\lambda}}}$
(E.274)
### E.2 Soft wall
For the soft-wall model, the Reggeized current form factor is given by
$\displaystyle\sum^{\rm
odd}_{j}\,\frac{1}{x^{j}}\,\left<P\right|\overline{q}\gamma^{\beta}\,\partial^{j-1}\,(1-\gamma_{5})q\left|P\right>=$
$\displaystyle-x^{j_{0}}\int_{-\infty}^{j_{0}}\frac{dj}{\pi}\left(\frac{1+e^{-i\pi}}{{\rm
sin}\,\pi
j}\right)x^{j-j_{0}}\,\text{Im}\left[\,2\times\frac{2}{3}\times\tilde{\kappa}^{(j-1)+\Delta(j)+1}\times\frac{\Gamma(\Delta(j)-2+a)}{\Gamma(\Delta(j)-2)}\times\frac{1}{g_{5}}\times\mathcal{V}^{\beta}_{L\bar{\Psi}\Psi}(p_{1}=p_{2}=p,k_{z}=0)\right]$
$\displaystyle=$
$\displaystyle-x^{j_{0}}\int_{-\infty}^{j_{0}}\frac{dj}{\pi}\left(\frac{1+e^{-i\pi}}{{\rm
sin}\,\pi
j}\right)x^{j-j_{0}}\,\text{Im}\left[\,2\times\frac{2}{3}\times\tilde{\kappa}^{(j-1)+\Delta(j)+1}\times\frac{\Gamma(\Delta(j)-2+a)}{\Gamma(\Delta(j)-2)}\times\frac{1}{g_{5}}\times
g_{5}F_{1}^{(LN)}(j,K=0)\times\bar{u}(p)\gamma^{\beta}u(p)\right]$
where $\mathcal{V}^{\beta}_{L\bar{\Psi}\Psi}(p_{1},p_{2},k_{z})$ is given by
(C), and $F_{1}^{(LN)}(j,K)$ is given by (C). Note that the bracket
$\displaystyle\bigg{[}2\times\frac{2}{3}\times\tilde{\kappa}^{(j-1)+\Delta(j)+1}\times\frac{\Gamma(\Delta(j)-2+a)}{\Gamma(\Delta(j)-2)}\times\frac{1}{g_{5}}\times
g_{5}F_{1}^{(LN)}(j,K)\bigg{]}$ (E.276)
is the spin-j form factor which reduces to the spin-1 form factor for $j=1$,
for the current operator
$2\times\tilde{J}_{L}^{\beta}(0)=\overline{q}\gamma^{\beta}(1-\gamma_{5})q$
sourced by $\frac{1}{2}\times\frac{3}{2}\times L_{\beta}^{0}(K,z\rightarrow
0)$ at the boundary. Also note that the momentum transfer is $k_{z}\equiv
q_{z}$ and that $-k^{2}=K^{2}\equiv Q^{2}$ with
$a={K^{2}}/{4\tilde{\kappa}^{2}}\equiv{Q^{2}}/{4\tilde{\kappa}^{2}}$. The
momentum of the in-coming nucleon is $p_{1}=p$, and the momentum of the out-
going nucleon is $p_{2}=p$, with $k=p_{2}-p_{1}\equiv q$.
Following the reasoning in Appendix C, we can evaluate the integral in (E.2)
with the result
$\displaystyle{\rm Im}\sum^{\rm
odd}_{j}\,\frac{1}{x^{j}}\,\left<P\right|\overline{q}\gamma^{\beta}\,\partial^{j-1}\,(1-\gamma_{5})q\left|P\right>\simeq$
$\displaystyle
e^{j_{0}\tau_{x}}\left[0\times(\sqrt{\lambda}/\pi)+i/i\right](\sqrt{\lambda}/2\pi)^{1/2}\;\tilde{\xi}\;\frac{e^{-\sqrt{\lambda}\tilde{\xi}^{2}/2\tau_{x}}}{\tau_{x}^{3/2}}\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\tau_{x}}\bigg{)}\right)\times\mathcal{G}_{5}(j_{0},x,Q=0)$
with
$\displaystyle\mathcal{G}_{5}(j_{0},x,Q=0)=$
$\displaystyle\bigg{(}\Gamma(\Delta(j)-2)\times
2\times\frac{2}{3}\times\tilde{\kappa}^{(j-1)+\Delta(j)+1}\times\frac{\Gamma(\Delta(j)-2+a)}{\Gamma(\Delta(j)-2)}\times\frac{1}{g_{5}}\times
g_{5}F_{1}^{(LN)}(j,Q)\times 2P^{\beta}\bigg{)}\bigg{|}_{j\rightarrow
j_{0},\,\Delta(j)\rightarrow 2,Q\rightarrow 0}$
Again, $j_{0}=1-\frac{1}{\sqrt{\lambda}}$, $\tau_{x}=\log[1/x]$,
$\bar{u}(p)\gamma^{\beta}u(p)=2p^{\beta}=2P^{\beta}$, and
$\tilde{\xi}-\pi/2=\gamma=0.55772.....$ is the Euler-Mascheroni constant.
Finally, comparing (E.2) to (E.260), we find
$\displaystyle
F_{3}(0,x)\approx\frac{1}{x^{1-1/\sqrt{\lambda}}}\times(\sqrt{\lambda}/2\pi)^{1/2}\;\tilde{\xi}\;\frac{e^{-\sqrt{\lambda}\tilde{\xi}^{2}/2\tau_{x}}}{\tau_{x}^{3/2}}\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\tau_{x}}\bigg{)}\right)\times\tilde{\mathcal{G}}_{5}(j_{0},x,0)$
(E.279)
with
$\displaystyle\tilde{\mathcal{G}}_{5}(j_{0},x,0)=$
$\displaystyle\bigg{(}\Gamma(\Delta(j)-2)\times
2\times\frac{2}{3}\times\tilde{\kappa}^{(j-1)+\Delta(j)+1}\times\frac{\Gamma(\Delta(j)-2+a)}{\Gamma(\Delta(j)-2)}\times\frac{1}{g_{5}}\times
g_{5}F_{1}^{(LN)}(j,Q)\bigg{)}\bigg{|}_{j\rightarrow
j_{0},\,\Delta(j)\rightarrow 2,Q\rightarrow 0}\,.$
## Appendix F Trace of Gamma matrices
Note that the Dirac traces do not depend on the specific form of the
$\gamma^{0},\gamma^{1},\gamma^{2},\gamma^{3}$ matrices but are completely
determined by the Clifford algebra
$\displaystyle\\{\gamma^{\mu},\gamma^{\nu}\\}\ \equiv\
\gamma^{\mu}\gamma^{\nu}\,+\,\gamma^{\nu}\gamma^{\mu}\ =\ 2\eta^{\mu\nu}\,,$
(F.281)
and some useful identities for carrying some of the Dirac traces of gamma
matrices above, are given by (note that
$\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ and it satisfies
$\gamma^{5}\gamma^{\mu}=-\gamma^{\mu}\gamma^{5}$)
$\displaystyle{\rm tr}(\gamma^{\mu}\gamma^{\nu})=4\eta^{\mu\nu}\,,$ (F.282)
$\displaystyle{\rm tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{5})=0\,,$ (F.283)
$\displaystyle{\rm
tr}(\gamma^{\alpha}\gamma^{\mu}\gamma^{\beta}\gamma^{\nu})=4\eta^{\alpha\mu}\eta^{\beta\nu}\
-\ 4\eta^{\alpha\beta}\eta^{\mu\nu}\ +\ 4\eta^{\alpha\nu}\eta^{\mu\beta}\,,$
(F.284)
$\displaystyle{\rm
tr}(\gamma^{\alpha}\gamma^{\mu}\gamma^{\beta}\gamma^{\nu}\gamma^{5})=-4i\epsilon^{\alpha\mu\beta\nu}\,,$
(F.285)
$\displaystyle{\rm
tr}(\gamma^{\alpha}\gamma^{\mu}\gamma^{\tilde{\nu}}\gamma^{\beta}\gamma^{\tilde{\mu}}\gamma^{\nu})$
$\displaystyle=4\eta^{\alpha\mu}\times\Bigl{(}\eta^{{\tilde{\nu}}\beta}\eta^{{\tilde{\mu}}\nu}\,-\,\eta^{{\tilde{\nu}}{\tilde{\mu}}}\eta^{\beta\nu}\,+\,\eta^{{\tilde{\nu}}\nu}\eta^{\beta{\tilde{\mu}}}\Bigr{)}$
(F.286) $\displaystyle\qquad-\
4\eta^{\alpha{\tilde{\nu}}}\times\Bigl{(}\eta^{\mu\beta}\eta^{{\tilde{\mu}}\nu}\,-\,\eta^{\mu{\tilde{\mu}}}\eta^{\beta\nu}\,+\,\eta^{\mu\nu}\eta^{\beta{\tilde{\mu}}}\Bigr{)}$
(F.287) $\displaystyle\qquad+\
4\eta^{\alpha\beta}\times\Bigl{(}\eta^{\mu{\tilde{\nu}}}\eta^{{\tilde{\mu}}\nu}\,-\,\eta^{\mu{\tilde{\mu}}}\eta^{{\tilde{\nu}}\nu}\,+\,\eta^{\mu\nu}\eta^{{\tilde{\nu}}{\tilde{\mu}}}\Bigr{)}$
(F.288) $\displaystyle\qquad-\
4\eta^{\alpha{\tilde{\mu}}}\times\Bigl{(}\eta^{\mu{\tilde{\nu}}}\eta^{\beta\nu}\,-\,\eta^{\mu\beta}\eta^{{\tilde{\nu}}\nu}\,+\,\eta^{\mu\nu}\eta^{{\tilde{\nu}}\beta}\Bigr{)}$
$\displaystyle\qquad+\
4\eta^{\alpha\nu}\times\Bigl{(}\eta^{\mu{\tilde{\nu}}}\eta^{\beta{\tilde{\mu}}}\,-\,\eta^{\mu\beta}\eta^{{\tilde{\nu}}{\tilde{\mu}}}\,+\,\eta^{\mu{\tilde{\mu}}}\eta^{{\tilde{\nu}}\beta}\Bigr{)}\,,$
$\displaystyle{\rm
tr}(\gamma^{\alpha}\gamma^{\mu}\gamma^{{\tilde{\nu}}}\gamma^{\beta}\gamma^{{\tilde{\mu}}}\gamma^{\nu}\gamma^{5})=$
$\displaystyle-$ $\displaystyle
4i(\eta^{\alpha\mu}\epsilon^{{\tilde{\nu}}\beta{\tilde{\mu}}\nu}-\eta^{\alpha{\tilde{\nu}}}\epsilon^{\mu\beta{\tilde{\mu}}\nu}+\eta^{{\tilde{\nu}}\mu}\epsilon^{\alpha\beta{\tilde{\mu}}\nu}-\eta^{{\tilde{\mu}}\nu}\epsilon^{\beta\alpha\mu{\tilde{\nu}}}$
(F.291) $\displaystyle+$
$\displaystyle\eta^{\beta\nu}\epsilon^{{\tilde{\mu}}\alpha\mu{\tilde{\nu}}}-\eta^{\beta{\tilde{\mu}}}\epsilon^{\nu\alpha\mu{\tilde{\nu}}})\,,$
and
$\displaystyle{\rm tr}(\gamma^{\nu_{1}}\cdots\gamma^{\nu_{n}}\gamma^{5})\ $
$\displaystyle=$ $\displaystyle\ 0\quad\forall\ {\rm odd}\ n\,,$
$\displaystyle{\rm tr}(\gamma^{\nu_{1}}\cdots\gamma^{\nu_{n}})\ $
$\displaystyle=$ $\displaystyle\ 0\quad\forall\ {\rm odd}\ n\,.$ (F.292)
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|
# Berezinskii-Kosterlitz-Thouless phase transitions with long-range couplings
Guido Giachetti<EMAIL_ADDRESS>SISSA and INFN Sezione di Trieste, Via
Bonomea 265, I-34136 Trieste, Italy Nicolò Defenu Institute for Theoretical
Physics, ETH Z$\ddot{u}$rich, Wolfgang-Pauli-Str. 27, 8093 Z$\ddot{u}$rich,
Switzerland Stefano Ruffo SISSA and INFN Sezione di Trieste, Via Bonomea
265, I-34136 Trieste, Italy Istituto dei Sistemi Complessi, Consiglio
Nazionale delle Ricerche, Via Madonna del Piano 10, I-50019 Sesto Fiorentino,
Italy Andrea Trombettoni Department of Physics, University of Trieste,
Strada Costiera 11, I-34151 Trieste, Italy SISSA and INFN Sezione di Trieste,
Via Bonomea 265, I-34136 Trieste, Italy CNR-IOM DEMOCRITOS Simulation Center,
Via Bonomea 265, I-34136 Trieste, Italy
###### Abstract
The Berezinskii-Kostelitz-Thouless (BKT) transition is the paradigmatic
example of a topological phase transition without symmetry-breaking, where a
quasi-ordered phase, characterized by a power law scaling of the correlation
functions at low temperature, is disrupted by the proliferation of topological
excitations above the critical temperature $T_{\rm BKT}$. In this letter, we
consider the effect of long-range decaying couplings $\sim r^{-2-\sigma}$ on
this phenomenon. After pointing out the relevance of this non trivial problem,
we discuss the phase diagram, which is far richer than the corresponding
short-range one. It features – for $7/4<\sigma<2$ – a quasi ordered phase in a
finite temperature range $T_{c}<T<T_{\rm BKT}$, which occurs between a
symmetry broken phase for $T<T_{c}$ and a disordered phase for $T>T_{\rm
BKT}$. The transition temperature $T_{c}$ displays unique universal features
quite different from those of the traditional, short-range XY model. Given the
universal nature of our findings, they may be observed in current experimental
realizations in $2D$ atomic, molecular and optical quantum systems.
## I Introduction
Two-dimensional interacting systems are well known not to display conventional
symmetry breaking transitions at finite temperature, due to the Hohenberg-
Mermin-Wagner theorem [1]. Yet, a phase transition may appear driven by
topological defects, according to the celebrated Berezinskii-Kosterlitz-
Thouless (BKT) mechanism [2]. In the presence of long-range interactions the
Hohenberg-Mermin-Wagner theorem no longer holds and local order parameters,
such as the magnetization [3], may have a non-zero expectation value. The
general question addressed by this Letter is the fate of the BKT transition
when the range of the interactions is increased. The Sak’s criterion [4]
provides an argument for understanding whether the long-range, power law
coupling $\sim 1/r^{d+\sigma}$ in the classical $O(N)$ model affects
criticality. It can be formulated as follows: at low momenta the short-range
(SR) and long-range (LR) critical two-points functions behave as
$p^{-2+\eta_{\rm sr}}\hskip 14.22636pt\text{vs}\hskip 14.22636ptp^{-\sigma}$
(1)
respectively, where $\eta_{\rm sr}$ is the anomalous dimension of the SR
limit. Therefore, one can define a critical value of the range of the
interactions, $\sigma_{*}=2-\eta_{\rm sr}$, such that, for
$\sigma>\sigma^{*}$, the critical behavior is not affected by LR. The validity
of Sak’s criterion for the classical $O(N)$ models has been the subject of
considerable scrutiny. Indeed, numerical evidences supporting (or rejecting)
the Sak’s result are notoriously hard to obtain [5, 6, 7]. Intense theoretical
investigations both via MC simulations [5, 8, 9], renormalization group (RG)
theory [10, 11, 12] and conformal bootstrap [13] appeared all to confirm the
validity of Sak’s conjecture for the LR-SR crossover so that it is fair to
conclude that the criterion has been a useful tool to understand the critical
behaviour of LR interacting systems [14, 15, 16, 17]. The criterion, is
believed to apply to all symmetry breaking transition in $d\geq 2$. The status
of the $d=2$ classical XY model is rather different, and only few results
(later commented) are known. The main reasons are
i) The Sak criterion cannot be straightforwardly applied, since in the SR
limit the critical behavior is not described by a single RG fixed point, but
rather by a whole line of fixed points with a temperature-dependent exponent
$\eta_{\rm sr}$.
ii) Numerically, the large number of non-vanishing couplings, coming form the
LR nature of the interaction, along with the logarithmic scaling typical of
$2D$ systems (the so-called “Texas state argument” [18]) make the study of the
$2D$ XY universality notoriously challenging.
iii) In the nearest-neighbours $2D$ XY model, the classical treatment takes
advantage of the duality construction [19], through which one can famously
relate the model to the Coulomb gas [20, 21] or the sine-Gordon model [22,
21]. However, this is no longer the case already for next-to-nearest-neighbors
couplings.
iv) It is known that in the SR limit, the physics of the $2D$ classical XY
model can be related to the one of the $1D$ quantum XXZ model via its transfer
matrix [23].
This approach is based on the mapping to hard-core bosons, and therefore to
the XXZ model, and cannot be straightforwardly applied to the the case of $XY$
LR interactions, as one should show the RG irrelevance of terms violating the
hard-core condition. Moreover, let us remark that $2D$ boson gas at finite
temperature with (isotropic) $1/r^{3}$ density-density interaction do exhibit
a BKT transition [24]; but this interaction corresponds to a quantum 1D XXZ in
which only in the $z-z$ interaction is long range.
v) Finally, we observe that the treatment of the SR XY model in $2D$ is very
much simplified by the introduction of the Villain model, [25, 26], which can
be mapped exactly onto the Coulomb gas, and shares the same universality class
of the SR $XY$ model. The physical reason of their connection in the SR
regime, is that the (gapped) amplitude fluctuations of the corresponding
$O(2)$ action are irrelevant [27]. Thus, once the periodic nature of the phase
is taken care of, all the relevant information is present in the theory.
However, in the LR regime the interplay between amplitude and phase
fluctuations cannot be neglected and it is not known whether they still share
the same universality class.
Despite these difficulties, the study of LR XY model is of great interest:
first, since its introduction, the BKT mechanism [28, 29, 30, 31] has been
found to quantitatively describe the universal scaling appearing in several
$2D$ systems with $U(1)$ symmetry, ranging from thin 4He films [32] to
quasi-2D layered superconductors [33, 34, 35, 36, 37], exciton-polariton
systems [38], cold atoms in $2D$ traps [39, 40] and $2D$ electron gases at the
interface between insulating oxides in artificial heterostructures [41, 42,
43]. Apart from these experimental realizations, topological defects are
expected to be relevant in several natural phenomena outside the condensed
matter realm, such as DNA tangling or stripe formation [44, 45, 46]. To
understand how $\sigma_{*}$ is modified, is then a crucial question in all the
cases in which a LR tail of the interaction can be added or tuned, especially
because the spin-wave interaction term, already present in the SR case, may
destroy, partially or totally, the topological nature of the phase transition.
Moreover, the physics of LR interacting systems has recently experienced a new
wave of interest, due to the current experimental realizations on atomic,
molecular and optical (AMO) systems. In particular, trapped ions [47, 48],
Rydberg gases [49] and optical cavities [50, 51] allowed the observation of
plenty of exotic equilibrium and dynamical phenomena induced by LR
interactions, including entanglement and correlations propagation [52, 53],
dynamical phase transitions [54, 55], time crystals [56, 57, 54] and defect
scaling [58, 59]. These experimental results stimulated an impressive
theoretical activity to characterize the equilibrium and dynamical critical
scaling induced by LR interactions in a wide variety of different systems [60,
61, 62, 63, 17, 64, 65].Despite this outpouring theoretical activity and the
long-standing relation between topological scaling and LR interactions, the
possible corrections induced by power-law decaying couplings to the
topological BKT scaling remain an open question, testable in experiments.
## II Model & Preliminaries
We consider a system of planar rotators on a $2D$ lattice of spacing $a$,
described by the Hamiltonian:
$\beta
H=\frac{1}{2}\sum_{\mathbf{i},\mathbf{j}}J_{|\mathbf{i}-\mathbf{j}|}\left[1-\cos(\theta_{\mathbf{j}}-\theta_{\mathbf{i}})\right]$
(2)
where $\mathbf{i},\mathbf{j}\in\mathbb{Z}^{2}$ and
$J_{|\mathbf{i}-\mathbf{j}|}$ has a power-law tail:
$J_{|\mathbf{i}-\mathbf{j}|}\sim\frac{g}{|\mathbf{i}-\mathbf{j}|^{2+\sigma}}$
for $|\mathbf{i}-\mathbf{j}|\gg 1$. The exponent $\sigma$ is assumed positive
in order to ensure additivity of the thermodynamic quantities [66]. For the
following arguments the specific form of the couplings is not important, as
long as that there are no frustration effects nor competing interactions.
Let us now summarize what we do know for sure about the LR XY model (2):
a) For $\sigma<2$, at low enough temperatures, the system magnetizes, as
rigorously proven in [3]. MC simulations at $\sigma=1$ indicate an order-
disorder transition and no BKT phase at finite temperature [67]. Moreover,one
could expect that For $\sigma\leq 1$ the critical exponents of the ferro-
paramagnetic transition are expected to be mean-field [11].
b) In agreement with a), the spin-wave theory in which the cosine is expanded
to the quadratic order, without imposing the periodicity, as in the original
Berezinskii calculation [28], does also magnetize for $\sigma<2$, since the
contribution of the spin fluctuations is of the form $\int d^{2}q/q^{\sigma}$
and thus infrared finite.
c) An upper bound for $\sigma_{*}$ has to be $\sigma_{*}=2$, i.e. for sure
there is BKT for $\sigma>2$, as one can deduce even from the Sak’s argument
since $\eta$ is positive. This result is supported by the self-consistent
harmonic calculation recently presented in [68], which anyway is unable to
make even qualitative predictions for $\sigma<2$.
## III Effective model
We decompose the coupling as
$J_{|\mathbf{i}-\mathbf{j}|}=J^{S}_{|\mathbf{i}-\mathbf{j}|}+g|\mathbf{i}-\mathbf{j}|^{-(2+\sigma)}$
where $J^{S}$ is a SR term taking into account the small-distances behavior of
the coupling. At low temperatures, the spin direction varies smoothly from
site to site and, as a consequence, we can expand the SR term for small phase
differences as $\cos(\theta(\mathbf{x}+\mathbf{r})-\theta(\mathbf{x}))\sim
1-|\nabla\theta|^{2}/2$. The same, however, it is not automatically true for
the LR term, since far-away pairs, whose phase difference is not necessarily
small, give a significant contribution to the Hamiltonian.
These considerations allow us to write a continuous version of the Hamiltonian
in Eq. (2) in terms of the field $\theta(\mathbf{x})$, namely the Euclidean
action
$S[\theta]=\frac{J}{2}\int d^{2}x|\nabla\theta|^{2}+S_{\mathrm{L}R},$ (3)
where the LR part can be written as
$S_{\mathrm{L}R}=-\frac{g}{2\gamma_{2,\sigma}}\int
d^{2}x(\cos\theta\,\nabla^{\sigma}\cos\theta+\sin\theta\,\nabla^{\sigma}\sin\theta),$
(4)
with
$\gamma_{2,\sigma}=2^{\sigma}\Gamma(\scriptstyle\frac{1+\sigma}{2}\displaystyle)\pi^{-1}|\Gamma(\scriptstyle-\frac{\sigma}{2}\displaystyle)|^{-1}$,
by using the definition of (bulk) fractional derivative given in Appendix A.
The first and the second term in Eq. (3) account for the short- and long-range
contributions respectively, with $J\sim 1/T$ and $g\sim 1/T$ being the
temperature dependent couplings. Notice that the result would be different for
a quantum $1D$ chain with LR interactions, where interactions are still SR
along the imaginary time axis [69].
If $g=0$, by following the usual duality procedure [26], one can take into
account the periodic nature of the field $\theta$ in Eq. (3) by isolating the
topological configurations and introducing the vortex fugacity
$y=\exp(-\varepsilon_{c})$, being $\varepsilon_{c}$ the corresponding core
energy. This, in turn, leads to the Kosteritz-Thouless RG equations [29, 30,
26, 70] (see [71, 72] for textbook presentations) which feature a line of
stable Gaussian fixed points for $y=0$ and $J>\frac{2}{\pi}$, describing the
power-law scaling observed in the low-temperature BKT phase. For $g$ small
enough, we expect to have then a continuum theory described by the three
parameters $J$, $g$ and $y$.
In order to explore the effects of LR interactions, we deform the traditional
BKT fixed-points theory with the non-local operator in the second term of Eq.
(3). Since only those fixed-points which are stable under topological
perturbation correspond to an infra-red (IR) limit of the SR BKT theory, we
can restrict ourselves to the region in which the topological excitations are
irrelevant ($J>\frac{2}{\pi}$). The relevance of the LR perturbation depends
on the scaling dimension $\Delta_{g}$ of the coupling $g$, which is defined
according to the asymptotic behavior $g_{\ell}\approx\exp(\Delta_{g}\ell)$ for
$\ell\gg 1$, where as usual in the BKT literature, we put $\ell=\ln(r/a)$. On
the other hand, due to the Gaussian nature of the measure,
$\left\langle\cos\left(\theta(\mathbf{x})-\theta(\mathbf{x^{\prime}})\right)\right\rangle=e^{-\frac{1}{2}\left\langle\left(\theta(\mathbf{x})-\theta(\mathbf{x^{\prime}})\right)^{2}\right\rangle}=|\mathbf{x}-\mathbf{x^{\prime}}|^{-\eta_{\rm
sr}(J)},$ (5)
where $\eta_{\rm sr}(J)=\frac{1}{2\pi J}$ is the exponent of the SR two-point
function, [29, 30, 26]. Following Eq. (5), the scaling dimension of the LR
term reads
$\displaystyle\Delta_{g}=2-\sigma-\eta_{\mathrm{sr}}(J)$ (6)
so that the LR perturbation becomes relevant only if $\sigma<2-\eta_{\rm
sr}(J)$, similarly to the traditional spontaneous symmetry breaking (SSB) case
[11], but with a temperature-dependent anomalous dimension. This confirms that
for $\sigma>2$ the LR perturbation is always irrelevant, as expected.
Let us now consider the case $\sigma<2$. There, the LR perturbation becomes
relevant at small temperatures, since $\eta_{\rm sr}\simeq 0$ for $T\simeq 0$.
Since $\eta_{sr}$ in Eq. (6) is the one of the SR unperturbed theory, we can
apply the results of the traditional BKT theory [73] as long as the LR
perturbation is not relevant. In particular we know that topologicaly
excitations are irrelevant for $\eta_{sr}<1/4$, so that in the range
$7/4<\sigma<2$, a subset of the BKT fixed points remains stable and we have
quasi-long-range order (qlro) for a certain temperature window. This result is
rather non trivial, since in SSB transitions the traditional Sak’s result [4]
predicts the irrelevance of LR couplings at all temperatures for
$\sigma>2-\eta_{\rm sr}$.
## IV RG Flow
These results may be confirmed by deriving the flow equations for the LR term
at the leading order in $g$ for $y=0$, obtaining (see Appendix B):
$\begin{split}\frac{dg_{\ell}}{d\ell}&=\Big{(}2-\sigma-\eta_{\rm
sr}(J_{\ell})\Big{)}g_{\ell}\\\
\frac{dJ_{\ell}}{d\ell}&=c_{\sigma}\eta_{sr}(J_{\ell})g_{\ell}\end{split}$ (7)
where $c_{\sigma}=\frac{\pi}{2}a^{2-\sigma}\int^{\infty}_{1}du\
u^{1-\sigma}\mathcal{J}_{0}(2\pi u)$ and $\mathcal{J}_{0}(x)$ is the zeroth
order Bessel function of the first kind. As shown in Appendix B the above
result is reliable as long as $a^{2-\sigma}g_{\ell}\ll J_{\ell}$ or,
equivalently, as long as $\frac{dJ}{d\ell}\ll J_{\ell}$. As expected, we see
that the flow equations (29) support a line of SR fixed points $g=0$ which
becomes unstable for $\eta_{\rm sr}(J)<2-\sigma$. As long as our hypothesis of
small $g$ holds, we can explicitly identify the form of the flow trajectories
of Eqs. (29):
$g_{\ell}(J)=\frac{\pi(2-\sigma)}{c_{\sigma}}\left[\left(J_{\ell}-J_{\sigma}\right)^{2}+k\right],$
(8)
where $k$ is a real number and $J_{\sigma}=\frac{1}{2\pi(2-\sigma)}$. The sign
of $k$ divides the trajectories which met the fixed point $g=0$ and those
which do not, the first ones ending at (starting from) the fixed point line
for $J\leq J_{\sigma}$ ($J>J_{\sigma}$). The separatrix corresponds to the
semi-parabola with $k=0$, $J\leq J_{\sigma}$. For $k>0$ $g\to\infty$, showing
the existence of a new low-temperature phase, where LR interactions are
relevant. The critical temperature $T_{c}$, below which this new phase
appears, is such that $\eta_{\rm sr}(J_{c})>2-\sigma$ .
Since, as in the traditional BKT calculation [29], Eqs. (29) were derived for
small $g$ and $y$, its use for $T<T_{c}$ is in principle not justified, since
LR interactions are relevant and $g_{\ell}$ grows indefinitely. However, let
us notice that the scaling of $g_{\ell}$ with $T$ for $T\rightarrow T_{c}^{-}$
can be reliably predicted from Eqs. (29), since in this limit the flow spends
a divergent amount of RG time $\ell$ in the vicinity of the line of fixed
points $g=0$. This scaling is derived in Appendix C. Moreover, we can guess
the infrared form of the action in the low temperature phase by observing that
the rigorous result of Ref. [3] implies that for $T<T_{c}$ the system displays
finite magnetization and, then, phase fluctuations are limited even at large
distances. Therefore, the expansion of the trigonometric function in Eq. (3)
is justified leading to an action of the form
$S_{g}=-\frac{\bar{g}}{2}\int d^{2}x\ \theta\nabla^{\sigma}\theta,$ (9)
where $\bar{g}=g\gamma_{2,\sigma}^{-1}$. Being the above action quadratic, the
properties of this exotic low temperature phase can be worked out: in
particular the scaling of the magnetization for $T\rightarrow T_{c}^{-}$ is
found to be (see Appendix C for details)
$\ln m\sim-e^{B(T_{c}-T)^{-1/2}}$ (10)
where $B$ is a non universal constant. Since all the derivatives of $m$ with
respect to $T$ vanish at $T=T_{c}$ (essential singularity), and since $m$ is
linked to the derivative of the free energy with respect to the external
field, we have that the phase transition between the ordered and disordered
phase is actually of infinite order. Moreover, the connected correlation
functions have a power-law decay for $T<T_{c}$ given by
$\left\langle\mathbf{S}(\mathbf{r})\cdot\mathbf{S}(\mathbf{0})\right\rangle_{c}\sim\frac{1}{r^{2-\sigma}}$,
where
$\mathbf{S}(\mathbf{r})=(\cos{\theta}_{\mathbf{r}},\sin{\theta}_{\mathbf{r}})$.
Figure 1: Sketch of the RG flow lines for $\frac{7}{4}<\sigma<2$ in the $y=0$
plane. The dashed red line is a possible realization of the physical
parameters line, from which the flow starts, as the temperature is varied. On
the right/left of the gray dotted line the vortex fugacity $y$ is
irrelevant/relevant ($\dot{y}_{\ell}/y_{\ell}\gtrless 0$). The two
separatrices (bold black lines) divide the flow in three regions: a high-
temperature region (orange, the flow ends up in the disordered phase), an
intermediate one (blue, the flow reaches a $g=0$ fixed point) and the low-
temperature region (green, the LR perturbation brings the system away from the
critical line).
We have so far assumed $y=0$; let us now consider the effect of topological
excitations. At leading order in both $g$ and $y$ the two perturbations remain
independent and, since the vortices contribute to the $J_{\ell}$ flow only
beyond leading order in $y$, Eqs.(29) are unchanged. Moreover, one has
$\frac{dy_{\ell}}{d\ell}=(2-\pi J_{\ell})y_{\ell}$ valid up to second order
terms in $y_{\ell}$ and $g_{\ell}$. Then, in agreement with what we stated
above, as long as $\frac{7}{4}<\sigma<2$, the temperature range $T$ between
$T_{c}$ and $T_{\rm BKT}$ of the line of fixed points $g=y=0$ remains stable
under both topological and LR perturbations. In the low-temperature phase
instead, it is natural to suppose $y$ to be irrelevant, due to the fact that a
non-negligible LR coupling increases the cost of, highly non-local topological
excitations. This idea is made rigorous in Appendix D where the interaction
energy between vortices in the low temperature phase is computed, and it is
shown that they cannot proliferate.
Summarizing, for $\sigma\in(7/4,2)$ we find three phases: i) an ordered phase
for $T<T_{c}$ with finite magnetization and temperature independent power-law
correlation functions ii) an intermediate BKT phase for $T_{c}<T<T_{\rm BKT}$,
where the magnetization vanishes and the exponent of the two-point correlation
function depends on $T$ iii) a disordered phase for $T>T_{\rm BKT}$. Due to
the LR character of the interactions, also the high-temperature phase displays
power-law decaying two-point functions
$\left\langle\mathbf{S}(\mathbf{r})\cdot\mathbf{S}(0)\right\rangle\sim
r^{-2-\sigma}$,[74, 75, 76]. As $\sigma\rightarrow 7/4^{+}$ the critical
temperature $T_{c}$ reaches $T_{BKT}$ from below. Therefore, for $\sigma<7/4$,
the whole BKT line fixed points becomes unstable either with respect to
topological or LR perturbations and the intermediate phase vanishes, leaving
only a SSB phase transition. However, our approach cannot reliably be used to
fully characterize this transition: as $T$ approaches $T_{c}$ from below, the
RG flow slows down close to the $g=0$, $J=J_{\sigma}$ fixed point. Since for
$\sigma<\frac{7}{4}$ $J_{\sigma}<J_{\rm BKT}$, $y$ grows indefinitely, away
from the $y\ll 1$ regime. Our results are summarized in Fig. 2.
Figure 2: Sketch of the possible phases of the model: ordered with
magnetization (solid black), BKT qlro (dashed light gray), disordered (dashed
dark gray). If $\sigma>2$ we find the usual SR phenomenology with a BKT phase
transition. For $\sigma<2$ an ordered phase appears at low-temperatures, the
BKT qlro phase disappearing for $\sigma<\frac{7}{4}$.
## Conclusions
We have shown that the introduction of long-range (LR) power decaying
couplings in the $2D$ XY model Hamiltonian produces a rich phase diagram,
different from the short-range (SR) case [29] and from the one of $O(N)$ LR
systems [4]. Remarkably, for $7/4<\sigma<2$, the system displays both BKT qlro
in the temperature interval $T_{c}<T<T_{\rm BKT}$ and actual long-range order
for $T<T_{c}$.
The introduction of complex interaction patterns in systems with $U(1)$
symmetry is known to generate exotic critical features, as in the anisotropic
$3D$ XY model [77], coupled XY planes [78], 2D systems with anisotropic
dipolar interactions [79, 80] or four-body interactions [81], and high-
dimensional systems with Lifshitz criticality [82, 83] The present work
constitutes a further milestone along this path, as it identifies a peculiar
critical behavior, namely a non-analytic exponential vanishing of the order
parameter, that eludes the current classification of universal scaling
behaviors [84].
Our predictions may be tested in several low dimensional AMO systems. It would
be interesting to perform extensive numerical simulations in order to observe
the scaling of the critical quantities, and especially the magnetization,
close to the low-temperature endpoint of the BKT line in the regime
$7/4<\sigma<2$. These simulations will be useful to classify this
unprecedented transition and to investigate possible corrections near the
$\sigma=7/4$ endpoint due to higher-order effects caused by spin-wave
excitations [85]. Further investigation is also needed to compare our results
with the LR diluted model studied in [86, 87]. In this model, at
$\sigma=1.875$, the numerical simulations presented in [87] do not find any
intermediate BKT region, but the general question whether the $2D$ LR diluted
XY model and the $2D$ LR non-diluted one have the same phase diagram remains
open.
Our results have also implications for LR quantum XXZ chains [69, 88, 89]. One
would need to perform the exact mapping of the classical $2D$ LR XY model to
an effective $1D$ quantum model, following the calculation presented in [23]
and valid for the classical $2D$ SR XY model. If the non-local/LR terms
violating the hard-core boson condition can be shown to be irrelevant, then
one could put in correspondence our phase diagram with that of the LR quantum
XXZ chains having LR couplings both for $x-y$ and $z-z$ terms [69]. This seems
to be confirmed by the similar structure of the RG flow equations of [69] with
our Eqs.(29) taken at low temperatures. If this is the case, then the two
lines, black and white, of Fig.1 of [69] would merge in a point, with the XY
phase disappearing, corresponding to our $\sigma=7/4$ point. Finally, we
mention that it would be interesting to study in detail the phase diagram of
the $2D$ LR Villain model for $\sigma<2$.
## V Acknowledgment
Valuable discussions with G. Parisi, F. Ricci-Tersenghi and A. Scardicchio are
gratefully acknowledged. N.D. and A.T. also acknowledges useful discussion
with M. Ibáñez Berganza. This work is supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC2181/1-390900948 (the Heidelberg STRUCTURES Excellence
Cluster). This work is supported by the CNR/HAS (Italy-Hungary) project
“Strongly interacting systems in confined geometries”. This work is part of
MUR-PRIN2017 project “Coarse-grained description for non-equilibrium systems
and transport phenomena (CONEST)” No. 201798CZL whose partial financial
support is acknowledged.
## VI Appendix
### VI.1 A. Defintion of the fractional Laplacian
Given a real parameter $\sigma\in(0,2)$, one can define the fractional
Laplacian of order $\sigma$ of a function
$f(\mathbf{x}):\,\mathbb{R}^{d}\rightarrow\mathbb{R}$ as:
$\nabla^{\sigma}f(\mathbf{x})\equiv\gamma_{d,\sigma}\int
d^{d}r\frac{f(\mathbf{x}+\mathbf{r})-f(\mathbf{x})}{r^{d+\sigma}},$ (11)
where
$\gamma_{d,\sigma}=\frac{2^{\sigma}\Gamma(\frac{d+\sigma}{2})}{\pi^{d/2}|\Gamma(-\frac{\sigma}{2})|}$
and $r=\mid\mathbf{r}\mid$. Another expression for this quantity can be
derived in terms of the Fourier transform, $f(\mathbf{q})$, of
$f(\mathbf{x})$:
$\nabla^{\sigma}f(\mathbf{x})=-\gamma_{d,\sigma}\int d^{d}q\ f(\mathbf{q})\
e^{i\mathbf{q}\cdot\mathbf{x}}\int
d^{d}r\frac{1-e^{i\mathbf{q}\cdot\mathbf{r}}}{r^{d+\sigma}}.$ (12)
Since
$\int
d^{d}r\frac{1-e^{i\mathbf{q}\cdot\mathbf{r}}}{r^{d+\sigma}}=\gamma_{d,\sigma}^{-1}\
q^{\sigma},$ (13)
we find the alternative definition:
$\nabla^{\sigma}f(\mathbf{x})=-\int d^{d}q\
q^{\sigma}f(\mathbf{q})e^{i\mathbf{q}\cdot\mathbf{x}}.$ (14)
In our case $d=2$ and we have to evaluate the quantity:
$\int
d^{2}x\int_{r>a}\frac{d^{2}r}{r^{2+\sigma}}[1-\cos\left(\theta(\mathbf{x})-\theta(\mathbf{x}+\mathbf{r})\right)]$
(15)
For $\sigma<2$, one can actually disregard the contribution coming from the
lattice spacing $a$, since it would just result in a correction of the short-
range term. Then, through trivial trigonometric manipulations we can write the
above expression as:
$\begin{split}&\int
d^{2}x\cos\theta(\mathbf{x})\int\frac{d^{2}r}{r^{2+\sigma}}[\cos\theta(\mathbf{x})-\cos\theta(\mathbf{x}+\mathbf{r})]\\\
+&\int
d^{2}x\sin\theta(\mathbf{x})\int\frac{d^{2}r}{r^{2+\sigma}}[\sin\theta(\mathbf{x})-\sin\theta(\mathbf{x}+\mathbf{r})].\end{split}$
(16)
Finally, using the definition (11) of the fractional derivative, we get
$-\gamma_{2,\sigma}^{-1}\int
d^{2}x\left(\cos\theta\nabla^{\sigma}\cos\theta+\sin\theta\nabla^{\sigma}\sin\theta\right)$
(17)
which justifies the alternative form of the long-range term given in the main
text as Eq.(4).
### VI.2 B. Renormalization group for $y=0$
We will now derive the set of RG equations (7) given in the main text, valid
for $y=0$. We then start form the action written in the form
$S[\theta]=\int
d^{2}x\left(\frac{J_{\ell}}{2}|\nabla\theta|^{2}+\frac{g_{\ell}}{2}\int_{r>a}\frac{d^{2}r}{r^{2+\sigma}}\left[1-\cos\left(\Delta_{\mathbf{r}}\theta(\mathbf{x})\right)\right]\right)$
(18)
where, as in the main text,
$\Delta_{\mathbf{r}}\theta(\mathbf{x})=\theta(\mathbf{x}+\mathbf{r})-\theta(\mathbf{x})$,
and compute the flux perturbatively around $g=0$. The field is split into fast
and slow modes with respect to the momentum cutoff $\Lambda=\frac{2\pi}{a}$,
namely $\theta=\theta^{>}+\theta^{<}$ with
$\begin{split}\theta^{<}(\mathbf{x})&=\int_{q<\Lambda
e^{-d\ell}}\frac{d^{2}q}{(2\pi)^{2}}\theta(\mathbf{q})e^{i\mathbf{q}\cdot\mathbf{x}}\\\
\theta^{>}(\mathbf{x})&=\int_{\Lambda>q>\Lambda
e^{-d\ell}}\frac{d^{2}q}{(2\pi)^{2}}\theta(\mathbf{q})e^{i\mathbf{q}\cdot\mathbf{x}},\end{split}$
(19)
where $\ell=\ln(r/a)$. If we assume the interacting long-range term in Eq.
(18) to be small with respect to the quadratic one, we can perform the
integration perturbatively. It is easy to see that this is possible if
$ga^{2-\sigma}<<J$. Under this assumption then we integrate out the fast
modes, expanding the partition function in cumulants of the non-Gaussian part
$S_{g}$:
$S_{\rm eff}[\theta^{<}]=S_{0}[\theta^{<}]+\left\langle
S_{g}\right\rangle_{>}+O(g^{2}).$ (20)
Writing
$\cos(\Delta_{\mathbf{r}}\theta)=\cos(\Delta_{\mathbf{r}}\theta^{>})\cos(\Delta_{\mathbf{r}}\theta^{<})+\sin(\Delta_{\mathbf{r}}\theta^{>})\sin(\Delta_{\mathbf{r}}\theta^{<})$,
one sees that only the first term will give a contribution. Then, up to
additive constants we have:
$\left\langle S_{g}\right\rangle_{>}=\frac{g_{\ell}}{2}\int
d^{2}x\int\frac{d^{2}r}{r^{2+\sigma}}\left\langle\cos(\Delta_{\mathbf{r}}\theta^{>})\right\rangle_{>}\left[1-\cos\left(\Delta_{\mathbf{r}}\theta^{<}\right)\right]$
(21)
(from now on we omit the $r>a$ condition in the integral over $r$). On the
other hand,
$\left\langle\cos(\Delta_{\mathbf{r}}\theta^{>})\right\rangle_{>}=e^{-\frac{1}{2}\left\langle\left(\theta(\mathbf{r})-\theta(0)\right)^{2}\right\rangle_{>}}$
and
$\frac{1}{2}\left\langle\left(\theta(\mathbf{r})-\theta(0)\right)^{2}\right\rangle_{>}=\int_{\Lambda>q>\Lambda
e^{-d\ell}}\frac{d^{2}q}{(2\pi)^{2}}\frac{1-\cos(\mathbf{q}\cdot\mathbf{r})}{J_{\ell}q^{2}}=\frac{d\ell}{2\pi
J_{\ell}}\Big{(}1-\mathcal{J}_{0}(\Lambda r)\Big{)},$ (22)
where $\mathcal{J}_{0}(x)$ is the zeroth-order Bessel function of the first
kind. Then, introducing $\eta_{\rm sr}(J)=\frac{1}{2\pi J_{\ell}}$, the
exponent of the correlations at the cutoff scale $\ell$, we have:
$\begin{split}\left\langle\cos(\Delta_{\mathbf{r}}\theta^{>})\right\rangle_{>}&=e^{-\eta_{\rm
sr}(J_{\ell})d\ell\left(1-\mathcal{J}_{0}(\Lambda r)\right)}\\\ &=1-\eta_{\rm
sr}(J_{\ell})d\ell+\eta_{\rm sr}(J_{\ell})d\ell\mathcal{J}_{0}(\Lambda
r)\end{split}$ (23)
up to second order corrections. The first two terms provide an anomalous
dimension of the coupling $g_{\ell+d\ell}=g_{\ell}e^{-\eta_{\rm
sr}(J_{\ell})d\ell}$, as expected, while the last one modifies the power-law
dependence on $r$ of the long-range term:
$\begin{split}\left\langle S_{g}\right\rangle_{>}=&\frac{1}{2}\int
d^{2}x\Biggl{\\{}\int\frac{d^{2}r}{r^{2+\sigma}}ge^{-\eta_{\rm
sr}(J_{\ell})d\ell}\left[1-\cos\left(\Delta_{\mathbf{r}}\theta^{<}\right)\right]\\\
+&g\eta_{\rm
sr}(J_{\ell})d\ell\int\frac{d^{2}r}{r^{2+\sigma}}\mathcal{J}_{0}(\Lambda
r)\left[1-\cos\left(\Delta_{\mathbf{r}}\theta^{<}\right)\right]\Biggr{\\}}.\end{split}$
(24)
Let us now examine the last term of the above equation. This can be seen as an
interaction term of the original $XY$ form. Since $\mathcal{J}_{0}(x)\sim
x^{-1/2}\cos(x-\pi/4)$ for large $x$, the new coupling decays faster than the
original and has an oscillating behavior, which provides a natural cutoff for
$r\sim\Lambda^{-1}$. It is then reasonable to approximate this with a short-
range coupling of the form $|\nabla\theta|^{2}$. The simplest way is to
replace
$1-\cos(\Delta_{r}\theta)\approx\frac{1}{2}(\mathbf{r}\cdot\nabla_{\mathbf{x}}\theta)^{2}$
and observe that
$\int\frac{d^{2}r}{r^{2+\sigma}}\mathcal{J}_{0}(\Lambda
r)(\mathbf{r}\cdot\nabla_{\mathbf{x}}\theta^{<})^{2}=\pi|\nabla_{\mathbf{x}}\theta^{<}|^{2}\int^{\Lambda^{-1}}_{a}drr^{1-\sigma}\mathcal{J}_{0}(\Lambda
r).$ (25)
For $\sigma>\frac{1}{2}$, we can neglect the cutoff and, with the substitution
$r=au$, we can express the correction in the action as
$\frac{c_{\sigma}}{2}(g_{\ell}a^{2-\sigma})\eta_{\rm sr}(J_{\ell})d\ell\int
d^{2}x|\nabla_{\mathbf{x}}\theta^{<}|^{2},$ (26)
where
$c_{\sigma}=\frac{\pi}{2}\int_{1}^{\infty}duu^{1-\sigma}\mathcal{J}_{0}(2\pi
u)>0$. The integral is actually ill-defined for $\sigma<\frac{1}{2}$ signaling
that our approximation breaks down. Let us notice however that the precise
value of the coefficient is not important for our analysis. Moreover, it
should be noticed that this entire procedure is only reliable for
$\sigma>7/4$, where part of the BKT fixed points line remains stable and
furnishes a viable expansion point, see the discussion in the main text. Up to
the first order in $g$, then the integration of the fast modes gives the
corrections:
$\begin{split}dg&=-\eta_{\rm sr}(J_{\ell})g_{\ell}d\ell\\\
dJ&=c_{\sigma}\eta_{\rm sr}(J_{\ell})(g_{\ell}a^{2-\sigma})d\ell.\end{split}$
(27)
In order to obtain a theory with the same cutoff scale, we have to do the
replacement $\mathbf{x}\rightarrow\mathbf{x}e^{-d\ell}$ in the action. This
modifies the couplings $g$, $J$ by their own bare length dimension, i.e.
$2-\sigma$ and $0$ respectively:
$\begin{split}dg&=(2-\sigma-\eta_{\rm sr}(J_{\ell}))g_{\ell}d\ell\\\
dJ&=c_{\sigma}\eta_{\rm
sr}(J_{\ell})(g_{\ell}a_{0}^{2-\sigma})d\ell.\end{split}$ (28)
In turn, one finally obtains the RG equations:
$\begin{split}\frac{dg}{d\ell}&=\left(2-\sigma-\eta_{\rm
sr}(J_{\ell})\right)g_{\ell}\\\ \frac{dJ}{d\ell}&=c_{\sigma}\eta_{\rm
sr}(J_{\ell})g_{\ell},\end{split}$ (29)
i.e. Eqs. (7) of the main text (we absorbed the constant ultraviolet cutoff
$a^{2-\sigma}$ in the definition of $c_{\sigma}$).
### VI.3 C. Magnetization in the low-temperature phase
We will now derive the scaling behavior (10) given in the main text for the
magnetization near $T_{c}$, for $T\to T_{c}^{-}$. We start from the Gaussian
theory, Eq. (9) of the main text, describing the low temperature phase of the
theory in the infrared (IR). Being the theory Gaussian, it is
$m=\left\langle\cos\theta(\mathbf{x})\right\rangle=e^{-\frac{1}{2}\left\langle\theta^{2}(\mathbf{x)}\right\rangle}$.
Being
$\left\langle\theta^{2}(\mathbf{x})\right\rangle=\int_{q<2\pi/a}\frac{d^{2}q}{(2\pi^{2})}\frac{1}{\bar{g}q^{\sigma}}\sim\frac{1}{\bar{g}a^{\sigma-2}}.$
(30)
we find
$m=e^{-A/\bar{g}},$ (31)
where $A$ is a non-universal constant. Now, from the flow equations (29), we
find:
$g_{\ell}=ge^{(2-\sigma)\ell}e^{-\int\eta_{\rm sr}(J_{\ell})d\ell},$ (32)
which is reliable as long as $g_{\ell}$ is small. Let us consider a trajectory
which runs very close to the separatrix which, according to Eq. (8) of the
main text, is described by the trajectory
$g=\frac{\pi(2-\sigma)}{c_{\sigma}}\left[(J-J_{\sigma})^{2}+k\right]$ with
$k\rightarrow 0^{+}$. Let us consider a point in the flow $\ell^{*}$ such that
$g(\ell^{*})$ is small and $J(\ell^{*})>J_{\sigma}$. Then:
$\int^{\ell^{*}}_{0}\eta_{\rm
sr}(J_{\ell})d\ell=\int^{\ell^{*}}_{J_{0}}\eta_{\rm sr}(J)\
\frac{dJ}{\dot{J}}=c_{\sigma}^{-1}\int^{\ell^{*}}_{J_{0}}\frac{dJ}{g(J)}=\pi(2-\sigma)\int^{J(\ell^{*})}_{J_{0}}\frac{dJ}{\left(J-J_{\sigma}\right)^{2}+k}$
(33)
By changing the value of the temperature, we have that $J_{0}$ crosses the
separatrix ($k\rightarrow 0^{+}$) for some $J_{c}<J_{\sigma}$ that corresponds
to the critical temperature $T_{c}$, and consequently $k\sim T_{c}-T$. Since
in this case the integration interval on $J$ contains the second order
singularity $J_{\sigma}$, we have that the integral diverges as $k^{-1/2}$ as
$k\rightarrow 0^{+}$. Then we have
$g_{\ell^{*}}\sim e^{-B(T-T_{c})^{-1/2}}$ (34)
where $B$ is a non universal constant. Since, as $k\rightarrow 0^{+}$, the
trajectories corresponding to different values of $k$ run close in the
parameter space, for large $g$ as well, we do not expect this scaling to be
modified in the non-perturbative region. Finally, exploiting Eq. (31), one has
the scaling:
$\ln m\sim-Ae^{B(T-T_{c})^{-1/2}}$ (35)
### VI.4 D. Irrelevance of topological excitations in the low-temperature
phase
We start from the quadratic action of Eq. (9) of the main text, which
describes the low temperature phase, we express it in terms of the Fourier
transform of $\mathbf{v}(\mathbf{x})=\nabla\theta$
$S_{g}=\bar{g}\int\frac{d^{2}q}{(2\pi)^{2}}\ q^{\sigma}\
|\mathbf{\theta}(\mathbf{q})|^{2}=\bar{g}\int\frac{d^{2}q}{(2\pi)^{2}}\
q^{\sigma-2}\ |\mathbf{v}(\mathbf{q})|^{2}$ (36)
We notice that circling around a topologically non-trivial region we have
$\oint\nabla\theta\cdot d\mathbf{r}=\oint\mathbf{v}\cdot d\mathbf{r}=2\pi
m_{enc}$ (37)
were $m_{enc}$ is the sum of all the topological charges $m_{i}$ enclosed in
the integration contour. This can be rephrased by saying that
$\nabla\times\mathbf{v}(\mathbf{x})=2\pi n(\mathbf{x})$, where
$n(\mathbf{x})=\sum_{j}m_{j}\delta(x-x_{j})$ is the vortex-density and $x_{j}$
correspond to the positions of the vortices. This can be further simplified if
we introduce the dual $\mathbf{u}(\mathbf{x})$ of $\mathbf{v}(\mathbf{x})$,
defined as $u_{j}=\epsilon_{jk}v_{k}$ where $\epsilon_{jk}$ is the fully
antisymmetric tensor of rank $2$. We then find the condition
$\nabla\cdot\mathbf{u}(\mathbf{x})=2\pi n(\mathbf{x})$ (38)
In turn, this can be solved in the Fourier space:
$\mathbf{u}(\mathbf{q})=\frac{2\pi\mathbf{q}}{q^{2}}n(\mathbf{q)}+\mathbf{u}_{\perp}(\mathbf{q})$
(39)
where $\mathbf{u}_{\perp}(\mathbf{q})$ is a generic function such that
$\mathbf{q}\cdot\mathbf{u}_{\perp}(\mathbf{q})=0$ and which represent the
topologically-trivial component of the field $\theta$. Now, since
$|\mathbf{v}(\mathbf{q})|^{2}=|\mathbf{u}(\mathbf{q})|^{2}=\frac{(2\pi)^{2}}{q^{2}}|n(\mathbf{q})|^{2}+|\mathbf{u}_{\perp}(\mathbf{q})|^{2}$
(40)
we have that the action $S_{g}$ splits into the sum on the non-topological and
topological part, the latter being:
$S_{\rm top}=\bar{g}\int d^{2}q\ q^{\sigma-4}|n(\mathbf{q})|^{2}$ (41)
Coming back to the real space we have:
$S_{\rm top}=\bar{g}\sum_{ij}m_{i}m_{j}G(\mathbf{r}_{i}-\mathbf{r}_{j})$ (42)
with $G(\mathbf{x})=\int d^{2}q\
q^{\sigma-4}e^{i\mathbf{q}\cdot\mathbf{x}}\sim L^{2-\sigma}-x^{2-\sigma}$, $L$
being the system size. The first term in $G$ gives raise to a term
proportional to
$L^{2-\sigma}\sum_{i,j}m_{i}m_{j}=L^{2-\sigma}\left(\sum_{i}m_{i}\right)^{2}$
which, in the thermodynamic limit, ensures the neutrality of the gas of
charges. We find then
$S_{\rm
top}\sim-\bar{g}\sum_{ij}m_{i}m_{j}|\mathbf{x}_{i}-\mathbf{x}_{j}|^{2-\sigma}$
(43)
As expected, this interaction is more binding than the logarithmic one for the
short-range case. A simple entropy-energy argument shows that the charges will
never unbound at any temperature: indeed the energetic cost of creating two
far apart vortices grows like $\bar{g}L^{2-\sigma}$ while the entropy as $\ln
L$ so that the free energy
$F\sim\ln L-\bar{g}TL^{2-\sigma}$ (44)
is always dominated by the interaction term for large enough $L$.
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|
††thanks: Present Address: Centre for Quantum Dynamics, Griffith University,
Brisbane Queensland 4111, Australia
# Heisenberg-Langevin approach to driven superradiance
Ori Somech Department of Chemical & Biological Physics, Weizmann Institute of
Science, Rehovot 7610001, Israel Yoav Shimshi Department of Chemical &
Biological Physics, Weizmann Institute of Science, Rehovot 7610001, Israel
Ephraim Shahmoon Department of Chemical & Biological Physics, Weizmann
Institute of Science, Rehovot 7610001, Israel
###### Abstract
We present an analytical approach for the study of driven Dicke superradiance
based on a Heisenberg-Langevin formulation. We calculate the steady-state
fluctuations of both the atomic-spin and the light-field operators. While the
atoms become entangled below a critical drive, exhibiting spin squeezing, we
show that the radiated light is in a classical-like coherent state whose
amplitude and spectrum are identical to those of the incident driving field.
Therefore, the nonlinear atomic system scatters light as a linear classical
scatterer. Our results are consistent with the recent theory of coherently
radiating spin states. The presented Heisenberg-Langevin approach should be
simple to generalize for treating superradiance beyond the permutation-
symmetric Dicke model.
## I Introduction
Superradiance describes the cooperative radiation of an ensemble of quantum
emitters into common photonic modes. A conceptually simple case that captures
the essence of cooperative radiation is that of Dicke superradiance, where all
the constituents of an ensemble of two-level atoms are coupled to the common
photonic modes in an identical manner, thus forming an effective “collective
spin” dipole Dicke ; mandel_wolf_1995 ; GH . Superradiance was observed both
in atoms HAR ; TOMs1 ; TOMs2 ; FLD ; BRWsr and artificial emitters MAJ and
plays a role in various quantum phenomena and technologies, ranging from phase
transitions KES ; EMAN to narrowband superradiant lasers HAK ; HOL ; TOM1 ;
MOL1 .
The situation wherein the atoms are additionally driven by a resonant laser
can be studied by a driven-dissipative master equation of the Dicke model.
Mean-field theory yields a second order phase transition of the steady-state
atomic population, or “magnetization”, as a function of the drive
DRUMMOND1978160 ; DRUMMOND1980 ; CAR ; LAW ; LAR ; BAR . Spin squeezing was
recently found in steady state by a numerical solution of the master equation
Alejandro ; yelin ; BAR ; REYt with a supporting analytical result obtained
in yelin . For the radiated light, intensity correlations $g^{(2)}$ were
calculated and found to exhibit bunching correlations above the phase-
transition point, but no correlations below it CAR . More recently, it was
found that the appearance of so-called coherently radiating spin states (CRSS)
as the steady-state of driven Dicke superradiance underlies these results CRSS
.
Here we present a simple analytical approach for driven superradiance based on
Heisenberg-Langevin (HL) equations. While this HL approach is in principle
equivalent to the master equation used previously, the HL equations are
natural for the direct analytical treatment of both spin and field
fluctuations via their operator-form solution. In particular, we account for
spin and field fluctuations around the mean field using the Holstein-Primakoff
approximation. For the spin fluctuations the operator-valued solutions are in
a Bogoliubov transformation form implying quantum correlations, as verified by
the subsequent analytical calculation of spin squeezing. For the field
operator, we find that the fluctuations are proportional to the vacuum field,
thus proving that the radiated field below the phase-transition point is in a
coherent state. We also calculate the two-time correlation of the field,
finding that the spectrum is delta-peaked at the incident-drive frequency.
Surprisingly, the light is thus scattered from the many-atom system as if the
latter is a linear system, although the atomic system is highly nonlinear, as
evident by its phase transition. We discuss the consistency and relation of
these results with the predictions of CRSS theory CRSS .
The paper is organized as follows. In Sec. II we derive the HL equations of
the driven Dicke model, focusing on a relevant cavity-scheme realization.
After recalling the mean-field solution in Sec. III, we treat spin
fluctuations and squeezing in Sec. IV. Sec. V is devoted to the analysis of
the radiated light. Finally, our conclusions are presented in Sec. VI.
## II Model
We begin with the derivation of the HL equations of motion that describe the
driven Dicke model, considering a system of atoms in a damped cavity as
realized in typical experiments HAR ; TOMs1 ; TOMs2 ; MAJ . Realizations of
Dicke physics exist also in other systems wherein many atoms are coupled to a
common photon bath, e.g. in waveguide QED Alejandro or even in an elongated
atomic ensemble in free space GH ; BRWsr ; however, the cavity case considered
here is conceptually the most straightforward one as it directly emphasizes a
single common photonic mode.
Figure 1: Cavity realization of driven superradiance. An atomic ensemble is
trapped inside a cavity, wherein all atoms (green dots) are identically
coupled to a cavity mode (lowering operator $\hat{c}$) and hence described by
a collective-spin dipole (lowering operator $\hat{J}_{-}$). The cavity field
is damped through its mirrors at rate $\kappa$ to the outside propagating
modes $\hat{b}_{k}$, which form the radiated field $\hat{E}$, and is driven by
a laser with Rabi-field amplitude $\Omega_{L}$. Here a single-sided cavity
scheme is presented, with one out-coupling mirror (right-hand side).
### II.1 System and Hamiltonian
We consider the system displayed in Fig. 1: $N$ two-level atoms are trapped
inside an optical cavity driven by external laser light thorough the cavity
mirrors. The atomic positions are such that all atoms are identically coupled
to the cavity mode (i.e. well within the cavity mode waist and at longitudinal
positions that are multiples of cavity wavelength apart). The Hamiltonian of
the atoms and the cavity is given by
$\hat{H}_{S}=\hbar\omega_{a}\hat{J}_{z}+\hbar\omega_{c}\hat{c}^{\dagger}\hat{c}+\hbar\left[\hat{c}^{\dagger}\left(g^{\ast}\hat{J}_{-}+\Omega_{L}e^{-i\omega_{L}t}\right)+\text{h.c.}\right].$
(1)
Here $\hat{c}$ is the boson lowering operator of the cavity mode of frequency
$\omega_{c}$, whereas
$\hat{J}_{\alpha}=(1/2)\sum_{n=1}^{N}\hat{\sigma}_{n}^{\alpha}$
($\alpha\in\\{x,y,z\\}$) are the collective-spin operators of the atomic
ensemble with $\hat{\sigma}_{n}^{\alpha}$ being the Pauli operator of a two-
level atom $n\in\\{1,...,N\\}$ with resonant frequency $\omega_{a}$. The
cavity is driven via its mirrors by an external laser of frequency
$\omega_{L}$ and amplitude $\Omega_{L}$, and is coupled to the atoms via the
dipole coupling $g$ identical to all atoms, where
$\hat{J}_{-}=\hat{J}_{x}-i\hat{J}_{y}=\sum_{n=1}^{N}\hat{\sigma}_{n}^{-}=\hat{J}_{+}^{{\dagger}}$
is the collective-spin lowering operator of the atoms and
$\hat{\sigma}_{n}^{-}=(\hat{\sigma}_{n}^{+})^{{\dagger}}$ the Pauli lowering
operator of atom $n$.
In addition, the cavity mode is coupled through its mirrors to a 1D continuum
of propagating photon modes characterized by the wavenumber $k$ and
corresponding boson modes $\hat{b}_{k}$ and frequencies $vk$ ($v$ being the
speed of light). The Hamiltonians describing this 1D photon reservoir and its
coupling to the system are given by, respectively (here, for one-sided cavity,
Fig. 1),
$\displaystyle\hat{H}_{R}$ $\displaystyle=$ $\displaystyle\sum_{k>0}\hbar
vk\hat{b}_{k}^{\dagger}\hat{b}_{k},$ $\displaystyle\hat{H}_{SR}$
$\displaystyle=$
$\displaystyle\hbar\sum_{k>0}\left(\eta\hat{b}_{k}^{\dagger}\hat{c}+\text{h.c.}\right),\quad\eta\equiv\sqrt{\frac{v}{L}\kappa},$
(2)
where the coupling constant $\eta$ is taken $k$-independent (consistent with
the Markov approximation) and $L$ is the quantization length of the 1D
continuum. The total Hamiltonian is given by
$\hat{H}=\hat{H}_{S}+\hat{H}_{R}+\hat{H}_{SR}$. We note that we neglect here
the direct spontaneous emission from atoms to photon modes in transverse
directions outside the cavity. For a dilute ensemble this is an individual-
atom process that is typically much slower than the relevant Dicke dynamics
discussed here.
### II.2 Heisenberg-Langevin equations
We begin with eliminating the reservoir modes $\hat{b}_{k}$ by inserting the
solution of their Heisenberg equations into the equation for $\hat{c}$,
obtaining within the usual Markov approximation SCU
$\dot{\tilde{c}}=\left(i\delta_{c}-\frac{\kappa}{2}\right)\tilde{c}-ig^{\ast}\tilde{J}_{-}-i\Omega_{L}+\hat{E}_{0}(t),\quad\delta_{c}=\omega_{L}-\omega_{c}.$
(3)
Here the system operators are already written in a rotated frame,
$\tilde{c}=\hat{c}e^{i\omega_{L}t}$ and
$\tilde{J}_{-}=\hat{J}_{-}e^{i\omega_{L}t}$, whereas the Langevin, vacuum
noise of the reservoir is given by
$\hat{E}_{0}(t)=-i\sum_{k}\eta^{\ast}e^{-i(vk-\omega_{L})t}\hat{b}_{k}(0)$,
satisfying (assuming an initial vacuum state)
$\langle\hat{E}_{0}(t)\hat{E}_{0}^{{\dagger}}(t^{\prime})\rangle=\kappa\delta(t-t^{\prime}).$
(4)
Next, we eliminate the cavity mode by assuming that its damping rate $\kappa$
to the 1D continuum is much faster than the typical time scale of variations
in $\tilde{J}_{-}$, i.e. $\kappa\gg|\dot{\tilde{J}}_{-}/\tilde{J}_{-}|$.
Within this coarse-grained dynamical picture and for times $t$ much longer
than $1/\kappa$, the elimination of $\tilde{c}$ is equivalent to setting
$\dot{\tilde{c}}=0$ in Eq. (3) and inserting the solution for $\tilde{c}$ into
the Heisenberg equations for atomic variables such as $\tilde{J}_{-}$ and
$\hat{J}_{z}$. Finally, we obtain (denoting
$\tilde{J}_{\mp}\rightarrow\hat{J}_{\mp}$ for simplicity)
$\displaystyle\dot{\hat{J}}_{-}$ $\displaystyle=$ $\displaystyle
i\delta\hat{J}_{-}+\left(\gamma-i2\Delta\right)\hat{J}_{z}\hat{J}_{-}-i2\hat{J}_{z}\left[\Omega+\hat{f}(t)\right],$
$\displaystyle\dot{\hat{J}}_{z}$ $\displaystyle=$
$\displaystyle-\gamma\hat{J}_{+}\hat{J}_{-}+i\hat{J}_{+}\left[\Omega+\hat{f}(t)\right]-i\left[\Omega^{*}+\hat{f}^{{\dagger}}(t)\right]\hat{J}_{-},$
with the laser-atom detuning $\delta=\omega_{L}-\omega_{a}$, the coefficients
$\gamma=\frac{|g|^{2}\kappa}{\delta_{c}^{2}+(\kappa/2)^{2}},\quad\Delta=\frac{-|g|^{2}\delta_{c}}{\delta_{c}^{2}+(\kappa/2)^{2}},\quad\Omega=\frac{-2g\Omega_{L}}{2\delta_{2}+i\kappa},$
(6)
and the effective Langevin, input-vacuum noise (filtered by the cavity),
$\hat{f}(t)\approx[2g/(\kappa-i2\delta_{c})]\hat{E}_{0}(t)$, satisfying
$\langle\hat{f}(t)\hat{f}^{{\dagger}}(t^{\prime})\rangle=\gamma\delta(t-t^{\prime}).$
(7)
Equations (LABEL:HL) form the HL equations of the driven Dicke model, with an
effective emission rate $\gamma$ of an atom to the outside modes via the
cavity, and an effective laser drive with Rabi frequency $\Omega$. The
collective shift $\Delta$ describes the resonant dipole-dipole interactions
between pairs of atoms LEH , corresponding to an effective Hamiltonian
$\hat{H}_{\text{dd}}=-\hbar\sum_{n}\sum_{m}\Delta_{nm}\hat{\sigma}_{n}^{+}\hat{\sigma}_{m}^{-}$.
Here the dipole-dipole kernel $\Delta_{nm}=\Delta$ is uniform for all atom
pairs $n$ and $m$ since all atoms are coupled identically to the mediating
cavity photon mode. In treatments of superradiance in free space, such
coherent dipole-dipole effects are often ignored in free-space GH whereas
they vanish in a waveguide QED superradiance scheme Alejandro . In the cavity
setting, they exist however if one allows for laser-cavity detuning
$\delta_{c}$ as seen in Eq. (6) for $\Delta$ and noted in Refs. BAR ; REYt .
We note that while this specific derivation was performed starting from the
damped-cavity model, equivalent HL equations (LABEL:HL) can be derived by
considering other models of photon continua to which all atoms are identically
coupled. Here the cavity mode effectively becomes a continuum due to its fast
damping rate $\kappa$.
### II.3 Equivalent master equation
The HL equations (LABEL:HL) are equivalent to the following master equation
for the density matrix of the atoms,
$\displaystyle\frac{d\hat{\rho}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[\hat{H}_{\text{eff
}},\hat{\rho}\right]+\gamma\left[\hat{J}_{-}\hat{\rho}\hat{J}_{+}-\frac{1}{2}\left(\hat{J}_{+}\hat{J}_{-}\hat{\rho}+\hat{\rho}\hat{J}_{+}\hat{J}_{-}\right)\right],$
$\displaystyle\hat{H}_{\text{eff}}$ $\displaystyle=$
$\displaystyle-\hbar\Delta\hat{J}_{+}\hat{J}_{-}-\hbar\left(\Omega\hat{J}_{+}+\Omega^{\ast}\hat{J}_{-}\right).$
(8)
Here we have already assumed that the laser drive is resonant with the atoms,
$\delta=\omega_{L}-\omega_{a}=0$. This master equation with $\Delta=0$ is a
typical starting point for the analysis of driven Dicke superradiance
presented in previous works DRUMMOND1978160 ; DRUMMOND1980 ; CAR ; LAW ; LAR ;
Alejandro ; yelin ; RABk , whereas the additional dipole-dipole term $\Delta$
is considered in Refs. BAR ; REYt . Here instead we will use the HL
formulation of Eq. (LABEL:HL), in order to derive analytical results for
fluctuations and correlations of atomic and photonic degrees of freedom. We
will use the master equation as a numerical verification of the one-time
correlation functions of the atoms. Since the total spin
$\hat{J}_{x}^{2}+\hat{J}_{y}^{2}+\hat{J}_{z}^{2}=j(j+1)$ is conserved under
the dynamics of Eqs.(LABEL:HL) and (8), the initial state sets the SU(2) spin
representation $j$. Assuming an initial ground state for the $N$ atoms, we
have $j=N/2$ and the Hilbert space that spans Eqs. (8) is of size $2j+1=N+1$
and can be easily solved numerically for reasonable $N$.
## III Mean-field solution
We begin with the mean-field solution of the model in steady state. To obtain
the mean-field equations, we take the average over the HL equations
(LABEL:HL), such that the Langevin vacuum-noise terms vanish, and perform the
factorization of operator products
$\langle\hat{J}_{\alpha}\hat{J}_{\beta}\rangle\approx\langle\hat{J}_{\alpha}\rangle\langle\hat{J}_{\beta}\rangle$
(with $\alpha,\beta\in\\{x,y,z\\}$). This factorization is justified for
$N\rightarrow\infty$ under the mean-field assumption that fluctuations of
observables are much smaller than their mean. It is important to note that
such a factorization does not mean that there are no correlations between the
atoms that comprise the collective spin $\hat{J}_{\alpha}$ DRUMMOND1978160 :
in fact, we see below that the atoms are entangled Alejandro ; yelin ; BAR ;
REYt . Considering the conservation of the total spin
$\hat{J}_{x}^{2}+\hat{J}_{y}^{2}+\hat{J}_{z}^{2}=j(j+1)$ with $j=N/2\gg 1$ and
taking a resonant drive $\delta=0$, the solution to the mean-field equations
becomes (see also BAR ),
$\displaystyle\langle\hat{J}_{z}\rangle=-\frac{N}{2}\sqrt{1-\frac{|\Omega|^{2}}{\Omega_{c}^{2}}},\quad\langle\hat{J}_{-}\rangle=-\frac{\Omega}{\Delta+i\gamma/2},$
(9)
with the critical driving field defined by
$\displaystyle\Omega_{c}=\Omega_{c}(\Delta)=\frac{N}{4}\sqrt{\gamma^{2}+4\Delta^{2}}.$
(10)
The steady-state population inversion (or “magnetization”)
$\langle\hat{J}_{z}\rangle$ thus exhibits a second order phase transition as a
function of the drive $\Omega$, where it vanishes at the critical value
$\Omega_{c}$. The latter increases with the strength of the dipole-dipole
shift $\Delta$ as seen in Eq. (10). For $|\Omega|>\Omega_{c}$ there exist
oscillatory solutions of the mean-field equations DRUMMOND1978160 which
nevertheless appear to decay to zero at long time scales, upon the
consideration of the full quantum problem BAR . Figure 2 displays
$\langle\hat{J}_{z}\rangle$ obtained by the exact numerical solutions of the
master equation (8) for $N=50$ and different values of $\Delta$. Very good
agreement with the mean-field expression (9) is exhibited when $|\Omega|$ is
not too close to the critical point $\Omega_{c}$. In particular, calculations
with different values of $\Delta$ all collapse to the same curve when $\Omega$
is scaled to the corresponding $\Omega_{c}(\Delta)$ from Eq. (10).
Disagreement between mean-field and numerical results is observed around
$\Omega_{c}$ due to the fact that the mean value of
$\langle\hat{J}_{z}\rangle$ near $\Omega_{c}$ becomes increasingly small while
fluctuations grow, in contradiction to the mean-field assumption. The second
order transition predicted by the mean-field solution in the thermodynamic
limit $N\rightarrow\infty$ then becomes smoother at finite $N$.
Figure 2: Population inversion $\langle\hat{J}_{z}\rangle$ of the collective
atomic system as function of the driving field $\Omega$. Results obtained by
the numerical solution of Eq. (8) with $N=50$ atoms and for different values
of the dipole-dipole shift $2\Delta/\gamma=0,1,2$ all collapse to the same
curve when $\Omega$ is scaled the critical field $\Omega_{c}(\Delta)$ from Eq.
(10). The red line represent the analytical mean-field solution from Eq. (9),
exhibiting a second order phase transition. The exact numerical solutions
agree with the mean-field result until they diverge away a bit before the
transition point due to the finite value of $N$.
The mean-field solution (9) can be also written as a mean of the spin vector
$\hat{\mathbf{J}}=(\hat{J}_{x},\hat{J}_{y},\hat{J}_{z})$ in a Bloch sphere,
$\displaystyle\langle\hat{\mathbf{J}}\rangle=\left(\begin{array}[]{c}\langle\hat{J}_{x}\rangle\\\
\langle\hat{J}_{y}\rangle\\\ \langle\hat{J}_{z}\rangle\\\
\end{array}\right)=-\frac{N}{2}\left(\begin{array}[]{c}\sin\theta\cos\phi\\\
\sin\theta\sin\phi\\\ \cos\theta\\\ \end{array}\right),$ (17)
with the angles in spherical coordinates given by
$\displaystyle\sin\theta=\frac{|\Omega|}{\Omega_{c}},\quad\phi=\mathrm{arg}(\Delta+i\gamma/2)-\mathrm{arg}(\Omega).$
(18)
For later purposes, it is instructive to introduce a rotated coordinate system
at which the mean spin vector is directed to the south pole of the Bloch
sphere and hence appears as a ground state in this rotated system. Spin
operators in the rotated system, described by the vector
$\hat{\mathbf{J}}^{\prime}=(\hat{J}^{\prime}_{x},\hat{J}^{\prime}_{y},\hat{J}^{\prime}_{z})$
are related to the original spin operators
$\hat{\mathbf{J}}=(\hat{J}_{x},\hat{J}_{y},\hat{J}_{z})$ via the rotation
matrix $\mathcal{R}$ as
$\displaystyle\hat{\mathbf{J}}^{\prime}=\mathcal{R}^{-1}\hat{\mathbf{J}},\quad\langle\hat{\mathbf{J}}^{\prime}\rangle=-\frac{N}{2}\left(\begin{array}[]{c}0\\\
0\\\ 1\\\ \end{array}\right),$ (22)
$\displaystyle\mathcal{R}=\left(\begin{array}[]{ccc}\cos\theta\cos\phi&-\sin\phi&\sin\theta\cos\phi\\\
\cos\theta\sin\phi&\cos\phi&\sin\theta\sin\phi\\\ -\sin\theta&0&\cos\theta\\\
\end{array}\right).$ (26)
As required, in the rotated system the mean spin vector
$\langle\hat{\mathbf{J}}^{\prime}\rangle$ points to the south pole, defining
the $-z^{\prime}$ axis as the mean spin direction.
## IV Spin fluctuations and squeezing
We now turn to the analysis of small fluctuations of spin variables around the
mean-field solution (Sec. IV A). This will allow us to estimate atomic
correlations such as spin squeezing (Sec. IV B), and later on also the
fluctuations in the scattered field (Sec. V).
### IV.1 Collective spin fluctuations in the Holstein-Primakoff approximation
We recall that within its representation in the rotated system (26), the mean
spin vector
$\hat{\mathbf{J}}^{\prime}=(\hat{J}^{\prime}_{x},\hat{J}^{\prime}_{y},\hat{J}^{\prime}_{z})$
is directed towards the axis $z^{\prime}$ and vanishes along the
$x^{\prime},y^{\prime}$ axes. In order to analyze fluctuations around this
mean, we first define the spin lowering operator in the rotated basis,
$\hat{J^{\prime}}_{-}=\hat{J}^{\prime}_{x}-i\hat{J}^{\prime}_{y}$, and re-
write the HL equations (LABEL:HL) in terms of the rotated-spin operators
$\hat{J}^{\prime}_{-},\hat{J}^{\prime}_{z}$ using the transformation
$\mathcal{R}$ from (26). As in the original basis, the HL in the rotated basis
are also nonlinear in their relevant variables,
$\hat{J}^{\prime}_{-},\hat{J}^{\prime}_{z}$; however, the linearization of the
equations for small fluctuations around the mean field is simpler in this
rotated basis. To this end, we use the Holstein-Primakoff transformation,
which is an exact representation of SU(2) spin operators (here of spin
$j=N/2$) in terms of a bosonic operator $\hat{a}$ (satisfying
$[\hat{a},\hat{a}^{{\dagger}}]=1$) ASA ,
$\hat{J}^{\prime}_{-}=\sqrt{N-\hat{a}^{\dagger}\hat{a}}\
\hat{a},\quad\hat{J}^{\prime}_{z}=\hat{a}^{\dagger}\hat{a}-\frac{N}{2}.$ (27)
We see that the limit $\hat{a}\rightarrow 0$ is that of the mean-field
solution (17), so that the vacuum of $\hat{a}$ is the mean field and $\hat{a}$
describes fluctuations on top of it. In line with the mean-field assumption,
we consider small fluctuations, $|\hat{a}|\sim O(1)\ll\sqrt{N}$, and expand
the nonlinear HL equation for $\hat{J}^{\prime}_{-}$ to leading orders in the
small parameter $1/\sqrt{N}$. This is achieved by the approximation
$\hat{J}^{\prime}_{-}\approx\sqrt{N}\
\hat{a},\quad\hat{J}^{\prime}_{z}\approx-\frac{N}{2},$ (28)
and the subsequent linearization of the HL equation to first orders of
$\hat{a}$ and the noise $\hat{f}$. Finally, we obtain the HL equation for the
spin fluctuations $\hat{a}$,
$\displaystyle\dot{\hat{a}}$ $\displaystyle=$
$\displaystyle-\left(N\frac{\gamma}{2}\cos\theta-
iN\Delta\frac{1+\cos^{2}\theta}{2}\right)\hat{a}-iN\Delta\frac{\sin^{2}{\theta}}{2}\hat{a}^{{\dagger}}$
(29) $\displaystyle+$ $\displaystyle
i\sqrt{N}\left[\frac{1+\cos\theta}{2}e^{i\phi}\hat{f}(t)-\frac{1-\cos\theta}{2}e^{-i\phi}\hat{f}^{{\dagger}}(t)\right].$
This yields coupled linear equations for $\hat{a}$ and $\hat{a}^{{\dagger}}$
whose solution in the steady state for times $t\gg(N\gamma\cos\theta/2)^{-1}$
is
$\displaystyle\hat{a}(t)$ $\displaystyle=$
$\displaystyle\sqrt{N}\left[\frac{1+\cos\theta}{2}e^{i\phi}\hat{B}(t)+\frac{1-\cos\theta}{2}e^{-i\phi}\hat{B}^{{\dagger}}(t)\right],$
$\displaystyle\hat{B}(t)$ $\displaystyle=$ $\displaystyle
i\int_{0}^{t}dt^{\prime}e^{-N\cos\theta\left[\frac{\gamma}{2}-i\Delta\right](t-t^{\prime})}\hat{f}(t^{\prime}).$
(30)
This operator-form solution, along with the correlation function (7) of the
Langevin vacuum-noise $\hat{f}(t)$, now allows to evaluate correlations of the
collective spin. In fact, even without performing specific calculations, the
operator solution itself is already quite insightful. We see that the lowering
operator of the spin fluctuation $\hat{a}$ exhibits a Bogoliubov
transformation form: it is a linear combination of the integrated vacuum noise
lowering operator $\hat{B}$ and its conjugate $\hat{B}^{{\dagger}}$, with
corresponding Bogoliubov coefficients proportional to $1+\cos\theta$ and
$1-\cos\theta$, respectively. Non-trivial, correlated fluctuations occur
whenever $\hat{a}$ contains the conjugate component $\hat{B}^{{\dagger}}$ (and
not only $\hat{B}$), requiring a non-vanishing coefficient $1-\cos\theta$.
Therefore, quantum correlations are expected to grow with the driving field
$|\Omega|/\Omega_{c}=\sin\theta>0$ , as seen explicitly below.
### IV.2 Spin squeezing
A particulary relevant characterization of collective-spin fluctuations is
provided by the spin squeezing parameter spinsqueezingreview ; KitaUeda . Spin
squeezing quantifies fluctuations of the spin vector perpendicular to its mean
direction, and is linked to the sensitivity of quantum-enhanced metrology with
collections of spins spinsqueezingparameter1 ; spinsqueezingparameter2 ; QSr
and their underlying pairwise entanglement LEW ; SOR . Within the rotated spin
representation from (26), where the mean is directed to $-z^{\prime}$, the
spin squeezing parameter is given by spinsqueezingparameter1 ;
spinsqueezingparameter2 ; spinsqueezingreview
$\xi^{2}=\mathrm{min}_{\varphi}\frac{\mathrm{Var}[\hat{J}^{\prime}_{\varphi}]N}{|\langle\hat{J}^{\prime}_{z}\rangle|^{2}},\quad\hat{J}^{\prime}_{\varphi}=\cos\varphi\hat{J}^{\prime}_{x}+\sin\varphi\hat{J}^{\prime}_{y},$
(31)
i.e. it is proportional to the minimal variance of the fluctuations along the
$x^{\prime}y^{\prime}$ plane. Spin squeezing exists for $\xi^{2}<1$, implying
that the collective-spin has improved phase sensitivity to rotations compared
to the standard quantum limit $\xi^{2}=1$ of an uncorrelated coherent spin
state.
Within our mean-field and small-fluctuations assumption, we use
$|\langle\hat{J}^{\prime}_{z}\rangle|\approx N/2$ and the bosonic
approximation (28) for $\hat{J}^{\prime}_{\mp}=\hat{J}^{\prime}_{x}\mp
i\hat{J}^{\prime}_{y}$, to obtain the spin squeezing parameter in terms of the
bosonic operators $\hat{a}$,
$\xi^{2}=1+2\langle\hat{a}^{\dagger}\hat{a}\rangle-2|\langle\hat{a}^{2}\rangle|.$
(32)
Using the solution for $\hat{a}$, Eq. (30), and the Langevin, vacuum-noise
correlation function (7), we then find
$\left|\langle
a^{2}\rangle\right|=\frac{1-\cos^{2}\theta}{4\cos\theta},\quad\langle
a^{\dagger}a\rangle=\frac{(1-\cos\theta)^{2}}{4\cos\theta},$ (33)
so that the spin squeezing parameter, Eq. (32), becomes
$\xi^{2}=\cos\theta=\sqrt{1-\frac{|\Omega|^{2}}{\Omega_{c}^{2}(\Delta)}}.$
(34)
We observe that the spin squeezing is determined by the ratio between the
driving field and the critical field, $|\Omega|/\Omega_{c}$. It depends on the
dipole-dipole interaction $\Delta$ through the critical field
$\Omega_{c}(\Delta)$ from Eq. (10). This generalizes the analytical result of
Ref. yelin , obtained for the case $\Delta=0$ using a master-equation
approach. When the drive is weak $|\Omega|/\Omega_{c}\rightarrow 0$, no spin
squeezing exists, $\xi^{2}=1$, since the system is in a coherent spin state
wherein all atoms are in the ground state. As the drive increases, population
in the atoms is created, such that collective emission is possible, building
entanglement and spin-squeezing correlations between the atoms, $\xi^{2}<1$.
At the critical point $|\Omega|/\Omega_{c}=1$ the spin-squeezing parameter
vanishes: this result is valid only at the limit $N\rightarrow\infty$ where it
does not contradict the Heisenberg limit $\xi^{2}\geqslant 1/N$
spinsqueezingreview . For finite $N$, our mean-field assumption of small
fluctuations breaks down as we approach the critical point, where fluctuations
become increasingly large (e.g.
$\mathrm{max}_{\varphi}\mathrm{Var}[\hat{J}^{\prime}_{\varphi}]\propto
1/\cos\theta$ diverges near the critical point).
It is instructive to compare the analytical result (34) to that obtained by an
exact numerical solution of the master equation for a finite $N$, as explained
above. In Fig. 3 we observe excellent agreement between the analytical and
numerical solutions up to a driving field somewhat below the critical point
$|\Omega|<\Omega_{c}(\Delta)$, above which the two solutions diverge away. As
in Fig. 2, the dependence on $\Delta$ is captured by plotting the numerical
solutions for different values of $\Delta$, which all collapse to the curve as
a function of the driving field $\Omega$ (e.g. taken real) scaled to the
corresponding critical field $\Omega_{c}(\Delta)$, as anticipated analytically
in Eq. (34). We observe that the exact solution obtains its optimal (minimal)
value for the squeezing $\xi^{2}$ close to the point where it begins to
diverge away from the analytical result. Therefore, this optimal value for
$\xi^{2}$ should improve (become smaller) with increasing $N$ yelin . The
scaling of the optimal $\xi^{2}$, being a finite-size effect, cannot be
accounted for by the above mean-field based results (valid for
$N\rightarrow\infty$). This scaling can be obtained analytically using CRSS
theory, yielding $\xi^{2}\sim N^{-1/3}$ CRSS .
Figure 3: Spin squeezing $\xi^{2}$ as function of the driving field $\Omega$.
Results obtained by the numerical solution of Eq. (8) with $N=50$ atoms and
for different values of the dipole-dipole shift $2\Delta/\gamma=0,1,2$ all
collapse to the same curve by when $\Omega$ is scaled the critical field
$\Omega_{c}(\Delta)$ from Eq. (10). The red line represent the analytical
solution from Eq. (34). The exact numerical solutions agree with the
analytical result until they diverge away close to the transition point, where
$\xi^{2}$ begins degrading (growing) with $\Omega$, see main text.
## V Radiated Light
So far we have treated the field degrees of freedom as a reservoir that
generates driven-dissipative dynamics of the atoms. However, superradiance is
essentially a scattering problem of an input coherent-state field off a
collective dipole $\hat{J}_{-}$ formed by the atoms. As such, the total field
exhibits the general form,
$\hat{E}(t)=\hat{E}_{\mathrm{free}}(t)+G\hat{J}_{-}.$ (35)
The first term is the freely propagating coherent-state field in the absence
of atoms, consisting of an average field and vacuum fluctuations. It may
include the influence of linear optical elements such as the cavity mirrors in
the cavity realization of Fig. 1. The second term is the field component
scattered by the atomic dipole $\hat{J}_{-}$, with a coupling coefficient $G$
(describing field propagation from the atoms to the detector). While the first
term exhibits non-correlated coherent-state statistics of the input field, the
second term may exhibit correlations generated by the nonlinearity of the
atoms QNLOr . In superradiance, the considered atomic system is clearly
nonlinear, as we have already seen that the population inversion
$\langle\hat{J}_{z}\rangle$ is a nonlinear function of the driving field
$\Omega$, see Eq. (9). Nevertheless, we show in the following that,
surprisingly, the scattered component of the field is also a coherent-state
field, linear in the input field. This holds for any driving field $\Omega$
smaller than the critical field $\Omega_{c}$.
Although this result is valid for any realization of superradiance, we focus
for concreteness on the cavity realization considered above. We define the
total observable field as the field propagating out of the cavity (in the
rotated frame $\omega_{L}$)
$\hat{E}(t)=-i\sum_{k>0}\eta^{\ast}\hat{b}_{k}(t)e^{i\omega_{L}t}-i\Omega_{L},$
(36)
where $-i\Omega_{L}$ is the average component of the input coherent field.
Using the same HL approach from Sec. II B, we solve for $\hat{E}(t)$ within
the coarse-grained dynamics at $t\gg 1/\kappa$, obtaining Eq. (35) with (see
Appendix),
$\displaystyle\hat{E}_{\mathrm{free}}(t)$ $\displaystyle=$
$\displaystyle\left(1+\chi\right)\left[\hat{E}_{0}(t)-i\Omega_{L}\right],\quad\chi=\frac{\kappa}{i\delta_{c}-\kappa/2},$
$\displaystyle G$ $\displaystyle=$ $\displaystyle-ig^{\ast}\chi.$ (37)
Here $\chi$ describes the linear response of the cavity to the input field
$\hat{E}_{0}(t)-i\Omega_{L}$ (vacuum + coherent drive), which interferes with
the input, yielding the factor $1+\chi$. Therefore, the atom-free field indeed
has the form of a coherent-state field composed of vacuum + average
components. In the following we will show that this turns out to be the case
also for the total field.
### V.1 Average field
Taking the average of Eq. (35), the vacuum term $\hat{E}_{0}$ does not
contribute so that the free-field component from Eq. (37) gives
$-i\Omega_{L}(1+\chi)$. For the scattered part we use $G=-ig^{\ast}\chi$ from
Eq. (37) and $\langle\hat{J}_{-}\rangle$ from Eq. (9) obtaining
$G\langle\hat{J}_{-}\rangle=i\chi\Omega_{L}$. The total average field then
becomes,
$\langle\hat{E}\rangle=-i\Omega_{L},$ (38)
equal to the incident average field. So, the average radiated field in
superradiance is linear in the incident-field amplitude even though the atomic
system is nonlinear, as discussed above.
### V.2 Field fluctuations
The HL approach allows us to gain direct access to field operators which
entail information on the quantum statistics of the field. We will use it here
to show that the fluctuating part of the radiated field is proportional to
vacuum fluctuations, thus proving that the radiated field is in a coherent
state. We first do this by solving for the operators directly, without the
need to infer the statistics from the calculation of correlations.
To this end, we focus on the scattered component of the field, $G\hat{J}_{-}$.
Using the transformation (26), we write $\hat{J}_{-}$ in terms of the rotated-
system spin operators as,
$\displaystyle\hat{J}_{-}$ $\displaystyle=$ $\displaystyle
e^{-i\phi}\left(\frac{\cos\theta+1}{2}\hat{J}^{\prime}_{-}+\frac{\cos\theta-1}{2}\hat{J}^{\prime}_{+}+\sin\theta\hat{J}^{\prime}_{z}\right)$
$\displaystyle\approx$ $\displaystyle
e^{-i\phi}\left(\frac{\cos\theta+1}{2}\sqrt{N}\hat{a}+\frac{\cos\theta-1}{2}\sqrt{N}\hat{a}^{{\dagger}}-\sin\theta\frac{N}{2}\right).$
In the second line we have used the Holstein-Primakoff linearization, Eq.
(28). Plugging in the solution for $\hat{a}$ from Eq. (30), we then obtain for
the fluctuating part of the field $\hat{E}$ from (35)
$\displaystyle\hat{\mathcal{E}}(t)\equiv\hat{E}-\langle\hat{E}\rangle=\left(1+\chi\right)\hat{E}_{0}(t)+GN\cos\theta\hat{B}(t).$
(40)
The first term describes the vacuum fluctuations $\propto\hat{E}_{0}$ of the
coherent free-field component from Eq. (37). The second term originates from
the fluctuating part of the scattered field $G\hat{J}_{-}$ from Eq.
(LABEL:Jm1) and is also essentially proportional to integrated vacuum
fluctuations $\hat{E}_{0}$ [noting that $\hat{B}$ in Eq. (30) is an integral
of $\hat{f}\propto\hat{E}_{0}$]. This proves that the total radiated field is
in a coherent state, comprised of vacuum fluctuations on top of a mean
coherent amplitude.
### V.3 Light squeezing vs. spin squeezing
Since the radiated field is a classical-like coherent state, it does not
exhibit any quantum correlations. We now show this explicitly for the case of
quantum squeezing correlations. Defining the quadrature operator of the
radiated field,
$\hat{X}_{\varphi}=e^{-i\varphi}\hat{E}+e^{i\varphi}\hat{E}^{{\dagger}}$, the
bosonic squeezing parameter of the field is given by
$\displaystyle\xi^{2}_{E}=\mathrm{min}_{\varphi}\frac{\mathrm{Var}[\hat{X}_{\varphi}]}{V_{0}}=1+\frac{2}{V_{0}}\left(\langle\hat{\mathcal{E}}^{{\dagger}}\hat{\mathcal{E}}\rangle-|\langle\hat{\mathcal{E}}^{2}\rangle|\right),$
(41)
with
$V_{0}\equiv[\hat{E},\hat{E}^{{\dagger}}]=[\hat{E}_{0},\hat{E}^{{\dagger}}_{0}]=\kappa\delta(t=0)$
being the vacuum-noise level. Squeezed quantum noise and correlations exist if
the quadrature noise can become lower than that of the vacuum, i.e. for
$\xi^{2}_{E}<1$. It is seen that this requires the existence of the phase-
dependent correlator $\langle\hat{\mathcal{E}}^{2}\rangle$. Similarly, spin
squeezing in Eq. (32) requires the existence of the phase-dependent correlator
of spin fluctuations $\langle\hat{a}^{2}\rangle$. For either of these
correlators to exist, the corresponding lowering operators $\hat{\mathcal{E}}$
and $\hat{a}$ then must contain a raising field-operator $\hat{B}^{{\dagger}}$
(equivalently, $\hat{E}_{0}^{{\dagger}}$) in addition to $\hat{B}$, since the
average is performed over the initial vacuum state. While the Bogoliubov
coefficient $1-\cos\theta$ in Eq. (30) indeed guarantees that $\hat{a}$
contains $\hat{B}^{{\dagger}}$ for any finite drive
$|\Omega|/\Omega_{c}=\sin\theta<1$, this is not the case for the field
fluctuations $\hat{\mathcal{E}}$: The transformation coefficients in Eq.
(LABEL:Jm1) from $\hat{a},\hat{a}^{{\dagger}}$ to
$\hat{J}_{-}\sim\hat{\mathcal{E}}$, which also depend on $\cos\theta$, lead to
an exact cancellation of the coefficient for $\hat{B}^{{\dagger}}$ in
$\hat{\mathcal{E}}$, as seen in Eq. (40). Therefore, for any drive strength
$|\Omega|/\Omega_{c}=\sin\theta<1$, spin squeezing exists while light
squeezing exactly cancels. This result is equivalent to the geometrical
interpretation given by the so-called dipole-projected squeezing CRSS .
### V.4 Spectrum
Having access to the field operator $\hat{E}(t)$, the HL approach also allows
to directly calculate two-time correlations and spectra. The spectrum of the
radiated field in a steady state time $t$ is given as usual by the Fourier
transform on the time-difference $\tau$ of the two-time correlation,
$\langle\hat{E}^{{\dagger}}(t)\hat{E}(t+\tau)\rangle$. Since this is a normal-
ordered correlator, the fluctuating part of $\hat{E}$ in Eq. (40) drops, as it
is proportional to the lowering operator $\hat{E}_{0}$. This trivially yields
$\langle\hat{E}^{{\dagger}}(t)\hat{E}(t+\tau)\rangle=|\langle\hat{E}\rangle|^{2}=|\Omega_{L}|^{2}$.
The spectrum of the radiated field is then a single delta peak at the incident
frequency $\omega_{L}$ (recalling we work in the laser-rotated frame). This
again shows that the collective atomic dipole scatters light as a linear
optical element even though the atomic population exhibits a strongly
nonlinear dependence on the drive $\Omega_{L}$.
## VI Conclusions
In this work we have presented a HL approach to driven Dicke superradiance in
steady state. The analytical results for steady-state spin squeezing agree and
generalize those obtained in Refs. Alejandro ; yelin ; BAR ; REYt .
Furthermore, our finding that the radiated field is in a coherent state
underlies previous results on uncorrelated photon statistics below the
transition CAR . These HL-based results are consistent with the formation of a
CRSS as described in CRSS . The HL approach is thus complementary to the CRSS
description of superradiance. On the one hand, it is based on the approximate
analysis of small fluctuations around the mean field for $N\rightarrow\infty$,
and did not yield the finite-size scalings with $N$ or the full atomic state
as in CRSS. But on the other hand, it is simpler to generalize for treating
superradiance beyond the permutation-symmetric Dicke case, e.g. by performing
the Holstein-Primakoff approximation for each individual atom separately,
while allowing for direct estimation of atom and field correlations.
*
## Appendix A Output field
Here we elaborate on the derivation of the general expression for the output
field, Eqs. (35) and (37), in the one-sided cavity scheme of Fig. 1. We begin
by defining the outside propagating field
$\displaystyle\hat{E}(x,t)=-i\sum_{k>0}\eta^{\ast}\hat{b}_{k}(t)e^{ikx}e^{i\omega_{L}t}-i\Omega_{L}e^{i\frac{\omega_{L}}{v}x}.$
(42)
Here $x$ is the propagation axis: in the one-sided scheme, $x=0$ denotes the
position of the outcoupling mirror (right-hand side mirror in Fig. 1), so that
$x<0$ denotes incoming left-propagating fields whereas $x>0$ denotes outgoing
right-propagating fields. The radiated field from Eq. (36) is then defined by
taking $x=0^{+}>0$. As in the derivation of the HL equations in Sec. II B, we
first formally solve the Heisenberg equations for $\hat{b}_{k}(t)$, obtaining
in the Markov approximation
$\displaystyle\hat{E}(x,t)$ $\displaystyle=$
$\displaystyle\hat{E}_{0}(x,t)-i\Omega_{L}e^{i\frac{\omega_{L}}{v}x}$ (43)
$\displaystyle-$ $\displaystyle
e^{i\frac{\omega_{L}}{v}x}\kappa\int_{0}^{t}dt^{\prime}\tilde{c}(t^{\prime})\delta(t-x/c-t^{\prime}).$
Here $\hat{E}_{0}(x,t)$ is the vacuum field from Eq. (4) with the exponentials
$e^{ikx}$ in the mode expansion $\hat{b}_{k}(0)$. For $x=0^{-}<0$, the Dirac
delta function does not contribute and we indeed obtain the input field
$\hat{E}_{0}(t)-i\Omega_{L}$. For $x=0^{+}>0$ we obtain
$\displaystyle\hat{E}(t)\equiv\hat{E}_{0}(0^{+},t)=\hat{E}_{0}(t)-i\Omega_{L}-\kappa\tilde{c}(t).$
(44)
Finally, inserting the coarse-grained solution for $\tilde{c}(t)$ [obtained
for simplicity by setting $\dot{\tilde{c}}=0$ in Eq. (3)], we arrive at Eqs.
(35) and (37).
###### Acknowledgements.
We acknowledge financial support from the Israel Science Foundation (ISF)
grant No. 2258/20, the ISF and the Directorate for Defense Research and
Development (DDR&D) grant No. 3491/21, the Center for New Scientists at the
Weizmann Institute of Science, the Council for Higher Education (Israel), and
QUANTERA (PACE-IN). This research is made possible in part by the historic
generosity of the Harold Perlman Family.
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|
# Qualitative quantum simulation of resonant tunneling and localization with
the shallow quantum circuits
P. Wang<EMAIL_ADDRESS>Department of Mathematics and Physics, North China
Electric Power University, 102206 Beijing, China
###### Abstract
In a circuit-based quantum computer, the computing is performed via the
discrete-time evolution driven by quantum gates. Accurate simulation of
continuous-time evolution requires a large number of quantum gates and
therefore suffers from more noise. In this paper, we find that shallow quantum
circuits are sufficient to qualitatively observe some typical quantum
phenomena in the continuous-time evolution limit, such as resonant tunneling
and localization phenomena. We study the propagation of a spin excitation in
Trotter circuits with a large step size. The circuits are formed of two types
of two-qubit gates, i.e. XY gates and controlled-$R_{x}$ gates, and single-
qubit $R_{z}$ gates. The configuration of the $R_{z}$ gates determines the
distribution of the spin excitation at the end of evolution. We demonstrate
the resonant tunneling with up to four steps and the localization phenomenon
with dozens of steps in Trotter circuits. Our results show that the circuit
depth required for qualitative observation of some significant quantum
phenomena is much smaller than that required for quantitative computation,
suggesting that it is feasible to apply qualitative observations to near-term
quantum computers. We also provide a way to use the physics laws to understand
the error propagation in quantum circuits.
###### pacs:
11.30.Er, 03.65.Nk, 03.65.-w, 42.82.Et
Keywords: Qualitative quantum simulation, shallow quantum circuits, resonant
tunneling, localization, error propagation
## 1 Introduction
Quantum computing can be used to investigate quantum systems as a universal
simulator [1, 2, 3]. In quantum mechanics, the time evolution of quantum
states is driven by a Hamiltonian and described by a unitary operator. In a
digital quantum computer, the computing is carried out by using a set of basic
quantum gates, and usually each gate is a single-qubit or two-qubit unitary
operator. The combination of these basic gates allows us to implement the
evolution operator of a multi-qubit system. A specific approach is Trotter-
Suzuki decomposition [4, 5, 6, 7], in which we approximate the continuous-time
evolution with a discrete-time evolution. For a general local-interaction
Hamiltonian, we can explicitly construct the evolution operator for a short
time, i.e. one time step, from quantum gates. By repetitive gates of one time
step for $N_{T}$ times, we realize the target time evolution. With a smaller
step size, the discrete-time evolution is closer to continuous-time evolution,
however, this requires a larger $N_{T}$, i.e. more quantum gates. Considering
a practical device [8], quantum computing is inaccurate due to decoherence and
imperfect control, and usually the error increases with the gate number [9,
10, 11]. Fault-tolerant quantum computing using quantum error correction is
able to remove the error but impractical using today’s technologies, because
of the large qubit overhand for encoding [12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25]. A family of practical methods have been developed to
mitigate errors, however the gate number is usually limited due to the finite
error rate on the physical level [26, 27, 35, 29, 30, 31, 32, 33, 34, 35, 36].
Therefore we can only realize the discrete-time Trotter evolution with a small
number of Trotter steps. This motivates researches on the effect of large step
size, i.e. few Trotter steps. Trotter errors induced by large step sizes in
digital quantum simulation have received extensive attention[37, 38, 39, 40,
41]. Some studies show that the Trotter step sizes can separate quantum
chaotic phase from localized phase and comparatively large Trotter steps can
retain controlled errors for local observables[42, 43].
In this paper, we are interested in, when the step size is large (or
equivalently the steps are few), whether some typical physical effects in the
limit of continuous-time can still be observed. The physical effects that we
focus on are the resonant tunneling phenomenon and the localization in
disordered systems [44, 45]. We find that, in a large-step-size Trotter
circuit, the resonant tunneling with $n$ resonant peaks can be observed in
circuits with $n+1$ Trotter steps. Experiments on an IBM quantum computer are
implemented to demonstrate the resonant tunneling with up to three peaks. We
also study the spin transport with the disordered configurations of the
$R_{z}$ gates (we will specify these gates later) using the large step size.
The numerical simulation of circuits with $15$ qubits and tens of Trotter
steps exhibits the localization in the disordered configuration. The results
indicate that shallow quantum circuits on near-term quantum computers are
sufficient to qualitatively simulate some significant physical phenomena. The
localization phenomenon of the spin excitation distribution implies that the
bit-flip error does not affect the measurement on distant qubits if the
configuration of the $R_{z}$ gates is disordered. These conclusions can be
generalized if we replace XY gates with controlled-$R_{x}$ gates, which can
transform one spin excitation into multiple spin excitations.
This paper is organized as follows. In Sec. 2, we discuss the quantum
transverse-field XY model and corresponding quantum circuits, the map between
the Hamiltonian of model and corresponding circuit is established in the limit
of small Trotter step size. In Sec. 3, we discuss the propagation of the spin
excitation, and compare resonant tunneling effects in circuits in the small-
step-size limit and the large-step-size limit. In Sec. 4, we investigate the
transport of the spin excitation in the ordered and disordered configurations
of single-qubit $R_{z}$ gates. Conclusion is given at the end of the paper.
Figure 1: (Color online) (a) The schematic diagram of quantum circuit, which
includes the initialization, $N_{T}$ Trotter steps, and measurements. The
preparation for the initial state is in the red dashed box including a NOT
gate (i.e. the X gate). The blue rectangle represents a layer of two-qubit
gates. The orange rectangles represent single-qubit $R_{z}$ gates. (b) The
schematic diagram of a layer of two-qubit gates. (c) The matrix
representations for two types of 2-qubit gates: XY gates and
controlled-$R_{x}$ gates.
## 2 Model
In this paper, we study the particle transport in the discrete-time evolution
in the quantum transverse-field XY model. The purpose of this study is to
investigate whether some typical quantum phenomena occurring in the
continuous-time limit can be qualitatively observed in discrete-time evolution
when the step size is large. The typical quantum phenomena we concerned here
including resonant tunneling and localization effect, which caused by
interference during the particle transport. Generally, the quantum transverse-
field XY model can be used to describe the propagation of spin excitations, or
particle transport when spin excitations can be treated as particles [46].
Additionally, in the quantum transverse-field XY model, the corresponding
discrete-time evolution operator can be mapped into a quantum circuit
according to the Trotter-Suzuki decomposition. Therefore, we investigate the
particle transport with the quantum transverse-field XY model
$H=\sum_{j}^{N-1}J_{j}H_{j,j+1}^{XY}+\sum_{j}^{N}V_{j}H_{j}^{Z},$ (1)
where $j$ is the lattice site, $N$ is the system size, $J_{j}$ is the
interaction strength and $V_{j}$ is the transverse-field strength.
$H_{j,j+1}^{XY}=(\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y})/4$
and $H_{j}^{Z}=\sigma_{j}^{z}/2$, where $\sigma_{j}^{i}$ $(i=x,y,z)$
represents the Pauli matrix at the $j$th site. In the case of $J_{j}<0$ and
$V_{j}<0$, the ground state of the transverse-field XY chain is
$\left|00...0\right\rangle$. In this work, we consider the time evolution of
the initial state $\left|\psi(0)\right\rangle=\left|10...0\right\rangle$ which
represents a spin excitation on the first site. The time evolution is in the
subspace of single spin excitation.
The time evolution operator $U(t)$ of quantum transverse-field XY model can be
approximated with a quantum circuit. According to Trotter-Suzuki
decomposition, $U(t)$ can be expanded approximately
$U(t)=e^{-iHt}\approx\left(\prod_{j=1}^{N-1}U_{j,j+1}^{XY}(J_{j}\tau)\prod_{j=1}^{N}U_{j}^{Z}(V_{j}\tau)\right)^{N_{T}},$
(2)
where $N_{T}$ is the number of Trotter steps, $\tau=t/N_{T}$ is the size of
each Trotter step. $U_{j,j+1}^{XY}(\theta_{j})=e^{-iH_{j,j+1}^{XY}\theta_{j}}$
and $U_{j}^{Z}\left(\phi_{j}\right)=e^{-iH_{j}^{Z}\phi_{j}}$, where
$\theta_{j}=J_{j}\tau$ and $\phi_{j}=V_{j}\tau$.
Figure 2: (Color online) (a1)-(d1) The circuit systems. The parameters of
$R_{z}$ gates are marked on the orange squares. For convenience, in (c1) and
(d1) we use single green rectangle to denote a layer of two-qubit gates.
(a2)-(d2) The quantum wells. The energy levels of the wells are denoted by the
horizontal lines in the wells. (a3)-(d3) The tight-binding chain. (a4)-(d4)
Numerical result of the discrete-time evolution. The solid curves represent
the probability of observing the spin excitation on the last qubit after the
time evolution. The curves with circles are the results obtaining on a IBM
quantum computer. The curves with light colors are the numerical result of the
continuous-time evolution.
The corresponding discrete-time evolution is realized with the quantum circuit
as shown in Fig. 1(a). The time evolution of each term, i.e. $U^{XY}$ and
$U^{Z}$, are two-qubit XY gate and single-qubit $R_{z}$ gate, respectively.
The circuit has $N$ qubits $\\{q_{1},q_{2},...,q_{N}\\}$ and $N_{T}$ Trotter
steps, and every Trotter step contains one layer of two-qubit XY gates and one
layer of single-qubit $R_{z}$ gates (We neglect $R_{z}$ gates in the last
Trotter step, because these gates does not have any effect on the distribution
of the spin excitation). In a quantum circuit, a NOT gate (i.e. the Pauli
$\sigma_{x}$ matrix) can flip the qubit $\left|0\right\rangle$ to
$\left|1\right\rangle$, therefore we prepare the initial state
$\left|\psi(0)\right\rangle$ by applying a NOT gate on the first qubit, i.e.
$\left|\psi(0)\right\rangle=\sigma_{1}^{x}\left|0\right\rangle^{N}=\left|10...0\right\rangle$
[see the dashed rectangle in Fig. 1(a)]. The matrix representation of XY gate
is shown in Fig. 1(c).
The behavior of a spin excitation under discrete-time evolution depends on
step size. When the Trotter step size is sufficiently small, the propagation
of a spin excitation in quantum circuit is equivalent to the particle
transport in continuous-time evolution. In this case, the propagation of spin
excitation can exhibit some typical physical phenomena in particle transport.
A natural question to ask is, in shallow quantum circuits with a large Trotter
step size, whether some physical phenomena during continuous-time evolution
nevertheless remains, so that we can observe these phenomena using fewer
quantum gates. In the following text, we show that we observe the resonant
tunneling and localization effect in shallow circuits even if the Trotter step
size is large.
## 3 Resonant tunneling for large Trotter step size
In this section, we study the resonance phenomenon related to the transport of
the spin excitation during the discrete-time evolution with a large step size.
We study the situation where the size of the circuit system is $N=2,3,4,5$
respectively, where the spin excitation is created by a NOT gate and
transmitted through the XY gate. In each one case, we qualitatively find the
resonance tunneling in the limit of continuous-time evolution and give the
corresponding minimum number of Trotter steps.
We also study the transport of a spin excitation through the
controlled-$R_{x}$ gate. In quantum computing, computational errors occur due
to the imperfect control and decoherence always error. The typical one is bit-
flip error corresponding to an unwanted NOT gate. So studying the transmission
of the spin excitation can help us understand the propagating of bit-flip
errors. In actual quantum circuits, errors are not only transmitted, but also
replicated. For example, a single-qubit error will become a multi-qubit error
after passing through the controlled-$R_{x}$ gate. This effect potentially has
a greater impact in quantum computing, because multi-qubit errors can lead to
the failure of quantum error correction. So in this section, we also study the
behavior of a spin excitation propagating through the controlled-$R_{x}$ gate
and find the resonance phenomenon similar to that propagating through the XY
gate.
### 3.1 Resonant tunneling in two-qubit circuit
We first discuss the case of $N=2$. The Hamiltonian of the transverse-field XY
model with 2 sites is
$H=J(\sigma_{1}^{x}\sigma_{2}^{x}+\sigma_{1}^{y}\sigma_{2}^{y})/4+V_{1}\sigma_{1}^{z}/2+V_{2}\sigma_{2}^{z}/2.$
(3)
According to the Trotter-Suzuki decomposition, the time evolution operator can
be approximated using a sequence of quantum gates
$U(t)=e^{-iHt}\approx[U_{1,2}^{XY}(J\tau)U_{1}^{Z}(V_{1}\tau)U_{2}^{Z}(V_{2}\tau)]^{N_{T}},$
(4)
where
$U_{j,j+1}^{XY}(J_{j}\tau)=e^{-i(\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y})J_{j}\tau}$,
$U_{j}^{Z}\left(V_{j}\tau\right)=e^{-i\sigma_{1}^{z}V_{j}\tau/2}$, and
$\tau=t/N_{T}$. The right side of ”$\approx$” in Eq. (4) represents the
discrete-time evolution. The accuracy of the approximation increases with
$N_{T}$. In Fig. 2(a1), we show the schematic diagram of the quantum circuit
corresponding to discrete-time evolution when $N_{T}=2$ (the last two $R_{z}$
gates have been ignored, and so do the other systems). As we can see, the
yellow squares represent that the qubits are initialized to
$\left|00\right\rangle,$ which is the ground state of $H$ in the case of
$V_{1},V_{2}<0$ and $J\ll V_{1}+V_{2}$. Following the initialization, a NOT
gate on the first qubit flips $\left|00\right\rangle$ into
$\left|10\right\rangle$, which represents that there is a spin excitation on
the first qubit. $\left|10\right\rangle$ is the initial state that we want to
prepare. Due to the symmetry of Hamiltonian, the spin excitation lies in the
subspace spanned by {$\left|10\right\rangle$, $\left|01\right\rangle$}. In
this single-spin-excitation subspace, we can regard a spin excitation as a
particle moving in a $2$-site tight-binding chain [see Fig. 2(a3)], and the
corresponding Hamiltonian is
$H=J\left(\left|10\right\rangle\left\langle
01\right|+\left|01\right\rangle\left\langle
10\right|\right)+V_{1}\left|10\right\rangle\left\langle
10\right|+V_{2}\left|01\right\rangle\left\langle 01\right|,$ (5)
where $\left|10\right\rangle$ or $\left|01\right\rangle$ represents a particle
in the first or second site respectively, $J$ is the tunneling strength,
$V_{1}$ and $V_{2}$ are the on-site potentials. In this chain system with
fixed parameters $J$, $V_{2}$ and variable $V_{1}$, resonance phenomenon can
be observed [47, 48, 49, 50, 51]: Assuming that a particle is on the first
site at $t=0$, the probability, which is denoted by $P_{2}(V_{1},t)$, of
observing the particle on the second site at any time $t$ reaches maximum when
$V_{1}=V_{2}$. We numerically simulate this phenomenon in a discrete-time
evolution with a large $N_{T}$ in Fig. 2(a4). We exhibit $P_{2}$ at $t=15$
(units of $1/J$, $J=0.1$) with the transparent lines for $V_{2}=0,-\pi/2$. As
expected, $P_{2}$ has one resonance peak at $V_{1}=V_{2}$. We can understand
this phenomenon more visually with the help of double-well system as shown in
Fig. 2(a2). Supposing that a particle is bounded in the left well at the
initial time, and then it will tunnel to the right well with a certain
probability. When the potential energies on both sides are equivalent (i.e.
$V_{1}=V_{2}$), the tunneling probability reaches maximization.
We wonder, when $N_{T}$ is small, whether we can qualitatively observe a
similar resonant effect as the large $N_{T}$ limit. Motivated by this, we
discuss the case of $N_{T}=2$. The parameters are redefined as
$\theta=J\tau,\phi=V_{1}\tau,$ and $\alpha=V_{2}\tau$ for convenience. Our
concern is the probability of finding spin excitation on the $2$nd qubit.
Figure 2(a1) shows two propagation paths of the spin excitation from the $1$st
to the $2$nd qubit. The blue path contributes $-i\sin\theta\cos\theta
e^{i(\alpha-\phi)/2}$ to the amplitude, the purple path contributes
$-i\sin\theta\cos\theta e^{-i(\alpha-\phi)/2}$ to the amplitude, so the final
state of the quantum circuit reads
$U_{1,2}^{XY}\left(\theta\right)U_{2}^{Z}\left(\alpha\right)U_{1}^{Z}\left(\phi\right)U_{1,2}^{XY}(\theta)\left|10\right\rangle=A_{10}\left|10\right\rangle+A_{01}\left|01\right\rangle,$
where
$A_{01}=-i\sin\theta\cos\theta(e^{-i(\alpha-\phi)/2}+e^{i(\alpha-\phi)/2})$.
The probability of spin excitation measured on the second qubit is
$P_{01}(\theta,\phi)=2\sin^{2}\theta\cos^{2}\theta\left(1+\cos(\alpha-\phi)\right).$
(6)
In Fig. 2(a4), we plot $P_{01}$ as function of $\phi$ with $\theta=\pi/2$ and
$\alpha=0,-\pi/2$. The accurate results (solid lines) computed using QuESTlink
coincide with the experimental outcomes (solid lines with point symbols)
computed using the IBM quantum device ”ibmq_rome”. The resonance peak is seen
near $\phi=\alpha$, which is related to the interference term
$2\sin^{2}\theta\cos^{2}\theta\cos(\alpha-\phi)$ in Eq. (6). $P_{2}$ coincides
well with $P_{01}$, both of them have only one peak and the position of the
peak is $\alpha=\phi$ (i.e. $V_{1}=V_{2}$). The above analysis indicates that
only two Trotter steps are required for the circuit to exhibit resonant
tunneling similar to the continuous-time evolution (i.e. $N_{T}$ is enough
large).
### 3.2 Multi-qubit system
In this section we discuss the multi-qubit quantum circuits with the system
sizes $N=3,4,5$ respectively.
We first discuss three-qubit system. The Hamiltonian of the transverse-field
XY chain with $3$ sites reads
$\displaystyle H$ $\displaystyle=$ $\displaystyle
J\sum_{j}^{2}(\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y})/4$
(7)
$\displaystyle+V_{1}\sigma_{1}^{z}/2+V_{2}\sigma_{2}^{z}/2+V_{1}\sigma_{3}^{z}/2.$
The corresponding time evolution can be approximated,
$\displaystyle U(t)$ $\displaystyle=$ $\displaystyle
e^{-iHt}\approx[U_{1,2}^{XY}(J\tau)U_{2,3}^{XY}(J\tau)U_{1}^{Z}(V_{1}\tau)\times$
(8) $\displaystyle U_{2}^{Z}(V_{2}\tau)U_{3}^{Z}(V_{1}\tau)]^{N_{T}},$
where
$U_{j,j+1}^{XY}(J_{j}\tau)=e^{-i(\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y})J_{j}\tau}$,
$U_{j}^{Z}\left(V_{j}\tau\right)=e^{-i\sigma_{1}^{z}V_{j}\tau/2}$,
$\tau=t/N_{T}$. The right side of above equation represents the discrete-time
evolution. The quantum circuit implementing the discrete-time evolution is
shown in Fig. 2(b1). We concern the time evolution of single spin excitation.
When the single spin excitation can be treated as a particle, the transverse-
field XY chain is equal to a $3$-site tight-binding chain [see Fig. 2(b3)],
$\displaystyle H$ $\displaystyle=$ $\displaystyle
J\left(\left|100\right\rangle\left\langle
010\right|+\left|010\right\rangle\left\langle
100\right|+\mathrm{h.c.}\right)+V_{1}\left|100\right\rangle\left\langle
100\right|$ (9) $\displaystyle+V_{2}\left|010\right\rangle\left\langle
010\right|+V_{1}\left|001\right\rangle\left\langle 001\right|.$
where $J$ is the coupling strength, the potentials on three sites are
$V_{1},V_{2}$ and $V_{1}$ respectively. In the case that $V_{1}$ is the only
variable parameter, the resonance phenomenon means that the probability,
$P_{3}(t,V_{1})$, of finding particle on the $3$th site reaches maximum at
$V_{1}=V_{2}$. We numerically simulate the discrete-time evolution with a
large $N_{T}$ and show the resonance phenomenon. The initial state is
$\left|100\right\rangle$. In Fig. 2(b4), we plot $P_{3}$ as the function of
$V_{1}$ at $t=22$ (units of $1/J$, $J=0.1$) with the transparent lines.
$P_{3}$ has one resonance peak at $V_{1}=V_{2}$. Similarly, we can consider
the resonance phenomenon with a triple-well system [see Fig. 2(b2)], whose
Hamiltonian can be written as Eq. (9). In the triple-well system, the
probability of the particle tunneling from the left well to the right well
reaches maximization at $V_{1}=V_{2},$ when the resonance occurs.
As for the small $N_{T}$, we find that only two Trotter steps are required for
the $3$-qubit circuit to exhibit resonant tunneling. The parameters of the
$3$-qubit circuit are redefined as
$J\tau=\theta,V_{1}\tau=\phi,V_{2}\tau=\alpha$. As shown the blue, red and
purple dashed lines in Fig. 2(b1), the spin excitation goes through three
paths. The blue and red paths contribute
$\cos\theta(-i\sin\theta)(-i\sin\theta)e^{-i\alpha/2}$ to the amplitude, and
the purple path contributes $-i\sin\theta\cos\theta
e^{i\left(\alpha/2-\phi\right)}\cos\theta(-i\sin\theta)$ to the amplitude. The
amplitude of the final state on the third qubit is
$A_{001}=-\sin^{2}\theta\cos\theta\left(2e^{-i\alpha/2}+\cos\theta
e^{i\left(\alpha/2-\phi\right)}\right).$ (10)
Accordingly, the probability of finding the spin excitation on the $3$rd qubit
is
$P_{001}(\theta,\phi)=\sin^{4}\theta\cos^{2}\theta(4+\cos^{2}\theta+4\cos\theta\cos\left(\alpha-\phi\right)).$
(11)
The interference term
$4\sin^{4}\theta\cos^{3}\theta\cos\left(\alpha-\phi\right)$ dominates the
resonant tunneling effect. In Fig. 2(b4), we plot $P_{001}$ as function of
$\phi$ with $\theta=\pi/2$ and $\alpha=0,-\pi/2$. The accurate results (solid
lines) computed using QuESTlink coincide with the experimental outcomes (solid
lines with point symbols) computed using the IBM quantum device ”ibmq_rome”.
As we can see, $P_{3}$ coincides well with $P_{001}$, both of them have only
one peak and the position of the peak is $\alpha=\phi$ (i.e. $V_{1}=V_{2}$).
The above analysis indicates that only two Trotter steps are required for the
$3$-qubit circuit to exhibit similar resonant tunneling in the continuous-time
limit.
We also study the four-qubit system. The Hamiltonian of the $4$-site
transverse-field XY chain we studied is
$\displaystyle H$ $\displaystyle=$ $\displaystyle
J_{1}\sum_{j=1,3}(\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y})/4+J_{2}(\sigma_{2}^{x}\sigma_{3}^{x}+\sigma_{2}^{y}\sigma_{3}^{y})/4$
(12)
$\displaystyle+V_{1}\sigma_{1}^{z}/2+V_{2}\sigma_{2}^{z}/2-V_{2}\sigma_{3}^{z}/2+V_{1}\sigma_{4}^{z}/2.$
The corresponding discrete-time evolution is in the form
$\displaystyle U(t)$ $\displaystyle=$ $\displaystyle
e^{-iHt}\approx[U_{1,2}^{XY}(J_{1}\tau)U_{2,3}^{XY}(J_{2}\tau)U_{3,4}^{XY}(J_{1}\tau)\times$
(13) $\displaystyle
U_{1}^{Z}(V_{1}\tau)U_{2}^{Z}(V_{2}\tau)U_{3}^{Z}(-V_{2}\tau)U_{3}^{Z}(V_{1}\tau)]^{N_{T}},$
where
$U_{j,j+1}^{XY}(J_{j}\tau)=e^{-i(\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y})J_{j}\tau}$,
$U_{j}^{Z}\left(V_{j}\tau\right)=e^{-i\sigma_{1}^{z}V_{j}\tau/2}$,
$\tau=t/N_{T}$. We plot the quantum circuit implementing the discrete-time
evolution in Fig. 2(c1). In the single-particle subspace, the equivalent
$4$-site tight-binding chain is
$\displaystyle H$ $\displaystyle=$ $\displaystyle
J_{1}\left(\left|1000\right\rangle\left\langle
0100\right|+\left|0010\right\rangle\left\langle
0001\right|\right)+J_{2}(\left|0100\right\rangle\left\langle 0010\right|$ (14)
$\displaystyle+\left|0010\right\rangle\left\langle
0100\right|)+V_{1}\left|1000\right\rangle\left\langle
1000\right|+V_{2}\left|0100\right\rangle\left\langle 0100\right|$
$\displaystyle-V_{2}\left|0010\right\rangle\left\langle
0010\right|+V_{1}\left|0001\right\rangle\left\langle 0001\right|.$
As we marked in Fig. 2(c4), the coupling strengths between neighboring sites
are $J_{1},J_{2}$, and $J_{1}$ respectively, and the on-site potentials on the
four sites are $V_{1},V_{2},-V_{2},V_{1}$ respectively. In the condition of
$J_{1}\ll J_{2},V_{2}$, we numerically simulate the discrete-time evolution of
one particle with a large $N_{T}$. In Fig. 2(c4), $P_{4}(V_{1},t)$, which is
the probability of finding the particle on the $4$th site, is plotted as
function of $V_{1}$ with the transparent lines. The cases of $V_{2}=10$ and
$V_{2}=20$ are studied when $J_{1}=1$, $t=3$ (units of $1/J_{1}$), $J_{2}=20$.
As we can see, the resonant peaks can be observed near
$\sqrt{J_{2}^{2}+V_{2}^{2}}$ and $-\sqrt{J_{2}^{2}+V_{2}^{2}}$, and the
distance between the resonance peaks varies when $V_{2},-V_{2}$ change. One
can observe the same resonance phenomenon in a triple-well system [see Fig.
2(c2)]. The energy levels of the left and right wells are $V_{1}$, the middle
well has two energy levels: $\sqrt{J_{2}^{2}+V_{2}^{2}}$ and
$-\sqrt{J_{2}^{2}+V_{2}^{2}}$. If there is a particle in the left well at the
initial moment, then we can detect this particle in the right well with a
certain probability. When $V_{1}$ is close to $\sqrt{J_{2}^{2}+V_{2}^{2}}$ or
$-\sqrt{J_{2}^{2}+V_{2}^{2}}$, the probability of finding particle in right
well reaches the maximum. The distance of the two peaks varies with $V_{2}$.
When $N_{T}$ is small, we find that only three Trotter steps are required for
the $4$-qubit circuit to qualitatively exhibit resonant tunneling. We
investigate the propagation of the spin excitation in the circuit. $P_{0001}$,
which is the probability of finding spin excitation on the $4$th qubit, is
shown (the solid lines) in Fig. 2(c1). The parameters are redefined as
$J_{1}\tau=\theta_{1},J_{2}\tau=\theta_{2},V_{1}\tau=\phi,V_{2}\tau=\alpha$.
With $\theta_{1}=\theta_{2}=\pi/1.5$, we plot two cases of $\alpha=\pi/4$ and
$\alpha=-\pi/1.5$. Compare Fig. 2(c4) with (a4) or (b4), we find that the
peaks are not at $\alpha$ and $-\alpha$. However, the distance between every
two peaks is changed when $\alpha$ is adjusted, which is the major
characteristic of the resonant tunneling effect.
Finally, we discuss the five-qubit system. The Hamiltonian of the $5$-site
transverse field XY chain we studied is
$\displaystyle H$ $\displaystyle=$ $\displaystyle
J_{1}\sum_{j=1,4}(\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y})/4$
(15)
$\displaystyle+J_{2}\sum_{j=2,3}(\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y})/4$
$\displaystyle+V_{1}\sigma_{1}^{z}/2+V_{2}\sigma_{2}^{z}/2-V_{2}\sigma_{4}^{z}/2+V_{1}\sigma_{5}^{z}/2.$
The quantum circuit implementing the discrete-time evolution is shown in Fig.
2(d1). In the single-particle subspace, the equivalent $5$-site tight-binding
chain is
$\displaystyle H$ $\displaystyle=$
$\displaystyle[J_{1}\left(\left|10000\right\rangle\left\langle
01000\right|+\left|00010\right\rangle\left\langle 00001\right|\right)$
$\displaystyle+J_{2}(\left|01000\right\rangle\left\langle
00100\right|+\left|00100\right\rangle\left\langle
00010\right|)+\mathrm{h.c.}]$
$\displaystyle+V_{1}\left|10000\right\rangle\left\langle
10000\right|+V_{2}\left|01000\right\rangle\left\langle 01000\right|$
$\displaystyle-V_{2}\left|00010\right\rangle\left\langle
00010\right|+V_{1}\left|00001\right\rangle\left\langle 0001\right|.$
The schematic diagram of Hamiltonian Eq. (3.2) is shown in Fig. 2(c4). We
numerically simulate the continuous-time evolution (implemented with the large
$N_{T}$). The system parameters are $J_{1}=0.1$, $t=40$ (units of $1/J_{1}$),
$J_{2}=20$, $V_{2}=10,20$. We focus on the probability $P_{5}(V_{1},t)$ (see
the transparent lines in Fig. 2(d4)) of finding particle on the $5$th site. As
we can see, $P_{5}(V_{1},t)$ will reach the maximum when $V_{1}\approx
0,\pm\sqrt{2J_{2}^{2}+V_{2}^{2}}$, and the distance of the peaks is affected
by $V_{2}$. The same resonance phenomenon can be observed in a triple-well
system (see Fig. 2(d2)). The energy levels on the left and right wells are
$V_{1}$, the middle well has three energy levels:
$\sqrt{2J_{2}^{2}+V_{2}^{2}}$, $0$, and $-\sqrt{2J_{2}^{2}+V_{2}^{2}}$. When
$V_{1}=\sqrt{J_{2}^{2}+V_{2}^{2}}$, $0$, or $-\sqrt{J_{2}^{2}+V_{2}^{2}}$, the
probability of finding particles in right well reaches the maximum. The
distances between three peaks vary with $V_{2}$.
When $N_{T}$ is small, we find that only four Trotter steps are required for
the circuit to qualitatively exhibit resonant tunneling. We investigate the
propagation of the spin excitation in the circuit shown in Fig. 2(d1). We
redefine parameters as
$J_{1}\tau=\theta_{1},J_{2}\tau=\theta_{2},V_{1}\tau=\phi,V_{2}\tau=\alpha,$
and denote the probability of finding spin excitation on the $5$th qubit as
$P_{00001}$. Figure 2(d4) exhibits $P_{00001}$ (the solid lines) as function
of $\phi$ by fixing $\alpha,\theta_{1}=\pi/3$ and $\theta_{2}=\pi/1.2$. We
compare two cases that $\alpha$ is $\pi/1.5,\pi/5$ respectively. The distance
between the peaks is changed when $\alpha$ is adjusted, which is the major
characteristic of the resonant tunneling effect.
### 3.3 CR model
Figure 3: (Color online) (a) 4-qubit circuit. The two-qubit XY gates are
replaced by controlled-$R_{x}$ gates. (b) The probability of finding the spin
excitation on the last qubit as function of $\phi$. (c) The probability of
observing the spin excitation on the last third of the qubits varies with
Trotter step $\eta$. (d) The probability distribution of spin excitation at
$\eta=10$. (c) and (d) use the same legend and drawing parameter
$\theta=\phi=\pi/2$. Figure 4: (Color online) (a) The IPRη varies with Trotter
number $\eta$, the drawing parameters are $N=15,N_{T}=80,\theta=\phi=\pi/2$.
The red and blue lines represent the ordered and disordered case respectively.
The horizontal lines are the average value of IPRη, i.e. IPRave. (b) IPRave as
function of the degree of randomness. For a fixed $R$, we have $20$ data. The
blue line is the average value of the $20$ data, and error bar is the
variance. (c) The physical quantity $P_{t}$ is plotted in disordered
configuration. (d) The probability distribution of spin excitation when
$\eta=10$. (c) and (d) share the same legend.
In this section, the propagation of the spin excitation through
controlled-$R_{x}$ gates is investigated. In the previous section, the spin
excitation is created by a NOT gate and propagated through the XY gate. Here,
we study the behavior of the spin excitation propagating through the
controlled-$R_{x}$ gates. The controlled-$R_{x}$ gate is expressed as
$U^{CR_{x}}=\left|0\right\rangle\left\langle 0\right|\otimes
I+\left|1\right\rangle\left\langle 1\right|\otimes e^{-i\theta\sigma_{x}/2},$
(17)
the matrix representation of $U^{CR_{x}}$ is shown in Fig. 1(c). Consider the
situation that a spin excitation on the first qubit passes through a
controlled-$R_{x}$ gate, we get the following equation
$U^{CR_{x}}\left|10\right\rangle=\cos\frac{\theta}{2}\left|10\right\rangle-i\sin\frac{\theta}{2}\left|11\right\rangle.$
(18)
The above equation indicates that after passing through a controlled-$R_{x}$
gate, the spin excitation becomes a two-qubit entangled state. From the view
point of propagation of the bit-flip error, this indicates that the
controlled-$R_{x}$ gate can transform a single-qubit error to a multi-qubit
error. We take the four-qubit circuit in Fig. 3(a) as a example to study the
behavior of spin excitation propagating through controlled-$R_{x}$ gates under
the discrete-time evolution with a large step size. We observed the
probability of finding the spin excitation on the $4$th qubit and denote the
probability as $P_{0001}$. In Fig. 3(b), $P_{0001}$ shows two resonant peaks,
the distance between the two resonant peaks is changed as the $\phi$ varies.
This indicates that even if the spin excitations are propagated by the
controlled-$R_{x}$ gate, when the step size is large we can qualitatively
observe the resonance phenomenon that occurs in the continuous-time limit.
## 4 Localization for large Trotter step size
In this section, we study whether the localization can be observed in the
discrete-time evolution when the Trotter step size is large. We still study
the transverse field XY model, but the parameters of a layer of $U^{Z}$ (i.e.
a layer of single-qubit $R_{z}$ gates) are random. We compare the probability
distribution of the spin excitation in different configurations of the
parameters of a layer of $R_{z}$ gates. The stronger the randomness of the
parameters, the higher the localization of the distribution of the spin
excitation, which means higher the probability of observing the spin
excitation near a specific qubit. In this study, the propagation of the bit-
flip error (i.e. an unwanted NOT gate) is similar to the transport of the spin
excitation. Therefore, the localization indicates that the bit-flip error may
be localized near a specific qubit and may not affect the measurement on
distant qubits.
The localization phenomenon is studied[52, 53, 54, 55, 56] during the
discrete-time evolution with a large step size. We begin with the Hamiltonian
in Eq. (1) with $J_{j}=J$. For convenience the parameters are defined as
$J\tau=\theta,V_{j}\tau=\phi_{j}.$ For the purpose of investigating
localization, one layer of parameters for $U^{Z}$ in Eq. (2) is
{$\phi_{j}$}$\equiv${$\phi_{1},-\phi_{2},\phi_{3},-\phi_{4},\cdots$}, where
$\phi_{j}=\phi+r_{j}$, $r_{j}\in[-R$, $R]$ is a random number. {$\phi_{j}$} is
the same for each layer of $U^{Z}$. {$\phi_{j}$} is ordered (periodic)
configuration when $R=0$ and disordered configuration when $R>0$. The inverse
participation ratio [57] (IPR) is a measure of localization and defined as
$IPR_{\eta}=\sum_{i=1}^{N}|P_{i}(\eta)|^{4},$ (19)
where
$\displaystyle P_{i}(\eta)$ $\displaystyle=$ $\displaystyle\left\langle
i\right.\left|\psi(\eta)\right\rangle,$
$\displaystyle\left|\psi(\eta)\right\rangle$ $\displaystyle=$
$\displaystyle(\prod_{j=1}^{N-1}U_{j,j+1}^{XY}(\theta_{j})\prod_{j=1}^{N}U_{j}^{Z}(\phi_{j}))^{\eta}\left|\psi(0)\right\rangle.$
(20)
$\left|\psi(\eta)\right\rangle$ represents the quantum state at the $\eta$th
Trotter step, $P_{i}(\eta)$ represents the corresponding amplitude at the
$i$th qubit. In general, IPRη varies from $1/N$ (system size) to $1$ and a
large value of IPRη means a stronger localization effect. The localization of
$\left|\psi(\eta)\right\rangle$ changes with $\eta$, thus the average IPR is
introduced to character the average level of localization during the whole
discrete-time evolution[42],
$IPR_{ave}=\frac{1}{N_{T}}\sum_{\eta=1}^{N_{T}}IPR_{\eta}.$ (21)
In Fig. 4(a), we plot $IPR_{\eta}$ (the solid lines) as function of $\eta$ for
the ordered ($R=0$) and disordered ($R=\pi/2$) configuration respectively. The
drawing parameters are $N=N_{T}=80,\theta=\phi=\pi/2$. $IPR_{\eta}$ for both
the ordered (the red lines) and disordered (the red lines) configuration show
a periodic-like behavior and is larger than $1/N$, which means
$\left|\psi(\eta)\right\rangle$ exhibits localization effect in both cases.
However, the average value (the horizontal line), i.e. $IPR_{ave}$, of the
blue line is larger than the red line, and the peaks of the blue line are
closer to $1$. This indicates stronger localization in the disordered case.
Furthermore, in Fig. 4(b), we show the $IPR_{ave}$ varying with the degree $R$
of the randomness. With the increase of disorder, the localization becomes
stronger.
In this study, the propagation of the bit-flip error (i.e. an unwanted NOT
gate) is similar to the transport of the spin excitation, thus the
localization implies that single bit-flip error propagated by disordered
{$U_{j}^{Z}(\phi_{j})$} does not affect the measurement on a distant qubit. To
illustrate this point, we propose a physical quantity
$P_{\mathrm{t}}\equiv\sum_{q_{i}=2N/3}^{N}p_{q_{i}}$ which is the average
probability of finding the spin excitation on the last third of the qubits,
where $p_{q_{i}}$ denotes the probability of finding the error on the $i$th
qubit. The smaller the $P_{\mathrm{t}}$, the shorter the distance the spin
excitation travels. As shown in Fig. 4(c), $P_{\mathrm{t}}$ is lower when
$r_{i}\neq 0$, which demonstrates that only a little probability is propagated
to the last several qubits. $P_{\mathrm{t}}$ is almost vanishing as the degree
of randomness keeps increasing. In Fig 4(d), we plot $p_{q_{i}}$ at $\eta=10$.
As we can see, more probabilities are propagated to the last few qubits for
$r_{i}=0$ and are localized at the first few qubits for $r_{i}\neq 0$. Figure
4(d) also show that the greater the degree of randomness, the stronger the
localization phenomenon. Above results indicates that the measurement is
almost unaffected by the bit-flip error on the first qubit for $r_{i}\neq 0$.
The above conclusion still holds if XY gates in the circuit are replaced by
controlled-$R_{x}$. As shown in Fig. 3(c), $P_{t}$ grows with the increasing
Trotter steps. However, $P_{\mathrm{t}}$ becomes lower when the random
perturbation is applied to a layer of $R_{z}$ gates, which means less
probabilities are propagated to the end of the circuit. With the increasing
degree of disorder, the inhibiting effect is more significant. In Fig. 3(d),
we plot the probability distribution of spin excitation when
$\eta=10,\theta=\phi=\pi/2$. These results show that, with a strong randomness
of {$\phi_{j}$}, the spin excitation will not be propagated to the farther
qubits. At the same time, this also shows that the randomness of the
parameters will inhibit the propagation of the bit-flip error and protect the
measurement on the distant qubits is not affected.
## 5 Conclusions
We study the transport of the spin excitation in the discrete-time evolution
using Trotter circuits with a large step size, and qualitatively observe the
quantum phenomena in the continuous-time limit, i.e. the resonant tunneling
and localization. We observe the resonance phenomenon during the transport of
the spin excitation in systems of sizes $N=2,3,4,5$ respectively. The
probability distribution of spin excitations propagating through several
Trotter steps agree qualitatively with that in the continuous-time limit. The
corresponding minimum number of Trotter steps is given for each system size.
In a Trotter circuit with random parameters of $R_{z}$ gates, we can observe
the localization phenomenon of the spin excitation distribution even with a
large step size. We study the spin excitation propagating through the XY gates
and also through the controlled-$R_{x}$ gates. Our research indicates that a
discrete-time quantum simulator with a large step size can qualitatively
demonstrate some physical phenomena in the continuous-time limit. Qualitative
observations require fewer quantum gates than quantitative calculations and
therefore are a promising application on near-term quantum computers. In
quantum computing, some errors, such as bit-flip errors, behave like spin
excitations, thus, our finding can be used to understand the propagating of
these errors in the quantum circuits.
I am grateful to Ying Li for the discussions and help in preparing the
manuscript. This work is supported by the National Natural Science Foundation
of China (Grants No. 11875050 and No. 12088101), NSAF (Grant No. U1930403),
and the Special Fund for Theoretical Physics of the National Natural Science
Foundation of China (Grant No. 12047547).
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|
# T-Zamfirescu and T-weak contraction mappings on cone metric spaces
José R. Morales and Edixon Rojas Department of Mathematics, Faculty of
Science, University of Los Andes, Mérida-5101, Venezuela<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.
The purpose of this paper is to obtain sufficient conditions for the existence
of a unique fixed point of T-Zamfirescu and T-weak contraction mappings in the
framework of complete cone metric spaces.
###### Key words and phrases:
Fixed point, complete cone metric space, $T-$zamficescu mapping, $T-$weak
contraction, subsequentially convergent.
###### 1991 Mathematics Subject Classification:
47H10, 46J10.
## 1\. Introduction
In 2007, Guang and Xiang [11] generalized the concept of metric space,
replacing the set of real numbers by an ordered Banach space and defined a
cone metric space. The authors there described the convergence of sequences in
cone metric spaces and introduced the completeness. Also, they proved some
fixed point theorems of contractive mappings on complete cone metric spaces.
Since then, fixed point theorems for different (classic) classes of mappings
on these spaces have been appeared, see for instance [1], [7], [8], [10],
[15], [16] and [17].
On the other hand, recently A. Beiranvand S. Moradi, M. Omid and H. Pazandeh
[5] introduced the $T-$contraction and $T-$contractive mappings and then they
extended the Banach contraction principle and the Edelstein’s fixed point
Theorem. S. Moradi [12] introduced the $T-$Kannan contractive mappings,
extending in this way the Kannan’s fixed point theorem [9]. The corresponding
version of $T$-contractive, $T$-Kannan mappings and $T-$Chalterjea
contractions on cone metric spaces was studied in [13] and [14] respectively.
In view of these facts, thereby the purpose of this paper is to study the
existence of fixed points of $T-$Zamficescu and $T-$weak contraction mappings
defined on a complete cone metric space $(M,d)$, generalizing consequently the
results given in [11] and [18].
## 2\. General framework
In this section we recall the definition of cone metric space and some of
their properties (see, [11]). The following notions will be useful for us in
order to prove the main results.
###### Definition 2.1.
Let $E$ be a real Banach space. A subset $P$ of $E$ is called a cone if and
only if:
(P1):
$P$ is closed, nonempty and $P\neq\\{0\\}$;
(P2):
$a,b\in\mathbb{R},\,\,a,b\geq 0,\,\,x,y\in P$ imply $ax+by\in P$;
(P3):
$x\in P$ and $-x\in P\Rightarrow x=0$. I.e., $P\cap(-P)=\\{0\\}$.
Given a cone $P\subset E,$ we define a partial ordering $\leq$ with respect to
$P$ by $x\leq y$ if and only if $y-x\in P$. We write $x<y$ to indicate that
$x\leq y$ but $x\neq y$, while $x\ll y$ will stand for
$y-x\in\operatorname{Int}P$. (interior of $P$.)
###### Definition 2.2.
Let $E$ be a Banach space and $P\subset E$ a cone. The cone $P$ is called
normal if there is a number $K>0$ such that for all $x,y\in E,\,\,0\leq x\leq
y$ implies $\|x\|\leq K\|y\|.$ The least positive number satisfying the above
is called the normal constant of $P.$
In the following, we always suppose that $E$ is a Banach space, $P$ is a cone
in $E$ with $\operatorname{Int}P\neq\emptyset$ and $\leq$ is partial ordering
with respect to $P$.
###### Definition 2.3 ([11]).
Let $M$ be a nonempty set. Suppose that the mapping $d:M\times
M\longrightarrow E$ satisfies:
(d1):
$0<d(x,y)$ for all $x,y\in M$, and $d(x,y)=0$ if and only if $x=y$;
(d2):
$d(x,y)=d(y,x)$ for all $x,y\in M$;
(d3):
$d(x,y)\leq d(x,z)+d(z,y)$ for all $x,y,z\in M$.
Then, $d$ is called a cone metric on $M$ and $(M,d)$ is called a cone metric
space.
Note that the notion of cone metric space is more general that the concept of
metric space.
###### Definition 2.4.
Let $(M,d)$ be a cone metric space. Let $(x_{n})$ be a sequence in $M$ and
$x\in M$.
* (i)
$(x_{n})$ converges to $x$ if for every $c\in E$ with $0\ll c$ there is an
$n_{0}$ such that for all $n>n_{0},\,\,d(x_{n},x)\ll c.$ We denote this by
$\displaystyle\lim_{n\rightarrow\infty}x_{n}=x$ or $x_{n}\rightarrow
x,\,\,(n\rightarrow\infty)$.
* (ii)
If for any $c\in E$ with $0\ll c$ there is an $n_{0}$ such that for all
$n,m\geq n_{0}$, $\;d(x_{n},x_{m})\ll c$, then $(x_{n})$ is called a Cauchy
sequence in $M$.
Let $(M,d)$ be a cone metric space. If every Cauchy sequence is convergent in
$M,$ then $M$ is called a complete cone metric space.
###### Lemma 2.1 ([11]).
Let $(M,d)$ be a cone metric space, $P\subset E$ a normal cone with normal
constant $K.$ Let $(x_{n}),\,\,(y_{n})$ be sequences in $M$ and $x,y\in M$.
* (i)
$(x_{n})$ converges to $x$ if and only if
$\displaystyle\lim_{n\rightarrow\infty}d(x_{n},x)=0$.
* (ii)
If $(x_{n})$ converges to $x$ and $(x_{n})$ converges to $y$, then $x=y$.
* (iii)
If $(x_{n})$ converges to $x$, then $(x_{n})$ is a Cauchy sequence.
* (iv)
$(x_{n})$ is a Cauchy sequence if and only if
$\displaystyle\lim_{n,m\rightarrow\infty}d(x_{n},x_{m})=0$.
* (v)
If $x_{n}\longrightarrow x$ and $y_{n}\longrightarrow
y,\,\,(n\rightarrow\infty)$, then $d(x_{n},y_{n})\longrightarrow d(x,y)$.
###### Definition 2.5.
Let $(M,d)$ be a cone metric space, $P$ a normal cone with normal constant $K$
and $T:M\longrightarrow M$. Then
* (i)
$T$ is said to be continuous, if
$\displaystyle\lim_{n\rightarrow\infty}x_{n}=x$ implies that
$\displaystyle\lim_{n\rightarrow\infty}T(x_{n})=T(x)$ for all $(x_{n})$ and
$x$ in $M$.
* (ii)
$T$ is said to be subsequentially convergent if we have, for every sequence
$(y_{n}),$ if $T(y_{n})$ is convergent, then $(y_{n})$ has a convergent
subsequence.
* (iii)
$T$ is said to be sequentially convergent if we have, for every sequence
$(y_{n}),$ if $T(y_{n})$ is convergent then $(y_{n})$ also is convergent.
Examples of cone metric spaces can be found for instance in [11], [17] and
references therein.
## 3\. Main Results
This section is devoted to give fixed point results for $T$-Zamfirescu and
$T$-weak contraction mappings on complete (normal) cone metric spaces, as well
as, their asymptotic behavior. First, we recall the following classes of
contraction type mappings:
###### Definition 3.1.
Let $(M,d)$ be a cone metric space and $T,S:M\longrightarrow M$ two mappings
* (i)
The mapping $S$ is called a $T-$Banach contraction, (TB - Contraction) if
there is $a\in[0,1)$ such that
$d(TSx,TSy)\leq ad(Tx,Ty)$
for all $x,y\in M$.
* (ii)
The mapping $S$ is called a $T-$Kannan contraction, (TK - Contraction) if
there is $b\in[0,1/2)$ such that
$d(TSx,TSy)\leq b[d(Tx,TSx)+d(y,TSy)]$
for all $x,y\in M$.
* (iii)
A mapping $S$ is said to be a Chatterjea contraction, (TC - Contraction) if
there is $c\in[0,1/2)$ such that
$d(TSx,TSy)\leq c[d(Tx,TSy)+d(Ty,TSx)]$
for all $x,y\in M.$
It is clear that if we take $T=I_{d}$ (the identity map) in the Definition 3.1
we obtain the definitions of Banach contraction, Kannan mapping ([9]) and
Chatterjea mapping ([6]).
Now, following the ideas of T. Zamfirescu [18] we introduce the notion of
$T-$Zamfirescu mappings.
###### Definition 3.2.
Let $(M,d)$ be a cone metric space and $T,S:M\longrightarrow M$ two mappings.
$S$ is called a $T-$Zamfirescu mapping, (TZ -mapping), if and only if, there
are real numbers, $0\leq a<1,\,\,0\leq b,c<1/2$ such that for all $x,y\in M,$
at least one of the next conditions are true:
($TZ_{1}$):
$d(TSx,TSy)\leq ad(Tx,Ty)$.
($TZ_{2}$):
$d(TSx,TSy)\leq b[d(Tx,TSx)+d(Ty,TSy)]$.
($TZ_{3}$):
$d(TSx,TSy)\leq c[d(Tx,TSy)+d(Ty,TSx)]$.
If in Definition 3.2 we take $T=I_{d}$ and $E=\mathbb{R}_{+}$ we obtain the
definition of T. Zamfirescu [18].
###### Lemma 3.1.
Let $(M,d)$ be a cone metric space and $T,S:M\longrightarrow M$ two mappings.
If $S$ is a $TZ-$mapping, then there is $0\leq\delta<1$ such that
(3.1) $d(TSx,TSy)\leq\delta d(Tx,Ty)+2\delta d(Tx,TSx)$
for all $x,y\in M$.
###### Proof.
If $S$ is a $TZ-$mapping, then at least one of $(TZ_{1})$, $(TZ_{2})$ o
$(TZ_{3})$ condition is true.
If $(TZ_{2})$ holds, then:
$\begin{array}[]{ccl}d(TSx,TSy)&\leq&b[d(Tx,TSx)+d(Ty,TSy)]\\\ \\\
&\leq&b[d(Tx,TSx)+d(Ty,Tx)+d(Tx,TSx)+d(TSx,TSy)]\end{array}$
thus,
$(1-b)d(TSx,TSy)\leq bd(Tx,Ty)+2bd(Tx,TSx).$
From the fact that $0\leq b<1/2$ we get:
$d(TSx,TSy)\leq\displaystyle\frac{b}{1-b}d(Tx,Ty)+\displaystyle\frac{2b}{1-b}d(Tx,TSx).$
with $\frac{b}{1-b}<1$. If $(TZ_{3})$ holds, then similarly we get
$d(TSx,TSy)\leq\displaystyle\frac{c}{1-c}d(Tx,Ty)+\displaystyle\frac{2c}{1-c}d(Tx,TSx).$
Therefore, denoting by
$\delta:=\max\left\\{a,\,\displaystyle\frac{b}{1-b},\,\displaystyle\frac{c}{1-c}\right\\}$
we have that $0\leq\delta<1$. Hence, for all $x,y\in M,$ the following
inequality holds:
$d(TSx,TSy)\leq\delta d(Tx,Ty)+2\delta d(Tx,TSx).$
∎
###### Remark 1.
Notice that inequality (3.1) in Lemma 3.1 can be replace by
$d(TSx,TSy)\leq\delta d(Tx,Ty)+2\delta d(Tx,TSy)$
for all $x,y\in M$.
###### Theorem 3.2.
Let $(M,d)$ be a complete cone metric space, $P$ be a normal cone with normal
constant $K$. Moreover, let $T:M\longrightarrow M$ be a continuous and one to
one mapping and $S:M\longrightarrow M$ a $T-$Zamfirescu continuous mapping.
Then
* (i)
For every $x_{0}\in M$,
$\displaystyle\lim_{n\rightarrow\infty}d(TS^{n}x_{0},TS^{n+1}x_{0})=0.$
* (ii)
There is $y_{0}\in M$ such that
$\displaystyle\lim_{n\rightarrow\infty}TS^{n}x_{0}=y_{0}.$
* (iii)
If $T$ is subsequentially convergent, then $(S^{n}x_{0})$ has a convergent
subsequence.
* (iv)
There is a unique $z_{0}\in M$ such that $Sz_{0}=z_{0}$.
* (v)
If $T$ is sequentially convergent, then for each $x_{0}\in M$ the iterate
sequence $(S^{n}x_{0})$ converges to $z_{0}$.
###### Proof.
* (i)
Since $S$ is a $T-$Zamfirescu mapping, then by Lemma 3.1, there exists
$0<\delta<1$ such that
$d(TSx,TSy)\leq\delta d(Tx,Ty)+2\delta d(Tx,TSx)$
for all $x,y\in M$.
Suppose $x_{0}\in M$ is an arbitrary point and the Picard iteration associated
to $S,$ $\;(x_{n})$ is defined by
$x_{n+1}=Sx_{n}=S^{n}x_{0},\qquad n=0,1,2,\ldots.$
Thus,
$d(TS^{n+1}x_{0},TS^{n}x_{0})\leq hd(TS^{n}x_{0},TS^{n-1}x_{0})$
where $h=\displaystyle\frac{\delta}{1-2\delta}<1$. Therefore, for all $n$ we
have
$d(TS^{n+1}x_{0},TS^{n}x_{0})\leq h^{n}d(TSx_{0},Tx_{0}).$
From the above, and the fact the cone $P$ is a normal cone we obtain that
$\|d(TS^{n+1}x_{0},TS^{n}x_{0})\|\leq Kh^{n}\|d(TSx_{0},Tx_{0})\|,$
taking limit $n\longrightarrow\infty$ in the above inequality we can conclude
that
$\displaystyle\lim_{n\rightarrow\infty}d(TS^{n+1}x_{0},TS^{n}x_{0})=0.$
* (ii)
Now, for $m,n\in\mathbb{N}$ with $m>n$ we get
$\begin{array}[]{ccl}d(TS^{m}x_{0},TS^{n}x_{0})&\leq&(h^{n}+\ldots+h^{m-1})d(TSx_{0},Tx_{0})\\\
\\\ &\leq&\displaystyle\frac{h^{n}}{1-h}d(TSx_{0},Tx_{0}).\end{array}$
Again; as above, since $P$ is a normal cone we obtain
$\displaystyle\lim_{n,m\rightarrow\infty}d(TS^{m}x_{0},TS^{n}x_{0})=0.$
Hence, the fact that $(M,d)$ is a complete cone metric space, imply that
$(TS^{n}x_{0})$ is a Cauchy sequence in $M$, therefore there is $y_{0}\in M$
such that
$\displaystyle\lim_{n\rightarrow\infty}TS^{n}x_{0}=y_{0}.$
* (iii)
If $T$ is subsequentially convergent, $(S^{n}x_{0})$ has a convergent
subsequence, so there is $z_{0}\in M$ and $(n_{k})_{k=1}^{\infty}$ such that
$\displaystyle\lim_{k\rightarrow\infty}S^{n_{k}}x_{0}=z_{0}.$
* (iv)
Since $T$ and $S$ are continuous mappings we obtain:
$\displaystyle\lim_{k\rightarrow\infty}TS^{n_{k}}x_{0}=Tz_{0},\qquad\displaystyle\lim_{k\rightarrow\infty}TS^{n_{k}+1}x_{0}=TSz_{0}$
therefore, $Tz_{0}=y_{0}=TSz_{0},$ and since $T$ is one to one, then
$Sz_{0}=z_{0}.$ So $S$ has a fixed point.
Now, suppose that $Sz_{0}=z_{0}$ and $Sz_{1}=z_{1}$.
$\begin{array}[]{ccl}d(TSz_{0},TSz_{1})&\leq&\delta d(Tz_{0},Tz_{1})+2\delta
d(Tz_{0},TSz_{0})\\\ \\\ d(Tz_{0},Tz_{1})&\leq&\delta
d(Tz_{0},Tz_{1})\end{array}$
from the fact that $0\leq\delta<1$ and that $T$ is one to one we obtain that
$z_{0}=z_{1}$.
* (v)
It is clear that if $T$ is sequentially convergent, then for each $x_{0}\in
M$, the iterate sequence $(S^{n}x_{0})$ converges to $z_{0}$.
∎
In 2003, V. Berinde (see, [2], [3]) introduced a new class of contraction
mappings on metric spaces, which are called weak contractions. We will extend
these kind of mappings by introducing a new function $T$ and we define it in
the framework of cone metric spaces.
###### Definition 3.3.
Let $(M,d)$ be a cone metric space and $T,S:M\longrightarrow M$ two mappings.
$S$ is called a $T-$weak contraction, (TW- Contraction,
$T_{(S,L)}-$Contraction), if there exist a constant $\delta\in(0,1)$ and some
$L\geq 0$ such that
$d(TSx,TSy)\leq\delta d(Tx,Ty)+Ld(Ty,TSx)$
for all $x,y\in M$.
It is clear that if we take $T=I_{d}$ and $E=\mathbb{R}_{+}$ then we obtain
the notion of Berinde [2].
Due to the symmetry of the metric, the $T-$weak contractive condition
implicitly include the following dual one:
$d(TSx,TSy)\leq\delta d(Tx,Ty)+Ld(Tx,TSy)$
for all $x,y\in M$.
The next proposition gives examples of $T-$weak contraction and it proof is
similar to the proof of Lemma 3.1.
###### Proposition 3.3.
Let $(M,d)$ be a cone metric space and $T,S:M\longrightarrow M$ two mappings.
* (i)
If $S$ is a TB - contraction, then $S$ is a $T-$weak contraction.
* (ii)
If $S$ is a TK - contraction, then $S$ is a $T-$weak contraction.
* (iii)
If $S$ is a TC - contraction, then $S$ is a $T-$weak contraction.
* (iv)
If $S$ is TZ - mapping, then $S$ is a $T-$weak contraction.
Now we have the following result:
###### Theorem 3.4.
Let $(M,d)$ be a complete cone metric space, $P$ a normal cone with normal
constant $K$. Let furthermore $T:M\longrightarrow M$ a continuous and one to
one mapping and $S:M\longrightarrow M$ a continuous $T-$weak contraction. Then
* (i)
For every $x_{0}\in M$,
$\displaystyle\lim_{n\rightarrow\infty}d(TS^{n}x_{0},TS^{n+1}x_{0})=0.$
* (ii)
There is $y_{0}\in M$ such that
$\displaystyle\lim_{n\rightarrow\infty}TS^{n}x_{0}=y_{0}.$
* (iii)
If $T$ is subsequentially convergent, then $(S^{n}x_{0})$ has a convergent
subsequence.
* (iv)
There is $z_{0}\in M$ such that
$Sz_{0}=z_{0}.$
* (v)
If $T$ is sequentially convergent, then for each $x_{0}\in M$ the iterate
sequence $(S^{n}x_{0})$ converges to $z_{0}$.
###### Proof.
Similar to the proof of Theorem 3.2. ∎
As we see in Theorem 3.2, a $T-$Zamfirescu mapping has a unique fixed point.
The next example shows that a $T-$weak contraction may has infinitely fixed
points.
###### Example 1 ([4]).
Let $M=[0,1]$ be the unit interval with the usual metric and
$T,S:M\longrightarrow M$ the identity maps, that is, $Tx=Sx=x$ for all $x\in
M$. Then, taking $0\leq a<1$ and $L\geq 1-a$ we obtain
$\begin{array}[]{ccl}d(TSx,TSy)&=&|TSx-TSy|\\\ \\\
|x-y|&\leq&a|x-y|+L|y-x|\end{array}$
which is valid for all $x,y\in[0,1]$. Thus the set of the fixed points $F_{S}$
of the map $S$ is the interval $[0,1]$. I.e.,
$F_{S}=\\{x\in[0,1]\,/\,Sx=x\\}=[0,1].$
It is possible to force the uniqueness of the fixed point of a $T-$weak
contraction by imposing an additional contractive condition, as is shown in
the next theorem.
###### Theorem 3.5.
Let $(M,d)$ be a complete cone metric space, $P$ be a normal cone with normal
constant $K$. Let furthermore $T:M\longrightarrow M$ a continuous and one to
one mapping and $S:M\longrightarrow M$ a $T-$weak contraction for which there
is $\theta\in(0,1)$ and some $L_{1}\geq 0$ such that
$d(TSx,TSy)\leq\theta d(Tx,Ty)+L_{1}d(Tx,TSx)$
for all $x,y\in M$. Then:
* (i)
For every $x_{0}\in M$
$\displaystyle\lim_{n\rightarrow\infty}d(TS^{n}x_{0},TS^{n+1}x_{0})=0.$
* (ii)
There is $y_{0}\in M$ such that
$\displaystyle\lim_{n\rightarrow\infty}TS^{n}x_{0}=y_{0}.$
* (iii)
It $T$ is subsequentially convergent, then $(S^{n}x_{0})$ has a convergent
subsequence.
* (iv)
There is a unique $z_{0}\in M$ such that
$Sz_{0}=z_{0}.$
* (v)
If $T$ is sequentially convergent, then for each $x_{0}\in M$ the iterate
sequence $(S^{n}x_{0})$ converges to $z_{0}.$
###### Proof.
Assume $S$ has two distinct fixed points $x^{*},y^{*}\in M.$ Then
$d(Tx^{*},Ty^{*})=d(TSx^{*},TSy^{*})\leq\theta
d(Tx^{*},Ty^{*})+L_{1}d(Tx^{*},TSx^{*})$
thus, we get
$d(Tx^{*},Ty^{*})\leq\theta
d(Tx^{*},Ty^{*})\Leftrightarrow(1-\theta)d(Tx^{*},Ty^{*})\leq 0.$
Therefore, $d(Tx^{*},Ty^{*})=0$. Since $T$ is one to one, then $x^{*}=y^{*}$.
The rest of the proof follows as the the proof of Theorem 3.2. ∎
## References
* [1] M. Abbas and B.E. Rhoades, Fixed and periodic results in cone metric space, Appl. Math. Lett., 22, (4), (2009), 511–515.
* [2] V. Berinde, Iterate Approximation of fixed points, lect. Notes Math., Vol 1912, (2nd ed.), Springer Verlag, Berlin, 2007.
* [3] V. Berinde, On the approximation of fixed points of weak contractive mappings, Carpathian J. Math, 19, (1), (2003), 7–22.
* [4] V. Berinde, On the convergence of the Ishikawa Iteration in the class of quasi contractive operators, Acta Math. Univ. Comenianae, 73, (1), (2004), 119–126.
* [5] A. Beiranvand, S. Moradi, M. Omid and H. Pazandeh, Two fixed point theorem for special mapping, arXiv:0903.1504v1 [math.FA].
* [6] S. K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci., 25, (1972), 727–730.
* [7] D. Ilić and V. Rakočević, Quasi-contraction on a cone metric space, Appl. Math. Lett., 22, (5), (2009), 728–731.
* [8] Z. Kadelburg, S. Radenović and V. Rakočević, Remarks on “Quasi-contraction on a cone metric space”, Appl. Math. Lett., (2009), doi:10.1016/j.aml.2009.06.003.
* [9] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60, (1968), 71–76.
* [10] M.S. Khan and M. Samanipour, Fixed point theorems for some discontinuous operators in cone metric space, Mathematica Moravica, Vol 12-2, (2008), 29–34.
* [11] Huan Long - Guang and Zhan Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332, (2007), 1468–1476.
* [12] S. Moradi, Kannan fixed point theorem on complete metric spaces and on generalized metric spaces depended on another function, arXiv:0903.1577v1 [math.FA].
* [13] J. Morales and E. Rojas, Cone metric spaces and fixed point theorems of $T-$contractive mappings, preprint, 2009.
* [14] J. Morales and E. Rojas, Cone metric spaces and fixed point theorems of $T-$Kannan contractive mappings, arXiv:0907.3949v1 [math.FA].
* [15] H.K. Pathak and N. Shahzad, Fixed points results for generalized quasicontraction mappings in abstract metric spaces, Nonlinear Analysis, (2009), doi:10.1016/j.na.2009.05.052.
* [16] V. Raja and S.M. Vaezpour, Some extension of Banach’s contraction principle in complete cone metric spaces, Fixed Point Theory and Applications, (2008), 11 p.
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|
# Artificial Concepts of Artificial Intelligence: Institutional Compliance and
Resistance in AI Startups
Amy A. Winecoff<EMAIL_ADDRESS>Princeton UniversitySherrerd
HallPrincetonNew JerseyUSA08544 and Elizabeth Anne Watkins
<EMAIL_ADDRESS>Princeton UniversitySherrerd HallPrincetonNew
JerseyUSA08544
(2022)
###### Abstract.
Scholars and industry practitioners have debated how to best develop
interventions for ethical artificial intelligence (AI). Such interventions
recommend that companies building and using AI tools change their technical
practices, but fail to wrangle with critical questions about the
organizational and institutional context in which AI is developed. In this
paper, we contribute descriptive research around the life of ”AI” as a
discursive concept and organizational practice in an understudied
sphere–emerging AI startups–and with a focus on extra-organizational pressures
faced by entrepreneurs. Leveraging a theoretical lens for how organizations
change, we conducted semi-structured interviews with 23 entrepreneurs working
at early-stage AI startups. We find that actors within startups both conform
to and resist institutional pressures. Our analysis identifies a central
tension for AI entrepreneurs: they often valued scientific integrity and
methodological rigor; however, influential external stakeholders either lacked
the technical knowledge to appreciate entrepreneurs’ emphasis on rigor or were
more focused on business priorities. As a result, entrepreneurs adopted hyped
marketing messages about AI that diverged from their scientific values, but
attempted to preserve their legitimacy internally. Institutional pressures and
organizational constraints also influenced entrepreneurs’ modeling practices
and their response to actual or impending regulation. We conclude with a
discussion for how such pressures could be used as leverage for effective
interventions towards building ethical AI.
organizational theory, artificial intelligence, industry practice, qualitative
methods, ethical systems
††journalyear: 2022††copyright: rightsretained††conference: Proceedings of the
2022 AAAI/ACM Conference on AI, Ethics, and Society; August 1–3, 2022; Oxford,
United Kingdom††booktitle: Proceedings of the 2022 AAAI/ACM Conference on AI,
Ethics, and Society (AIES’22), August 1–3, 2022, Oxford, United Kingdom††doi:
10.1145/3514094.3534138††isbn: 978-1-4503-9247-1/22/08††ccs: Social and
professional topics Socio-technical systems††ccs: Applied computing Sociology
## 1\. Introduction
Academic researchers, advocacy groups, and technology companies have created
guidelines and tools for developing ethical artificial intelligence (AI)
(Morley et al., 2020; Fish and Stark, 2021). This research is intended to
ameliorate the considerable negative social impacts produced in AI systems,
such as how AI models encode racial and gender biases (Caliskan et al., 2017;
Noble, 2018; Sweeney, 2013; Rekabsaz and Schedl, 2020; Bolukbasi et al.,
2016), worsen disordered eating and body dysmorphia (Karizat et al., 2021),
and magnify inequality (Obermeyer et al., 2019; Dastin, 2018; Angwin et al.,
2016). However, the real world-utility of available interventions for ethical
AI remains unclear.
As with any research intended for real-world applications, robust
consideration of the context of implementation is critical. An emerging body
of research has begun to recognize that effective change demands non-technical
strategies to contend with organizational context (Rakova et al., 2021), such
as the conditions inside technology firms which might influence, or even
prevent, the effectiveness of interventions for ethical AI. Tight development
timelines, lack of formal organizational processes, and challenging internal
stakeholder dynamics shape how real-world companies can move in the direction
of more ethical AI development (Holstein et al., 2019; Madaio et al., 2020,
2021; Rakova et al., 2021; Hopkins and Booth, 2021). Studies of AI ethics in
organizational contexts often focus on interventions such as model fairness
(Holstein et al., 2019; Madaio et al., 2020, 2021) and model interpretability
techniques (Bhatt et al., 2020; Hong et al., 2020; Kaur et al., 2020).
However, such studies have largely been constrained to organizations that are
mature enough to consider specific AI ethical interventions to begin with.
AI startups constitute a growing portion of the technology sector (Tricot,
2021). As a result, these companies and the ethical practices they embrace are
likely to play a significant role in the impact of future technology on
society. Only a handful of studies have characterized ethical AI development
at smaller firms (Vakkuri et al., 2020; Hopkins and Booth, 2021). This
emerging area of research has begun to illuminate the unique challenges
nascent firms must address when attempting to adopt responsible, transparent,
and accountable AI practices. For example, as with more mature companies,
small firms must navigate complex dynamics amongst stakeholders like clients,
investors, and regulators but unlike more established organizations, they must
do so under significant resource constraints that threaten their very
existence (Hopkins and Booth, 2021). Therefore, the ethical AI practices they
are able to adopt are necessarily limited.
While existing research has illustrated the organizational constraints to
ethical AI, especially intra-organizational dynamics, less is known about how
the inter-organizational or field-level dynamics shape firms’ capacity to
develop ethical approaches. The field-level, i.e., ”institutional” dynamics
and market-based pressures that impact an organization’s chances of survival
inevitably alter the structures and practices firms adopt (Meyer and Rowan,
1977; Salancik and Pfeffer, 1978; DiMaggio and Powell, 1983; Oliver, 1991). By
behaving in ways that conform with institutional expectations, emerging
organizations can improve their social and cultural fitness; however,
institutional expectations sometimes conflict with each other and also with
economic pressures. As a result, emerging organizations such as AI startups
must skillfully navigate a complex gauntlet of social, cultural, and economic
challenges. How these field-level dynamics factor into the ethical choices of
startups, such as their decisions around the use of AI, is an under-explored
area of research.
Here, we contend that before effective ethical AI practices for startups can
be developed, an understanding of the inter-organizational and institutional
dynamics these firms face must be developed. Building on recent scholarship
that takes a contextual and organizational approach to ethical AI, we engage
in descriptive research around the life of ”AI” as a discursive concept and
organizational practice that is situated within an institutional context.
Instead of focusing on ethical practices directly, we take a step back to ask
fundamental questions about the forces that shape the very nature of how
entrepreneurs define, build, and talk about AI itself.
To that end, we ask two research questions:
RQ1: What institutional pressures influence how startup entrepreneurs define,
discuss, and build AI?
RQ2: When do entrepreneurs comply with, avoid, or resist these pressures?
To address these questions, we conducted semi-structured interviews with 23
individuals working at early-stage startups across a range of industry
domains. In our interviews, we focused on the financial, regulatory, and
normative pressures AI startups encounter. Using abductive analysis, we
illustrate how AI entrepreneurs both comply with and resist institutional
pressures through the technological and business practices they employ. We
find that AI entrepreneurs’ face a tension between the expectations of
technology entrepreneurship, which rewards rapid development and optimistic
promises about technology’s potential, and entrepreneurs’ own values of
scientific integrity, which prioritize meticulous practices and encourages
skepticism. This tension was further heightened by external stakeholders’
unrealistic expectations about the potential of AI, particularly when such
stakeholders had limited technical knowledge. We also find that whereas AI
entrepreneurs saw privacy regulation as beneficial and aligned with their own
values of autonomy, they held less uniformly positive views of other AI
regulatory processes such as those employed by the Food and Drug
Administration (FDA) in approving AI medical devices. Drawing from our
theoretical motivations, we conclude with a discussion of how our results
point to both constraints and opportunities for future research on ethical
interventions for AI startups.
## 2\. Related Works
Organizational dynamics are a significant source of influence on the
effectiveness of interventions for ethical and responsible AI. Practitioners
in well-resourced organizations have expressed aspirations for ethics-
supportive structures as such cross-team integration, risk-anticipation
frameworks, and firm-level mission and values (Rakova et al., 2021). Our own
research builds on these recent findings by addressing inter-organizational
and institutional conditions (i.e., external pressures) that are likely to be
sources of change for organizations, especially emerging organizations such as
AI startups, which typically lack formal mechanisms for addressing ethical
concerns (Vakkuri et al., 2020; Hopkins and Booth, 2021). In this section we
provide an overview of the relevant frameworks we draw from in organizational
theory to explain how organizations adopt procedures and adapt over time to
institutional pressures.
### 2.1. Resource Dependency and Institutionalism: How Organizations Change
Organizational theory offers different frameworks to understand how
organizations change. In the early days of the discipline, the dominant
paradigm was of rationality: theorists described organizations as rational
systems, machines for achieving a goal in the market, and that all
organizational decisions were imbued with this same mechanical, systematic
precision. Within this school were early 1900s thinkers like the German
sociologist Max Weber, with his focus on bureaucracies as structural
realizations of rational authority (Weber, 1978), and American mechanical
engineer Frederick Winslow Taylor, with his focus on bringing ”scientific”
methods to management to wring ever-greater ”efficiency” out of a labor force
(Taylor, 1919).
Starting in the 1970s, however, the field took a relational turn, recognizing
that organizations do not operate in a vacuum of rationality but instead
within complex ecologies of other actors. Two schools of thought–resource
dependency theory and institutionalism–both address how firms seek to mitigate
external pressures and uncertainty within their organizational ecosystems.
Resource dependency theorists focus on organizations as their unit of
analysis, i.e. the ”meso” or ”middle” level of institutional change (bigger
than ”micro”, or individual people, but not as large as ”macro”, or field-
level norms or systems). They examine the interactions between these units,
using this perspective to analyze how organizations strategically seek to
manage resources and mitigate dependencies on their exchange partners
(Salancik and Pfeffer, 1978). In doing so, organizations improve their fitness
within the market. Resource dependency theory has been recently used to
analyze the precarity of firms operating within complex supply chains, as they
”require networks to accommodate the interdependencies in product and service
flows, resource flows, and information flows” (Olan et al., 2022).
The theory of institutionalism looks at the ”macro” level, foregrounding
patterns taking place at the level of entire organizational fields or social
orders. Institutionalism focuses on unconscious social and cultural
expectations, contending that these influences lead to widespread changes in
multiple organizations, shaping fields of industry (DiMaggio and Powell, 1983;
Meyer and Rowan, 1977; Zucker, 1977; Van Wijk et al., 2019). A critical
component of institutional theory examines how new organizations establish
legitimacy, where their actions are perceived as ”desirable, proper, or
appropriate within some socially constructed system of norms, values, beliefs,
and definitions” (Suchman, 1995). Amidst technological and market uncertainty,
new firms improve their odds of survival by accruing legitimacy from an
audience of stakeholders in the field including funding entities, regulatory
bodies, and competitor companies. In their pursuit of legitimacy,
organizations change over time, increasingly reflecting the established norms
and practices of the field (DiMaggio and Powell, 1983). Recent scholarship has
used this lens to examine organizational changes around the implementation of
novel technologies and practices such as how news publishers choose to
implement novel cybersecurity tools (Watkins and Anderson, 2019) and how AI
governance and accountability (i.e. algorithmic impact assessment) may become
more widespread (Selbst, 2021).
Both frameworks have limitations in their explanatory power; resource
dependency theory under-emphasizes sociocultural forces and institutionalism
under-emphasizes organizations’ instrumental actions (Scott, 2013). Oliver
(Oliver, 1991) argues that neither institutionalism nor resource dependency
can adequately capture the complexity of organizational action and evolution
within an institutional context. She theorizes that in addition to exercising
agency in response to market demands, organizational actors can also respond
strategically to institutional pressures through a variety of compliance and
resistance tactics. Oliver details five core strategies organizations adopt in
response to institutional pressures varying from passive compliance to active
resistance. On the compliance end of the spectrum, she places ”acquiescence”
in which organizations conform to institutional expectations. She describes
”avoidance” and ”compromise” as partial compliance strategies in which
organizations, for example, attempt to disguise their non-conformity with
institutional norms or attempt to balance the sometimes conflicting pressures
of institutional stakeholders. On the resistance end of the spectrum, she
places ”defiance,” in which organizations actively and openly reject
institutional norms, and ”manipulation,” in which organizations attempt to co-
opt, supplant, or control institutional pressures.
Oliver also theorizes organizations’ likelihood of engaging in compliance or
resistance strategies will depend on a variety of institutional conditions.
The greater the perceived benefit of institutional conformity, the likelier
organizations are to comply with such pressures. On the other hand, competing
stakeholder expectations, multiple conflicting institutional norms, or
conflicting organizational goals and institutional pressures are likelier to
engender resistance. Moreover, when organizations are coerced into conformity
via legal means, they are likelier to resist as compared to when institutional
conformity is effected through diffuse institutional norms that are adopted
voluntarily by organizations. Through the exertion of skillful agency,
organizations can both gain legitimacy through selective compliance with
institutional norms, while also maintaining the practices they adopt in the
service of market fitness (Oliver, 1991; Fligstein, 1997; Suddaby and
Greenwood, 2005).
In our current work, we draw on Oliver’s framework (Oliver, 1991) to
conceptualize how AI startups attempt to manage institutional and market
pressures through a variety of compliance and resistance strategies. We
consider the conditions that contribute to startups’ responses to
institutional and inter-organizational pressures and discuss how such
strategies might interact with or constrain the ethicality of applied AI.
## 3\. Methods
In this section, we describe our methods for gathering data to answer our
research questions. We elected to conduct qualitative interviews, as
interviews are an ideal method for better understanding actors’ cognitive
interpretations of their social reality, and for accessing their own
explanations of their behavioral practices within that social reality.
### 3.1. Participant Recruitment & Sampling
We recruited participants, which we refer to here as ”entrepreneurs,” from US-
based, early-stage startups that involve a significant AI, machine learning,
or predictive analytics component. Our focus on early-stage startups was
driven by theory (Palinkas et al., 2015); because emerging organizations face
many threats to their survival, they are heavily dependent on other
organizations. As a result, inter-organizational and institutional dynamics
are likely to factor significantly into their behavior. We define “early-
stage” as companies that had raised less than $50M in funding from any source
or who were at or before the Series B stage. As our interviews progressed and
patterns related to both regulatory pressures, especially privacy and the FDA,
and related to funding, especially VC and crowdfunding, we began targeting our
recruitment efforts towards additional startups that would further illuminate
these trends. We recruited participants through a variety of methods
including: 1) posts to AI and technology related listservs, message boards,
social media, and Slack groups; 2) messages to general company contact email
addresses or to specific individuals through email or LinkedIn; and 3) through
our own direct or indirect professional contacts. In total, we interviewed 23
entrepreneurs from 20 different companies. Our entrepreneurs’ companies came
from a variety of industry domains including healthcare (n=7), business
analytics (n=6), fitness and wellness (n=5), design and engineering services
(n=2), aviation (n=1), social planning (n=1), and agriculture (n=1). A
breakdown of the self-described demographic, educational, and professional
characteristics of our sample are available in the Supplementary Materials
111https://arxiv.org/abs/2203.01157. Participants were sent a $25 gift
certificate in exchange for their participation.
### 3.2. Interview Protocol
At the beginning of the interview session, we described to our entrepreneur
participants our practices for protecting their privacy and confidentiality,
then read them a verbal consent script, and gave them an opportunity to ask
any questions. After providing consent, the first author asked questions based
on the interview instrument (provided in the Supplementary Materials). In
general the interview instrument was designed to surface data relevant to
several core areas related to our research questions, including questions
about the overarching aims of the company and the entrepreneur’s role within
it, followed by questions tailored to entrepreneur’s area of expertise. For
example, in cases where entrepreneurs were involved in the company’s AI
development, we asked additional technical questions about data collection,
choice of models, evaluation criteria, and infrastructure. In most cases, we
asked entrepreneurs about their companies’ existing sources of financing and
their plans for fundraising. We also asked entrepreneurs for their personal
definition of AI. Lastly, we asked entrepreneurs about the social or ethical
implications of their company. Audio recordings of the interviews were sent to
a third party service for transcription, which were then verified by the first
author. Our study design and protocol were approved by the Princeton
University Institutional Review Board (IRB).
### 3.3. Data Analysis
We adopted an abductive approach to our analysis (Timmermans and Tavory, 2012;
Tavory and Timmermans, 2014), which allowed us to iterate between deductive
analysis guided by relevant theory and inductive analysis guided by emergent
patterns in our data. To facilitate this process, the first author initially
selected 11 transcripts that contained discussions of theoretically-meaningful
themes or that illustrated common patterns across participants. In a
preliminary analysis phase, both authors read each of these transcripts and
applied descriptive codes (Saldana, 2021). After discussing these codes in
detail, transcripts were re-coded using line-by-line in-vivo codes (i.e.,
using participants’ own words) in an effort to better preserve entrepreneurs’
perspectives (Saldana, 2021). For example, one participant discussed the
drawbacks of venture capital (VC), relating VC financing to rocket fuel:
”There are actually very few businesses where rocket fuel is the right thing
[P13].” This excerpt was tagged with the in-vivo code ”rocket fuel.” In-vivo
codes were then aggregated into groupings of similar topics. For example,
codes related to VC funding were grouped with the ”rocket fuel” quote. The
authors subsequently discussed the in-vivo codes as well as relevant theory
and chose to focus the next analysis phase on five core themes: 1) the ”AI
hype cycle” or how ”buzz” surrounding AI drives external stakeholders’
interest in AI companies; 2) practices surrounding the scientific legitimacy
of entrepreneurs’ AI approaches; 3) pressures to raise funds or secure
clients; 4) the impact of regulations; or 5) entrepreneurs’ own personal
beliefs and ethical values that relate to their companies. These themes all
appeared in multiple interviews with participants and had direct relevance to
the study’s theoretical focus on institutional and organizational theory. All
23 transcripts were then coded according to the five themes. The authors
frequently discussed transcripts and code applications to achieve consensus.
We did not measure inter-rater reliability. Inter-rater reliability is a
methodologically unhelpful tool for interpretive research, when codes comes
out of the collaborative process between researchers and consultation with
literature, and not emergent ground-up from data (McDonald et al., 2019).
## 4\. Results
### 4.1. Organizational Responses to Financial Pressures
The need to signal legitimacy to sources of financial support (e.g., investors
and clients) constituted a significant vector of influence on how
entrepreneurs defined, spoke about, and developed practices for AI. A tension
that arose repeatedly in our interviews derived from a conflict between
institutional values, specifically the values of science as a practice, and
the values of technology entrepreneurship. Whereas scientific practice values
systematic, methodological approaches paired with conservative interpretations
of findings, technology entrepreneurship values rapid innovation and
aspirational visions that extend beyond current technological reality, i.e.,
the ”fake it ’til you make it” Silicon Valley culture. In a variety of ways,
AI entrepreneurs attempted to mitigate the conflicts between the values of
entrepreneurship and science by decoupling their external rhetoric from their
day-to-day practices.
One way this decoupling manifested was through the use of the idea of ”AI”
itself. Entrepreneurs leveraged the concept of ”AI” as a symbol of their
technological proficiency even though they personally harbored disdain for the
technical ambiguity of the concept. According to our entrepreneurs, ”AI” had
no precise technical meaning and was instead employed as an operational tool
to signal legitimacy to resource-rich external stakeholders [P1, P2, P5, P6,
P7, P9, P10, P11, P13, P14, P15, P18, P21] rather than as an accurate
descriptor of what their companies actually do. In other words, ”It’s just a
buzz word [P6]” for primarily marketing benefit.
Entrepreneurs described a widespread belief that companies benefited from
marketing themselves as ”AI” companies regardless of the nature of their
underlying technology. They expressed frustration with their peers who ”got to
use the hype term [P18]” without employing any technical practices that
entrepreneurs judged as legitimate. Entrepreneurs described feeling annoyance
with these AI imposters, but nevertheless admitted to employing the same
marketing tactics themselves. Faced with a competitive landscape in which
startups’ technical and business value cannot be objectively verified,
entrepreneurs leverage the institutional expectations around the legitimacy of
AI because it ”gives credibility that we’re on the cutting edge of stuff
[P2].”
Despite the prevalence of this narrative in our interviews, only rarely were
entrepreneurs able to provide specific explanations or concrete examples of
how the abstract idea of AI yielded a tangible benefit. One entrepreneur,
however, pinpointed investors’ fear of missing out on deals as a key driver of
the ”AI hype cycle [P6]”:
> I think [the AI space] feels very confusing to [investors], but they also
> feel like there’s every signal that it’s super lucrative. […] The key thing
> that keeps all of the subordinates [up at night], the ones whose job it is
> to go find those deals and make sure their bosses don’t miss any great deals
> […] is a version of the world, where you passed on Lyft. And then your boss
> comes back to you five years later and is like, ”I would’ve made a billion
> dollars off of Lyft. […] What’s wrong with you? [P10]”
Entrepreneurs emphasized that although some investors and clients have AI
expertise, most do not have the technical background required to adequately
evaluate an AI solution or were simply ”totally disinterested in the technical
details [P6].” As a result, entrepreneurs face institutional pressures to
describe their technology using homogenized, hyped language about AI even
though their underlying algorithmic approaches were heterogeneous and often
carefully devised. Through their instrumental use of hyped AI messaging, AI
startups engage in what Oliver (Oliver, 1991) refers to as ”concealment.”
Externally startups affect the appearance of compliance with institutional
expectations surrounding technology entrepreneurship even as their internal
practices diverged, often significantly, from this affectation; ”AI” became a
discursive tool for avoiding institutional pressures via a process of ”window
dressing” (Oliver, 1991).
In contrast to their external messaging, internally, entrepreneurs sought to
achieve high standards of scientific rigor and validity. For example,
entrepreneurs emphasized their rigorous data selection and curation processes
[P4, P5, P18], described checks on the validity of their systems [P4],
designed algorithm evaluations to appropriately assess the performance of
their systems [P2, P4, P5, P9], and even employed independent validations by
academic collaborators to ensure that their models had good generalizability
[P5]. Entrepreneurs also described translating methods and findings from the
academic literature in their products [P7, P17, P15] and employing scientific
subject matter experts either directly on their development teams or
indirectly through advisors or boards of directors [P7, P17, P18].
Yet these scientific priorities could engender serious conflicts with the
priorities of users, clients, and investors. For example, when entrepreneurs
attempted to use scientific legitimacy as a differentiator in pitches, this
attempt was sometimes regarded by investors or potential clients as confusing
or unconvincing. Two entrepreneurs described remarkably similar experiences,
in which presentations about the scientific merits of their technologies were
dismissed by investors as being merely ”science projects [P5]” or ”high school
projects [P3],” having little business relevance, which from one
entrepreneur’s perspective, felt ”like an anti-science trivialization of what
scientists do [P5].”
External stakeholders’ beliefs about the potential of AI also created a
barrier for entrepreneurs to be readily transparent in their external
messaging about their models’ methodological strengths and weaknesses.
Instead, dovetailing a finding briefly touched upon by (Hopkins and Booth,
2021), we observed institutional pressure around how ”quality” in AI ought to
be reported [P4, P5, P14, P21], specifically in external stakeholders’
arbitrary notions of what constitutes ”good” model accuracy. Though an
algorithm’s accuracy may seem objective, in practice, accuracy metrics involve
many subjective choices. For example, the practical applicability of a measure
of an algorithm’s accuracy is contingent upon its mathematical formulation
(e.g., area under the curve (AUC), F1, sensitivity, etc.) as well as
contextual relevance (e.g., the severity of a false positive versus a false
negative for a medical test) and which data are selected. In our interviews,
entrepreneurs felt that in order to obtain the resources necessary for their
companies to survive, they needed to engage in rhetorical messaging that
complied with stakeholders’ expectations about model performance, even if
these metrics were not the best reflection of the task at hand, nor a valid
reflection of their algorithms’ capabilities.
> How it’s measured is we have to make sure it’s 90% or above […]. So if we
> need to switch from top 3 accuracy to top 5, just people seeing a 9, they
> don’t even think about what it’s measuring … People just have artificial
> concepts of what’s good and what’s bad [P4].
Pressure to present model metrics that have the right ”psychological effect
[P14]” on outside stakeholders was in conflict with entrepreneurs’ desire to
adopt methodologically rigorous AI approaches internally. Strikingly, one
explained that his attempts to include diverse training data in the service of
higher out-of-sample generalization damaged his company’s credibility when his
models’ performance was compared to competitors who use less realistic data.
> Ultimately, our results aren’t going to be as stellar as a lot of others
> because now we have to account for […] all the variability within the data
> set whereas, if we’re just focused on one homogeneous data set, our accuracy
> stats will be higher. So, that has been one sticky, difficult point in terms
> of head-to-head comparisons [P5]
In the service of survival, entrepreneurs sometimes conformed at least
superficially to the pressures of stakeholder expectations, adapting their
external messaging over time to provide a level of scientific detail that was
persuasive to the target audience. In this way, they again engage in what
Oliver refers to as concealment tactics. However, these pressures did not
entirely undermine their desire to externally project the methodological rigor
they prioritized internally. Instead, entrepreneurs chose to target their
products to specific stakeholders who would be more receptive to messages
about the product’s scientific credibility or its technical utility within a
domain. For example, one entrepreneur highlighted that having extensive
scientific references available on their product’s website attracted desirable
early users:
> ”[…] most marketers are like, ”I don’t think that sells the product,” but we
> disagree. […] I’m not sure it makes it so it’s a blockbuster of a product,
> but it brings in the right type of people for your product […]. It brings in
> good early adopters, anyways [P7]”
In this public commitment to scientific integrity, entrepreneurs engage in a
form of ”defiance” (Oliver, 1991). Oliver hypothesizes that defiance is more
likely when the perceived cost of resistance to institutional norms is low and
”when they can demonstrate the rationality or righteousness of their own
alternative convictions or conduct.” AI entrepreneurs who externally project
their strongly-held personal values of the scientific process may do so
because they can promote science as a virtue while still attracting science-
inclined external stakeholders.
Entrepreneurs recognized that in order for their companies to grow, their
products must eventually translate into market success and that their
companies’ investors were ultimately motivated by whether or not the company
would ”provide liquidity [P17]” on their investment. Nevertheless,
entrepreneurs demonstrated a wide range of compliance and resistance tactics
when it came to their decisions regarding financial backing. They described
pursuing a variety of strategies to fund their businesses including revenue
[P8, P13, P21], friends and family raises [P1, P10], grants from government or
private entities [P2, P12, P17], angel investors [P4, P5, P6, P7, P13, P14,
P15, P17, P21], debt financing [P12], VC financing [P4, P10, P14, P16, P17,
P18, P20], and crowdfunding [P7, P9, P13, P21, P23]. Even still, several
entrepreneurs pointed out that VC is regarded as a default financing path:
> We looked at the VC funding route in the beginning because that’s what
> you’re told to do, right? That’s how you get funded. This is the path. You
> go pre-seed, it’s angel investors, after that it’s VCs, and then you go
> through the Series process [P12].
For some, capital from VC firms formed a cornerstone of their strategy for
building their business [P4, P10, P16, P17,P18, P20]. These entrepreneurs saw
investors not only as a source of financial capital, but also of valuable
industry domain and business expertise as well as a mechanism for accessing
important professional networks. Entrepreneurs viewed the fit between the
needs of the company and the expertise of investors as a critical component of
establishing a productive relationship.
> So, we have a number of investors, and the asks really change based on
> business needs. So it’s really, what do we really need today, this week,
> this month, that can help us take the business to the next level, and who do
> we have as investors that we can ask for help in those areas? [P17].
However, entrepreneurs did not blindly acquiesce to the demands of investors.
Where their own goals conflicted with the goals of investors, VC or otherwise,
they would sometimes decline further involvement. For example, one
entrepreneur described ending early conversations with an investor because
their desired exit strategy was not consistent with her own goal to eventually
take the company public [P1]. In another case, an entrepreneur described
evaluating potential investors based on their alignment with the company’s
ethical values:
> We are trying to raise capital from investors that have the same kind of
> values and mindset with us and people who are not afraid to lose certain
> revenue or sales just to follow the same values. We had clients asking us to
> do things that we said, you know what? No. No, this is not something we feel
> comfortable with doing [P23].
One entrepreneur noted that because of intense investor interest in the field
of AI, instead of entrepreneurs doggedly pursuing financiers, ”they find you
[P10].” In a resource-rich environment, entrepreneurs have more latitude to
resist institutional pressures arising from dependencies on investors. To some
extent, we saw this resistance in how entrepreneurs described choosing
specific investors; however, we saw even greater resistance amongst
entrepreneurs who expressed hesitancy about pursuing VC financing [P7, P9,
P12, P13, P16, P21]. In these cases, VCs were seen as having financial goals
that conflicted with entrepreneurs’ long-term business or product objectives.
One entrepreneur noted that unlike traditional software, AI software
development typically requires specialized algorithm expertise,
infrastructure, expensive data labeling, and continuous model performance
monitoring; however, if VC firms’ valuation of AI companies is based on their
knowledge of traditional software startups, they may impose timelines or key
performance indicators that undermine what entrepreneurs believe to be
methodologically-sound practices in AI development. Recognizing that VCs’
profit motives and responsibility to their limited partners (LPs) could
conflict with their own goals, entrepreneurs often described actively avoiding
VC financing.
> That was my whole experience [at a previous startup]. The VCs wanted to hype
> things up, get a lot of press, make a splash, so they could raise the next
> round at a higher valuation and look good to their LPs, which was actually
> contrary to what we needed to do for the slow growth to build the business
> [P13]
Instead of VC funding, several entrepreneurs described using crowdfunding to
finance their businesses [P7, P9, P13, P21, P23]. Crowdfunding and other
financing vehicles that provide investors with little direct control over the
companies’ behavior were viewed by entrepreneurs as a way to maintain autonomy
over their businesses, control progress towards the product vision, and to
maintain the equity of current employees. Moreover, although prior literature
has indicated that the disclosure requirements of crowdfunding platforms can
create a risk for companies’ subsequent financing prospects (Blaseg et al.,
2021), entrepreneurs valued the transparency associated with public
disclosures on crowdfunding platforms and the opportunity to directly engage
with potential crowd investors. The choice of crowdfunding over the ”default”
VC financing path constitutes another form of organizational defiance; these
entrepreneurs challenge the culturally dominant mode of startup funding by
choosing financing paths that they felt would better serve their long-term
objectives. As predicted by Oliver (Oliver, 1991), startups are able to engage
in resistance because doing so does not compromise their chances of survival
since they can rely on alternatives to VC to fund their businesses.
### 4.2. Organizational Responses to Regulatory Pressures
In addition to financial pressures, entrepreneurs also described their
compliance with and resistance to pressures in the form of regulation,
especially regulations surrounding privacy and the Food and Drug
Administration (FDA) approval processes for applications of AI in medicine. On
the whole, entrepreneurs viewed privacy protections as normatively good, a
competitive necessity, or even a competitive advantage compared to industry
peers who are slower to adopt privacy-protecting practices. One entrepreneur
described how his company had chosen to temporarily avoid adapting to the
”landscape of privacy and privacy laws [that are getting] a lot more strict
[P19]” by operating selectively in markets that were subject to less stringent
privacy regulation (e.g., in the US versus European Union where GDPR is
applicable), but even this entrepreneur noted that circumvention was only
tenable in the short term. Yet in contrast to (Hopkins and Booth, 2021), most
entrepreneurs in our study did not express resistance to or subversion of
privacy regulations but openly endorsed them as well as the ethical values
underlying them, such as personal autonomy. In their discussions about
privacy, entrepreneurs sometimes contrasted their own beliefs and policies
with those of large technologies such as Facebook, whose privacy-related
behaviors they generally regarded as reprehensible [P9, P6, P23].
Unlike privacy regulation, the FDA approval processes for AI in healthcare was
viewed less favorably. Entrepreneurs who discussed the FDA viewed these
regulatory requirements as unnecessarily onerous and in some cases,
unscientific [P2, P4, P5, P7, P16]. A theme that arose in our interviews with
multiple entrepreneurs was that the lack of standardization in the FDA
approval process for AI-enabled healthcare products created a serious burden
for startups with an unclear upside. One entrepreneur from a more mature
startup described how challenges posed by the FDA approval process contributed
to her company’s decision to eventually pivot away from the AI products she
felt had life-saving potential to a core product and business model that she
thought was more profitable but ultimately less useful to society. She
highlighted the opportunity and financial costs of pursuing FDA approval as
well as the difficulty of providing evidence to meet FDA performance
standards:
> Whenever you’re thinking about a rule-out device–in our case, [the finding
> that a medical test is normal]–that means you rule out every possible thing,
> it’s statistically insanely hard to do. And in order to get approved, you
> would literally have to do better than humanly possible [P16].
All but one entrepreneur [P18] who discussed relevant FDA regulation found the
ad-hoc approval process to be legally or financially arduous; however, there
was less consensus on whether the FDA’s model performance standards were
unreasonable. One entrepreneur noted:
> The data that needs to be provided in order to get clearance in our opinion
> is relatively low, but it does take a lot of money and other things to get
> [P5]
Similar to this perspective, another noted that if the financial demands for
securing FDA approval were lower, they would begin reallocating their research
and development efforts to undergo FDA approval as soon as possible since
their model performance was already strong.
Entrepreneurs either implied or stated explicitly that the onerous FDA
approval process stifled innovation, but they also noted that such regulations
were necessary to protect users from harmful products. Another entrepreneur
noted that regulatory approval could even be beneficial to his businesses
because ”customers are much more receptive to the FDA stamp than they are to
stats [P5]”. Still, the perception that the FDA approval process was ”a whole
monster [P4]” that was often not worth pursuing motivated AI entrepreneurs to
attempt to operate in regulatory gray areas or exploit loopholes so that they
could continue to pursue technological or product objectives. Such
entrepreneurs described using other, non-regulatory avenues to demonstrate
their legitimacy such as publishing their model details and performance in
academic journals or technical whitepapers. It is important to note that
entrepreneurs’ opposition to the idiosyncratic FDA approval process was not
merely a matter of logistical difficulty; they also viewed the discretionary
nature of the FDA review process as an opportunity for established companies
to be unfairly advantaged:
> It’s also, I think, unfair how the FDA, […] they have existing relationships
> with Pfizer and Johnson & Johnson. I get it. But they’re obnoxiously hard on
> startups because they’re not known [P16]
Thus, entrepreneurs’ personal values only partially aligned with the
institutional pressures from the FDA–they value the consumer protection
intent, but decry its consequences for innovation and believe it to be at odds
with a fair, competitive marketplace. In contrast, entrepreneurs were more
likely to adopt privacy preserving practices, which were consistent with their
own normative beliefs, even in cases when companies were not yet subject to
strict privacy regulation. In other words, consistent with Oliver’s (Oliver,
1991) prediction about legal coercion engendering less compliance than
institutionally diffuse norms, there appears to be an association between
internal adoption of institutional rules derived from regulation when such
rules aligned with field-level values.
### 4.3. Organizational Responses to Technological Pressures
Institutional theorists posit that emerging organizations can bolster their
own legitimacy by adopting the values, structures, and practices of
established organizations (Meyer and Rowan, 1977; DiMaggio and Powell, 1983).
That is, organizations can improve their chances of survival by mimicking what
incumbent organizations already do. Yet this idea is in tension with the
Silicon Valley notion that the most legitimate innovations are those that
”disrupt” existing ways of operating (Hogarth, 2017; Geiger, 2020). So-called
radical innovations are those that break from or are discontinuous with prior
scientific and engineering practices whereas incremental innovations are those
that build upon and extend the existing technological paradigm (Colombo et
al., 2015). In our interviews with entrepreneurs, their rhetoric typically
suggested they viewed deep learning models as constitutive of radical
innovation in that they distinguished deep learning models from other machine
learning techniques; however, their implementations of deep learning
models–typically via transfer learning–were, in contrast, fundamentally
incremental.
Despite the murkiness around definitions of AI more broadly, entrepreneurs
often held deep learning out as distinct, which sometimes manifested in how
entrepreneurs defined AI. Many entrepreneurs provided definitions that
contrasted algorithms’ capabilities with human capabilities [P6, P7, P14,
P16], that differentiated between general and narrow intelligence [P5, P6, P7,
P15, P23], or that described high-level processes that are applicable to any
AI model [P1, P2, P4,P7, P8, P15, P18, P19, P20, P21, P22, P17]. However,
several entrepreneurs implicated deep learning specifically in their
definitions [P2, P9, P13, P11], using deep learning as a threshold of ”real”
AI:
> The most concise answer I can give you is just deep learning. That is almost
> the new cutoff for AI, in my mind at least [P9].
Even for those that did not equate deep learning with AI, the ways
entrepreneurs discussed deep learning relative to other AI approaches suggest
that they consider such techniques separate or superior. For example, some
described using ”machine learning and deep learning [P14]” or explained that
their companies constrained their models to ”classifiers and regression [P3]”
instead of ”doing any deep learning or anything nutty like that [P11]” as
though deep learning models were not a subset of machine learning techniques.
In one case, we interviewed the chief technology officer of a company that
intended to develop an AI-enabled solution but that had not yet begun data
collection. Even in the absence of any empirical evidence to support his
conclusion or expertise in deep learning, he had preemptively concluded that
simple, linear techniques would be insufficient to achieve the high accuracy
he hoped to obtain with deep learning methods [P14]. Thus, entrepreneurs
distinguish the ”magic [P13]” of deep learning from other ”rudimentary data
science [P11]” techniques.
Yet in conflict with widespread framing of deep learning as ”magic” and
perhaps radical innovation, most entrepreneurs’ implementations of deep
learning constituted a more incremental form of development that draws on the
scientific products of researchers in the AI community. Many entrepreneurs
described relying heavily on transfer learning for their deep learning
applications. Transfer learning is technique where models that are initially
trained with massive datasets for one task can be adapted for related tasks
with much lower data requirements. The use of transfer learning can reduce the
computational and data costs associated with training models from scratch
while still affording entrepreneurs ”the accuracy that we feel we need from
that model [P9]”. Through the use of pretrained models initially developed by
AI researchers in academia or at large technology companies, AI startups
”build upon the state of the art, all the advancements that are being driven
by the Googles of the world [P13]”. In a paradoxical way, entrepreneurs’
discursive distinction of deep learning techniques complies with the
institutional pressures to seek rapid and disruptive innovation, even though
the pretrained models used for transfer learning coordinate practices of AI
entrepreneurs, possibly down to the specific pretrained models they employ.
Entrepreneurs’ rhetoric surrounding the distinction of deep learning, which
conforms with institutional expectations about the utility of disruptive
technology, acts to conceal their use of publicly available, incremental
technologies that while not fundamentally disruptive in a scientific or
technical sense, are nevertheless sufficient to meet AI startups’ needs.
Interestingly, the disconnect between entrepreneurs’ rhetoric and practices in
deep learning did not appear to be driven by attempts to appeal to external
stakeholders who would be unlikely to appreciate the difference between deep
learning and other machine learning techniques, but potentially to signal
status to other AI startups or industry peers or to bolster their sense of the
company’s legitimacy internally.
### 4.4. Organizational Responses to Normative Pressures
As an organizational field becomes more institutionalized, professionalization
is enacted through education, membership in professional bodies, and other
aspects of professional culture. These professional mechanisms can drive
organizations within that field to adopt similar norms and values, which
become embedded in their organizational practices (DiMaggio and Powell, 1983).
Recent scholarship on professional norms within the AI research community has
found that pervasive professional norms include efficiency, universality, and
impartiality (Scheuerman et al., 2021). In our interviews, we observed
instances where entrepreneurs articulated personal ethical values that were
either distinct from, or resistant to, professional norms.
The demands of operating within the fast moving technology industry constrain
the extent to which industry practitioners can fully realize ethical values
into substantive practices (Vakkuri et al., 2020; Metcalf and Moss, 2019).
Consistent with these findings, some entrepreneurs hoped to incorporate their
personal ethical values into their product or business model in the future,
but had yet to make much tangible progress towards those ideals [P1, P2, P10,
P13]. Yet, in other cases, entrepreneurs took a strong stance on ethical
issues and described how they built these values into their technology and
organizational cultures. For example, several entrepreneurs described how
their algorithms [P21, P23, P19, P6] or data practices arose from normative
beliefs about the ethics of privacy protection [P2, P9, P11, P14].
Racial bias also came up repeatedly in our interviews, but did not always
inform product or business decisions. In a handful of cases, startups’ AI
approaches had been explicitly designed to ensure that their algorithms would
perform equally well across demographic groups [P4, P21, P23]. Similarly, some
entrepreneurs had designed non-algorithmic elements of their products to
prevent racial bias [P17, P12]. In several cases, entrepreneurs’ motivations
for developing algorithms that perform well across racial groups were not only
based on personal value systems, but also based on the belief that fair
algorithms realized market value. For example, several entrepreneurs pointed
out that in order to serve international clients and diverse users, it was
important for AI-enabled products to be equitable. On the other hand, some
entrepreneurs were aware of the types of racial biases that can be reproduced
by AI algorithms [P6, P7, P9, P11, P22], but either thought that race was
irrelevant to their models [P6, P11] or that racial biases were only a
priority in high-stakes contexts such as healthcare and finance [P9, P11].
Although less common than algorithmic strategies, several entrepreneurs
touched on how they promoted racial and gender equity within their companies
[P12, P22]. Drawing from his own experience of racial marginalization, one
entrepreneur noted:
> A fundamental shift in power from straight white men to the rest of the
> world is really something that needs to happen. […] I want to be able to
> show people that look like me, that they can also use things, and they can
> also build something that’s great and can also help build those communities
> [P22].
Even outside of explicit interview questions about ethics or social impact,
entrepreneurs often espoused values related to democratization and expanded
access to technology [P1, P8, P17, P16, P19, P4, P2] (e.g., ”democratizing
access to data [P17]”). Entrepreneurs described wanting to provide financially
valuable expertise or insights to other businesses, especially small
businesses and startups [P1, P19], to provide needed services to emerging
economies [P16], or to empower users to take on tasks that are more typically
performed by specialized professionals [P4, P8, P16]. In line with these
values, entrepreneurs were critical of insider cultures, implicating ”old
boys’” networks [P6, P7, P22] or ”traditional male VC [P12]” in gatekeeping
behaviors related to client acquisition, external financing, or in other ways
that affected their businesses’ success. Entrepreneurs’ skepticism of
centralization and insider culture was also manifested through their choices
about funding. Whereas VC served as a stand-in for centralized power,
crowdfunding was viewed as consistent with the ideal of democratization since
crowd investors do not need to meet the same financial accreditation standards
required to invest in a VC fund.
Entrepreneurs’ normative values reflect a mix of the techno-libertarian
leanings of Silicon Valley that have been documented elsewhere (Metcalf and
Moss, 2019; Lenhard, 2021; Dahlberg, 2010; Hütten, 2019) as well as beliefs in
social equity and fairness. Sometimes these values conflicted, as is
highlighted by the tension entrepreneurs expressed around racial bias in AI;
they believe that all users should be treated equally, but under the same
resource-constrained system that encourages developers to ”move fast and break
things” (Vardi, 2018), they do not always prioritize development around that
belief.
## 5\. Discussion
Our current study adds to the growing literature on organizational challenges
to ethical AI by describing how broader inter-organizational and institutional
forces shape the practices of AI startups. In this section, we discuss both
the theory-based as well as pragmatic contributions of our research. This
discussion is structured along the same categorical lines of our findings,
discussing in turn financial, regulatory, technological, and normative
pressures.
### 5.1. Financial Pressures
A central tension recurred between entrepreneurs’ desire to preserve the
scientific integrity of their AI approaches and the demands of technology
entrepreneurship that often ran counter to this desire. As one entrepreneur
noted, ”the value in the technology that you use doesn’t necessarily even have
to come from the technology [P9]”. In contrast to purely scientific
enterprises, the import and meaning of novel technologies is not entirely
determined by scientific inventiveness or rigor, but is also constructed
within an economic, social, and cultural context.
The demands of external stakeholders with power to affect the financial
outcomes of AI startups exerted influence over the narratives entrepreneurs
constructed about the benefits of their technology. In response to
stakeholders’ expectations of “silver bullet [P21]” AI solutions,
entrepreneurs tended to adapt their external messaging accordingly, but they
did not necessarily alter their internal practices. In this way, entrepreneurs
engaged in a resistance strategy of concealment (Oliver, 1991), decoupling the
symbolic and homogeneous marketing tactics they adopted to accrue legitimacy
from business partners, from the substantive and often heterogeneous
approaches they employed internally.
That entrepreneurs placed a strong value on scientific integrity points
towards an ethical opportunity within the startup ecosystem. As several
entrepreneurs themselves pointed out, models with inequitable outcomes are
necessarily less valid since they do not generalize well. Moreover, they are
less able to realize business value since they cannot meet the expectations of
diverse clientele. Thus, entrepreneurs’ values of scientific legitimacy might
act as a ”value lever” (Shilton, 2013) through which principles of AI ethics
can be imported into AI startups. On the other hand, external stakeholders’
tendency to treat decontextualized accuracy metrics as a superficial indicator
for AI quality is suggestive of a risk for institutionalization of AI ethical
ideals. For example, the “80% rule” for establishing disparate impact, which
has often been imported into AI fairness research without regard for its
original legal nuance, may have already created an artificial standard within
the research community (Watkins et al., 2022). This metric as a target could
create further ethical risk if stakeholders in the AI startup ecosystem also
adopt it without considering its relevance and caveats within context. Our
observations around the use of ”AI” as a marketing ”buzz word,” reflect recent
concerns around ”AI as snake oil” (Kaltheuner, 2021), and the exploitation of
AI’s vague definition as a loose umbrella term. Thus, strategies that ensure
that AI ethics constitute more than an ethical Potemkin facade are needed
(Cihon et al., 2021).
Consistent with prior literature (Willoughby, 2008), entrepreneurs also
demonstrated more heterogeneity in financing strategies than the culturally
dominant VC-startup narrative would suggest. While some entrepreneurs
conformed with institutional pressures to pursue VC funding and found benefits
beyond financial capital in their partnerships with VCs, others actively
avoided VC funding. This opposition was sometimes based on philosophical
opposition to VC as antithetical to democratic ideals and other times informed
by entrepreneurs’ personal experiences of VCs driving startups away from sound
technological and business practices. Some evidence suggests that
entrepreneurs’ avoidance of VCs could harm their companies’ growth potential
(Baum and Silverman, 2004; Bertoni et al., 2011), but other evidence shows
that the benefits of VC do not always extend to profitability (Rosenbusch et
al., 2013). Moreover, even if VC does improve the financial outcomes of
companies on the whole, this financial benefit does not necessarily redound to
founders themselves since their stake in the company is significantly diluted
by VC investment (Florin, 2005). Thus, resistance to the institutional norm of
VC financing could be conceptualized as economically rational as well as in
line with entrepreneurs’ desire to retain control over their businesses since
VCs sometimes use their power to replace the founding team with professional
executives (Hellmann and Puri, 2002).
Entrepreneurs’ desire to match their financing strategies with their business
goals and normative values presents an opportunity for ethical practices. Even
if AI ethics interventions are seen as antithetical to profit goals as
demonstrated in (von Zahn et al., 2021), entrepreneurs may be able to preserve
AI ethical ideas by matching with investors who share these priorities,
especially if public scrutiny around the ethical implications of investor
strategies increases (International, 2021). It is important to note, however,
that the entrepreneurs who are able to exercise more discretion in terms of
when they seek funding and from whom they seek it are likely already
advantaged in the entrepreneurial ecosystem, as one of our entrepreneurs
himself noted: ”I know we have the luxury that we could decline money. I know
that that is a luxury [P6]”. Black and Latinx founders (Crunchbase, 2020) and
female founders (Teare, 2021) secure less financing than other founders, and
as a result are likely to have fewer options when attempting to find financing
partners who prioritize ethical objectives. Thus, selective matching between
entrepreneurs and investors could also further magnify inequality.
### 5.2. Regulatory Pressures
With respect to regulatory pressures, entrepreneurs typically endorsed privacy
regulations but expressed more frustration with FDA regulations. While privacy
regulations were perceived as aligning with the values of personal freedom and
autonomy, which have been documented in other research on technology sector
actors (Metcalf and Moss, 2019; Lenhard, 2021), FDA regulations were seen as a
barrier to innovation and entrepreneurial autonomy and a mechanism through
which industry insiders receive favor from other institutional actors. These
contrasting results support both theory and evidence that a mismatch between
an organizational field’s normative values and coercive regulatory pressures
will result in less meaningful internalization of policies (Oliver, 1991;
Scott, 2013). However, an alternative resource-based explanation is also
possible. Privacy regulations are likely to apply uniformly, but FDA
regulations are idiosyncratic, and therefore require more expertise and
financial resources to navigate. Regardless of the cause, as legislators
debate proposals to further regulate AI, they should take care to consider
what negative, second-order effects regulations might have on AI startups.
Greater engagement with AI entrepreneurs could improve both policy and its
adoption within startups since active participation from business owners has
been shown to increase regulatory compliance (Malesky and Taussig, 2017).
### 5.3. Technical Pressures
The resource constraints of startups also fed into our findings regarding the
use of deep learning amongst AI startups. Both entrepreneurs who did use deep
learning and those who did not tended to discuss the use of deep learning
techniques with a reverence not afforded to other algorithmic approaches. Yet
deep learning startups most often developed their technology on top of
preexisting pretrained models, especially those developed to perform natural
language processing and computer vision tasks. As with most scientific
advancements, applications developed through transfer learning are incremental
innovations, inextricably tied to established approaches developed by a
broader community of researchers and practitioners. That is, the use of deep
learning in most AI startups is not a radical departure from the dominant
machine learning paradigm, but an endorsement of it. This is not to say that
deep learning applications developed through transfer learning are not
valuable, creative, or innovative. Experts have implicated transfer learning
specifically in the acceleration of AI discovery (Ng, 2016).
The widespread use of pretrained models does raise ethical questions. Word and
image embeddings derived from models trained on human data often encode human-
like biases such as gender, racial, and other harmful stereotypes (Bolukbasi
et al., 2016; Steed and Caliskan, 2021). How entrepreneurs adapt pretrained
models to their applications may obviate transmission of harmful biases from
pretrained models to industry applications; however, some researchers have
suggested that the biases of pretrained models, if not mitigated for their
contextual application, could further propagate harms (Steed and Caliskan,
2021). Thus, our finding supports the call for research to better understand
not only the negative social impacts of models developed in academic contexts,
but also how these impacts are attenuated or magnified by their applications
in industry through transfer learning (Narayanan, 2021).
### 5.4. Normative Pressures
As we have already discussed, entrepreneurs’ beliefs played a significant role
in how they developed their technologies and their business practices. In some
cases, AI entrepreneurs espoused libertarian ideals, such as individual
autonomy and personal responsibility. Much of AI ethics research focuses on
establishing the fairness of model outputs or mitigating unfairness in model
predictions. In this way, AI ethics interventions often center on equity in
outcomes. In contrast, entrepreneurs expressed valuing democratization, which
emphasizes the importance of equality of access, rather than equity in
outcomes. This distinction points to further risks for translating the
technology and ideas developed in AI ethics research contexts into AI startups
or the technology industry on the whole. Institutional pressures deriving from
the technology industry will likely interact or conflict with the values
embedded in ethical AI interventions. Designers of ethical interventions
should consider the normative context in which they are intended to apply.
Otherwise, they could be dismissed as irrelevant by practitioners or be
employed in ways other than how they were designed, which itself constitutes
an ethical risk.
## 6\. Future Research
Our findings on the significant influence exerted by institutional pressures
on AI startups, and the variance in entrepreneurs’ decision-making around
compliance, avoidance, and resistance, open a number of potential research
pathways. First, as mentioned above, more research is needed to better
understand how the social impacts of AI models may be exacerbated through
transfer learning in industry settings. Second, identifying and interviewing
other stakeholders in this sector would allow us to analyze the interactive
dimension of these field-level dynamics, yielding data about how investors,
regulators, competitors, and customers participate in and contribute to
complex system dynamics of institutionalism in AI. Findings around the
alignment between regulatory pressure and normative pressure, further, suggest
that such alignments lead to better take-up within organizations, and so
collaboration with policy researchers could lead to the design of AI policy
better positioned to act as an effective guardrail against the harms of such
systems.
## 7\. Limitations
Our study design presents several limitations which may influence our
findings. First, a limitation of the interview instrument was its exploratory
nature. Due to the broad scope of our research questions, themes could not be
identified prior to the study, but rather were identified in our data as a set
of findings. As a result, we were unable to reach depth within particular
themes, nor did we find that we reached theoretical saturation for any
thematic category. Instead, the exploratory nature of this study identifies
pathways for future research opportunities. Second, our recruitment and
sampling strategies also present limitations. Our sample size was relatively
small, and cannot – and is not intended to be – generalizable to the larger
population of AI startup entrepreneurs. Within qualitative research, sample
size requirements are a subject of debate, and are always a reconciliation
between research interests and goals, access to participants, and maintaining
rigor. We used theoretical sampling, which is intended to ensure that there
are enough participants to surface ”a range of concepts and characteristics
that are deemed critical for emergent findings,” (Dworkin, 2012; Glaser and
Strauss, 1967), which we determined was achieved with our sample.
## 8\. Conclusion
On the whole, our research shows that although institutional forces do shape
AI startups’ beliefs and practices surrounding AI, they do not dictate them
entirely. As a result, while future interventions to support ethical AI should
be mindful of the organizational contexts for which they are intended, they
also should not assume that startup practitioners have no agency to act in the
service of ethical values. Even if the ethical practices adopted by startups
at their outset evolve over time in response to shifting market demands,
founders typically have a lasting influence on startups’ trajectories, even
after they leave the company (Sahaym et al., 2016). As a result, though
startups face more resource constraints than the more mature companies that
have been the focus on most applied AI ethics research, they also may be an
ideal stage for ethical interventions.
## 9\. Acknowledgments
We thank Ranjit Singh, Arvind Narayanan, and Alex Hanna for helpful feedback
on our manuscript and Pedro Gomes for guidance on our research questions. We
gratefully acknowledge financial support from the Schmidt DataX Fund at
Princeton University made possible through a major gift from the Schmidt
Futures Foundation.
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## Appendix A Interview Instrument
The interview instrument we used to loosely structure our interview is below.
We note that the primary questions we asked participants from the ideals and
values section of our instrument were about social and ethical implications.
Typically, we asked follow up questions based on participants’ responses to
this main question rather than other questions in this instrument.
Background
Can you tell me a little bit about your company?
[if not already answered] What is the problem your company is trying to solve?
What is your role in the company?
Can you walk me through what you did at work on a specific day recently?
Product & AI
How do you define AI?
How does your company use/want to use machine learning, artificial
intelligence, or predictive analytics?
Why did you/your team decide ML/AI was the right approach?
What was your/your team’s experience in AI/ML before starting this company?
Funding
How is your business funded?
Can you describe your experience trying to secure funding?
[If funded]
How do you typically interact with your funders? What happens in these
interactions?
Have you discussed how your company uses AI with your funders?
What do you think your funders think about AI?
[If not funded]
Once you do secure funding, how do you anticipate you will interact with your
funders?
Have you discussed how your company uses AI with potential funders?
What do you think potential funders think about AI?
What is your company’s exit strategy?
Ideals & Values
What would you say are the core values of your company?
What do you think differentiates a successful startup from an unsuccessful
one?
Are there other companies that you think are good examples for your own
company to follow?
What role does AI/machine learning play in the technology industry as a whole?
Has your team ever discussed the ethical or social implications of the AI you
use in your product?
Demographics & Background
What is your title at your company?
What is your gender?
What is your race?
What is your age?
What is your educational background?
## Appendix B Participant Characteristics
Participants’ self-described demographic characteristics and company roles are
listed in Table 1. Note that some participants listed more than one race, and
many listed more than one role. The methods through which participants were
recruited are listed in 2. The breakdown of participants by industry are
available in Table 3
| n
---|---
Race |
White/Caucasian | 14
Asian | 2
South Asian/Indian | 5
Black/African American | 2
Middle Eastern | 1
|
Gender |
Male | 17
Female | 6
|
Education (Highest) |
High School | 1
Bachelor’s | 7
Master’s | 9
PhD | 4
MD (or equivalent) | 3
|
Role |
C-Suite (e.g., CEO, CTO) | 11
Founder/Co-Founder | 15
Division Head | 3
Other | 4
Table 1. Participant Demographics | n
---|---
1st degree contact of authors | 1
2nd degree contact of authors | 5
Slack groups | 4
In-person networking event | 4
University alumni message board | 2
Cold contact | 7
Table 2. Recruitment Methods | n
---|---
Healthcare | 7
Aviation | 1
Fitness & Wellness | 5
Business Intelligence & Analytics | 6
Social Planning | 1
Design & Engineering Services | 2
Agriculture | 1
Table 3. Participant Industry
## Appendix C Supporting Quotes
Scholars have hotly debated whether the goals of open science that have been
gaining traction in the quantitative sciences are also relevant for
qualitative research (Pratt et al., 2020; Reinhart, 2016; Kapiszewski and
Karcher, 2021; Seale, 1999; Rhoads, 2020; Murphy et al., 2021). Unlike
quantitative research, qualitative human subjects often participate in
research on the condition of anonymity, which precludes complete transparency.
Moreover, qualitative work is often premised on the idea that the
interpretation lens the researcher brings to research is itself a valuable
component of any qualitative scientific pursuit. Here, we attempt to create a
balance between these values by offering our own interpretation of our
findings in the body of the article and offering as much transparency as
possible without compromising our participants’ anonymity through supporting
quotes. We omit quotes from the supplement that that directly or indirectly
could identify participants. To reduce the possibility that participants could
be identified by patterns across their quotes, we do not provide a participant
identifier for each quote, but we include the participants whose quotes are
listed within each section. To align with the organization of the main
article, quotes are organized according to financial pressures, regulatory
pressures, and normative pressures and are in no particular order within
sections.
#### C.0.1. Financial Pressures
Quotes are derived from P1, P2, P5, P6, P7, P8, P9, P10, P11, P12, P13, P14,
P15, P17, P18, P20, P21.
* •
I think it’s because there’s a little bit of a sense of AI being a like
magical silver bullet type solution. AI is just like this loosely defined
thing that if you give to somebody, it could potentially make them more money
or give you better insights or something like that, that from a more public
perspective, as far as a company saying that they are an AI based company
providing a service that may not use AI at all as better investment and also
people, it shows or it signals that the solution could be more scalable than
it is in its current fashion.”
* •
We’re going to talk about our machine learning algorithms, because from a
marketing standpoint, it connotes this next generation high tech, God, it has
to be good.
* •
And I think part of the biggest issue, and this may not be unspoken, is that
everyone and their brother wants to have AI in their product, especially in
healthcare right now, because it’s the buzzword du jour. So basically, if
you’re in the physical space, you want two things: you want to say your
product has AI and you want to say your product has a robot.
* •
Because [AI is] the sexy thing for investors. And quite frankly, coming from
an engineering perspective of what I know of AI and what I know of machine
learning, I actually think a lot of it is overblown and a lot of things that
are called AI is not actually AI. It’s actually machine learning or a learning
algorithm that is kind of tweaked and people are bringing up AI just to say
they have it.
* •
Just including a small amount of AI gives you a marketing edge.
* •
Plus it’s a feedback loop, I guess. That’s what you see in industry, everyone
doing this. And getting good results by saying that, ”Look, we use AI.” It
almost seems like, ”Why wouldn’t you?” I guess, whether or not you have it.
While I did mention it as a pet peeve of mine it’s also, I guess,
understandable. Especially for people in startups, is a very competitive
space. So you’re trying to get every edge you can.
* •
It is similar thing as saying ”blockchain.” You’re well aware, but it’s a
super common thing with startups, that startups are doing. Just trying to
catch people’s attention with the hottest new tech.
* •
When I was doing research, it was just super focused on like, ”Here’s this new
thing that I did that’s novel. And it’s state of the art and it gets 0.001%
better accuracy than this other guy’s thing. It’s not reproducible, but it’s
AI, and it’s really cool.” So I don’t know. The field is super legitimate. How
do I say this? You end up getting a lot of people who are just trying to ride
that wave of legitimacy and not contribute anything substantial.
* •
I think the basics of it is – a large dataset, train a machine learning model,
you can predict many things. I think that’s permeated a lot of just public
understanding, scientific popular science. I think that basic equation
resonates with people, even if they don’t really care to understand okay
exactly how does a neural network work or what is an LSTM or something like
that. They might not care about those specifics but they probably see the
results in their daily lives. I think there’s been a lot of remarkable changes
in the last 5 or 10 years with real products that people are using AI,
improving them. And so, I think they can appreciate its ability, while not
necessarily caring to delve too deep on the technical side.
* •
But AI, machine learning, they’re buzzwords, they make us sound smart, like
we’ve got access to something because to them, they don’t know any of this
stuff […] INTERVIEWER: Who is that a buzzword for? Who’s impressed by the word
AI or machine learning? PARTICIPANT: It just puts us in a box of, and I’m out
here in Silicon Valley – so it’s the nerds, the data people, they’re in touch
with tech, they associate us with the tech industry
* •
When it comes to funders, I think AI is a buzzword that everybody likes
hearing on the VC side, big data, machine learning, those types of things. But
I think from a marketing perspective, we’re marketing to work with
collaborators and hospitals or potential customers, it’s really focusing on
the clinical impact, that’s the most important.
* •
At least, to me it’s pretty obvious that machine learning has transformed a
lot of different tech sectors. The larger scale ones certainly, from Google to
Facebook. But down to even more specialized sectors. So I think there’s this…
I think the feeling is that AI has the capability of transforming or paradigm
shifting different sectors, and if you get in on it early you can be part of
that wave. And I think it’s certainly been a very successful methodology in
many fields across many domains. So it has a lot of demonstrated success.
* •
[…] We have these hype cycles for AI throughout history of at least the last
80 years, 70, 80 years. And if you look back into the history of what we now
understand as artificial intelligence, we have these just incredible claims
what will be possible tomorrow or at least next year. And then we had these AI
winters and all of that again. And I think sometimes in the last years, the
last 10, 15 years, people started to recognize, ”Hey, it starts working.” And
then again, all these claims we came up with, we will have robot butlers at
home and automatically driving cars and all that. And people really jumped on
it. But I think it was the first time that some of the promises get fulfilled.
To what degree, is another topic, but that you could, as a non-technical
academic person involved, see that is something happening. So you have Siri on
your phone, or if you have this incredible Google voice assistant that is
completely AI driven. So when the people first recognized it and then said,
”Hey, if I can use something that’s smart for my business case, I will make a
gazillion dollars.” And everyone jumped on this early AI thing. And now
companies think, ”Hey, I must do something with AI.”
* •
Five years ago, if you had an AI startup, you’d just get stupid money without
any proof.
* •
Because hype is nothing logical. So you can see it if you really look at
startup financing, lots of the larger VCs moved away from funding AI
companies, or solidly AI companies. Now the hype is biotech, of course,
everyone wants to be the next Pfizer. And I think that’s how humans work.
* •
I guess the one dirty little truth is that I care a little about the fact that
it’s AI per se, but it gets a lot of resonance and interest when I use the
word AI as opposed to machine learning, as opposed to algorithm.
* •
Well, at a certain point, it certainly resonated with investors. Maybe we’re
at a certain point in society now where it’s almost overused, and so there’s a
certain backlash against just the general use of the word AI, but certainly
for a period there, no matter what business you’re in, in society, you had to
drop the word AI in order to seem relevant and to seem like you’re doing
something important. Ultimately, again, from a fundraising perspective, from a
customer perspective, so there’s different stages of adoption of different
technologies that generalize across a lot of different sectors of the economy.
AI or whatever it is, is relatively early in [my company’s domain], and so
now, we’re at kind of like what’s called like the early adopters’ phase, but I
think there’s an even more extreme form of that which is some [businesses that
could be clients] want to adopt. It seems like they have an incentive to adopt
AI for the sake of the fact that it’s AI, and that they can then now say that
they’re using AI
* •
It’s a check box. INTERVIEWER: What does that mean that it’s a checkbox?
PARTICIPANT: You have it or you don’t. It’s binary. They really don’t give a
rip of how good it is. They just want to say that you have it because it’s a
marketing buzz.
* •
I mean, [investors believe you have AI] because you say it and you can talk
about the algorithm. I think the vast majority don’t dig in because I think
the vast majority truly don’t understand it. I mean, there are some investors
that… if you want to maybe get to talk to them, there are some that focus just
on AI, and they have experts who know it inside out because they came from
that space. And they’re going to be savvy enough to know the difference. There
are a lot of investors who don’t. And so they’re trying to catch up with the
next big thing in tech and they’re just following whatever the buzz is.
* •
I want to say I have machine learning and AI because it makes me sound like
I’m on the cutting edge.
* •
I don’t think it’s a specific message. […] I mean, listen to CNBC, look at the
investors in startups. And I think if you scan the vast majority of startups,
anything related with tech is going to talk about their AI engine or their
machine learning engine. […] But the fact is at the end of the day, regardless
of what you’re doing, the end goal is that you’re meeting an unmet need, and
AI and machine learning is just a way that you’re getting there. So saying
that you’re using it in your product is the way that you solve the problem.
And right now, because it’s the buzz du jour and everyone wants to do it to
say, ”Yeah, we’re on the cutting edge.” So that’s why you’d see more of it. To
be fair, there are some investors that see it as, okay, this is cutting edge.
And with multiple evolutions of this, we are going to get to that point where
this starts to overtake humans in terms of their function and intelligence.
And that’ll happen at some point in the future. But I believe that with 95% of
the uses of AI and machine learning, it’s just a way to solve a particular
problem and meet a need. And it’s just a slightly better algorithm.
* •
I have my own personal definition of AI. I’ll tell you. I think it’s just the
marketing term for the ability to do massive amounts of equations in order to
make predictions. I don’t view it as actual artificial intelligence. I view it
as a marketing term in order to … The ability to use very clever algorithms to
do massive amounts of statistical calculations to make better predictions.
That’s what I view as artificial intelligence.
* •
I think when we talk to most of our customers, they’re quite aware of AI and
how this stuff works, because our target is a mostly technical audience. We
end up having this sort of conversation with mostly people from tech companies
who are likely to be our customers, and they’ve a fair understanding of AI and
how it could work. [..] Yeah, we rarely mention, the AI does it. We are AI
powered, that’s sort of understood in most cases.
* •
We speak about AI value but we don’t mention AI so much.
* •
Because most of my experience with that is in the B2B space, where everyone is
somewhat technically inclined, or at least people making the decisions are
familiar with the general typology of what’s out there. And they understand at
least and in broad details, about the benefits that AI can offer a business.
But in terms of users… Just giving an example of, when I tell new people that
I meet what my job is, what I do, they’re a lot of times, ”What’s that? I have
no idea what that is.” It’s definitely not as strong of a marketing tool
directly to users.
* •
So we’re not fundraising right now, still bootstrapping. So I don’t know if it
will help or perhaps even the opposite, be not helpful anymore. When we
fundraise, I think with, to be honest, most customers don’t recognize it.
* •
If you have something that you say is a predictor and your name is something
.ai, you lose a lot of credibility when your predictors are not quality. So it
was just this growing pains. It happened really early on when there’s probably
like a couple hundred users. We adapted to it pretty quickly but it’s
something that the real … When I see it every day in my cloud customers, it’s
a real concern.
* •
I would say from the perspective of founders that would be [a signal] to
investors and clients as well, that’s definitely not at what we do simply
because this is something that the founder, [Founder Name] went to school for
and really has a passion about computer vision. And so he wanted a true
computer vision solution moving from the get-go. But yeah, even without the AI
moniker, [compared to] before we were considered [not an AI] company, before
we [developed our AI features…] there’s more attraction from investors if you
are labeled as a AI company.
* •
…one of the things I’ve learned about what makes this distinct is that when
you’re fundraising, or if you’re even thinking about fundraising for something
even remotely related to AI/ML and the market potential is as big as it is and
you’re at a time right now where the liquidity is really, really high in the
market, they find you. [… ] It’s a very different kind of power dynamic.
* •
there oftentimes questions [from investors…] it’s just like, ”Okay, how does
your AI work?” So then we have to describe the backend processes. We have
questions about where does the data come from? So that’s an often question. I
think those are the two most common questions.
* •
INTERVIEWER: So you said that you hate that AI as in your name. Then why is it
in your name? PARTICIPANT: Pretty much because we want to ride the hype cycle
too. So let’s be honest here, of course. And I don’t see that negative as I
might’ve sounded. It helps, labels help people.
* •
[…] there are lots of already set up AI frameworks that you can just connect
to and apply to your own products and then you have an AI product. Or other
cases that I see personally most often testing different tools as a marketer
is that lots of tools are automating something, and then they say, ”Oh, I have
an AI.”
* •
I actually wouldn’t classify any of it [as AI]. Anything I see today I think
artificial intelligence is not applied because none of the algorithms are
self-aware. So I actually think that the whole… You see this a lot in tech.
I’ll use autonomous driving as an example, right now where it’s like, ”Oh, I
got full autonomous driving.” Bullshit. That’s two decades away. I mean,
anyone who really can look at those algorithms and see what’s happening
without having some sort of sensors in the road, we’re so far away from that
and all these edge cases to get there. I mean, same thing with AI. People hear
of AI and they think of robots or the character in the Marvel movies who’s
actually a fully humanoid, thinking being. No. And I could be wrong on this
because it’s not my area of expertise, but I just feel that there’s so much…
Right now it’s one of those buzzwords that everybody’s jumping on to say they
have it, but very few actually do. In reality, what it is are better
algorithms to figure things out. And to be fair, I mean, there are some things
that are really good, like the voice recognition. And if you think about the
AI like with Siri and Amazon’s Alexa and the amount of language processing
that’s happening to pull things out, I mean, that’s very impressive. But at
the end of the day, they’re not self-aware. Yet, I hope. I mean, I don’t know.
But at the end of the day, it’s just a really good algorithm. And so I think a
lot of this people are glomming on to that futuristic view of it. And it’s
that next big, futuristic thing that they can do. And I think it ranges from,
oh yeah, we’re using machine learning and AI. On the one hand it could be just
it’s an adaptive algorithm for something fairly simple that’s looking at a
relationship with two variables all the way up to now we actually have a whole
platform like Siri and it’s a different thing. And that entire continuum
contains AI. So everyone wants to say, ”Yep, I got AI. I’m just like Apple or
I’m just like Amazon.”
* •
I’m so bad with names, but if you look into the acquisitions of Salesforce
from the last three years, and I think it was 12 companies, 11 claim to do AI.
And Salesforce bought them and with big marketing, ”Hey, we bought another AI
company to do our Einstein platform,” I think it’s called, ”To make it better,
smarter, faster.” And after half a year, if you looked into it, they just
discontinued all the companies because you find a little press release or if
you know someone who works at Salesforce in San Francisco, and once again,
they couldn’t do what they claimed it could do. So I know that it’s very
episodical. INTERVIEWER: Yeah, but it sounds like then from your experience, a
lot of the companies that are claiming to do AI are telling their customers,
they can do AI, ultimately those solutions are failing. Is that accurate?
PARTICIPANT: I don’t know if they are failing, but at least they’re not
succeeding with AI technology I’d say.
* •
[…] Artificial general intelligence. They think of something that actually can
think like a human being can think, but an AI model doesn’t actually think
right? It’s just, it can maybe make you, fool you into thinking one day, but
it’s actually not really intelligence. It could do massive amounts of
calculations and use that to make predictions. It’s not the same thing as
actually, intelligence. You, as a human being, I could suddenly tell you
something completely unrelated and you could apply what you learned to figure
that out. Is there an AI model on the planet that can really do that? No.
Could you walk and run, and suddenly I teach you about, tell you a little bit
about what a movie is, and watch a movie, and you’d make comments on a movie?
That’s real intelligence. A human being has intelligence. What’s amazing about
the human brain isn’t so much that it can do things, one thing at a time. It’s
the fact that it can take things it’s learned in those things and apply them
to completely different situations and come up with its own ideas. It’s not
like an AI suddenly is going to come up with a brand new statistical [domain
of application] model for you. That requires, still, human intelligence.
That’s what I view as true intelligence. That idea that creativity, that an
idea suddenly pops in your head and then you can implement it and come up with
something new that does not exist. […] To me, that’s real intelligence.
* •
I tend never to use the word artificial intelligence. Internally I think we
almost exclusively use the word machine learning. I think the FDA now kind of
labels some of it as AI so I use it a little bit more now publicly but there
was a time where I probably only used the word machine learning because AI
feels so amorphous.
* •
I think absolutely, when anybody hears the term artificial intelligence,
they’re thinking of HAL from 2001. They’re thinking of all the science fiction
novels they’ve ever read, or all the movies they’ve ever watched, or the
Terminator from Terminator 2. I think they’re thinking of something that
mimics human behavior, that has a consciousness. INTERVIEWER: Some people
would think that evoking something that, or using a term that evokes The
Terminator would be a bad thing, would not be a good marketing tactic.
PARTICIPANT: I think that’s what they think. It’s like it can actually replace
human intelligence. But they can’t. I do not see any AI model that’s even
close to human intelligence right now. Even a child’s intelligence. Even a
child’s intelligence, I do not see it.
* •
So we’ve been around [for several years]. I think there was some pitch
competitions in the beginning when we were a small startup and I remember
there was a few other companies that were pitching and a few of them were
using the term AI and I remember just listening and realizing I don’t think
there’s any AI at all happening. I can’t quite remember the application but I
remember being kind of annoyed that they got to use the hype term whereas I
didn’t really think there was any of that happening. I think that’s probably
less so now. I think a lot of companies really are using machine learning more
than they were five years ago because if you have a lot of data, that’s the
right thing to do.
* •
So I definitely think that in the same vein as like crypto and blockchain, as
buzzwords, as companies will spin up the idea of providing a service that can
be automated with artificial intelligence and what they wind up doing is doing
a bunch of manual work to make it seem like they can provide that service. But
once they go to scale, it doesn’t scale very well because you’re still doing a
lot of things manually and not doing a AI data driven approach first, because
from day one, we started with a, our proprietary model that we began training
to make sure that we weren’t like, ”All right, well, submit us [the raw data]
and then we’ll process it and then give it back to you.” That kind of thing,
because it’s just not a scalable solution.
* •
But in terms of why we decided to include a bunch of real AI, deep learning,
all that kind of stuff, it’s the only solution that will enable our grander
vision.
* •
Personally, it’s a pet peeve of mine, that some companies will call their
solution AI. And it’s like, ”Okay. You used a random forest. Congratulations.”
But no. There’s that. You can get value. The value in the technology that you
use doesn’t necessarily even have to come from the technology. Just saying you
use AI. In the same way that if you’re interviewing, saying that there’s like,
”Oh yeah. I know like ML, I know these ML frameworks, whatever.” Even if you
don’t really. It’s a huge bonus. I think the value we deliver is like, ”We do
use AI/ML technologies. Here is exactly where we use them. And then here’s how
it’s making your life easier.” And being able to actually substantiate that
with results from our application, I think it’s more than just a marketing
edge and it substantiates the claim.
* •
On [a recent date], I did a pitch event. It was a virtual pitch event with
about five or six investors and an audience as well. It was a three-minute
pitch. I ran through a slide deck, talked about the core technologies and the
products that we’re building. After that, there’s was a two-minute QA where
the investors asked me different questions. So from that, four out of five of
the investors reached out to me to say, hey, I’m interested in what you’re
doing. So I said, hey, thanks for your interest. I’m not currently raising
funds, but I’m more than happy to keep you informed about the progress we’re
doing. Here’s some materials you can read over. We’ll reach out to you every
month with our progress.
* •
The reason we use the word is just it sounds cool and people like that in the
marketplace. Like, ”Oh, you’re going to do AI on my data.” I’m like, ”Well
we’re going to do machine learning on your data.” I don’t really have the need
to do AI where I would with these kind of things, right? It’s not that
complex. I’m sure someone could. Like if you’re trying to say, ”Hey, I’m
looking at this data and maybe can I impute whether you have an illness or
something?” That probably would take something a little bit more nuanced and
AI-centric. But that’s not what we’re trying to do.
* •
So our overall strategy is to take as little investment as possible. And
there’s two sides to that. One is to keep the burn as low as possible, two is
to get to revenue as quickly as possible. And we began with seed investors. We
actually did [a University angels funding event], that was some of our first
money in. We had great experience with a crowdfunding platform. Easiest
fundraising I’ve ever done in my life. We raised a [an approximate sum] in
$10,000 checks in one day. One day, I couldn’t believe it. And it made me look
at these things differently. It allows you to control your destiny a lot more.
* •
There’s angel investors. There’s VCs. It depends. I can’t pick the kinds of
investors that come to these [pitch] events. I only know the panel maybe after
I sign up, but it’s a good mix. It’s fairly classified angel investors,
accredited angel investors. Only those accredited angel investors can actually
invest in startups. There’s a key distinction. Not everybody can give you
money. I have to say no to a lot of people, actually, because we don’t have
enough of a net worth to do business. But if they’re accredited investors or
venture capitalists, we’ll keep them on our Rolodex.
* •
I always try to speak to the dumbest person in the room. Based on the
questions, I’ll go into the level of detail that’s appropriate. I never like
to start up here because you just lose the audience immediately. The biggest
barrier to data science is being able to tell the story. I think a lot of data
scientists really are not good at that. So if you have really technical
founders, they don’t know how to relate their product to laymen, and so I
always try to be cognizant of the fact that I’m talking to people that are not
industry experts. They’re experts at determining value propositions, and so I
need to be able to accurately say what my value proposition is.
* •
So we’re a venture backed company, so we’ve raised a Series A in the past, and
as a company we have to continue raising. So our funders are typically other
venture capitalists, we talk to some strategics, so other big companies in the
area that are really interested in what we’re doing, but kind of want to play
an observer role. So we talk to them and then we also try to make use of
government funding as well, so there’s some kind of programs available for
non-dilutive funding for grants. So we make use of those three avenues of
funding.
* •
I think we certainly want to make sure any VCs that we work with are well-
aligned with us and can provide added value. We’re not just looking for source
of funding but also some support and oftentimes VCs or funders, equity funders
will take a board seat, and so we want to make sure we work well with them. So
there certainly is a mutual vetting process any time that there’s a potential
relationship in the works. But in general, the VCs that we work with tend to
be in the healthcare space and be from backgrounds that could help us, they
have a lot of contacts or they know how to build these types of devices or how
I think through pricing models, commercial strategy, things like that.
* •
So we’re funded through investors. So [Company Name] has done a seed round,
raised about [an approximate sum of money]. And so we’re very lean. I have
done venture backed businesses. And I actually try and avoid that business
model because I think oftentimes, the VC’s goals are at cross purposes with
building a long term successful business.
* •
And [at a previous startup] the VCs wanted to hype things up, get a lot of
press, make a splash, so they could raise the next round at a higher valuation
and look good to their LPs, which was actually contrary to what we needed to
do for the slow growth to build the business. In [current startup name], our
goal is to get to revenue and cashflow positive as quickly as possible. In
many ways, VCs, they won’t say this directly, but as an entrepreneur, once you
have revenue, it’s problematic from a fundraising standpoint. Because before
you have revenue, you’re all about promise and potential. Once you have
revenue, there are metrics. And the question becomes, why aren’t you growing
faster? And it’s very rare that something comes along with the true hockey
stick growth that VCs are looking for. So that actually puts you in this mode
where you want to go on the hype, as long as you can, put off revenue as far
as you can, which puts the entrepreneur in a defensive position, because the
only option then is to raise VC money for the next round, and of course, the
VCs want you to spend more and more, because that’s what their metrics to
their LPs look like. So I found that for many, many businesses, it’s not the
right model. There certainly are cases where if you talk to Peter Wendell at
Sierra, he’ll say, ”I sell rocket fuel.” If you’re not going to Mars, you
don’t need rocket fuel. There are actually very few businesses where rocket
fuel is the right thing. So I’m mindful of that, and that’s something I’ve
learned. INTERVIEWER: It sounds like that can create a vicious cycle for the
entrepreneur in terms of interrupting good product development. PARTICIPANT:
Totally. And the burn gets higher and higher, your runway gets shorter and
shorter, the expectations diverge from reality faster and faster. Yet the
entrepreneur is in a situation where, I’d feel this very viscerally where I’d
been one thing to my customers, they care about what I’m doing today. And my
investors would care about where’s this going to be in five years? And as that
gap got bigger and bigger, that’s a huge source of stress.
* •
VCs are doing just fine these days. And it’s also like, where people are in
their careers and their experience levels, there will always be a pipeline of
people coming fresh out of college early in their career, where they want that
rocket fuel. And they don’t realize that it’s only going to work out for 5
percent of them. And as you get later in your career, you realize there’s a
lot more options than VCs. There’s other ways to build a business. Whereas
when you start out, that seems like the, maybe it’s the way we’re taught, but
that seems to me like the only option. I didn’t even know there were other
options besides that, to build a business.
* •
I personally do not want to pursue funding just because then it will be like a
real job. Like I’ll be in debt to someone and someone else will influence what
we’re doing. If I didn’t work every day, that might be interesting, but it’s
not interesting right now. It’s like an endless stream of people who want me
to go present and talk to people but it’s like if I do that, what’s the
benefit to me? It could be useful to solve some problems, but I think it adds
a lot of complexity to where it’s like all of a sudden, ”Eh,” like I’m having
to spend several hours a day on this as opposed to if I’m working hard at [my
main job], this stuff goes on pause. If [my main job] stuff is chill, I can
work nights and weekends on this. That’s the other thing is the more
interesting more for me is to have an equity partner who is really good at
user experience and maybe one who is really good at data science as opposed to
someone who just has deep pockets, right? Then it’s all we’re doing this for
our passions and what our shared interests are as opposed to trying to get
rich. Maybe some day it will be something I could sell to someone if it has a
large user base and has a proven track record of profitability, that’s
probably interesting. But until then I don’t need somebody to give me a bunch
of money.
* •
The best investors and the most helpful ones have really been staying up to
date with what we’re doing, and we have a number of asks that we ask of our
investors and we have great investors who will follow through with that. And
it really depends on who they are, so if we’re looking for advisors in a
specific, like [one aspect of the industry domain], we know who to ask. If
we’re looking for advisors in brand building or marketing, there’s people to
ask those questions. Or how do we think about press, something like that. And
so, it’s really folks that are engaged and willing to help in the areas where
they’re able to.
* •
I think there’s not really a formula for it, it’s kind of a feel of would I
want this person on the board and likewise on their end, it’s do I have enough
belief in this company that I think they’re really going to succeed and I want
to work with these founders or work with the management team.
* •
We did it pretty simple, we put up a website, said what we have as a product
and looked who signed up and then went from there. And yeah, we had both
pretty well-paying jobs before, so we didn’t have to charge a lot, we wanted
just to get iteration speed and did that. And then we somehow noticed, ”Hey,
we now have X customers and the company is running itself more or less.” And
yeah, so we stumbled into it.
* •
So, in our case, we raised pre-seed. So not yet to seed. I think that’s still
something we’re working on. So for the pre-seed, we applied for an accelerator
program. We actually applied for a few. And then luckily we had a choice at
the end, so that was great. But yeah, we basically just applied for a few
accelerator programs, because we felt like we wanted to go through a program
to get mentors on top of just the funding for the early stage, and this is
what they did. That’s kind of was our focus.
* •
And because I think Silicon Valley has this idea of, you’ve got to grow like a
rocket ship speed and huge margins.
* •
So we had some great conversations; a few investors who actually were
interested in joining, but I didn’t feel like they were in line with our own
vision, so we didn’t agree to proceed there. INTERVIEWER: What about them made
you feel like they weren’t aligning with your vision? PARTICIPANT: Yeah, so
like one of the very prominent examples is that a VC’s focus specifically to
just do the seed round and then make sure that the startup doesn’t get
additional investment and just exits, so basically sells the company right
after, which is not what we want to do. I want to develop our technology to
have a [the fully developed product] for everyone, and then go for an IPO.
* •
But I also don’t think we would have built ML if we didn’t have VC money. And
I think you’re right. We would have probably gone to more stable, ”Let’s build
a good [core services business] with good software. And once that is on sure
footing, then let’s maybe some special projects.”
* •
This is something we actually see with our own customer base. We recently had
a fundraising round, and instead of going through VCs, we’ve generated … It’s
everybody who’s invested in us uses our product. It’s very simple. You come in
here, you see the recommendations we get. You try them out and they work, and
you’re like, ”Oh, wow. This is legit stuff. This is very, very legit stuff.”
You know?
* •
What maximizes efficiency, and what’s something that people are willing to pay
for? And oftentimes it’s efficiency. I’m so cynical. I don’t think it’s
quality.
* •
So the biggest thing about ML is like, oh, take a step back. Software is
pretty cheap to make and run. Other than your engineers, the most expensive
part. But for machine learning, you have to label a lot of data. You have to
label thousands of images for your training, for your validation. And then you
have to pay [specialists to label your data]. That’s what’s hard. You have to
pay people who are [specialists] otherwise then you get bad ML.
* •
So angels, we consider any individual that’s not part of an institutional
fund. And the strategic part of that, are really people who have subject
matter expertise in something having to do with our business. […]. And so, we
have angels that really have that diversity of expertise in those areas.
* •
INTERVIEWER: What do you think motivates the questions they ask you about your
modeling approach or data approaches? PARTICIPANT: Revenue generation
opportunities. INTERVIEWER: Can you tell me more about that? PARTICIPANT: I
would say first and foremost, investors are investing in a company because
they believe that it could grow and scale, and ultimately exit and provide
liquidity. So, the questions or conversations that we’re having from a
business operations, whatever standpoint, is all thinking about how are we
building the best business that we can, so that we can grow as quickly as we
can to create some sort of liquidity for investors? And so, I would say that’s
the basis of the questions.
* •
I would say both because institutional investors that want to invest in AI
companies that are very mindful of how they’re applying AI and doing the
thinking for the investors essentially being like, look we are aware of all of
the problems at large and upcoming regulation and all of the confusion around
it. And we want to stay ahead of it and educate people about it and do it in
as transparently as possible manner to make sure that people are comfortable
with the solution before it is deployed en mass.
* •
We’re VC backed, which means that we believe that we can IPO at some point. If
the IPO doesn’t happen, then there’s a number of strategic exit opportunities
that would make sense for this kind of company.
* •
We’ve had private investment and then we also have a crowd funding round that
we raised. I think we’re almost at [a sum less than 1M], anyway it’s on [a
publicly available site…]. But that has a lot of our publicly available
information as to share count a number of investors and all that kind of
stuff.
* •
So we went the route of equity crowd funding, so it’s a little bit different
than like Kickstarter or something like that. Yeah. And so one of the main
benefits is we’re like, ”Hey, if you invest in the company, then you own a
share account. And so there’s potential for you to have a return on that
investment.” And that’s the major selling point of doing equity crowd funding
like this. And was it really only possible because of newer federal laws that
allow that kind of funding. The main benefit was just a little bit more
transparency to individual people. And we learned a lot from it. There’s a lot
of people that we talk to that are business sales, engineering, whatever. And
they’re like, ”Okay, we’ll take this to our leadership and talk about doing
business with you, but at a personal level we would like to invest.”
* •
Yeah, I would say the downside of it is just that funding, it trickles in and
it’s less lump sum payments, but it really just depends. The main downside
that we face through this equity crowdfund funding is through the company that
|
# Reduced Dirac Equation And Lamb Shift As An Off-mass-shell
Effect In Quantum Electrodynamics
Ni Guang-jiong a,b<EMAIL_ADDRESS>a Department of Physics, Fudan University,
Shanghai, 200433, China
b Department of Physics, Portland State University, Portland, OR97207, U. S.
A. Xu Jianjun<EMAIL_ADDRESS>Department of Physics, Fudan University,
Shanghai, 200433, China Lou Senyuec,d<EMAIL_ADDRESS>c Department of
Physics, Shanghai Jiao Tong University, Shanghai, 200030, China
d Department of Physics, Ningbo University, Ningbo 315211, China
###### Abstract
Based on the precision experimental data of energy-level differences in
hydrogenlike atoms, especially the $1S-2S$ transition of hydrogen and
deuterium, the necessity of introducing a reduced Dirac equation with reduced
mass as the substitution of original electron mass is stressed. Based on new
cognition about the essence of special relativity, we provide a reasonable
argument for reduced Dirac equation to have two symmetries, the invariance
under the (newly defined) space-time inversion and that under the pure space
inversion, in a noninertial frame. By using reduced Dirac equation and within
the framework of quantum electrodynamics in covariant form, the Lamb shift can
be evaluated (at one-loop level) as the radiative correction on a bound
electron staying in an off-mass-shell state–a new approach eliminating the
infrared divergence. Hence the whole calculation, though with limited
accuracy, is simplified, getting rid of all divergences and free of ambiguity.
Keywords: Reduced Dirac Equation, Lamb shift, off-mass-shell
PACC: 0365, 1110G, 1220D
## I Introduction
As is well known, the Dirac equation for electron in a hydrogenlike atom is
usually treated as a one-body equation with the nucleus being an inert core
having infinite mass and exerting a potential $V(r)=-\frac{Z\alpha}{r}\ \
(\hbar=c=1)$ on the electron. Then the rigorous solution of energy levels
reads[1]:
$\displaystyle E_{nj}$ $\displaystyle=$ $\displaystyle m_{e}f(n,j)$ (1)
$\displaystyle f(n,j)$ $\displaystyle=$
$\displaystyle\left[1+\frac{(Z\alpha)^{2}}{(n-\beta)^{2}}\right]^{-\frac{1}{2}}$
(2) $\displaystyle\beta$ $\displaystyle=$ $\displaystyle
j+\frac{1}{2}-\sqrt{(j+\frac{1}{2})^{2}-(Z\alpha)^{2}}$ (3)
where $j$ is the total angular momentum. The expansion of $f(n,j)$ to the
power of $(Z\alpha)^{6}$ is given as 1
$\begin{array}[]{l}f(n,j)=1-\frac{(Z\alpha)^{2}}{2n^{2}}-\frac{(Z\alpha)^{4}}{2n^{3}}\left(\frac{1}{j+\frac{1}{2}}-\frac{3}{4n}\right)\\\
-\frac{(Z\alpha)^{6}}{8n^{3}}\left[\frac{1}{(j+\frac{1}{2})^{3}}+\frac{3}{n(j+\frac{1}{2})^{2}}+\frac{5}{2n^{3}}-\frac{6}{n^{2}(j+\frac{1}{2})}\right]+\cdots\end{array}$
(4)
Obviously, besides the rest energy of the electron given by the first term,
the second term has exactly the form of Bohr energy level except that the mass
$m_{e}$ must be replaced by the reduced mass
$\mu=\frac{m_{e}m_{N}}{m_{e}+m_{N}}\equiv\frac{m_{e}m_{N}}{M}$ (5)
with $m_{N}$ being the mass of the nucleus and $M=m_{e}+m_{N}$.
However, as discussed in Refs.1 and 2 , the concept of reduced mass in
relativistic quantum mechanics (RQM) is ambiguous to some extent. Beginning
from 1950’s, a number of authors have been devoting a great effort at the
level of two-body RQM and that of quantum electrodynamics (QED) to take
account of the recoil effect 3 ; 4 ; 1 , incorporating their results in a
compact form (to order of $\alpha^{4}$):
$E=M+\mu[f(n,j)-1]-\frac{{\mu}^{2}}{2M}[f(n,j)-1]^{2}+\frac{(Z\alpha)^{4}{\mu}^{3}}{2n^{3}m_{N}^{2}}\left[\frac{1}{j+\frac{1}{2}}-\frac{1}{l+\frac{1}{2}}\right](1-{\delta}_{l0})$
(6)
A comprehensive review on the theory of hydrogenlike atoms can be found in
Ref.27 (27). The aim of this paper is two-fold: First, based on the
experimental data of hydrogen $1S-2S$ transition frequency 5 and its isotope
shift of hydrogen and deuterium 6 , we stress the necessity of the
introduction of reduced mass $\mu$ (section II) before we are able to argue
the reasonableness of introducing a ”reduced Dirac equation” with $\mu$ as the
substitution of $m_{e}$ (section III). Second, based on above conception, we
will present a calculation of Lamb Shift (LS) as an off-mass-shell effect by
performing the evaluation of self-energy diagrams of electron (section IV) and
photon (section V) as well as the vertex function (section VI) at the one-loop
level of QED in covariant form. The new insight of our calculation is focused
on the regularization renormalization method (RRM). As initiated by J-F Yang 7
and elaborated in a series of papers (8a ; 8b ; 9 (9, 24, 25, 26) and
references therein), we can get rid of all ultra violet divergences in the
calculation of quantum field theory (QFT). Furthermore, in this paper, we will
be able to get rid of the annoying infrared divergence in the vertex function
by treating the electron moving off its mass-shell to certain extent which is
fixed through the evaluation of self-energy diagram or by the Virial theorem.
Based on above improvements, the one-loop calculation yields values of LS in a
simple but semi-quantitative way (section VII and VIII). Although the accuracy
is limited at one-loop level, we hope our approach could be served as a new
starting point for calculations at high-loop order to get accurate results at
a comparably low labor cost. The final section IX and Appendix will contain a
summary and discussion.
## II The $1S-2S$ Transition of Atomic Hydrogen and Deuterium
In the last decade, thanks to remarkable advances in high resolution laser
spectroscopy and optical frequency metrology, the $1S-2S$ two-photon
transition in atomic hydrogen $H$ (or deuterium $D$) with its natural
linewidth of only $1.3Hz$ had been measured to a very high precision. In 1997,
Udem et al.determined the $1S-2S$ interval of $H$ being 5
$f^{(H)}(1S-2S)=2466061413187.34(84)\ \ kHz$ (7)
Even four years earlier, Schmidt-Kalar et al. measured the isotope-shift of
the $1S-2S$ transition of $H$ and $D$ to an accuracy of $3.7\times 10^{-8}$6 ,
giving (as quoted in 10 (10)):
$\Delta f\equiv f^{(D)}(2S-1S)-f^{(H)}(2S-1S)=670994337(22)\ \ kHz$ (8)
(In 1998, Huber et al.measured a more accurate data 28 (28):
$670994334.64(15)\ kHz$). which is of the order of $10^{-4}$ in comparison
with Eq.(7). As pointed out in Ref.6 , this $671\ GHz$ isotope-shift can be
ascribed almost entirely to the different masses of proton ($p$) and deuteron
($d$). And the nuclear volume effects become important because the QED effects
cancel considerably in the isotope shift.
Here, we wish to emphasize that in the first approximation, both experimental
data (7) and (8) can be well accounted for by simply resorting to Eq.(1) with
$m_{e}$ replaced by the reduced mass
$\mu_{H}=\frac{m_{e}m_{p}}{m_{e}+m_{p}},\ \ \
\mu_{D}=\frac{m_{e}m_{d}}{m_{e}+m_{d}}$ (9)
for $H$ and $D$ respectively.
Indeed, adopting the following updated values 10 (10, 11, 12, 13)
$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle(137.03599944)^{-1},\ \ \
{\alpha}^{2}=0.532513542\times 10^{-4}$ (10) $\displaystyle{\alpha}^{4}$
$\displaystyle=$ $\displaystyle 0.283570673\times 10^{-8},\ \ \
{\alpha}^{6}=0.151005223\times 10^{-12}$ (11) $\displaystyle m_{e}$
$\displaystyle=$ $\displaystyle 0.51099906\ \ MeV=1.2355897\times 10^{20}\ \
Hz$ (12) $\displaystyle R_{\infty}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\alpha}^{2}m_{e}=3.28984124\times 10^{15}\ \ Hz$
(13) $\displaystyle\frac{m_{p}}{m_{e}}$ $\displaystyle=$ $\displaystyle
1836.1526665$ (14)
and denoting
$\displaystyle\frac{m_{e}}{m_{p}}$ $\displaystyle=$ $\displaystyle
b_{H}=5.446170255\times 10^{-4},\ \ \ \frac{1}{1+b_{H}}=0.999455679$ (15)
$\displaystyle\frac{m_{e}}{m_{d}}$ $\displaystyle=$ $\displaystyle
b_{D}=2.724436319\times 10^{-4},\ \ \ \frac{1}{1+b_{D}}=0.99972763$ (16)
we can calculate the energy difference of $2S$ and $1S$ of $H$ through Eq.(1)
with $m_{e}$ replaced by $\mu_{H}$ (the superscript RDE refers to the reduced
Dirac equation)
$\displaystyle\Delta E^{RDE}_{H}(2S-1S)$ $\displaystyle=$
$\displaystyle\mu_{H}[f(2,1/2)-f(1,1/2)]$ (17) $\displaystyle=$
$\displaystyle\frac{m_{e}}{1+b_{H}}(1.996950159\times 10^{-5})$
$\displaystyle=$ $\displaystyle 1.2355897\times 10^{20}\times
0.999455679\times 1.996950159\times 10^{-5}$ $\displaystyle=$ $\displaystyle
2.466067984\times 10^{15}\ \ Hz$
which is only a bit larger than the experimental data Eq.(7) with accuracy
$3\times 10^{-6}$. However, a more stringent test of RDE should be the isotope
shift of $H$ and $D$. We have
$\frac{1}{1+b_{D}}-\frac{1}{1+b_{H}}=(b_{H}-b_{D})-(b^{2}_{H}-b^{2}_{D})+(b^{3}_{H}-b^{3}_{D})+\cdots=2.719511528\times
10^{-4}$ (18)
$\Delta E^{RDE}_{D-H}=(\mu_{D}-\mu_{H})[f(2,1/2)-f(1,1/2)]=6.7101527879\times
10^{11}\ \ Hz$ (19)
which has only a discrepancy larger than the experimental data, Eq.(8) by
$20.941\ MHz$ with accuracy $3\times 10^{-5}$. Of course, it is still not
satisfied in an analysis of high precision 6 . Let us resort to the Eq.(6),
where the third term does provide a further modification:
$\displaystyle-$
$\displaystyle\frac{1}{2}m_{e}[\frac{b_{D}}{(1+b_{D})^{3}}-\frac{b_{H}}{(1+b_{H})^{3}}]\\{[f(2,1/2)-1]^{2}-[f(1,1/2)-1]^{2}\\}$
(20) $\displaystyle=$
$\displaystyle\frac{1}{2}m_{e}[(b_{H}-b_{D})-3(b^{2}_{H}-b^{2}_{D})+\cdots](-6.646361554\times
10^{-10})=-11.176\ MHz$
which brings the discrepancy between the theory and experimental down to less
than $10\ MHz$.
Although the detail explanation for this discrepancy remains quite
complicated6 , the above comparison is enough to convince us that the
inevitable appearance of reduced mass in the RDE or Eq.(6) is by no means a
simple fortune. It must have a deep reason from a theoretical point of view.
Notice further that once the conditions $m_{e}\ll m_{p}$ and $m_{e}\ll m_{d}$
hold, the difference of spin between $p$ and $d$ seems not so important. So in
next section, we will strive to justify the reduced Dirac equation on a
reasonable basis. Of course, it is still an approximate one, but seems much
better than the original Dirac equation when dealing with hydrogenlike atoms.
## III Reduced Mass and Reduced Dirac Equation
Consider a system of two particles with rest masses $m_{1}$ and $m_{2}$. Their
coordinates in the center-of-mass (CM) system are ${\mathbf{r}}_{1}$ and
${\mathbf{r}}_{2}$ respectively, as shown in Fig.1. If there is a potential
$V(r)=V(|{\mathbf{r}_{1}-r_{2}}|)$ between them, two equations
$m_{1}\ddot{\mathbf{r}}_{1}=-\nabla_{r}V(r)$ and
$m_{2}\ddot{\mathbf{r}}_{2}=\nabla_{r}V(r)$ will reduce to one:
$\mu\frac{d^{2}{\mathbf{r}}}{dt^{2}}=-\nabla_{r}V(r),\ \ \
(\mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}})$ (21)
At first sight, the definition of center-of -mass (CM) in classical mechanics
$m_{1}r_{1}=m_{2}r_{2}$ becomes doubtful in the theory of special relativity
(SR) because the mass is no longer a constant. But actually, we can still
introduce the coordinate of CM in the laboratory coordinate system (LCS) (with
${\mathbf{r}}^{\prime}_{1}$ and ${\mathbf{r}}^{\prime}_{2}$ being the
coordinates of $m_{1}$ and $m_{2}$):
$R=\frac{1}{M}(m_{1}{\mathbf{r}}^{\prime}_{1}+m_{2}{\mathbf{r}}^{\prime}_{2})=(X,Y,Z),\
\ \ (M=m_{1}+m_{2})$ (22)
and the relative coordinate of $m_{1}$ and $m_{2}$
(${\mathbf{r}}_{i}={\mathbf{r}}^{\prime}_{i}-{\mathbf{R}},\,i=1,2$) :
${\mathbf{r}}={\mathbf{r}}^{\prime}_{1}-{\mathbf{r}}^{\prime}_{2}={\mathbf{r}}_{1}-{\mathbf{r}}_{2}=(x,y,z)$
(23)
Here the motion of CM in the LCS is assumed to be slow and so
$\frac{\partial}{\partial
x^{\prime}_{1}}=\frac{m_{1}}{M}\frac{\partial}{\partial
X}+\frac{\partial}{\partial x},\ \ \ \frac{\partial}{\partial
x^{\prime}_{2}}=\frac{m_{2}}{M}\frac{\partial}{\partial
X}-\frac{\partial}{\partial x}$ (24)
Notice that the momentum $\mathbf{P}$ of CM and the relative momentum
${\mathbf{p}}_{r}$ becomes operator in quantum mechanics (QM) without explicit
dependence on mass:
${\mathbf{P}}=-i\hbar\nabla_{\mathbf{R}},\ \ \
{\mathbf{p}}_{r}=-i\hbar\nabla_{\mathbf{r}}$ (25)
Thus the momenta of $m_{1}$ and $m_{2}$ in laboratory coordinate system (LCS)
read:
${\mathbf{p}}^{\prime}_{1}=-i\hbar\nabla_{{\mathbf{r}}^{\prime}_{1}}=\frac{m_{1}}{M}{\mathbf{P}}+{\mathbf{p}}_{r},\
\ \
{\mathbf{p}}^{\prime}_{2}=-i\hbar\nabla_{{\mathbf{r}}^{\prime}_{2}}=\frac{m_{2}}{M}{\mathbf{P}}-{\mathbf{p}}_{r}$
(26)
Since the center-of-mass coordinate system (CMCS) is also an inertial frame
which can be transformed from the LCS via a linear Lorentz transformation, it
is defined by the condition that $\mathbf{P}=0$ in CMCS. In other words, CMCS
is defined by the condition ${\mathbf{p}}_{1}+{\mathbf{p}}_{2}=0$, or from
Eq.(26):
${\mathbf{p}}^{\prime}_{1}=-i\hbar\nabla_{{\mathbf{r}}^{\prime}_{1}}={\mathbf{p}}_{r},\
\ \
{\mathbf{p}}^{\prime}_{2}=-i\hbar\nabla_{{\mathbf{r}}^{\prime}_{2}}=-{\mathbf{p}}_{r}$
(27)
Evidently, the above definition of CMCS remains valid in the realm of
relativistic QM (RQM) even the exact meaning of CM seems obscure to some
extent due to the conjugation relation of $a$ particle’s position and its
momentum, see Fig.1.
Now, from Eq.(27), it is natural to replace $\mathbf{p}_{1}$ and
$\mathbf{p}_{2}$ by ${\mathbf{p}}_{r}$, reducing the two-particle degrees of
freedom to one. In the meantime, the origin of CMCS is discarded, it is
substituted by the position of $m_{2}$
(${\mathbf{r}}={\mathbf{r}}_{1}-{\mathbf{r}}_{2}$). We will call the system
associated with $\mathbf{r}$ the relative motion coordinate system (RMCS),
which should be viewed as a deformation of CMCS. The transformation from CMCS
to RMCS is by no means a linear one. Rather, the origin of RMCS ($m_{2}$) is
moving non-uniformly in the CMCS. Therefore, while rest masses $m_{1}$ and
$m_{2}$ remain the same in both LCS and CMCS, they reduce to one mass
$\mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$ for $m_{1}$ in RMCS (or for $m_{2}$ if
$m_{1}$ is chosen as the origin of RMCS).
Let us express the total energy $E=E_{1}+E_{2}$ in CMCS in terms of $p_{r}$
and reduced mass $\mu$ ($\mu=\frac{m_{1}m_{2}}{M},\ M=m_{1}+m_{2}$), where
$E_{1}=\sqrt{m_{1}^{2}+p_{1}^{2}}=\sqrt{m_{1}^{2}+p_{r}^{2}},\ \ \
E_{2}=\sqrt{m_{2}^{2}+p_{2}^{2}}=\sqrt{m_{2}^{2}+p_{r}^{2}}$ (28)
Treating all $p_{1},p_{2}$ and $p_{r}$ being $c$-numbers, we have
$E^{2}=(E_{1}+E_{2})^{2}=M^{2}+\frac{M}{\mu}p_{r}^{2}+\frac{1}{4\mu^{2}}p_{r}^{4}(4-\frac{M}{\mu})+\cdots$
(29)
where the expansion in $p_{r}$ is kept to the order of $p_{r}^{4}$. Two
extreme cases will be considered separately:
A. $m_{2}\gg m_{1},\ \ \mu\lesssim m_{1},\ \ M\gg\mu$:
$\displaystyle E^{2}$ $\displaystyle=$ $\displaystyle
M^{2}\left[1+\frac{1}{\mu
M}p_{r}^{2}-\frac{1}{4M\mu^{3}}p_{r}^{4}(1-\frac{4\mu}{M})+\cdots\right]$
$\displaystyle E$ $\displaystyle=$ $\displaystyle M\left[1+\frac{1}{2\mu
M}p_{r}^{2}-\frac{1}{8M\mu^{3}}p_{r}^{4}(1-\frac{3\mu}{M})+\cdots\right]$ (30)
$\displaystyle=$ $\displaystyle
M+\frac{1}{2\mu}p_{r}^{2}-\frac{1}{8\mu^{3}}p_{r}^{4}+\cdots$
$\displaystyle\simeq$ $\displaystyle M-\mu+\sqrt{\mu^{2}+p_{r}^{2}}\simeq
m_{2}+(m_{1}-\mu)+\sqrt{\mu^{2}+p_{r}^{2}}$
$E^{\prime}\equiv E-m_{2}=(m_{1}-\mu)+\sqrt{\mu^{2}+p_{r}^{2}}$ (31)
B. $m_{1}=m_{2}=m,\ \ \mu=\frac{m}{2},\ \ M=2m=4\mu$ Then to the accuracy of
$p_{r}^{4}$, we have :
$\displaystyle E^{2}$ $\displaystyle=$ $\displaystyle
M^{2}+\frac{M}{\mu}p_{r}^{2}=4m^{2}+4p_{r}^{2}$ $\displaystyle E$
$\displaystyle=$ $\displaystyle
2m+\frac{1}{2\mu}p_{r}^{2}-\frac{1}{32\mu^{3}}p_{r}^{4}+\cdots$ $\displaystyle
E^{\prime}$ $\displaystyle\equiv$ $\displaystyle
E-M=\frac{1}{2\mu}p_{r}^{2}-\frac{1}{32\mu^{3}}p_{r}^{4}\simeq\frac{1}{2\mu}p_{r}^{2},\
\ \ (\mbox{if}\ \ p_{r}^{2}\ll\mu^{2})$ (32)
It is interesting to see that after introducing $\mu$ and $p_{r}$, the energy
$E^{\prime}$ in RMCS looks quite ”relativistic” in the case A whereas it looks
rather ”non-relativistic” in the case B even both of them are derived from the
relativistic expressions, Eq.(28), approximately.
Since the RMCS is not an inertial system, the original mass of $m_{1}$ in CM
changes abruptly to $\mu$ as shown in Eq.(31). How can we derive the reduced
Dirac equation (RDE) in RMCS? Fortunately, we already found a basic symmetry,
the space-time inversion symmetry, which not only serves as the essence of
special relativity (SR), but also goes beyond it to derive the original Dirac
equation and the tachyon theory for neutrinos 14 (14, 15, 16, 17). Based on
this symmetry, we are going to derive the equation in RQM for case either A or
B respectively.
Let us consider case B ($m_{1}\simeq m_{2}$) first. The motivation is stemming
from the success of using the Schrödinger equation to heavy-quarkoniums like
$c\bar{c}$ and $b\bar{b}$ in particle physics (18 (18), see also 15 (15) §9.5
D). Ignoring the spin of both $m_{1}$ and $m_{2}$, we assume the coupling
equations in laboratory system for the two-particle system as:
$\left\\{\begin{array}[]{ll}i\hbar\frac{\partial\varphi}{\partial
t}&=(m_{1}+m_{2})c^{2}\varphi+V(|{\mathbf{r}^{\prime}_{1}-\mathbf{r}^{\prime}_{2}}|)(\varphi+\chi)-(\frac{\hbar^{2}}{2m_{1}}\nabla^{2}_{\mathbf{r}^{\prime}_{1}}+\frac{\hbar^{2}}{2m_{2}}\nabla^{2}_{\mathbf{r}^{\prime}_{2}})(\varphi+\chi)\\\
i\hbar\frac{\partial\chi}{\partial
t}&=-(m_{1}+m_{2})c^{2}\chi-V(|{\mathbf{r}^{\prime}_{1}-\mathbf{r}^{\prime}_{2}}|)(\varphi+\chi)+(\frac{\hbar^{2}}{2m_{1}}\nabla^{2}_{\mathbf{r}^{\prime}_{1}}+\frac{\hbar^{2}}{2m_{2}}\nabla^{2}_{\mathbf{r}^{\prime}_{2}})(\varphi+\chi)\end{array}\right.$
(33)
where $\varphi=\varphi({\mathbf{r}^{\prime}_{1},\mathbf{r}^{\prime}_{2}},t)$
and $\chi=\chi({\mathbf{r}^{\prime}_{1},\mathbf{r}^{\prime}_{2}},t)$ are
hidden ”particle” and ”antiparticle” fields of the two-particle system (From
now on, the ${\mathbf{r}}^{\prime}_{i}(i=1,2)$ is the flowing coordinate of
”fields” in QM, i.e., that of ”fictitious point particles”. See Fig.1).
Eq.(33) remains invariant under the (newly defined) space-time inversion
(${\mathbf{r}^{\prime}_{1}\rightarrow-\mathbf{r}^{\prime}_{1},\mathbf{r}^{\prime}_{2}\rightarrow-\mathbf{r}^{\prime}_{2}},t\rightarrow-t$):
$\left\\{\begin{array}[]{ll}\varphi({-\mathbf{r}^{\prime}_{1},-\mathbf{r}^{\prime}_{2}},-t)&\longrightarrow\chi({\mathbf{r}^{\prime}_{1},\mathbf{r}^{\prime}_{2}},t)\\\
\chi({-\mathbf{r}^{\prime}_{1},-\mathbf{r}^{\prime}_{2}},-t)&\longrightarrow\varphi({\mathbf{r}^{\prime}_{1},\mathbf{r}^{\prime}_{2}},t)\end{array}\right.$
(34)
$V({-\mathbf{r}^{\prime}_{1},-\mathbf{r}^{\prime}_{2}},-t)\longrightarrow
V({\mathbf{r}^{\prime}_{1},\mathbf{r}^{\prime}_{2}},t)$ (35)
Note that, however, the time $t$ is not contained in $V$ explicitly. Eq.(35)
merely means that both $m_{1}$ and $m_{2}\ (m_{1}\approx m_{2})$ transform
into their antiparticles under the space-time inversion. Actually, the hidden
antiparticle field $\chi$ enhances in nearly equal strength in $m_{1}$ and
$m_{2}$ when fictitious particles’ velocities increase with the enhancement of
attractive potential $V(r)$.
After introducing the CM coordinate
${\mathbf{R}}=\frac{1}{M}(m_{1}{\mathbf{r}^{\prime}}_{1}+m_{2}{\mathbf{r}^{\prime}}_{2})$,
($M=m_{1}+m_{2}$) and relative coordinate
$\mathbf{r}=\mathbf{r}^{\prime}_{1}-\mathbf{r}^{\prime}_{2}$, and setting
$\varphi=\Phi+i\frac{\hbar}{Mc^{2}}\dot{\Phi},\ \ \ \
\chi=\Phi-i\frac{\hbar}{Mc^{2}}\dot{\Phi}$ (36)
we find ($\mu=\frac{m_{1}m_{2}}{M}$)
$\ddot{\Phi}-c^{2}\nabla^{2}_{R}\Phi-c^{2}\frac{M}{\mu}\nabla^{2}_{r}\Phi+\frac{1}{\hbar^{2}}(M^{2}c^{4}+2VMc^{2})\Phi=0$
(37)
Its stationary solution reads
$\Phi({\mathbf{R},\mathbf{r}},t)=\psi(\mathbf{r})\exp\left[\frac{i}{\hbar}({\mathbf{P}\cdot\mathbf{R}}-Et)\right]$
(38)
where $E$ is the total energy of the system while $\mathbf{P}$ the momentum of
CM. The reduced ”one-body” equation for $\psi(\mathbf{r})$ turns out to be:
111With Eq.(35), Eq.(37) is invariant under the space-time inversion (${\bf
r}\to-{\bf r},t\to-t$). Equivalently, under the mass inversion ($m_{1}\to-
m_{1},m_{2}\to-m_{2}$), Eq.(37) and Eq.(39) remain invariant in the sense that
not only $\mu\to-\mu,M\to-M$, but also $V({\bf r})\to-V({\bf
r}),\varepsilon\to-\varepsilon$. Notice that, however, the simultaneous
inversion of $m_{1}$ and $m_{2}$ implies $m_{1}\simeq m_{2}$, so both
particles change under their mutual interaction $V({\bf r})$ simultaneously.
Here $V$, being the ”internal potential energy” of two-body system, was called
as a ”scalar potential”. We see that either the invariance under the space-
time inversion or that under the mass inversion is capable of showing the
particle-antiparticle symmetry (i.e., relativistic nature) of a system
essentially.
$\left\\{\begin{array}[]{ll}&\left[-\dfrac{\hbar^{2}}{2\mu}\nabla^{2}_{\mathbf{r}}+V(\mathbf{r})\right]\psi(\mathbf{r})=\varepsilon\psi(\mathbf{r})\\\\[11.38109pt]
&\varepsilon=\dfrac{1}{2Mc^{2}}(E^{2}-M^{2}c^{4}-{\mathbf{P}}^{2}c^{2})\end{array}\right.$
(39)
We set $\mathbf{P}=0$ (i.e.turn to CMCS) and denote the binding energy
$B=Mc^{2}-E$, yielding:
$B=Mc^{2}\left[1-(1+\frac{2\varepsilon}{Mc^{2}})^{1/2}\right]=-\varepsilon+\frac{1}{2}\frac{\varepsilon^{2}}{Mc^{2}}-\cdots$
(40)
Notice that although Eq.(39) looks like a ”non-relativistic” stationary
Schrödinger equation, it is essentially relativistic. This can be seen from
its remarkable property that the eigenvalue $\varepsilon$ has a lower bound
$-\frac{1}{2}Mc^{2}$, corresponding to $E_{\mbox{min}}=0\ (B_{\mbox{max}}=M)$!
An example is: consider ”positronium” composed of $e^{+}$ and $e^{-}$ with
charge $Ze$ and $-Ze$ respectively. Once when the ”fictitious charge number”
$Z$ increases from $1$ to $Z_{max}=(\frac{4}{\alpha^{2}})^{1/4}=16.555$, the
whole bound system would have lowest ground energy $E_{min}=0$! So Eq.(39) is
really a relativistic QM equation capable of giving a nonperturbative solution
under the strong coupling.
Eq.(39) provides a justification (realization) of conjecture Eq.(32) relevant
to case B ($m_{1}\simeq m_{2}$) where the spin of both particles is merely of
second importance.
Now let us turn to case A where $m_{2}\gg m_{1}$, taking the spin of $m_{1}$
into account but ignoring that of $m_{2}$ as before. Based on the experience
in case B, also because of great difficulty to derive the equation starting
from the laboratory system for this case A, we directly introduce the reduced
Dirac equation (RDE) in the RMCS as a pair of coupled equations of two-
component spinors $\varphi({\mathbf{r}},t)$ and $\chi({\mathbf{r}},t)$,
($c=\hbar=1$)
$\left\\{\begin{array}[]{ll}i\dot{\varphi}&=i{\mathbf{\sigma}}_{1}\cdot\nabla_{\mathbf{r}}\chi+\mu\varphi+V({\mathbf{r}})\varphi\\\
i\dot{\chi}&=i{\mathbf{\sigma}}_{1}\cdot\nabla_{\mathbf{r}}\varphi-\mu\chi+V({\mathbf{r}})\chi\end{array}\right.$
(41)
with $\mu$ replacing $m_{1}$. Here $\mathbf{\sigma}_{1}$ are Pauli matrices
acting on the spin space of particle $m_{1}$. Eq.(41) is invariant under the
space-time inversion $({\mathbf{r}\rightarrow-\mathbf{r}},t\rightarrow-t),\
\varphi(-{\mathbf{r}},-t)\rightarrow\chi({\mathbf{r}},t),\
\chi(-{\mathbf{r}},-t)\rightarrow\varphi({\mathbf{r}},t)$ whereas we assume
$V(-{\mathbf{r}},-t)\longrightarrow-V({\mathbf{r}},t)$ (42)
here in contrast to Eq.(35) for the case B. 222For a hydrogenlike atom,
$V(r)=-\frac{Ze^{2}}{r}$ does not contain time $t$ explicitly. Eq.(42) merely
means that under the space-time inversion, the electron transforms into a
position whereas the nucleus remains unchanged. See point (a) of section IX.
Previously, the $V$ in Eq.(42) was called as a ”vector potential”, meaning the
”potential energy” of the electron in an ”external field” of nucleus. Note
that, formally, Eq.(41) remains invariant under a mass inversion as
$\mu\to-\mu,\phi\to\chi,\chi\to\phi$ ($V({\bf r})$ remains unchanged) in a
noninertial frame $RMCS$. Actually, since $m_{1}=m_{e}\to-m_{e}$, but
$m_{2}=m_{N}\to m_{N},\mu\to-\mu(1+\frac{2m_{e}}{M})$. So Eq.(41) has an
inaccuracy up to $\frac{2m_{e}}{M}$ ($<1.1\times 10^{-3}$ for $H$).
The reasons are as follows: (a) Eq.(41) should degenerate into the original
Dirac equation when $m_{2}\rightarrow\infty$,
$\mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}\rightarrow m_{1}$. (b) Since now $m_{2}\gg
m_{1}$ (but $m_{2}\neq\infty$), $m_{1}$ is moving much faster than $m_{2}$ in
the CMCS. Hence the antiparticle field $\chi$ enhances much appreciably in
$m_{1}$ than that in $m_{2}$, a situation totally different from that in the
case B where $m_{1}\approx m_{2}$. (c) If instead of Eq.(42), we still assume
$V(-{\mathbf{r}},-t)\longrightarrow V({\mathbf{r}},t)$ like Eq.(35) and change
the sign before $V(\mathbf{r})$ in the second equation of Eq.(41) to keep its
invariance under the space-time inversion, then we would get an equation which
would lead to a reversed fine-structure of atom (e.g., the $P_{1/2}$ state
would lie above the $P_{3/2}$ state), a wrong prediction obviously excluded by
experiments.
However, one kind of invariance is not enough to fix an equation. Indeed, the
beauty of Dirac equation or RDE is hidden in two symmetries: besides the
symmetry of space-time inversion, it has another left-right (parity) symmetry.
To see it, we define
$\xi=\frac{1}{\sqrt{2}}(\varphi+\chi),\ \ \
\eta=\frac{1}{\sqrt{2}}(\varphi-\chi)$ (43)
and recast Eq.(41) into:
$\left\\{\begin{array}[]{ll}i\dot{\xi}&=i{\mathbf{\sigma}}_{1}\cdot\nabla_{\mathbf{r}}\xi+\mu\eta+V(\mathbf{r})\xi\\\
i\dot{\eta}&=-i{\mathbf{\sigma}}_{1}\cdot\nabla_{\mathbf{r}}\eta+\mu\xi+V(\mathbf{r})\eta\end{array}\right.$
(44)
which is invariant under a pure space inversion
(${\mathbf{r}\rightarrow-\mathbf{r}},t\rightarrow t$) if assuming
$\xi({-\mathbf{r}},t)\rightarrow\eta({\mathbf{r}},t),\ \
\eta({-\mathbf{r}},t)\rightarrow\xi({\mathbf{r}},t),\ \
V(-{\mathbf{r}})\rightarrow V({\mathbf{r}})=V(r)$ (45)
The parity invariance of Dirac equation or RDE has a far-reaching consequence
that the Dirac particle is always a subluminal one. By contrast, once the
parity is violated to maximum, a superluminal particle (tachyon) will emerge.
Interestingly enough, any theory capable of treating particle and antiparticle
on an equal footing must respect to the common basic symmetry—the invariance
of space-time inversion. The new insight of this section is this symmetry can
be applied even in a noninertial frame—the RMCS. Of course, the validity of
RDE can only be verified by experiments as discussed in section II, although
it is still an approximate description of nature like any other theory in
physics. For further discussion, see section IX.
## IV Self-Energy Correction of a Bound Electron in Atom
In our understanding, one important reason why the calculations of QED for
electron in a hydrogenlike atom is so complicated lies in the fact that while
calculations are performed in the CMCS, the center of potential (the nucleus
with mass $m_{2}=m_{N}$) undergoes a complex motion. So the recoil effect
interwinds with the high-loop correction of QED, as discussed in many chapters
of the books 1 and 2 . We will try to find an alternative approach by
adopting the RDE and doing calculation in the RMCS. Let us begin with the
Feynman diagram integral (FDI) of electron self-energy at one-loop level,
adopting the Bjorken-Drell metric and rationalized Gaussian units with
electron charge $-e(e>0)$, see Fig.2(a) (8a ).
$-i\Sigma(p)=(ie)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{g_{\mu\nu}}{ik^{2}}\gamma^{\mu}\frac{i}{\not\\!\\!p-\not\\!\\!k-\mu}\gamma^{\nu}$
(46)
Here a free electron with reduced mass $\mu$ is moving at a four-dimensional
momentum $p$, whose spatial component is just the relative momentum
$\mathbf{p}_{r}$ discussed in the previous section, $k$ is the momentum of
virtual photon. As usual, a Feynman parameter $x$ will bring Eq.(46) into
$-i\Sigma(p)=-e^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{N}{D}$ (47)
$\frac{1}{D}=\int^{1}_{0}\frac{dx}{[k^{2}-2p\cdot kx+(p^{2}-\mu^{2})x]^{2}},\
\ \ N=-2(\not\\!\\!p-\not\\!\\!k)+4\mu$ (48)
($\not\\!\\!p=p^{\mu}\gamma_{\mu},\ p\cdot k=p^{\mu}k_{\mu}$). A shift in
momentum integration:$k\rightarrow K=k-xp$ recast Eq.(47) into
$-i\Sigma(p)=-e^{2}\int^{1}_{0}dx[-2(1-x)\not\\!\\!p+4\mu]I$ (49)
with a logarithmically divergent integral (in Minkowski momentum space):
$I=\int\frac{d^{4}K}{(2\pi)^{4}}\frac{1}{[K^{2}-M^{2}]^{2}},\ \ \
M^{2}=p^{2}x^{2}+(\mu^{2}-p^{2})x$ (50)
Our new regularization-renormalization method (RRM) is based on a cognition
that the virtual process in the self-energy diagram does provide a radiative
correction to the electron mass but only when the electron is off the mass
shell, i.e., $p^{2}\not=\mu^{2}$. When it is on the mass shell,
$p^{2}=\mu^{2}$, the appearance of a divergent integral like $I$ in Eq.(50) is
essentially a warning on the fact that to calculate the mass of electron is
beyond the ability of perturbative QED.
Let us consider the converse: if $\Sigma(p)$ does modify the electron mass
$\mu$ to some extent, it must comes from the divergent integral $I$. However,
the latter is a dimensionless number, we can change the unit of $M$ (and $k$)
at our disposal without any change in the value of $I$. So any real change of
$\mu$ (on the mass shell) is incredible. The deeper reason lies in a
”principle of relativity” in epistemology: everything is moving and becomes
recognizable only in relationship with other things. What we can understand is
either no mass scale or two mass scales, but never one mass scale. For
instance, in the famous Gross-Neveu model 19 (19), a massive fermion is
created only in accompanying with the change (phase transition) of its
environment (vacuum) which provides another mass scale (a standard weight).
Another example is just the change of electron mass from $m_{e}$ to $\mu_{H}$
in a hydrogen atom due to the coexistence of atom nucleus—the proton, this
change is also a nonperturbative effect.
Therefore, we expected too much in the past. There is no way to evaluate
Eq.(50) unambiguously or pick out some finite and fixed modification on the
mass $\mu$. What we can do is to separate the valuable information carried by
Eq.(50) from an arbitrary constant which will be introduced by a simple trick
and then fixed by the experimental data of $\mu$. We will see the information
telling us exactly how the value of $I$ changes when the electron is moving
off the mass shell.
To handle Eq.(50), we perform a differentiation with respect to the mass-
square parameter $M^{2}$, then the integration with respect to $K$ becomes
convergent, yielding:
$\frac{\partial I}{\partial M^{2}}=\frac{-i}{(4\pi)^{2}}\frac{1}{M^{2}}$ (51)
which tells us that while the exact value of $I$ remains obscure, its change
linked with $M^{2}$ has a definite meaning. So we reintegrate Eq.(51) with
respect to $M^{2}$ and arrive at
$I=\frac{-i}{(4\pi)^{2}}(\ln
M^{2}+C_{1})=\frac{-i}{(4\pi)^{2}}\ln\frac{M^{2}}{\mu_{2}^{2}}$ (52)
where an arbitrary constant $C_{1}=-\ln\mu_{2}^{2}$ is introduced ($\mu_{2}$
should not be confused with the reduced mass $\mu$). Further integration with
respect to Feynman parameter $x$ leads to
$\begin{array}[]{ll}\Sigma(p)&=A+B\not\\!\\!p\\\
A&=\frac{\alpha}{\pi}\mu\left[2-2\ln\frac{\mu}{\mu_{2}}+\frac{(\mu^{2}-p^{2})}{p^{2}}\ln\frac{(\mu^{2}-p^{2})}{\mu^{2}}\right]\\\
B&=\frac{\alpha}{4\pi}\left[2\ln\frac{\mu}{\mu_{2}}-3-\frac{(\mu^{2}-p^{2})}{p^{2}}\left[1+\frac{(\mu^{2}+p^{2})}{p^{2}}\ln\frac{(\mu^{2}-p^{2})}{\mu^{2}}\right]\right]\end{array}$
(53)
Using the chain approximation, we can derive the modification of electron
propagator as
$\frac{i}{\not\\!\\!p-\mu}\rightarrow\frac{i}{\not\\!\\!p-\mu}\frac{1}{1-\frac{\Sigma(p)}{\not\\!\\!p-\mu}}=\frac{iZ_{2}}{\not\\!\\!p-\mu_{R}}$
(54)
where
$Z_{2}=\frac{1}{1-B}$ (55)
is the renormalization factor for wave function of electron and
$\mu_{R}=\frac{\mu+A}{1-B}$ (56)
is the renormalized mass of $\mu$. The increment of mass reads
$\delta\mu=\mu_{R}-\mu=\frac{A+\mu B}{1-B}$ (57)
For a free electron (in the atom), the mass-shell condition $p^{2}=\mu^{2}$
should lead to
$\delta\mu|_{p^{2}=\mu^{2}}=\frac{\alpha\mu}{4\pi}(5-6\ln\frac{\mu}{\mu_{2}})=0$
(58)
as discussed above333We will keep the same mass symbol $\mu$ through out high-
loop calculations of QED and reconfirm (renormalize) it at every step by
experiment. Just like one has to reconfirm his plane ticket before his
departure from the airport, he must use the same name through out his entire
journey 8b .. So we must set $\mu_{2}=\mu e^{-5/6}$ which in turn fixes
$Z_{2}|_{p^{2}=\mu^{2}}=\frac{1}{1+\frac{\alpha}{3\pi}}\approx
1-\frac{\alpha}{3\pi}$ (59)
However, the above evaluation further provides us with important knowledge of
$\delta\mu$ when electron is moving off the mass-shell. Consider the similar
diagram in Fig.(2b), we can set on an average meaning that
$p^{2}=\mu^{2}(1-\zeta)$ (60)
with $\zeta>0$, which implies from Eq.(57) with Eq.(53) that 20 (20):
$\delta\mu=\frac{\alpha\mu}{4\pi}\frac{(-\zeta+2\zeta\ln\zeta)}{1+\alpha/3\pi}$
(61)
where some terms of the order of $\zeta^{2}$ or $\zeta^{2}\ln\zeta$ are
neglected since $\zeta\ll 1$. Eq.(61) establishes the correspondence between
the mass modification $\delta\mu$ and the parameter $\zeta$ describing the
off-mass-shell extent of electron in the bound state. For a hydrogenlike atom,
we may ascribe $\delta\mu$ to the (minus) binding energy of electron in the
Bohr theory:
$\delta\mu=\varepsilon_{n}=-\frac{Z^{2}\alpha^{2}}{2n^{2}}\mu$ (62)
Then Eq.(61) gives the value of $\zeta$ for fixed values $Z$ and $n$. We will
see from the vertex function that these values of $\zeta$ are crucial to the
calculation of Lamb shift (sections VII and VIII).
## V Photon Self-energy
As discussed in various text books 21 (21, 22, 23), we encounter the FDI of
vacuum polarization Fig.2(c) as 8a :
$\Pi_{\mu\nu}(q)=-(-ie)^{2}Tr\int\frac{d^{4}\bar{p}}{(2\pi)^{4}}\gamma_{\mu}\frac{i}{\not\\!\\!\bar{p}-m}\gamma_{\nu}\frac{i}{\not\\!\\!\bar{k}+\not\\!\\!q-m}$
(63)
Here $q$ is the momentum transfer along the photon line and $m$ the mass of
electron. Introducing the Feynman parameter $x$ as in previous section and
performing a shift in momentum integration: $\bar{p}\rightarrow K=\bar{p}+xq$,
we get:
$\Pi_{\mu\nu}(q)=-4e^{2}\int_{0}^{1}dx(I_{1}+I_{2})$ (64)
where
$I_{1}=\int\frac{d^{4}K}{(2\pi)^{4}}\frac{2K_{\mu}K_{\nu}-g_{\mu\nu}K^{2}}{(K^{2}-M^{2})^{2}}$
(65)
with
$M^{2}=m^{2}+q^{2}(x^{2}-x)$ (66)
is quadratically divergent while
$I_{2}=\int\frac{d^{4}K}{(2\pi)^{4}}\frac{(x^{2}-x)(2q_{\mu}q_{\nu}-g_{\mu\nu}q^{2})+m^{2}g_{\mu\nu}}{(K^{2}-M^{2})^{2}}$
(67)
is only logarithmically divergent like that in Eq.(50). An elegant way to
handle $I_{1}$, Eq.(65), is modifying $M^{2}$ to
$M^{2}(\sigma)=m^{2}+q^{2}(x^{2}-x)+\sigma$ (68)
and differentiating $I_{1}$ with respect to $\sigma$. After integration with
respect to $K$, we reintegrate it with respect to $\sigma$ twice, arriving at
the limit $\sigma\rightarrow 0$:
$I_{1}=\frac{ig_{\mu\nu}}{(4\pi)^{2}}\left\\{[m^{2}+q^{2}(x^{2}-x)]\ln\frac{m^{2}+q^{2}(x^{2}-x)}{\mu_{3}^{2}}+C_{2}\right\\}$
(69)
with two arbitrary constant: $C_{1}=-\ln\mu_{3}^{2}$ and $C_{2}$. Combining
$I_{1}$ and $I_{2}$ together, we find:
$\Pi_{\mu\nu}(q)=\frac{8ie^{2}}{(4\pi)^{2}}(q_{\mu}q_{\nu}-g_{\mu\nu}q^{2})\int_{0}^{1}dx(x^{2}-x)\ln\frac{m^{2}+q^{2}(x^{2}-x)}{\mu_{3}^{2}}-\frac{4ie^{2}}{(4\pi)^{2}}g_{\mu\nu}C_{2}$
(70)
The continuity equation of current induced in the vacuum polarization 21 (21)
$q^{\mu}\Pi_{\mu\nu}(q)=0$ (71)
is ensured by the factor $(q_{\mu}q_{\nu}-g_{\mu\nu}q^{2})$. So we set
$C_{2}=0$. Consider the scattering between two electrons via the exchange of a
photon with momentum transfer $q\rightarrow 0$. Adding the contribution of
$\Pi_{\mu\nu}(q)$ to the tree diagram amounts to modify the charge square:
$e^{2}\longrightarrow e^{2}_{R}=Z_{3}e^{2},\ \ \
Z_{3}=1+\frac{\alpha}{3\pi}(\ln\frac{m^{2}}{\mu_{3}^{2}}-\frac{q^{2}}{5m^{2}}+\cdots)$
(72)
As in Ref.8b , we will set $\mu_{3}=m$ so that at the Thomson
limit:$\lim_{q\rightarrow 0}e_{R}^{2}=e^{2}$. However, for the purpose of
calculating Lamb shift (LS) below, the second term in the parenthesis of
$Z_{3}$ is important because for a bound state it contributes a term of
effective potential (adding to Coulomb potential), called the Uehling
potential (23 (23),p.253):
$-\frac{4\alpha^{2}}{15m^{2}}\delta(\mathbf{r})$ (73)
## VI The Off-Mass-Shell Vertex Function
Consider an electron (see Fig.2(d)) moving in a hydrogen atom, its momentum
changes from $p$ to $p^{\prime}$ via the scattering by the proton and an
exchange of virtual photon with momentum $k$. The FDI at one-loop level reads
$\Lambda_{\mu}(p^{\prime},p)=(-ie)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{-i}{k^{2}}\gamma_{\nu}\frac{i}{\not\\!\\!p^{\prime}-\not\\!\\!k^{\prime}-\mu}\gamma_{\mu}\frac{i}{\not\\!\\!p-\not\\!\\!k-\mu}\gamma^{\nu}$
(74)
However, different from 8b and many other literatures, not only the reduced
mass $\mu$ (instead of $m$) of electron is used, but also a new approach will
be adopted. We assume that the electron is moving off-mass-shell in the sense
of (as in section IV):
$p^{2}=p^{\prime 2}=\mu^{2}(1-\zeta)$ (75)
We still have
$p^{\prime}-p=q,\ \ \ p\cdot q=-\frac{1}{2}q^{2}$ (76)
Introducing Feynman parameters $u=x+y$ and $v=x-y$, we perform a shift in the
momentum integration:$k\rightarrow K=k-(p+q/2)u-(q/2)v$, thus
$\Lambda_{\mu}=-ie^{2}[I_{3}\gamma_{\mu}+I_{4}]$ (77)
$\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}du\int_{-u}^{u}dv\int\frac{d^{4}K}{(2\pi)^{4}}\frac{K^{2}}{(K^{2}-M^{2})^{3}}$
(78) $\displaystyle M^{2}$ $\displaystyle=$
$\displaystyle[\mu^{2}(1-\zeta)-\frac{q^{2}}{4}]u^{2}+\frac{q^{2}}{4}v^{2}+\zeta\mu^{2}u$
(79) $\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}du\int_{-u}^{u}dv\int\frac{d^{4}K}{(2\pi)^{4}}\frac{A_{\mu}}{(K^{2}-M^{2})^{3}}$
(80) $\displaystyle A_{\mu}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!(4\\!\\!-\\!\\!4u\\!\\!-\\!\\!2u^{2})\mu^{2}(1\\!\\!-\\!\\!\zeta)\gamma_{\mu}\\!+\\!2i(u^{2}\\!\\!-\\!\\!u)\mu
q^{\nu}\\!\sigma_{\mu\nu}\\!-\\!(2\\!-\\!2u\\!+\\!\frac{u^{2}}{2}\\!-\\!\frac{v^{2}}{2})q^{2}\gamma_{\mu}\\!-\\!(2\\!+\\!2u)v\mu
q_{\mu}$ (81)
Set $K^{2}=K^{2}-M^{2}+M^{2}$, then $I_{3}=I^{\prime}_{3}-\frac{i}{32\pi^{2}}$
and $I^{\prime}_{3}$ is only logarithmically divergent and so can be treated
as in previous sections, yielding:
$I^{\prime}_{3}=\frac{-i}{(4\pi)^{2}}\int_{0}^{1}du\int_{-u}^{u}dv\ln\frac{M^{2}}{\mu_{1}^{2}}$
(82)
with $\mu_{1}$ an arbitrary constant.
However, unlike Ref.8b where the calculation was conducted on the mass-shell,
now the off-mass-shell integration in Eq.(82) can be performed in the
approximation that $\frac{Q^{2}}{4\mu^{2}}\ll 1$ and $\zeta\ll 1$
($Q^{2}=-q^{2}$, $Q$ is the three-dimensional momentum transfer) which will be
enough to calculate the Lamb shift (LS). Denoting
$a=[\mu^{2}(1-\zeta)+\frac{Q^{2}}{4}]u^{2}+\zeta\mu^{2}u,\ \ \
b=\frac{Q^{2}}{4}$ (83)
we will perform the integration with respect to $v$ and $u$ rigorously:
$\int_{-u}^{u}dv\ln(a-bv^{2})=2u[\ln\mu^{2}+\ln
u+\ln[(1-\zeta)u+\zeta]-4u+2\sqrt{\frac{4a}{Q^{2}}}\ln\frac{1+\sqrt{Q^{2}/4a}u}{1-\sqrt{Q^{2}/4a}u}$
(84)
Expanding the last term and keeping only up to the order of $\zeta$ and
$Q^{2}/4\mu^{2}$, we obtain
$\int_{0}^{1}du\int_{-u}^{u}dv\ln(a-bv^{2})\simeq\ln\mu^{2}-1+\zeta+\frac{Q^{2}}{6\mu^{2}}(1-\zeta)$
(85)
To our great pleasure, throughout the evaluation of $I_{4}$, there is no any
infrared divergence which would appear in previous literatures when
integrating with respect to $u$ with lower limit zero. To avoid the infrared
divergence, e.g., in 8b , a cutoff was introduced at the lower limit. Now the
infrared divergence disappears due to the existence of off-mass-shell
parameter $\zeta$. For example, we encounter the following integral, in which
no cutoff is needed ($\lambda=(1-\zeta)+Q^{2}/4\mu^{2}\sim 1$):
$\int_{0}^{1}\frac{du}{u+\zeta/\lambda}=\frac{\zeta}{\lambda}-\ln\frac{\zeta}{\lambda}$
(86)
Hence, after elementary but tedious calculation, we find:
$\begin{array}[]{l}\Lambda_{\mu}(p^{\prime},p)=\frac{\alpha}{4\pi}[\frac{11}{2}-\ln\frac{\mu^{2}}{\mu_{1}^{2}}-3\zeta+4(1+\zeta)\ln\zeta]\gamma_{\mu}+\frac{\alpha}{4\pi}\frac{Q^{2}}{\mu^{2}}\gamma_{\mu}(\frac{1}{6}+\frac{1}{2}\zeta+\frac{4}{3}\ln\zeta+2\zeta\ln\zeta)\\\
+i\frac{\alpha}{4\pi}\frac{q^{\nu}}{\mu}\sigma_{\mu\nu}(1+3\zeta+2\zeta\ln\zeta)\end{array}$
(87)
## VII Calculation of Lamb Shift as an Off-Mass-Shell Effect at One-Loop
Level
There are three parts in Eq.(87). The first part in combination with the
vertex $\gamma_{\mu}$ at tree level provides a renormalization factor as
$Z_{1}^{-1}=1+\frac{\alpha}{4\pi}[\frac{11}{2}-\ln\frac{\mu^{2}}{\mu_{1}^{2}}-3\zeta+4(1+\zeta)\ln\zeta]$
(88)
Further combination with $Z_{2}$ in Eq.(55) and $Z_{3}$ in Eq.(72) leads to a
renormalized charge (at one-loop level, see Fig.2):
$e_{R}=\frac{Z_{2}}{Z_{1}}Z_{3}^{1/2}e$ (89)
However the Ward identity implies that 21 (21, 22, 23)
$Z_{1}=Z_{2}$ (90)
Therefore
$\alpha_{R}=\frac{e_{R}^{2}}{4\pi}=Z_{3}\alpha$ (91)
Note that Ward identity holds not only for an electron on the mass-shell, but
also for off-mass-shell case. Hence for every bound state in hydrogenlike atom
with a definite value of $\zeta$ ($Z_{1}$ and $Z_{2}$ are functions of
$\zeta$), the arbitrary constant $\mu_{1}$ in Eq.(88) plays a flexible role to
guarantee the validity of Eq.(90) (other two constants $\mu_{2}$ and $\mu_{3}$
had been fixed in Eq.(58) and (72) respectively). For further discussion, see
section IX.
The second part of Eq.(87) contains $Q^{2}\gamma_{\mu}$. Just like the Uehling
potential in Eq.(72) (with $q^{2}=-Q^{2}$), it contributes an effective
potential of $\delta$ function type as
$\frac{\alpha^{2}}{\mu^{2}}[-\frac{1}{6}-\frac{1}{2}\zeta-\frac{4}{3}\ln\zeta-2\zeta\ln\zeta]\delta(\mathbf{r})$
(92)
Finally, the third part of Eq.(87) amounts to a modification of electron
magnetic moment in the atom, the gyromagnetic ratio of electron reads:
$g=2[1+\frac{\alpha}{2\pi}(1+3\zeta+2\zeta\ln\zeta)]$ (93)
We will call the anomalous part of magnetic moment
$a=\frac{\tilde{\alpha}}{2\pi}$,
$\tilde{\alpha}=\alpha(1+3\zeta+2\zeta\ln\zeta)$. The radiative correction on
the magnetic moment of an electron has two consequences. One is a modification
to the L-S coupling in a hydrogenlike atom (with charge number $Z$) 21 (21,
22):
$H_{LS}^{rad}=2(\frac{\tilde{\alpha}}{2\pi})\frac{\alpha
Z}{4\mu^{2}r^{3}}{\mathbf{\sigma}\cdot L}$ (94)
Here the electron mass has been modified from $m$ (see, e.g., 15 (15)) to
$\mu$ which can be derived from the reduced Dirac equation.
Another consequence of anomalous magnetic moment of electron exhibits itself
as an additional potential of $\delta$ function type like Eq.(73)21 (21, 22)
$\frac{Z\alpha\tilde{\alpha}}{2\mu^{2}}\delta(\mathbf{r})$ (95)
Note that Eqs.(94) and (95) are only effective to states with $L\not=0$ and
$S$ state with $L=0$ respectively.
Adding the results of Eqs.(94), (95) and the sum of Eqs.(73) and (92)
multiplied by $Z$ together to get all radiative corrections (at one-loop
level) on electron in the hydrogenlike atom, then we get the effective
potential as
$\begin{array}[]{ll}V_{eff}^{rad}&=\frac{Z\alpha^{2}}{\mu^{2}}[-\frac{4}{3}\ln\zeta-\frac{1}{2}\zeta-2\zeta\ln\zeta-\frac{1}{6}-\frac{4}{15}\frac{\mu^{2}}{m^{2}}+\frac{1}{2}(1+3\zeta+2\zeta\ln\zeta)]\delta(\mathbf{r})\\\
&+\frac{Z\alpha^{2}}{4\pi\mu^{2}r^{3}}(1+3\zeta+2\zeta\ln\zeta){\mathbf{\sigma}\cdot
L}\\\
&\simeq\frac{Z\alpha^{2}}{\mu^{2}}[-\frac{4}{3}\ln\zeta+\frac{1}{15}+\zeta-\zeta\ln\zeta]\delta(\mathbf{r})+\frac{Z\alpha^{2}}{4\pi\mu^{2}r^{3}}(1+3\zeta+2\zeta\ln\zeta){\mathbf{\sigma}\cdot
L}\end{array}$ (96)
where we take $\mu^{2}/m^{2}\approx 1$ in the Uehling potential to make the
formula simpler for a semi-quantitative calculation. Eq.(96) leads to the
energy modification of a bound state (with quantum numbers $n,l,j$) in a
hydrogenlike atom:
$\delta({\mathbf{r}})\longrightarrow|\psi_{ns}(0)|^{2}=\frac{Z^{3}\alpha^{3}}{\pi
n^{3}}\mu^{3},\ \ \ (l=0)$ (97)
$\Delta E^{rad}=\Delta E^{rad}(ns)+\Delta E^{rad}_{LS}$ (98)
$\Delta E^{rad}(ns)=\frac{Z^{4}\alpha^{3}}{\pi
n^{3}}R_{y}[\frac{8}{3}\ln\frac{1}{\zeta}+\frac{2}{15}+2\zeta(1-\ln\zeta)]\delta_{l0}$
(99)
$\Delta E^{rad}_{LS}=\frac{Z^{4}\alpha^{3}}{\pi
n^{3}}R_{y}\frac{1+\zeta(3+2\ln\zeta)}{l(2l+1)(l+1)}\left\\{\begin{array}[]{ll}&l,\
\ \ \ \ \ (j=l+1/2)\\\ -&(l+1),\ \ \ (j=l-1/2)\end{array}\right.$ (100)
where
$R_{y}=\frac{1}{2}\alpha^{2}\mu=\frac{\mu}{m}R_{\infty}$ (101)
## VIII Energy-Level Difference in Hydrogenlike Atom: Theory vs. Experiment
We will study some energy-level differences near the ground state of
hydrogenlike atoms, where precise experimental data are available.
Theoretically, the energy level is fixed primarily by the formula derived from
the reduced Dirac equation (RDE), i.e., Eq.(1) with $m_{e}$ substituted by
$\mu_{A}$ where the subscript $A$ refers to atom $H$, $D$ or $He^{+}$, et
al..:
$\displaystyle E_{A}^{RDE}$ $\displaystyle=$
$\displaystyle\mu_{A}[f(n,j)-1]=\frac{m_{e}}{1+b_{A}}[f(n,j)-1]$ (102)
$\displaystyle=$ $\displaystyle\frac{1}{1+b_{A}}(1.2355897\times
10^{20})[-\frac{(Z\alpha)^{2}}{2n^{2}}-\frac{(Z\alpha)^{4}}{3n^{3}}(\frac{1}{j+1/2}-\frac{3}{4n})-\cdots]\
\ Hz$
Further recoil corrections Eq.(6) derived by previous authors will be divided
into two terms:
$\displaystyle\Delta E_{A}^{recoil-1}(n,j)$ $\displaystyle=$
$\displaystyle-\frac{\mu^{2}_{A}}{2M_{A}}[f(n,j)-1]^{2}=-\frac{m_{e}b_{A}}{2(1+b_{A})^{3}}[f(n,j)-1]^{2}$
(103) $\displaystyle\Delta E_{A}^{recoil-2}(n,j,l)$ $\displaystyle=$
$\displaystyle\frac{(Z\alpha)^{4}\mu^{3}_{A}}{2n^{3}m_{N}^{(A)^{2}}}(\frac{1}{j+\frac{1}{2}}-\frac{1}{l+\frac{1}{2}})(1-\delta_{l0})$
(104)
Next comes the radiative correction calculated by QED at one-loop level,
Eq.(98):
$\Delta
E_{A}^{rad}(n,j,l)=\frac{1}{1+b_{A}}\frac{Z^{4}}{n^{3}}(\frac{\alpha^{3}}{\pi}R_{\infty})[(-\frac{8}{3}\ln\zeta+\frac{2}{15}+2\zeta(1-\ln\zeta))\delta_{l0}+\frac{1+\zeta(3+2\ln\zeta)}{2l+1}C_{jl}(1-\delta_{l0})]$
(105)
where
$C_{jl}=\left\\{\begin{array}[]{ll}&\frac{1}{l+1},\ \ \ j=l+\frac{1}{2}\\\
&-\frac{1}{l},\ \ \ j=l-\frac{1}{2}\end{array}\right.$ (106)
Finally, the finite nucleus size (NS) with radius $r_{N}^{(A)}$ brings a
correction 10 (10):
$\begin{array}[]{ll}\Delta
E_{A}^{NS}(n,j)&=\frac{4}{3}(\frac{\mu_{A}}{m_{e}})^{3}\frac{Z^{4}}{n^{3}}(\frac{r_{N}^{(A)}}{a_{\infty}})^{2}R_{\infty}\delta_{l0}\\\
&=(\frac{1}{1+b_{A}})^{3}\frac{Z^{4}}{n^{3}}(4.386454987\times
10^{7})[\frac{r_{N}^{(A)}(fm)}{5.2917725}]^{2}\delta_{l0}\ \ Hz\end{array}$
(107)
As explained in Eq.(61) with Eq.(62), the value of off-mass-shell parameter
$\zeta$ in Eq.(105) can be calculated from the electron self-energy at one-
loop level:
$\frac{Z^{2}\alpha}{n^{2}}=\frac{1}{2\pi}\frac{(\zeta^{<S>}-2\zeta^{<S>}\ln\zeta^{<S>})}{1+\alpha/3\pi}$
(108)
where the superscript $<S>$ refers to ”self-energy”. However, we may derive
the value of $\zeta$ in an alternative way. Divide the square average of four-
dimensional momentum $p$ into two parts:
$<p^{2}>=<E^{2}>-<{\mathbf{p}}^{2}>$ (109)
where
$<E^{2}>=E^{2}=(\mu-B)^{2}\simeq\mu^{2}-2\mu B,$ (110)
since the binding energy
$B=\frac{Z^{2}\alpha^{2}}{2n^{2}}\mu\ll\mu$ (111)
The square average of three-dimensional momentum $\mathbf{p}$,
$<{\mathbf{p}}^{2}>$, can be evaluated by the Virial theorem (e.g., 15 (15)).
In a Coulomb field, an electron has potential energy $V=-\frac{Ze^{2}}{4\pi
r}$ and kinetic energy $T=\frac{1}{2\mu}{\mathbf{p}}^{2}$. Then
$\displaystyle<{\mathbf{p}}^{2}>$ $\displaystyle=$ $\displaystyle
2\mu<T>=2\mu[-B-<V>]=2\mu B$ $\displaystyle<p^{2}>$ $\displaystyle=$
$\displaystyle\mu^{2}-4\mu B=\mu^{2}(1-\frac{4B}{\mu})$ (112)
Comparing Eq.(112) with $<p^{2}>=\mu^{2}(1-\zeta^{<V>})$, we find
$\zeta^{<V>}=\frac{4B}{\mu}=\frac{2Z^{2}\alpha^{2}}{n^{2}}$ (113)
where the superscript $<V>$ refers to ”Virial theorem”. Table 1 gives the
values of $\zeta^{<S>}$ and $\zeta^{<V>}$ with their logarithm values as well
as two kinds of ”average”,
$\zeta^{<S+V>}=\frac{1}{2}(\zeta^{<S>}+\zeta^{<V>})$ and
$\zeta^{<SV>}=\sqrt{\zeta^{<S>}\zeta^{<V>}}$, to be used in Eq.(105).
Table 1. Off-mass-shell parameter $\zeta$ and $\ln\zeta$ |
---|---
$\frac{Z^{2}}{n^{2}}$ | $\zeta^{<S>}\times 10^{4}$ | -$\ln\zeta^{<S>}$ | $\zeta^{<V>}\times 10^{6}$ | -$\ln\zeta^{<V>}$ | $\zeta^{<S+V>}\times 10^{5}$ | -$\ln\zeta^{<S+V>}$ | $\zeta^{<SV>}\times 10^{5}$ | $-\ln\zeta^{<SV>}$
$\frac{1}{16}$ | $1.546093458$ | $8.77461$ | $\frac{\alpha^{2}}{8}=6.6564192$ | 11.91992886 | $8.0632$ | 9.425609 | 3.2080284 | 10.34727
$\frac{1}{4}$ | $7.446539697$ | 7.20259 | $\frac{\alpha^{2}}{2}=26.6256771$ | 10.5336345 | $38.5639$ | 7.860609 | 14.0808 | 8.86816225
1 | $37.73719345$ | 5.57969 | $2\alpha^{2}=106.502$ | 9.147340142 | $194.011$ | 6.2450103 | 63.39626 | 7.36351521
Now we are in a position to discuss a number of cases:
(a) The so-called classic Lamb shift of hydrogen atom was measured
experimentally as 10 (10):
$L_{H}^{exp}(2S-2P)\equiv E_{H}(2S_{1/2})-E_{H}(2P_{1/2})=1057.845\ \ MHz$
(114)
Theoretically, in this case ($b_{H}=5.446170255\times 10^{-4},\
r_{N}^{H}=r_{p}=0.862fm$), Eqs.(102) and (103) make no contributions while
Eqs.(104) and (107) only contribute
$\Delta
E_{H}^{recoil-2}(2S_{1/2}-2P_{1/2})=-E_{H}^{recoil-2}(2,1/2,1)=-2.16156\ \
kHz$ (115)
and
$\Delta E_{H}^{NS}(2S-2P)=0.14525347\ \ MHz$ (116)
respectively. The dominant contribution comes from Eq.(105). If using
$\zeta^{<S>}$, we obtain
$\begin{array}[]{l}\Delta
E_{H}^{Rad<S>}(2S-2P)=\frac{1}{1+b_{H}}\frac{1}{8}(4.06931316\times
10^{8})[-\frac{8}{3}\ln\zeta^{<S>}+\frac{7}{15}+3\zeta^{<S>}-\frac{4}{3}\zeta^{<S>}\ln\zeta^{<S>}]\\\
=1000.6567\ MHz\end{array}$ (117)
If we use another three values of $\ln\zeta$ in Table 1, we get
$\displaystyle\Delta E_{H}^{Rad<V>}(2S-2P)$ $\displaystyle=$ $\displaystyle
1451.7912\ \ MHz$ (118) $\displaystyle\Delta E_{H}^{Rad<S+V>}(2S-2P)$
$\displaystyle=$ $\displaystyle 1089.6513\ \ MHz$ (119) $\displaystyle\Delta
E_{H}^{Rad<SV>}(2S-2P)$ $\displaystyle=$ $\displaystyle 1226.0871\ \ MHz$
(120)
It seems that Eq.(117) is smaller whereas Eq.(118) too large. So as an
empirical rule in our semiquantitative calculation, we may use Eq.(119) to get
$L_{H}^{theor.}(2S-2P)=1089.651+0.145-0.002=1089.794\ \ MHz$ (121)
which is larger than Eq.(114) by $3\%$.
(b) The Lamb shift of $He^{+}$ atom has been measured as (quoted from 27
(27)):
$L_{He^{+}}^{exp}(2S-2P)=14041.13(17)\ \ MHz$ (122)
Similar to the case of hydrogen atom but with $Z=2$ and
$b_{He^{+}}=\frac{m_{e}}{m_{\alpha}}=0.0001371$, we find
$\displaystyle\Delta E_{He^{+}}^{Rad<S>}(2S-2P)$ $\displaystyle=$
$\displaystyle 1.252680693\times 10^{10}\ \ Hz$ $\displaystyle\Delta
E_{He^{+}}^{Rad<V>}(2S-2P)$ $\displaystyle=$ $\displaystyle 2.023083608\times
10^{10}\ \ Hz$ $\displaystyle\Delta E_{He^{+}}^{Rad<S+V>}(2S-2P)$
$\displaystyle=$ $\displaystyle 1.369980830\times 10^{10}\ \ Hz$
$\displaystyle\Delta E_{He^{+}}^{Rad<SV>}(2S-2P)$ $\displaystyle=$
$\displaystyle 1.636521214\times 10^{10}\ \ Hz$ (123)
As in the case of $H$ atom, we take the $<S+V>$ scheme and add
$\displaystyle\Delta E_{He^{+}}^{recoil-2}(2S-2P)$ $\displaystyle=$
$\displaystyle-2.165\ \ kHz$ (124) $\displaystyle\Delta
E_{He^{+}}^{NS}(2S-2P)$ $\displaystyle=$ $\displaystyle 4.514\ \ MHz$ (125)
($r_{\alpha}\simeq 1.2fm$), to find the theoretical value:
$L_{He^{+}}^{theor.}(2S-2P)=13704.220\ \ MHz$ (126)
which is smaller than Eq.(122) by $2.41\%$.
(c) The following energy-level difference is related to the ”hyper Lamb shift
(HLS)” 10 (10):
$\Delta_{H}^{exp}\equiv
E_{H}(4S)-E_{H}(2S)-\frac{1}{4}[E_{H}(2S)-E_{H}(1S)]=4797.338(10)\ \ MHz$
(127)
Theoretically, now Eq.(102) makes the main contribution:
$\Delta E_{H}^{RDE}[(4S)-\frac{5}{4}(2S)+\frac{1}{4}(1S)]=3923.95\ \ MHz$
(128)
(The notation in parenthesis is self-evident). Eq.(103) and Eq.(105)
contribute
$\Delta E_{H}^{recoil-1}[(4S)-\frac{5}{4}(2S)+\frac{1}{4}(1S)]=-4.186\ \ MHz$
(129)
and
$\displaystyle\Delta E_{H}^{Rad<S>}$ $\displaystyle=$ $\displaystyle
451.229097\ \ MHz$ $\displaystyle\Delta E_{H}^{Rad<S+V>}$ $\displaystyle=$
$\displaystyle 529.288296\ \ MHz$ $\displaystyle\Delta E_{H}^{Rad<SV>}$
$\displaystyle=$ $\displaystyle 675.907131\ \ MHz$ $\displaystyle\Delta
E_{H}^{Rad<V>}$ $\displaystyle=$ $\displaystyle 903.266275\ \ MHz$ (130)
respectively. Adding a small contribution from Eq.(107)
$\Delta E_{H}^{NS}[(4S)-\frac{5}{4}(2S)+\frac{1}{4}(1S)]=0.1270967854\ \ MHz$
(131)
we get
$\displaystyle\Delta_{H}^{Theor.<S>}$ $\displaystyle=$ $\displaystyle
4371.120197\ \ MHz$ $\displaystyle\Delta_{H}^{Theor.<S+V>}$ $\displaystyle=$
$\displaystyle 4449.179396\ \ MHz$ $\displaystyle\Delta_{H}^{Theor.<SV>}$
$\displaystyle=$ $\displaystyle 4595.798231\ \ MHz$
$\displaystyle\Delta_{H}^{Theore.<V>}$ $\displaystyle=$ $\displaystyle
3923.95-4.186+903.266275+0.1271=4823.1574\ \ MHz$ (132)
The $<V>$ scheme is only larger than Eq.(127) by $0.54\%$. All other schemes
would be too small. So we guess that for $S$ states $<V>$ scheme is better
than $<S>$ scheme.
(d) The following energy-level difference was also measured as 10 (10):
${\Delta^{\prime}}_{H}^{exp}\equiv
E_{H}(4D_{5/2})-E_{H}(2S)-\frac{1}{4}[E_{H}(2S)-E_{H}(1S)]=6490.144(24)\ \
MHz$ (133)
Theoretically, Eq.(102) also makes the main contribution:
$\Delta E_{H}^{RDE}[(4D_{5/2})-\frac{5}{4}(2S)+\frac{1}{4}(1S)]=5747.92\ \
MHz$ (134)
while
$\Delta E_{H}^{recoil-1}[(4D_{5/2})-\frac{5}{4}(2S)+\frac{1}{4}(1S)]=-4.18611\
\ MHz$ (135)
$\Delta E_{H}^{recoil-2}[(4D_{5/2})]=\alpha^{4}m_{e}(5.446170255\times
10^{-4})^{2}(\frac{1}{3}-\frac{2}{5})=-6.9283\ \ kHz$ (136)
are all small, we will have
$\displaystyle{\Delta^{\prime}}E_{H}^{rad<S>}[(4D_{5/2})-\frac{5}{4}(2S)+\frac{1}{4}(1S)]$
$\displaystyle=$ $\displaystyle 302.088631\ \ MHz$
$\displaystyle{\Delta^{\prime}}E_{H}^{rad<V>}[(4D_{5/2})-\frac{5}{4}(2S)+\frac{1}{4}(1S)]$
$\displaystyle=$ $\displaystyle 700.843464\ \ MHz$
$\displaystyle{\Delta^{\prime}}E_{H}^{rad<S+V>}[(4D_{5/2})-\frac{5}{4}(2S)+\frac{1}{4}(1S)]$
$\displaystyle=$ $\displaystyle 369.124660\ \ MHz$
$\displaystyle{\Delta^{\prime}}E_{H}^{rad<SV>}[(4D_{5/2})-\frac{5}{4}(2S)+\frac{1}{4}(1S)]$
$\displaystyle=$ $\displaystyle 500.131264\ \ MHz$ (137)
Finally, the nucleus size effect gives
${\Delta^{\prime}}E_{H}^{NS}[(4D_{5/2})-\frac{5}{4}(2S)+\frac{1}{4}(1S)]=11.62027752\times
10^{5}(\frac{1}{4}-\frac{5}{4}\times\frac{1}{8})=0.10894\ \ MHz$ (138)
In sum, we have
$\displaystyle{\Delta^{\prime}}_{H}^{<S>}$ $\displaystyle=$
$\displaystyle{\Delta^{\prime}}E_{H}^{RDE}+{\Delta^{\prime}}E_{H}^{recoil-1}+{\Delta^{\prime}}E_{H}^{recoil-2}+{\Delta^{\prime}}E_{H}^{rad<S>}+{\Delta^{\prime}}E_{H}^{rad<NS>}=6045.925\
\ MHz$ $\displaystyle{\Delta^{\prime}}_{H}^{<V>}$ $\displaystyle=$
$\displaystyle 6444.679\ \ MHz$ $\displaystyle{\Delta^{\prime}}_{H}^{<S+V>}$
$\displaystyle=$ $\displaystyle 6112.961\ \ MHz$
$\displaystyle{\Delta^{\prime}}_{H}^{<SV>}$ $\displaystyle=$ $\displaystyle
6243.967\ \ MHz$ (139)
which are smaller than the experimental value (133) by $6.8\%,\ 0.7\%,\ 5.8\%$
and $3.8\%$ respectively. .
(e) Experimentally, the combination of Eq.(127) with Eq.(133) yields:
${\Delta^{\prime\prime}}_{H}^{exp}\equiv E(4D_{5/2})-E(4S_{1/2})=1692.806\ \
MHz$ (140)
Then, theoretically, we have
$\displaystyle{\Delta^{\prime\prime}}_{H}^{RDE}(4D_{5/2}-4S)$ $\displaystyle=$
$\displaystyle 1.823886903\times 10^{9}\ \ Hz$ (141)
$\displaystyle{\Delta^{\prime\prime}}_{H}^{recoil-1}(4D_{5/2}-4S)$
$\displaystyle=$ $\displaystyle 1.1008\ \ Hz$ (142)
$\displaystyle{\Delta^{\prime\prime}}_{H}^{recoil-2}(4D_{5/2}-4S)$
$\displaystyle=$ $\displaystyle-6.9283\ \ kHz$ (143)
$\displaystyle{\Delta^{\prime\prime}}_{H}^{NS}(4D_{5/2}-4S)$ $\displaystyle=$
$\displaystyle-0.0181605862\ \ MHz$ (144)
and
$\displaystyle{\Delta^{\prime\prime}}_{H}^{rad<S>}(4D_{5/2}-4S)$
$\displaystyle=$ $\displaystyle-149.1404661\ \ MHz$ (145)
$\displaystyle{\Delta^{\prime\prime}}_{H}^{rad<V>}(4D_{5/2}-4S)$
$\displaystyle=$ $\displaystyle-202.4228107\ \ MHz$ (146)
$\displaystyle{\Delta^{\prime\prime}}_{H}^{rad<S+V>}(4D_{5/2}-4S)$
$\displaystyle=$ $\displaystyle-160.1636366\ \ MHz$ (147)
$\displaystyle{\Delta^{\prime\prime}}_{H}^{rad<SV>}(4D_{5/2}-4S)$
$\displaystyle=$ $\displaystyle-175.7758676\ \ MHz$ (148)
Altogether, we have
$\displaystyle{\Delta^{\prime\prime}}_{H}^{theore.<S>}(4D_{5/2}-4S)$
$\displaystyle=$ $\displaystyle 1674.721349\ \ MHz$
$\displaystyle{\Delta^{\prime\prime}}_{H}^{theore.<S+V>}(4D_{5/2}-4S)$
$\displaystyle=$ $\displaystyle 1663.716339\ \ MHz$
$\displaystyle{\Delta^{\prime\prime}}_{H}^{theore.<SV>}(4D_{5/2}-4S)$
$\displaystyle=$ $\displaystyle 1648.104108\ \ MHz$
$\displaystyle{\Delta^{\prime\prime}}_{H}^{theore.<V>}(4D_{5/2}-4S)$
$\displaystyle=$ $\displaystyle 1621.439105\ \ MHz$ (149)
which are smaller than Eq.(140) by $1.1\%,\ 1.7\%,\ 2.6\%$ and $4.2\%$
respectively.
(f) It’s time to go back to the precision data of $2S-1S$ transition in
hydrogen atom as discussed in section II. Rewrite Eq.(7) as (see also 43
(43)):
$\Delta E_{H}^{exp}(2S-1S)=2.46606141318734\times 10^{15}\ \ Hz$ (150)
Theoretically, we have [see Eq.(17)]:
$\Delta E_{H}^{RDE}(2S-1S)=2.466067984\times 10^{15}\ \ Hz$ (151)
$\Delta E_{H}^{recoil-1}(2S-1S)=22.32598676\ \ MHz$ (152)
$\displaystyle\Delta E_{H}^{rad<S>}(2S-1S)$ $\displaystyle=$
$\displaystyle-5142.081146\ \ MHz$ $\displaystyle\Delta
E_{H}^{rad<S+V>}(2S-1S)$ $\displaystyle=$ $\displaystyle-5765.958928\ \ MHz$
$\displaystyle\Delta E_{H}^{rad<SV>}(2S-1S)$ $\displaystyle=$
$\displaystyle-6835.535314\ \ MHz$ $\displaystyle\Delta E_{H}^{rad<V>}(2S-1S)$
$\displaystyle=$ $\displaystyle-8541.095068\ \ MHz$ (153)
$\Delta E_{H}^{NS}(2S-1S)=11.62027752\times
10^{5}(\frac{1}{8}-1)=-1.016774283\ \ MHz$ (154)
If taking the value of $\Delta E_{H}^{rad}(2S-1S)$, we get
$\displaystyle\Delta E_{H}^{theore.<S>}(2S-1S)$ $\displaystyle=$
$\displaystyle 2.466062836\times 10^{15}\ \ Hz$ $\displaystyle\Delta
E_{H}^{theore.<S+V>}(2S-1S)$ $\displaystyle=$ $\displaystyle 2.466062239\times
10^{15}\ \ Hz$ $\displaystyle\Delta E_{H}^{theore.<SV>}(2S-1S)$
$\displaystyle=$ $\displaystyle 2.466061169\times 10^{15}\ \ Hz$
$\displaystyle\Delta E_{H}^{theore.<V>}(2S-1S)$ $\displaystyle=$
$\displaystyle 2.466059464\times 10^{15}\ \ Hz$ (155)
They are larger than Eq.(150) by $1450\ MHz,\ 826\ MHz$ and smaller than
Eq.(150) by $244\ MHz,\ 1949\ MHz$ respectively. Or, their discrepancies are
$+5.9\times 10^{-7},\ +3.3\times 10^{-7},\ -1.0\times 10^{-7},\ -7.9\times
10^{-7}$, respectively. This discrepancy is basically stemming from the
uncertainty in the calculation of $\Delta E_{H}^{rad}(2S-1S)$.
(g) Let us turn to the isotope-shift of $2S-1S$ transition. Rewrite Eq.(8) as
$\Delta E_{D-H}^{exp}(2S-1S)=6.70994337\times 10^{11}\ \ Hz$ (156)
Theoretically, rewrite Eqs.(19) and (20) as
$\Delta E_{D-H}^{RDE}(2S-1S)=6.7101527879\times 10^{11}\ \ Hz$ (157)
and
$\Delta E_{D-H}^{recoil-1}(2S-1S)=-11.176\ \ MHz$ (158)
$\displaystyle\Delta E_{D-H}^{rad<S>}(2S-1S)$ $\displaystyle=$
$\displaystyle-1.399158\ \ MHz$ $\displaystyle\Delta E_{D-H}^{rad<V>}(2S-1S)$
$\displaystyle=$ $\displaystyle-2.324028\ \ MHz$ $\displaystyle\Delta
E_{D-H}^{rad<S+V>}(2S-1S)$ $\displaystyle=$ $\displaystyle-1.568915\ \ MHz$
$\displaystyle\Delta E_{D-H}^{rad<SV>}(2S-1S)$ $\displaystyle=$
$\displaystyle-1.859945\ \ MHz$ (159)
$\Delta E_{D-H}^{NS}(2S-1S)=-5.11384949\ \ MHz$ (160)
Altogether, we find [using $<V>$ scheme in Eq.(159)]:
$\Delta E_{D-H}^{theore.<V>}(2S-1S)=6.709966701\times 10^{11}\ \ Hz$ (161)
which is larger than Eq.(156) by $2.333\ MHz$ or only $3.5\times 10^{-6}$.
Evidently, even Eq.(157) solely deviates from Eq.(156) by $3\times 10^{-5}$
only. And as expected, the different schemes for $\Delta E_{D-H}^{rad}(2S-1S)$
have little influence on the theoretical value, because any one of Eq.(159) is
much smaller than the nucleus size effect Eq.(160)
($r_{N}^{D}=r_{d}=2.115fm$).
(h) Finally, the so-called absolute Lamb-shift of $1S$ state in hydrogen atom
was determined by Weitz et al.10 (10) from the measured value Eq.(127) or
(133). In our notation, using Eq.(133), we will write it as follows:
$\begin{array}[]{ll}L_{H}(1S)=&4\\{{\Delta^{\prime}}_{H}^{exp}-\Delta
E_{H}^{RDE}[(4D_{5/2})-\frac{5}{4}(2S)+\frac{1}{4}(1S)]-\Delta
E_{H}^{recoil-1}[(4D_{5/2})-\frac{5}{4}(2S)+\frac{1}{4}(1S)]\\\ &-\Delta
E_{H}^{recoil-2}(4D_{5/2})+\frac{5}{4}L_{H}(2S)-L_{H}(4D_{5/2})\\}\end{array}$
(162)
Here the Lamb shift of $2S$ state $L_{H}(2S)$ can be determined from the
experimental value of Eq.(114) with $L_{H}(2P_{1/2})$ being calculated from
Eq.(105):
$L_{H}(2S)=L_{H}^{exp}(2S-2P)-\Delta
E_{H}^{recoil-2}(2S-2P_{1/2})+L_{H}(2P_{1/2})=1040.901\ \ MHz$ (163)
And $L_{H}(4D_{5/2})$ can also be calculated from Eq.(105), so
$L_{H}(1S)=8188.478\ \ MHz$ (164)
which is in agreement with $8172.874(60)\ MHz$ given by 10 (10) within an
accuracy $\lesssim 0.2\%$. If we use Eq.(127) to derive $L_{H}(1S)$, we would
have to calculate $L_{H}(4S)$ which is much larger than $L_{H}(4D_{5/2})$ and
its derivation from Eq.(105) seems not reliable. Similarly, the theoretical
value of $L_{H}(1S)$ turns out to be
$L_{H}^{theore.}(1S)=\Delta E_{H}^{rad}(1S)+\Delta E_{H}^{NS}(1S)$ (165)
with $\Delta E_{H}^{NS}(1S)=0.14525347\ MHz$. However, the value of $\Delta
E_{H}^{rad}(1S)$ strongly depends on the scheme we used in Eq.(105), which
must be narrowed in a high-loop calculation. The theoretical prediction was
given in 27 (27) as:
$L_{H}^{theor.}(1S)=8172754(14)(32)\ kHz$ (166)
Further discussions can be found in Refs. 5 ; 44 (44, 45).
## IX Summary and Discussion
The remarkable progress of the experimental research on energy-level
differences in hydrogenlike atoms has been making this field an ideal
theoretical laboratory for physics:
(a) The inevitable and successful use of reduced Dirac equation (RDE) to
hydrogenlike atoms, especially to the isotope-shift of $2S-1S$ transition as
reflected by Eqs.(156) through (161), is by no means an accidental fortune. It
implies that the argument in section III for introducing RDE, Eq.(41), is
correct to a high accuracy. In particular, the basic principle of invariance
under space-time inversion Eq.(42) (with original mass $m$) could remain valid
even for a noninertial frame. This implication has a far-reaching consequence
that a generalization at the above symmetry to a localized curved space-time
may be served as a possible road to quantize the general theory of relativity
16 (16).
However, there are two realizations of potential $V$ under the space-time
inversion, Eq.(35)(”scalar” type) and Eq.(42)(”vector” type). While Eq.(42)
does dominant in an atom like $H$ with $m_{p}\gg m_{e}$, the remaining
discrepancy of $2.333\ MHz$ between theory and experiment [Eq.(161) versus
Eq.(156)] strongly hints that an important and subtle effect had been ignored.
(To consider the contribution of the deuteron polarizability merely accounts
for about $20\ kHz$ 6 ). We think what neglected must be a tiny excitation of
antiparticle field in the nucleus due to its interaction with electron in the
CMCS. So when we reduce the degrees of freedom of two-body system from two to
one, the RDE should be modified to take account of the tiny mixture of
”scalar” potentials (see the page note after Eq.(42)). We don’t know how to
improve $RDE$ yet. However, an experimental evidence for the above conjecture
could be the following prediction: The discrepancy between present theory
(with RDE) and experiment must be smaller for the isotope shift in $2S-1S$
transition of atoms ${}^{4}He$ and ${}^{3}He$ than that of atoms $H$ and $D$.
Recently, by using Dirac’s method, Marsch rigorously solved the hydrogen atom
as a two-Dirac particle system bound by Coulomb force 34 (34). His solutions
are composed of positive and negative pairs, corresponding respectively to
hydrogen and anti-hydrogen as expected. However, surprisingly, in the hydrogen
spectrum, besides the normal type-1 solution with reduced mass $\mu$, there is
another anomalous type$-2$ solution with energy levels:
${E^{\prime}}_{n}=Mc^{2}-2\mu c^{2}+\frac{1}{2}\mu
c^{2}(\frac{\alpha}{n})^{2}+\cdots\ (n=1,2,\ldots)$ and ”strange enough, the
type$-2$ ground state $(n=1)$ does not have lowest energy but the continuum
$(n=\infty)$”. In our opinion, based on what we learnt from the Dirac equation
and RDE, these anomalous solutions imply a positron moving in the field of
proton. So all discrete states with energy ${E^{\prime}}_{n}$ are actually
unbound, they should be and can be ruled out in physics either by the ”square
integrable condition” or the ”orthogonality criterion” acting on their
rigorous wave functions (for one-body Dirac equation, see 35 (35), also
p.$28-31$, $50$ of 36 (36)). On the other hand, all continuum states
($n=\infty$) with energies lower than $Mc^{2}-2\mu c^{2}$ correspond to
scattering wave functions with negative phase shifts , showing the repulsive
force between positron and proton. (see 37 (37), section 1.5 in 36 (36) or
section 9.5 of 15 (15)). Marsch’s discovery precisely reflects two things: (a)
the negative energy state of a particle just describes its antiparticle state.
(b) The Coulomb potential allows a complete set of solutions comprising of two
symmetric sectors,hydrogen and antihydrogen.In the hydrogen sector, the proton
remains unchanged regardless of the changing process of electron into positron
under the Coulomb interaction.
The above particle-antiparticle symmetry (including Eq.(42) showing the
unequal treatment between electron and nucleus), together with the parity
symmetry, is hidden in the Dirac’s four-component theory in covariant form so
they were overlooked to some extent in the past. The advantage or flexibility
of two-component noncovariant form of Dirac equation or RDE (as discussed in
this paper) lies in the fact that the above two symmetries become accurate and
so easily to be extended (or violated) in an explicit manner. For
completeness, let us stress again that for antiparticle, one should use the
momentum and energy operators being ${\mathbf{p}}_{c}=i\nabla$ and
$E_{c}=-i\frac{\partial}{\partial t}$ versus ${\mathbf{p}}=-i\nabla$ and
$E=i\frac{\partial}{\partial t}$ for particle as required by the space-time
inversion symmetry. The historical mission of the conception to imagine the
positron as a ”hole” in the sea of negative energy electrons is already over.
Since the CPT invariance had been further verified 39 (39), the relation
between a particle $|a\rangle$ and its antiparticle $|\bar{a}\rangle$ is well-
established as: 444To our knowledge, the correct definition, Eq.(167), was
first given by T. D. Lee and C. S. Wu at Ann. Rev. Nucl. Sci. 15, 381(1965).
See also G. J. Ni at J. Fudan Univ. (Natural Science) No.3-4, 125(1974).
$|\bar{a}\rangle=CPT|a\rangle$ (167)
with their wave-functions (in free motion) being respectively:
$\langle{\bf x},t|a\rangle\sim\exp[\frac{i}{\hbar}({\bf p}\cdot{\bf x}-Et)]$
(168) $\langle{\bf x},t|\bar{a}\rangle\sim\exp[-\frac{i}{\hbar}({\bf
p}\cdot{\bf x}-Et)]$ (169)
Note that in Eqs.(168) and (169), they have the same momentum $\bf p$ and
positive energy $E$. Either a newly defined space-time inversion (${\bf
x}\to-{\bf x},\,t\to-t$) or a simple change of $i\to-i$ will transform
Eq.(168) into Eq.(169) (or vice versa).
(b) Throughout this paper, the electron bound in an atom is just treated like
a stationary ”ball” with nucleus at its center and having a (Bohr) radius
($\sim 1/\alpha m_{e}$). However, it is in an off-mass-shell state (In some
sense, our atom model is just the opposite to J. J. Thomson’s atom model 100
years ago). In fact, the electron’s mass is reduced suddenly from $m_{e}$ to
$\mu$ in the RMCS when it is captured by a nucleus at the far remote orbit
with quantum number $n\longrightarrow\infty$ and further reduced to
$\mu+\delta\mu\simeq\mu-\frac{Z^{2}\alpha^{2}}{2n^{2}}\mu$ until $n$
decreasing to the lowest limit $n=1$. The Lamb shift should be viewed as a
further modification on the mass of an off-mass-shell electron due to
radiative correction.
Notice that the parameter $Q^{2}$ in the vertex function, Eq.(87), means the
square of (three-dimensional) momentum transfer when a free electron is on its
mass-shell and collides with some other particle as discussed in Ref.8b . By
contrast, now $Q^{2}$ exhibits itself as an effective potential of
$\delta$-function type exerted by the nucleus to the bound (and so off-mass-
shell) electron as shown by Eq.(92). To bind an electron to a nucleus is a
nonperturbative effect. Hence we can understand why the discrepancy between
$\zeta^{<S>}$ (calculated by perturbative QED at one-loop order) and
$\zeta^{<V>}$ (evaluated via nonperturbative Virial theorem) is so large.
Fortunately, they lead to discrepancies in the calculated values of Lamb shift
being not so large as shown in Section VIII. When $\zeta^{<V>}$ or
$\zeta^{<S+V>}$ (or $\zeta^{<SV>}$) is substituted into the Eq.(105) which is
derived from perturbative ($L=1$) theory, we should always be aware of some
theoretical inconsistency in such a semi-empirical treatment. But as a whole,
we believe that the concept of Lamb shift as an off-mass-shell effect in
covariant QED is basically correct.
(c) For a free on-mass-shell electron, its charge square $e_{R}^{2}$ will
increase with the increase of $Q^{2}$ as shown by Eq.(72) (with
$\mu_{3}=m_{e},\ q^{2}=-Q^{2}$) and was calculated in detail in 8b ,
coinciding with the experimental data. Note that, however, the Ward identity
$Z_{1}=Z_{2}$ had been used. An interesting question arises for a bound
electron: as its $e_{R}^{2}$ is not a function of $Q^{2}$, will $e_{R}^{2}$
change with the variation of the quantum number $n$? To answer this question,
let us put Ward identity aside for a while and write down the renormalized
$\alpha_{R}=\frac{e_{R}^{2}}{4\pi}$ as
$\alpha_{R}=\frac{Z_{2}^{2}}{Z_{1}^{2}}Z_{3}\alpha\longrightarrow\frac{Z_{2}^{2}}{Z_{1}^{2}}\alpha$
(170)
Let us work in the CMCS, so $Z_{2}=\frac{1}{1-B}$ and $B$ is shown in Eq.(53)
but with $\mu$ replaced by $m_{e}=m$. Similarly, $Z_{1}$ is given by the first
part of Eq.(87) with $\mu\longrightarrow m$:
$Z_{1}\simeq
1+\frac{\alpha}{4\pi}[\frac{11}{2}-\ln\frac{m^{2}}{\mu_{1}^{2}}-3\xi+4(1+\xi)\ln\xi]$
(171)
where the off-mass-shell parameter $\xi$ in CMCS is defined by
$p^{2}=m^{2}(1-\xi)=m^{2}(1-\eta-\zeta^{\prime})=m^{2}(1-\eta)-m^{2}\zeta^{\prime}=\mu^{2}-m^{2}\zeta^{\prime}=\mu^{2}(1-\zeta)$
(172)
with
$m^{2}(1-\eta)=\mu^{2},\ \ \eta=1-\frac{\mu^{2}}{m^{2}},\ \
\zeta^{\prime}=\frac{\mu^{2}}{m^{2}}\zeta$ (173)
and $\zeta$ is exactly that in Eq.(88). If we ignore the dependence of $(1-B)$
on $\zeta$, Eq.(170) would give ($\zeta\ll 1$):
$\alpha_{R}=\alpha[1+\frac{2\alpha}{\pi}\ln(1+\frac{\zeta}{\eta})]$ (174)
after renormalization by adjusting the arbitrary constant $\mu_{1}$ so that
$\alpha_{R}|_{\zeta\rightarrow 0}=\alpha$ (175)
which connects to the Thomson limit $\alpha_{R}|_{Q\rightarrow 0}=\alpha$ for
a free electron continuously but not smoothly. Then for two lowest bound
states with $n=1$ and $n=2$, we would have (in $<V>$ scheme):
$\alpha_{R}|_{n=1}\simeq\alpha(1.000433832),\ \
\alpha_{R}|_{n=2}\simeq\alpha(1.0001123)$ (176)
This would modify the Bohr energy level in hydrogenlike atom A to
$\tilde{E}_{A}^{Bohr}(n)=-\frac{Z^{2}\alpha_{n}^{2}}{2n^{2}}\mu_{A}$ (177)
and make an extra contribution to the isotope-shift as
$\Delta\tilde{E}_{D-H}^{Bohr}(2S-1S)\simeq 726\ \ MHz$ (178)
which is definitely excluded by the experiment. Hence the above consideration
from Eq.(170) till Eq.(178) is wrong. We learn concretely once again that the
Ward identity $Z_{1}=Z_{2}$ is valid not only for an electron on its mass-
shell, but also for off-mass-shell case. Thus we use the same value of
$\alpha$ throughout the whole calculation.
(d) In Ref.8b , using our RRM and new renormalization group equation (RGE) for
QCD derived from it and keeping all masses of 6 quarks ($m_{c}=1.031\;GeV$,
$m_{b}=4.326\;GeV$, $m_{t}=175\;GeV$, $m_{s}=200\;MeV$, $m_{u}=8\;MeV$,
$m_{d}=10\;MeV$), we calculated the strong coupling constant
$\alpha_{s_{i}}(Q)$ for $i=u,d,s,c,b$ respectively. Their running curves
(starting from the common renormalization point $\alpha_{s}(M_{Z})=0.118$)
follow the trend of experimental data (as shown on p.158 of 39 (39)) quite
well but separate at the low $Q$ region. Interesting enough, each of them
rises to a maximum $\alpha_{s_{i}}^{max}$ and then suddenly drops to zero at
$Q=0$ corresponding to a threshold energy scale $E_{i}^{th}$ which could be
explained as the excitation energy scale for breaking the quark pair. For
example, we find $E_{b}^{th}=1.13\;GeV$ which is just the hadronization energy
scale of Upsilon $\Upsilon(b\bar{b})$ against its dissociation into two
bosons. Experimentally, $M(\Upsilon(4s))-M(\Upsilon)=1.12\;GeV$ and
$\Upsilon(4s)\to B^{+}B^{-}$ or $B^{0}\bar{B}^{0}$. similarly,
$E_{c}^{th}=0.398\;GeV$ while $M(\psi(3770))-M(\psi(3097))=673\;MeV$ and
$\psi(3770)\to D^{+}D^{-}$ or $D^{0}\bar{D}^{0}$. it seems that
$E_{s}^{th}\sim 90\,MeV$ and $E_{u,d}^{th}\sim 0.4\,MeV$ are not so reliable
but still reasonable.
Actually, our calculation on QCD is backed by that on QED. In [8]b, using our
RRM and improved RGE, keeping all contributions from 9 charged leptons and
quarks we were able to calculate the running fine-structure constant
$\alpha_{R}(Q)$ from the renormalization point
$\alpha_{R}(Q)|_{Q=0}=\alpha=(137.036)^{-1}$ until it coincides with the
experimental value of $\alpha_{exp}(M_{Z})=(128.89)^{-1}$. We fitted quark’s
masses as mentioned above and found no further room left for extra charged
elementary particles (say, of 4th generation).
(e) In 1989, we had estimated the upper and lower bounds on Higgs mass $M_{H}$
by using a nonperturbative approach in QFT — the Gaussian effective potential
(GEP) method, yielding 40 (40):
$76\;GeV<M_{H}<170\;GeV$ (179)
Like many other authors, we were bothered a lot by divergences. After a deeper
study on the $\lambda\phi^{4}$ model by using our new RRM 8a , we restudied
this problem by combination GEP with RRM, yielding24 (24):
$M_{H}=138\;GeV$ (180)
This is not a upper or lower bound but a prediction based on the input of
experimental data:
$\begin{array}[]{l}M_{W}=80.359\;GeV,\;M_{Z}=91.1884\;GeV,\alpha^{-1}=\dfrac{4\pi}{g^{2}\sin^{2}\theta_{W}}=128.89,\\\
\sin^{2}\theta_{W}=\dfrac{{g^{\prime}}^{2}}{g^{2}+{g^{\prime}}^{2}}=0.2317\end{array}$
(181)
where $\theta_{W}$ is the weak mixing (Weinberg) angle. Because of getting rid
of all divergences, our calculation is well under control at every step. As
now the search for Higgs particle becomes so urgent experimentally but the
theoretical estimation about its mass still remains uncertain39 (39), we think
our approach with the prediction (180) deserves to be reconsidered.
(f) Moreover, our RRM can be used in $D+1$ space-time without limitation on
the space dimension $D$. A detailed analysis of sinh(sine)-Gordon models with
$D=1,2$ and $3$ (also using GEPM) is given by Ref.25 (25). Another example is
again the Lamb shift but calculated by QED in noncovariant form and by using
RRM similar to that in this paper, see Appendix (26 (26), see also the
Appendix 9A in 15 (15)). The theoretical value (A.20) seems better than (121),
showing that for dealing with the Lamb shift, the noncovariant theory may be
more suitable than the covariant one at least in the lowest order.
(g) Previously, the theories for Lamb shift or generally for calculating
energy levels in hydrogenlike atoms are rather complicated as reviewed in refs
1 ; 2 and 27 (27), some of them have been discussed in the Appendix of this
paper. For further clarity, let us try to summarize the main obstacles, or
challenges in four points:
(1) Different masses of nuclei must be taken into account;
(2) Relativistic effects of the electron (not nucleus) are important;
(3) In calculating radiative corrections, the divergence becomes severer and
severer with the increase of loop number;
(4) Since nuclei’s properties are different from one atom to another, to treat
each atom as a two-body system individually would be a daunting task, it
couldn’t be rigorous eventually too. This can be clearly seen from the recent
work by Marsch 34 (34).
Facing these challenges and learning from lessons and experiences of previous
authors, we see that the clue point is to replace the electron mass $m$ by
reduced mass $\mu$ and work in the noninertial frame (RMCS) throughout the
entire calculation. As is well known, this can be handled in nonrelativistic
QM by a mathematical trick but is impossible in relativistic case. So what we
need is a new understanding on the essence of special relativity — the
invariance (of theory) under the (newly defined) space-time inversion in one
inertial frame. Then we are able to claim the same invariance in the RMCS with
$\mu$ replacing $m$ for establishing the RDE, ignoring a small centripetal
acceleration of the nucleus in $CMCS$ (see page note after Eq.(42)). The
approximation in $RDE$ is some price paid for the much bigger gain—improving
the original Dirac equation (unable to treat different nuclei) and avoiding
the confusion in QED calculation because of the entanglement of two frames:
CMCS with RMCS ( i.e., the radiative corrections are entangled with the recoil
effect as we can see from previous literatures). In some senses, we jump over
obstacles (1) ,(2) and (4)at the least labor cost (by constructing RDE). In
the meantime, we hope RDE would help to ease the difficulty in point (3). And
it’s a great pleasure to see that the essential correctness of our
understanding has been validated by Marsch’s work as well as puzzles raised in
his paper 34 (34). Please see also Ref 41 (41).
As to challenge (3), only after we puzzled over the ”divergence” for decades,
could we suddenly realize that we misread its implication as a ”large number”.
Rather, it means the ”uncertainty”. Let us look at the calculation in section
IV again. Previously, many authors treated the divergent integral I in Eq.(50)
by different tricks of regularization , arriving at Eq.(56). Because both A
and B are divergent, it was thought that the original mass ($\mu$ here) does
receive some radiative corrections (via the self-energy diagrams in Fig 2(a)
and (b)) and becomes a ”renormalized” mass ($\mu_{R}$ here). The latter should
be the observed mass in experiment or physical mass (of electron). So the
original mass was called as the ”bare mass”, which was written into the
Lagrangian density as an input parameter of QFT. Then in constructing Feynman
diagrams of certain perturbative calculation, one needs to further introduce
some (divergent) ”counter terms” for cancelling the divergence stemming from
the bare mass. Based on that understanding, the renormalization factor for
wavefunction, $Z_{2}$ in Eq.(55), would be a divergent quantity too (in sharp
contrast to here Eq.(59) being a fixed and finite number). Previously, In
Eq.(72), while the $e_{R}$ on the left handed side is the observed charge
which should be finite, the $e$ on the right handed side was regarded as a
”bare charge” which, together with the $Z_{3}$, was a divergent quantity. (see
Fig. 7.8 in 23 (23). By contrast, here both $Z_{3}$ and $e$ are finite.
Actually, here $e$ is defined as the physical charge observed at the Thomson
limit in experiment).
In our opinion, the reason why we encountered so many superfluous troubles in
the past is because we overlooked what Bethe said in 194729 (29). Please read
his words quoted after Eq. (A.2) in the Appendix. Let us explain via our
Eq.(46). The (reduced) mass ($\mu$) already contains some contributions from
self-energy diagrams like Fig. 2(a) and (b). When we evaluate the (divergent)
integral, Eq.(50), trying to find the radiative corrections on the electron,
the latter is bound to confuse with that already contained in the mass. In
other words, the dividing line between them is blurred inevitably. The
emergence of explicit divergence is essentially a warning that the effect you
want to evaluate has been entangled with the mass itself, rendering both of
them uncertain. Hence the aim of so-called renormalization is nothing but to
redraw the dividing line between them such that the values of mass
(reconfirmed by the experiment) and the new effect (e.g., the mass increment
when the electron is moving off-mass-shell, Eq.(61)) can be clearly separated.
In short, what we have been learning in the past decade is: At the level of
QM, in the Hamiltonian like Eq. (A.1), the parameters $m$ and $e$ can be
regarded as well-defined. But they are not so at the level of QFT. Once the
calculation is made beyond the tree level, i.e., with loop number $L\geq 1$,
the divergence occurs and the meaning of parameters becomes obscure
immediately.
We need to reconfirm all parameters contained in the Lagrangian density before
they can be linked with experiments. In this sense, a model of QFT is at most
an ”effective field theory”. According to the above point of view, we believe
that our RRM just provides a natural way to carry out these processes of
reconfirmation 8a , getting rid of divergences and ambiguities. Please see
also Ref 42 (42).
(h) Last, but not least, during the learning and teaching of graduate courses
on QFT for decades, we have been sharing the joy and puzzle with our students
all the time. We hope that the presentation of this paper could be useful as a
teaching reference to render the QFT course more understandable, interesting
and attractive.
## Acknowledgements
We thank S.Q. Chen, S.S. Feng, R.T. Fu, P.T. Leung, W.F. Lu, X.T. Song, F.
Wang, H.B. Wang, J. Yan, G.H. Yang and J.F. Yang for close collaboration and
helpful discussions. We are also indebted to referees whose comments provided
us opportunities to improve our manuscript.
## Appendix: Comparison Between Noncovariant and Covariant Theories for Lamb
Shift
1\. To our knowledge, the precision theory for Lamb shift was based on a
combination of noncovariant (nonrelativistic or old-fashioned) QED with
covariant (or relativistic) QED as discussed in Ref.27 (27). As explained
clearly by Sakurai in Ref.21 (21), in perturbative QFT of noncovariant form,
all virtual particles are ”on-mass-shell”. Here we wish to emphasize that a
rigorous reconfirmation procedure of mass parameter was often overlooked in
previous literatures. The theory for hydrogenlike atom begins with a
Hamiltonian:
$H_{0}=\frac{1}{2m}{\mathbf{p}}^{2}+\frac{1}{2m_{N}}{\mathbf{p}}^{2}-\frac{Z\alpha}{r}$
$None$
(${\mathbf{p}}=-i\nabla$, see Eq.(34) in 27 (27)). As Bethe 29 (29) first
pointed out that the effect of electron’s interaction with the vector
potential $\mathbf{A}$ of radiation field (see 21 (21),15 (15))
$H_{int}^{(1)}=\frac{e}{mc}{\mathbf{A}}\cdot{\mathbf{p}}$ $None$
should properly be regarded as already included in the observed mass $m_{obs}$
of the electron, which is denoted by $m$ in (A1). However, once a concrete
calculation is made with (A2) being taken into account, the divergence emerges
immediately. What does it mean? Mathematicians teach us that there are three
implications for a divergence:
(a)It is a dimensionless number; (b)It is a large number; (c)It is uncertain.
While we physicists often emphasized the point (b), we didn’t pay enough
attention to the points (a) and (c). We often talked about a quadratically (or
linearly) divergent integral without noticing that it has a dimension (say,
mass dimension) and thus meaningless in mathematics unless a mass parameter
(say, $m$) in the integral is already fixed as a mass ”unit” so that the
integral can be divided by $m^{2}$ (or $m$) to become dimensionless.
Alternatively, a logarithmically divergent integral is dimensionless and thus
unaffected by the choice of unit [like Eq.(50), see also Eq.(A6) below], it
just implies an uncertainty waiting to be fixed. The implication of
uncertainty of a divergence will never vanish even after we introduced a
cutoff by hand to curb it. For example, in a pioneering paper to explain the
Lamb shift, Welton (30 (30), see section 9.6B in Ref.15 (15)) encountered an
integral $I=\int_{\omega_{min}}^{\omega_{max}}\frac{d\omega}{\omega}$ with
$\omega$ being the (angular) frequency of virtual photon (vacuum fluctuation).
He simply set $\omega_{min}\sim mZ\alpha=Z/a$, ($a$ is Bohr radius) and
$\omega_{max}\sim m$ so that $I\simeq\ln(1/Z\alpha)=4.92$ (for $Z=1$) which
leads to an estimation of Lamb shift $L_{H}^{theor.}(2S_{1/2}-2P_{1/2})\simeq
668\ MHz$. If instead of Bohr radius, the lower cutoff is provided by the
electron binding energy, one should get $I\simeq\ln(Z\alpha)^{-2}$ and
$L_{H}^{theor.}\simeq 1336\ MHz$. (see Eq.(30) in 27 (27)). The above
arbitrariness just reflects what essential in a divergent integral is not its
large magnitude ($\ln(Z\alpha)^{-1}$ is merely of the order of 10) but its
uncertainty. So what important in handling the integral is not to curb (or to
hide) its divergence but let the divergence exhibits itself as some arbitrary
constants explicitly (as shown in section IV-VI). We will show later how to do
this way for noncovariant QED.
2\. While Eqs.(A1) and (A2) only describe a spinless particle, the electron
has spin which endows it with the relativistic nature as shown by
Eqs.(41)-(45). For two-particle system, based on Bethe-Salpeter equation, an
effective Dirac equation (EDE) was derived as shown by Eq.(23) in 27 (27).
When the electromagnetic field interaction is taken into account, the Breit
potential $V_{Br}$ was derived as shown by Eq.(35) in 27 (27). Then the total
Breit Hamiltonian reads (Eq.(36) in 27 (27)):
$H_{Br}=H_{0}+V_{Br}$ $None$
However, the electron mass $m^{\prime}$ (in our notation here) appeared in EDE
or $V_{Br}$ should be that in the Dirac equation, also that in the definition
of reduced mass $\mu=\frac{m^{\prime}m_{N}}{m^{\prime}+m_{N}}$, eventually
$m^{\prime}$ could be identified with the observed mass $m_{obs}$, which is
not equal to the $m$ in Eq.(A1). This is because besides (A2) there is an
extra interaction due to electron spin with the radiation field:
$H_{int}^{(2)}=\frac{ge\hbar}{4\mu
c}{\mathbf{\sigma}}\cdot\nabla\times{\mathbf{A}}$ $None$
($g=2\times 1.0011596522$ is the gyromagnetic ratio of electron, see
Eq.(9A.15) in 15 (15)). The difference between $m$ and $m^{\prime}$ will be
calculated in (A16) below. It turns out to be of the order of $\alpha m$ and
cannot be ignored at the level of QED, especially for the explanation of Lamb
shift. We guess this must be one of the reasons why all calculations based on
Eq.(A3) became so complicated.
3\. In noncovariant theory, the leading contribution to the Lamb shift comes
from the one-photon electron self-energy. The nomenclature here is different
from that in the covariant theory. Roughly speaking, so-called electron self-
energy often corresponds to the vertex function in covariant theory (Fig.2(d)
in this paper) or to Figs.8 and 11 in Ref.27 (27) and its evaluations have
extended over 50 years 31 (31). More precisely, it is identified with the
radiative insertions in the electron line and the Dirac form factor
contribution. Further contributions from the Pauli form factor and the vacuum
polarization 27 (27) will add to a theoretical value of classic Lamb shift
being $1050.559\ MHz$. If taking more high-order corrections into account, the
theoretical value coincides with the experimental value $1057.845\ MHz$ rather
accurately (see Table 20 in 27 (27)). However, the above calculation looks
quite complicated due to two reasons: (a) The difficulty of dealing with two
masses in two coordinate systems, the electron mass $m$ and the reduced mass
$\mu$; (b) The introduction of an auxiliary parameter $\sigma\
[m(Z\alpha)\gg\sigma\gg m(Z\alpha)^{2}]$ to separate the radiative photon
integration region into two parts. In the low momentum region, the Bethe
Logarithm 32 (32) in noncovariant form makes the main contribution. In the
high momentum region, the evaluation is resorting to some relativistic
covariant form 22 (22). Then two expressions are matched together to get the
correct result. It seems to us that the matching trick used is doubtful
because both ultraviolet and infrared divergences were ambiguously handled by
some cutoff which missed the main point of renormalization—to reconfirm the
mass parameter in the presence of radiative corrections as shown in section IV
(covariant form) or below.
4\. A simple calculation for Lamb shift in noncovariant form was proposed in
Ref.26 (26) (see also Appendix 9A of Ref.15 (15)). Consider the self-energy
diagram of an electron with reduced mass $\mu$ and (three-dimensional)
momentum $\mathbf{p}$ in the RMCS of a hydrogenlike atom. Similar to Fig.2(a),
but also different in the virtual state, now a photon has energy
$\omega_{k}=k=|{\mathbf{k}}|$ while the electron has momentum
${\mathbf{q}}={\mathbf{p}}-{\mathbf{k}}$ and energy
$\varepsilon_{q}=\frac{1}{2\mu}q^{2}$. The electron in plane-wave state
$|{\mathbf{p}}>$ has two interactions with the radiative field at each vertex
as shown by (A2) ($m\rightarrow\mu$) and (A4), acquiring an increase in energy
respectively (see FIG. 3):
$\Delta
E_{p}^{(j)}=\sum_{i}\frac{|<i|H_{int}^{(j)}|{\mathbf{p}}>|^{2}}{\varepsilon_{p}-\varepsilon_{i}},\quad(j=1,2)$
$None$
Here $\varepsilon_{i}=\varepsilon_{q}+\omega_{k}$ is the energy of the
intermediate virtual state $|i>$. Simple evaluation leads to
$\Delta E_{p}^{(1)}=-\frac{\alpha
p}{\pi\mu}\int_{-1}^{1}d\eta(1-\eta^{2})I,\quad
I=\int_{0}^{\infty}\frac{dk}{k+\xi}$ $None$
where $\eta=\cos\theta$ with $\theta$ being the angle between $\mathbf{k}$ and
$\mathbf{p}$, $\xi=2(\mu-p\eta)$. Like Eq.(50), we take partial derivative of
the divergent integral $I$ with respect to $\xi$ (then the integration of $k$)
and integrate back to $I$ again, yielding:
$\Delta E_{p}^{(1)}=b_{1}^{(1)}p^{2}+b_{2}^{(1)}p^{4}+\cdots$ $None$
$b_{1}^{(1)}=\frac{\alpha}{\pi\mu}(\frac{4}{3}\ln
2+\frac{4}{3}\ln\mu-\frac{4}{3}C_{1})$ $None$
$b_{2}^{(1)}=\frac{\alpha}{\pi\mu^{3}}(-\frac{2}{15})$ $None$
Note that the term $b_{1}^{(1)}p^{2}$ will combine with the kinetic energy
$\frac{1}{2\mu}p^{2}$ of a (”spinless”) electron, they are indistinguishable.
The appearance of an arbitrary constant $C_{1}$ precisely reflects the fact
that we cannot find the reduced mass via the valuation of $\Delta E_{p}^{(1)}$
in perturbation theory. So we must choose $b_{1}^{(1)}=0$ to reconfirm the
value of $\mu$ (which is still not the final observed mass, see below).
Similar evaluation on $H_{int}^{(2)}$ (of the real electron with spin $1/2$)
which would induce the spin flip process between states
$|{\mathbf{p}},\pm\frac{1}{2}>$ and $|{\mathbf{q}},\pm\frac{1}{2}>$, leads to
$\Delta E_{p}^{(2)}=\frac{1}{2}\sum_{i,s_{z}=\pm
1/2}\frac{|<i|H_{int}^{(2)}|{\mathbf{p}},s_{z}>|^{2}}{\varepsilon_{p}-\varepsilon_{i}}=-\frac{\alpha
g^{2}}{8\pi\mu}\int_{-1}^{1}d\eta J$
$J=\int_{0}^{\infty}\frac{k^{2}dk}{k+\xi}$ $None$
Being a quadratically divergent integral, $J$ needs partial derivative of
third order with respect to $\xi$, yielding:
$\Delta E_{p}^{(2)}=b_{0}^{(2)}+b_{1}^{(2)}p^{2}+b_{2}^{(2)}p^{4}+\cdots$
$None$
$b_{0}^{(2)}=\frac{g^{2}}{4}\frac{\alpha\mu}{\pi}[4(\ln
2+\ln\mu)-4C_{2}-\frac{2C_{3}}{\mu}-\frac{C_{4}}{\mu^{2}}]$ $None$
$b_{1}^{(2)}=\frac{g^{2}}{4}\frac{\alpha}{\pi\mu}(\frac{4}{3}\ln
2+2+\frac{4}{3}\ln\mu-\frac{4}{3}C_{2})$ $None$
$b_{2}^{(2)}=\frac{g^{2}}{4}\frac{\alpha}{\pi\mu^{3}}(-\frac{1}{15})$ $None$
Let’s manage to fix three arbitrary constants $C_{2},C_{3}$ and $C_{4}$.
First, the term $b_{1}^{(2)}p^{2}$ should be combined with
$\frac{1}{2\mu}p^{2}$ term. Since $\mu$ is already fixed, further modification
on $\mu$ due to electron spin should be finite and fixed. So the only possible
choice of $C_{2}$ is to cancel $\ln\mu$ which is ambiguous in dimension:
$C_{2}=\ln\mu$, yielding
$b_{1}^{(2)}=\frac{\beta}{2\mu},\quad\beta=\frac{g^{2}\alpha}{2\pi}(\frac{4}{3}\ln
2+2)$ $None$
Then the dimensional constants $C_{3}$ and $C_{4}$ must be chosen such that
$b_{0}^{(2)}=0$, implying that the starting point of this theory is the
nonrelativistic Hamiltonian $H_{0}$ in Eq.(A1) without rest energy term while
both masses of the nucleus and the electron (with spin) are fixed by
experiments. Hence now $\mu$ acquires a modification via $b_{1}^{(2)}p^{2}$
term and becomes an observable one:
$\mu\longrightarrow\mu_{obs}=\frac{\mu}{1+\beta}$ $None$
However, we have to consider the relativistic energy of electron shown in
Eq.(30), where the term $(-\frac{1}{8\mu^{3}}p^{4})$ goes beyond Eq.(A1). Yet
the modification of $\mu$ shown as (A16) does induce a corresponding change
$-\frac{1}{8}(\frac{1}{\mu_{obs}^{3}}-\frac{1}{\mu^{3}})p^{4}$, which should
be regarded as an invisible ”background” and subtracted from the $p^{4}$ term
induced by radiative corrections. (The relativistic correction is brought in
via the RDE as discussed in section VIII). As a whole, the combination of
contributions from $H_{int}^{(1)}$ and $H_{int}^{(2)}$ leads to
$b_{1}=b_{1}^{(1)}+b_{1}^{(2)}=b_{1}^{(2)}$ $None$
and a ”renormalized” $b_{2}$:
$b_{2}^{R}=b_{2}^{(1)}+b_{2}^{(2)}+\frac{1}{8\mu^{3}}(3\beta+3\beta^{2}+\beta^{3})\simeq\frac{\alpha}{\pi\mu_{obs}^{3}}(1.99808)$
$None$
Here we only keep the lowest approximation at the last step. Hence the
electron self-energy-diagram contributes a radiative correction to the energy
level of the stationary state $|Z,n,l>$ in a hydrogenlike atom:
$\Delta E^{rad}(Z,n,l)=\langle
Z,n,l|b_{2}^{R}p^{4}|Z,n,l\rangle=[\frac{8n}{2l+1}-3]\frac{b_{2}^{R}Z^{4}\alpha^{4}}{n^{4}}\mu^{4}_{obs}$
$None$
This form, together with contributions from the vacuum polarization and
nuclear size effect, gives a theoretical value for classic Lamb shift:
$L_{H}^{theor.}(2S_{1/2}-2P_{1/2})\approx 1056.52\ MHz$ $None$
which is smaller than the experimental value by $0.13\%$. Despite its
approximation involved, the above method clearly shows that so-called
renormalization is nothing but a reconfirmation process of mass. We must
reconfirm the mass before it could be modified via radiative corrections.
Either ”skipping over the first step” or ”combining two steps into one” is not
allowed.
5\. In noncovariant theory, the (three-dimensional) momentum $\mathbf{p}$ is
combined with the reduced mass $\mu$ to form a kinetic energy term
$\frac{1}{2\mu}{\mathbf{p}}^{2}$ on the mass shell. Once the energy is
modified whereas $\mathbf{p}$ is conserved at the vertex, $\mu$ is bound to be
modified. On the other hand, in covariant theory, the electron energy turns to
a component of four-dimensional momentum $p$ and the latter is conserved at
the vertex. So the (reduced) mass $\mu$ cannot be modified on the mass shell
$(p^{2}=\mu^{2})$. Therefore, the renormalization as some reconfirmation has
different meaning in covariant theory versus that in noncovariant theory. We
guess this is why the matching procedure of these two formalisms into one
theory for Lamb shift proves so difficult.
6\. Every theory in physics is not only a discovery of natural law, but also
an invention of human being 33 (33). Hence the comparison among various
theories, in many cases, is not about a problem of being right or wrong.
Rather, it’s about a choice of simplicity, harmony (self-consistency) and
beauty. Only time can tell.
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Figure 1: A hydrogenlike atom in quantum mechanical description. The nucleus
with mass $m_{2}$ occupies a small sphere with radius $r_{N}$ (greatly
exaggerated in the diagram) while the electron with mass $m_{1}$ spreads over
a larger sphere with radius $R_{e}$ (i.e.atomic radius). Their common center
is the atom’s center of mass (CM). The wavefunction $\psi({\bf r})e^{-iEt}$
with ${\bf r}={\bf r}_{1}-{\bf r}_{2}$ shows the electron’s amplitude under a
”fictitious measurement” 15 (15), during which the electron and nucleus shrink
into two ”fictitious point particles ” located at ${\bf r}_{1}$ and ${\bf
r}_{2}$ simultaneously. The Coulomb potential $V(r)=-\frac{Ze^{2}}{r}$ between
them is a static one. The probability to find the electron at ${\bf r}$ is
$|\psi({\bf r})|^{2}$ while that to find its momentum being ${\bf p}$ is
$|\phi({\bf p})|^{2}$ with $\phi({\bf p})$ being the Fourier transform of
$\psi({\bf r})$.
Figure 2: Four Feynman diagrams at one-loop level (in covariant form). (a) and
(b) are self-energy diagrams of the electron. (c) is vacuum polarization. (d)
is vertex function. Solid lines and wavy lines refer to electron and photon
respectively, while X denotes the nucleus. Here $p,q$ and $k$ are four-
dimensional momenta.
Figure 3: The electron self-energy (radiative correction) diagram at one-loop
level of perturbative QCD in noncovariant form. $H_{int}^{(1)}$ (A.2) or
$H_{int}^{(2)}$ (A.4) is inserted into two vertices. Here $\bf p,q$ and $\bf
k$ are three dimensional momenta.
|
# Large deviation principle for geometric and topological functionals and
associated point processes
Christian<EMAIL_ADDRESS>[ Takashi
<EMAIL_ADDRESS>[ Department of Mathematics
Aarhus University
Ny Munkegade, 118, 8000, Aarhus C, Denmark, Department of Statistics
Purdue University
West Lafayette, 47907, USA,
###### Abstract
We prove a large deviation principle for the point process associated to
$k$-element connected components in ${\mathbb{R}}^{d}$ with respect to the
connectivity radii $r_{n}\to\infty$. The random points are generated from a
homogeneous Poisson point process or the corresponding binomial point process,
so that $(r_{n})_{n\geq 1}$ satisfies $n^{k}r_{n}^{d(k-1)}\to\infty$ and
$nr_{n}^{d}\to 0$ as $n\to\infty$ (i.e., sparse regime). The rate function for
the obtained large deviation principle can be represented as relative entropy.
As an application, we deduce large deviation principles for various
functionals and point processes appearing in stochastic geometry and topology.
As concrete examples of topological invariants, we consider persistent Betti
numbers of geometric complexes and the number of Morse critical points of the
min-type distance function.
60F10,
60D05, 60G55, 55U10,
Large deviation principle,
point process,
stochastic geometry,
stochastic topology,
persistent Betti number,
Morse critical point,
###### keywords:
[class=MSC]
###### keywords:
, and
## 1 Introduction
The objective of this paper is to examine large deviation behaviors of a point
process associated to a configuration of random points generated by a
homogeneous Poisson point process or the corresponding binomial point process.
The _Donsker–Varadhan large deviation principle (LDP)_ (see [14])
characterizes the limiting behavior of a family of probability measures
$(\mu_{n})_{n\geq 1}$ on a measurable space $(\mathcal{X},\mathcal{B})$. We
say that $(\mu_{n})_{n\geq 1}$ satisfies an LDP with rate $a_{n}$ and _rate
function_ $I(x)$, if for any $A\in\mathcal{B}$,
${-\inf_{x\in\text{int}(A)}I(x)}\leq\liminf_{n\to\infty}\frac{1}{a_{n}}\log\mu_{n}(A)\leq\limsup_{n\to\infty}\frac{1}{a_{n}}\log\mu_{n}(A)\leq-\inf_{x\in\bar{A}}I(x),$
(1.1)
where $\text{int}(A)$ denotes the interior of $A$ and $\bar{A}$ the closure of
$A$.
Let $\mathcal{P}_{n}$ be a homogeneous Poisson point process with intensity
$n$ on the unit cube $[0,1]^{d}$, $d\geq 2$, and $(r_{n})_{n\geq 1}$ be a
sequence of positive numbers decreasing to $0$ as $n\to\infty$. We put our
focus on the spatial distribution of $k$-tuples
${\mathcal{Y}}\subset\mathcal{P}_{n}$, which consist of components with
respect to $r_{n}$. More specifically, given a parameter $t\in(0,\infty)$, we
consider the following point process in $({\mathbb{R}}^{d})^{k}$:
$\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k}s_{n}({\mathcal{Y}},\mathcal{P}_{n};t)\,\delta_{r_{n}^{-1}({\mathcal{Y}}-\ell({\mathcal{Y}}))},$
(1.2)
where $\delta_{\bf z}$ denotes the Dirac measure at ${\bf
z}=(z_{1},\dots,z_{k})\in({\mathbb{R}}^{d})^{k}$ and $\ell({\mathcal{Y}})$ is
a “center” point of ${\mathcal{Y}}$ such as the left most point (in the
lexicographic ordering). Moreover, $s_{n}$ denotes an indicator function,
requiring that the diameter of $k$-tuples
${\mathcal{Y}}\subset\mathcal{P}_{n}$ be at most a constant multiple of
$r_{n}$, and further, such $k$-tuples must be distant at least $r_{n}t$ from
all the remaining points in $\mathcal{P}_{n}$. In this setup, our main theorem
(i.e., Theorem 2.1) aims to describe the LDP for the process (1.2) in the so-
called _sparse regime_ : $nr_{n}^{d}\to 0$ as $n\to\infty$.
As a primary application, we also provide the LDP for geometric and
topological statistics possessing the structure as component counts. Given
$t\in(0,\infty)$ and measurable functions $H\mathrel{\mathop{\mathchar
58\relax}}({\mathbb{R}}^{d})^{k}\to[0,\infty)$ that are symmetric in the $k$
arguments of ${\mathbb{R}}^{d}$, we define
$G_{n}({\mathcal{Y}},\mathcal{P}_{n};t)\mathrel{\mathop{\mathchar
58\relax}}=H(r_{n}^{-1}{\mathcal{Y}})\,{\mathbbm{1}}\big{\\{}\|y-z\|\geq
r_{n}t,\ \text{for all }y\in{\mathcal{Y}}\text{ and
}z\in\mathcal{P}_{n}\setminus{\mathcal{Y}}\big{\\}},$ (1.3)
where ${\mathbbm{1}}\\{\cdot\\}$ denotes an indicator function and $\|\cdot\|$
is the Euclidean norm. We then consider the point process
$\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k}\delta_{G_{n}({\mathcal{Y}},\mathcal{P}_{n};t)}.$
(1.4)
The process (1.4) is associated to scaled $k$-tuples
$r_{n}^{-1}{\mathcal{Y}}$, satisfying the geometric condition implicit in $H$,
with the additional restriction that geometric objects generated by
$r_{n}^{-1}{\mathcal{Y}}$ be distant at least $t$ from all the remaining
points of $r_{n}^{-1}\mathcal{P}_{n}$. A more precise and general setup for
the process (1.4) is given in Section 3.
With an appropriate choice of scaling regimes, the large deviations of point
processes as in (1.4) have been studied by Sanov’s theorem and its variant
[16, 37, 17]. In recent times, the process (1.4) also found applications in
stochastic geometry [13, 26, 9]. In particular, the authors of [9] explored
the Poisson process approximation of the point process for general stabilizing
functionals, including (1.4) as its special case, by deriving the rate of
convergence in terms of the Kantorovich-Rubinstein distance. Furthermore, [27]
discussed large deviations of the probability distribution of (1.4) under the
$M_{0}$-topology, with the assumption that the connectivity radii are even
smaller than those considered in [13, 9].
As another main application, we also examine the LDP for statistics of the
form
$\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k}G_{n}({\mathcal{Y}},\mathcal{P}_{n};t).$
(1.5)
A variety of functionals in stochastic geometry can be considered as such
statistics; see [5, 9, 26, 22]. Moreover, with an appropriate choice of $H$,
the statistics (1.5) can be used to investigate the behavior of topological
invariants of a geometric complex [6, 8, 20, 30, 38, 28]. Along this line of
research, Section 4 below discusses the LDP for persistent Betti numbers and
the number of Morse critical points of the min-type distance function. Loosely
speaking, the persistent Betti number is a quantifier for topological
complexity, capturing the creation / destruction of topological cycles. The
Morse critical points under our consideration provide a good approximation of
homological changes in the geometric complex.
The large deviation behavior of the processes (1.2), (1.4), and (1.5) heavily
depends on how rapidly the sequence $(r_{n})_{n\geq 1}$ decays to $0$ as
$n\to\infty$. In the existing literature (not necessarily relating to large
deviations), the configuration of geometric objects (determined by $H$ in
(1.3)) splits into multiple regimes [31, 20, 30]. In the context of random
geometric complexes (resp. random geometric graphs), if $nr_{n}^{d}\to 0$,
called the _sparse regime_ , the spatial distribution of complexes (resp.
graphs) is sparse, so that they are mostly observed as isolated components. In
the critical phase $nr_{n}^{d}\to c\in(0,\infty)$, called the _critical
regime_ for which $r_{n}$ decreases to $0$ at a slower rate than the sparse
regime, the complex (resp. graph) begins to coalesce, forming much larger
components. Finally, the case when $nr_{n}^{d}\to\infty$ is the _dense regime_
, for which the complex (resp. graph) is even more connected and may even
consist of a single giant component.
Among the regimes described above, the present paper focuses on the sparse
regime. In this case, there have been a number of studies on the “average”
behavior and the likely deviations from the “average” behavior of geometric
functionals and topological invariants, such as subgraph/component counts and
Betti numbers. More specifically, a variety of strong laws of large numbers
and central limit theorems for these quantities have already been established.
The readers may refer to the monograph [31] for the limit theorems for
subgraph/component counts, while the works in [20, 30, 38] provide the limit
theorems for topological invariants.
In contrast, however, there have been very few attempts made at examining the
large deviation behaviors of these quantities, especially from the viewpoints
of an LDP. In fact, even for a simple edge count in a random geometric graph,
determining the rate $a_{n}$ and the rate function $I(x)$ in (1.1) is a highly
non-trivial problem ([11]). Although there are several works (e.g., [3, 33,
39]) that deduced concentration inequalities for subgraph counts, the length
power functionals, and Betti numbers, these papers were not aimed to derive
the LDP. As a consequence, the obtained upper bounds in these studies do not
seem to be tight. Furthermore, [35] studied the LDP for the functional of
spatial point processes satisfying a weak dependence condition characterized
by a radius of stabilization. One of the major assumptions in their study is
that the contribution of any particular vertex must be uniformly bounded (see
condition (L1) therein). However, our functionals and point processes may not
fulfill such conditions. More importantly, the study in [35] treated only the
critical regime, whereas the main focus of our study is the sparse regime. One
of the benefits of studying the sparse regime is that geometric and
topological objects have a relatively simpler structure in the limit, which
will allow us to explicitly identify the structure of a rate function. In some
cases, we may even solve the variational problem for the rate function; see
Remark 3.6. We also note that [36] provides a framework to establish LDPs when
the uniform boundedness assumption in [35] is replaced by the finiteness of
suitably scaled logarithmic moment generating functions (see Equ. (2.5)
therein). However, the LDPs in [36] are designed only for real-valued random
variables, whereas our main results include LDPs for random measures, such as
Theorems 2.1 and 3.1 below. As a final remark, we note that the work in [19],
which is still under preparation, seems to utilize the ideas of [36] to derive
the LDP for Betti numbers and persistence diagrams of cubical complexes.
The remainder of this paper is structured as follows. Section 2 gives a
rigorous description of the LDP for the point process in (1.2). In Section 3,
we shall deduce the LDP for the processes in (1.4) and (1.5). Additional
examples on persistent Betti numbers and the number of Morse critical points
will be offered in Section 4. All the proofs are deferred to Section 5. For
the proof of our main theorems, we partition the unit cube $[0,1]^{d}$ into
multiple smaller sub-cubes and consider a family of point processes restricted
to each of these small sub-cubes. Subsequently, with the aid of the main
theorem in [13], we establish weak convergence of such point processes in
terms of the total variation distance. We then exploit Cramér’s theorem in
Polish spaces [14, Theorem 6.1.3] to identify the structure of a rate
function. After that, Proposition 2.2 ensures that the rate function can be
represented in terms of relative entropy. The required approximation argument
relies on the standard technique on the maximal coupling ([21, Lemma 4.32]).
For the proof of the theorems in Section 3, we shall utilize an extension of
the contraction principle, provided in [14, Theorem 4.2.23].
Before concluding the Introduction, we comment on our setup and assumptions.
First, we assume that the Poisson point process $\mathcal{P}_{n}$ is
homogeneous. It is clear, however, that in many applications (e.g., [1, 29]),
it is important to understand geometric and topological effects of lack of
homogeneity. A possible starting point for introducing inhomogeneity is to
look at point clouds arising from “inhomogeneous” Poisson point processes. In
this case, there is no doubt that a new and more involved machinery must be
developed to analyze such data; a detailed discussion will be postponed to a
future publication. Another possible extension seems to investigate the
regimes other than the sparse case. At least for the critical case (i.e.,
$nr_{n}^{d}\to c\in(0,\infty)$) however, it is impossible to directly
translate our proof techniques. In particular, Lemma 5.3 does not hold
anymore; see Remark 5.4 for more details on this point. Finally, the LDPs in
Section 4 are proven only when the dimensions of the Euclidean space, as well
as those of topological invariants, are small enough. This is due to the fact
that our proof techniques can apply only to a lower-dimensional case. It is
still unclear whether the LDP holds in higher-dimensions; this is also left as
a future topic of our research.
## 2 Model and main results
Let $\mathcal{P}_{n}$ be a homogeneous Poisson point process on $[0,1]^{d}$,
$d\geq 2$, with intensity $n$. Choose an integer $k\geq 2$, which will remain
fixed for the remainder of this section, and let
$\mathsf{diam}(x_{1},\dots,x_{k})\mathrel{\mathop{\mathchar
58\relax}}=\max_{1\leq i<j\leq k}\|x_{i}-x_{j}\|,\ \ \
x_{i}\in{\mathbb{R}}^{d}.$
Taking a sequence $(r_{n})_{n\geq 1}$ decaying to $0$ as $n\to\infty$, we
focus on the _sparse regime_ :
$\rho_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=n^{k}r_{n}^{d(k-1)}\to\infty,\ \ \ nr_{n}^{d}\to 0,\ \ \text{ as
}n\to\infty.$ (2.1)
For a subset ${\mathcal{Y}}$ of $k$ points in ${\mathbb{R}}^{d}$, a finite set
$\mathcal{Z}\supset{\mathcal{Y}}$ in ${\mathbb{R}}^{d}$ and $t\in(0,\infty)$,
we define
$c({\mathcal{Y}},\mathcal{Z};t)\mathrel{\mathop{\mathchar
58\relax}}={\mathbbm{1}}\big{\\{}\|y-z\|\geq t,\ \text{for all
}y\in{\mathcal{Y}}\text{ and }z\in\mathcal{Z}\setminus{\mathcal{Y}}\big{\\}}.$
We then define a scaled version of $c$ by
$c_{n}({\mathcal{Y}},\mathcal{Z};t)\mathrel{\mathop{\mathchar
58\relax}}=c(r_{n}^{-1}{\mathcal{Y}},r_{n}^{-1}\mathcal{Z};t)={\mathbbm{1}}\big{\\{}\|y-z\|\geq
r_{n}t,\ \text{for all }y\in{\mathcal{Y}}\text{ and
}z\in\mathcal{Z}\setminus{\mathcal{Y}}\big{\\}}.$ (2.2)
and also, for a fixed $L\in(t,\infty)$,
$s_{n}({\mathcal{Y}},\mathcal{Z};t)\mathrel{\mathop{\mathchar
58\relax}}=c_{n}({\mathcal{Y}},\mathcal{Z};t)\,{\mathbbm{1}}\big{\\{}\mathsf{diam}({\mathcal{Y}})\leq
r_{n}L\big{\\}}.$ (2.3)
Let $M_{+}\big{(}({\mathbb{R}}^{d})^{k}\big{)}$ be the space of Radon measures
on $({\mathbb{R}}^{d})^{k}$. For a finite set
${\mathcal{Y}}\subset{\mathbb{R}}^{d}$ of $k$ points in general position,
denote by $\ell({\mathcal{Y}})$ a _center point_ of ${\mathcal{Y}}$. For
example, it may represent the left most point of ${\mathcal{Y}}$ in the
lexicographic ordering. Another way of defining it is that one may set
$\ell({\mathcal{Y}})$ to be a center of the unique $(k-2)$-dimensional sphere
containing ${\mathcal{Y}}$. In either case, we write
$\overline{\mathcal{Y}}\mathrel{\mathop{\mathchar
58\relax}}={\mathcal{Y}}-\ell({\mathcal{Y}})$.
The primary objective of this section is to describe the LDP for the point
process
$\xi_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k}s_{n}({\mathcal{Y}},\mathcal{P}_{n};t)\,\delta_{r_{n}^{-1}\overline{\mathcal{Y}}},\
\ n\geq 1.$ (2.4)
The process $\xi_{k,n}$ counts scaled $k$-tuples
$r_{n}^{-1}\overline{\mathcal{Y}}$, that are locally concentrated in the sense
of $\mathsf{diam}({\mathcal{Y}})\leq r_{n}L$, under an additional restriction
that $r_{n}^{-1}{\mathcal{Y}}$ be distant at least $t$ from all the remaining
points of $r_{n}^{-1}\mathcal{P}_{n}$. Notice that
$\mathsf{diam}({\mathcal{Y}})\leq r_{n}L$ indicates
$\|r_{n}^{-1}\overline{\mathcal{Y}}\|\leq C$ for some constant
$C\in(0,\infty)$. Thus, one can fix a compact subset $E$ of
$({\mathbb{R}}^{d})^{k}$ such that the process (2.4) can be viewed as an
element of $M_{+}(E)$. Because of this restriction, $M_{+}(E)$ is now
equivalent to the space of _finite_ measures on $E$. Write
$\mathcal{M}_{+}(E)$ for the Borel $\sigma$-field generated by weak topology
on $M_{+}(E)$.
The proofs of the main results below are deferred to Section 5.1.
###### Theorem 2.1.
The sequence $(\xi_{k,n})_{n\geq 1}$ satisfies an LDP in the weak topology
with rate $\rho_{k,n}$ and rate function
$\begin{split}\Lambda_{k}^{*}(\rho)&\mathrel{\mathop{\mathchar
58\relax}}=\sup_{f\in C_{b}(E)}\Big{\\{}\int_{E}f({\bf
x})\rho(\operatorname{d\\!}{\bf x})\\\
&\qquad\qquad-\frac{1}{k!}\int_{({\mathbb{R}}^{d})^{k-1}}\big{(}e^{f(\overline{({\bf
0}_{d},{\bf y})})}-1\big{)}\,{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{d},{\bf y})\leq L\big{\\}}\operatorname{d\\!}{\bf y}\Big{\\}},\ \ \rho\in
M_{+}(E),\end{split}$ (2.5)
where ${\bf 0}_{d}=(0,\dots,0)\in{\mathbb{R}}^{d}$, ${\bf
y}\in({\mathbb{R}}^{d})^{k-1}$ and $C_{b}(E)$ is the collection of continuous
and bounded functions on $E$.
The rate function (2.5) can be associated to the notion of _relative entropy_.
Writing $\lambda_{m}$ for Lebesgue measure on $({\mathbb{R}}^{d})^{m}$, we
define
$\tau_{k}(A)=\frac{1}{k!}\,\lambda_{k-1}\big{\\{}{\bf
y}\in({\mathbb{R}}^{d})^{k-1}\mathrel{\mathop{\mathchar
58\relax}}\mathsf{diam}({\bf 0}_{d},{\bf y})\leq L,\,\overline{({\bf
0}_{d},{\bf y})}\in A\big{\\}},\ \ A\subset E.$ (2.6)
For a measure $\rho\in M_{+}(E)$,
$H_{k}(\rho|\tau_{k})\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\int_{E}\log\frac{\operatorname{d\\!}\rho}{\operatorname{d\\!}\tau_{k}}({\bf
x})\rho(\operatorname{d\\!}{\bf x})-\rho(E)+\tau_{k}(E)&\text{ if
}\rho\ll\tau_{k},\\\ \infty&\text{ otherwise, }\end{cases}$ (2.7)
denotes the relative entropy of $\rho$ with respect to $\tau_{k}$. In the
special case when $\rho$ and $\tau_{k}$ are probability measures, (2.7)
reduces to the relative entropy defined for the space of probability measures;
see, for example, [14, Equ. (6.2.8)]. In our setup, however, $\rho$ and
$\tau_{k}$ are not necessarily probability measures, so we need to extend the
definition of relative entropy as in (2.7).
###### Proposition 2.2.
Under the setup of Theorem 2.1, we have that
$\Lambda_{k}^{*}(\rho)=H_{k}(\rho|\tau_{k}),\ \ \ \rho\in M_{+}(E).$
Finally, we also prove the analog of Theorem 2.1 when the Poisson point
process is replaced by a binomial point process. To make this precise, we put
$\xi_{k,n}^{\mathsf{B}}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=k}s_{n}({\mathcal{Y}},\mathcal{B}_{n};t)\,\delta_{r_{n}^{-1}\overline{\mathcal{Y}}},\
\ n\geq 1,$
where $\mathcal{B}_{n}=\\{X_{1},\dots,X_{n}\\}$ is a binomial point process
consisting of $n$ i.i.d. uniform random vectors in $[0,1]^{d}$.
###### Corollary 2.3.
The sequence $(\xi_{k,n}^{\mathsf{B}})_{n\geq 1}$ satisfies an LDP in the weak
topology with rate $\rho_{k,n}$ and rate function
$\Lambda_{k}^{*}=H_{k}(\cdot|\tau_{k})$.
## 3 Large deviation principles for geometric and topological functionals
In this section, we provide the LDP for point processes relating more directly
to geometric and topological functionals. Choose integers $k\geq 2$ and $m\geq
1$, which remain fixed throughout the section. Define
$H\mathrel{\mathop{\mathchar
58\relax}}=(h^{(1)},\dots,h^{(m)})\mathrel{\mathop{\mathchar
58\relax}}({\mathbb{R}}^{d})^{k}\to[0,\infty)^{m}$ to be a non-negative
measurable function satisfying the following conditions:
(H1) $H$ is symmetric with respect to permutations of variables in
${\mathbb{R}}^{d}$.
(H2) $H$ is translation invariant:
$H(x_{1},\dots,x_{k})=H(x_{1}+y,\dots,x_{k}+y),\ \ \
x_{i},y\in{\mathbb{R}}^{d}.$
(H3) $H$ is locally determined:
$H(x_{1},\dots,x_{k})={\bf 0}_{m}\ \ \text{whenever
}\mathsf{diam}(x_{1},\dots,x_{k})>L,$
where $L>0$ is a constant and ${\bf 0}_{m}=(0,\dots,0)\in{\mathbb{R}}^{m}$.
(H4) For every ${\bf a}=(a_{1},\dots,a_{m})\in{\mathbb{R}}^{m}$,
$\int_{({\mathbb{R}}^{d})^{k-1}}e^{\langle{\bf a},H({\bf 0}_{d},{\bf
y})\rangle}\Big{(}{\mathbbm{1}}\big{\\{}H({\bf 0}_{d},{\bf y})\neq{\bf
0}_{m}\big{\\}}+\sum_{1\leq i\leq j\leq m}h^{(i)}({\bf 0}_{d},{\bf
y})h^{(j)}({\bf 0}_{d},{\bf y})\Big{)}\operatorname{d\\!}{\bf y}<\infty,$
where ${\bf y}\in({\mathbb{R}}^{d})^{k-1}$ and $\langle\cdot,\cdot\rangle$
denotes the Euclidean inner product.
(H5) For every ${\bf a}=(a_{1},\dots,a_{m})\in{\mathbb{R}}^{m}\setminus\\{{\bf
0}_{m}\\}$, it holds that
$\int_{({\mathbb{R}}^{d})^{k-1}}\big{|}\sum_{i=1}^{m}a_{i}h^{(i)}({\bf
0}_{d},{\bf y})\big{|}\operatorname{d\\!}{\bf y}>0$.
If $h^{(i)}$ are all bounded functions, we can drop condition (H4) because it
can be implied by (H3). Moreover, note that (H3) remains true even when $L$ is
increased. Hence, we may assume that $L$ is larger than a fixed value $t>0$.
Taking up a sequence $(r_{n})_{n\geq 1}$ in (2.1), we define a scaled version
of $H$ by
$H_{n}(x_{1},\dots,x_{k})\mathrel{\mathop{\mathchar
58\relax}}=H(r_{n}^{-1}x_{1},\dots,r_{n}^{-1}x_{k}),\ \ \ n=1,2,\dots.$ (3.1)
For a $k$-point subset ${\mathcal{Y}}\subset{\mathbb{R}}^{d}$, a finite subset
$\mathcal{Z}\supset{\mathcal{Y}}$ in ${\mathbb{R}}^{d}$ and ${\bf
t}=(t_{1},\dots,t_{m})\in[0,\infty)^{m}$, define
$c({\mathcal{Y}},\mathcal{Z};{\bf t})\mathrel{\mathop{\mathchar
58\relax}}=\big{(}c({\mathcal{Y}},\mathcal{Z};t_{i})\big{)}_{i=1}^{m},$ (3.2)
and also,
$G({\mathcal{Y}},\mathcal{Z};{\bf t})\mathrel{\mathop{\mathchar
58\relax}}=H({\mathcal{Y}})\odot c({\mathcal{Y}},\mathcal{Z};{\bf t}),$
where $\odot$ means the Hadamard product; that is, for $A=(a_{ij})$ and
$B=(b_{ij})$, we have $A\odot B=(a_{ij}b_{ij})$. We can define the scaled
version of these functions by
$c_{n}({\mathcal{Y}},\mathcal{Z};{\bf t})\mathrel{\mathop{\mathchar
58\relax}}=c(r_{n}^{-1}{\mathcal{Y}},r_{n}^{-1}\mathcal{Z};{\bf t}),\ \ \text{
and }\ \ G_{n}({\mathcal{Y}},\mathcal{Z};{\bf t})\mathrel{\mathop{\mathchar
58\relax}}=H_{n}({\mathcal{Y}})\odot c_{n}({\mathcal{Y}},\mathcal{Z};{\bf
t}).$ (3.3)
For notational convenience, the $i$th entries of $H_{n}({\mathcal{Y}})$ and
$G_{n}({\mathcal{Y}},\mathcal{Z};{\bf t})$ are denoted respectively as
$h_{n}^{(i)}({\mathcal{Y}})$ and
$g_{n}^{(i)}({\mathcal{Y}},\mathcal{Z};t_{i})$.
Henceforth, our aim is to explore the large deviations of the process
$U_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k}\delta_{G_{n}({\mathcal{Y}},\mathcal{P}_{n};{\bf
t})}\in M_{+}(E^{\prime}),$ (3.4)
where $E^{\prime}=[0,\infty)^{m}\setminus\\{{\bf 0}_{m}\\}$. Although the
process (3.4) aggregates the contributions of a certain score function over
all $k$-tuples, it is not a pure $U$-statistic since this score function
depends also on the remaining points of $\mathcal{P}_{n}$. Finally, as an
analog of (2.6), we define
$\tau_{k}^{\prime}(A)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{k!}\,\lambda_{k-1}\big{\\{}{\bf
y}\in({\mathbb{R}}^{d})^{k-1}\mathrel{\mathop{\mathchar 58\relax}}H({\bf
0}_{d},{\bf y})\in A\big{\\}},\ \ \ A\subset E^{\prime}.$
The proofs of all the theorems and propositions below are deferred to Sections
5.2 and 5.3.
###### Theorem 3.1.
The sequence $(U_{k,n})_{n\geq 1}$ satisfies an LDP in the weak topology with
rate $\rho_{k,n}$ and rate function
$\begin{split}\bar{\Lambda}_{k}^{*}(\rho)&\mathrel{\mathop{\mathchar
58\relax}}=\sup_{f\in C_{b}(E^{\prime})}\Big{\\{}\int_{E^{\prime}}f({\bf
x})\rho(\operatorname{d\\!}{\bf x})\\\
&\qquad\qquad-\frac{1}{k!}\int_{({\mathbb{R}}^{d})^{k-1}}\big{(}e^{f(H({\bf
0}_{d},{\bf y}))}-1\big{)}\,{\mathbbm{1}}\big{\\{}H({\bf 0}_{d},{\bf
y})\neq{\bf 0}_{m}\big{\\}}\operatorname{d\\!}{\bf y}\Big{\\}},\ \ \rho\in
M_{+}(E^{\prime}),\end{split}$ (3.5)
where ${\bf y}\in({\mathbb{R}}^{d})^{k-1}$. Furthermore,
$\bar{\Lambda}_{k}^{*}(\rho)=H_{k}^{\prime}(\rho|\tau_{k}^{\prime}),\ \
\rho\in M_{+}(E^{\prime}),$ (3.6)
where $H_{k}^{\prime}$ is the relative entropy in $M_{+}(E^{\prime})$ with
respect to $\tau_{k}^{\prime}$; that is, if $\rho\ll\tau_{k}^{\prime}$,
$H_{k}^{\prime}(\rho|\tau_{k}^{\prime})\mathrel{\mathop{\mathchar
58\relax}}=\int_{E^{\prime}}\log\frac{\operatorname{d\\!}\rho}{\operatorname{d\\!}\tau_{k}^{\prime}}({\bf
x})\rho(\operatorname{d\\!}{\bf
x})-\rho(E^{\prime})+\tau_{k}^{\prime}(E^{\prime})$ (3.7)
and $H_{k}^{\prime}(\rho|\tau_{k}^{\prime})=\infty$ otherwise.
As in Corollary 2.3, we can extend Theorem 3.1 to the case of a binomial
input. We define
$U_{k,n}^{\mathsf{B}}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=k}\delta_{G_{n}({\mathcal{Y}},\mathcal{B}_{n};{\bf
t})}.$
###### Corollary 3.2.
The sequence $(U_{k,n}^{\mathsf{B}})_{n\geq 1}$ satisfies an LDP in the weak
topology with rate $\rho_{k,n}$ and rate function
$\bar{\Lambda}_{k}^{*}=H_{k}^{\prime}(\cdot|\tau_{k}^{\prime})$.
Subsequently, we also consider statistics of the form
$T_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k}G_{n}({\mathcal{Y}},\mathcal{P}_{n};{\bf
t}).$ (3.8)
One can describe the corresponding LDP as follows. We define
$\mathcal{D}=\\{A\subset E^{\prime}\mathrel{\mathop{\mathchar
58\relax}}\bar{A}=\overline{\text{int}(A)}\\}$.
###### Theorem 3.3.
For every measurable $A\in\mathcal{D}$, we have as $n\to\infty$,
$\frac{1}{\rho_{k,n}}\log\mathbb{P}\big{(}\rho_{k,n}^{-1}T_{k,n}\in
A\big{)}\to-\inf_{{\bf x}\in A}I_{k}({\bf x}),$ (3.9)
where $I_{k}$ is a rate function given as
$I_{k}({\bf x})=\sup_{{\bf a}\in{\mathbb{R}}^{m}}\Big{\\{}\langle{\bf a},{\bf
x}\rangle-\frac{1}{k!}\,\int_{({\mathbb{R}}^{d})^{k-1}}\big{(}e^{\langle{\bf
a},H({\bf 0}_{d},{\bf y})\rangle}-1\big{)}\operatorname{d\\!}{\bf
y}\Big{\\}},\ \ {{\bf x}=(x_{1},\dots,x_{m})\in{\mathbb{R}}^{m}},$ (3.10)
with ${\bf y}\in({\mathbb{R}}^{d})^{k-1}$.
Again, we can extend the above result to the case of binomial point processes.
We define
$T_{k,n}^{\mathsf{B}}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=k}G_{n}({\mathcal{Y}},\mathcal{B}_{n};{\bf
t}).$
###### Corollary 3.4.
For every measurable $A\in\mathcal{D}$, we have as $n\to\infty$,
$\displaystyle\frac{1}{\rho_{k,n}}\log\mathbb{P}\big{(}\rho_{k,n}^{-1}T_{k,n}^{\mathsf{B}}\in
A\big{)}\to-\inf_{{\bf x}\in A}I_{k}({\bf x}).$
The rate function $I_{k}$ satisfies the following properties.
###### Proposition 3.5.
$(i)$ $I_{k}$ is continuously differentiable on $E^{\prime}$.
$(ii)$ $I_{k}$ is strictly convex on $E^{\prime}$.
$(iii)$ $I_{k}({\bf x})=0$ if and only if ${\bf x}=(\mu_{1},\dots,\mu_{m})$,
where
$\mu_{i}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{k!}\,\int_{({\mathbb{R}}^{d})^{k-1}}h^{(i)}({\bf
0}_{d},{\bf y})\operatorname{d\\!}{\bf y},\ \ \ i=1,\dots,m,$
with ${\bf y}\in({\mathbb{R}}^{d})^{k-1}$.
###### Remark 3.6.
If $m=1$ and $h^{(1)}$ is an indicator function, one can explicitly solve the
variational problem in (3.10), to get that
$I_{k}(x)=x\log(x/\mu_{1})-x+\mu_{1}$. In this case, $I_{k}$ coincides with a
rate function in the LDP for $\big{(}n^{-1}\sum_{i=1}^{n}X_{i}\big{)}_{n\geq
1}$, where $(X_{i})$ are i.i.d. Poisson random variables with mean $\mu_{1}$.
###### Example 3.7 (Čech complex component counts).
We consider an application to the Čech complex component counts. Let
$\check{C}({\mathcal{X}},r)$ be the Čech complex on a point set
${\mathcal{X}}=\\{x_{1},\dots,x_{m}\\}\subset{\mathbb{R}}^{d}$ with
connectivity radius $r>0$. Namely,
* •
The $0$-simplices of $\check{C}({\mathcal{X}},r)$ are the points in
${\mathcal{X}}$.
* •
The $p$-simplex $\\{x_{i_{0}},\dots,x_{i_{p}}\\}\subset{\mathcal{X}}$ with
$1\leq i_{0}<\dots<i_{p}\leq m$, belongs to $\check{C}({\mathcal{X}},r)$ if
$\bigcap_{\ell=0}^{p}B(x_{i_{\ell}},r/2)\neq\emptyset$, where $B(x,r)$ is the
$d$-dimensional closed ball of radius $r$ centered at $x\in{\mathbb{R}}^{d}$.
We then explore the LDP for
$S_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=\Big{(}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+1}{\mathbbm{1}}\big{\\{}\check{C}({\mathcal{Y}},r_{n}t_{i})\cong\Gamma_{i}\big{\\}}\,c_{n}({\mathcal{Y}},\mathcal{P}_{n};t_{i})\Big{)}_{i=1}^{m},$
(3.11)
where $\Gamma_{i}$ is a _connected_ Čech complex on $k+1$ vertices, and
$\cong$ means isomorphism between simplicial complexes, and $c_{n}$ is defined
in (2.2). Assume that $\Gamma_{i}\not\cong\Gamma_{j}$ for $i\neq j$. Then, the
$i$th entry of $S_{k,n}$ represents the number of components isomorphic to
$\Gamma_{i}$ in the Čech complex $\check{C}(\mathcal{P}_{n},r_{n}t_{i})$. In
this setting, it is easy to check that the function
$H=(h^{(1)},\dots,h^{(m)})$ with
$h^{(i)}(x_{1},\dots,x_{k+1})={\mathbbm{1}}\Big{\\{}\check{C}\big{(}\\{x_{1},\dots,x_{k+1}\\},t_{i}\big{)}\cong\Gamma_{i}\Big{\\}},\
\ \ (x_{1},\dots,x_{k+1})\in({\mathbb{R}}^{d})^{k+1},$
satisfies conditions (H1)–(H5).
Finally, we also define $S_{k,n}^{\mathsf{B}}$ to be the statistics analogous
to (3.11), that are generated by the binomial point process $\mathcal{B}_{n}$.
The following corollary can be obtained as a direct application of the above
results.
###### Corollary 3.8.
Assume that $\rho_{k+1,n}\to\infty$ and $nr_{n}^{d}\to 0$ as $n\to\infty$.
Then, for any measurable $A\in\mathcal{D}$, as $n\to\infty$,
$\displaystyle\frac{1}{\rho_{k+1,n}}\log\mathbb{P}\big{(}\rho_{k+1,n}^{-1}S_{k,n}\in
A\big{)}\to-\inf_{{\bf x}\in A}I_{k+1}({\bf x}),$
$\displaystyle{\frac{1}{\rho_{k+1,n}}\log\mathbb{P}\big{(}\rho_{k+1,n}^{-1}S_{k,n}^{\mathsf{B}}\in
A\big{)}\to-\inf_{{\bf x}\in A}I_{k+1}({\bf x}).}$
Here, $I_{k+1}$ is the rate function from (3.10). The unique minimizer of
$I_{k+1}$ equals $(\mu_{1},\dots,\mu_{m})$, where
$\mu_{i}=\frac{1}{(k+1)!}\,\int_{({\mathbb{R}}^{d})^{k}}{\mathbbm{1}}\big{\\{}\check{C}\big{(}\\{{\bf
0}_{d},{\bf y}\\},t_{i}\big{)}\cong\Gamma_{i}\big{\\}}\operatorname{d\\!}{\bf
y},\ \ i=1,\dots,m,$
with ${\bf y}\in({\mathbb{R}}^{d})^{k}$.
###### Example 3.9 (Component counts in a random geometric graph).
We next deal with the component counts in a random geometric graph
$G(\mathcal{P}_{n},r_{n}t_{i})$ with radius $r_{n}t_{i}$, $i=1,\dots,m$.
Define the function $H=(h^{(1)},\dots,h^{(m)})$ by
$h^{(i)}(x_{1},\dots,x_{k})={\mathbbm{1}}\Big{\\{}G\big{(}\\{x_{1},\dots,x_{k}\\},t_{i}\big{)}\cong\Gamma_{i}\Big{\\}},\
\ \ (x_{1},\dots,x_{k})\in({\mathbb{R}}^{d})^{k},$
where $\Gamma_{i}$ is a _connected_ graph on $k$ vertices and $\cong$ denotes
a graph isomorphism. Assume that $\Gamma_{i}\not\cong\Gamma_{j}$ for $i\neq
j$. Then, the collection of component counts defined by
$S_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=\Big{(}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k}{\mathbbm{1}}\big{\\{}G({\mathcal{Y}},r_{n}t_{i})\cong\Gamma_{i}\big{\\}}\,c_{n}({\mathcal{Y}},\mathcal{P}_{n};t_{i})\Big{)}_{i=1}^{m},$
satisfies the following LDP.
###### Corollary 3.10.
Assume that $\rho_{k,n}\to\infty$ and $nr_{n}^{d}\to 0$ as $n\to\infty$. Then,
for any measurable $A\in\mathcal{D}$, as $n\to\infty$,
$\displaystyle\frac{1}{\rho_{k,n}}\log\mathbb{P}\big{(}\rho_{k,n}^{-1}S_{k,n}\in
A\big{)}\to-\inf_{{\bf x}\in A}I_{k}({\bf x}),$
$\displaystyle{\frac{1}{\rho_{k,n}}\log\mathbb{P}\big{(}\rho_{k,n}^{-1}S_{k,n}^{\mathsf{B}}\in
A\big{)}\to-\inf_{{\bf x}\in A}I_{k}({\bf x}).}$
Once again, $I_{k}$ is the rate function with its unique minimizer
$(\mu_{1},\dots,\mu_{m})$ given by
$\mu_{i}=\frac{1}{k!}\,\int_{({\mathbb{R}}^{d})^{k-1}}{\mathbbm{1}}\big{\\{}G\big{(}\\{{\bf
0}_{d},{\bf y}\\},t_{i}\big{)}\cong\Gamma_{i}\big{\\}}\operatorname{d\\!}{\bf
y},\ \ i=1,\dots,m,$
where ${\bf y}\in({\mathbb{R}}^{d})^{k-1}$.
## 4 Applications in stochastic topology
In this section, we elucidate how to apply our main results to derive LDPs for
two key quantities in stochastic topology, namely _persistent Betti numbers_
(Section 4.1) and _Morse critical points_ (Section 4.2). In both cases, we
assume $d=2$ and $k=3$ in the notation of Section 2. In other words, we work
with 1-dimensional topological features generated by $3$ random points in the
plane. Although the proposed methods can also be applied to $0$-dimensional
features in the plane, we focus only on $1$-dimensional quantities, because
they are much more non-trivial than $0$-dimensional ones. We conjecture that
our findings can be generalized, at least partially, to higher-dimensional
cases. The remark after Theorem 4.1 below explains why such extensions are
challenging.
### 4.1 Persistent Betti number of alpha complex
First, we deal with the persistent Betti numbers in the alpha complex. For
readers not familiar with algebraic topology, we briefly discuss conceptual
ideas behind persistent Betti numbers. We suggest [10, 18] as a good
introductory reading, while a more rigorous coverage of algebraic topology is
in [25].
The Betti numbers $(\beta_{k})_{k\geq 0}$ are fundamental invariants of
topological spaces counting the number of $k$-dimensional cycles (henceforth
we call it $k$-cycle) as the boundary of a $(k+1)$-dimensional body. In the
$3$-dimensional space, $\beta_{1}$ and $\beta_{2}$ can be viewed as the number
of loops and cavities, respectively. Figure 1 illustrates a sphere in
${\mathbb{R}}^{3}$, which encompasses one central cavity; therefore,
$\beta_{2}=1$. The Betti number $\beta_{1}$ of this sphere is zero; even if we
wind a closed loop around the sphere, the loop ultimately vanishes as it moves
upward (or downward) along the sphere until the pole. Figure 1 also
illustrates a torus in ${\mathbb{R}}^{3}$, for which there are two distinct
non-contractible loops; therefore, $\beta_{1}=2$. Moreover, the torus has a
cavity, meaning that $\beta_{2}=1$.
Figure 1: Illustration of a sphere (left) and torus (right).
We consider the case when the (persistent) Betti numbers are built on the
alpha complex. The alpha complex is similar, in nature, to the Čech complex
(see Example 3.7), but it has a more natural geometric realization. Given a
finite set ${\mathcal{X}}$ of points in ${\mathbb{R}}^{d}$ and
$r\in[0,\infty]$, the alpha complex $\alpha({\mathcal{X}},r)$ is defined as a
collection of subsets $\sigma\subset{\mathcal{X}}$ such that
$\bigcap_{x\in\sigma}\big{(}B(x,r/2)\cap V_{x}\big{)}\neq\emptyset$, where
$V_{x}$ is a Voronoi cell of $x$; that is,
$V_{x}=\big{\\{}y\in{\mathbb{R}}^{d}\mathrel{\mathop{\mathchar
58\relax}}\|y-x\|\leq\inf_{z\in{\mathcal{X}}}\|y-z\|\\}$. Clearly, it holds
that $\alpha({\mathcal{X}},r)\subset\check{C}({\mathcal{X}},r)$. Moreover, we
have inclusions
$\alpha({\mathcal{X}},r)\subset\alpha({\mathcal{X}},r^{\prime})$ for all
$r\leq r^{\prime}$, which indicates that $\alpha({\mathcal{X}},r)$ is a
subcomplex of the Delaunay complex
$\text{Del}({\mathcal{X}})\mathrel{\mathop{\mathchar
58\relax}}=\alpha({\mathcal{X}},\infty)$. This property plays a crucial role
in our analysis; see Remark 4.2.
Returning to the setup in Section 2, we consider the filtration induced by a
collection of alpha complexes over a scaled Poisson point process
$r_{n}^{-1}\mathcal{P}_{n}$,
$\big{(}\alpha(r_{n}^{-1}\mathcal{P}_{n},t),\,t\geq
0\big{)}=\big{(}\alpha(\mathcal{P}_{n},r_{n}t),\,t\geq 0\big{)}.$ (4.1)
Topological changes in the filtration (4.1) can be captured by the $k$th
persistent Betti number $\beta_{k,n}(s,t)$ for $0\leq s\leq t\leq\infty$.
Loosely speaking, $\beta_{k,n}(s,t)$ represents the number of $k$-cycles, that
appear in (4.1) before time $r_{n}s$ and remain alive at time $r_{n}t$. More
formally, $\beta_{k,n}(s,t)$ is defined as
$\beta_{k,n}(s,t)=\text{dim}\frac{Z_{k}\big{(}\alpha(\mathcal{P}_{n},r_{n}s)\big{)}}{Z_{k}\big{(}\alpha(\mathcal{P}_{n},r_{n}s)\big{)}\cap
B_{k}\big{(}\alpha(\mathcal{P}_{n},r_{n}t)\big{)}},$
where $Z_{k}(\cdot)$ is the $k$th cycle group of an alpha complex and
$B_{k}(\cdot)$ denotes its $k$th boundary group. Here, the homology
coefficients are taken from an arbitrary field. In the special case $s=t$,
$\beta_{k,n}(s,t)$ reduces to the ordinary $k$th Betti number.
As mentioned before, in the following we restrict ourselves to a lower-
dimensional case $d=2$ and $k=3$ (in the notation of Section 2). For
$(x_{1},x_{2},x_{3})\in({\mathbb{R}}^{2})^{3}$ and $r>0$, define
$\displaystyle h_{r}(x_{1},x_{2},$ $\displaystyle
x_{3})\mathrel{\mathop{\mathchar
58\relax}}={\mathbbm{1}}\Big{\\{}\beta_{1}\big{(}\alpha(\\{x_{1},x_{2},x_{3}\\},r)\big{)}=1\Big{\\}}$
(4.2) $\displaystyle={\mathbbm{1}}\bigg{\\{}\Big{\\{}\bigcap_{j=1,\,j\neq
j_{0}}^{3}B(x_{j},r/2)\neq\emptyset\ \text{for all
}j_{0}\in\\{1,2,3\\}\Big{\\}}\cap\Big{\\{}\bigcap_{j=1}^{3}B(x_{j},r/2)=\emptyset\Big{\\}}\bigg{\\}}.$
For a subset ${\mathcal{Y}}$ of $3$ points in ${\mathbb{R}}^{2}$, a finite set
$\mathcal{Z}\supset{\mathcal{Y}}$ in ${\mathbb{R}}^{2}$, and $0\leq s\leq
t<\infty$,
$\displaystyle g_{r_{n}s,r_{n}t}({\mathcal{Y}},\mathcal{Z})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=h_{r_{n}s}({\mathcal{Y}})\,h_{r_{n}t}({\mathcal{Y}})\,{\mathbbm{1}}\big{\\{}\|y-z\|\geq
r_{n}t,\text{ for all }y\in{\mathcal{Y}}\text{ and
}z\in\mathcal{Z}\setminus{\mathcal{Y}}\big{\\}}$
$\displaystyle=h_{r_{n}s}({\mathcal{Y}})\,h_{r_{n}t}({\mathcal{Y}})\,{\mathbbm{1}}\big{\\{}\alpha({\mathcal{Y}},r_{n}t)\text{
is a connected component of }\alpha(\mathcal{Z},r_{n}t)\big{\\}}.$
Note that $g_{r_{n}s,r_{n}t}({\mathcal{Y}},\mathcal{Z})=1$ if and only if a
set ${\mathcal{Y}}$ in ${\mathbb{R}}^{2}$ with $|{\mathcal{Y}}|=3$ forms a
single $1$-cycle before time $r_{n}s$, such that this $1$-cycle remains alive
at time $r_{n}t$ and isolated from all the remaining points in $\mathcal{Z}$
at that time. The proof of the next theorem is presented in Section 5.4.
###### Theorem 4.1.
Assume that $\rho_{3,n}\to\infty$ and $nr_{n}^{2}\to 0$ as $n\to\infty$. Then,
for every $0\leq s_{i}\leq t_{i}<\infty$, $i=1,\dots,m$, with
$(s_{i},t_{i})\neq(s_{j},t_{j})$ for $i\neq j$, and $A\in\mathcal{D}$,
$\frac{1}{\rho_{3,n}}\log\mathbb{P}\Big{(}\big{(}\rho_{3,n}^{-1}\beta_{1,n}(s_{i},t_{i}),\,i=1,\dots,m\big{)}\in
A\Big{)}\to-\inf_{{\bf x}\in A}I_{3}({\bf x}),\ \ \text{as }n\to\infty,$ (4.3)
where $I_{3}$ is the rate function from (3.10). The unique minimizer of
$I_{3}$ equals $(\mu_{1},\dots,\mu_{m})$, where
$\mu_{i}=\frac{1}{6}\,\int_{({\mathbb{R}}^{2})^{2}}h_{s_{i}}({\bf 0}_{2},{\bf
y})\,h_{t_{i}}({\bf 0}_{2},{\bf y})\operatorname{d\\!}{\bf y},\ \
i=1,\dots,m,$
with ${\bf 0}_{2}=(0,0)\in{\mathbb{R}}^{2}$ and ${\bf
y}\in({\mathbb{R}}^{2})^{2}$.
Moreover, define $\beta_{1,n}^{\mathsf{B}}(s,t)$ to be the first-order
persistent Betti number generated by the binomial point process. Then,
$(\beta_{1,n}^{\mathsf{B}})_{n\geq 1}$ satisfies the same LDP as (4.3).
###### Remark 4.2 (Extensions to higher-dimensions).
In the proof of Theorem 4.1, we shall exploit the Morse inequality, which
enables to bound the first-order persistent Betti number by the number of
$1$-simplex counts (i.e., edge counts). A key property for dealing with the
corresponding exponential moments is that the $1$-simplex count in the planar
Delaunay triangulation grows at most linearly in the number of vertices (see
Equ. (5.47) for details). Unfortunately, this property breaks down in higher-
dimensions ([2]); this is the reason why our discussion must be restricted to
a special case $d=2$, $k=3$. Notice, however, that one needs this assumption
only for showing (5.47); the rest of our analyses holds true for general $d$
and $k$. For the LDP of higher-order persistent Betti numbers in a higher-
dimensional space, we need to develop a new machinery that does not rely on
the Morse inequality. More concretely, we need to detect the parameters $d$
and $k$, such that the (persistent) Betti number grows at most linearly in the
number of vertices.
### 4.2 Morse critical points of min-type distance functions
The objective of this example is to deduce the LDP for the number of Morse
critical points of a certain min-type distance function. The behavior of Morse
critical points of such distance functions have been intensively investigated
in the context of central limit theorems and the Poisson process approximation
[6, 8, 9, 38]. In addition to its intrinsic interests, this concept has served
as a practical quantifier of homological changes in random geometric
complexes, especially in the field of Topological Data Analysis ([6, 7]).
Once again, we treat only the special case $d=2$ and $k=3$ (in the notation of
Section 2). Given a homogeneous Poisson point process $\mathcal{P}_{n}$ on
$[0,1]^{2}$ with intensity $n$, we define min-type distance functions by
$d_{\mathcal{P}_{n}}(x)\mathrel{\mathop{\mathchar
58\relax}}=\min_{y\in\mathcal{P}_{n}}\|x-y\|,\ \ \ x\in{\mathbb{R}}^{2}.$
Though $d_{\mathcal{P}_{n}}$ is not differentiable, one can still define the
notion of Morse critical points in the following sense. A point
$c\in{\mathbb{R}}^{2}$ is said to be a Morse critical point of
$d_{\mathcal{P}_{n}}$ with index $2$ if there exists a subset
${\mathcal{Y}}\subset\mathcal{P}_{n}$ of three points such that
$(i)$ The points in ${\mathcal{Y}}$ are in general position.
$(ii)$ $d_{\mathcal{P}_{n}}(c)=\|c-y\|$ for all $y\in{\mathcal{Y}}$ and
$d_{\mathcal{P}_{n}}(c)<\min_{z\in\mathcal{P}_{n}\setminus{\mathcal{Y}}}\|c-z\|$.
$(iii)$ The interior of the convex hull spanned by the points in
${\mathcal{Y}}$, denoted $\text{conv}^{\circ}({\mathcal{Y}})$, contains $c$.
By the Nerve theorem (see, e.g., Theorem 10.7 in [4]), for each $r>0$ the
sublevel set $d_{\mathcal{P}_{n}}(-\infty,r]$ is homotopy equivalent to a Čech
complex $\check{C}(\mathcal{P}_{n},2r)$. For this reason, the number of
critical points of $d_{\mathcal{P}_{n}}$ with index $2$ whose critical values
are less than $r_{n}$, behaves very similarly to the first-order Betti number
of $\check{C}(\mathcal{P}_{n},2r_{n})$ (see [6]). A similar analysis was
conducted for the case that distributions are supported on a closed manifold
embedded in the ambient Euclidean space [8]. Moreover, [38] studied the case
when random points are sampled from a stationary point process.
Given a point set ${\mathcal{Y}}\subset{\mathbb{R}}^{2}$ with
$|{\mathcal{Y}}|=3$ in general position, let $\gamma({\mathcal{Y}})$ be the
center of a unique $1$-dimensional sphere containing ${\mathcal{Y}}$, and
define
$\mathcal{R}({\mathcal{Y}})=d_{\mathcal{P}_{n}}\big{(}\gamma({\mathcal{Y}})\big{)}$.
If $\gamma({\mathcal{Y}})$ gives a critical point of $d_{\mathcal{P}_{n}}$
(with index $2$), then $\mathcal{R}({\mathcal{Y}})$ represents its critical
value. Additionally, $\mathcal{U}({\mathcal{Y}})$ denotes an open ball in
${\mathbb{R}}^{2}$ with radius $\mathcal{R}({\mathcal{Y}})$ centered at
$\gamma({\mathcal{Y}})$. Given $t_{i}\in[0,\infty)$, $i=1,\dots,m$, with
$t_{i}\neq t_{j}$ for $i\neq j$, we define
$N_{n}\mathrel{\mathop{\mathchar
58\relax}}=\Big{(}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=3}{\mathbbm{1}}\big{\\{}\gamma({\mathcal{Y}})\in\text{conv}^{\circ}({\mathcal{Y}}),\
\mathcal{R}({\mathcal{Y}})\leq r_{n}t_{i},\
\mathcal{U}({\mathcal{Y}})\cap\mathcal{P}_{n}=\emptyset\big{\\}}\Big{)}_{i=1}^{m}.$
(4.4)
In particular, the $i$th entry of $N_{n}$ represents the number of Morse
critical points of index $2$ with critical values less than or equal to
$r_{n}t_{i}$.
The result below is essentially a consequence of Theorem 3.3 and Proposition
3.5. Unlike Theorem 4.1, we do not provide the result for the version of
binomial point processes; the required extension actually involves more
complicated machinery. A formal proof is given in Section 5.4.
###### Theorem 4.3.
Assume that $\rho_{3,n}\to\infty$ and $nr_{n}^{2}\to 0$ as $n\to\infty$. Then,
for every measurable $A\in\mathcal{D}$,
$\frac{1}{\rho_{3,n}}\,\log\mathbb{P}\big{(}\rho_{3,n}^{-1}N_{n}\in
A\big{)}\to-\inf_{{\bf x}\in A}I_{3}({\bf x}),\ \ \text{as }n\to\infty,$
where $I_{3}$ is the rate function from (3.10). The unique minimizer of
$I_{3}$ equals $(\mu_{1},\dots,\mu_{m})$, where
$\mu_{i}=\frac{1}{6}\,\int_{({\mathbb{R}}^{2})^{2}}{\mathbbm{1}}\big{\\{}\gamma({\bf
0}_{2},{\bf y})\in\text{conv}^{\circ}({\bf 0}_{2},{\bf y}),\,\mathcal{R}({\bf
0}_{2},{\bf y})\leq t_{i}\big{\\}}\operatorname{d\\!}{\bf y},\ \ i=1,\dots,m,$
with ${\bf y}\in({\mathbb{R}}^{2})^{2}$.
## 5 Proofs
As preparation, we partition the unit cube $[0,1]^{d}$ into $\rho_{k,n}$ sub-
cubes $Q_{1},\dots,Q_{\rho_{k,n}}$ of volume $\rho_{k,n}^{-1}$. To avoid
notational complication we assume that $\rho_{k,n}$ takes only positive
integers for all $n\in{\mathbb{N}}$. This assumption applies to many of the
sequences and functions throughout this section. In particular, we set
$Q_{1}=[0,\rho_{k,n}^{-1/d}]^{d}$. Given a finite set ${\mathcal{Y}}$ of $k$
points in ${\mathbb{R}}^{d}$, we take $\ell({\mathcal{Y}})$ to be the left
most point of ${\mathcal{Y}}$ in the lexicographic ordering. As discussed in
Section 2, one may set $\ell({\mathcal{Y}})$ to be a center of the unique
$(k-2)$-dimensional sphere containing ${\mathcal{Y}}$. In this case, however,
the description will be slightly more involved. Hence, for ease of
description, we prefer to define $\ell({\mathcal{Y}})$ as the left most point.
In the below, $C^{*}$ denotes a generic constant, which is independent of $n$
and may vary between and within the lines.
### 5.1 Proofs of Theorem 2.1, Proposition 2.2, and Corollary 2.3
The proof of Theorem 2.1 can be completed via Propositions 5.1 and 5.6 below.
Proposition 5.1 is aimed to give the LDP for
$\eta_{k,n}=\frac{1}{\rho_{k,n}}\sum_{\ell=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}},\,|{\mathcal{Y}}|=k}s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t)\,\delta_{r_{n}^{-1}\overline{\mathcal{Y}}}=\mathrel{\mathop{\mathchar
58\relax}}\frac{1}{\rho_{k,n}}\sum_{\ell=1}^{\rho_{k,n}}\eta_{k,n}^{(\ell)}\in
M_{+}(E),$ (5.1)
where $\mathcal{P}_{n}|_{Q_{\ell}}$ denotes the restriction of
$\mathcal{P}_{n}$ to the cube $Q_{\ell}$. Setting up a “blocked” point process
as in (5.1) is a standard approach in the literature ([36, 35]). The process
(5.1) is of course different from the original process $(\xi_{k,n})_{n\geq
1}$. In fact, if geometric/topological objects exist spreading over multiple
cubes in $(Q_{\ell})_{\ell\geq 1}$, then these objects are counted possibly by
$(\xi_{k,n})_{n\geq 1}$, whereas they are never counted by
$(\eta_{k,n})_{n\geq 1}$. Despite such a difference, one may justify in
Proposition 5.6 that the difference between $(\xi_{k,n})_{n\geq 1}$ and
$(\eta_{k,n})_{n\geq 1}$ is exponentially negligible in terms of the total
variation distance. By virtue of this proposition, as well as Theorem 4.2.13
in [14], our task can be reduced to prove the LDP for $(\eta_{k,n})_{n\geq
1}$.
The main benefit of working with $(\eta_{k,n})_{n\geq 1}$ is that it breaks
down into a collection of i.i.d. point processes
$(\eta_{k,n}^{(\ell)})_{\ell\geq 1}$. Note that
$(\eta_{k,n}^{(\ell)})_{\ell\geq 1}$ are defined in the space $M_{p}(E)$ of
_finite_ point measures on $E$. We here equip $M_{p}(E)$ with a Borel
$\sigma$-field $\mathcal{M}_{p}(E)\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{M}_{+}(E)\cap M_{p}(E)$. Because of this decomposition,
one can clarify the correspondence between (5.1) and the process
$\zeta_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\rho_{k,n}}\sum_{\ell=1}^{\rho_{k,n}}\zeta_{k}^{(\ell)}\in
M_{+}(E),\ \ \ n=1,2,\dots,$ (5.2)
where $(\zeta_{k}^{(\ell)})_{\ell\geq 1}$ are i.i.d. Poisson random measures
on $E$ with intensity measure $\tau_{k}$. Write
$(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{P}^{\prime})$ for the
probability space on which (5.2) is defined, and $\mathbb{E}^{\prime}$ denotes
the corresponding expectation.
In Lemma 5.2 below, we first prove that (5.2) satisfies the required LDP in
Theorem 2.1. After that, Lemmas 5.3 and 5.5 are aimed to establish exponential
equivalence between (5.1) and (5.2). Now, we formally state one of the main
results of this section.
###### Proposition 5.1.
The sequence $(\eta_{k,n})_{n\geq 1}$ fulfills an LDP in the weak topology
with rate $\rho_{k,n}$ and the rate function $\Lambda_{k}^{*}$ in (2.5).
Since the proof of Proposition 5.1 is rather long, we divide it into several
lemmas. Combining these lemmas can conclude Proposition 5.1.
###### Lemma 5.2.
The sequence $(\zeta_{k,n})_{n\geq 1}$ satisfies an LDP in the weak topology
with rate $\rho_{k,n}$ and rate function $\Lambda_{k}^{*}$.
###### Proof.
It follows from Theorem 5.1 in [34] that for every $f\in C_{b}(E)$,
$\displaystyle\Lambda_{k}(f)$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\log\mathbb{E}^{\prime}\Big{[}e^{\zeta_{k}^{(1)}(f)}\Big{]}=\frac{1}{k!}\int_{({\mathbb{R}}^{d})^{k-1}}\big{(}e^{f(\overline{({\bf
0}_{d},{\bf y})})}-1\big{)}\,{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{d},{\bf y})\leq L\big{\\}}\operatorname{d\\!}{\bf y}.$
According to Cramér’s theorem in Polish spaces in [14, Theorem 6.1.3] (see
also [14, Corollary 6.2.3]), it turns out that $(\zeta_{k,n})_{n\geq 1}$
satisfies a _weak_ LDP in $M_{+}(E)$ with rate function $\Lambda_{k}^{*}$.
In order to extend this to a _full_ LDP, we need to demonstrate that
$(\zeta_{k,n})_{n\geq 1}$ is exponentially tight in the space $M_{+}(E)$. The
proof is analogous to that in Lemma 6.2.6 of [14]. Since $\tau_{k}$ is tight
in the space $E$, for every $\ell\geq 1$ there exists a compact subset
$\Gamma_{\ell}\subset E$, such that
$\tau_{k}(E\setminus\Gamma_{\ell})\leq\ell/(e^{2\ell^{2}}-1)$. Define
$K_{\ell}\mathrel{\mathop{\mathchar 58\relax}}=\big{\\{}\nu\in
M_{+}(E)\mathrel{\mathop{\mathchar
58\relax}}\nu(E\setminus\Gamma_{\ell})\leq\ell^{-1},\,\nu(E)\leq\ell\big{\\}}.$
By Portmanteau’s theorem for weak convergence, one can deduce that $K_{\ell}$
is weakly closed in $M_{+}(E)$. For $m\geq 1$, let
$L_{m}\mathrel{\mathop{\mathchar
58\relax}}=\bigcap_{\ell=m}^{\infty}K_{\ell}$. Obviously, $L_{m}$ is weakly
closed in $M_{+}(E)$. Furthermore, Prohorov’s theorem (see, e.g., Theorem
A2.4.I in [12]) ensures that $L_{m}$ is relatively compact. Since $L_{m}$ is
now compact in $M_{+}(E)$, the exponential tightness of $(\zeta_{k,n})_{n\geq
1}$ follows from
$\limsup_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}^{\prime}\big{(}\zeta_{k,n}\in
M_{+}(E)\setminus L_{m}\big{)}\leq-\big{(}m-\tau_{k}(E)(e-1)\big{)},$ (5.3)
for every $m\geq 1$. By the union bound,
$\mathbb{P}^{\prime}\big{(}\zeta_{k,n}\in M_{+}(E)\setminus
L_{m}\big{)}\leq\sum_{\ell=m}^{\infty}\Big{\\{}\mathbb{P}^{\prime}\big{(}\zeta_{k,n}(E\setminus\Gamma_{\ell})>\ell^{-1}\big{)}+\mathbb{P}^{\prime}\big{(}\zeta_{k,n}(E)>\ell\big{)}\Big{\\}}.$
By Markov’s inequality and the fact that
$\zeta_{k}^{(1)}(E\setminus\Gamma_{\ell})$ is Poisson distributed with mean
$\tau_{k}(E\setminus\Gamma_{\ell})$,
$\displaystyle\sum_{\ell=m}^{\infty}\mathbb{P}^{\prime}\big{(}\zeta_{k,n}(E\setminus\Gamma_{\ell})>\ell^{-1}\big{)}\leq\sum_{\ell=m}^{\infty}e^{-2\rho_{k,n}\ell}\Big{(}\mathbb{E}^{\prime}\big{[}e^{2\ell^{2}\zeta_{k}^{(1)}(E\setminus\Gamma_{\ell})}\big{]}\Big{)}^{\rho_{k,n}}$
(5.4)
$\displaystyle=\sum_{\ell=m}^{\infty}e^{-2\rho_{k,n}\ell+\rho_{k,n}\tau_{k}(E\setminus\Gamma_{\ell})(e^{2\ell^{2}}-1)}\leq\sum_{\ell=m}^{\infty}e^{-\rho_{k,n}\ell}\leq
2e^{-\rho_{k,n}m}.$
By the similar calculation,
$\sum_{\ell=m}^{\infty}\mathbb{P}^{\prime}\big{(}\zeta_{k,n}(E)>\ell\big{)}\leq
2e^{-(m-\tau_{k}(E)(e-1))\rho_{k,n}}.$ (5.5)
Putting (5.4) and (5.5) together gives (5.3), as desired. ∎
By virtue of Lemma 5.2, our goal is now to show that $(\eta_{k,n})_{n\geq 1}$
in (5.1) satisfies the same LDP as $(\zeta_{k,n})_{n\geq 1}$. As a first step,
we claim that for every $\ell\geq 1$, the total variation distance between the
laws of $\eta_{k,n}^{(\ell)}$ and $\zeta_{k}^{(\ell)}$ tends to $0$ as
$n\to\infty$. Write $\mathcal{L}(\xi)$ for the probability law of a random
element $\xi$.
###### Lemma 5.3.
For every $\ell\geq 1$,
$d_{\mathsf{TV}}\big{(}\mathcal{L}(\eta_{k,n}^{(\ell)}),\mathcal{L}(\zeta_{k}^{(\ell)})\big{)}\mathrel{\mathop{\mathchar
58\relax}}=\sup_{A\in\mathcal{M}_{p}(E)}\big{|}\mathbb{P}(\eta_{k,n}^{(\ell)}\in
A)-\mathbb{P}^{\prime}(\zeta_{k}^{(\ell)}\in A)\big{|}\to 0,\ \ \text{as
}n\to\infty.$ (5.6)
###### Remark 5.4.
One of the possible extensions of Lemma 5.3 seems to consider the case, for
which $(r_{n})_{n\geq 1}$ belongs to the critical regime, i.e., $nr_{n}^{d}\to
c\in(0,\infty)$ as $n\to\infty$. In this case, however, the asymptotic
behaviors of (5.1) and (5.7) below are essentially different. More concretely,
in the critical regime (5.10) no longer holds, since the last term in (5.11)
does not vanish as $n\to\infty$.
###### Proof.
We begin with defining a sequence of random measures,
$\widetilde{\eta}_{k,n}=\frac{1}{\rho_{k,n}}\sum_{\ell=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}},\,|{\mathcal{Y}}|=k}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\mathcal{Y}})\leq
r_{n}L\big{\\}}\,\delta_{r_{n}^{-1}\overline{\mathcal{Y}}}=\mathrel{\mathop{\mathchar
58\relax}}\frac{1}{\rho_{k,n}}\sum_{\ell=1}^{\rho_{k,n}}\widetilde{\eta}_{k,n}^{(\ell)},\
\ \ n\geq 1.$ (5.7)
Note that $(\widetilde{\eta}_{k,n}^{(\ell)})_{\ell\geq 1}$ are i.i.d. point
processes in the space $M_{p}(E)$. We first show that
$d_{\mathsf{TV}}\big{(}\mathcal{L}(\widetilde{\eta}_{k,n}^{(\ell)}),\mathcal{L}(\zeta_{k}^{(\ell)})\big{)}\to
0,\ \ \text{as }n\to\infty.$ (5.8)
According to Theorem 3.1 in [13], (5.8) follows if one can verify that
$\sup_{A\mathrel{\mathop{\mathchar 58\relax}}\text{Borel in
}E}\Big{|}\mathbb{E}\big{[}\widetilde{\eta}_{k,n}^{(\ell)}(A)\big{]}-\mathbb{E}^{\prime}\big{[}\zeta^{(\ell)}_{k}(A)\big{]}\Big{|}\to
0,\ \ \ n\to\infty,$
and
$\displaystyle\gamma_{n}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\max_{1\leq p\leq
k-1}n^{2k-p}\int_{(Q_{\ell})^{2k-p}}{\mathbbm{1}}\big{\\{}\mathsf{diam}(x_{1},\dots,x_{k})\leq
r_{n}L\big{\\}}$ (5.9)
$\displaystyle\qquad\qquad\qquad\qquad\times{\mathbbm{1}}\big{\\{}\mathsf{diam}(x_{1},\dots,x_{p},x_{k+1},\dots,x_{2k-p})\leq
r_{n}L\big{\\}}\operatorname{d\\!}{\bf x}\to 0,\ \ n\to\infty.$
By the multivariate Mecke formula for Poisson point processes (see, e.g.,
Chapter 4 in [23]),
$\mathbb{E}\big{[}\widetilde{\eta}_{k,n}^{(\ell)}(A)\big{]}=\frac{n^{k}}{k!}\,\int_{(Q_{\ell})^{k}}{\mathbbm{1}}\big{\\{}\mathsf{diam}(x_{1},\dots,x_{k})\leq
r_{n}L,\ r_{n}^{-1}\overline{(x_{1},\dots,x_{k})}\in
A\big{\\}}\operatorname{d\\!}{\bf x}.$
Performing the change of variables $x_{i}=x+r_{n}y_{i-1}$, $i=1,\dots,k$ (with
$y_{0}\equiv 0$),
$\displaystyle\mathbb{E}\big{[}\widetilde{\eta}_{k,n}^{(\ell)}(A)\big{]}$
$\displaystyle=\frac{\rho_{k,n}}{k!}\,\int_{Q_{\ell}}\int_{({\mathbb{R}}^{d})^{k-1}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{d},{\bf y})\leq L,\,\overline{({\bf 0}_{d},{\bf y})}\in A\big{\\}}$
$\displaystyle\qquad\qquad\qquad\qquad\times\prod_{i=1}^{k-1}{\mathbbm{1}}\\{x+r_{n}y_{i}\in
Q_{\ell}\\}\operatorname{d\\!}{\bf y}\operatorname{d\\!}x.$
On the other hand,
$\mathbb{E}^{\prime}\big{[}\zeta^{(\ell)}_{k}(A)\big{]}=\tau_{k}(A)=\frac{\rho_{k,n}}{k!}\,\int_{Q_{\ell}}\int_{({\mathbb{R}}^{d})^{k-1}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{d},{\bf y})\leq L,\ \overline{({\bf 0}_{d},{\bf y})}\in
A\big{\\}}\operatorname{d\\!}{\bf y}\operatorname{d\\!}x.$
Thus, as $n\to\infty$,
$\displaystyle\sup_{A}\Big{|}\mathbb{E}\big{[}\widetilde{\eta}_{k,n}^{(\ell)}(A)\big{]}-\mathbb{E}^{\prime}\big{[}\zeta^{(\ell)}_{k}(A)\big{]}\Big{|}$
$\displaystyle\leq\frac{\rho_{k,n}}{k!}\,\int_{Q_{\ell}}\int_{({\mathbb{R}}^{d})^{k-1}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{d},{\bf y})\leq L\big{\\}}$
$\displaystyle\qquad\qquad\times\Big{(}1-\prod_{i=1}^{k-1}{\mathbbm{1}}\\{x+r_{n}y_{i}\in
Q_{\ell}\\}\Big{)}\operatorname{d\\!}{\bf y}\operatorname{d\\!}x\to 0.$
As for (5.9), the change of variables $x_{i}=x+r_{n}y_{i-1}$, $i=1,\dots,2k-p$
(with $y_{0}\equiv 0$), yields that
$\displaystyle\gamma_{n}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\max_{1\leq p\leq
k-1}n^{2k-p}r_{n}^{d(2k-p-1)}\int_{Q_{\ell}}\int_{({\mathbb{R}}^{d})^{2k-p-1}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{d},y_{1},\dots,y_{k-1})\leq L\big{\\}}$
$\displaystyle\qquad\times{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{d},y_{1},\dots,y_{p-1},y_{k},\dots,y_{2k-p-1})\leq
L\big{\\}}\prod_{i=1}^{2k-p-1}{\mathbbm{1}}\\{x+r_{n}y_{i}\in
Q_{\ell}\\}\operatorname{d\\!}{\bf y}\operatorname{d\\!}x$
$\displaystyle\leq\max_{1\leq p\leq
k-1}(nr_{n}^{d})^{k-p}\int_{({\mathbb{R}}^{d})^{2k-p-1}}\prod_{i=1}^{2k-p-1}{\mathbbm{1}}\big{\\{}\|y_{i}\|\leq
L\big{\\}}\operatorname{d\\!}{\bf y}\to 0,\ \ \ n\to\infty.$
Thus, (5.8) has been established, and it remains to verify that
$d_{\mathsf{TV}}\big{(}\mathcal{L}(\widetilde{\eta}_{k,n}^{(\ell)}),\mathcal{L}(\eta_{k,n}^{(\ell)})\big{)}\to
0,\ \ \ n\to\infty.$ (5.10)
Noting that $\widetilde{\eta}_{k,n}^{(\ell)}$ and $\eta_{k,n}^{(\ell)}$ are
both defined in the same probability space, we have
$d_{\mathsf{TV}}\big{(}\mathcal{L}(\widetilde{\eta}_{k,n}^{(\ell)}),\mathcal{L}(\eta_{k,n}^{(\ell)})\big{)}=\frac{1}{2}\sup_{\|f\|_{\infty}\leq
1}\Big{|}\mathbb{E}\big{[}f(\widetilde{\eta}_{k,n}^{(\ell)})\big{]}-\mathbb{E}\big{[}f(\eta_{k,n}^{(\ell)})\big{]}\Big{|},$
where the supremum is taken over all $f\mathrel{\mathop{\mathchar
58\relax}}M_{p}(E)\to{\mathbb{R}}$ with
$\|f\|_{\infty}\mathrel{\mathop{\mathchar 58\relax}}=\text{esssup}_{x\in
M_{p}(E)}|f(x)|\leq 1$. Whenever
$f(\widetilde{\eta}_{k,n}^{(\ell)})-f(\eta_{k,n}^{(\ell)})$ is non-zero, there
exists a subset ${\mathcal{Y}}$ of $k$ points in
$\mathcal{P}_{n}|_{Q_{\ell}}$, such that $\mathsf{diam}({\mathcal{Y}})\leq
r_{n}L$ and $c_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t)=0$. This
implies that
$\displaystyle
d_{\mathsf{TV}}\big{(}\mathcal{L}(\widetilde{\eta}_{k,n}^{(\ell)}),\mathcal{L}(\eta_{k,n}^{(\ell)})\big{)}$
$\displaystyle\leq\frac{1}{2}\sup_{\|f\|_{\infty}\leq
1}\mathbb{E}\bigg{[}\,\big{|}\,f(\widetilde{\eta}_{k,n}^{(\ell)})-f(\eta_{k,n}^{(\ell)})\,\big{|}$
$\displaystyle\qquad\times{\mathbbm{1}}\bigg{\\{}\bigcup_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}}}\Big{\\{}\mathsf{diam}({\mathcal{Y}})\leq
r_{n}L,\,c_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t)=0\Big{\\}}\bigg{\\}}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\mathcal{Y}})\leq
r_{n}L\big{\\}}\,\big{(}1-c_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t)\big{)}\bigg{]}.$
Writing
$\mathcal{B}\big{(}\\{x_{1},\dots,x_{k}\\};r\big{)}\mathrel{\mathop{\mathchar
58\relax}}=\bigcup_{i=1}^{k}B(x_{i},r)$ and appealing to the Mecke formula for
Poisson point processes, the rightmost term above is equal to
$\begin{split}&\frac{n^{k}}{k!}\int_{(Q_{\ell})^{k}}{\mathbbm{1}}\big{\\{}\mathsf{diam}(x_{1},\dots,x_{k})\leq
r_{n}L\big{\\}}\Big{(}1-e^{-n\text{vol}\big{(}\mathcal{B}(\\{x_{1},\dots,x_{k}\\};r_{n}t)\cap
Q_{\ell}\big{)}}\Big{)}\operatorname{d\\!}{\bf x}\\\
&=\frac{\rho_{k,n}}{k!}\int_{Q_{\ell}}\int_{({\mathbb{R}}^{d})^{k-1}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{d},{\bf y})\leq L\big{\\}}\prod_{i=1}^{k-1}{\mathbbm{1}}\\{x+r_{n}y_{i}\in
Q_{\ell}\\}\\\
&\qquad\qquad\qquad\times\Big{(}1-e^{-n\text{vol}\big{(}\mathcal{B}(\\{x,x+r_{n}y_{1},\dots,x+r_{n}y_{k-1}\\};r_{n}t)\cap
Q_{\ell}\big{)}}\Big{)}\operatorname{d\\!}{\bf y}\operatorname{d\\!}x\\\
&\leq\frac{1}{k!}\int_{({\mathbb{R}}^{d})^{k-1}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{d},{\bf y})\leq
L\big{\\}}\Big{(}1-e^{-nr_{n}^{d}\text{vol}\big{(}\mathcal{B}(\\{{\bf
0}_{d},{\bf y}\\};t)\big{)}}\Big{)}\operatorname{d\\!}{\bf y}.\end{split}$
(5.11)
Since $nr_{n}^{d}\to 0$ as $n\to\infty$, the last integral tends to $0$ as
$n\to\infty$. Now, (5.10) is obtained and the proof of (5.6) is completed. ∎
By the standard argument on the maximal coupling (see [21, Lemma 4.32]), for
every $\ell\geq 1$, there exists a coupling
$(\hat{\eta}_{k,n}^{(\ell)},\hat{\zeta}^{(\ell)}_{k})$ on some probability
space
$(\hat{\Omega}_{\ell},\hat{\mathcal{F}}_{\ell},\hat{\mathbb{P}}_{\ell})$, such
that $\hat{\eta}_{k,n}^{(\ell)}\stackrel{{\scriptstyle
d}}{{=}}\eta_{k,n}^{(\ell)}$, $\hat{\zeta}^{(\ell)}_{k}\stackrel{{\scriptstyle
d}}{{=}}\zeta^{(\ell)}_{k}$, and
$\hat{\mathbb{P}}_{\ell}\big{(}\hat{\eta}_{k,n}^{(\ell)}\neq\hat{\zeta}^{(\ell)}_{k}\big{)}=d_{\mathsf{TV}}\big{(}\mathcal{L}(\eta_{k,n}^{(\ell)}),\mathcal{L}(\zeta_{k}^{(\ell)})\big{)}\to
0,\ \ \ n\to\infty,$ (5.12)
where the last convergence follows from Lemma 5.3. Define
$\hat{\Omega}=\prod_{\ell=1}^{\infty}\hat{\Omega}_{\ell}$,
$\hat{\mathcal{F}}=\bigotimes_{\ell=1}^{\infty}\hat{\mathcal{F}}_{\ell}$,
$\hat{\mathbb{P}}=\bigotimes_{\ell=1}^{\infty}\hat{\mathbb{P}}_{\ell}$, and
$\hat{\mathbb{E}}$ is the corresponding expectation; then
$(\hat{\eta}_{k,n}^{(\ell)},\hat{\zeta}^{(\ell)}_{k})_{\ell\geq 1}$ becomes a
sequence of i.i.d. random vectors under $\hat{\mathbb{P}}$. Define
$\hat{\eta}_{k,n}$ and $\hat{\zeta}_{k,n}$ analogously to (5.1) and (5.2).
According to [14, Theorem 4.2.13], if one can show that
$(\hat{\eta}_{k,n})_{n\geq 1}$ and $(\hat{\zeta}_{k,n})_{n\geq 1}$ are
exponentially equivalent under a coupled probability $\hat{\mathbb{P}}$ (in
terms of the total variation distance), then it will be concluded that
$(\eta_{k,n})_{n\geq 1}$ and $(\zeta_{k,n})_{n\geq 1}$ exhibit the same LDP. A
more precise statement is given as follows.
###### Lemma 5.5.
For every $\delta>0$,
$\lim_{n\to\infty}\frac{1}{\rho_{k,n}}\log\hat{\mathbb{P}}\Big{(}d_{\mathsf{TV}}\big{(}\hat{\eta}_{k,n},\hat{\zeta}_{k,n}\big{)}\geq\delta\Big{)}=-\infty.$
(5.13)
###### Proof.
By Markov’s inequality, we have, for any $a>0$,
$\displaystyle\hat{\mathbb{P}}\Big{(}d_{\mathsf{TV}}\big{(}\hat{\eta}_{k,n},\hat{\zeta}_{k,n}\big{)}\geq\delta\Big{)}$
$\displaystyle\leq\hat{\mathbb{P}}\Big{(}\sum_{\ell=1}^{\rho_{k,n}}d_{\mathsf{TV}}\big{(}\hat{\eta}_{k,n}^{(\ell)},\hat{\zeta}^{(\ell)}_{k}\big{)}\geq\delta\rho_{k,n}\Big{)}$
$\displaystyle\leq
e^{-a\delta\rho_{k,n}}\Big{(}\hat{\mathbb{E}}\Big{[}e^{ad_{\mathsf{TV}}\big{(}\hat{\eta}_{k,n}^{(1)},\,\hat{\zeta}_{k}^{(1)}\big{)}}\Big{]}\Big{)}^{\rho_{k,n}}.$
Hence, (5.13) follows if we can show that, for every $a>0$,
$\lim_{n\to\infty}\hat{\mathbb{E}}\Big{[}e^{ad_{\mathsf{TV}}\big{(}\hat{\eta}_{k,n}^{(1)},\,\hat{\zeta}_{k}^{(1)}\big{)}}\Big{]}=1.$
(5.14)
Because of (5.12), we get that
$d_{\mathsf{TV}}\big{(}\hat{\eta}_{k,n}^{(1)},\,\hat{\zeta}_{k}^{(1)}\big{)}\stackrel{{\scriptstyle\hat{\mathbb{P}}}}{{\to}}0$
as $n\to\infty$. It thus remains to demonstrate that
$\limsup_{n\to\infty}\hat{\mathbb{E}}\Big{[}e^{ad_{\mathsf{TV}}\big{(}\hat{\eta}_{k,n}^{(1)},\,\hat{\zeta}_{k}^{(1)}\big{)}}\Big{]}<\infty,\
\ \text{ for all }a>0.$ (5.15)
In fact, (5.15) ensures the uniform integrability for the convergence (5.14).
By the Cauchy–Schwarz inequality,
$\displaystyle\hat{\mathbb{E}}\Big{[}e^{ad_{\mathsf{TV}}\big{(}\hat{\eta}_{k,n}^{(1)},\,\hat{\zeta}_{k}^{(1)}\big{)}}\Big{]}$
$\displaystyle\leq\Big{\\{}\mathbb{E}\Big{[}e^{2a\eta_{k,n}^{(1)}(E)}\Big{]}\Big{\\}}^{1/2}\Big{\\{}\mathbb{E}^{\prime}\Big{[}e^{2a\zeta_{k}^{(1)}(E)}\Big{]}\Big{\\}}^{1/2}$
$\displaystyle=\Big{\\{}\mathbb{E}\big{[}e^{2a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t)}\big{]}\Big{\\}}^{1/2}\Big{\\{}\exp\big{\\{}\tau_{k}(E)(e^{2a}-1)\big{\\}}\Big{\\}}^{1/2}.$
Now we need to verify that for all $a>0$,
$\limsup_{n\to\infty}\mathbb{E}\Big{[}e^{a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t)}\Big{]}<\infty.$
(5.16)
For the proof of (5.16), we consider the diluted family of cubes
$G\mathrel{\mathop{\mathchar
58\relax}}=\big{\\{}4Lr_{n}z+[0,tr_{n}/\sqrt{d}]^{d}\subset
Q_{1}\mathrel{\mathop{\mathchar 58\relax}}z\in{\mathbb{Z}}^{d}\big{\\}}$
(5.17)
(recall that we have taken $L>t$). Then, $Q_{1}$ can be covered by at most
$(4L\sqrt{d}/t)^{d}$ many translates of $G$. Denote these translates as
$G_{1},\dots,G_{(4L\sqrt{d}/t)^{d}}$ (with $G_{1}\equiv G$). Let
$b_{n}\mathrel{\mathop{\mathchar 58\relax}}=\rho_{k,n}^{-1}/(4Lr_{n})^{d}$
denote the number of cubes (of side length $tr_{n}/\sqrt{d}$) that are
contained in $G$. As mentioned at the beginning of Section 5, we assume,
without loss of generality, that $(4L\sqrt{d}/t)^{d}$ and $b_{n}$ take only
positive integers. Suppose ${\mathcal{Y}}$ is a set of $k$ points with
$\mathsf{diam}({\mathcal{Y}})\leq r_{n}L$, then there is a unique
$j\in\\{1,\dots,(4L\sqrt{d}/t)^{d}\\}$, so that the left most point
$\ell({\mathcal{Y}})$ belongs to one of the cubes in $G_{j}$. Hence,
$\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t\big{)}=\sum_{j=1}^{(4L\sqrt{d}/t)^{d}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t\big{)}{\mathbbm{1}}\Big{\\{}\ell({\mathcal{Y}})\in\bigcup_{J\in
G_{j}}J\Big{\\}}.$ (5.18)
Write $G_{1}=\\{J_{1},\dots,J_{b_{n}}\\}$ with
$J_{1}=\big{[}0,tr_{n}/\sqrt{d}\big{]}^{d}$. By (5.18), Hölder’s inequality,
and the spatial independence and homogeneity of $\mathcal{P}_{n}$, we need to
demonstrate that
$\limsup_{n\to\infty}\bigg{(}\mathbb{E}\Big{[}e^{a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t){\mathbbm{1}}\\{\ell({\mathcal{Y}})\in
J_{1}\\}}\Big{]}\bigg{)}^{b_{n}}<\infty.$ (5.19)
A key observation for the proof of (5.19) is that there is at most a single
$k$-point subset ${\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}$ such that
$c_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t)=1$ and
$\ell({\mathcal{Y}})\in J_{1}$. In fact, if there are two distinct point sets
${\mathcal{Y}},{\mathcal{Y}}^{\prime}\subset\mathcal{P}_{n}|_{Q_{1}}$ with
$\ell({\mathcal{Y}})\in J_{1}$ and $\ell({\mathcal{Y}}^{\prime})\in J_{1}$,
then it must be that
$c_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t)=c_{n}({\mathcal{Y}}^{\prime},\mathcal{P}_{n}|_{Q_{1}};t)=0$.
It turns out from this observation that
$\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t\big{)}{\mathbbm{1}}\big{\\{}\ell({\mathcal{Y}})\in
J_{1}\big{\\}}$
is a $\\{0,1\\}$-valued random variable; hence,
$\displaystyle\bigg{(}\mathbb{E}\Big{[}e^{a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t){\mathbbm{1}}\\{\ell({\mathcal{Y}})\in
J_{1}\\}}\Big{]}\bigg{)}^{b_{n}}$
$\displaystyle=\bigg{(}1+(e^{a}-1)\mathbb{P}\Big{(}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t\big{)}{\mathbbm{1}}\big{\\{}\ell({\mathcal{Y}})\in
J_{1}\big{\\}}=1\Big{)}\bigg{)}^{b_{n}}$
$\displaystyle\leq\bigg{(}1+(e^{a}-1)\mathbb{E}\Big{[}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\mathcal{Y}})\leq
r_{n}L,\,\ell({\mathcal{Y}})\in J_{1}\big{\\}}\Big{]}\bigg{)}^{b_{n}}.$
Repeating the same calculations as before, it is not hard to see that
$\mathbb{E}\Big{[}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\mathcal{Y}})\leq
r_{n}L,\,\ell({\mathcal{Y}})\in
J_{1}\big{\\}}\Big{]}=C^{*}\big{(}\rho_{k,n}r_{n}^{d}+o(1)\big{)}.$
Now, we obtain
$\bigg{(}1+(e^{a}-1)\mathbb{E}\Big{[}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\mathcal{Y}})\leq
r_{n}L,\,\ell({\mathcal{Y}})\in J_{1}\big{\\}}\Big{]}\bigg{)}^{b_{n}}\to
e^{C^{*}(e^{a}-1)/(4L)^{d}}<\infty,$
as desired. ∎
Now that the proof of Proposition 5.1 has been completed, we next verify that
the difference between $(\xi_{k,n})_{n\geq 1}$ and $(\eta_{k,n})_{n\geq 1}$ in
terms of the total variation distance, is exponentially negligible.
###### Proposition 5.6.
For every $\delta>0$,
$\lim_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\big{(}d_{\mathsf{TV}}(\xi_{k,n},\eta_{k,n})\geq\delta\big{)}=-\infty.$
(5.20)
###### Proof.
For $\ell=1,\dots,\rho_{k,n}$, define a collection of points in $Q_{\ell}$
that are distance at most $r$ from the boundary of $Q_{\ell}$:
$Q^{\partial}_{\ell}(r)\mathrel{\mathop{\mathchar 58\relax}}=\big{\\{}x\in
Q_{\ell}\mathrel{\mathop{\mathchar 58\relax}}\inf_{y\in\partial
Q_{\ell}}\|x-y\|\leq r\big{\\}},\ \ \ r>0.$
For a subset $A\subset E$, we discuss two distinct cases for which a $k$-point
set ${\mathcal{Y}}\subset[0,1]^{d}$ with $\mathsf{diam}({\mathcal{Y}})\leq
r_{n}L$, makes different contributions to $\xi_{k,n}(A)$ and $\eta_{k,n}(A)$.
The first case is that the point set ${\mathcal{Y}}$ “crosses a boundary”
between two neighboring sub-cubes, i.e., there exist distinct $\ell_{1}$ and
$\ell_{2}$ such that
${\mathcal{Y}}\cap Q_{\ell_{1}}\neq\emptyset,\ \ \ {\mathcal{Y}}\cap
Q_{\ell_{2}}\neq\emptyset.$
Then, ${\mathcal{Y}}$ must be contained in
$\bigcup_{\ell=1}^{\rho_{k,n}}Q^{\partial}_{\ell}\big{(}(L+t)r_{n}\big{)}$
such that
${\mathcal{Y}}\cap\bigcup_{\ell=1}^{\rho_{k,n}}Q_{\ell}^{\partial}(tr_{n})\neq\emptyset$.
Furthermore, ${\mathcal{Y}}$ may increase the value of $\xi_{k,n}(A)$, while
the value of $\eta_{k,n}(A)$ is unchanged. The second case is that there are
two neighboring sub-cubes $Q_{\ell_{1}}$ and $Q_{\ell_{2}}$, together with two
point sets ${\mathcal{Y}}_{i}\subset Q_{\ell_{i}}$, $i=1,2$, of cardinality
$k$ with $\mathsf{diam}({\mathcal{Y}}_{i})\leq r_{n}L$, such that
$\inf_{y_{i}\in{\mathcal{Y}}_{i},\,i=1,2}\|y_{1}-y_{2}\|\leq r_{n}t$. It then
holds that ${\mathcal{Y}}_{i}\subset
Q^{\partial}_{\ell_{i}}\big{(}(L+t)r_{n}\big{)}$ with ${\mathcal{Y}}_{i}\cap
Q_{\ell}^{\partial}(tr_{n})\neq\emptyset$ for $i=1,2$. Moreover,
${\mathcal{Y}}_{i}$ may increase the value of $\eta_{k,n}(A)$, but the value
of $\xi_{k,n}(A)$ is unchanged. Putting these observations together, we
conclude that
$\displaystyle d_{\mathsf{TV}}(\xi_{k,n},\eta_{k,n})$
$\displaystyle\leq\frac{1}{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{\bigcup_{\ell=1}^{\rho_{k,n}}Q_{\ell}^{\partial}((L+t)r_{n})};t\big{)}{\mathbbm{1}}\Big{\\{}{\mathcal{Y}}\cap\bigcup_{\ell=1}^{\rho_{k,n}}Q_{\ell}^{\partial}(tr_{n})\neq\emptyset\Big{\\}}$
(5.21)
$\displaystyle\qquad\qquad+\frac{1}{\rho_{k,n}}\sum_{\ell=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q^{\partial}_{\ell}((L+t)r_{n})};t\big{)}{\mathbbm{1}}\big{\\{}{\mathcal{Y}}\cap
Q_{\ell}^{\partial}(tr_{n})\neq\emptyset\big{\\}}.$
Now, (5.20) will follow if one can show that for every $\delta>0$,
$\displaystyle\lim_{n\to\infty}\frac{1}{\rho_{k,n}}\,\log\mathbb{P}\Big{(}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{\bigcup_{\ell=1}^{\rho_{k,n}}Q_{\ell}^{\partial}((L+t)r_{n})};t\big{)}$
(5.22)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times{\mathbbm{1}}\Big{\\{}{\mathcal{Y}}\cap\bigcup_{\ell=1}^{\rho_{k,n}}Q_{\ell}^{\partial}(tr_{n})\neq\emptyset\Big{\\}}\geq\delta\rho_{k,n}\Big{)}=-\infty,$
and
$\lim_{n\to\infty}\frac{1}{\rho_{k,n}}\,\log\mathbb{P}\Big{(}\sum_{\ell=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q^{\partial}_{\ell}((L+t)r_{n})};t\big{)}{\mathbbm{1}}\big{\\{}{\mathcal{Y}}\cap
Q_{\ell}^{\partial}(tr_{n})\neq\emptyset\big{\\}}\geq\delta\rho_{k,n}\Big{)}=-\infty.$
(5.23)
The proof techniques for (5.22) and (5.23) are similar, so we show only
(5.22). To begin, for each $1\leq j\leq d$, denote a collection of ordered
$j$-tuples by
$\mathcal{I}_{j}=\big{\\{}{\bm{\ell}}=(\ell_{1},\dots,\ell_{j})\mathrel{\mathop{\mathchar
58\relax}}1\leq\ell_{1}<\dots<\ell_{j}\leq d\big{\\}}.$
For ${\bm{\ell}}=(\ell_{1},\dots,\ell_{j})\in\mathcal{I}_{j}$, we define a
collection of disjoint hyper-rectangles by
$\displaystyle J$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\bigg{\\{}\Big{(}\rho_{k,n}^{-1/d}z+\big{[}0,\rho_{k,n}^{-1/d}\big{]}^{\ell_{1}-1}\times\big{[}-(L+t)r_{n},(L+t)r_{n}\big{]}\times\big{[}0,\rho_{k,n}^{-1/d}\big{]}^{\ell_{2}-\ell_{1}-1}$
(5.24)
$\displaystyle\qquad\times\big{[}-(L+t)r_{n},(L+t)r_{n}\big{]}\times\big{[}0,\rho_{k,n}^{-1/d}\big{]}^{\ell_{3}-\ell_{2}-1}\times\dots\times\big{[}-(L+t)r_{n},(L+t)r_{n}\big{]}$
$\displaystyle\qquad\times\big{[}0,\rho_{k,n}^{-1/d}\big{]}^{\ell_{j}-\ell_{j-1}-1}\times\big{[}-(L+t)r_{n},(L+t)r_{n}\big{]}\times\big{[}0,\rho_{k,n}^{-1/d}\big{]}^{d-\ell_{j}}\Big{)}\cap[0,1]^{d}\mathrel{\mathop{\mathchar
58\relax}}z\in{\mathbb{Z}}_{+}^{d}\bigg{\\}}.$
By construction, all the rectangles in $J$ are contained in
$\bigcup_{\ell=1}^{\rho_{k,n}}Q_{\ell}^{\partial}((L+t)r_{n})$. Moreover,
since the number of rectangles in $J$ is $\rho_{k,n}$, one can enumerate $J$
in a way that $J=(I_{p,n}^{\bm{\ell}},\,p=1,\dots,\rho_{k,n})$. Then, one can
offer the following bound:
$\displaystyle\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{\bigcup_{\ell=1}^{\rho_{k,n}}Q_{\ell}^{\partial}((L+t)r_{n})};t\big{)}{\mathbbm{1}}\Big{\\{}{\mathcal{Y}}\cap\bigcup_{\ell=1}^{\rho_{k,n}}Q_{\ell}^{\partial}(tr_{n})\neq\emptyset\Big{\\}}$
(5.25)
$\displaystyle\leq\sum_{j=1}^{d}\sum_{{\bm{\ell}}\in\mathcal{I}_{j}}\sum_{p=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{I_{p,n}^{\bm{\ell}}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{I_{p,n}^{\bm{\ell}}};t\big{)}.$
Owing to this bound, it remains to show that for every $j\in\\{1,\dots,d\\}$,
${\bm{\ell}}\in\mathcal{I}_{j}$, and $\delta>0$,
$\lim_{n\to\infty}\frac{1}{\rho_{k,n}}\,\log\mathbb{P}\Big{(}\sum_{p=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{I_{p,n}^{\bm{\ell}}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{I_{p,n}^{\bm{\ell}}};t\big{)}\geq\delta\rho_{k,n}\Big{)}=-\infty.$
(5.26)
Since $(I_{p,n}^{\bm{\ell}},\,p=1,\dots,\rho_{k,n})$ are disjoint, the spatial
independence of $\mathcal{P}_{n}$ ensures that
$\bigg{(}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{I_{p,n}^{\bm{\ell}}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{I_{p,n}^{\bm{\ell}}};t\big{)},\,p=1,\dots,\rho_{k,n}\bigg{)}$
are i.i.d. random variables. Hence, we have for every $a>0$,
$\displaystyle\mathbb{P}\Big{(}\sum_{p=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{I_{p,n}^{\bm{\ell}}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{I_{p,n}^{\bm{\ell}}};t\big{)}\geq\delta\rho_{k,n}\Big{)}$
$\displaystyle\quad\leq
e^{-a\delta\rho_{k,n}}\bigg{(}\mathbb{E}\Big{[}\exp\Big{\\{}a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{I_{1,n}^{\bm{\ell}}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{I_{1,n}^{\bm{\ell}}};t\big{)}\Big{\\}}\Big{]}\bigg{)}^{\rho_{k,n}}.$
For the proof of (5.26), it is now sufficient to show that for every $a>0$,
$\mathbb{E}\bigg{[}\exp\Big{\\{}a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{I_{1,n}^{\bm{\ell}}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{I_{1,n}^{\bm{\ell}}};t\big{)}\Big{\\}}\bigg{]}\to
1,\ \ \text{as }n\to\infty.$ (5.27)
By the same argument as that for (5.16), one can see that for every $a>0$,
$\limsup_{n\to\infty}\mathbb{E}\bigg{[}\exp\Big{\\{}a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{I_{1,n}^{\bm{\ell}}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{I_{1,n}^{\bm{\ell}}};t\big{)}\Big{\\}}\bigg{]}<\infty,$
which implies the required uniform integrability. Now, (5.27) will follow if
we can verify that
$\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{I_{1,n}^{\bm{\ell}}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{I_{1,n}^{\bm{\ell}}};t\big{)}\stackrel{{\scriptstyle\mathbb{P}}}{{\to}}0,\
\ \text{as }n\to\infty.$ (5.28)
To show this, we have as $n\to\infty$,
$\displaystyle\mathbb{E}\Big{[}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{I_{1,n}^{\bm{\ell}}}}s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{I_{1,n}^{\bm{\ell}}};t\big{)}\Big{]}\leq\mathbb{E}\Big{[}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{I_{1,n}^{\bm{\ell}}}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\mathcal{Y}})\leq
r_{n}L\big{\\}}\Big{]}$
$\displaystyle\qquad\qquad=\frac{n^{k}}{k!}\,\int_{(I_{1,n}^{\bm{\ell}})^{k}}{\mathbbm{1}}\big{\\{}\mathsf{diam}(x_{1},\dots,x_{k})\leq
r_{n}L\big{\\}}\operatorname{d\\!}{\bf x}$
$\displaystyle\qquad\qquad=\frac{\rho_{k,n}}{k!}\,\int_{I_{1,n}^{\bm{\ell}}}\int_{({\mathbb{R}}^{d})^{k-1}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{d},{\bf y})\leq
L\big{\\}}\prod_{i=1}^{k-1}{\mathbbm{1}}\big{\\{}x+r_{n}y_{i}\in
I_{1,n}^{\bm{\ell}}\big{\\}}\operatorname{d\\!}{\bf y}\operatorname{d\\!}x$
$\displaystyle\qquad\qquad\leq\frac{\rho_{k,n}}{k!}\,\text{vol}(I_{1,n}^{\bm{\ell}})\int_{({\mathbb{R}}^{d})^{k-1}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{d},{\bf y})\leq L\big{\\}}\operatorname{d\\!}{\bf
y}=\mathcal{O}\big{(}(nr_{n}^{d})^{k/d}\big{)}\to 0.$
Hence (5.28) has been established, as desired. ∎
Before concluding this subsection, we present the proof of Proposition 2.2.
###### Proof of Proposition 2.2.
Our proof is closely related to Theorem 5.4 in [32]. Our goal here is to show
that, for every $f\in C_{b}(E)$,
$\int_{E}\big{(}e^{f({\bf x})}-1\big{)}\tau_{k}(\operatorname{d\\!}{\bf
x})=\sup_{\rho\in M_{+}(E)}\big{\\{}\langle\rho,f\rangle-
H_{k}(\rho|\tau_{k})\big{\\}},$ (5.29)
where $\langle\rho,f\rangle\mathrel{\mathop{\mathchar
58\relax}}=\int_{E}f({\bf x})\rho(\operatorname{d\\!}{\bf x})$. Given $f\in
C_{b}(E)$, we define
$\operatorname{d\\!}\rho=e^{f}\operatorname{d\\!}\tau_{k}$. Then, $\rho\in
M_{+}(E)$ with $\rho\ll\tau_{k}$. By (2.7), it is elementary to calculate that
$\langle\rho,f\rangle-H_{k}(\rho|\tau_{k})=\int_{E}\big{(}e^{f({\bf
x})}-1\big{)}\tau_{k}(\operatorname{d\\!}{\bf x})$, which has shown that the
left hand side in (5.29) is bounded by the right hand side.
Next, let us take $\nu\in M_{+}(E)$ with $\nu\ll\tau_{k}$, say with density
$\varphi$. Then, by Jensen’s inequality,
$\int_{E}\log\bigg{(}\frac{e^{f({\bf x})}}{\varphi({\bf
x})}\bigg{)}\,\frac{\nu(d{\bf
x})}{\nu(E)}\leq\log\bigg{(}\int_{E}\frac{e^{f({\bf x})}}{\varphi({\bf
x})}\frac{\nu(d{\bf x})}{\nu(E)}\bigg{)}=\log\bigg{(}\int_{E}e^{f({\bf
x})}\frac{\tau_{k}(d{\bf x})}{\nu(E)}\bigg{)}.$
Hence, by (2.7) and the elementary inequality: $1+\log x\leq x$ for $x>0$,
$\langle\nu,f\rangle-
H_{k}(\nu|\tau_{k}){\leq\nu(E)\bigg{\\{}1+\log\bigg{(}\int_{E}e^{f({\bf
x})}\frac{\tau_{k}(dx)}{\nu(E)}\bigg{)}\bigg{\\}}-\tau_{k}(E)}\leq\int_{E}\big{(}e^{f({\bf
x})}-1\big{)}\tau_{k}(\operatorname{d\\!}{\bf x}).$
Now (5.29) is obtained, and the rest of the argument after (5.29) is
essentially the same as Theorem 5.4 in [32], so we will omit it. ∎
Finally, we prove Corollary 2.3. We can deduce its assertion from Theorem 2.1
by showing that for every $\varepsilon_{0}>0$,
$\lim_{n\to\infty}\rho_{k,n}^{-1}\log\mathbb{P}\big{(}d_{\mathsf{TV}}(\xi_{k,n},\xi_{k,n}^{\mathsf{B}})\geq\varepsilon_{0}\big{)}=-\infty.$
###### Proof of Corollary 2.3.
Similarly as in the proof of Lemma 5.5, we consider a family of diluted cubes.
In the current setting, we take
$G\mathrel{\mathop{\mathchar
58\relax}}=\big{\\{}8Lr_{n}z+[0,4Lr_{n}]^{d}\subset[0,1]^{d}\mathrel{\mathop{\mathchar
58\relax}}z\in{\mathbb{Z}}^{d}\big{\\}}.$ (5.30)
Then, $[0,1]^{d}$ can be covered by $2^{d}$ translates of $G$. As before, we
write $G=\\{J_{1},\dots,J_{b_{n}^{\prime}}\\}$ with
$J_{1}=\big{[}0,4Lr_{n}\big{]}^{d}$, where
$b_{n}^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=(8Lr_{n})^{-d}$ denotes
the number of cubes that are contained in $G$. Since there are at most
finitely many translates of $G$, it suffices to prove that as $n\to\infty$,
$\displaystyle\rho_{k,n}^{-1}\log\mathbb{P}\bigg{(}\sup_{A\subset
E}\Big{|}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k}s_{n}({\mathcal{Y}},\mathcal{P}_{n};t){\mathbbm{1}}\Big{\\{}\ell({\mathcal{Y}})\in\bigcup_{i=1}^{b_{n}^{\prime}}J_{i}\Big{\\}}\,\delta_{r_{n}^{-1}\overline{\mathcal{Y}}}(A)$
$\displaystyle\qquad\qquad\qquad-\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=k}s_{n}({\mathcal{Y}},\mathcal{B}_{n};t){\mathbbm{1}}\Big{\\{}\ell({\mathcal{Y}})\in\bigcup_{i=1}^{b_{n}^{\prime}}J_{i}\Big{\\}}\,\delta_{r_{n}^{-1}\overline{\mathcal{Y}}}(A)\Big{|}\geq\varepsilon_{0}\rho_{k,n}\bigg{)}\to-\infty$
(recall that $\ell({\mathcal{Y}})$ is the left most point of ${\mathcal{Y}}$
in the lexicographic ordering).
We say that $J_{i}$ is an _$n$ -bad cube_ if one of the following events
happens.
* •
There exists a $k$-element subset ${\mathcal{Y}}\subset\mathcal{P}_{n}$ with
$\ell({\mathcal{Y}})\in J_{i}$, $s_{n}({\mathcal{Y}},\mathcal{P}_{n};t)=1$,
such that ${\mathcal{Y}}\not\subset\mathcal{B}_{n}$ or
$s_{n}\big{(}{\mathcal{Y}},\mathcal{B}_{n};t\big{)}=0$ holds.
* •
There exists a $k$-element subset ${\mathcal{Y}}\subset\mathcal{B}_{n}$ with
$\ell({\mathcal{Y}})\in J_{i}$, $s_{n}({\mathcal{Y}},\mathcal{B}_{n};t)=1$,
such that ${\mathcal{Y}}\not\subset\mathcal{P}_{n}$ or
$s_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n};t\big{)}=0$ holds.
In this setting, the key observation is that there exists a constant $M>0$,
such that
$\displaystyle\sup_{A\subset
E}\Big{|}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k}s_{n}({\mathcal{Y}},\mathcal{P}_{n};t){\mathbbm{1}}\Big{\\{}\ell({\mathcal{Y}})\in\bigcup_{i=1}^{b_{n}^{\prime}}J_{i}\Big{\\}}\,\delta_{r_{n}^{-1}\overline{\mathcal{Y}}}(A)$
$\displaystyle\qquad\qquad\qquad-\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=k}s_{n}({\mathcal{Y}},\mathcal{B}_{n};t){\mathbbm{1}}\Big{\\{}\ell({\mathcal{Y}})\in\bigcup_{i=1}^{b_{n}^{\prime}}J_{i}\Big{\\}}\,\delta_{r_{n}^{-1}\overline{\mathcal{Y}}}(A)\Big{|}$
$\displaystyle\leq M\sum_{i=1}^{b_{n}^{\prime}}{\mathbbm{1}}\\{J_{i}\text{ is
$n$-bad}\\}.$
Thus, it suffices to show that for every $\varepsilon_{0}>0$,
$\lim_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\Big{(}\sum_{i=1}^{b_{n}^{\prime}}{\mathbbm{1}}\\{J_{i}\text{
is $n$-bad}\\}\geq\varepsilon_{0}\rho_{k,n}\Big{)}=-\infty.$
For $0<\varepsilon\leq 1$, let $\mathcal{P}_{n}^{(\varepsilon)}$ be a
homogeneous Poisson point process on $[0,1]^{d}$ with intensity
$n\varepsilon$, independent of $\mathcal{P}_{n}$. Then,
$\mathcal{P}_{n}^{(\varepsilon,\mathsf{a})}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{P}_{n}\cup\mathcal{P}_{n}^{(\varepsilon)}$ represents an
_augmented_ version of $\mathcal{P}_{n}$ with intensity $n(1+\varepsilon)$.
Moreover, $\mathcal{P}_{n}^{(\varepsilon,\mathsf{t})}$ denotes a _thinned_
version of $\mathcal{P}_{n}$, obtained by removing points with probability
$\varepsilon$. Notice that
$\mathcal{P}_{n}^{(\varepsilon,\mathsf{a})}\stackrel{{\scriptstyle
d}}{{=}}\mathcal{P}_{n(1+\varepsilon)}$ and
$\mathcal{P}_{n}^{(\varepsilon,\mathsf{t})}\stackrel{{\scriptstyle
d}}{{=}}\mathcal{P}_{n(1-\varepsilon)}$. In this setting, we introduce the
event
$F_{n,\varepsilon}\mathrel{\mathop{\mathchar
58\relax}}=\big{\\{}\mathcal{P}_{n}^{(\varepsilon,\mathsf{t})}\subset\mathcal{B}_{n}\subset\mathcal{P}_{n}^{(\varepsilon,\mathsf{a})}\big{\\}},$
and note by the Poisson concentration bound from [31, Lemma 1.2],
$\displaystyle\limsup_{n\to\infty}\rho_{k,n}^{-1}\log\mathbb{P}\big{(}F_{n,\varepsilon}^{c}\big{)}\leq-C^{*}\lim_{n\to\infty}\rho_{k,n}^{-1}n=-\infty.$
(5.31)
The key advantage of the event $F_{n,\varepsilon}$ is that it allows to
simplify the property of being $n$-bad. Indeed, if $J_{i}$ is $n$-bad and
$F_{n,\varepsilon}$ holds, then $J_{i}$ becomes _$(n,\varepsilon)$ -special_,
in the sense that
$T\cap(\mathcal{P}_{n}^{(\varepsilon,\mathsf{a})}\setminus\mathcal{P}_{n}^{(\varepsilon,\mathsf{t})})\neq\emptyset,\
\ \text{ and }\ \ \mathcal{P}_{n}^{(\varepsilon,\mathsf{a})}(T)\geq k,$
where
$T\mathrel{\mathop{\mathchar
58\relax}}=\big{\\{}x\in{\mathbb{R}}^{d}\mathrel{\mathop{\mathchar
58\relax}}\inf_{y\in J_{i}}\|x-y\|\leq 2r_{n}L\big{\\}},\ r>0.$
Since we work with diluted cubes in (5.30), one can see that
$\big{(}{\mathbbm{1}}\\{J_{i}\text{ is
}(n,\varepsilon)\text{-special}\\}\big{)}_{i=1}^{b_{n}^{\prime}}$ are i.i.d.
Bernoulli random variables. Thus, the number of $(n,\varepsilon)$-special
cubes is a binomial random variable with $b_{n}^{\prime}$ trials and success
probability $p_{n,\varepsilon}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{P}\big{(}J_{1}\text{ is $(n,\varepsilon)$-special}\big{)}.$
Then, one can bound $p_{n,\varepsilon}$ as follows:
$\displaystyle p_{n,\varepsilon}$
$\displaystyle\leq\mathbb{P}\big{(}T\cap(\mathcal{P}_{n}^{(\varepsilon,\mathsf{a})}\setminus\mathcal{P}_{n}^{(\varepsilon,\mathsf{t})})\neq\emptyset\,\big{|}\,\mathcal{P}_{n}^{(\varepsilon,\mathsf{a})}(T)=k\big{)}\mathbb{P}\big{(}\mathcal{P}_{n}^{(\varepsilon,\mathsf{a})}(T)=k\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\mathbb{P}\big{(}\mathcal{P}_{n}^{(\varepsilon,\mathsf{a})}(T)\geq
k+1\big{)}$ $\displaystyle\leq
2k\varepsilon\mathbb{P}\big{(}\mathcal{P}_{n(1+\varepsilon)}(T)=k\big{)}+\mathbb{P}\big{(}\mathcal{P}_{n(1+\varepsilon)}(T)\geq
k+1\big{)}$ $\displaystyle\leq
C^{*}\big{(}\varepsilon(nr_{n}^{d})^{k}+(nr_{n}^{d})^{k+1}\big{)}.$
In conclusion, if one takes sufficiently small $\varepsilon>0$, then
$b_{n}^{\prime}p_{n,\varepsilon}\leq
C^{*}b_{n}^{\prime}\big{(}\varepsilon(nr_{n}^{d})^{k}+(nr_{n}^{d})^{k+1}\big{)}\leq\varepsilon_{0}\rho_{k,n}$
for large $n$ enough. Therefore, the binomial concentration inequality [31,
Lemma 1.1] gives that
$\displaystyle\limsup_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\big{(}\text{Bin}(b_{n}^{\prime},p_{n,\varepsilon})\geq\varepsilon_{0}\rho_{k,n}\big{)}$
$\displaystyle\leq-\lim_{n\to\infty}\frac{\varepsilon_{0}}{2}\,\log\bigg{\\{}\frac{\varepsilon_{0}\rho_{k,n}}{C^{*}b_{n}^{\prime}(\varepsilon(nr_{n}^{d})^{k}+(nr_{n}^{d})^{k+1})}\bigg{\\}}$
$\displaystyle=-\frac{\varepsilon_{0}}{2}\log\Big{\\{}\frac{(8L)^{d}\varepsilon_{0}}{C^{*}\varepsilon}\Big{\\}}.$
The last term tends to $-\infty$ as $\varepsilon\to 0$. Hence, combining this
result with (5.31) concludes the proof of Corollary 2.3. ∎
### 5.2 Proofs of Theorem 3.1 and Corollary 3.2
First we point out that the proof of Corollary 3.2 is almost identical to that
of Corollary 2.3, so we skip it here. For the proof of Theorem 3.1, we define
$V_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k}s_{n}({\mathcal{Y}},\mathcal{P}_{n};t)\,\delta_{H_{n}({\mathcal{Y}})}\in
M_{+}(E^{\prime}).$
The key step of our proof is to show that the assertions of Theorem 3.1 still
hold even when $(U_{k,n})_{n\geq 1}$ is replaced by $(V_{k,n})_{n\geq 1}$. To
clarify our presentation, we state this step as a separate proposition.
###### Proposition 5.7.
The sequence $(V_{k,n})_{n\geq 1}$ satisfies an LDP in the weak topology with
rate $\rho_{k,n}$ and the rate function $\bar{\Lambda}_{k}^{*}$ defined in
(3.5).
###### Proof.
The process $(V_{k,n})_{n\geq 1}$ has structure very similar to that of
$(\xi_{k,n})_{n\geq 1}$ in (2.4); thus, the proof techniques for Proposition
5.7 are parallel to those for Theorem 2.1. More precisely, as an analog of
(5.1), we define
$W_{k,n}=\frac{1}{\rho_{k,n}}\sum_{\ell=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}},\,|{\mathcal{Y}}|=k}s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t)\,\delta_{H_{n}({\mathcal{Y}})}\in
M_{+}(E^{\prime}).$
It then follows from the same argument as Proposition 5.1 that
$(W_{k,n})_{n\geq 1}$ satisfies an LDP in the weak topology with rate
$\rho_{k,n}$ and the rate function $\bar{\Lambda}_{k}^{*}$. Subsequently, by
repeating the proof of Proposition 5.6, one can also establish that for every
$\delta>0$,
$\lim_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\big{(}d_{\mathsf{TV}}(V_{k,n},W_{k,n})\geq\delta\big{)}=-\infty.$
This concludes the proof of Proposition 5.7. ∎
###### Proof of Theorem 3.1.
We now have to show that $(U_{k,n})_{n\geq 1}$ exhibits the same LDP as
$(V_{k,n})_{n\geq 1}$ above. We take, without loss of generality, $0<t_{1}\leq
t_{2}\leq\dots\leq t_{m}<\infty$ for time parameters of $(U_{k,n})_{n\geq 1}$,
whereas we fix the parameter of $(V_{k,n})_{n\geq 1}$ at $t=t_{1}$. In this
setup, we need to verity that for every $\delta>0$,
$\lim_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\big{(}{d_{\mathsf{TV}}(U_{k,n},V_{k,n})}\geq\delta\big{)}=-\infty.$
It is straightforward to see that
${d_{\mathsf{TV}}(U_{k,n},V_{k,n})}\leq\frac{1}{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k}s_{n}({\mathcal{Y}},\mathcal{P}_{n};t_{1})\big{(}1-c_{n}({\mathcal{Y}},\mathcal{P}_{n};t_{m})\big{)}.$
By virtue of the partition of $[0,1]^{d}$ into multiple sub-cubes
$Q_{1},\dots,Q_{\rho_{k,n}}$ as in the proof of Proposition 5.1, what need to
be shown are
$\lim_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\Big{(}\sum_{\ell=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}}}s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t_{1})\big{(}1-c_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t_{m})\big{)}\geq\delta\rho_{k,n}\Big{)}=-\infty,$
(5.32)
and
$\displaystyle\lim_{n\to\infty}\frac{1}{\rho_{k,n}}\log\,$
$\displaystyle\mathbb{P}\Big{(}\,\Big{|}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}s_{n}({\mathcal{Y}},\mathcal{P}_{n};t_{1})\big{(}1-c_{n}({\mathcal{Y}},\mathcal{P}_{n};t_{m})\big{)}$
$\displaystyle\qquad-\sum_{\ell=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}}}s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t_{1})\big{(}1-c_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t_{m})\big{)}\Big{|}\geq\delta\rho_{k,n}\Big{)}=-\infty,$
for every $\delta>0$. Of the last two conditions, the latter can be
established by the same argument as Proposition 5.6. By Markov’s inequality,
(5.32) will follow if we can show that for every $a>0$,
$\mathbb{E}\Big{[}e^{a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t_{1})(1-c_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t_{m}))}\Big{]}\to
1,\ \ n\to\infty.$ (5.33)
Repeating the same calculations as in (5.11), while using an obvious bound
$s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t_{1})\leq{\mathbbm{1}}\big{\\{}\mathsf{diam}({\mathcal{Y}})\leq
r_{n}L\big{\\}}$, as well as the Mecke formula for Poisson point processes,
$\displaystyle\mathbb{E}\Big{[}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t_{1})(1-c_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t_{m}))\Big{]}$
$\displaystyle\leq\frac{1}{k!}\,\int_{({\mathbb{R}}^{d})^{k-1}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{d},{\bf y})\leq
L\big{\\}}\Big{(}1-e^{-nr_{n}^{d}\text{vol}(\mathcal{B}(\\{{\bf 0}_{d},{\bf
y}\\};t_{m}))}\Big{)}\operatorname{d\\!}{\bf y}\to 0,\ \ \text{as
}n\to\infty.$
Since we have already shown the uniform integrability by (5.16), we now obtain
(5.33), as desired.
Finally, by the same proof as Proposition 2.2, we can obtain (3.6). ∎
### 5.3 Proofs of Theorem 3.3, Corollary 3.4, and Proposition 3.5
In this section, we first prove Proposition 3.5, because the proof of Theorem
3.3 makes use of Proposition 3.5.
###### Proof of Proposition 3.5.
For every ${\bf x}=(x_{1},\dots,x_{m})\in E^{\prime}$, one can choose ${\bf
a}=(a_{1},\dots,a_{m})\in{\mathbb{R}}^{m}$ so that
$u_{i}({\bf x},{\bf a})\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{k!}\int_{({\mathbb{R}}^{d})^{k-1}}h^{(i)}({\bf 0}_{d},{\bf
y})e^{\sum_{r=1}^{m}a_{r}h^{(r)}({\bf 0}_{d},{\bf y})}\operatorname{d\\!}{\bf
y}-x_{i}=0,\ \ i=1,\dots,m,$
with ${\bf y}\in({\mathbb{R}}^{d})^{k-1}$. By condition (H4), $u_{i}$ is
continuously differentiable with respect to $({\bf x},{\bf a})$. Furthermore,
(H5) ensures that the matrix $\big{(}\partial u_{i}/\partial
a_{j}\big{)}_{i,j=1}^{m}$ is positive definite. Thus, by the implicit function
theorem, there exist continuously differentiable functions
$\alpha_{r}\mathrel{\mathop{\mathchar 58\relax}}E^{\prime}\to{\mathbb{R}}$,
$r=1,\dots,m$, such that
$I_{k}({\bf x})=\sum_{i=1}^{m}\alpha_{i}({\bf
x})x_{i}-\frac{1}{k!}\,\int_{({\mathbb{R}}^{d})^{k-1}}\Big{(}e^{\sum_{i=1}^{m}\alpha_{i}({\bf
x})h^{(i)}({\bf 0}_{d},{\bf y})}-1\Big{)}\operatorname{d\\!}{\bf y},$ (5.34)
and
$x_{i}=\frac{1}{k!}\int_{({\mathbb{R}}^{d})^{k-1}}h^{(i)}({\bf 0}_{d},{\bf
y})e^{\sum_{r=1}^{m}\alpha_{r}({\bf x})h^{(r)}({\bf 0}_{d},{\bf
y})}\operatorname{d\\!}{\bf y},\ \ i=1,\dots,m.$
In particular, (5.34) implies that $I_{k}$ is continuously differentiable on
$E^{\prime}$.
For Part $(ii)$, we show that the Hessian of $I_{k}$, denoted
$\mathcal{H}(I_{k})$, is positive definite. First it is elementary to check
that $\partial I_{k}({\bf x})/\partial x_{j}=\alpha_{j}({\bf x})$ for
$j=1,\dots,m$. Using this, it is not hard to calculate that
$\mathcal{H}(I_{k})=A^{-1}$ where $A=(A_{ij})$ is given by
$A_{ij}=\frac{1}{k!}\,\int_{({\mathbb{R}}^{d})^{k-1}}h^{(i)}({\bf 0}_{d},{\bf
y})h^{(j)}({\bf 0}_{d},{\bf y})e^{{\sum_{r=1}^{m}\alpha_{r}({\bf
x})h^{(r)}({\bf 0}_{d},{\bf y})}}\operatorname{d\\!}{\bf y}.$
We conclude from condition (H5) that $A$ is positive definite; thus, so is
$A^{-1}$ as required. Finally, if we set $0=\partial I_{k}({\bf x})/\partial
x_{j}$ for every $j=1,\dots,m$, then $\alpha_{j}({\bf x})=0$, and so,
$x_{j}=\mu_{j}$ and $I_{k}({\bf x})=0$. As $I_{k}$ is strictly convex,
$(\mu_{1},\dots,\mu_{m})$ is a unique minimizer of $I_{k}$. ∎
The proof of Theorem 3.3 is based on an extension of the contraction
principle, provided in [14, Theorem 4.2.23]. We begin with defining a map
$F\mathrel{\mathop{\mathchar 58\relax}}M_{+}(E^{\prime})\to[0,\infty)^{m}$ by
$F(\rho)=\big{(}\int_{E^{\prime}}x_{i}\rho(\operatorname{d\\!}{\bf
x})\big{)}_{i=1}^{m}$. Then, $F(U_{k,n})=T_{k,n}/\rho_{k,n}$, where $U_{k,n}$
and $T_{k,n}$ are defined in (3.4) and (3.8), respectively. Since $F$ is not
continuous in the weak topology, we need to introduce a family of continuous
maps: $F_{K}(\rho)\mathrel{\mathop{\mathchar
58\relax}}=\big{(}\int_{E^{\prime}}s_{i}^{(K)}({\bf
x})\rho(\operatorname{d\\!}{\bf x})\big{)}_{i=1}^{m}$, where
$s_{i}^{(K)}(x_{1},\dots,x_{m})=\begin{cases}x_{i}&\text{ if }0\leq x_{i}\leq
K,\\\ -K^{2}(x_{i}-K)+K&\text{ if }K\leq x_{i}\leq K+K^{-1},\\\ 0&\text{ if
}x_{i}\geq K+K^{-1}.\end{cases}$
Clearly, $s_{i}^{(K)}$ is continuous and bounded on $E^{\prime}$, and
consequently, $F_{K}$ becomes continuous in the weak topology.
In order to apply [14, Theorem 4.2.23], we need to demonstrate the following
auxiliary results.
###### Lemma 5.8.
$(i)$ Let $\delta>0$. Then,
$\limsup_{K\to\infty}\limsup_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\Big{(}\big{\|}F(U_{k,n})-F_{K}(U_{k,n})\big{\|}>\delta\Big{)}=-\infty,$
$(ii)$ Let $a>0$. Then,
$\limsup_{K\to\infty}\sup_{\rho\in
M_{+}(E^{\prime}),\,\bar{\Lambda}_{k}^{*}(\rho)\leq
a}\big{\|}F(\rho)-F_{K}(\rho)\big{\|}=0.$
Assuming the assertions of Lemma 5.8 temporarily, we first explain how to
conclude the proof of Theorem 3.3.
###### Proof of Theorem 3.3.
Note that one can conclude from Theorem 3.1, [14, Theorem 4.2.23], and Lemma
5.8 that $(T_{k,n}/\rho_{k,n})_{n\geq 1}$ satisfies an LDP with rate
$\rho_{k,n}$ and the rate function
$\inf_{\nu\in M_{+}(E^{\prime}),\,F(\nu)={\bf
x}}H_{k}^{\prime}(\nu|\tau_{k}^{\prime}),\ \ \ {\bf x}\in{\mathbb{R}}^{m},$
where $H_{k}^{\prime}$ is a relative entropy given at (3.7). It thus remains
to show that
$\inf_{\nu\in M_{+}(E^{\prime}),\,F(\nu)={\bf
x}}H_{k}^{\prime}(\nu|\tau_{k}^{\prime})=I_{k}({\bf x}),\ \ \ {\bf
x}\in{\mathbb{R}}^{m}.$ (5.35)
If $x_{i}<0$ for some $i\in\\{1,\dots,m\\}$, both sides above are equal to
infinity. So from onward, we consider ${\bf x}\in E^{\prime}$ and prove first
that $\inf_{\nu\in M_{+}(E^{\prime}),\,F(\nu)={\bf
x}}H_{k}^{\prime}(\nu|\tau_{k}^{\prime})\geq I_{k}({\bf x})$. Fix $\nu\in
M_{+}(E^{\prime})$ such that $F(\nu)={\bf x}$. For every ${\bf
a}\in{\mathbb{R}}^{m}$, we set $f({\bf z})=\langle{\bf a},{\bf z}\rangle$ for
${\bf z}\in E^{\prime}$. Approximating $f$ via a sequence of continuous and
bounded functions, we have
$\displaystyle H_{k}^{\prime}(\nu|\tau_{k}^{\prime})$
$\displaystyle=\bar{\Lambda}_{k}^{*}(\nu)\geq\langle\nu,f\rangle-\frac{1}{k!}\int_{({\mathbb{R}}^{d})^{k-1}}\big{(}e^{f(H({\bf
0}_{d},{\bf y}))}-1\big{)}\,{\mathbbm{1}}\big{\\{}H({\bf 0}_{d},{\bf
y})\neq{\bf 0}_{m}\big{\\}}\operatorname{d\\!}{\bf y}$
$\displaystyle=\langle{\bf a},{\bf
x}\rangle-\frac{1}{k!}\int_{({\mathbb{R}}^{d})^{k-1}}\big{(}e^{\langle{\bf
a},H({\bf 0}_{d},{\bf y})\rangle}-1\big{)}\operatorname{d\\!}{\bf y};$
thus, $H_{k}^{\prime}(\nu|\tau_{k}^{\prime})\geq I_{k}({\bf x})$ holds. Next,
for every ${\bf x}\in E^{\prime}$, there exists ${\bf
a}_{0}\in{\mathbb{R}}^{m}$ such that
$I_{k}({\bf x}){=\langle{\bf a}_{0},{\bf
x}\rangle-\frac{1}{k!}\int_{({\mathbb{R}}^{d})^{k-1}}\big{(}e^{\langle{\bf
a}_{0},H({\bf 0}_{d},{\bf y})\rangle}-1\big{)}\operatorname{d\\!}{\bf
y}}=\langle{\bf a}_{0},{\bf x}\rangle-\int_{E^{\prime}}\big{(}e^{\langle{\bf
a}_{0},{\bf z}\rangle}-1\big{)}\tau_{k}^{\prime}(\operatorname{d\\!}{\bf z}),$
where $x_{i}=\int_{E^{\prime}}z_{i}e^{\langle{\bf a}_{0},{\bf
z}\rangle}\tau_{k}^{\prime}(\operatorname{d\\!}{\bf z})$ for $i=1,\dots,m$.
Define $\nu(A)\mathrel{\mathop{\mathchar 58\relax}}=\int_{A}e^{\langle{\bf
a}_{0},{\bf z}\rangle}\tau_{k}^{\prime}(\operatorname{d\\!}{\bf z})$,
$A\subset E^{\prime}$. It then follows that
$\int_{E^{\prime}}z_{i}\nu(\operatorname{d\\!}{\bf z})=x_{i}$ for
$i=1,\dots,m$; equivalently, $F(\nu)={\bf x}$. Moreover,
$H_{k}^{\prime}(\nu|\tau_{k}^{\prime})=\int_{E^{\prime}}\log\frac{\operatorname{d\\!}\nu}{\operatorname{d\\!}\tau_{k}^{\prime}}({\bf
z})\nu(\operatorname{d\\!}{\bf
z})-\nu(E^{\prime})+\tau_{k}^{\prime}(E^{\prime})=\langle{\bf a}_{0},{\bf
x}\rangle-\int_{E^{\prime}}\big{(}e^{\langle{\bf a}_{0},{\bf
z}\rangle}-1\big{)}\tau_{k}^{\prime}(\operatorname{d\\!}{\bf z})=I_{k}({\bf
x}).$
This implies $\inf_{\nu\in M_{+}(E^{\prime}),\,F(\nu)={\bf
x}}H_{k}^{\prime}(\nu|\tau_{k}^{\prime})\leq I_{k}({\bf x})$, and thus, the
proof of (5.35) has been completed.
Finally, Proposition 3.5 $(i)$ ensures that $I_{k}$ is continuous; therefore,
the LDP for $(T_{k,n}/\rho_{k,n})$ implies the convergence in (3.9).
∎
Now, we present the proof of Lemma 5.8.
###### Proof of Lemma 5.8 $(i)$.
Note first that the result follows from
$\limsup_{K\to\infty}\limsup_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\Big{(}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}g_{n}^{(i)}({\mathcal{Y}},\mathcal{P}_{n};t_{i})\,{\mathbbm{1}}\big{\\{}h_{n}^{(i)}({\mathcal{Y}})>K\big{\\}}\geq\delta\rho_{k,n}\Big{)}=-\infty,$
(5.36)
for each $i\in\\{1,\dots,m\\}$. In the sequel, we prefer to use the notation
$G_{n}$ and $H_{n}$, instead of $g_{n}^{(i)}$ and $h_{n}^{(i)}$, with the
assumption $m=1$, so that the range of $G_{n}$ and $H_{n}$ is $[0,\infty)$.
Additionally, we drop the subscript $i$ from $t_{i}$. Then, (5.36) can be
rephrased as
$\limsup_{K\to\infty}\limsup_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\Big{(}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}G_{n}({\mathcal{Y}},\mathcal{P}_{n};t)\,{\mathbbm{1}}\big{\\{}H_{n}({\mathcal{Y}})>K\big{\\}}\geq\delta\rho_{k,n}\Big{)}=-\infty.$
(5.37)
Clearly, (5.37) is implied by
$\limsup_{K\to\infty}\limsup_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\Big{(}\sum_{\ell=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}}}G_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t)\,{\mathbbm{1}}\big{\\{}H_{n}({\mathcal{Y}})>K\big{\\}}\geq\delta\rho_{k,n}\Big{)}=-\infty,$
(5.38)
and
$\displaystyle\limsup_{K\to\infty}\limsup_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\bigg{(}\,\Big{|}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}G_{n}({\mathcal{Y}},\mathcal{P}_{n};t)\,{\mathbbm{1}}\big{\\{}H_{n}({\mathcal{Y}})>K\big{\\}}$
(5.39)
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\sum_{\ell=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}}}G_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t)\,{\mathbbm{1}}\big{\\{}H_{n}({\mathcal{Y}})>K\big{\\}}\Big{|}\geq\delta\rho_{k,n}\bigg{)}=-\infty,$
for every $\delta>0$. As for (5.38), Markov’s inequality yields that
$\displaystyle\frac{1}{\rho_{k,n}}\log\mathbb{P}\Big{(}\sum_{\ell=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}}}G_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t)\,{\mathbbm{1}}\big{\\{}H_{n}({\mathcal{Y}})>K\big{\\}}\geq\delta\rho_{k,n}\Big{)}$
$\displaystyle\leq-a\delta+\log\mathbb{E}\Big{[}e^{a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}G_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t)\,{\mathbbm{1}}\\{H_{n}({\mathcal{Y}})>K\\}}\Big{]}.$
It is thus sufficient to demonstrate that for every $a>0$,
$\limsup_{K\to\infty}\limsup_{n\to\infty}\mathbb{E}\Big{[}e^{a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}G_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t){\mathbbm{1}}\\{H_{n}({\mathcal{Y}})>K\\}}\Big{]}=1.$
(5.40)
Although the proof techniques for (5.40) are mostly the same as those for
(5.16), we still need to update it because $G_{n}$ in (5.40) is not
necessarily an indicator function. Using the same logic as in the proof of
(5.16), our goal is to prove that for every $a>0$,
$\limsup_{K\to\infty}\limsup_{n\to\infty}\bigg{(}\mathbb{E}\Big{[}e^{a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}G_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t){\mathbbm{1}}\\{H_{n}({\mathcal{Y}})>K,\,\ell({\mathcal{Y}})\in
J_{1}\\}}\Big{]}\bigg{)}^{b_{n}}\leq 1,$
where $J_{1}=\big{[}0,tr_{n}/\sqrt{d}\big{]}^{d}$ and
$b_{n}=\rho_{k,n}^{-1}/(4Lr_{n})^{d}$. Observe, once again, that there exists
at most a single $k$-point set ${\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}$
satisfying $c_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t)=1$ and
$\ell({\mathcal{Y}})\in J_{1}$. Therefore, as $n\to\infty$,
$\displaystyle\mathbb{E}\Big{[}e^{a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}G_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t){\mathbbm{1}}\\{H_{n}({\mathcal{Y}})>K,\,\ell({\mathcal{Y}})\in
J_{1}\\}}\Big{]}$ (5.41) $\displaystyle\leq
1+\mathbb{E}\Big{[}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}e^{aH_{n}({\mathcal{Y}})}{\mathbbm{1}}\big{\\{}H_{n}({\mathcal{Y}})>K,\,\ell({\mathcal{Y}})\in
J_{1}\big{\\}}\Big{]}$
$\displaystyle=1+\frac{n^{k}}{k!}\int_{([0,1]^{d})^{k}}e^{aH_{n}(x_{1},\dots,x_{k})}{\mathbbm{1}}\big{\\{}H_{n}(x_{1},\dots,x_{k})>K,\,\ell(x_{1},\dots,x_{k})\in
J_{1}\big{\\}}\operatorname{d\\!}{\bf x}$
$\displaystyle=1+\frac{\rho_{k,n}}{k!}\int_{[0,1]^{d}}\int_{({\mathbb{R}}^{d})^{k-1}}e^{aH({\bf
0}_{d},{\bf y})}{\mathbbm{1}}\big{\\{}H({\bf 0}_{d},{\bf
y})>K,\,\ell(x,x+r_{n}y_{1},\dots,x+r_{n}y_{k-1})\in J_{1}\big{\\}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\prod_{i=1}^{k-1}{\mathbbm{1}}\big{\\{}x+r_{n}y_{i}\in[0,1]^{d}\big{\\}}\operatorname{d\\!}{\bf
y}\operatorname{d\\!}x$ $\displaystyle\leq
1+C^{*}\rho_{k,n}r_{n}^{d}\int_{({\mathbb{R}}^{d})^{k-1}}e^{aH({\bf
0}_{d},{\bf y})}{\mathbbm{1}}\big{\\{}H({\bf 0}_{d},{\bf
y})>K\big{\\}}\operatorname{d\\!}{\bf y}.$
Here, the last integral is finite due to property (H4). Now, it follows from
(5.41) that
$\displaystyle\limsup_{K\to\infty}\limsup_{n\to\infty}\bigg{(}\mathbb{E}\Big{[}e^{a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}G_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}};t){\mathbbm{1}}\\{H_{n}({\mathcal{Y}})>K,\,\ell({\mathcal{Y}})\in
J_{1}\\}}\Big{]}\bigg{)}^{b_{n}}$
$\displaystyle\leq\limsup_{K\to\infty}\exp\Big{\\{}\frac{C^{*}}{(4L)^{d}}\int_{({\mathbb{R}}^{d})^{k-1}}e^{aH({\bf
0}_{d},{\bf
y})}{\mathbbm{1}}\big{\\{}H_{n}({\mathcal{Y}})>K\big{\\}}\operatorname{d\\!}{\bf
y}\Big{\\}}=1$
Next, turning to condition (5.39), we deduce an inequality analogous to
(5.21):
$\displaystyle\Big{|}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}G_{n}({\mathcal{Y}},\mathcal{P}_{n};t)\,{\mathbbm{1}}\big{\\{}H_{n}({\mathcal{Y}})>K\big{\\}}-\sum_{\ell=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}}}G_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}};t)\,{\mathbbm{1}}\big{\\{}H_{n}({\mathcal{Y}})>K\big{\\}}\Big{|}$
$\displaystyle\leq\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}}G_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{\bigcup_{\ell=1}^{\rho_{k,n}}Q_{\ell}^{\partial}((L+t)r_{n})};t\big{)}{\mathbbm{1}}\Big{\\{}{\mathcal{Y}}\subset\bigcup_{\ell=1}^{\rho_{k,n}}Q_{\ell}^{\partial}\big{(}(L+t)r_{n}\big{)}\Big{\\}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{\ell=1}^{\rho_{k,n}}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q^{\partial}_{\ell}((L+t)r_{n})}}G_{n}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q^{\partial}_{\ell}((L+t)r_{n})};t\big{)}.$
Proceeding the same argument as those after (5.21), while using property (H4),
we can get (5.39). Now, the proof of Lemma 5.8 $(i)$ is completed. ∎
###### Proof of Lemma 5.8 $(ii)$.
Suppose, for contradiction, that we can choose $a>0$ and $\delta>0$ such that
$\limsup_{K\to\infty}\sup_{\rho\in
M_{+}(E^{\prime}),\,\bar{\Lambda}_{k}^{*}(\rho)\leq
a}\big{\|}F(\rho)-F_{K}(\rho)\big{\|}>\delta.$ (5.42)
For $K^{\prime}\geq K+K^{-1}$, define
$F_{K,K^{\prime}}(\rho)\mathrel{\mathop{\mathchar
58\relax}}=\Big{(}\int_{E^{\prime}}u_{i}^{(K,K^{\prime})}({\bf
x})\rho(\operatorname{d\\!}{\bf x})\Big{)}_{i=1}^{m},\ \ \ \rho\in
M_{+}(E^{\prime}),$
where
$u_{i}^{(K,K^{\prime})}(x_{1},\dots,x_{m})=\begin{cases}0&\text{ if
}x_{i}\in[0,K]\cup[K^{\prime}+(K^{\prime})^{-1},\infty),\\\
(K^{2}+1)(x_{i}-K)&\text{ if }K\leq x_{i}\leq K+K^{-1},\\\ x_{i}&\text{ if
}K+K^{-1}\leq x_{i}\leq K^{\prime},\\\
-(K^{\prime})^{2}(x_{i}-K^{\prime})+K^{\prime}&\text{ if }K^{\prime}\leq
x_{i}\leq K^{\prime}+(K^{\prime})^{-1}.\end{cases}$
Then, $F_{K,K^{\prime}}$ is continuous in the weak topology, and further,
$\big{\|}F_{K,K^{\prime}}(\rho)\big{\|}\nearrow\big{\|}F(\rho)-F_{K}(\rho)\big{\|},\
\ \text{as }K^{\prime}\uparrow\infty.$ (5.43)
By Lemma 5.8 $(i)$ and (5.43), we can choose $K_{1}>0$ so that
$\limsup_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\Big{(}\big{\|}F_{K,K^{\prime}}(U_{k,n})\big{\|}>\delta\Big{)}<-a$
(5.44)
for any $K^{\prime}$ and $K$, with $K^{\prime}\geq K+K^{-1}>K\geq K_{1}$. By
virtue of the continuity of $F_{K,K^{\prime}}$ together with (5.44), one can
apply the contraction principle to the LDP for $(U_{k,n})_{n\geq 1}$ (see
Theorem 3.1), to obtain that
$\inf_{\rho\in
M_{+}(E^{\prime}),\,\|F_{K,K^{\prime}}(\rho)\|>\delta}\bar{\Lambda}_{k}^{*}(\rho)>a,$
for all $K^{\prime}\geq K+K^{-1}>K\geq K_{1}$.
Next, let us turn to (5.42) and fix $K\geq K_{1}$, so that $\sup_{\rho\in
M_{+}(E^{\prime}),\,\bar{\Lambda}_{k}^{*}(\rho)\leq
a}\big{\|}F(\rho)-F_{K}(\rho)\big{\|}>\delta$. Then, we can choose $\nu\in
M_{+}(E^{\prime})$ with $\bar{\Lambda}_{k}^{*}(\nu)\leq a$ and
$\big{\|}F(\nu)-F_{K}(\nu)\big{\|}>\delta$. Furthermore, (5.43) implies that
there exists $K^{\prime}\geq K+K^{-1}$ with
$\big{\|}F_{K,K^{\prime}}(\nu)\big{\|}>\delta$. Since this is contradiction,
we have established Lemma 5.8 $(ii)$, as desired. ∎
Finally, we prove Corollary 3.4. The proof is very similar to that of
Corollary 2.3. Hence, we focus only on some of its key differences. The goal
is to show that for every $\varepsilon_{0}>0$,
$\lim_{n\to\infty}\rho_{k,n}^{-1}\log\mathbb{P}\big{(}\|T_{k,n}-T_{k,n}^{\mathsf{B}}\|\geq\varepsilon_{0}\rho_{k,n}\big{)}=-\infty.$
###### Proof of Corollary 3.4.
As in the proof of Lemma 5.8 $(i)$, we may put the assumption $m=1$ and use
the notation $G_{n}$ and $H_{n}$ (instead of $g_{n}^{(1)}$ and $h_{n}^{(1)}$).
For $K>0$, we decompose $T_{k,n}^{\mathsf{B}}=T_{\leq
K,k,n}^{\mathsf{B}}+T_{>K,k,n}^{\mathsf{B}}$, where
$T_{\leq K,k,n}^{\mathsf{B}}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=k}G_{n}({\mathcal{Y}},\mathcal{B}_{n};t){\mathbbm{1}}\\{H_{n}({\mathcal{Y}})\leq
K\\},$
and
$T_{>K,k,n}^{\mathsf{B}}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=k}G_{n}({\mathcal{Y}},\mathcal{B}_{n};t){\mathbbm{1}}\\{H_{n}({\mathcal{Y}})>K\\}.$
Similarly, we write $T_{k,n}=T_{\leq K,k,n}+T_{>K,k,n}$. Since all the
summands in $T_{\leq K,k,n}$ and $T_{\leq K,k,n}^{\mathsf{B}}$ are bounded by
$K$, a simple repetition of the proof of Corollary 2.3 can yield that
$\frac{1}{\rho_{k,n}}\log\mathbb{P}\big{(}\,|\,T_{\leq K,k,n}-T_{\leq
K,k,n}^{\mathsf{B}}\,|\geq\varepsilon_{0}\rho_{k,n}\big{)}\to-\infty,\ \ \
n\to\infty,$
for every $K>0$. Additionally, the proof of Lemma 5.8 $(i)$ has already shown
that
$\limsup_{K\to\infty}\limsup_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}(T_{>K,k,n}\geq\varepsilon_{0}\rho_{k,n})=-\infty;$
so it remains to verify that
$\limsup_{K\to\infty}\limsup_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\big{(}T_{>K,k,n}^{\mathsf{B}}\geq\varepsilon_{0}\rho_{k,n}\big{)}=-\infty.$
(5.45)
Using the same diluted cubes from the proof of Corollary 2.3, (5.45) will
follow, provided that
$\displaystyle\limsup_{K\to\infty}\limsup_{n\to\infty}\frac{1}{\rho_{k,n}}\log\mathbb{P}\Big{(}\sum_{i=1}^{b_{n}^{\prime}}\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=k}G_{n}({\mathcal{Y}},\mathcal{B}_{n};t)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times{\mathbbm{1}}\big{\\{}H_{n}({\mathcal{Y}})>K,\,\ell({\mathcal{Y}})\in
J_{i}\big{\\}}\geq\varepsilon_{0}\rho_{k,n}\Big{)}=-\infty.$
Observe again that there exists a constant $M>0$ such that
$\displaystyle\begin{split}&\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=k}G_{n}({\mathcal{Y}},\mathcal{B}_{n};t){\mathbbm{1}}\\{H_{n}({\mathcal{Y}})>K,\,\ell({\mathcal{Y}})\in
J_{i}\\}\\\ &\quad\leq
M\max_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=k}\Big{(}H_{n}({\mathcal{Y}}){\mathbbm{1}}\\{H_{n}({\mathcal{Y}})>K,\,\ell({\mathcal{Y}})\in
J_{i}\\}\Big{)}.\end{split}$ (5.46)
Moreover, as in the proof of Corollary 2.3, we may work under the event
$\big{\\{}\mathcal{B}_{n}\subset\mathcal{P}_{n}^{(1,\mathsf{a})}\big{\\}}$,
where
$\mathcal{P}_{n}^{(1,\mathsf{a})}=\mathcal{P}_{n}\cup\mathcal{P}_{n}^{(1)}\stackrel{{\scriptstyle
d}}{{=}}\mathcal{P}_{2n}$. By Markov’s inequality and (5.46), we have, for
every $a>0$,
$\displaystyle\mathbb{P}\Big{(}\Big{\\{}\sum_{i=1}^{b_{n}^{\prime}}\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=k}G_{n}({\mathcal{Y}},\mathcal{B}_{n};t){\mathbbm{1}}\\{H_{n}({\mathcal{Y}})>K,\,\ell({\mathcal{Y}})\in
J_{i}\\}\geq\varepsilon_{0}\rho_{k,n}\Big{\\}}\cap\big{\\{}\mathcal{B}_{n}\subset\mathcal{P}_{n}^{(1,\mathsf{a})}\big{\\}}\Big{)}$
$\displaystyle\quad\leq\mathbb{P}\Big{(}\sum_{i=1}^{b_{n}^{\prime}}M\max_{{\mathcal{Y}}\subset\mathcal{P}_{n}^{(1,\mathsf{a})},\,|{\mathcal{Y}}|=k}\Big{(}H_{n}({\mathcal{Y}}){\mathbbm{1}}\\{H_{n}({\mathcal{Y}})>K,\,\ell({\mathcal{Y}})\in
J_{i}\\}\Big{)}\geq\varepsilon_{0}\rho_{k,n}\Big{)}$ $\displaystyle\quad\leq
e^{-a\varepsilon_{0}\rho_{k,n}}\big{(}\mathbb{E}\big{[}e^{aZ_{n}}\big{]}\big{)}^{b_{n}^{\prime}},$
where
$Z_{n}\mathrel{\mathop{\mathchar
58\relax}}=M\max_{{\mathcal{Y}}\subset\mathcal{P}_{2n},\,|{\mathcal{Y}}|=k}\Big{(}H_{n}({\mathcal{Y}}){\mathbbm{1}}\\{H_{n}({\mathcal{Y}})>K,\,\ell({\mathcal{Y}})\in
J_{1}\\}\Big{)}.$
Now, proceeding as in (5.41),
$\displaystyle\mathbb{E}\big{[}e^{aZ_{n}}\big{]}$ $\displaystyle\leq
1+\mathbb{E}\Big{[}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{2n},\,|{\mathcal{Y}}|=k}e^{aMH_{n}({\mathcal{Y}})}{\mathbbm{1}}\big{\\{}H_{n}({\mathcal{Y}})>K,\,\ell({\mathcal{Y}})\in
J_{1}\big{\\}}\Big{]}$ $\displaystyle\leq
1+C^{*}\rho_{k,n}r_{n}^{d}\int_{({\mathbb{R}}^{d})^{k-1}}e^{aMH({\bf
0}_{d},{\bf y})}{\mathbbm{1}}\big{\\{}H({\bf 0}_{d},{\bf
y})>K\big{\\}}\operatorname{d\\!}{\bf y}$
$\displaystyle\leq\exp\Big{\\{}C^{*}\rho_{k,n}r_{n}^{d}\int_{({\mathbb{R}}^{d})^{k-1}}e^{aMH({\bf
0}_{d},{\bf y})}{\mathbbm{1}}\big{\\{}H({\bf 0}_{d},{\bf
y})>K\big{\\}}\operatorname{d\\!}{\bf y}\Big{\\}}.$
This, together with property (H4), implies that
$\displaystyle\limsup_{K\to\infty}\limsup_{n\to\infty}\frac{1}{\rho_{k,n}}\log\Big{\\{}e^{-a\varepsilon_{0}\rho_{k,n}}\Big{(}\mathbb{E}\big{[}e^{aZ_{n}}\big{]}\Big{)}^{b_{n}^{\prime}}\Big{\\}}$
$\displaystyle\quad\leq-a\varepsilon_{0}+C^{*}\limsup_{K\to\infty}\int_{({\mathbb{R}}^{d})^{k-1}}e^{aMH({\bf
0}_{d},{\bf y})}{\mathbbm{1}}\big{\\{}H({\bf 0}_{d},{\bf
y})>K\big{\\}}\operatorname{d\\!}{\bf y}=-a\varepsilon_{0}.$
Since $a$ is arbitrary, letting $a\to\infty$ concludes the entire proof. ∎
### 5.4 Proofs of Theorem 4.1 and Theorem 4.3
###### Proof of Theorem 4.1 (Poisson input).
We first deal with the case of a Poisson input. Since the function
$\big{(}h_{s_{i}}(x_{1},x_{2},x_{3})h_{t_{i}}(x_{1},x_{2},x_{3})\big{)}_{i=1}^{m}$
defined at (4.2) satisfies conditions (H1)–(H5), a direct application of
Theorem 3.3 yields that as $n\to\infty$,
$\frac{1}{\rho_{3,n}}\,\log\mathbb{P}\bigg{(}\Big{(}\rho_{3,n}^{-1}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=3}g_{r_{n}s_{i},r_{n}t_{i}}({\mathcal{Y}},\mathcal{P}_{n}),\,i=1,\dots,m\Big{)}\in
A\bigg{)}\to-\inf_{{\bf x}\in A}I_{3}({\bf x}).$
To complete the proof we show that for every $0\leq s\leq t<\infty$ and
$\delta>0$,
$\lim_{n\to\infty}\frac{1}{\rho_{3,n}}\,\log\mathbb{P}\bigg{(}\beta_{1,n}(s,t)-\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=3}g_{r_{n}s,r_{n}t}({\mathcal{Y}},\mathcal{P}_{n})\geq\delta\rho_{3,n}\bigg{)}=-\infty.$
By definition, $\beta_{1,n}(s,t)$ represents the number of persistent
$1$-cycles in the region $[0,s]\times[t,\infty]$ of the first-order
persistence diagram. In particular, $\beta_{1,n}(s,t)$ accounts for subsets of
$3$ points in ${\mathbb{R}}^{2}$ that form a single $1$-cycle before time
$r_{n}s$, such that this 1-cycle remains alive at time $r_{n}t$ and isolated
from all the remaining points in $\mathcal{P}_{n}$ at that time. Note that
these subsets of $3$ points in ${\mathbb{R}}^{2}$ are counted by
$\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=3}g_{r_{n}s,r_{n}t}({\mathcal{Y}},\mathcal{P}_{n})$
as well. Thus, all the remaining points in $[0,s]\times[t,\infty]$ of the
first-order persistence diagram are associated to the 1-cycles on connected
components of size greater than or equal to $4$ at time $r_{n}t$. From this
point of view,
$\beta_{1,n}(s,t)-\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=3}g_{r_{n}s,r_{n}t}({\mathcal{Y}},\mathcal{P}_{n})$
can be bounded by the first-order Betti number at time $r_{n}t$, associated
only to connected components of size greater than or equal to $4$. Moreover,
this Betti number is further bounded by the corresponding $1$-simplex counts
(i.e., edge counts). By Lemma 2.1 in [15], one can bound such $1$-simplex
counts by three times the number of vertices that are contained in connected
components of size greater than or equal to $4$. In conclusion, we have that
$\beta_{1,n}(s,t)-\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=3}g_{r_{n}s,r_{n}t}({\mathcal{Y}},\mathcal{P}_{n})\leq
3\sum_{i=4}^{\infty}i\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}({\mathcal{Y}},\mathcal{P}_{n}),$
(5.47)
where
$v_{n}^{(i)}({\mathcal{Y}},\mathcal{P}_{n})={\mathbbm{1}}\big{\\{}\alpha({\mathcal{Y}},r_{n}t)\text{
is a connected component of }\alpha(\mathcal{P}_{n},r_{n}t)\big{\\}}.$
As in the proof of Proposition 5.1, we partition $[0,1]^{2}$ into sub-cubes
$Q_{1},\dots,Q_{\rho_{3,n}}$ of volume $\rho_{3,n}^{-1}$ so that
$Q_{1}=[0,\rho_{3,n}^{-1/2}]^{2}$. We claim that
$\sum_{i=4}^{\infty}i\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}({\mathcal{Y}},\mathcal{P}_{n})\leq\sum_{{\bf
z}\in\\{0,\pm
1\\}^{2}}\sum_{\ell=1}^{\rho_{3,n}}\sum_{i=4}^{\infty}i\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{(Q_{\ell}+3r_{n}t{\bf
z})\cap[0,1]^{2}}}v_{n}^{(i)}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{(Q_{\ell}+3r_{n}t{\bf
z})\cap[0,1]^{2}}\big{)}.$ (5.48)
To show (5.48), suppose there exists a connected component ${\mathcal{Y}}$ of
cardinality at least $4$, whose points are counted by the left hand side of
(5.48). If ${\mathcal{Y}}$ is contained in one of the cubes in
$(Q_{\ell})_{\ell=1}^{\rho_{3,n}}$, all the points in ${\mathcal{Y}}$ can also
be counted by the statistics on the right hand side of (5.48) with ${\bf
z}=(0,0)$. If we observe a subset $\mathcal{Z}\subset{\mathcal{Y}}\cap
Q_{\ell}$ for some $\ell$ with $|\mathcal{Z}|\leq 3$, such that $\mathcal{Z}$
itself forms a connected component within $Q_{\ell}$, with respect to the
process $\mathcal{P}_{n}|_{Q_{\ell}}$, then the points in $\mathcal{Z}$ will
be missed from the above statistics with ${\bf z}=(0,0)$. Even in that case,
however, all the points in $\mathcal{Z}$ can eventually be counted by the
statistics in (5.48) with other choice of ${\bf z}\in\\{0,\pm
1\\}^{2}\setminus(0,0)$. Since there are at most finitely many choices of
${\bf z}$, the entire proof will be complete, provided that for any
$\delta>0$,
$\lim_{n\to\infty}\frac{1}{\rho_{3,n}}\,\log\mathbb{P}\bigg{(}\sum_{\ell=1}^{\rho_{3,n}}\sum_{i=4}^{\infty}i\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}}\big{)}\geq\delta\rho_{3,n}\bigg{)}=-\infty.$
(5.49)
For the proof of (5.49) we apply Markov’s inequality to obtain that
$\displaystyle\frac{1}{\rho_{3,n}}\,\log\mathbb{P}\bigg{(}\sum_{\ell=1}^{\rho_{3,n}}\sum_{i=4}^{\infty}i\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{\ell}},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q_{\ell}}\big{)}\geq\delta\rho_{3,n}\bigg{)}$
$\displaystyle\leq-a\delta+\log\mathbb{E}\bigg{[}\exp\Big{\\{}a\sum_{i=4}^{\infty}i\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}}\big{)}\Big{\\}}\bigg{]}.$
From this, it suffices to show that for every $a>0$,
$\mathbb{E}\bigg{[}\exp\Big{\\{}a\sum_{i=4}^{\infty}i\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}}\big{)}\Big{\\}}\bigg{]}\to
1,\ \ \text{as }n\to\infty.$ (5.50)
To show this, we first claim that
$\sum_{i=4}^{\infty}i\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}}\big{)}\stackrel{{\scriptstyle\mathbb{P}}}{{\to}}0,\
\ \ n\to\infty.$ (5.51)
Taking an expectation of (5.51),
$\displaystyle\sum_{i=4}^{\infty}i\,\mathbb{E}\Big{[}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}}\big{)}\Big{]}\leq\sum_{i=4}^{\infty}i\,\mathbb{E}\Big{[}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}},\,|{\mathcal{Y}}|=i}u_{n}^{(i)}({\mathcal{Y}})\Big{]},$
where $u_{n}^{(i)}({\mathcal{Y}})\mathrel{\mathop{\mathchar
58\relax}}={\mathbbm{1}}\big{\\{}\alpha({\mathcal{Y}},r_{n}t)\text{ is
connected}\big{\\}}.$ By the Mecke formula for Poisson point processes, along
with the customary change of variables, the right hand side of the above is
equal to
$\displaystyle\sum_{i=4}^{\infty}i\,\frac{n^{i}}{i!}\,\int_{(Q_{1})^{i}}u_{n}^{(i)}(x_{1},\dots,x_{i})\operatorname{d\\!}{\bf
x}$
$\displaystyle=\sum_{i=4}^{\infty}\frac{n^{i}r_{n}^{2(i-1)}}{(i-1)!}\int_{Q_{1}}\int_{({\mathbb{R}}^{2})^{i-1}}{\mathbbm{1}}\big{\\{}\alpha\big{(}\\{{\bf
0}_{2},y_{1},\dots,y_{i-1}\\},t\big{)}\text{ is
connected}\big{\\}}\prod_{\ell=1}^{i-1}{\mathbbm{1}}\\{x+r_{n}y_{\ell}\in
Q_{1}\\}\operatorname{d\\!}{\bf y}\operatorname{d\\!}x$
$\displaystyle\leq\sum_{i=4}^{\infty}\frac{(nr_{n}^{2})^{i-3}}{(i-1)!}\,\int_{({\mathbb{R}}^{2})^{i-1}}{\mathbbm{1}}\Big{\\{}\alpha\big{(}\\{{\bf
0}_{2},y_{1},\dots,y_{i-1}\\},t\big{)}\text{ is
connected}\Big{\\}}\operatorname{d\\!}{\bf y}.$
Because of an elementary fact that there exist at most $i^{i-2}$ spanning
trees on $i$ vertices,
$\int_{({\mathbb{R}}^{2})^{i-1}}{\mathbbm{1}}\Big{\\{}\alpha\big{(}\\{{\bf
0}_{2},y_{1},\dots,y_{i-1}\\},t\big{)}\text{ is
connected}\Big{\\}}\operatorname{d\\!}{\bf y}\leq i^{i-2}(t^{2}\pi)^{i-1}.$
Referring this bound back into the above and appealing to Stirling’s formula,
i.e., $(i-1)!\geq\big{(}(i-1)/e\big{)}^{i-1}$ for large $i$, we conclude that
as $n\to\infty$,
$\displaystyle\sum_{i=4}^{\infty}i\,\mathbb{E}\Big{[}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}},\,|{\mathcal{Y}}|=i}u_{n}^{(i)}({\mathcal{Y}})\Big{]}$
$\displaystyle\leq\sum_{i=4}^{\infty}\frac{(nr_{n}^{2})^{i-3}}{(i-1)!}\,i^{i-2}(t^{2}\pi)^{i-1}\leq
C^{*}\sum_{i=4}^{\infty}\big{(}et^{2}\pi nr_{n}^{2}\big{)}^{i-3}\to 0.$
To conclude (5.50) from (5.51), one needs to show the uniform integrability:
for every $a>0$,
$\limsup_{n\to\infty}\mathbb{E}\bigg{[}\exp\Big{\\{}a\sum_{i=4}^{\infty}i\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}}\big{)}\Big{\\}}\bigg{]}<\infty.$
(5.52)
For the proof, we consider the family $G$ of diluted cubes of side length
$8r_{n}$ that are contained in $Q_{1}=\big{[}0,\rho_{3,n}^{-1/2}\big{]}^{2}$.
Then, the total number of such cubes in $G$ is
$c_{n}\mathrel{\mathop{\mathchar 58\relax}}=\rho_{3,n}^{-1}/(8r_{n})^{2}$,
which is assumed without loss of generality to be integer-valued for every
$n$. Observe now that for any connected component of size greater than or
equal to $4$, there exist ${\bf z}=(z_{1},z_{2})\in\big{\\{}0,\pm 1,\dots,\pm
4\big{\\}}^{2}$ and $J\in G$ such that $J+r_{n}{\bf z}\subset Q_{1}$ and
$\big{|}\mathcal{P}_{n}\cap(J+r_{n}{\bf z})\big{|}\geq 4$. In conclusion, we
have
$\displaystyle\sum_{i=4}^{\infty}i\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}\big{(}{\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}}\big{)}$
$\displaystyle\leq\sum_{{\bf z}\in\\{0,\pm 1,\dots,\pm 4\\}^{2}}\sum_{J\in
G,\,J+r_{n}{\bf z}\subset Q_{1}}\big{|}\mathcal{P}_{n}\cap(J+r_{n}{\bf
z})\big{|}\,{\mathbbm{1}}\Big{\\{}\big{|}\mathcal{P}_{n}\cap(J+r_{n}{\bf
z})\big{|}\geq 4\Big{\\}}.$
Now, according to Hölder’s inequality as well as the homogeneity of
$\mathcal{P}_{n}$, (5.52) follows from
$\limsup_{n\to\infty}\mathbb{E}\Big{[}e^{a\sum_{J\in G}|\mathcal{P}_{n}\cap
J|\,{\mathbbm{1}}\\{|\mathcal{P}_{n}\cap J|\geq 4\\}}\Big{]}<\infty,$
for every $a>0$. Writing $G=\\{J_{1},\dots,J_{c_{n}}\\}$ with
$J_{1}=[0,8r_{n}]^{2}$, we have that
$\mathbb{E}\Big{[}e^{a\sum_{J\in G}|\mathcal{P}_{n}\cap
J|\,{\mathbbm{1}}\\{|\mathcal{P}_{n}\cap J|\geq
4\\}}\Big{]}=\bigg{\\{}\Big{(}\mathbb{E}\Big{[}e^{a\,|\mathcal{P}_{n}\cap
J_{1}|\,{\mathbbm{1}}\\{|\mathcal{P}_{n}\cap J_{1}|\geq
4\\}}\Big{]}\Big{)}^{1/(\rho_{3,n}r_{n}^{2})}\bigg{\\}}^{1/64}$
It is elementary to calculate that
$\mathbb{E}\Big{[}e^{a\,|\mathcal{P}_{n}\cap
J_{1}|\,{\mathbbm{1}}\\{|\mathcal{P}_{n}\cap J_{1}|\geq 4\\}}\Big{]}\leq
1+\sum_{\ell=4}^{\infty}e^{a\ell}\mathbb{P}\big{(}|\mathcal{P}_{n}\cap
J_{1}|=\ell\big{)}\leq 1+C^{*}(nr_{n}^{2})^{4}.$
Since $\rho_{3,n}r_{n}^{2}=(nr_{n}^{2})^{3}$, one can obtain that
$\displaystyle\limsup_{n\to\infty}\Big{(}\mathbb{E}\Big{[}e^{t\,|\mathcal{P}_{n}\cap
J_{1}|\,{\mathbbm{1}}\\{|\mathcal{P}_{n}\cap J_{1}|\geq
4\\}}\Big{]}\Big{)}^{1/(\rho_{3,n}r_{n}^{2})}\leq\limsup_{n\to\infty}\big{(}1+C^{*}(nr_{n}^{2})^{4}\big{)}^{1/(nr_{n}^{2})^{3}}=1.$
∎
###### Proof of Theorem 4.1 (binomial input).
In the case of a binomial input, instead of (5.47), we deduce that
$\beta_{1,n}^{\mathsf{B}}(s,t)-\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=3}g_{r_{n}s,r_{n}t}({\mathcal{Y}},\mathcal{B}_{n})\leq
3\sum_{i=4}^{n}i\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}({\mathcal{Y}},\mathcal{B}_{n}).$
As in the proofs of Corollaries 2.3 and 3.4, we may work under the event
$\\{\mathcal{B}_{n}\subset\mathcal{P}_{n}^{(1,\mathsf{a})}\\}$, where
$\mathcal{P}_{n}^{(1,\mathsf{a})}=\mathcal{P}_{n}\cup\mathcal{P}_{n}^{(1)}\stackrel{{\scriptstyle
d}}{{=}}\mathcal{P}_{2n}$. Then, for every $\delta>0$,
$\displaystyle\begin{split}&\frac{1}{\rho_{3,n}}\log\mathbb{P}\bigg{(}\Big{\\{}\sum_{i=4}^{n}i\sum_{{\mathcal{Y}}\subset\mathcal{B}_{n},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}({\mathcal{Y}},\mathcal{B}_{n})\geq\delta\rho_{3,n}\Big{\\}}\cap\\{\mathcal{B}_{n}\subset\mathcal{P}_{n}^{(1,\mathsf{a})}\\}\bigg{)}\\\
&\leq\frac{1}{\rho_{3,n}}\log\mathbb{P}\Big{(}\sum_{i=4}^{\infty}i\sum_{{\mathcal{Y}}\subset\mathcal{P}_{2n},\,|{\mathcal{Y}}|=i}v_{n}^{(i)}({\mathcal{Y}},\mathcal{P}_{2n})\geq\delta\rho_{3,n}\Big{)}.\end{split}$
(5.53)
Here, we used that the double sum counts the number of vertices in connected
components of size at least 4. Moreover, adding further points in
$\mathcal{P}_{n}^{(1,\mathsf{a})}\setminus\mathcal{B}_{n}$ increases the
number of points in the components associated to $\mathcal{B}_{n}$. This is
clear for the Čech complex and follows by the nerve lemma for the alpha
complex. Repeating the same argument as in the Poisson input case, one can
show that the right-hand side in (LABEL:e:binomial.betti.to.Poisson) goes to
$-\infty$ as $n\to\infty$. ∎
###### Proof of Theorem 4.3.
We start by formulating the function $H\mathrel{\mathop{\mathchar
58\relax}}=(h^{(1)},\dots,h^{(m)})\mathrel{\mathop{\mathchar
58\relax}}({\mathbb{R}}^{2})^{3}\to[0,\infty)^{m}$ in the current setup:
$h^{(i)}(x_{1},x_{2},x_{3})\mathrel{\mathop{\mathchar
58\relax}}={\mathbbm{1}}\big{\\{}\gamma(x_{1},x_{2},x_{3})\in\text{conv}^{\circ}(x_{1},x_{2},x_{3}),\,\mathcal{R}(x_{1},x_{2},x_{3})\leq
t_{i}\big{\\}}.$
Clearly, $H$ satisfies conditions (H1)–(H5). In particular, fix a constant $L$
determined by property (H3). Define the scaled version $H_{n}$ of $H$ as in
(3.1). For a $3$-point subset ${\mathcal{Y}}\subset{\mathbb{R}}^{2}$ and a
finite subset $\mathcal{Z}\supset{\mathcal{Y}}$ in ${\mathbb{R}}^{2}$, define
$c({\mathcal{Y}},\mathcal{Z})\mathrel{\mathop{\mathchar
58\relax}}=\big{(}{\mathbbm{1}}\big{\\{}\mathcal{U}({\mathcal{Y}})\cap\mathcal{Z}=\emptyset\big{\\}}\big{)}_{i=1}^{m}.$
(5.54)
In contrast to (3.2), the function (5.54) does not involve a time parameter
${\bf t}=(t_{1},\dots,t_{m})$. Furthermore, unlike (3.3),
$c_{n}({\mathcal{Y}},\mathcal{Z})\mathrel{\mathop{\mathchar
58\relax}}=c(r_{n}^{-1}{\mathcal{Y}};r_{n}^{-1}\mathcal{Z})=c({\mathcal{Y}},\mathcal{Z})$
does not depend on $n\geq 1$. Finally we define
$s_{n}({\mathcal{Y}},\mathcal{Z})\mathrel{\mathop{\mathchar
58\relax}}=c_{n}({\mathcal{Y}},\mathcal{Z})\,{\mathbbm{1}}\big{\\{}\mathsf{diam}({\mathcal{Y}})\leq
r_{n}L\big{\\}}$ as in (2.3).
The required large deviations in Theorem 4.3 can be deduced by an application
of Theorem 3.3 to the number of Morse critical points in (4.4). Before doing
so, however, one must properly modify the proof of Theorem 2.1 under the setup
of Theorem 4.3. After that, one must also modify the proofs of Theorems 3.1
and 3.3; however, the required modification for these two theorems seems to be
sufficiently simple, so we focus our attention only to Theorem 2.1. In the
below, we discuss two specific points. First, one has to modify the
calculation in (5.11) as follows: as $n\to\infty$,
$\displaystyle\frac{n^{3}}{6}\int_{(Q_{\ell})^{3}}{\mathbbm{1}}\big{\\{}\mathsf{diam}(x_{1},x_{2},x_{3})\leq
r_{n}L\big{\\}}\Big{(}1-e^{-n\text{vol}\big{(}\mathcal{U}(x_{1},x_{2},x_{3})\cap
Q_{\ell}\big{)}}\Big{)}\operatorname{d\\!}{\bf x}$
$\displaystyle=\frac{\rho_{3,n}}{6}\int_{Q_{\ell}}\int_{({\mathbb{R}}^{2})^{2}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{2},{\bf y})\leq L\big{\\}}\prod_{i=1}^{2}{\mathbbm{1}}\\{x+r_{n}y_{i}\in
Q_{\ell}\\}$
$\displaystyle\qquad\qquad\qquad\qquad\times\Big{(}1-e^{-n\text{vol}\big{(}\mathcal{U}(x,x+r_{n}y_{1},x+r_{n}y_{2})\cap
Q_{\ell}\big{)}}\Big{)}\operatorname{d\\!}{\bf y}\operatorname{d\\!}x$
$\displaystyle\leq\frac{1}{6}\int_{({\mathbb{R}}^{2})^{2}}{\mathbbm{1}}\big{\\{}\mathsf{diam}({\bf
0}_{2},{\bf y})\leq L\big{\\}}\Big{(}1-e^{-nr_{n}^{2}\mathcal{R}({\bf
0}_{2},{\bf y})^{2}\pi}\Big{)}\operatorname{d\\!}{\bf y}\to 0.$
Second, we also need to modify the proof of (5.16). Specifically, we need to
show that
$\limsup_{n\to\infty}\mathbb{E}\Big{[}e^{a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}})}\Big{]}<\infty.$
For this purpose, we again consider the diluted families of cubes
$G_{1},G_{2},\dots$ as in (5.17) (with $t\equiv 1$). Then, as an analog of
(5.19), our task is reduced to showing that for every $a>0$,
$\limsup_{n\to\infty}\bigg{(}\mathbb{E}\Big{[}e^{a\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n}|_{Q_{1}}}s_{n}({\mathcal{Y}},\mathcal{P}_{n}|_{Q_{1}}){\mathbbm{1}}\\{\ell({\mathcal{Y}})\in
J_{1}\\}}\Big{]}\bigg{)}^{b_{n}}<\infty,$ (5.55)
where $J_{1}=[0,r_{n}/\sqrt{2}]^{2}$ and
|
Tractable and Intractable Entailment Problems in Separation Logic
# Tractable and Intractable Entailment Problems in Separation Logic with
Inductively Defined Predicates
Mnacho Echenim This work has been partially funded by the the French National
Research Agency (ANR-21-CE48-0011). Nicolas Peltier1
Univ. Grenoble Alpes CNRS Grenoble INP LIG 38000 Grenoble France
###### Abstract
We establish various complexity results for the entailment problem between
formulas in Separation Logic (SL) with user-defined predicates denoting
recursive data structures. The considered fragments are characterized by
syntactic conditions on the inductive rules that define the semantics of the
predicates. We focus on so-called $\mathtt{P}$-rules, which are similar to
(but simpler than) the so-called bounded treewidth fragment of SL studied by
Iosif et al. [14]. In particular, for a specific fragment where predicates are
defined by so-called _$\mathtt{loc}$ -deterministic_ inductive rules, we
devise a sound and complete cyclic proof procedure running in polynomial time.
Several complexity lower bounds are provided, showing that any relaxing of the
provided conditions makes the problem intractable.
###### keywords:
Separation logic, Inductive reasoning, Decision procedures, Cyclic proofs
††terms: def=††terms: _
#### ACM Computing Classification System:
$\bullet$ Theory of computation, Logic, Automated reasoning $\bullet$ Theory
of computation, Logic, Separation logic
## 1 Introduction
Separation Logic [16, 22] (SL) is widely used in verification to reason on
programs manipulating pointer-based data structures. It forms the basis of
several automated static program analyzers such as Smallfoot [1], Infer [3]
(Facebook) or SLAyer [2] (Microsoft Research) and several correctness proofs
were carried out by embedding SL in interactive theorem provers such as Coq
[23], see for instance [17]. SL uses a special connective $*$, called
separating conjunction, modeling heap compositions and allowing for concise
and natural specifications. More precisely, atoms in SL are expressions of the
forms $x\mapsto(y_{1},\dots,y_{n})$, where $x,y_{1},\dots,y_{n}$ are variables
denoting locations (i.e., memory addresses), asserting that location $x$ is
allocated and refers to the tuple (record) $(y_{1},\dots,y_{n})$. The special
connective $\phi*\psi$ asserts that formulas $\phi$ and $\psi$ hold on
disjoint parts of the memory. Recursive data structures may then be described
by considering predicates associated with inductive rules, such as:
$\begin{array}[]{llllll}{\mathtt{ls}}(x,y)&\Leftarrow&x\mapsto
y&{\mathtt{tree}}(x)&\Leftarrow&x\mapsto()\\\
{\mathtt{ls}}(x,y)&\Leftarrow&x\mapsto
z*{\mathtt{ls}}(z,y)&{\mathtt{tree}}(x)&\Leftarrow&x\mapsto(y,z)*{\mathtt{tree}}(y)*{\mathtt{tree}}(z)\end{array}$
where ${\mathtt{ls}}(x,y)$ denotes a nonempty list segment and
${\mathtt{tree}}(x)$ denotes a tree. For the sake of genericity, such rules
are not built-in but may be provided by the user. Due to the expressive power
of such inductive definitions, the input language is usually restricted in
this context to so-called symbolic heaps, i.e., existentially quantified
conjunctions and separating conjunctions of atoms (dismissing for instance
universal quantifications, negations and separating implications). Many
problems in verification require to solve entailment problems between such SL
formulas, for instance when these formulas denote pre- or post-conditions of
programs. Unfortunately, the entailment problem between symbolic heaps is
undecidable in general [20], but it is decidable if the considered inductive
rules satisfy the so-called PCE conditions (standing for Progress,
Connectivity and Establishment) [14]. However even for the PCE fragment the
complexity of the entailment problem is still very high; more precisely, this
problem is $2$-ExpTime-complete [9, 10, 18]. Less expressive fragments have
thus been considered, for which more efficient algorithms were developed. In
[15] a strict subclass of PCE entailments is identified with an ExpTime
complexity based on a reduction to the language inclusion problem for tree
automata [7]. In [12], an algorithm is developed to handle various kinds of
(possibly nested) singly linked lists based on a reduction to the membership
problem for tree automata. The complexity of the procedure is dominated by the
boolean satisfiability and unsatisfiability tests, that are NP and co-NP
complete, respectively. A polynomial proof procedure has been devised for the
specific case of singly linked lists [8]. In [5], the tractability result is
extended to more expressive fragments, with formulas defined on some unique
nonlinear compositional inductive predicate with distinguished source,
destination, and static parameters. The compositional properties satisfied by
the considered predicate (as originally introduced in [13]) ensure that the
entailment problem can be solved efficiently. Recently [19] introduced a
polynomial-time cyclic proof to solve entailment problem efficiently, under
some condition on the inductive rules.
In the present paper, we study the complexity of the entailment problem for a
specific fragment that is similar to the PCE fragment, but simpler. The
fragment inherits most of the conditions given in [14] and admits an
additional restriction that is meant to ensure that entailment problems can be
solved in a more efficient way111At the cost, of course, of a loss of
expressivity.: every predicate is bound to allocate exactly one of its
parameters (forbidding for instance predicates denoting doubly linked list
segments from $x$ to $y$, as both $x$ and $y$ would be allocated). This means
that the rules do not allow for multiple pointers into a data structure
(whereas multiple pointers out of the structure are allowed). We first show
that this additional restriction is actually not sufficient to ensure
tractability. More precisely, we establish several lower-bound complexity
results for the entailment problem under various additional hypotheses.
Second, we define a new class of inductive definitions for which the
entailment problem can be solved in polynomial time, based mainly on the two
following additional restrictions: (i) the arity of the predicates is bounded;
and (ii) the rules defining the same predicate do not overlap, in a sense that
will be formally defined below. Both conditions are rather natural
restrictions in the context of programming. Indeed, the number of parameters
is usually small in this context. Also, data structures are typically defined
using a finite set of free constructors, which yields inductive definitions
that are trivially non-overlapping.
If Condition $(i)$ is not satisfied, then the complexity is simply
exponential. In contrast with other polynomial-time algorithms, the formulas
we consider may contain several inductive predicates, and these predicates are
possibly non-compositional (in the sense of [12]). The algorithm for testing
entailment is defined as a sequent-like cyclic proof procedure, with standard
unfolding and decomposition rules, together with a specific strategy ensuring
efficiency and additional syntactic criteria to detect and dismiss non-
provable sequents. Our approach is close to that of [19], in the sense that
the two procedures use cyclic proof procedures with non-disjunctive
consequents, however the conditions on the rules are completely different: our
definition allows for multiple inductive rules with mutually recursive
definitions, yielding richer recursive data structures. On the other hand, the
SHLIDe rules in [19] support ordering and equality relations on non-
addressable values, whereas the predicate we consider are purely spatial.
Moreover, the base cases of the rules in [19] correspond to empty heaps, which
are forbidden in our approach.
To provide some intuition on what can and cannot be expressed in the fragment
we consider, we provide some examples (formal definitions will be given
later); consider the predicate $\mathtt{P}$ defined by the following rules,
which encode a combination of lists and trees, possibly looping on an initial
element $y$, and ending with an empty tuple:
$\mathtt{P}(x,y)$ | $\Leftarrow$ | $x\mapsto(\mathtt{list},u)*\mathtt{P}(u,y)$
---|---|---
$\mathtt{P}(x,y)$ | $\Leftarrow$ | $x\mapsto(\mathtt{tree},u_{1},u_{2})*\mathtt{P}(u_{1},y)*\mathtt{P}(u_{2},y)$
$\mathtt{P}(x,y)$ | $\Leftarrow$ | $x\mapsto(\mathtt{loop},y)$
$\mathtt{P}(x,y)$ | $\Leftarrow$ | $x\mapsto()$
The constants $\mathtt{list},\mathtt{tree}$ and $\mathtt{loop}$ may be viewed
as constructors for the data structure. This predicate does not fall in the
scope of the fragment considered in [19] since it involves a definition with
several inductive rules, but it falls in the scope of the fragment considered
in the present paper. Our restrictions require that the definition must be
deterministic, in the sense that there can be no overlap between distinct
rules. This is the case here, as the tuples $(\mathtt{list},u)$,
$(\mathtt{tree},u_{1},u_{2})$ and $(\mathtt{loop},y)$ are pairwise distinct
(not unifiable), but replacing for instance the constant $\mathtt{loop}$ by
$\mathtt{list}$ in the third rule would not be possible, as the resulting rule
would overlap with the first one (both rules could allocate the same heap
cell). As explained above, a key limitation of our fragment (compared to that
of [14]) is that it does not allow predicates allocating several parameters,
such as the following predicate ${\mathtt{dllseg}}(x,y,z,u)$ defining a doubly
linked list segment from $x$ to $z$ (each cell points to a pair containing the
previous and next element and $y$ and $u$ denote the previous and next element
in the list, respectively):
${\mathtt{dllseg}}(x,y,z,u)$ | $\Leftarrow$ | $(x\mapsto(y,x^{\prime})*{\mathtt{dllseg}}(x^{\prime},x,z,u))\wedge x\not\approx z$
---|---|---
${\mathtt{dllseg}}(x,y,z,u)$ | $\Leftarrow$ | $x\mapsto(y,u)\wedge x\approx z$
Other definitions of ${\mathtt{dllseg}}$ are possible, but none would fit in
with our restrictions: in every case, both $x$ (the beginning of the list) and
$z$ (its end) must be eventually allocated, which is not permitted in the
fragment we consider. On the other hand, the following predicate, defining a
doubly linked list, ending with $()$, can be defined ($y$ denotes the previous
element in the list):
${\mathtt{dll}}(x,y)$ | $\Leftarrow$ | $x\mapsto(y,z)*{\mathtt{dll}}(z,x)$
---|---|---
${\mathtt{dll}}(x,y)$ | $\Leftarrow$ | $x\mapsto()$
The rest of the paper is organised as follows. In Section 2, the syntax and
semantics of the logic are defined. The definitions are mostly standard,
although we consider a multisorted framework, with a special sort
$\mathtt{loc}$ denoting memory locations and additional sorts for data or
constructors. We then introduce a class of inductive definitions called
$\mathtt{P}$-rules. In Section 3, various lower bounds on the complexity of
the entailment problem for SL formulas with $\mathtt{P}$-rules are established
which allow one to motivate additional restrictions on the inductive rules.
These lower bounds show that all the restrictions are necessary to ensure that
the entailment problem is tractable. This leads to the definition of the
notion of a $\mathtt{loc}$-deterministic set of rules, that is a subset of
$\mathtt{P}$-rules for which entailment can be decided in polynomial time. The
proof procedure is defined in Section 4. For the sake of readability and
generality we first define generic inference rules and establish their
correctness, before introducing a specific strategy to further restrict the
application of the rules that is both complete and efficient. Section 5
contains all soundness, completeness and complexity results and Section 6
concludes the paper.
## 2 Definitions
### 2.1 Syntax
We use a multisorted framework, which is essentially useful to distinguish
locations from data. Let $\mathfrak{S}$ be a set of sorts, containing a
special sort $\mathtt{loc}$, denoting memory locations. Let
$\mathcal{V}_{{\mathtt{s}}\in\mathfrak{S}}$ be a family of countably infinite
disjoint sets of variables of sort ${\mathtt{s}}$, with
$\mathcal{V}\overset{\text{\tiny\it
def}}{=}\bigcup_{{\mathtt{s}}\in\mathfrak{S}}\mathcal{V}_{\mathtt{s}}$. Let
$\mathcal{C}_{{\mathtt{s}}\in\mathfrak{S}}$ be a family of disjoint sets of
constant symbols of sort ${\mathtt{s}}$, also disjoint from $\mathcal{V}$,
with $\mathcal{C}\overset{\text{\tiny\it
def}}{=}\bigcup_{{\mathtt{s}}\in\mathfrak{S}}\mathcal{C}_{\mathtt{s}}$. The
set of terms of sort ${\mathtt{s}}$ is ${\mathtt{s}}\overset{\text{\tiny\it
def}}{=}\mathcal{V}_{\mathtt{s}}\cup\mathcal{C}_{\mathtt{s}}$, and we let
$\bigcup_{{\mathtt{s}}\in\mathfrak{S}}{\mathtt{s}}$. Constants are especially
useful in our framework to denote constructors in data structures. To simplify
technicalities, we assume that there is no constant of sort $\mathtt{loc}$,
i.e., $\mathcal{C}_{\mathtt{loc}}=\emptyset$.
An equation (resp. a disequation) is an expression of the form $t\approx s$
(resp. $t\not\approx s$) where $t,s\in{\mathtt{s}}$ for some
${\mathtt{s}}\in\mathfrak{S}$. The set of pure formulas ${\cal F}_{P}$ is the
set of formulas of the form $e_{1}\wedge\dots\wedge e_{n}$, where every
expression $e_{i}$ is either an equation or a disequation. Such formulas are
considered modulo contraction, e.g., a pure formula $\xi\wedge\xi$ is
considered identical to $\xi$, and also modulo associativity and commutativity
of conjunction. We denote by $\bot$ (false) any formula of the form
$t\not\approx t$. If $n=0$, then $\bigwedge_{i=1}^{n}e_{i}$ may be denoted by
$\top$ (true). If $(t_{1},\dots,t_{n})$ and $(s_{1},\dots,s_{m})$ are vectors
of terms, then $(t_{1},\dots,t_{n})\approx(s_{1},\dots,s_{m})$ denotes the
formula $\bot$ if either $n\not=m$ or $n=m$ and there exists
$i\in\\{1,\dots,n\\}$ such that $s_{i}$ and $t_{i}$ are of different sorts;
and denotes $\bigwedge_{i=1}^{n}t_{i}\approx s_{i}$ otherwise.
Let $\mathcal{P}$ be a set of predicate symbols. Each symbol in $\mathcal{P}$
is associated with a unique profile of the form
$({\mathtt{s}}_{1},\dots,{\mathtt{s}}_{n})$ with $n\geq 1$,
${\mathtt{s}}_{1}=\mathtt{loc}$ and ${\mathtt{s}}_{i}\in\mathfrak{S}$, for all
$i\in\\{2,\dots,n\\}$. We write $p:{\mathtt{s}}_{1},\dots,{\mathtt{s}}_{n}$ to
denote a symbol with profile $({\mathtt{s}}_{1},\dots,{\mathtt{s}}_{n})$ and
we write $p:{\mathtt{s}}_{1},\dots,{\mathtt{s}}_{n}\in\mathcal{P}$ to state
that $p$ is a predicate symbol of profile
${\mathtt{s}}_{1},\dots,{\mathtt{s}}_{n}$ in $\mathcal{P}$. A spatial atom
$\alpha$ is either a points-to atom $x\mapsto(t_{1},\dots,t_{n})$ with
$x\in\mathcal{V}_{\mathtt{loc}}$ and $t_{1},\dots,t_{n}\in,ora{\em
predicateatom}oftheform$p(x,t_1,…,t_n)$,where$p$isapredicateofprofile$loc,s_1,…,s_n$in$P$,theterm$x$isavariablein$V_loc$and$t_i
∈s_i$forall$i ∈{ 1,…, n}$.Inbothcases,thevariable$x$iscalledthe{\em
root}of$α$andisdenotedby$root(α)$.\par Thesetof{\em
spatialformulas}$F_S$isthesetofformulasoftheform$β_1 * …*
β_n$,whereeveryexpression$β_i$isaspatialatom.If$n = 0$then$β_1 * …*
β_n$isdenotedby$emp$.Thenumber$n$ofoccurrencesofspatialatomsinaspatialformula$ϕ=
β_1 * …* β_n$isdenotedby$len(ϕ)$.Wewrite$ϕ⊑ψ$if$ψ$isoftheform$ϕ*
ϕ’$,moduloassociativityandcommutativityof$*$.Thesetof(nonquantified){\em
symbolicheaps}$F_H$isthesetofexpressionsoftheform$ϕ⋏ξ$,where$ϕ∈F_S$and$ξ∈F_P$.Notethatforclarityweuse$⋏$todenoteconjunctionsbetweenspatialandpureformulasand$∧$todenoteconjunctionsoccurringwithinpureformulas.If$ξ=
⊤$,then$ϕ⋏ξ$maybewritten$ϕ$(i.e.,anyspatialformulamaybeviewedasasymbolicheap).Foranyformula$λ,—λ—$denotesthesizeof$λ$(whichisdefinedinductivelyasusual).Notethat$⊤$isnotasymbolicheap(but$emp⋏⊤$isasymbolicheap).\par
Wedenoteby$V(β)$(resp.\ $V_s(β)$)thesetofvariables(resp.\
ofvariablesofsort$s$)occurringinavariableorformula$β$.A{\em
substitution}isasort-preservingtotalmappingfrom$V$to$. We denote by
$\mathit{dom}(\sigma)$ the set of variables such that $\sigma(x)\not=x$, and
by $\mathit{codom}(x)$ the set $\\{\sigma(x)\mid x\in\mathit{dom}(\sigma)\\}$.
The substitution $\sigma$ such that $\sigma(x_{i})=y_{i}$ for all
$i=1,\dots,n$ and $\mathit{dom}(\sigma)\subseteq\\{x_{1},\dots,x_{n}\\}$ is
denoted by $\replall{}{x_{i}}{y_{i}}{i=1,\dots,n}$. For any expression $\beta$
and substitution $\sigma$, we denote by $\beta\sigma$ the expression obtained
from $\beta$ by replacing every variable $x$ by $\sigma(x)$. A unifier of two
expressions or tuples of expressions $\beta$ and $\beta^{\prime}$ is a
substitution $\sigma$ such that $\beta\sigma=\beta^{\prime}\sigma$.
An inductive rule is an expression of the form
$p(x_{1},\dots,x_{n})\Leftarrow\lambda$, where
$p:{\mathtt{s}}_{1},\dots,{\mathtt{s}}_{n}\in\mathcal{P}$, $x_{1},\dots,x_{n}$
are pairwise distinct variables of sorts
${\mathtt{s}}_{1},\dots,{\mathtt{s}}_{n}$ respectively and $\lambda\in{\cal
F}_{H}$. The set of variables in
$\mathcal{V}(\lambda)\setminus\\{x_{1},\dots,x_{n}\\}$ are the existential
variables of the rule. Let $\mathfrak{R}$ be a set of inductive rules. We
write $p(t_{1},\dots,t_{n})\Leftarrow_{\mathfrak{R}}\lambda$ iff
$\mathfrak{R}$ contains (up to a renaming, and modulo AC) a rule of the form
$p(y_{1},\dots,y_{n})\Leftarrow\gamma$ and
$\lambda=\replall{\gamma}{y_{i}}{t_{i}}{i=1,\dots,n}$. We assume by renaming
that $\gamma$ contains no variable in $\left\\{t_{1},\dots,t_{n}\right\\}$. We
write $p(t_{1},\dots,t_{n})\rightsquigarrow_{\mathfrak{R}}E$ if $E$ is the set
of symbolic heaps $\lambda$ such that
$p(t_{1},\dots,t_{n})\Leftarrow_{\mathfrak{R}}\lambda$. Note that if
$\mathfrak{R}$ is finite then $E$ is finite up to a renaming of variables not
occurring in $\\{t_{1},\dots,t_{n}\\}$. Note also that the considered logic
does not allow for negations (hence entailment is not reducible to
satisfiability) or separating implications, as this would make satisfiability
undecidable (see for instance [21]).
The symbol $\subseteq_{m}$ denotes the inclusion relation between multisets.
With a slight abuse of notations, we will sometimes identify sequences with
sets when the order and number of repetitions is not important, for instance
we may write $\bm{x}\subseteq\bm{y}$ to state that every element of $\bm{x}$
occurs in $\bm{y}$.
In the present paper, we shall consider entailment problems between symbolic
heaps.
### 2.2 Semantics
We assume for technical convenience that formulas are interpreted over a fixed
universe and that constants are interpreted as pairwise distinct elements. Let
$\mathfrak{U}_{{\mathtt{s}}\in\mathfrak{S}}$ be pairwise disjoint countably
infinite sets and let $\mathfrak{U}\overset{\text{\tiny\it
def}}{=}\bigcup_{{\mathtt{s}}\in\mathfrak{S}}\mathfrak{U}_{\mathtt{s}}$. We
assume that an injective function is given, mapping every constant
$c\in\mathcal{C}_{\mathtt{s}}$ to an element of $\mathfrak{U}_{{\mathtt{s}}}$,
denoted by $\dot{c}$.
A heap is a partial finite function from $\mathfrak{U}_{\mathtt{loc}}$ to
$\mathfrak{U}^{*}$, where $\mathfrak{U}^{*}$ denotes as usual the set of
finite sequences of elements of $\mathfrak{U}$. An element
${\ell}\in\mathfrak{U}_{\mathtt{loc}}$ is allocated in a heap $\mathfrak{h}$
if ${\ell}\in\mathit{dom}(\mathfrak{h})$. Two heaps $\mathfrak{h}$ and
$\mathfrak{h}^{\prime}$ are disjoint if
$\mathit{dom}(\mathfrak{h})\cap\mathit{dom}(\mathfrak{h}^{\prime})=\emptyset$,
in which case $\mathfrak{h}\uplus\mathfrak{h}^{\prime}$ denotes their disjoint
union. We write $\mathfrak{h}\subseteq\mathfrak{h}^{\prime}$ if there is a
heap $\mathfrak{h}^{\prime\prime}$ such that
$\mathfrak{h}^{\prime}=\mathfrak{h}\uplus\mathfrak{h}^{\prime\prime}$. For
every heap $\mathfrak{h}$, we denote by $\mathit{ref}(\mathfrak{h})$ the set
of elements $\ell\in\mathfrak{U}_{\mathtt{loc}}$ such that there exists
$\ell_{0},\dots,\ell_{n}$ with $\ell_{0}\in\mathit{dom}(\mathfrak{h})$,
$\mathfrak{h}(\ell_{0})=(\ell_{1},\dots,\ell_{n})$ and $\ell=\ell_{i}$ for
some $i=0,\dots,n$. We write $\ell\rightarrow_{\mathfrak{h}}\ell^{\prime}$ if
$(\ell,\ell^{\prime})\in\mathfrak{U}_{\mathtt{loc}}^{2}$,
$\ell\in\mathit{dom}(\mathfrak{h})$,
$\mathfrak{h}(\ell)=(\ell_{1},\dots,\ell_{n})$ and $\ell^{\prime}=\ell_{i}$,
for some $i=1,\dots,n$.
###### Proposition 2.1
Let $\mathfrak{h},\mathfrak{h}^{\prime}$ be two heaps such that
$\mathfrak{h}\subseteq\mathfrak{h}^{\prime}$. For all
$\ell,\ell^{\prime}\in\mathfrak{U}_{\mathtt{loc}}$, if
$\ell\rightarrow_{\mathfrak{h}}^{*}\ell^{\prime}$ then
$\ell\rightarrow_{\mathfrak{h}^{\prime}}^{*}\ell^{\prime}$.
###### Proof 2.2
By definition of the relation $\rightarrow_{\mathfrak{h}}$ we have
$\rightarrow_{\mathfrak{h}}\subseteq\rightarrow_{\mathfrak{h}^{\prime}}$, thus
$\rightarrow_{\mathfrak{h}}^{*}\subseteq\rightarrow_{\mathfrak{h}^{\prime}}^{*}$
A store $\mathfrak{s}$ is a total function mapping every term in
${\mathtt{s}}$ to an element in $\mathfrak{U}_{\mathtt{s}}$ such that
$\mathfrak{s}(c)=\dot{c}$, for all $c\in\mathcal{C}$ (note that this entails
that $\mathfrak{s}$ is injective on $\mathcal{C}$). A store $\mathfrak{s}$ is
injective on a multiset of variables $V$ if
$\\{x,y\\}\subseteq_{m}V\implies\mathfrak{s}(x)\not=\mathfrak{s}(y)$. When a
store is injective on a multiset of variables, this entails that the latter is
a set, i.e., that contains at most one occurrence of each variable. For any
$V\subseteq\mathcal{V}$, and for any store $\mathfrak{s}$, a store
$\mathfrak{s}^{\prime}$ is an associate of $\mathfrak{s}$ w.r.t. $V$ if
$\mathfrak{s}(x)=\mathfrak{s}^{\prime}(x)$ holds for all $x\not\in V$.
###### Definition 2.3
An SL-structure is a pair $(\mathfrak{s},\mathfrak{h})$, where $\mathfrak{s}$
is a store and $\mathfrak{h}$ is a heap.
The satisfiability relation on SL-formulas is defined inductively as follows:
###### Definition 2.4
An SL-structure $(\mathfrak{s},\mathfrak{h})$ validates a formula (pure
formula, spatial formula, or symbolic heap) $\lambda$ modulo a set of
inductive rules $\mathfrak{R}$, written
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\lambda$, if one of the
following conditions holds:
* •
$\lambda=\mathit{emp}$ and $\mathfrak{h}=\emptyset$;
* •
$\lambda=(t\approx s)$ and $\mathfrak{s}(t)=\mathfrak{s}(s)$;
* •
$\lambda=(t\not\approx s)$ and $\mathfrak{s}(t)\not=\mathfrak{s}(s)$;
* •
$\lambda=x\mapsto(t_{1},\dots,t_{k})$ and
$\mathfrak{h}=\\{(\mathfrak{s}(x),\mathfrak{s}(t_{1}),\dots,\mathfrak{s}(t_{k}))\\}$;
* •
either $\lambda=\lambda_{1}\wedge\lambda_{2}$ or
$\lambda=\lambda_{1}\curlywedge\lambda_{2}$, and
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\lambda_{i}$ for all
$i=1,2$;
* •
$\lambda=\lambda_{1}*\lambda_{2}$ and there exist disjoint heaps
$\mathfrak{h}_{1}$ and $\mathfrak{h}_{2}$ such that
$\mathfrak{h}=\mathfrak{h}_{1}\uplus\mathfrak{h}_{2}$ and
$(\mathfrak{s},\mathfrak{h}_{i})\models_{\mathfrak{R}}\lambda_{i}$ for all
$i=1,2$;
* •
$\lambda=p(t_{1},\dots,t_{n})$ and
$p(t_{1},\dots,t_{n})\Leftarrow_{\mathfrak{R}}\gamma$, where there exists an
associate $\mathfrak{s}^{\prime}$ of $\mathfrak{s}$ w.r.t. the set
$\mathcal{V}(\gamma)\setminus\mathcal{V}(\lambda)$ such that
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}\gamma$.
An $\mathfrak{R}$-model of a formula $\lambda$ is a structure
$(\mathfrak{s},\mathfrak{h})$ such that
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\lambda$. A formula is
satisfiable (w.r.t. $\mathfrak{R}$) if it admits at least one
$\mathfrak{R}$-model.
###### Remark 2.5
Note that a formula $x\mapsto(t_{1},\dots,t_{k})$ asserts not only that $x$
refers to $(t_{1},\dots,t_{k})$ but also that $x$ is the only allocated
location. This fits with usual definitions (see, e.g., [16]). The assertions
are meant to describe elementary heaps, which can be combined afterwards using
the connective $*$. Simply asserting that $x$ refers to $(t_{1},\dots,t_{k})$
could be done in full SL using the following formula:
$x\mapsto(t_{1},\dots,t_{k})*\top$, but such a formula is not a symbolic heap
and is thus outside of the fragment we consider in the present paper.
We write $(\mathfrak{s},\mathfrak{h})\models\lambda$ instead of
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\lambda$ if $\lambda$
contains no predicate symbol, since the relation is independent of
$\mathfrak{R}$ in this case. Similarly, if $\lambda$ is a pure formula then
the relation does not depend on the heap, thus we may simply write
$\mathfrak{s}\models\lambda$.
###### Remark 2.6
Note that there is no symbolic heap that is true in every structure. For
instance $\mathit{emp}\curlywedge\top$ is true only in structures with an
empty heap.
###### Proposition 2.7
Let $(\mathfrak{s},\mathfrak{h})$ be a structure and let $\sigma$ be a
substitution. If
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\lambda\sigma$ then we have
${(\mathfrak{s}\circ\sigma,\mathfrak{h})\models_{\mathfrak{R}}\lambda}$.
###### Proof 2.8
By an immediate induction on the satisfiability relation.
In the present paper, we shall consider inductive rules of a particular form,
defined below.
###### Definition 2.9
An inductive rule is a $\mathtt{P}$-rule if it is of the form
$p(x_{1},\dots,x_{n})\Leftarrow
x_{1}\mapsto(y_{1},\dots,y_{k})*q_{1}(z_{1},\bm{u}_{1})*\dots
q_{m}(z_{m},\bm{u}_{m})\curlywedge\xi$
possibly with $m=0$, where:
1. 1.
$\xi$ is a conjunction of disequations of the form $u\not\approx v$, where
$u\in\\{x_{1},\dots,x_{n},y_{1},\dots,y_{k}\\}$ and
$v\in\\{y_{1},\dots,y_{k}\\}\setminus\\{x_{1},\dots,x_{n}\\}$;
2. 2.
$\\{z_{1},\dots,z_{m}\\}=(\\{y_{1},\dots,y_{k}\\}\setminus\\{x_{1},\dots,x_{n}\\})\cap\mathcal{V}_{\mathtt{loc}}$,
and $z_{1},\dots,z_{m}$ are pairwise distinct;
3. 3.
All the elements of $\bm{u}_{i}$ occurs in
$\\{x_{1},\dots,x_{n}\\}\cup\\{y_{1},\dots,y_{k}\\}\cup\mathcal{C}$.
The predicate symbol $p$ is called the head of the rule.
###### Example 2.10
The rules associated with ${\mathtt{ls}}$ and ${\mathtt{tree}}$ in the
introduction are $\mathtt{P}$-rules, as well as the following rules.
Intuitively ${\mathtt{als}}(x,y)\curlywedge x\not\approx y$ denotes an acyclic
list (${\mathtt{als}}(x,y)$ is thus “quasi-acyclic”, in the sense that it may
loop only on the first element). Note that the constraint $x\not\approx y$
cannot be added to the right-hand side of the rules because the obtained rule
would not be a $\mathtt{P}$-rule, hence it must be added in the formula. The
atom ${\mathtt{dll}}(x,y)$ denotes a doubly linked list starting at $x$, with
the convention that each element of the list points to a pair containing the
previous and next elements. The parameter $y$ denotes the element before $x$
and the last element points to the empty tuple $()$. The atom
${\mathtt{tptr}}(x,y,z)$ denotes a binary tree in which every node refers to
its two successors and to its parent and brother nodes (the parameter $y$ and
$z$ denote the brother and parent nodes, respectively). Leaves point to $()$.
${\small\begin{array}[]{llll}{\mathtt{als}}(x,y)&\Leftarrow&(x\mapsto(z)*{\mathtt{als}}(z,y))\curlywedge
y\not\approx z&\text{\tt\% (quasi-)acyclic list}\\\
{\mathtt{als}}(x,y)&\Leftarrow&x\mapsto(y)\\\
{\mathtt{tll}}(x,y)&\Leftarrow&x\mapsto(y,z)*{\mathtt{tree}}(z)&\text{\tt\%
binary trees with}\\\
{\mathtt{tll}}(x,y)&\Leftarrow&(x\mapsto(z,u)*{\mathtt{tll}}(z,y)*{\mathtt{tree}}(u))&\text{\tt\%
leftmost leaf $y$}\\\ &&\qquad\curlywedge(y\not\approx z)&\\\
{\mathtt{dll}}(x,y)&\Leftarrow&x\mapsto(y,z)*{\mathtt{dll}}(z,x)&\text{\tt\%
doubly linked lists}\\\ {\mathtt{dll}}(x,y)&\Leftarrow&x\mapsto()\\\
{\mathtt{tptr}}(x,y,z)&\Leftarrow&x\mapsto(u,v,y,z)*{\mathtt{tptr}}(u,v,x)&\text{\tt\%
binary trees with}\\\ &&\qquad*{\mathtt{tptr}}(v,u,x)&\text{\tt\% pointers to
brother}\\\ {\mathtt{tptr}}(x,y,z)&\Leftarrow&x\mapsto()&\text{\tt\% and
parent nodes}\\\ \end{array}}$
The following rules are not $\mathtt{P}$-rules (if all variables are of sort
$\mathtt{loc}$):
$\begin{array}[]{llll}p(x)&\Leftarrow&x\mapsto(z)&\text{Condition 2
violated}\\\ p(x)&\Leftarrow&{\mathtt{ls}}(x,z)*p(z)&\text{No points-to
atom}\\\ q(x,y)&\Leftarrow&x\mapsto(z)\curlywedge y\approx
z&\text{\text{Condition 2 violated}}\\\
q(x,y)&\Leftarrow&{\mathtt{ls}}(x,y)&\text{No points-to atom}\\\
{\mathtt{als}}(x,y)&\Leftarrow&(x\mapsto(z)*{\mathtt{als}}(z,y))\curlywedge
x\not\approx y&\text{Condition $1$ violated}\\\
{\mathtt{als}}(x,y)&\Leftarrow&x\mapsto(y)\curlywedge x\not\approx
y&\text{Condition $1$ violated}\\\ \end{array}$
###### Remark 2.11
As evidenced by the rules in Example 2.10, the tuple $()$ is frequently used
as a base case, to end a data structure. This departs from standard
conventions in SL in which a non-allocated constant $\mathtt{nil}$ is
frequently used instead. We avoid considering constants of sort $\mathtt{loc}$
in our framework because this would complicate definitions: one would have to
keep track of allocated and non allocated constants and/or to add syntactic
conditions on the formulas and rules to ensure that such constants are never
allocated.
Note that $\mathtt{P}$-rules are progressing and connected (in the sense of
[14]): every rule allocates exactly one location –the first parameter of the
predicate– and the first parameter of every predicate in the body of the rule
occurs in the right-hand side of the (necessarily unique) points-to atom of
the rule. They are not necessarily established (again in the sense of[14]) as
non-allocated existential variables are allowed provided they are not of sort
$\mathtt{loc}$.
###### Example 2.12
The following (non established, in the sense of [14]) rules, denoting list
segments with unallocated elements are $\mathtt{P}$-rules iff
$u\not\in\mathcal{V}_{\mathtt{loc}}$:
$\begin{array}[]{lllll}{\mathtt{ls}}(x,y)&\Leftarrow
x\mapsto(u,y)&\quad{\mathtt{ls}}(x,y)&\Leftarrow
x\mapsto(u,z)*{\mathtt{ls}}(z,y)\end{array}$
The heap of any model of ${\mathtt{ls}}(x,y)$ is of the form
$\\{\ell_{i}\mapsto(u_{i},\ell_{i+1})\mid i\in\\{1,\dots,n\\}\\}$, where
$u_{1},\dots,u_{n}$ denote arbitrary elements (of a sort distinct from
$\mathtt{loc}$).
$\mathtt{P}$-Rules containing no variable of a sort distinct from
$\mathtt{loc}$ are established. $\mathtt{P}$-Rules also differ from PCE rules
in that every predicate allocates exactly one of its parameters, namely the
first one (the other allocated locations are associated with existential
variables). In other words, there may be only one “entry point” to the
structure allocated by a predicate, namely its root. For instance the rule
$p(x,y)\Leftarrow x\mapsto(y)*q(y)$ (along with another rule for symbol $q$,
e.g., $q(y)\Leftarrow y\mapsto()$) is PCE but it is not a $\mathtt{P}$-rule,
whereas $p(x)\Leftarrow x\mapsto(y)*q(y)$ is a $\mathtt{P}$-rule. Such a
restriction makes the entailment problem easier to solve because it rules out
data structures that can be constructed in different orders (for instance
doubly linked lists with a reference to the end of the list).
We introduce some useful notations and measures on sets of $\mathtt{P}$-rules.
For every set of $\mathtt{P}$-rules $\mathfrak{R}$, we denote by
$\mathcal{P}(\mathfrak{R})$ the set of predicate symbols occurring in a rule
in $\mathfrak{R}$. We define:
$\mathit{ar}_{\mathit{max}}(\mathfrak{R})=\max\\{n\mid
p:{\mathtt{s}}_{1},\dots,{\mathtt{s}}_{n}\in\mathcal{P}(\mathfrak{R})\\}$
and
$\mathit{record}_{\mathit{max}}(\mathfrak{R})=\max\\{k\mid
p(x_{1},\dots,x_{n})\Leftarrow(x\mapsto(t_{1},\dots,t_{k})*\phi)\curlywedge\xi\in\mathfrak{R}\\}$
The numbers $\mathit{ar}_{\mathit{max}}(\mathfrak{R})$ and
$\mathit{record}_{\mathit{max}}(\mathfrak{R})$ respectively denote the maximum
arity of the predicate symbols in $\mathfrak{R}$ and the maximum number of
record fields in a points-to atom occurring in $\mathfrak{R}$. The width of
$\mathfrak{R}$ is defined as follows:
$\mathit{width}(\mathfrak{R})\overset{\text{\tiny\it
def}}{=}\max(\mathit{ar}_{\mathit{max}}(\mathfrak{R}),\mathit{record}_{\mathit{max}}(\mathfrak{R}))$.
We make two additional assumptions about the considered set of rules: we
assume that every predicate is productive (Assumption 2.2) and that no
parameter is useless (Assumption 2.16). More precisely, the set of productive
predicate symbols is inductively defined as follows: $p\in\mathcal{P}$ is
productive w.r.t. a set of inductive rules $\mathfrak{R}$ if $\mathfrak{R}$
contains a rule $p(\bm{x})\Leftarrow\lambda$ such that all the predicate
symbols occurring in $\lambda$ are productive. In particular, a rule with no
predicate in its right-hand side is always productive (base case). Productive
rules can easily be computed using a straightforward least fixpoint algorithm.
###### Example 2.13
Let $\mathfrak{R}=\\{p(x)\Leftarrow q(x),q(x)\Leftarrow p(x),r(x)\Leftarrow
x\mapsto(y)*p(y)\\}$. The predicates $p,q,r$ are not productive.
It is easy to check that every formula containing at least one non-productive
predicate symbol is unsatisfiable. Indeed, if a predicate symbol $p$ is non-
productive, then an atom $p(\bm{x})$ cannot be unfolded into a formula
containing no predicate symbol. This justifies the following:
For all sets of $\mathtt{P}$-rules $\mathfrak{R}$, we assume that all the
predicate symbols are productive w.r.t. $\mathfrak{R}$.
For all predicates $p:{\mathtt{s}}_{1},\dots,{\mathtt{s}}_{n}$, the set
$\mathit{out}_{\mathfrak{R}}(p)$ denotes the least set of indices $i$ in
$\\{1,\dots,n\\}$ such that ${\mathtt{s}}_{i}=\mathtt{loc}$ and there exists a
rule $p(x_{1},\dots,x_{n})\Leftarrow\lambda$ in $\mathfrak{R}$ such that
$\lambda$ contains either a points-to atom $x_{1}\mapsto(t_{1},\dots,t_{k})$
where $x_{i}\in\\{t_{1},\dots,t_{k}\\}$ or a predicate atom
$q(t_{1},\dots,t_{m})$ with $t_{j}=x_{i}$, for some
$j\in\mathit{out}_{\mathfrak{R}}(q)$. Intuitively,
$\mathit{out}_{\mathfrak{R}}(p)$ denote the set of “out-going” nodes of the
structures corresponding to $p$, i.e., the set of parameters corresponding to
locations that can be referred to but not necessarily allocated.
###### Example 2.14
Consider the following rules:
$p(x,y,z)\Leftarrow x\mapsto(x,y)$ | $p(x,y,z)\Leftarrow x\mapsto(x,u)*q(u,z,z)$ | $q(x,y,z)\Leftarrow x\mapsto(y)$
---|---|---
Then $\mathit{out}_{\mathfrak{R}}(p)=\\{1,2\\}$ and
$\mathit{out}_{\mathfrak{R}}(q)=\\{2\\}$.
###### Proposition 2.15
Let $\mathfrak{R}$ be a set of $\mathtt{P}$-rules. If
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}p(t_{1},\dots,t_{n})$ and
the index $i\neq 1$ is such that $i\not\in\mathit{out}_{\mathfrak{R}}(p)$
(i.e., $i$ is not an outgoing parameter of $p$) and
${\mathtt{s}}_{i}=\mathtt{loc}$, then the entailment
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}p(t_{1},\dots,t_{i-1},s,t_{i+1},t_{n})$
holds for all terms $s$.
###### Proof 2.16
By an induction on the satisfiability relation. By hypothesis
$p(t_{1},\dots,t_{n})\Leftarrow_{\mathfrak{R}}\gamma$, where $\gamma$ is of
the form $t_{1}\mapsto(t_{1}^{\prime},\ldots,t_{k}^{\prime})*\gamma^{\prime}$,
and there exists an associate $\mathfrak{s}^{\prime}$ of $\mathfrak{s}$ w.r.t.
$\mathcal{V}(\gamma)\setminus\mathcal{V}(\lambda)$ such that
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}\gamma$. By
hypothesis $i\neq 1$, and since $i\not\in\mathit{out}_{\mathfrak{R}}(p)$,
$t_{i}$ cannot occur in
$\left\\{t_{1}^{\prime},\ldots,t_{k}^{\prime}\right\\}$. This entails that
$p(t_{1},\dots,t_{i-1},s,t_{i+1},t_{n})\Leftarrow_{\mathfrak{R}}t_{1}\mapsto(t_{1}^{\prime},\ldots,t_{k}^{\prime})*\gamma^{\prime\prime}$.
If $\gamma^{\prime}$ contains a predicate $q(s_{1},\ldots,s_{m})$ and there
exists an index $j$ such that $s_{j}=t_{i}$, then we cannot have $j=1$ because
$t_{i}\notin\left\\{t_{1}^{\prime},\ldots,t_{k}^{\prime}\right\\}$ and the
rule under consideration is a $\mathtt{P}$-rule. Since
$i\not\in\mathit{out}_{\mathfrak{R}}(p)$ by hypothesis, $j$ cannot belong to
$\mathit{out}_{\mathfrak{R}}(q)$ and by induction, we deduce that
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}t_{1}\mapsto(t_{1}^{\prime},\ldots,t_{k}^{\prime})*\gamma^{\prime\prime}$,
hence the result.
Proposition 2.15 states that if
$i\not\in\mathit{out}_{\mathfrak{R}}(p)\cup\\{1\\}$ and
${\mathtt{s}}_{i}=\mathtt{loc}$, then the semantics of $p(t_{1},\dots,t_{n})$
does not depend on $t_{i}$, thus the $i$-th argument of $p$ is redundant and
can be removed. This justifies the following: For all sets of
$\mathtt{P}$-rules $\mathfrak{R}$ and for all predicate symbols
$p:\mathtt{loc},{\mathtt{s}}_{1},\dots,{\mathtt{s}}_{n}\in\mathcal{P}$, we
assume that $\mathit{out}_{\mathfrak{R}}(p)\supseteq\\{2\leq i\leq
n\mid{\mathtt{s}}_{i}=\mathtt{loc}\\}$.
###### Definition 2.17
For any formula $\lambda$, we write $x\rightarrow_{\lambda}y$ if
$x,y\in\mathcal{V}_{\mathtt{loc}}$ and $\lambda$ contains an atom
$p(t_{1},\dots,t_{n})$ (resp. $t_{1}\mapsto(t_{2},\dots,t_{n})$) such that
$t_{1}=x$ and $t_{i}=y$, for some $i\in\mathit{out}_{\mathfrak{R}}(p)$ (resp.
for some $i\in\\{2,\dots,n\\}$).
A structure $(\mathfrak{s},\mathfrak{h})$ is called a $\rightarrow$-compatible
model of a formula $\lambda$ iff
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\lambda$ and for every
$x,y\in\mathcal{V}_{\mathtt{loc}}$,
$\mathfrak{s}(x)\rightarrow_{\mathfrak{h}}^{*}\mathfrak{s}(y)\implies
x\rightarrow_{\lambda}^{*}y$.
Intuively, $x\rightarrow_{\lambda}y$ states that the formula $\lambda$
allocates an edge from $x$ to $y$.
###### Definition 2.18
A sequent is an expression of the form
$\lambda\vdash_{\mathfrak{R}}^{V}\gamma$, where $\lambda,\gamma$ are symbolic
heaps, $V$ is a multiset of variables of sort $\mathtt{loc}$ and
$\mathfrak{R}$ is a finite set of inductive rules. If $V=\emptyset$ then the
sequent is written $\lambda\vdash_{\mathfrak{R}}\gamma$. A sequent is
equality-free if $\lambda$ and $\gamma$ contain no atoms of the form $u\approx
v$. A counter-model of a sequent $\lambda\vdash_{\mathfrak{R}}^{V}\gamma$ is a
structure $(\mathfrak{s},\mathfrak{h})$ such that:
* •
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\lambda$ and
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}\gamma$,
* •
$\forall x\in V,\,\mathfrak{s}(x)\not\in\mathit{dom}(\mathfrak{h})$,
* •
$\mathfrak{s}$ is injective on the multiset $V$.
A sequent is valid iff it has no counter-model.
## 3 Lower Bounds
We establish various lower bounds for the validity problems for sequents
$\lambda\vdash_{\mathfrak{R}}\gamma$, where $\mathfrak{R}$ satisfies some
additional conditions. These lower bounds will motivate the additional
restrictions that are imposed to devise a polynomial-time proof procedure.
Checking the validity of sequents $\lambda\vdash_{\mathfrak{R}}\gamma$ where
$\mathfrak{R}$ is a set of $\mathtt{P}$-rules is actually undecidable in
general. This result can established using an argument similar to the one used
in [11] to prove the undecidability of PCE entailments modulo theories; it is
not given here for the sake of conciseness, and because the goal of this paper
is to investigate tractable cases. The undecidability proof relies on the
existence of variables of a sort distinct from $\mathtt{loc}$. If such
variables are forbidden, then the rules are PCE hence entailment is decidable
[14], but we still get an ExpTime lower bound:
###### Proposition 3.1
Checking the validity of sequents $\lambda\vdash_{\mathfrak{R}}\gamma$ is
ExpTime-hard, even if $\mathfrak{R}$ is a set of $\mathtt{P}$-rules and all
the variables occurring in $\lambda\vdash_{\mathfrak{R}}\gamma$ are of sort
$\mathtt{loc}$.
###### Proof 3.2
The proof is by a straightforward reduction from the inclusion problem for
languages accepted by tree automata (see [6]). Indeed, a tree automaton
$(Q,V,\\{q_{0}\\},R)$ can be straightforwardly encoded as a set of
$\mathtt{P}$-rules, where each rule $q\rightarrow f(q_{1},\dots,q_{n})$ in $R$
is encoded by an inductive rule of the form ${q(x)\Leftarrow
x\mapsto(f,x_{1},\dots,x_{n})}*\scaleobj{1.5}{*}_{i=1}^{n}q_{i}(x_{i})$. Each
function symbol $f$ is considered as a constant of a sort
${\mathtt{s}}\not=\mathtt{loc}$, and a term $f(t_{1},\dots,t_{n})$ is
represented as a heap
$\mathfrak{h}_{1}\uplus\dots\uplus\mathfrak{h}_{n}\uplus\\{(\ell_{0},\ell_{1},\dots,\ell_{n})\\}$,
where $\ell_{0},\dots,\ell_{n}$ are pairwise distinct locations and
$\mathfrak{h}_{1},\dots,\mathfrak{h}_{n}$ are disjoint representations of
$t_{i}$ with ${\ell_{0}\not\in\mathit{dom}(\mathfrak{h}_{i})}$, for
$i=1,\dots,n$. It is straightforward to verify that the language accepted by
$(Q,V,\\{q_{0}\\},R)$ is included in that of
$(Q^{\prime},V,\\{q_{0}^{\prime}\\},R^{\prime})$ iff the sequent
$q_{0}(x)\vdash_{\mathfrak{R}}q_{0}^{\prime}(x)$ is valid.
Since the inclusion problem is polynomial for top-down deterministic tree
automata [6], it is natural to further restrict the considered rules to make
them deterministic, in the following sense:
###### Definition 3.3
A set of $\mathtt{P}$-rules $\mathfrak{R}$ is deterministic if for all pairs
of distinct rules of the form
$p(\bm{x}_{i})\Leftarrow(y_{i}\mapsto\bm{t}_{i}*\phi_{i})\curlywedge\xi_{i}$
(where $i=1,2$) occurring in $\mathfrak{R}$, the formula
$\bm{x}_{1}\approx\bm{x}_{2}\wedge\bm{t}_{1}\approx\bm{t}_{2}\wedge\xi_{1}\wedge\xi_{2}$
is unsatisfiable (we assume by renaming that the rules share no variable).
For instance the rules associated with the predicate ${\mathtt{ls}}$ in the
introduction are not deterministic, whereas the rules associated with
${\mathtt{tree}}$ are deterministic, as well as all those in Example 2.10. For
the predicate ${\mathtt{ls}}$, the formula $x\approx x^{\prime}\wedge y\approx
z$ is satisfiable, whereas for the predicate ${\mathtt{tree}}$, the formula
$x\approx x^{\prime}\wedge()\approx(y,z)$ is unsatisfiable (in both cases the
variable $x$ is renamed by $x^{\prime}$ in the second rule).
The following proposition shows that the restriction to deterministic sets of
$\mathtt{P}$-rules is still not sufficient to obtain a tractable validity
problem:
###### Proposition 3.4
Checking the validity of sequents $\lambda\vdash_{\mathfrak{R}}\gamma$ is
PSpace-hard, even if $\mathfrak{R}$ is a deterministic set of
$\mathtt{P}$-rules and all variables in $\lambda\vdash_{\mathfrak{R}}\gamma$
are of sort $\mathtt{loc}$.
###### Proof 3.5
Let “$w\in E$” be any problem in PSpace. By definition, there exists a Turing
machine $M=(Q,\Sigma,B,\Gamma,\delta,q_{0},F)$ accepting exactly the words in
$E$ and a polynomial $R$ such that $M$ runs in space $R(n)$ on all words
$w\in\Sigma^{n}$. The set $Q$ denotes the set of states of $M$, $\Sigma$ is
the input alphabet, $B$ is the blank sumbol, $\Gamma$ is the tape alphabet,
$\delta$ is the transition function, $q_{0}$ is the initial state and $F$ is
the set of final states. We shall reduce the problem “$w\in E$” to the
entailment problem, for a sequent fulfilling the conditions above. Consider a
word $w$ of length $n$, and let $N=R(n)$. Assume that $\mathcal{C}$ contains
all the elements in $\Gamma$. We consider $\mathit{card}(Q)\cdot N$ predicates
$q^{i}$ of arity $N+3$, for all $q\in Q$ and $i\in\\{1,\dots,N\\}$, associated
with the following rules:
$\displaystyle q^{i}(x,y_{1},\dots,y_{N},u,v)$
$\displaystyle\Leftarrow\begin{multlined}x\mapsto(x^{\prime},u,v,a)*p^{i+\mu}(x^{\prime},y_{1},\dots,y_{i-1},b,y_{i+1},\dots,y_{N},y_{i},a)\\\
{\text{if $q\not\in F$ and $\delta$ contains a rule
$(q,a)\rightarrow(p,b,\mu)$}}\\\ {\text{ with $i+\mu\in\\{1,\dots,N\\}$}}\\\
\end{multlined}x\mapsto(x^{\prime},u,v,a)*p^{i+\mu}(x^{\prime},y_{1},\dots,y_{i-1},b,y_{i+1},\dots,y_{N},y_{i},a)\\\
{\text{if $q\not\in F$ and $\delta$ contains a rule
$(q,a)\rightarrow(p,b,\mu)$}}\\\ {\text{ with $i+\mu\in\\{1,\dots,N\\}$}}\\\ $
$\displaystyle q^{i}(x,y_{1},\dots,y_{N},u,v)$ $\displaystyle\Leftarrow
x\mapsto(x,u,v,B)\text{,\ if $q\in F$.}$
Intuitively, $q$ is the state of the machine, the arguments
$y_{1},\dots,y_{N}$ denote the tape (that is of length $N$ by hypothesis) and
$i$ denotes the position of the head on the tape. The constants $a,b$ denote
the symbols read and written on the tape, respectively, $p$ is the final state
of the transition rule and the integer $\mu$ denotes the move, i.e., an
element of $\\{-1,0,+1\\}$, so that $i+\mu$ is the final position of the head
on the tape. Note that at this point the inductive rule does not test whether
the symbol $a$ is indeed identical to the symbol at position $i$, namely
$y_{i}$. Instead, it merely stores both $y_{i}$ and $a$ within the next tuple
of the heap, by passing them as parameters to $p^{i+\mu}$. The arguments $u$
and $v$ are used to encode respectively the symbol read on the tape at the
previous state and the symbol that was expected. By definition of the above
rules, it is clear that $q^{i}(x,y_{1},\dots,y_{N},B,B)$ holds if the heap is
a list of tuples $(x_{j},u_{j},v_{j},a_{j})$, for $j=1,\ldots,k$, linked on
the first argument (the last element loops on itself). The heap encodes a
“candidate run” of length $k$ of $M$, i.e., a run for which one does not
check, when applying a transition $(p,a)\rightarrow(q,b,\mu)$, that the symbol
read on the tape is identical to the expected symbol $a$. The symbols
$u_{i},v_{i},a_{i}$ stored at each node are precisely the symbols that are
read ($u_{i}$) and expected ($v_{i}$) at the previous step, respectively,
along with the symbol $(a_{i}$) that is expected at the current step (this
last symbol is added to ensure that the rules are deterministic). Note that
for $i=1$ there is no previous step and for $i=k$ no symbol is read since the
state is final; thus by convention, $a_{k}$ is set to $B$ (see the last rule
of $q^{i}$). Furthermore, $u_{1},v_{1}$ will also be set to $B$ by invoking
the initial state predicate with $B$ as last and before last argument (see the
definition of the sequent below). To check that the list corresponds to an
actual run of $M$, it thus suffices to check that $u_{i}=v_{i}$ holds for all
$i=1,\dots,k$.
The right-hand side of the sequent will allocate all structures not satisfying
this condition. To this purpose, we associate with each state $r\in Q$ two
predicate symbols $r_{0}$ and $r_{1}$ defined by the following rules (where
$i,j\in\\{0,1\\}$):
$\displaystyle r_{i}(x)$
$\displaystyle\Leftarrow\begin{multlined}x\mapsto(x^{\prime},a,b,c)*r_{j}(x^{\prime})\\\
{\text{for all $r\not\in F$, $a,b,c\in\Gamma$, where $j=1$ iff either $i=1$ or
$a\not=b$;}}\end{multlined}x\mapsto(x^{\prime},a,b,c)*r_{j}(x^{\prime})\\\
{\text{for all $r\not\in F$, $a,b,c\in\Gamma$, where $j=1$ iff either $i=1$ or
$a\not=b$;}}$ $\displaystyle r_{i}(x)$ $\displaystyle\Leftarrow
x\mapsto(x,a,b,B)\text{\quad if $a,b\in\Gamma$, $r\in F$ and either $i=1$ or
$a\not=b$.}$
Intuitively, the index $i$ in predicate $r_{i}$ is equal to $1$ iff a faulty
location has been encountered (i.e., a tuple $(x^{\prime},a,b)$ with
$a\not=b$).
Note that the number of rules is polynomial w.r.t. $N$, since the machine $M$
is fixed. Also, the obtained set of rules is deterministic, because $M$ is
deterministic and the expected symbol is referred to by the location allocated
by each predicate symbol $q^{i}$, thus the tuples $(x^{\prime},u,v,a)$
corresponding to distinct rules associated with the same symbol $q^{i}$ cannot
be unifiable.
Let $w_{1}.\dots,w_{N}=w.B^{N-n}$ be the initial tape (where $w$ is completed
by blank symbols $B$ to obtain a word of length $N$). It is clear that the
sequent $q_{0}^{1}(x,w_{1},\dots,w_{N},B,B)\vdash_{\mathfrak{R}}r_{0}(x)$ is
valid iff all the “candidate runs” of $M$ fulfill the conditions of the right-
hand side, i.e., falsify at least one equality between read and expected
symbols. Thus
$q_{0}^{1}(x,w_{1},\dots,w_{N},B,B)\vdash_{\mathfrak{R}}r_{0}(x)$ is valid iff
$M$ does not accept $w$.
In view of this result, it is natural to investigate the complexity of the
entailment problem when the maximal arity of the predicates is bounded.
However, this is still insufficient to get a tractable problem, as the
following lemma shows.
###### Lemma 3.6
Checking the validity of sequents $\lambda\vdash_{\mathfrak{R}}\gamma$ is co-
Np-hard, even if $\mathit{width}(\mathfrak{R})\leq 4$ (i.e., if the symbols
and tuples are of arity at most $4$) and $\mathfrak{R}$ is a deterministic set
of $\mathtt{P}$-rules.
###### Proof 3.7
The proof is by a reduction from the complement of the $3$-coloring problem,
that is well-known to be NP-complete. Let $G=(V,E)$ be a graph, where
$V=\\{v_{1},\dots,v_{n}\\}$ is a finite set of vertices and $E$ is a set of
undirected edges, i.e., a set of unordered pairs of vertices. Let
$\mathtt{Colors}=\\{a,b,c\\}$ be a set of colors, with
$\mathit{card}(\mathtt{Colors})=3$. We recall that a solution of the
$3$-coloring problem is a function $f:V\rightarrow\mathtt{Colors}$ such that
$(x,y)\in E\implies f(x)\not=f(y)$. We assume, w.l.o.g., that all vertices
occur in at least one edge. We consider two distinct sorts $\mathtt{loc}$ and
${\mathtt{s}}$. We assume, w.l.o.g., that
$V\cup\mathtt{Colors}\subseteq\mathcal{V}_{{\mathtt{s}}}$ (i.e., $a,b,c$, as
well as the set of vertices in $G$, are variables) and
$V\cap\mathtt{Colors}=\emptyset$.
Let $E=\\{(x_{i},y_{i})\mid i=1,\dots,m\\}$ (where the edges are ordered
arbitrarily) and let $u_{1},\dots,u_{m+1}$ be pairwise distinct variables of
sort $\mathtt{loc}$. Let $\phi$ be the formula: $\scaleobj{1.5}{*}_{i=1}^{m}\
u_{i}\mapsto(x_{i},y_{i},u_{i+1})*u_{m+1}\mapsto()$. Let $p$ and $q$ be
predicate symbols associated with the following rules:
$\begin{array}[]{lll}p(u,a,b,c)&\Leftarrow&u\mapsto(v,v,u^{\prime})*q(u^{\prime})\\\
p(u,a,b,c)&\Leftarrow&u\mapsto(v_{1},v_{2},u^{\prime})*v_{1}\not\approx
v_{2}*v_{1}\not\approx a*v_{1}\not\approx b*v_{1}\not\approx
c*q(u^{\prime})\\\
p(u,a,b,c)&\Leftarrow&u\mapsto(d,v_{2},u^{\prime})*v_{2}\not\approx
a*v_{2}\not\approx b*v_{2}\not\approx c*q(u^{\prime})\\\ &&\text{\quad for all
$d\in\\{a,b,c\\}$}\\\
p(u,a,b,c)&\Leftarrow&u\mapsto(d_{1},d_{2},u^{\prime})*p(u^{\prime},a,b,c)\\\
&&\text{\quad for all $d_{1},d_{2}\in\\{a,b,c\\}$ where $d_{1}\not=d_{2}$}\\\
q(u)&\Leftarrow&u\mapsto(v_{1},v_{2},u^{\prime})*q(u^{\prime})\\\
q(u)&\Leftarrow&u\mapsto()\end{array}$
Intuitively, any model $(\mathfrak{s},\mathfrak{h})$ of $\phi$ encodes a
candidate solution of the $3$-coloring problem, where each variable
$z\in\\{x_{i}\mid i=1,\dots,m\\}\cup\\{y_{i}\mid i=1,\dots,m\\}$ is mapped to
an element $\mathfrak{s}(z)$ in $\mathfrak{U}_{{\mathtt{s}}}$. The heap
$\mathfrak{h}$ is a list of tuples linked on the last element and containing a
tuple
$(\mathfrak{s}(u_{i}),\mathfrak{s}(x_{i}),\mathfrak{s}(y_{i}),\mathfrak{s}(u_{i+1}))$
for all $(x_{i},y_{i})\in E$. To check that this candidate solution indeed
fulfills the required properties, one has to verify that all the pairs
$(\mathfrak{s}(x_{i}),\mathfrak{s}(y_{i}))$ are composed of distinct elements
in $\\{a,b,c\\}$.
By definition of the rules for predicate $p$, $(\mathfrak{s},\mathfrak{h})$ is
a model of $p(u_{1},a,b,c)$ iff the list contains a pair
$(\mathfrak{s}(x),\mathfrak{s}(y))$ such that one of the following holds:
* •
$\mathfrak{s}(x)=\mathfrak{s}(y)$ (first rule of $p$),
* •
$\mathfrak{s}(x)\not=\mathfrak{s}(y)$ and
$\mathfrak{s}(x)\not\in\\{\mathfrak{s}(a),\mathfrak{s}(b),\mathfrak{s}(c)\\}$
(second rule of $p$),
* •
$\mathfrak{s}(x)\in\\{\mathfrak{s}(a),\mathfrak{s}(b),\mathfrak{s}(c)\\}$ and
$\mathfrak{s}(y)\not\in\\{\mathfrak{s}(a),\mathfrak{s}(b),\mathfrak{s}(c)\\}$
(third rule of $p$).
After the cell corresponding to this faulty pair is allocated, $q$ is invoked
to allocate the remaining part of the list. Thus $p(u_{1},a,b,c)$ holds iff
the model does not encode a solution of the $3$-coloring problem, either
because $\mathfrak{s}(x_{i})=\mathfrak{s}(y_{i})$ for some $(x_{i},y_{i})\in
E$ or because one of the variables is mapped to an element distinct from
$a,b,c$ – note that by the above assumption, each of these variables occurs in
the list. Consequently $\phi\vdash_{\mathfrak{R}}p(u_{1},a,b,c)$ admits a
counter-model if there exists a model of $\phi$ that does not satisfy
$p(u_{1},a,b,c)$, i.e., iff the $3$-coloring problem admits a solution.
The results above motivate the following definition, that strengthens the
notion of a deterministic set of rules.
###### Definition 3.8
A set of $\mathtt{P}$-rules $\mathfrak{R}$ is $\mathtt{loc}$-deterministic if
it is deterministic and all the disequations occurring in the rules in
$\mathfrak{R}$ are of the form $x\not\approx y$ with
$x,y\in\mathcal{V}_{\mathtt{loc}}$.
The intuition behind $\mathtt{loc}$-deterministic rules is that, to get an
efficient proof procedure, we have to restrict the amount of equational
reasoning needed to establish the validity of the sequents. Disequations
between locations are relatively easy to handle because (by definition of
$\mathtt{P}$-rules) all existential variables of sort $\mathtt{loc}$ must be
pairwise distinct (as they are allocated in distinct atoms). However, dealing
with disequations between data is much more difficult, as evidenced by the
proof of Lemma 3.6. Thus we restrict such disequations to those occurring in
the initial sequent.
The rules associated with ${\mathtt{als}}$, ${\mathtt{tree}}$,
${\mathtt{tll}}$, ${\mathtt{tptr}}$ or ${\mathtt{dll}}$ in the Introduction
and in Example 2.10 are $\mathtt{loc}$-deterministic. In contrast, the
following rules are deterministic, but not $\mathtt{loc}$-deterministic (where
$u,v$ denote variables of some sort distinct from $\mathtt{loc}$):
$\begin{array}[]{llllll}p(x,u)&\Leftarrow&x\mapsto(v)\curlywedge v\not\approx
u&\quad p(x,u)&\Leftarrow&x\mapsto(u)\end{array}$
Rules that are $\mathtt{loc}$-deterministic are well-suited to model
constructor-based data structures used in standard programming languages; for
instance, lists could be represented as follows (where $\mathtt{cons}$ is a
constant symbol denoting a constructor and $y$ is a variable of some sort
distinct from $\mathtt{loc}$, denoting data stored in the list):
$\begin{array}[]{llllll}{\mathtt{ls}}(x)\Leftarrow&x\mapsto(\mathtt{cons},y,z)*{\mathtt{ls}}(z)&\quad{\mathtt{ls}}(x)\Leftarrow&x\mapsto()\end{array}$
We end this section by establishing a key property of deterministic set of
rules, namely the fact that every spatial formula $\phi$ is precise, in the
sense of [4]: it is fulfilled on at most one subheap within a given structure.
###### Lemma 3.9
Let $\mathfrak{R}$ be a deterministic set of rules. For every spatial formula
$\phi$, for every store $\mathfrak{s}$ and for every heap $\mathfrak{h}$ there
exists at most one heap $\mathfrak{h}^{\prime}$ such that
$\mathfrak{h}^{\prime}\subseteq\mathfrak{h}$ and
$(\mathfrak{s},\mathfrak{h}^{\prime})\models_{\mathfrak{R}}\phi$.
###### Proof 3.10
The proof is by induction on the satisfiability relation
$\models_{\mathfrak{R}}$. Note that by hypothesis $\phi$ is a spatial formula,
hence contains no occurrences of $\approx$, $\not\approx$, $\curlywedge$ or
$\wedge$. Assume that there exist two heaps
$\mathfrak{h}_{1}^{\prime},\mathfrak{h}_{2}^{\prime}$ such that
$\mathfrak{h}_{i}^{\prime}\subseteq\mathfrak{h}$ and
$(\mathfrak{s},\mathfrak{h}_{i}^{\prime})\models_{\mathfrak{R}}\phi$ (for
$i=1,2$). We show that $\mathfrak{h}_{1}^{\prime}=\mathfrak{h}_{2}^{\prime}$.
* •
If $\phi=\mathit{emp}$ then necessarily $\mathfrak{h}_{i}=\emptyset$ for
$i=1,2$ thus $\mathfrak{h}_{1}^{\prime}=\mathfrak{h}_{2}^{\prime}$.
* •
If $\phi=y_{0}\mapsto(y_{1},\dots,y_{n})$ then by Definition 2.4 we have
$\mathfrak{h}_{i}^{\prime}=\\{(\mathfrak{s}(y_{0}),\dots,\mathfrak{s}(y_{n}))\\}$
for $i=1,2$ thus $\mathfrak{h}_{1}^{\prime}=\mathfrak{h}_{2}^{\prime}$.
* •
If $\phi=\phi_{1}*\phi_{2}$ then for all $i=1,2$ there exist two disjoint
heaps $\mathfrak{h}_{i}^{j}$ (for $j=1,2$) such that
$\mathfrak{h}_{i}^{\prime}=\mathfrak{h}_{i}^{1}\uplus\mathfrak{h}_{i}^{2}$ for
$i=1,2$ and
$(\mathfrak{s},\mathfrak{h}_{i}^{j})\models_{\mathfrak{R}}\phi_{j}$, for
$i,j\in\\{1,2\\}$. Since
$\mathfrak{h}_{i}^{j}\subseteq\mathfrak{h}_{i}^{\prime}\subseteq\mathfrak{h}$
we get by the induction hypothesis $\mathfrak{h}_{1}^{j}=\mathfrak{h}_{2}^{j}$
for $j=1,2$. Therefore
$\mathfrak{h}_{1}^{1}\uplus\mathfrak{h}_{1}^{2}=\mathfrak{h}_{2}^{1}\uplus\mathfrak{h}_{2}^{2}$,
i.e., $\mathfrak{h}_{1}^{\prime}=\mathfrak{h}_{2}^{\prime}$.
* •
If $\phi$ is a predicate atom of root $x$, then for $i=1,2$ we have
$\phi\Leftarrow_{\mathfrak{R}}\lambda_{i}$, and there exists an associate
$\mathfrak{s}_{i}$ of $\mathfrak{s}$ w.r.t.
$\mathcal{V}(\psi_{i})\setminus\mathcal{V}(\phi)$ such that
$(\mathfrak{s}_{i},\mathfrak{h}_{i}^{\prime})\models_{\mathfrak{R}}\lambda_{i}$.
Since $\mathfrak{R}$ is a set of $\mathtt{P}$-rules, $\lambda_{i}$ is of the
form $(x_{i}\mapsto\bm{y}_{i}*\phi_{i})\curlywedge\xi_{i}$ and there exist
disjoint heaps $\mathfrak{h}_{i}^{j}$ (for $j=1,2$) such that the following
conditions are satisfied: (i)
$\mathfrak{h}_{i}^{\prime}=\mathfrak{h}_{i}^{1}\uplus\mathfrak{h}_{i}^{2}$;
(ii)
$(\mathfrak{s}_{i},\mathfrak{h}_{i}^{1})\models_{\mathfrak{R}}x_{i}\mapsto(\bm{y}_{i})$;
(iii) $(\mathfrak{s}_{i},\mathfrak{h}_{i}^{2})\models_{\mathfrak{R}}\phi_{i}$;
(iv) and $\mathfrak{s}_{i}\models\xi_{i}$. Furthermore, $x_{i}$ must be the
root of $\phi$, thus $x_{1}=x_{2}=x$. For $i=1,2$ we have
$\mathfrak{h}_{i}^{1}=\\{(\mathfrak{s}(x_{i}),\mathfrak{s}_{i}(\bm{y}_{i}))\\}$,
and since $\mathfrak{h}_{i}^{1}\subseteq\mathfrak{h}$, necessarily
$\mathfrak{s}_{1}(\bm{y}_{1})=\mathfrak{s}_{2}(\bm{y}_{2})$ and
$\mathfrak{h}_{1}^{1}=\mathfrak{h}_{2}^{1}$. The heap $\mathfrak{h}_{i}^{2}$
is the restriction of $\mathfrak{h}_{i}^{\prime}$ to the locations distinct
from $\mathfrak{s}(x)$. We distinguish two cases.
* –
Assume that the inductive rules applied on $\phi$ to respectively derive
$\lambda_{1}$ and $\lambda_{2}$ are different. We may assume by
$\alpha$-renaming that
$(\mathcal{V}(\lambda_{1})\setminus\mathcal{V}(\phi))\cap(\mathcal{V}(\lambda_{2})\setminus\mathcal{V}(\phi))=\emptyset$,
which entails that there exists a store $\mathfrak{s}^{\prime}$ that coincides
with $\mathfrak{s}_{i}$ on $\mathcal{V}(\lambda_{i})$ (since
$\mathfrak{s}_{1}$ and $\mathfrak{s}_{2}$ coincide on the variables in
$\mathcal{V}(\phi)$). Then we have
$\mathfrak{s}^{\prime}\models\bm{y}_{1}\approx\bm{y}_{2}$ (since
$\mathfrak{s}_{1}(\bm{y}_{1})=\mathfrak{s}_{2}(\bm{y}_{2})$) and
$\mathfrak{s}^{\prime}\models\xi_{i}$ (since
$\mathfrak{s}_{i}\models\xi_{i}$), which entails that the formula
$\bm{y}_{1}\approx\bm{y}_{2}\wedge\xi_{1}\wedge\xi_{2}$ is satisfiable,
contradicting the fact that $\mathfrak{R}$ is deterministic.
* –
Assume that the same rule is used to derive both $\lambda_{1}$ and
$\lambda_{2}$. We may assume in this case (again by $\alpha$-renaming) that
the vector of variables occurring in $\lambda_{1}$ and $\lambda_{2}$ are the
same, so that $\bm{y}_{1}=\bm{y}_{2}$ and $\phi_{1}=\phi_{2}$. Since
$\mathfrak{R}$ is a set of $\mathtt{P}$-rules, all variables $z$ in
$\mathcal{V}(\phi_{i})\setminus\mathcal{V}(\phi)$ occur in $\bm{y}_{i}$. As
$\mathfrak{s}_{1}(\bm{y}_{1})=\mathfrak{s}_{2}(\bm{y}_{2})$, this entails that
$\mathfrak{s}_{1}(z)=\mathfrak{s}_{2}(z)$ holds for all such variables, thus
$\mathfrak{s}_{1}=\mathfrak{s}_{2}$. Consequently,
$(\mathfrak{s}_{1},\mathfrak{h}_{i}^{2})\models_{\mathfrak{R}}\phi_{1}$, for
all $i=1,2$ with
$\mathfrak{h}_{i}^{2}\subseteq\mathfrak{h}_{i}^{\prime}\subseteq\mathfrak{h}$.
By the induction hypothesis this entails that
$\mathfrak{h}_{1}^{2}=\mathfrak{h}_{2}^{2}$, thus
$\mathfrak{h}_{1}^{\prime}=\mathfrak{h}_{2}^{\prime}$.
###### Example 3.11
Lemma 3.9 does not hold if the rules are not deterministic. For instance, the
formula ${\mathtt{ls}}(x,y)$ (with the rules given in the introduction) has
two models $(\mathfrak{s},\mathfrak{h})$ and
$(\mathfrak{s},\mathfrak{h}^{\prime})$ where $\mathfrak{h}^{\prime}$ is a
strict subheap of $\mathfrak{h}$: $\mathfrak{s}(x)=\ell_{1}$,
$\mathfrak{s}(y)=\ell_{2}$,
$\mathfrak{h}=\\{\ell_{1}\mapsto(\ell_{2}),\ell_{2}\mapsto(\ell_{2})\\}$ and
$\mathfrak{h}^{\prime}=\\{\ell_{1}\mapsto(\ell_{2})\\}$. Intuitively, the
formula ${\mathtt{ls}}(y,y)$ (which is useful to derive ${\mathtt{ls}}(x,y)$)
can be derived by any of the two rules of ${\mathtt{ls}}$, yielding two
different models. In contrast ${\mathtt{als}}(x,y)$ (with the rules of Example
2.10) has only one model with the store $\mathfrak{s}$ and a heap included in
$\mathfrak{h}$, namely $(\mathfrak{s},\mathfrak{h}^{\prime})$.
## 4 Proof Procedure
From now on, we consider a fixed $\mathtt{loc}$-deterministic set of
$\mathtt{P}$-rules $\mathfrak{R}$, satisfying Assumptions 2.2 and 2.16. For
technical convenience, we also assume that $\mathfrak{R}$ is nonempty and that
every constant in $\mathcal{C}$ occurs in a rule in $\mathfrak{R}$.
### 4.1 Some Basic Properties of $\mathtt{P}$-Rules
We begin by introducing some definitions and deriving straightforward
consequences of the definition of $\mathtt{P}$-rules. We shall denote by
$\mathit{alloc}(\lambda)$ the multiset of variables allocated by a formula
$\lambda$:
###### Definition 4.1
For every formula $\lambda$, we denote by $\mathit{alloc}(\lambda)$ the
multiset of variables $x$ such that $\lambda$ contains a spatial atom with
root $x$.
Lemma 4.2 states that the variables in $\mathit{alloc}(\lambda)$ are
necessarily allocated in every model of $\lambda$, which entails (Corollary
4.4) that they must be associated with pairwise distinct locations. Moreover,
a formula distinct from $\mathit{emp}$ has at least one root, hence allocates
at least one variable (Corollary 4.6).
###### Lemma 4.2
Let $\lambda$ be a formula. If
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\lambda$ and
$x\in\mathit{alloc}(\lambda)$ then
$\mathfrak{s}(x)\in\mathit{dom}(\mathfrak{h})$.
###### Proof 4.3
By hypothesis, $\lambda$ is of the form $(\alpha*\phi)\curlywedge\xi$ where
$\alpha$ is a spatial atom with root $x$. Thus
$(\mathfrak{s},\mathfrak{h})\models\alpha*\phi$ and there exist disjoint heaps
$\mathfrak{h}_{1},\mathfrak{h}_{2}$ such that
$(\mathfrak{s},\mathfrak{h}_{1})\models_{\mathfrak{R}}\alpha$,
$(\mathfrak{s},\mathfrak{h}_{2})\models_{\mathfrak{R}}\phi$, and
$\mathfrak{h}=\mathfrak{h}_{1}\uplus\mathfrak{h}_{2}$. Since the root of
$\alpha$ is $x$, $\alpha$ is either of the form $x\mapsto\bm{y}$ or of the
form $p(x,\bm{y})$ where $p\in\mathcal{P}$. In the former case, it is clear
that
$\mathfrak{s}(x)\in\mathit{dom}(\mathfrak{h}_{1})\subseteq\mathit{dom}(\mathfrak{h})$
since $(\mathfrak{s},\mathfrak{h}_{1})\models_{\mathfrak{R}}\alpha$. In the
latter case, we have $\alpha\Leftarrow_{\mathfrak{R}}\lambda^{\prime}$ and
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}\lambda^{\prime}$,
where $\mathfrak{s}^{\prime}$ is an associate of $\mathfrak{s}$ w.r.t.
$\mathcal{V}(\lambda^{\prime})\setminus\mathcal{V}(\alpha)$. Since
$\mathfrak{R}$ is a set of $\mathtt{P}$-rules, necessarily $\lambda^{\prime}$
contains a points-to atom of the form $x\mapsto\bm{z}$, which entails that
$\mathfrak{s}^{\prime}(x)\in\mathit{dom}(\mathfrak{h}_{1})$, hence
$\mathfrak{s}(x)\in\mathit{dom}(\mathfrak{h})$.
###### Corollary 4.4
Let $\lambda$ be a formula and let $(\mathfrak{s},\mathfrak{h})$ be an
$\mathfrak{R}$-model of $\lambda$. If
$\\{x,y\\}\subseteq_{m}\mathit{alloc}(\lambda)$ then
$\mathfrak{s}(x)\not=\mathfrak{s}(y)$. In particular, if
$\mathit{alloc}(\lambda)$ contains two occurrences of the same variable $x$
then $\lambda$ is unsatisfiable.
###### Proof 4.5
By definition, $\lambda$ is of the form
$(\alpha_{1}*\alpha_{2}*\phi)\curlywedge\xi$, where $\alpha_{1}$ and
$\alpha_{2}$ are spatial atoms of roots $x$ and $y$, respectively, with
$\mathit{alloc}(\alpha_{1})=\\{x\\}$ and $\mathit{alloc}(\alpha_{2})=\\{y\\}$.
If $\lambda$ admits a model $(\mathfrak{s},\mathfrak{h})$, then there exists
disjoint heaps $\mathfrak{h}_{1},\mathfrak{h}_{2},\mathfrak{h}^{\prime}$ such
that
$\mathfrak{h}=\mathfrak{h}_{1}\uplus\mathfrak{h}_{2}\uplus\mathfrak{h}^{\prime}$,
$(\mathfrak{s},\mathfrak{h}_{i})\models_{\mathfrak{R}}\alpha_{i}$ (for
$i=1,2$) and $(\mathfrak{s},\mathfrak{h}^{\prime})\models_{\mathfrak{R}}\phi$.
By Lemma 4.2 we have $\mathfrak{s}(x)\in\mathit{dom}(\mathfrak{h}_{1})$ and
$\mathfrak{s}(y)\in\mathit{dom}(\mathfrak{h}_{2})$, thus
$\mathfrak{s}(x)\not=\mathfrak{s}(y)$ since $\mathfrak{h}_{1}$ and
$\mathfrak{h}_{2}$ are disjoint.
###### Corollary 4.6
Let $\phi$ be a spatial formula and let $(\mathfrak{s},\mathfrak{h})$ be an
$\mathfrak{R}$-model of $\phi$. If $\phi\not=\mathit{emp}$ then
$\mathfrak{h}\not=\emptyset$.
###### Proof 4.7
Since $\phi\not=\mathit{emp}$, necessarily $\phi$ contains at least one atom
$\alpha$, thus $\mathit{root}(\alpha)\in\mathit{alloc}(\phi)$. Then the result
follows immediately from Lemma 4.2.
Corollary 4.4 motivates the following definition, which provides a simple
syntactic criterion to identify some formulas that cannot be satisfiable, due
to the fact that the same variable is allocated twice.
###### Definition 4.8
A formula $\lambda$ is heap-unsatisfiable if $\mathit{alloc}(\lambda)$
contains two occurrences of the same variable. Otherwise, it is heap-
satisfiable.
The next proposition states that every location that is referred to in the
heap of some model of $\lambda$ must be reachable from one of the roots of
$\lambda$. This follows from the fact that, by definition of
$\mathtt{P}$-rules, the set of allocated locations has a tree-shaped
structure: the root of each atom invoked in an inductive rule must be
connected to the location allocated by the rule (see Condition 2 in Definition
2.9).
###### Proposition 4.9
Let $\lambda$ be a symbolic heap and let $(\mathfrak{s},\mathfrak{h})$ be an
$\mathfrak{R}$-model of $\lambda$. For every
$\ell\in\mathit{ref}(\mathfrak{h})$, there exists
$x\in\mathit{alloc}(\lambda)$ such that
$\mathfrak{s}(x)\rightarrow_{\mathfrak{h}}^{*}\ell$.
###### Proof 4.10
The proof is by induction on the satisfiability relation. We establish the
result also for spatial formulas and pure formulas.
* •
If $\lambda=\mathit{emp}$ or is $\lambda$ is a pure formula then
$\mathit{ref}(\lambda)=\emptyset$ hence the proof is immediate.
* •
If $\lambda=y_{0}\mapsto(y_{1},\dots,y_{n})$, then
$\mathfrak{h}=\\{(\mathfrak{s}(y_{0}),\dots,\mathfrak{s}(y_{n}))\\}$, by
Definition 2.4, thus $\mathit{ref}(\mathfrak{h})=\\{\mathfrak{s}(y_{i})\mid
i=0,\dots,n,\text{and $y_{i}$ is of sort $\mathtt{loc}$}\\}$ and
$\mathfrak{s}(y_{0})\rightarrow_{\mathfrak{h}}^{*}\mathfrak{s}(y_{i})$, for
all $i=1,\dots,n$ such that $y_{i}$ is of sort $\mathtt{loc}$. By Definition
4.1 $\mathit{alloc}(\lambda)=\\{y_{0}\\}$, thus the proof is completed.
* •
If $\lambda=\phi\curlywedge\xi$, where $\phi$ is a spatial formula and $\xi$
is a pure formula distinct from $\top$, then we have
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\phi$ and
$\mathit{alloc}(\lambda)=\mathit{alloc}(\phi)$, hence the result follows
immediately from the induction hypothesis.
* •
If $\lambda=\phi_{1}*\phi_{2}$, then there exist two disjoint heaps
$\mathfrak{h}_{1},\mathfrak{h}_{2}$ such that
$\mathfrak{h}=\mathfrak{h}_{1}\uplus\mathfrak{h}_{2}$ and
$(\mathfrak{s},\mathfrak{h}_{i})\models_{\mathfrak{R}}\phi_{i}$, for all
$i=1,2$. If $\ell\in\mathit{ref}(\mathfrak{h})$ then necessarily
$\ell\in\mathit{ref}(\mathfrak{h}_{i})$ for some $i=1,2$. By the induction
hypothesis, we deduce that there exists $x\in\mathit{alloc}(\phi_{i})$ such
that $\mathfrak{s}(x)\rightarrow_{\mathfrak{h}_{i}}^{*}\ell$. Since
$\mathit{alloc}(\phi_{i})\subseteq_{m}\mathit{alloc}(\phi)$ and
$\rightarrow_{\mathfrak{h}_{i}}^{*}\subseteq\rightarrow_{\mathfrak{h}}^{*}$ by
Proposition 2.1, we obtain the result.
* •
If $\lambda$ is a predicate atom, then
$\lambda\Leftarrow_{\mathfrak{R}}\gamma$ and
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}\gamma$ for some
formula $\gamma$ and some associate $\mathfrak{s}^{\prime}$ of $\mathfrak{s}$
w.r.t. $\mathcal{V}(\gamma)\setminus\mathcal{V}(\lambda)$. Let
$\ell\in\mathit{ref}(\mathfrak{h})$. By the induction hypothesis, there exists
$x\in\mathit{alloc}(\gamma)$ such that
$\mathfrak{s}(x)\rightarrow_{\mathfrak{h}}^{*}\ell$. By definition, $x$ is the
root of some atom $\alpha$ in $\gamma$. If $\alpha$ is a points-to atom, then
since $\lambda\Leftarrow_{\mathfrak{R}}\gamma$ is an instance of a rule in
$\mathfrak{R}$ and all rules are $\mathtt{P}$-rules, $x$ must be the root of
$\lambda$; in this case $x\in\mathit{alloc}(\lambda)$ and the proof is
completed. Otherwise, $x$ is the root of a spatial atom in $\gamma$, and,
because all rules are $\mathtt{P}$-rules, $\gamma$ must contain an atom of the
form $y_{0}\mapsto(y_{1},\dots,y_{n})$, such that
$y_{0}=\mathit{root}(\lambda)$ and $y_{i}=x$, for some $i=1,\dots,n$. Since
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}\gamma$, we have
$(\mathfrak{s}(y_{0}),\dots,\mathfrak{s}(y_{n}))\in\mathfrak{h}$, hence
$\mathfrak{s}(y_{0})\rightarrow_{\mathfrak{h}}\mathfrak{s}(x)$. Using the fact
that $\mathfrak{s}(x)\rightarrow_{\mathfrak{h}}^{*}\ell$, we deduce that
$\mathfrak{s}(y_{0})\rightarrow_{\mathfrak{h}}^{*}\ell$, hence the proof is
completed since $\mathit{alloc}(\lambda)=\\{y_{0}\\}$.
The next lemma asserts a key property of the considered formulas: all the
locations occurring in the heap of a model of some formula $\phi$ are either
allocated or associated with a variable that is free in $\phi$. This follows
from the definition of $\mathtt{P}$-rules: all variables of sort
$\mathtt{loc}$ that are existentially quantified in an inductive rule must be
allocated (at the next recursive call). Recall that spatial formulas contain
no quantifiers.
###### Lemma 4.11
Let $\phi$ be a spatial formula and let $(\mathfrak{s},\mathfrak{h})$ be an
$\mathfrak{R}$-model of $\phi$. Then the following inclusion holds:
${\mathit{ref}(\mathfrak{h})\subseteq\mathit{dom}(\mathfrak{h})\cup\mathfrak{s}(\mathcal{V}(\phi))}$.
###### Proof 4.12
The proof is by induction on the relation $\models_{\mathfrak{R}}$. Note that
as $\phi$ is spatial, it contains no occurrence of $\approx$, $\not\approx$,
$\curlywedge$ and $\wedge$.
* •
If $\phi$ is of the form $y_{0}\mapsto(y_{1},\dots,y_{n})$ then by definition
$\mathfrak{h}=\\{(\mathfrak{s}(y_{0}),\dots,\mathfrak{s}(y_{n}))\\}$ and
$\mathit{ref}(\mathfrak{h})=\\{\mathfrak{s}(y_{0}),\dots,\mathfrak{s}(y_{n})\\}=\mathfrak{s}(\mathcal{V}(\phi))$.
* •
If $\phi=\mathit{emp}$ then $\mathfrak{h}=\emptyset$ thus
$\mathit{ref}(\mathfrak{h})=\emptyset$ and the proof is immediate.
* •
If $\phi$ is a predicate atom then we have
$\phi\Leftarrow_{\mathfrak{R}}\psi\curlywedge\xi$, and
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}\psi$, for some
associate $\mathfrak{s}^{\prime}$ of $\mathfrak{s}$ w.r.t.
$\mathcal{V}(\psi\curlywedge\xi)\setminus\mathcal{V}(\phi)$. Let
$\ell\in\mathit{ref}(\mathfrak{h})\setminus\mathit{dom}(\mathfrak{h})$. By the
induction hypothesis, $\ell=\mathfrak{s}^{\prime}(x)$ for some variable
$x\in\mathcal{V}(\psi)$. If $x\in\mathcal{V}(\phi)$ then necessarily
$\mathfrak{s}^{\prime}(x)=\mathfrak{s}(x)$, thus
$\ell\in\mathfrak{s}(\mathcal{V}(\phi))$ and the proof is completed.
Otherwise, by Definition 2.9, since all the rules in $\mathfrak{R}$ are
$\mathtt{P}$-rules; $x$ occurs as the root of some predicate atom in $\psi$,
i.e., $x\in\mathit{alloc}(\psi)$. By Lemma 4.2 we deduce that
$\mathfrak{s}^{\prime}(x)\in\mathit{dom}(\mathfrak{h})$, i.e.,
$\ell\in\mathit{dom}(\mathfrak{h})$ which contradicts our assumption.
* •
If $\phi$ is of the form $\phi_{1}*\phi_{2}$ then there exist disjoint heaps
$\mathfrak{h}_{1},\mathfrak{h}_{2}$ such that
$(\mathfrak{s},\mathfrak{h}_{i})\models_{\mathfrak{R}}\phi_{i}$ (for $i=1,2$)
and $\mathfrak{h}=\mathfrak{h}_{1}\uplus\mathfrak{h}_{2}$. Let
$\ell\in\mathit{ref}(\mathfrak{h})\setminus\mathit{dom}(\mathfrak{h})$.
Necessarily we have $\ell\in\mathit{ref}(\mathfrak{h}_{i})$, for some $i=1,2$
and $\ell\not\in\mathit{dom}(\mathfrak{h}_{i})$, hence by the induction
hypothesis we deduce that $\ell=\mathfrak{s}(x)$, for some
$x\in\mathcal{V}(\phi_{i})$. Since
$\mathcal{V}(\phi_{i})\subseteq\mathcal{V}(\phi)$, the proof is completed.
### 4.2 A Restricted Entailment Relation
We introduce a simple syntactic criterion, used in the inference rules of
Section 4.3, that is sufficient to ensure that a given pure formula $\xi$
holds in every counter-model of a sequent with left-hand side $\lambda$ and
multiset of variables $V$. The idea is to test that $\xi$ either occurs in
$\lambda$, is trivial, or is a disequation entailed by the fact that the
considered store must be injective on $\mathit{alloc}(\lambda)\cup V$ (using
Definition 2.18 and Corollary 4.4). Lemma 4.15 states that the relation
satisfies the expected property.
###### Definition 4.13
Let $\lambda$ be a symbolic heap, $\xi$ be a pure formula and let $V$ be a
multiset of variables. We write $\lambda\triangleright_{V}\xi$ if for every
atom $\zeta$ occurring in $\xi$, one of the following conditions holds:
1. 1.
$\zeta$ occurs in $\lambda$;
2. 2.
either $\zeta=(t\approx t)$ for some variable $t$, or $\zeta=(t_{1}\not\approx
t_{2})$ and $t_{1},t_{2}$ are distinct constants;
3. 3.
$\zeta=(x_{1}\not\approx x_{2})$ modulo commutativity, and one of the
following holds: $\\{x_{1},x_{2}\\}\subseteq_{m}\mathit{alloc}(\lambda)$,
($x_{1}\in\mathit{alloc}(\lambda)$ and $x_{2}\in V$) or
$\\{x_{1},x_{2}\\}\subseteq_{m}V$. This is equivalent to stating that
$\\{x_{1},x_{2}\\}\subseteq_{m}\mathit{alloc}(\lambda)+V$ where
$\mathit{alloc}(\lambda)+V$ denotes as usual the union of the multisets
$\mathit{alloc}(\lambda)$ and $V$.
###### Example 4.14
Consider the symbolic heap $\lambda=(p(x,y)*q(z))\curlywedge x\not\approx y$,
and let $V=\\{u\\}$. We have $\lambda\triangleright_{V}x\not\approx y\wedge
x\not\approx z\wedge x\not\approx u$. Indeed, $x$ and $z$ are necessarily
distinct since they are allocated by distinct atoms $p(x,y)$ and $q(z)$ (as,
by definition of the $\mathtt{P}$-rules, every predicate allocates it first
parameter) $x$ cannot be equal to $u$ as $u\in V$ and $V$ is intended to
denote a set of non-allocated variables (see Definition 2.18) and $x$ is
allocated, and $x$ cannot be equal to $y$ as the disequation $x\not\approx y$
occurs in $\lambda$.
###### Lemma 4.15
Let $\lambda$ be a symbolic heap and let $\xi$ be a pure formula such that
$\lambda\triangleright_{V}\xi$. For every structure
$(\mathfrak{s},\mathfrak{h})$, if
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\lambda$,
$\mathfrak{s}(V)\cap\mathit{dom}(\mathfrak{h})=\emptyset$ and $\mathfrak{s}$
is injective on $V$ then $\mathfrak{s}\models_{\mathfrak{R}}\xi$.
###### Proof 4.16
We show that $(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\zeta$, for all
atoms $\zeta$ in $\xi$. We consider each case in Definition 4.13 separately:
1. 1.
If $\zeta$ occurs in $\lambda$ then since
$(\mathfrak{s},\mathfrak{h})\models\gamma$, necessarily
$\mathfrak{s}\models_{\mathfrak{R}}\zeta$.
2. 2.
If $\zeta=(t\approx t)$ then it is clear that $\mathfrak{s}\models\zeta$. If
$\zeta=(t_{1}\not\approx t_{2})$ and $t_{1},t_{2}$ are distinct constants then
$\mathfrak{s}(t_{1})\not=\mathfrak{s}(t_{2})$ since all stores are injective
on constants by definition.
3. 3.
If $\zeta=x_{1}\not\approx x_{2}$ and
$\\{x_{1},x_{2}\\}\subseteq_{m}\mathit{alloc}(\lambda)$ then by Corollary 4.4,
we get $\mathfrak{s}(x_{1})\not=\mathfrak{s}(x_{2})$ since
$\mathfrak{s}(x_{1})$ and $\mathfrak{s}(x_{2})$ must be allocated in disjoint
heaps. Thus $\mathfrak{s}\models_{\mathfrak{R}}\zeta$. If
$x_{1}\in\mathit{alloc}(\phi)$ and $x_{2}\in V$ then we have
$\mathfrak{s}(x_{1})\in\mathit{dom}(\mathfrak{h})$ by Lemma 4.2, and since
$\mathfrak{s}(V)\cap\mathit{dom}(\mathfrak{h})=\emptyset$, we deduce that
$\mathfrak{s}\models_{\mathfrak{R}}\zeta$. Finally, if
$\\{x_{1},x_{2}\\}\subseteq_{m}V$ then
$\mathfrak{s}(x_{1})\not=\mathfrak{s}(x_{2})$, because $\mathfrak{s}$ is
injective on $V$ by hypothesis.
### 4.3 Inference Rules
We consider the rules represented in Figure 1. The rules apply modulo AC, they
are intended to be applied bottom-up: a rule is applicable on a sequent
$\lambda\vdash_{\mathfrak{R}}^{V}\gamma$ if there exists an instance of the
rule the conclusion of which is $\lambda\vdash_{\mathfrak{R}}^{V}\gamma$. We
recall that an inference rule is sound if the validity of the premises entails
the validity of the conclusion, and invertible if the converse holds.
###### Remark 4.17
The application conditions given in Figure 1 are the most general ones
ensuring that the rules are sound. Additional application conditions will be
provided afterwards (see Definition 4.43) to obtain a proof procedure that
runs in polynomial time. The latter conditions are rather complex, and in our
opinion introducing them at this point could hinder the understanding of the
rules.
$\repl{\phi}{x}{t}\curlywedge\repl{\xi}{x}{t}\vdash_{\mathfrak{R}}^{\repl{V}{x}{t}}\repl{\gamma}{x}{t}$
R: if $x\in\mathcal{V}$ $\phi\curlywedge(x\approx
t\wedge\xi)\vdash_{\mathfrak{R}}^{V}\gamma$
$\lambda\vdash_{\mathfrak{R}}^{V}\psi\curlywedge\zeta$ E: if
$\lambda\triangleright_{V}\zeta^{\prime}$
$\lambda\vdash_{\mathfrak{R}}^{V}\psi\curlywedge(\zeta\wedge\zeta^{\prime})$
$\phi_{1}\curlywedge\xi_{1}\vdash_{\mathfrak{R}}^{V\cup\mathit{alloc}(\phi_{2})}\psi_{1}\curlywedge\zeta_{1}$
$\phi_{2}\curlywedge\xi_{2}\vdash_{\mathfrak{R}}^{V\cup\mathit{alloc}(\phi_{1})}\psi_{2}\curlywedge\zeta_{2}$
S: if $\phi_{1}\not=\mathit{emp}$ and $\phi_{2}\not=\mathit{emp}$
$(\phi_{1}*\phi_{2})\curlywedge(\xi_{1}\wedge\xi_{2})\vdash_{\mathfrak{R}}^{V}(\psi_{1}*\psi_{2})\curlywedge(\zeta_{1}\wedge\zeta_{2})$
$(\phi^{\prime}_{1}*\phi)\curlywedge(\xi^{\prime}_{1}\wedge\xi)\vdash_{\mathfrak{R}}^{V}\gamma$
$\dots$
$(\phi^{\prime}_{n}*\phi)\curlywedge(\xi^{\prime}_{n}\wedge\xi)\vdash_{\mathfrak{R}}^{V}\gamma$
U: if
$p(\bm{t})\rightsquigarrow_{\mathfrak{R}}\\{\phi^{\prime}_{i}\curlywedge\xi^{\prime}_{i}\mid
i=1,\dots,n\\}$
$(p(\bm{t})*\phi)\curlywedge\xi\vdash_{\mathfrak{R}}^{V}\gamma$
$(x\mapsto(y_{1},\dots,y_{k})*\phi)\curlywedge\xi\vdash_{\mathfrak{R}}^{V}(x\mapsto(y_{1},\dots,y_{k})*\psi^{\prime}\sigma*\psi)\curlywedge\zeta$
I: if all the conditions below hold:
$(x\mapsto(y_{1},\dots,y_{k})*\phi)\curlywedge\xi\vdash_{\mathfrak{R}}^{V}(p(x,\bm{z})*\psi)\curlywedge\zeta$
(i)
$p(x,\bm{z})\Leftarrow_{\mathfrak{R}}(x\mapsto(u_{1},\dots,u_{k})*\psi^{\prime})\curlywedge\zeta^{\prime}$;
(ii) $\sigma$ is a substitution such that
$\mathit{dom}(\sigma)\subseteq\\{u_{1},\dots,u_{k}\\}\setminus(\\{x\\}\cup\bm{z})$;
(iii) $\sigma(u_{i})=y_{i}$, for all $i\in\\{1,\dots,k\\}$; and (iv)
$(x\mapsto(y_{1},\dots,y_{k})*\phi)\curlywedge\xi\triangleright_{V}\zeta^{\prime}\sigma$
$\phi\curlywedge\xi\vdash_{\mathfrak{R}}^{V}\gamma$ W: if one of the
conditions below holds:
$\phi\curlywedge(\xi\wedge\xi^{\prime})\vdash_{\mathfrak{R}}^{V}\gamma$
(i) $\xi^{\prime}=x\not\approx y$ and $\phi\triangleright_{V}x\not\approx y$;
or (ii) $\xi^{\prime}=\bigwedge_{i=1}^{n}x\not\approx y_{i}$ and
$x\not\in\mathcal{V}(\phi)\cup\mathcal{V}(\xi)\cup\mathcal{V}(\gamma)\cup\\{y_{1},\dots,y_{n}\\}$
$\repl{\phi}{x}{y}\curlywedge\repl{\xi}{x}{y}\vdash_{\mathfrak{R}}^{\repl{V}{x}{y}}\repl{\gamma}{x}{y}$
$\phi\curlywedge(\xi\wedge x\not\approx y)\vdash_{\mathfrak{R}}^{V}\gamma$ C:
if $x,y\in\mathcal{V}_{\mathtt{loc}}$
$\phi\curlywedge\xi\vdash_{\mathfrak{R}}^{V}\gamma$
$\lambda\vdash_{\mathfrak{R}}^{V}\gamma$ V: if
$x\not\in\mathcal{V}(\lambda)\cup\mathcal{V}(\gamma)$
$\lambda\vdash_{\mathfrak{R}}^{V\cup\\{x\\}}\gamma$
Figure 1: The Inference rules
We provide some examples and explanations concerning the inference rules.
###### Example 4.18
Rules R (replacement) and E (elimination) handle equational reasoning. For
instance, given the sequent $(p(x,u)*p(y,u))\curlywedge(u\approx
v)\vdash_{\mathfrak{R}}^{\emptyset}(p(x,u)*p(y,v))\curlywedge(x\not\approx
y)$, one may first apply R, yielding:
$p(x,u)*p(y,u)\vdash_{\mathfrak{R}}^{\emptyset}(p(x,u)*p(y,u))\curlywedge(x\not\approx
y)$. As $\\{x,y\\}\subseteq_{m}\mathit{alloc}(p(x,u)*p(y,x))$, E applies,
which yields the trivially valid sequent
$p(x,u)*p(y,u)\vdash_{\mathfrak{R}}^{\emptyset}p(x,u)*p(y,u)$.
###### Example 4.19
Rule S (separation) applies on the sequent
$p(x)*p(y)\vdash_{\mathfrak{R}}^{\emptyset}q(x)*r(y)$, yielding
$p(x)\vdash_{\mathfrak{R}}^{\\{y\\}}q(x)$ and
$p(y)\vdash_{\mathfrak{R}}^{\\{x\\}}r(y)$ (as $\\{x\\}=\mathit{alloc}(p(x))$
and $\\{y\\}=\mathit{alloc}(p(y))$). The addition of $y$ (resp. $x$) to the
variables associated with the sequent allows us to keep track of the fact that
these variables cannot be allocated in $p(x)$ (resp. $p(y)$) as they are
already allocated in the other part of the heap. Note that the rule also
yields $p(x)\vdash_{\mathfrak{R}}^{\\{y\\}}r(y)$ and
$p(y)\vdash_{\mathfrak{R}}^{\\{x\\}}q(x)$, however as we shall see later (see
Definition 4.34) the latter premises cannot be valid and this application can
be dismissed.
###### Example 4.20
Rules U (unfolding) and I (imitation) both unfold inductively defined
predicate symbols. U unfolds predicates occurring on the left-hand side of a
sequent, yielding one premise for each inductive rule. In contrast, I applies
on the right-hand side and selects one rule in a non-deterministic way
(provided it fulfills the rule application condition), yielding exactly one
premise. Let $\mathfrak{R}$ be the following set of rules, where
$\mathtt{a},\mathtt{b}$ are constant symbols and $z,z_{1},z_{2}$ are variables
of the same sort as $\mathtt{a}$ and $\mathtt{b}$:
$p(x)$ | $\Leftarrow$ | $x\mapsto(\mathtt{a},x)$
---|---|---
$p(x)$ | $\Leftarrow$ | $x\mapsto(\mathtt{b},x)$
$q(x)$ | $\Leftarrow$ | $x\mapsto(z,y)$
$q(x)$ | $\Leftarrow$ | $x\mapsto(z_{1},z_{2},x)$
Rule U applies on $p(x,y)\vdash_{\mathfrak{R}}^{\emptyset}q(x)$, yielding
$x\mapsto(\mathtt{a},x)\vdash_{\mathfrak{R}}^{\emptyset}q(x)$ and
$x\mapsto(\mathtt{b},x)\vdash_{\mathfrak{R}}^{\emptyset}q(x)$. Then I applies
on both sequents, with the respective substitutions $\\{y\leftarrow
x,z\leftarrow\mathtt{a}\\}$ and $\\{y\leftarrow x,z\leftarrow\mathtt{b}\\}$,
yielding
$x\mapsto(\mathtt{a},x)\vdash_{\mathfrak{R}}^{\emptyset}x\mapsto(\mathtt{a},x)$
and
$x\mapsto(\mathtt{b},x)\vdash_{\mathfrak{R}}^{\emptyset}x\mapsto(\mathtt{b},x)$.
Note that I cannot be applied with the inductive rule $q(x)\Leftarrow
x\mapsto(z_{1},z_{2},x)$, as $(\mathtt{a},x)$ and $(\mathtt{b},x)$ are not
instances of $(z_{1},z_{2},x)$.
###### Example 4.21
Rules W (weakening) and V (variable weakening) allow to get rid of
irrelevant information, which is essential for termination. For instance one
may deduce $p(x)\vdash_{\mathfrak{R}}^{\emptyset}q(x)$ from
$p(x)\curlywedge(u\not\approx v\wedge u\not\approx
w)\vdash_{\mathfrak{R}}^{\emptyset}q(x)$. Indeed, if the former sequent admits
a counter-model, then one gets a counter-model of the latter one by
associating $u,v,w$ with pairwise distinct elements.
###### Example 4.22
Rule C performs a case analysis on $x\approx y$. It is essential to allow
further applications of Rule I in some cases. Consider the sequent
$u\mapsto(x,x)*p(x)\vdash_{\mathfrak{R}}^{\emptyset}q(u,y)$, with the rules
$q(u,y)\Leftarrow u\mapsto(y,z)*p(z)$, and
$q(u,y)\Leftarrow(u\mapsto(z,z)*p(z))\curlywedge z\not\approx y$. Rule I does
not apply because there is no substitution $\sigma$ with domain $\\{z\\}$ such
that $(x,x)=(y,z)\sigma$, and
${u\mapsto(x,x)*p(x)\not\\!\triangleright_{V}x\not\approx y}$. The rule C
yields $u\mapsto(y,y)*p(y)\vdash_{\mathfrak{R}}^{\emptyset}q(u,y)$ and
$(u\mapsto(x,x)*p(x))\curlywedge x\not\approx
y\vdash_{\mathfrak{R}}^{\emptyset}q(u,y)$. Then the rule I applies on both
sequents, yielding the premisses
${u\mapsto(y,y)*p(y)\vdash_{\mathfrak{R}}^{\emptyset}u\mapsto(y,y)*p(y)}$ and
${(u\mapsto(x,x)*p(x))\curlywedge x\not\approx
y\vdash_{\mathfrak{R}}^{\emptyset}u\mapsto(x,x)*p(x)}$.
We have the following facts, which can be verified by an inspection of the
inference rules:
###### Proposition 4.23
Consider an equality-free sequent $\lambda\vdash_{\mathfrak{R}}^{V}\gamma$
that is the conclusion of an inference rule, of which
$\lambda^{\prime}\vdash_{\mathfrak{R}}^{V^{\prime}}\gamma^{\prime}$ is a
premise.
1. 1.
No rule can introduce any equality to
$\lambda^{\prime}\vdash_{\mathfrak{R}}^{V^{\prime}}\gamma^{\prime}$.
2. 2.
If $x\in V^{\prime}\setminus V$, then the inference rule is either V or C.
3. 3.
If $\mathit{alloc}(\gamma^{\prime})\subsetneq\mathit{alloc}(\gamma)$ then the
inference rule is either S or C.
4. 4.
The only inference rules that can introduce new variables to $\gamma^{\prime}$
are I and C.
5. 5.
No rule introduces a disequation between terms of a sort distinct from
$\mathtt{loc}$.
6. 6.
The only rule that introduces a predicate atom to the right-hand side of a
premise is I.
7. 7.
If $v\in(\mathit{alloc}(\gamma)\cup
V)\setminus(\mathit{alloc}(\gamma^{\prime})\cup V^{\prime})$, then the
inference rule must be C.
###### Proof 4.24
The first six facts are straightforward to verify. Fact 7 holds because Rule
R cannot apply if no equality occurs; if rule S is applied then the variables
in $\mathit{alloc}(\gamma)\setminus\mathit{alloc}(\gamma^{\prime})$ must occur
in $V^{\prime}$; Rule I deletes a predicate atom but introduces a points-to
atom with the same root and rule V cannot be applied on variables occurring
in $\mathcal{V}{\gamma}$.
We establish additional basic properties of the inference rules. All the rules
are sound and invertible, except for rule Sthat is only sound. The results
follow easily from the semantics, except for the invertibility of I, which
crucially depends on the fact that rules are deterministic. Indeed, I unfolds
one atom on the right-hand side by selecting one specific inductive rule. In
our case, at most one rule can be applied, which ensures that equivalence is
preserved. This is a crucial point because otherwise one would need to
consider disjunctions of formulas on the right-hand side of the sequent (one
disjunct for each possible rule), which would make the procedure much less
efficient.
###### Example 4.25
Consider the sequent
$x\mapsto(y)*y\mapsto(z)\vdash_{\mathfrak{R}}^{\emptyset}p(x,z)$, with the
rules
$p(x,z)$ | $\Leftarrow$ | $x\mapsto(y)*q(y,z)$ | $p(x,z)$ | $\Leftarrow$ | $x\mapsto(y)*q^{\prime}(y,y)$
---|---|---|---|---|---
$q(y,z)$ | $\Leftarrow$ | $y\mapsto(z)\curlywedge y\not\approx z$ | $q^{\prime}(y,z)$ | $\Leftarrow$ | $y\mapsto(z)$
It is clear that the rules are not deterministic, as there is an overlap
between the two rules associated with $p$. Applying rule I on the above
sequent yields either
$x\mapsto(y)*y\mapsto(z)\vdash_{\mathfrak{R}}^{\emptyset}x\mapsto(y)*q(y,z)$
or
$x\mapsto(y)*y\mapsto(z)\vdash_{\mathfrak{R}}^{\emptyset}x\mapsto(y)*q^{\prime}(y,y)$.
None of these two possible premises is valid, although the initial sequent is
valid. This shows that I is not invertible (although it is still sound) when
$\mathfrak{R}$ is not deterministic. The intuition is that it is not possible
to decide which rule must be applied on $p$ before deciding whether $z$ is
equal to $y$ or not.
###### Lemma 4.26
The rules R, E, U, W, V and C and I are sound and invertible. More
specifically, for all heaps $\mathfrak{h}$, the conclusion of the rule admits
a counter-model $(\mathfrak{s},\mathfrak{h})$ iff one of the premises admits a
counter-model $(\mathfrak{s}^{\prime},\mathfrak{h})$.
###### Proof 4.27
We consider each rule separately (we refer to the definitions of the rules for
notations) and establish the equivalence of the lemma. We recall (Definition
2.18) that a counter-model of a sequent is a structure
$(\mathfrak{s},\mathfrak{h})$ that validates the antecedent, falsifies the
conclusion, and is such that the store is injective on $V$ and no variable in
$V$ is allocated.
* R
Assume that
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\phi\curlywedge(x\approx
t\wedge\xi)$, that
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}\gamma$, that
$\mathfrak{s}(V)\cap\mathit{dom}(\mathfrak{h})=\emptyset$ and that
$\mathfrak{s}$ is injective on $V$. Then we have
$\mathfrak{s}(x)=\mathfrak{s}(t)$, thus
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\repl{\phi}{x}{t}\curlywedge\repl{\xi}{x}{t}$
and
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}\repl{\gamma}{x}{t}$.
For all $y\in\repl{V}{x}{t}$, we have either $y\in V$ and
$\mathfrak{s}(y)\not\in\mathit{dom}(\mathfrak{h})$, or $x\in V$ and $y=t$,
thus
$\mathfrak{s}(y)=\mathfrak{s}(t)=\mathfrak{s}(x)\not\in\mathit{dom}(\mathfrak{h})$.
Finally, assume (for the sake of contradiction) that
$\\{u,v\\}\subseteq\repl{V}{x}{t}$ with $\mathfrak{s}(u)=\mathfrak{s}(v)$.
Then there exist variables $u^{\prime},v^{\prime}$ such that
$\\{u^{\prime},v^{\prime}\\}\subseteq V$, with $\repl{u^{\prime}}{x}{t}=u$ and
$\repl{v^{\prime}}{x}{t}=v$. If $u^{\prime}$ and $v^{\prime}$ are both equal
to $x$ or both distinct from $x$ then we have
$\mathfrak{s}(u^{\prime})=\mathfrak{s}(v^{\prime})$, which contradicts the
fact that $\mathfrak{s}$ is injective on $V$. If $u^{\prime}=x$ and
$v^{\prime}\neq x$, then we have
$\mathfrak{s}(v^{\prime})=\mathfrak{s}(t)=\mathfrak{s}(x)$, again
contradicting the fact that $\mathfrak{s}$ is injective on $V$. The proof is
similar when $u^{\prime}\neq x$ and $v^{\prime}=x$. Consequently,
$(\mathfrak{s},\mathfrak{h})$ is also a counter-model of the premise.
Conversely, assume that
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\repl{\phi}{x}{t}\curlywedge\repl{\xi}{x}{t}$;
$\mathfrak{s}(\repl{V}{x}{t})\cap\mathit{dom}(\mathfrak{h})=\emptyset$; the
store $\mathfrak{s}$ is injective on $\repl{V}{x}{t}$; and
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}\repl{\gamma}{x}{t}$. We
consider the store $\mathfrak{s}^{\prime}$ such that
$\mathfrak{s}^{\prime}(x)=\mathfrak{s}(t)$ and
$\mathfrak{s}^{\prime}(y)=\mathfrak{s}(y)$ if $y\not=x$. It is clear that
$\mathfrak{s}^{\prime}\models_{\mathfrak{R}}x\approx t$,
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}\phi\curlywedge\xi$
and $(\mathfrak{s}^{\prime},\mathfrak{h})\not\models_{\mathfrak{R}}\gamma$.
For all $y\in V$, if $y\not=x$ then $y\in\repl{V}{x}{t}$ and
$\mathfrak{s}^{\prime}(y)=\mathfrak{s}(y)\not\in\mathit{dom}(\mathfrak{h})$;
otherwise $y=x$ and $t\in\repl{V}{x}{t}$, thus
$\mathfrak{s}^{\prime}(y)=\mathfrak{s}^{\prime}(x)=\mathfrak{s}(t)\not\in\mathit{dom}(\mathfrak{h})$.
There only remains to check that $\mathfrak{s}^{\prime}$ is injective on $V$,
which is done by contradiction: if this is not the case then there exists
$\\{u,v\\}\subseteq_{m}V$ such that
$\mathfrak{s}^{\prime}(u)=\mathfrak{s}^{\prime}(v)$. Hence
$\\{\repl{u}{x}{t},\repl{v}{x}{t}\\}\subseteq_{m}\repl{V}{x}{t}$, and we have
$\mathfrak{s}^{\prime}(\repl{u}{x}{t})=\mathfrak{s}(\repl{u}{x}{t})$ and
$\mathfrak{s}^{\prime}(\repl{v}{x}{t})=\mathfrak{s}(\repl{v}{x}{t})$. This
contradicts the fact that $\mathfrak{s}$ is injective on $\repl{V}{x}{t}$.
* E
Let $(\mathfrak{s},\mathfrak{h})$ be a structure such that
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\lambda$;
$\mathfrak{s}(V)\cap\mathit{dom}(\mathfrak{h})=\emptyset$; the store
$\mathfrak{s}$ is injective on $V$; and
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}\psi\curlywedge(\zeta\wedge\zeta^{\prime})$.
By the application condition of the rule we have
$\lambda\triangleright_{V}\zeta^{\prime}$, thus by Lemma 4.15, we deduce that
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\zeta^{\prime}$. Therefore
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}\psi\curlywedge\zeta$.
Conversely, it is clear that any counter-model of
$\lambda\vdash_{\mathfrak{R}}^{V}\psi\curlywedge\zeta$ is a counter-model of
$\lambda\vdash_{\mathfrak{R}}^{V}\psi\curlywedge(\zeta\wedge\zeta^{\prime})$.
* U
Assume that
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}(p(\bm{t})*\phi)\curlywedge\xi$,
$\mathfrak{s}(V)\cap\mathit{dom}(\mathfrak{h})=\emptyset$, $\mathfrak{s}$ is
injective on $V$ and
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}\gamma$. By definition,
$\mathfrak{s}\models\xi$, and there exist disjoint heaps
$\mathfrak{h}_{1},\mathfrak{h}_{2}$ such that
${(\mathfrak{s},\mathfrak{h}_{1})\models_{\mathfrak{R}}p(\bm{t})}$,
${(\mathfrak{s},\mathfrak{h}_{2})\models_{\mathfrak{R}}\phi}$ and
$\mathfrak{h}=\mathfrak{h}_{1}\uplus\mathfrak{h}_{2}$. We get that
$p(\bm{t})\Leftarrow_{\mathfrak{R}}\phi^{\prime}\curlywedge\xi^{\prime}$ and
$(\mathfrak{s}^{\prime},\mathfrak{h}_{1})\models_{\mathfrak{R}}\phi^{\prime}\curlywedge\xi^{\prime}$,
for some associate of $\mathfrak{s}$ w.r.t.
$\mathcal{V}(\phi^{\prime}\curlywedge\xi^{\prime})\setminus\mathcal{V}(p(\bm{t}))$.
We assume that
${\mathcal{V}(\phi^{\prime}\curlywedge\xi^{\prime})\cap{\mathcal{V}((p(\bm{t})*\phi)\curlywedge\xi)\subseteq\bm{t}}}$
(by $\alpha$-renaming). We get $\mathfrak{s}^{\prime}\models\xi^{\prime}$ and
$(\mathfrak{s}^{\prime},\mathfrak{h}_{1}\uplus\mathfrak{h}_{2})\models_{\mathfrak{R}}\phi^{\prime}*\phi$,
hence
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}(\phi^{\prime}*\phi)\curlywedge(\xi^{\prime}\wedge\xi)$.
Furthermore, the formula $\phi^{\prime}\curlywedge\xi^{\prime}$ occurs in
$\\{\phi_{i}^{\prime}\curlywedge\xi_{i}^{\prime}\mid i=1,\dots,n\\}$, up to a
renaming of the variables not occurring in $\bm{t}$, by definition of
$\rightsquigarrow_{\mathfrak{R}}$. Thus $(\mathfrak{s}^{\prime},\mathfrak{h})$
is a counter-model of a sequent
$(\phi^{\prime}_{i}*\phi)\curlywedge(\xi^{\prime}_{i}\wedge\xi)\vdash_{\mathfrak{R}}^{V}\gamma$,
for some $i=1,\dots,n$.
Conversely, let $(\mathfrak{s},\mathfrak{h})$ be a structure such that
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}(\phi^{\prime}_{i}*\phi)\curlywedge(\xi^{\prime}_{i}\wedge\xi)$,
$\mathfrak{s}(V)\cap\mathit{dom}(\mathfrak{h})=\emptyset$, $\mathfrak{s}$ is
injective on $V$ and
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}\gamma$. We deduce that
$\mathfrak{s}\models\xi^{\prime}_{i}\wedge\xi$ and there exist disjoint heaps
$\mathfrak{h}_{1},\mathfrak{h}_{2}$ such that
$(\mathfrak{s},\mathfrak{h}_{1})\models_{\mathfrak{R}}\phi^{\prime}_{i}$ and
$(\mathfrak{s},\mathfrak{h}_{2})\models_{\mathfrak{R}}\phi$. By the
application condition of the rule we have
$p(\bm{t})\Leftarrow_{\mathfrak{R}}\phi^{\prime}_{i}\curlywedge\xi^{\prime}_{i}$,
thus $(\mathfrak{s},\mathfrak{h}_{1})\models_{\mathfrak{R}}p(\bm{t})$ by
definition of the semantics of predicate atoms (since $\mathfrak{s}$ is an
extension of itself). Consequently,
$(\mathfrak{s},\mathfrak{h}_{1}\uplus\mathfrak{h}_{2})\models_{\mathfrak{R}}p(\bm{t})*\phi$,
hence
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}(p(\bm{t})*\phi)\curlywedge\xi$,
and $(\mathfrak{s},\mathfrak{h})$ is a counter-model of
$(p(\bm{t})*\phi)\curlywedge\xi\vdash_{\mathfrak{R}}^{V}\gamma$.
* W
Assume that
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\phi\curlywedge\xi$,
$\mathfrak{s}(V)\cap\mathit{dom}(\mathfrak{h})=\emptyset$, $\mathfrak{s}$ is
injective on $V$ and
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}\gamma$. If
$\xi^{\prime}$ is $x\not\approx y$, with $\phi\triangleright_{V}x\not\approx
y$, then, by Lemma 4.15, we have $\mathfrak{s}(x)\not=\mathfrak{s}(y)$, thus
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\phi\curlywedge(\xi\wedge\xi^{\prime})$,
hence $(\mathfrak{s},\mathfrak{h})$ is a counter-model of the conclusion of
the rule. Assume that $\xi^{\prime}=\bigwedge_{i=1}^{n}x\not\approx y_{i}$
with
$x\not\in\mathcal{V}(\phi)\cup\mathcal{V}(\xi)\cup\mathcal{V}(\gamma)\cup\\{y_{1},\dots,y_{n}\\}$.
Let $\mathfrak{s}^{\prime}$ be a store such that
$\mathfrak{s}^{\prime}(y)=\mathfrak{s}(y)$ if $y\not=x$ and
$\mathfrak{s}^{\prime}(x)$ is an arbitrarily chosen location not occurring in
$\mathit{dom}(\mathfrak{h})\cup\\{\mathfrak{s}(y_{i})\mid i=1,\dots,n\\}$
(such a location exists since $\mathfrak{h}$ is finite and
$\mathfrak{U}_{\mathtt{loc}}$ is infinite). By definition, we have
$\mathfrak{s}^{\prime}(V)\cap\mathit{dom}(\mathfrak{h})=\emptyset$,
$\mathfrak{s}^{\prime}\models\bigwedge_{i=1}^{n}x\not\approx y_{i}$,
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}\phi\curlywedge\xi$
(since $\mathfrak{s}^{\prime}$ and $\mathfrak{s}$ coincide on
$\mathcal{V}(\phi)\cup\mathcal{V}(\xi)$) and
$(\mathfrak{s}^{\prime},\mathfrak{h})\not\models_{\mathfrak{R}}\gamma$ (since
$\mathfrak{s}^{\prime}$ and $\mathfrak{s}$ coincide on $\mathcal{V}(\gamma)$).
Thus $(\mathfrak{s}^{\prime},\mathfrak{h})$ is a counter-model of
$\phi\curlywedge(\xi\wedge\bigwedge_{i=1}^{n}x\not\approx
y_{i})\vdash_{\mathfrak{R}}^{V}\gamma$.
Conversely, it is clear that any counter-model of
$\phi\curlywedge(\xi\wedge\xi^{\prime})\vdash_{\mathfrak{R}}^{V}\gamma$ is a
counter-model of $\phi\curlywedge\xi\vdash_{\mathfrak{R}}^{V}\gamma$.
* C
Let $(\mathfrak{s},\mathfrak{h})$ such that
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\phi\curlywedge\xi$,
$\mathfrak{s}(V)\cap\mathit{dom}(\mathfrak{h})=\emptyset$, $\mathfrak{s}$ is
injective on $V$ and
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}\gamma$. We distinguish
two cases.
* –
If $\mathfrak{s}(x)=\mathfrak{s}(y)$ then
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\repl{\phi}{x}{y}$,
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\repl{\xi}{x}{y}$, and
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}\repl{\gamma}{x}{y}$.
Moreover,
$\mathfrak{s}(\repl{V}{x}{y})\cap\mathit{dom}(\mathfrak{h})=\mathfrak{s}(V)\cap\mathit{dom}(\mathfrak{h})=\emptyset$,
and $\mathfrak{s}$ must be injective on $\repl{V}{x}{y}$, hence
$(\mathfrak{s},\mathfrak{h})$ is a counter-model of
$\repl{\phi}{x}{y}\curlywedge\repl{\xi}{x}{y}\vdash_{\mathfrak{R}}^{\repl{V}{x}{y}}\repl{\lambda}{x}{y}$.
* –
Otherwise, we have $\mathfrak{s}\models x\not\approx y$, thus
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\phi\curlywedge(\xi\wedge
x\not\approx y)$, and $(\mathfrak{s},\mathfrak{h})$ is a counter-model of
$\phi\curlywedge(\xi\wedge x\not\approx y)\vdash_{\mathfrak{R}}^{V}\lambda$.
Conversely, if $(\mathfrak{s},\mathfrak{h})$ is a counter-model of
$\phi\curlywedge(\xi\wedge x\not\approx y)\vdash_{\mathfrak{R}}^{V}\gamma$
then it is clear that it is also a counter-model of
$\phi\curlywedge\xi\vdash_{\mathfrak{R}}^{V}\gamma$. If
$(\mathfrak{s},\mathfrak{h})$ is a counter-model of
$\repl{\phi}{x}{y}\curlywedge\repl{\xi}{x}{y}\vdash_{\mathfrak{R}}^{\repl{V}{x}{y}}\repl{\gamma}{x}{y}$,
then consider the store $\mathfrak{s}^{\prime}$ such that
$\mathfrak{s}^{\prime}(x)=\mathfrak{s}(y)$ and
$\mathfrak{s}^{\prime}(z)=\mathfrak{s}(z)$ if $z\not=x$. By definition,
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}\phi\curlywedge\xi$
and $(\mathfrak{s}^{\prime},\mathfrak{h})\not\models_{\mathfrak{R}}\gamma$.
The set
${\mathfrak{s}^{\prime}(V)\cap\mathit{dom}(\mathfrak{h})=\mathfrak{s}(\repl{V}{x}{y})\cap\mathit{dom}(\mathfrak{h})}$
is empty, and $\mathfrak{s}^{\prime}$ is injective on $V$, since
$\mathfrak{s}$ is injective on $\repl{V}{x}{y}$ and by definition
$\mathfrak{s}^{\prime}(u)=\mathfrak{s}(\repl{u}{x}{y})$, for all variables
$u$. Therefore $(\mathfrak{s}^{\prime},\mathfrak{h})$ is a counter-model of
$\phi\curlywedge\xi\vdash_{\mathfrak{R}}^{V}\gamma$.
* V
Let $(\mathfrak{s},\mathfrak{h})$ be a counter-model of
$\lambda\vdash_{\mathfrak{R}}^{V}\gamma$. Then
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\lambda$ and
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}\gamma$. Let
$\mathfrak{s}^{\prime}$ be a store such that
$\mathfrak{s}^{\prime}(x)\not\in\mathit{dom}(\mathfrak{h})\cup\mathfrak{s}(V)$
and $\mathfrak{s}^{\prime}(y)=\mathfrak{s}(y)$, if $y\not=x$. Since
$x\not\in\mathcal{V}(\lambda)\cup\mathcal{V}(\gamma)$ we have
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}\lambda$, and
$(\mathfrak{s}^{\prime},\mathfrak{h})\not\models_{\mathfrak{R}}\gamma$.
Moreover $\mathfrak{s}^{\prime}(x)\not\in\mathit{dom}(\mathfrak{h})$ and
$\mathfrak{s}^{\prime}$ is injective on $V\cup\\{x\\}$ (since $\mathfrak{s}$
is injective on $V$ and $\mathfrak{s}^{\prime}(x)\not\in V$), thus
$(\mathfrak{s}^{\prime},\mathfrak{h})$ is a counter-model of
$\lambda\vdash_{\mathfrak{R}}^{V\cup\\{x\\}}\gamma$.
Conversely, it is clear that any counter-model of
$\lambda\vdash_{\mathfrak{R}}^{V\cup\\{x\\}}\gamma$ is also a counter-model of
$\lambda\vdash_{\mathfrak{R}}^{V}\gamma$.
* I
Let $(\mathfrak{s},\mathfrak{h})$ be a counter-model of
$(x\mapsto(y_{1},\dots,y_{k})*\phi)\curlywedge\xi\vdash_{\mathfrak{R}}^{V}(p(x,\bm{z})*\psi)\curlywedge\zeta$.
By definition
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}(x\mapsto(y_{1},\dots,y_{k})*\phi)\curlywedge\xi$,
$(\mathfrak{s},\mathfrak{h})\not\models_{\mathfrak{R}}(p(x,\bm{z})*\psi)\curlywedge\zeta$,
the set ${\mathfrak{s}(V)\cap\mathit{dom}(\mathfrak{h})}$ is empty and
$\mathfrak{s}$ is injective on $V$. By the application conditions of the rule
and Lemma 4.15, we have $\mathfrak{s}\models\zeta^{\prime}\sigma$. Assume for
the sake of contradiction that $(\mathfrak{s},\mathfrak{h})$ is not a counter-
model of
$(x\mapsto(y_{1},\dots,y_{k})*\phi)\curlywedge\xi\vdash_{\mathfrak{R}}^{V}(x\mapsto(y_{1},\dots,y_{k})*\psi^{\prime}\sigma*\psi)\curlywedge\zeta$.
This entails that
${(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}(x\mapsto(y_{1},\dots,y_{k})*\psi^{\prime}\sigma*\psi)\curlywedge\zeta}$,
hence $\mathfrak{s}\models\zeta$, and there exist disjoint heaps
$\mathfrak{h}_{1},\mathfrak{h}_{2}$ and $\mathfrak{h}_{3}$ such that
$\mathfrak{h}=\mathfrak{h}_{1}\uplus\mathfrak{h}_{2}\uplus\mathfrak{h}_{3}$,
$(\mathfrak{s},\mathfrak{h}_{1})\models_{\mathfrak{R}}x\mapsto(y_{1},\dots,y_{k})$,
$(\mathfrak{s},\mathfrak{h}_{2})\models_{\mathfrak{R}}\psi^{\prime}\sigma$ and
$(\mathfrak{s},\mathfrak{h}_{3})\models_{\mathfrak{R}}\psi$. Let
$\lambda=x\mapsto(u_{1},\dots,u_{k})*\psi^{\prime})\curlywedge\zeta^{\prime}$,
so that $p(x,\bm{z})\Leftarrow_{\mathfrak{R}}\lambda$. Since all the rules in
$\mathfrak{R}$ are $\mathtt{P}$-rules, necessarily
$\mathcal{V}(\lambda)\setminus\mathcal{V}(p(x,\bm{z}))\subseteq\\{u_{1},\dots,u_{k}\\}$.
Let $\mathfrak{s}^{\prime}$ be the store defined by:
$\mathfrak{s}^{\prime}(y)=\mathfrak{s}(\sigma(y))$, for all $y\in\mathcal{V}$.
By the application condition of the rule, $\mathit{dom}(\sigma)$ is a subset
of $\\{u_{1},\dots,u_{k}\\}\setminus(\left\\{x\right\\}\cup\bm{z})$, hence
$\mathfrak{s}^{\prime}$ coincides with $\mathfrak{s}$ on all variables in
$x,\bm{z}$. We deduce that $\mathfrak{s}^{\prime}$ is an associate of
$\mathfrak{s}$ w.r.t.
$\\{u_{1},\dots,u_{k}\\}\setminus\mathcal{V}(p(x,\bm{z}))$. We have
${(\mathfrak{s},\mathfrak{h}_{1})\models_{\mathfrak{R}}x\mapsto(y_{1},\dots,y_{k})}$,
with $\sigma(x)=x$ and $\sigma(u_{i})=y_{i}$, thus
$(\mathfrak{s},\mathfrak{h}_{1})\models_{\mathfrak{R}}(x\mapsto(u_{1},\dots,u_{k}))\sigma$.
Moreover, we also have
$(\mathfrak{s},\mathfrak{h}_{2})\models_{\mathfrak{R}}\psi^{\prime}\sigma$,
hence
$(\mathfrak{s},\mathfrak{h}_{1}\uplus\mathfrak{h}_{2})\models_{\mathfrak{R}}\lambda\sigma$.
By Proposition 2.7, we get
$(\mathfrak{s}^{\prime},\mathfrak{h}_{1}\uplus\mathfrak{h}_{2})\models_{\mathfrak{R}}\lambda$.
Since $p(x,\bm{z})\Leftarrow_{\mathfrak{R}}\lambda$, we deduce that
$(\mathfrak{s},\mathfrak{h}_{1}\uplus\mathfrak{h}_{2})\models_{\mathfrak{R}}p(x,\bm{z})$,
hence $(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}p(x,\bm{z})*\psi$.
Thus
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}(p(x,\bm{z})*\psi)\curlywedge\zeta$,
which contradicts our hypothesis.
Conversely, assume that
$(x\mapsto(y_{1},\dots,y_{k})*\phi)\curlywedge\xi\vdash_{\mathfrak{R}}^{V}(p(x,\bm{z})*\psi)\curlywedge\zeta$
is valid, and let $(\mathfrak{s},\mathfrak{h})$ be an $\mathfrak{R}$-model of
$(x\mapsto(y_{1},\dots,y_{k})*\phi)\curlywedge\xi$ such that
$\mathfrak{s}(V)\cap\mathit{dom}(\mathfrak{h})=\emptyset$ and $\mathfrak{s}$
is injective on $V$. We deduce that
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}(p(x,\bm{z})*\psi)\curlywedge\zeta$,
thus $\mathfrak{s}\models\zeta$, and there exist disjoint heaps
$\mathfrak{h}_{1},\mathfrak{h}_{2}$ such that
$(\mathfrak{s},\mathfrak{h}_{1})\models_{\mathfrak{R}}(p(x,\bm{z})$ and
$(\mathfrak{s},\mathfrak{h}_{2})\models_{\mathfrak{R}}\psi$. This entails that
there exists a symbolic heap $\lambda$ and a associate $\mathfrak{s}^{\prime}$
of $\mathfrak{s}$ w.r.t.
$\mathcal{V}(\lambda)\setminus\mathcal{V}(p(x,\bm{z}))$ such that
$p(x,\bm{z})\Leftarrow_{\mathfrak{R}}\lambda$, and
$(\mathfrak{s}^{\prime},\mathfrak{h}_{1})\models_{\mathfrak{R}}\lambda$. Since
the rules in $\mathfrak{R}$ are $\mathtt{P}$-rules, $\lambda$ is of the form
$(x\mapsto(v_{1},\dots,v_{m})*\psi^{\prime\prime})\curlywedge\zeta^{\prime\prime}$.
Moreover, it is clear that
$\mathfrak{h}(\mathfrak{s}(x))=(\mathfrak{s}(y_{1}),\dots,\mathfrak{s}(y_{k}))$,
so that $m=k$ and $\mathfrak{s}^{\prime}(v_{i})=\mathfrak{s}(y_{i})$, for all
$i=1,\dots,k$. Let $\sigma^{\prime}$ be the substitution mapping every
variable in $\\{v_{1},\dots,v_{k}\\}$ not occurring in $x,\bm{z}$ to the first
variable $y_{j}$ such that $\mathfrak{s}^{\prime}(v_{i})=\mathfrak{s}(y_{j})$.
By definition, we have
$\mathfrak{s}^{\prime}=\mathfrak{s}\circ\sigma^{\prime}$, thus by Proposition
2.7, we get
$(\mathfrak{s},\mathfrak{h}_{1})\models_{\mathfrak{R}}\lambda\sigma^{\prime}$.
Assume for the sake of contradiction that the inductive rule used to derive
$\lambda$ is distinct from the one used to derive the formula
$\lambda^{\prime}=(x\mapsto(u_{1},\dots,u_{k})*\psi^{\prime})\curlywedge\zeta^{\prime}$
in the application condition of the rule. We may assume by renaming that
$\mathcal{V}(\lambda^{\prime})\cap\mathcal{V}(\lambda)\subseteq\mathcal{V}(p(x,\bm{z})$.
Let $\mathfrak{s}^{\prime\prime}$ be a store coinciding with
$\mathfrak{s}^{\prime}$ on all constants and on all variables in
$\mathcal{V}(\lambda)$, and such that, for all variables
$y\in\mathcal{V}(\lambda^{\prime})\setminus\mathcal{V}(p(x,\bm{z}))$,
$\mathfrak{s}^{\prime\prime}(y)=\mathfrak{s}(\sigma(y))$. Since
$\mathfrak{s}^{\prime}\models\zeta^{\prime\prime}$ we have
$\mathfrak{s}^{\prime\prime}\models\zeta^{\prime\prime}$. By the application
condition of the rule $\sigma(u_{i})=y_{i}$, thus
$\mathfrak{s}^{\prime\prime}\models u_{i}\approx v_{i}$, for all
$i=1,\dots,k$. Since $\mathfrak{s}\models\xi$ and
$\xi\models\zeta^{\prime}\sigma$ (still by the application condition of the
rule), we get $\mathfrak{s}\models\zeta^{\prime}\sigma$, and by Proposition
2.7 we deduce that $\mathfrak{s}\models\zeta^{\prime}$. Thus
$(u_{1},\dots,u_{k})\approx(v_{1},\dots,v_{k})\wedge\zeta^{\prime}\wedge\zeta^{\prime\prime}$
is satisfiable, which contradicts the fact that $\mathfrak{R}$ is
deterministic.
This entails that the rules applied to derive $\lambda$ and $\lambda^{\prime}$
are the same, and by renaming we may assume in this case that
$(u_{1},\dots,u_{k})=(v_{1},\dots,v_{k})$ (which entails that
${(x\mapsto(v_{1},\dots,v_{k}))\sigma}={x\mapsto(y_{1},\dots,y_{k})}$), with
$\psi^{\prime}=\psi^{\prime\prime}$ and $\zeta^{\prime}=\zeta^{\prime\prime}$.
Using the fact that $\sigma(u_{i})=y_{i}$ and
$\mathfrak{s}^{\prime}(v_{i})=\mathfrak{s}(y_{i})$, for all $i=1,\dots,k$, it
is easy to check that $\mathfrak{s}^{\prime}(y)=\mathfrak{s}(\sigma(y))$, for
all variables $y$. Since
$(\mathfrak{s}^{\prime},\mathfrak{h}_{2})\models_{\mathfrak{R}}\lambda$ we get
by Proposition 2.7,
$(\mathfrak{s},\mathfrak{h}_{2})\models_{\mathfrak{R}}\lambda\sigma$, i.e.,
$(\mathfrak{s},\mathfrak{h}_{2})\models_{\mathfrak{R}}(x\mapsto(y_{1},\dots,y_{k})*\psi^{\prime}\sigma)$.
Therefore
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}(x\mapsto(y_{1},\dots,y_{k})*\psi^{\prime}\sigma*\psi)\curlywedge\zeta$.
Rule S is sound but in contrast with the other rules, it is not invertible in
general (intuitively, there is no guarantee that the decomposition of the
left-hand side of the sequent corresponds to that of the right-hand side).
###### Example 4.28
Consider the (valid) sequent
$x\mapsto(y)*y\mapsto(x)\vdash_{\mathfrak{R}}^{\emptyset}p(x,y)*p(y,x)$ with
the rule $p(u,v)\Leftarrow u\mapsto(v)$. Rule S applies, yielding the valid
premises $x\mapsto(y)\vdash_{\mathfrak{R}}^{\emptyset}p(x,y)$ and
$y\mapsto(x)\vdash_{\mathfrak{R}}^{\emptyset}p(y,x)$. However, since the rules
apply modulo commutativity of $*$ we may also get the premises:
$x\mapsto(y)\vdash_{\mathfrak{R}}^{\emptyset}p(y,x)$ and
$y\mapsto(x)\vdash_{\mathfrak{R}}^{\emptyset}p(x,y)$ which are not valid.
###### Lemma 4.29
Rule S is sound. More specifically, if $(\mathfrak{s},\mathfrak{h})$ is a
counter-model of the conclusion, then one of the premises admits a counter-
model $(\mathfrak{s},\mathfrak{h}^{\prime})$, where $\mathfrak{h}^{\prime}$ is
a proper subheap of $\mathfrak{h}$.
###### Proof 4.30
Let $(\mathfrak{s},\mathfrak{h})$ be an $\mathfrak{R}$-model of
$(\phi_{1}*\phi_{2})\curlywedge(\xi_{1}\wedge\xi_{2})$, where
$\mathfrak{s}(V)\cap\mathit{dom}(\mathfrak{h})=\emptyset$ and $\mathfrak{s}$
is injective on $V$. Assume that the premises admit no counter-model of the
form $(\mathfrak{s},\mathfrak{h}^{\prime})$ with
$\mathfrak{h}^{\prime}\subset\mathfrak{h}$. By definition, there exist
disjoint heaps $\mathfrak{h}_{1},\mathfrak{h}_{2}$ such that
$\mathfrak{h}=\mathfrak{h}_{1}\uplus\mathfrak{h}_{2}$, and
$(\mathfrak{s},\mathfrak{h}_{i})\models_{\mathfrak{R}}\phi_{i}$, for $i=1,2$.
Since $\phi_{i}\not=\mathit{emp}$, $\phi_{i}$ contains at least one predicate
atom, with a root $x_{i}$. By Lemma 4.2, necessarily
$\mathfrak{s}(x_{i})\in\mathit{dom}(\mathfrak{h}_{i})$, so $\mathfrak{h}_{i}$
is not empty and $\mathfrak{h}_{i}\subset\mathfrak{h}$ for $i=1,2$. Still by
Lemma 4.2,
$\mathfrak{s}(\mathit{alloc}(\phi_{i}))\subseteq\mathit{dom}(\mathfrak{h}_{i})$,
and since $\mathfrak{h}_{1}$ and $\mathfrak{h}_{2}$ are disjoint, we deduce
that
$\mathfrak{s}(\mathit{alloc}(\phi_{3-i}))\cap\mathit{dom}(\mathfrak{h}_{i})=\emptyset$
for $i=1,2$. Thus
$\mathfrak{s}(V\cup\mathit{alloc}(\phi_{3-i}))\cap\mathit{dom}(\mathfrak{h}_{i})=\emptyset$.
By Corollary 4.4 $\mathfrak{s}$ is injective on
$\mathit{alloc}(\phi_{1}*\phi_{2})$, hence since $\mathfrak{s}$ is injective
on $V$, we deduce that $\mathfrak{s}$ is injective on
$V\cup\mathit{alloc}(\phi_{3-i})$. Since $(\mathfrak{s},\mathfrak{h}_{i})$
cannot be a counter-model of the premises because
$\mathfrak{h}_{i}\subset\mathfrak{h}$, this entails that
$(\mathfrak{s},\mathfrak{h}_{i})\models_{\mathfrak{R}}\psi_{i}\curlywedge\zeta_{i}$
for $i=1,2$, thus
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}(\psi_{1}*\psi_{2})\curlywedge(\zeta_{1}\wedge\zeta_{2})$.
### 4.4 Axioms and Anti-Axioms
We define two sets of syntactic criteria on sequents that allow to quickly
conclude that such sequents are respectively valid or non-valid. This will be
useful to block the application of the inference rules in these cases. Axioms
(i.e., necessarily valid sequents) are defined as follows.
###### Definition 4.31
An axiom is a sequent that is of one of the following forms modulo AC:
1. 1.
$\phi\curlywedge(\xi\wedge\xi^{\prime})\vdash_{\mathfrak{R}}^{V}\phi\curlywedge\xi$;
2. 2.
$\phi\curlywedge(\xi\wedge x\not\approx x)\vdash_{\mathfrak{R}}^{V}\gamma$;
3. 3.
$\phi\curlywedge\xi\vdash_{\mathfrak{R}}^{V}\gamma$ where $\phi$ is heap-
unsatisfiable;
4. 4.
$\phi\curlywedge\xi\vdash_{\mathfrak{R}}^{V}\gamma$ where either
$\mathit{alloc}(\phi)\cap V\not=\emptyset$ or $V$ contains two occurrences of
the same variable.
Intuitively, a sequent is valid if the right-hand side is a trivial
consequence of the left-hand side, if the left-hand side is (trivially)
unsatisfiable, or if $V$ contains a variable that is allocated by the left-
hand side or two occurrences of the same variable (since by hypothesis
counter-models must be injective on $V$).
###### Lemma 4.32
All axioms are valid.
###### Proof 4.33
We consider each case separately (using the same notations as in Definition
4.31):
1. 1.
It is clear that every model of $\phi\curlywedge(\xi\wedge\xi^{\prime})$ is a
model of $\phi\curlywedge\xi$.
2. 2.
By definition, $\phi\curlywedge(\xi\wedge x\not\approx x)$ has no model, hence
$\phi\curlywedge(\xi\wedge x\not\approx x)\vdash_{\mathfrak{R}}^{V}\gamma$ has
no counter-model.
3. 3.
If $\phi$ is heap-unsatisfiable then $\mathit{alloc}(\phi)$ contains two
occurrences of the same variable, which by Corollary 4.4, entails that $\phi$
has no model. Thus $\phi\curlywedge\xi\vdash_{\mathfrak{R}}^{V}\gamma$ has no
counter-model.
4. 4.
Let $(\mathfrak{s},\mathfrak{h})$ be a counter-model of
$\phi\curlywedge\xi\vdash_{\mathfrak{R}}^{V}\gamma$. By definition
$\mathfrak{s}$ is injective on $V$ hence we cannot have
$\\{x,x\\}\subseteq_{m}V$. Also, by definition, if $x\in V$ then we cannot
have $\mathfrak{s}(x)\in\mathit{dom}(\mathfrak{h})$, and if
$x\in\mathit{alloc}(\phi)$ then if $x\in\mathit{alloc}(\phi)\cap V$ then
$\mathfrak{s}(x)\in\mathit{dom}(\mathfrak{h})$ by Lemma 4.2. We conclude that
it is impossible to have $x\in\mathit{alloc}(\phi)\cap V$ either.
We also introduce the notion of an anti-axiom, which is a sequent satisfying
some syntactic conditions that prevent it from being valid.
###### Definition 4.34
A sequent $\phi\curlywedge\xi\vdash_{\mathfrak{R}}^{V}\psi\curlywedge\zeta$ is
an anti-axiom if it is not an axiom, $\xi$ contains no equality, $\zeta=\top$
and one of the following conditions holds:
1. 1.
$\mathit{alloc}(\psi)\not\subseteq\mathit{alloc}(\phi)$;
2. 2.
$\psi=\mathit{emp}$ and $\phi\not=\mathit{emp}$.
3. 3.
There exists a variable
$x\in\mathit{alloc}(\phi)\setminus\mathit{alloc}(\psi)$, such that
$y\not\rightarrow_{\phi}^{*}x$ holds, for all $y\in\mathit{alloc}(\psi)$;
4. 4.
$V\cap(\mathcal{V}(\phi)\setminus\mathcal{V}(\psi))$ is not empty.
5. 5.
$\mathcal{V}_{\mathtt{loc}}(\phi)\setminus(\mathcal{V}(\psi)\cup\mathit{alloc}(\phi))$
is not empty.
We provide examples illustrating every case in Definition 4.34:
###### Example 4.35
The following sequents (where $p$ is some arbitrary predicate) are anti-
axioms:
$\begin{array}[]{llllcllll}1:&p(x,y)&\vdash_{\mathfrak{R}}^{\emptyset}&p(y,x)&&2:&p(x,y)&\vdash_{\mathfrak{R}}^{\emptyset}&\mathit{emp}\\\
3:&p(x,y)*p(z,y)&\vdash_{\mathfrak{R}}^{\emptyset}&q(x,y)&&4:&p(x,y)&\vdash_{\mathfrak{R}}^{\\{y\\}}&r(x)\\\
5:&p(x,y)&\vdash_{\mathfrak{R}}^{\emptyset}&r(x)\end{array}$
Intuitively, $1$ cannot be valid because there exist models of $p(x,y)$ in
which $y$ is not allocated whereas $y$ is allocated in all models of $p(y,x)$
Note that by Assumption 2.2, all predicates are productive, hence $p(x,y)$
admits at least one model. Furthermore, a predicate cannot allocate any of its
arguments other than the root, for instance rules of the form
$p(x,y)\Leftarrow x\mapsto(y)*p(y,x)$, indirectly allocating $y$, are not
allowed. $2$ cannot be valid because the models of $p(x,y)$ allocate at least
$x$. For $3$, assuming that all variables are associated with distinct
locations, one can construct a model of $p(x,y)*p(z,y)$ in which there is no
path from $x$ to $z$ and by Proposition 4.9 all locations occurring in the
heap of any model of $q(x,y)$ must be reachable from $x$. For $4$ and $5$, we
can construct a counter-model by considering any structure in which $y$ occurs
in the heap but is not allocated, and by Lemma 4.11, all the locations
occurring in the heap of any model of $r(x)$ must be allocated.
To show that all anti-axioms admit counter-models, we use the following lemma,
which will also play a key rôle in the completeness proof. It states that all
the formulas that are heap-satisfiable admit a model satisfying some
particular properties:
###### Lemma 4.36
Let $\phi$ be a spatial formula, containing a variable $x$ of sort
$\mathtt{loc}$. Let $\mathfrak{s}$ be a store that is injective on
$\mathit{alloc}(\phi)$. Let $U$ be an infinite subset of
$\mathfrak{U}_{\mathtt{loc}}$ such that
${U\cap\mathfrak{s}(\mathcal{V}_{\mathtt{loc}})}=\emptyset$. If $\phi$ is
heap-satisfiable, then it admits an $\mathfrak{R}$-model of the form
$(\mathfrak{s},\mathfrak{h})$, where
$\mathfrak{s}(x)\in\mathit{ref}(\mathfrak{h})$, the set
$\mathit{dom}(\mathfrak{h})$ is a subset of
$U\cup\mathfrak{s}(\mathit{alloc}(\phi))$ and
${\mathit{ref}(\mathfrak{h})\subseteq U\cup\mathfrak{s}(\mathcal{V}(\phi))}$.
Moreover, if $\mathfrak{s}$ is injective then $(\mathfrak{s},\mathfrak{h})$ is
a $\rightarrow$-compatible model of $\phi$.
###### Proof 4.37
The proof is by induction on the formulas. Note that we cannot have
$\phi=\mathit{emp}$ since $x\in\mathcal{V}(\phi)$.
* •
If $\phi=y_{0}\mapsto(y_{1},\dots,y_{n})$ then we set:
$\mathfrak{h}\overset{\text{\tiny\it
def}}{=}\\{(\mathfrak{s}(y_{0}),\dots,\mathfrak{s}(y_{n}))\\}$. By definition,
${(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\phi}$. Moreover,
$x\in\\{y_{0},\dots,y_{n}\\}$, and
$\mathit{ref}(\mathfrak{h})=\\{\mathfrak{s}(y_{i})\mid i=0,\dots,n\text{\ and\
}y_{i}\in\mathcal{V}_{\mathtt{loc}}\\}$, hence
$\mathfrak{s}(x)\in\mathit{ref}(\mathfrak{h})$ and
$\mathit{ref}(\mathfrak{h})\subseteq\mathfrak{s}(\mathcal{V}(\phi))$.
Furthermore, $\mathit{dom}(\mathfrak{h})=\\{\mathfrak{s}(y_{0})\\}$ and
$\mathit{alloc}(\phi)=\\{y_{0}\\}$ thus
$\mathit{dom}(\mathfrak{h})\subseteq\mathfrak{s}(\mathit{alloc}(\phi))$.
Finally, assume that $\mathfrak{s}$ is injective. We have by definition
$\rightarrow_{\mathfrak{h}}=\\{(\mathfrak{s}(y_{0}),\mathfrak{s}(y_{i}))\mid
i=1,\dots,n\\}$, thus if
$\mathfrak{s}(u)\rightarrow_{\mathfrak{h}}^{*}\mathfrak{s}(v)$ for
$u,v\in\mathcal{V}(\lambda)$ then we must have either
$\mathfrak{s}(u)=\mathfrak{s}(v)$, so that $u=v$ because $\mathfrak{s}$ is
injective, in which case it is clear that $u\rightarrow_{\phi}^{*}v$; or
$\mathfrak{s}(u)=\mathfrak{s}(y_{0})$ and
$\mathfrak{s}(v)=\mathfrak{s}(y_{i})$ for some $i=1,\dots,n$. Since
$\mathfrak{s}$ is injective this entails that $u=y_{0}$, $v=y_{i}$, thus
$u\rightarrow_{\phi}v$ by definition of $\rightarrow_{\phi}$.
* •
Assume that $\phi=p(y_{0},\dots,y_{n})$ is a predicate atom. Then, since by
Assumption 2.2 every predicate symbol is productive, there exists a symbolic
heap $\gamma$ such that $\phi\Leftarrow_{\mathfrak{R}}\gamma$. If $x=y_{0}$
then since the rules in $\mathfrak{R}$ are $\mathtt{P}$-rules, $\gamma$
contains a points-to-atom with root $x$. Otherwise, by Assumption 2.16,
$x=y_{i}$ for some $i\in\mathit{out}_{\mathfrak{R}}(p)$, hence there exists a
rule application $\phi\Leftarrow_{\mathfrak{R}}\gamma$ such that $x$ occurs in
some predicate atom in $\gamma$. Thus in both cases we may assume that $x$
occurs in a spatial atom in $\gamma$. Note that $\gamma$ must be heap-
satisfiable, since all considered rules are $\mathtt{P}$-rules and by
Definition 2.9 the roots of the predicate symbols in $\gamma$ are pairwise
distinct existential variables, thus also distinct from the root $y_{0}$ of
the points-to atom. Furthermore, $\gamma$ is of the form
$\psi\curlywedge\zeta$, where $x\in\mathcal{V}(\psi)$ and $\zeta$ is a
conjunction of disequations $u\not\approx v$, with $u\not=v$.
Let $\mathfrak{s}^{\prime}$ be an associate of $\mathfrak{s}$ w.r.t.
$\mathcal{V}(\gamma)\setminus\mathcal{V}(\phi)$ mapping the variables in
$\mathcal{V}(\gamma)\setminus\mathcal{V}(\phi)$ to pairwise distinct locations
in $U$. Since by hypothesis $U\cap\mathfrak{s}(\mathcal{V})=\emptyset$,
$\mathfrak{s}^{\prime}$ is injective on any set of the form
$E\cup(\mathcal{V}(\gamma)\setminus\mathcal{V}(\phi))$ when $\mathfrak{s}$ is
injective on $E$. Let $U^{\prime}\overset{\text{\tiny\it
def}}{=}U\setminus\mathfrak{s}^{\prime}(\mathcal{V}(\gamma)\setminus\mathcal{V}(\phi))$.
By the induction hypothesis, there exists a heap $\mathfrak{h}$ such that
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}\psi$, with
$\mathfrak{s}^{\prime}(x)\in\mathit{ref}(\mathfrak{h})$,
${\mathit{dom}(\mathfrak{h})\subseteq
U^{\prime}\cup\mathfrak{s}^{\prime}(\mathit{alloc}(\psi))}$ and
$\mathit{ref}(\mathfrak{h})\subseteq
U^{\prime}\cup\mathfrak{s}^{\prime}(\mathcal{V}(\psi))$. Now if $\mathfrak{s}$
is injective then (as $\mathfrak{s}^{\prime}$ is also injective in this case),
$(\mathfrak{s}^{\prime},\mathfrak{h})$ is a $\rightarrow$-compatible
$\mathfrak{R}$-model of $\psi$. We show that $(\mathfrak{s},\mathfrak{h})$
fulfills all the properties of the lemma.
* –
Since $\mathfrak{s}^{\prime}$ maps the variables in
$\mathcal{V}(\gamma)\setminus\mathcal{V}(\phi)$ to pairwise distinct locations
in $U$ and ${\mathfrak{s}(\mathcal{V})\cap U}$ is $\emptyset$, necessarily
$\mathfrak{s}^{\prime}\models_{\mathfrak{R}}\zeta$, thus
$(\mathfrak{s}^{\prime},\mathfrak{h})\models_{\mathfrak{R}}\gamma$ which
entails that $(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\phi$. We also
have $\mathfrak{s}(x)=\mathfrak{s}^{\prime}(x)\in\mathit{ref}(\mathfrak{h})$.
* –
Let $\ell\in\mathit{ref}(\mathfrak{h})$. We show that $\ell\in
U\cup\mathfrak{s}(\mathcal{V}(\phi))$. By the induction hypothesis, we have
$\ell\in U^{\prime}\cup\mathfrak{s}^{\prime}(\mathcal{V}(\psi))$. If $\ell\in
U^{\prime}\subseteq U$ then the proof is completed, otherwise we have
$\ell=\mathfrak{s}^{\prime}(y)$ for some $y\in\mathcal{V}(\psi)$. If
$y\in\mathcal{V}(\phi)$ then $\mathfrak{s}(y)=\mathfrak{s}^{\prime}(y)$ thus
$\ell\in\mathfrak{s}(\mathcal{V}(\phi))$. Otherwise, we must have
$y\in\mathcal{V}(\gamma)\setminus\mathcal{V}(\phi)$, thus
$\mathfrak{s}^{\prime}(y)\in U$ by definition of $\mathfrak{s}^{\prime}$ and
the result holds.
* –
Let $\ell\in\mathit{dom}(\mathfrak{h})$, we show that $\ell\in
U\cup\mathfrak{s}(\mathit{alloc}(\phi))$. By the induction hypothesis we have
$\ell\in U^{\prime}\cup\mathfrak{s}^{\prime}(\mathit{alloc}(\psi))$. If
$\ell\in U^{\prime}\subseteq U$ then the proof is completed. Otherwise,
$\ell=\mathfrak{s}^{\prime}(y)$ with $y\in\mathit{alloc}(\psi)$. Since the
rules in $\mathfrak{R}$ are $\mathtt{P}$-rules, $y$ is either the root $y_{0}$
of $\phi$, in which case we have $y\in\mathit{alloc}(\phi)$ and
$\mathfrak{s}(y)=\mathfrak{s}^{\prime}(y)$, thus the result holds; or $y$
occurs in $\mathcal{V}(\gamma)\setminus\mathcal{V}(\phi)$, in which that we
have $\mathfrak{s}^{\prime}(y)\in U$, by definition of
$\mathfrak{s}^{\prime}$.
* –
There remains to show that $(\mathfrak{s},\mathfrak{h})$ is a
$\rightarrow$-compatible $\mathfrak{R}$-model of $\phi$, in the case where
$\mathfrak{s}$ is injective. Assume that
$\mathfrak{s}(u)\rightarrow_{\mathfrak{h}}^{*}\mathfrak{s}(v)$. If
$\mathfrak{s}(u)=\mathfrak{s}(v)$ then $u=v$ by injectivity of $\mathfrak{s}$,
hence $u\rightarrow_{\mathfrak{h}}^{*}v$. Otherwise, we must have
$\\{\mathfrak{s}(u),\mathfrak{s}(v)\\}\subseteq\mathit{ref}(\mathfrak{h})\subseteq
U^{\prime}\cup\mathfrak{s}^{\prime}(\mathcal{V}(\psi))$, and since
$U^{\prime}\subseteq U$ and $U\cap\mathfrak{s}(\mathcal{V})=\emptyset$, we
have
$\\{\mathfrak{s}(u),\mathfrak{s}(v)\\}\subseteq\mathfrak{s}^{\prime}(\mathcal{V}(\psi))$.
We also have
$\mathfrak{s}^{\prime}(\mathcal{V}(\gamma)\setminus\mathcal{V}(\phi))\subseteq
U$, which entails that
$\\{\mathfrak{s}(u),\mathfrak{s}(v)\\}\subseteq\mathfrak{s}^{\prime}(\mathcal{V}(\phi))$.
Since $\mathfrak{s}^{\prime}$ is an associate of $\mathfrak{s}$ w.r.t.
$\mathcal{V}(\gamma)\setminus\mathcal{V}(\phi)$, $\mathfrak{s}$ and
$\mathfrak{s}^{\prime}$ coincide on all variables in $\mathcal{V}(\phi)$ and
we deduce that
$\\{\mathfrak{s}(u),\mathfrak{s}(v)\\}\subseteq\mathfrak{s}(\mathcal{V}(\phi))$.
Because $\mathfrak{s}$ is injective, this entails that
$u,v\in\mathcal{V}(\phi)$, so that $\mathfrak{s}(u)=\mathfrak{s}^{\prime}(u)$
and $\mathfrak{s}(v)=\mathfrak{s}^{\prime}(v)$. By hypothesis
$(\mathfrak{s}^{\prime},\mathfrak{h})$ is a $\rightarrow$-compatible
$\mathfrak{R}$-model of $\psi$, and we deduce that $u\rightarrow_{\psi}^{*}v$.
Since $u\not=v$, necessarily $u\in\mathit{alloc}(\psi)$ (by definition of
$\rightarrow_{\psi}$), and since the rules in $\mathfrak{R}$ are
$\mathtt{P}$-rules, and $u\in\mathcal{V}(\phi)$, this entails that
$u=\mathit{roots}(\phi)$. Since $v\in\mathcal{V}(\phi)$, by Assumption 2.16 we
have $u\rightarrow_{\phi}^{*}v$.
* •
Assume that $\phi=\phi_{1}*\phi_{2}$, with $\phi_{i}\not=\mathit{emp}$. Let
$U_{1},U_{2}$ be disjoint infinite subsets of $U$. Let $\\{x_{1},x_{2}\\}$ be
some arbitrary chosen variables such that $x_{i}\in\mathcal{V}(\phi_{i})$ and
$x\in\\{x_{1},x_{2}\\}$ (it is easy to check that such a pair of variables
always exists). By the induction hypothesis, there exist heaps
$\mathfrak{h}_{i}$ such that $(\mathfrak{s},\mathfrak{h}_{i})\models\phi_{i}$
where $\mathfrak{s}(x_{i})\in\mathit{ref}(\mathfrak{h}_{i})$,
${\mathit{dom}(\mathfrak{h}_{i})\subseteq
U_{i}\cup\mathfrak{s}(\mathit{alloc}(\phi_{i}))}$ and
$\mathit{ref}(\mathfrak{h}_{i})\subseteq
U_{i}\cup\mathfrak{s}(\mathcal{V}(\phi_{i}))$. Moreover, if $\mathfrak{s}$ is
injective, then $(\mathfrak{s},\mathfrak{h}_{i})$ is an
$\rightarrow$-compatible $\mathfrak{R}$-model of $\phi_{i}$. We first show
that $\mathfrak{h}_{1}$ and $\mathfrak{h}_{2}$ are disjoint. Assume for the
sake of contradiction that
$\ell\in\mathit{dom}(\mathfrak{h}_{1})\cap\mathit{dom}(\mathfrak{h}_{2})$.
Since $U_{1}\cap U_{2}=\emptyset$, necessarily $\ell=\mathfrak{s}(y_{i})$ (for
$i=1,2$), with $y_{i}\in\mathit{alloc}(\phi_{i})$. Since $\mathfrak{s}$ is
injective on $\mathit{alloc}(\phi)$, we deduce that $y_{1}=y_{2}$. We have
$\\{y_{1},y_{2}\\}\subseteq_{m}\mathit{alloc}(\phi)$, hence $\phi$ is heap-
unsatisfiable, which contradicts the hypotheses of the lemma. Thus
$\mathfrak{h}_{1}$ and $\mathfrak{h}_{2}$ are disjoint.
Let $\mathfrak{h}=\mathfrak{h}_{1}\uplus\mathfrak{h}_{2}$. We have
$\mathfrak{s}(x)\in\\{\mathfrak{s}(x_{1}),\mathfrak{s}(x_{2})\\}\subseteq\mathit{ref}(\mathfrak{h}_{1})\cup\mathit{ref}(\mathfrak{h}_{2})=\mathit{ref}(\mathfrak{h})$.
Moreover,
$\mathit{dom}(\mathfrak{h})=\mathit{dom}(\mathfrak{h}_{1})\cup\mathit{dom}(\mathfrak{h}_{2})\subseteq
U_{1}\cup
U_{2}\cup\mathfrak{s}(\mathit{alloc}(\phi_{1}))\cup\mathfrak{s}(\mathit{alloc}(\phi_{2})){\subseteq
U\cup\mathfrak{s}(\mathit{alloc}(\phi))}$, and
$\mathit{ref}(\mathfrak{h})=\mathit{ref}(\mathfrak{h}_{1})\cup\mathit{ref}(\mathfrak{h}_{2})\subseteq
U_{1}\cup
U_{2}\cup\mathfrak{s}(\mathcal{V}(\phi_{i}))\cup\mathfrak{s}(\mathcal{V}(\phi_{2})){\subseteq
U\cup\mathfrak{s}(\mathcal{V}(\phi))}$. Furthermore,
$(\mathfrak{s},\mathfrak{h})\models_{\mathfrak{R}}\phi_{1}*\phi_{2}=\phi$.
There only remains to prove that $(\mathfrak{s},\mathfrak{h})$ is a
$\rightarrow$-compatible $\mathfrak{R}$-model of $\phi$ when $\mathfrak{s}$ is
injective. Assume that this is not the case, and let $u,v$ be variables such
that $\mathfrak{s}(u)\rightarrow_{\mathfrak{h}}^{*}\mathfrak{s}(v)$ and
$u\not\rightarrow_{\phi}^{*}v$. This entails that $u\not=v$. By definition,
there exist $\ell_{0},\dots,\ell_{m}$ such that $\ell_{0}=\mathfrak{s}(u)$,
$\ell_{m}=\mathfrak{s}(v)$, and $\forall
i=1,\dots,m,\,\ell_{i-1}\rightarrow_{\mathfrak{h}}\ell_{i}$. We assume,
w.l.o.g., that $m$ is miminal, i.e., that there is no sequence
$\ell_{0}^{\prime},\dots,\ell_{k}^{\prime}$ and no variables $x_{0},x_{k}$
such that $k<m$, $\ell_{0}^{\prime}=\mathfrak{s}(x_{0})$,
$\ell_{k}^{\prime}=\mathfrak{s}(x_{k})$ and
$x_{0}\not\rightarrow_{\phi}^{*}x_{k}$. We may also assume, by symmetry, that
$\ell_{0}\in\mathit{dom}(\mathfrak{h}_{1})$. If all the locations
$\ell_{1},\dots,\ell_{m-1}$ occur in $\mathit{dom}(\mathfrak{h}_{1})$ then
$\mathfrak{s}(u)\rightarrow_{\mathfrak{h}_{1}}^{*}\mathfrak{s}(v)$, thus
$u\rightarrow_{\phi_{i}}^{*}v$ because $(\mathfrak{s},\mathfrak{h}_{i})$ is an
|
# Exponential asymptotics and the generation of free-surface flows by
submerged point vortices
Josh Shelton<EMAIL_ADDRESS>Philippe H. Trinh<EMAIL_ADDRESS>Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
( [Draft])
###### Abstract
There has been significant recent interest in the study of water waves coupled
with non-zero vorticity. Here, we derive analytical approximations for the
exponentially-small free-surface waves generated by one or several submerged
point vortices when driven at low Froude numbers. The vortices are fixed in
place, and a boundary-integral formulation in the arclength along the surface
allows the study of nonlinear waves and strong point vortices. We demonstrate
that for a single point vortex, techniques in exponential asymptotics
prescribe the formation of waves in connection with the presence of Stokes
lines originating from the vortex. When multiple point vortices are placed
within the fluid, trapped waves may occur, which are confined to lie between
the vortices. We also demonstrate that for the two-vortex problem, the
phenomena of trapped waves occurs for a countably infinite set of values of
the Froude number. This work will form a basis for other asymptotic
investigations of wave-structure interactions where vorticity plays a key role
in the formation of surface waves.
## 1 Introduction
In this paper we study the nonlinear flow of an ideal fluid past a submerged
point vortex. The two-dimensional inviscid and incompressible fluid of
infinite depth is assumed to be irrotational everywhere, with the exception of
the point vortices themselves. For a flow in the complex
$z=x+\mathrm{i}y$-plane, with a vortex at $z=z^{*}$, the complex potential
behaves as
$f=\phi+\mathrm{i}\psi\sim cz-\frac{\mathrm{i}\Gamma}{2\pi}\log{(z-z^{*})},$
(1)
where $\Gamma$ is the circulation of the vortex, and the background flow is of
speed $c$. The non-dimensional system is then characterised by two key
parameters: $\Gamma_{\text{c}}=\Gamma/(cH)$, relating vortex strength(s) to
inertial effects, and the Froude number, $F=c/\sqrt{gH}$, relating inertial
effects to gravitational effects. Here, $H$ is the depth of the point vortex
and $g$ is the constant acceleration due to gravity.
The study of such vortex-driven potential flows is complicated by the
following fact. Typically, the formulation of the free surface system would
involve the inversion of the velocity potential and streamfunction $\phi(x,y)$
and $\psi(x,y)$, which are both a function of the domain $x$ and $y$. In
expressing these instead as $x(\phi,\psi)$ and $y(\phi,\psi)$, we note that
the free surface is a streamline along which $\psi$ takes constant values, and
so $x(\phi)$ and $y(\phi)$ represent the free surface. However, near the point
vortex, the local behaviour (1), can not be inverted analytically to give
$z(f)$. This motivated the work of Forbes (1985), who re-formulated the
boundary-integral formulation in terms of a free-surface arclength, $s$, and a
more complex set of governing equations results.
The imposition of a uniform stream as $x\to-\infty$ results in the generation
of downstream free-surface waves, as shown in figure 1(a). As hinted in the
preliminary numerical investigations of Forbes (1985), the wave amplitude
tends to zero as $F\to 0$. In this work, we confirm this behaviour and
demonstrate, both numerically and analytically, that the amplitude is
exponentially-small in the low-Froude limit. For instance, the amplitude
versus $1/F^{2}$ graph shown in figure 2 demonstrates the fit between our
asymptotic predictions of §3 and numerical results of §4. We note that this
theory is nonlinear in the vortex strength, $\Gamma_{\text{c}}$, and the
assumption of small $\Gamma_{\text{c}}$ need not apply.
The purpose of this paper is to thus characterise the formation of water waves
using the framework of exponential asymptotics. We show that these
exponentially-small waves smoothly switch-on as the fluid passes beyond the
vortex, resulting in oscillations as $x\to\infty$ in the far field. When two
submerged vortices are considered, the waves switched-on due to each of the
vortices may be out of phase with one another and cancel for certain values of
the Froude number. This yields trapped waves between the vortices, and a free
surface whose derivative decays to zero as $x\to\infty$. A trapped wave
solution is depicted in figure 1(b). This phenomenon of trapped waves has
previously been studied for obstructions both within the fluid, and for flows
of finite depth past lower topography. For instance, both Gazdar (1973) and
Vanden-Broeck & Tuck (1985) detected these numerically for flows over a
specified lower topography. More recent works, such as those by Dias & Vanden-
Broeck (2004), Hocking et al. (2013), and Holmes et al. (2013), have focused
on detecting parameter values for which these trapped wave solutions occur in
various formulations.
Figure 1: The two physical regimes of underlying point vortices considered
within this paper are shown. In (a), a single point vortex with circulation
$\Gamma$ is placed within the fluid. In (b), two point vortices, each with
circulation $\Gamma$, are located at the same depth within the fluid. These
solutions have been computed using the numerical scheme detailed in §4. Figure
2: The amplitude, $\bar{y}$, of the free-surface waves is shown for
$\log(\bar{y})$ vs $1/F^{2}$ for the analytical (line) and numerical (dots)
solutions of §3 and §4. These have a fixed value of the nondimensional vortex
strength, $\Gamma_{\text{c}}=0.25$. The graph confirms exponential smallness
of the waves. The solid line has a gradient of $\approx 0.7395$, computed
using the exponential asymptotic theory of §4.
The work in this paper provides a first step towards extending many of the
existing ideas and techniques of exponential asymptotics, previously developed
for purely gravity- or capillary-driven waves (e.g. Chapman & Vanden-Broeck
2002, 2006) to wave phenomena with vortices. As noted above, because the
governing equations require an alternative formulation (originally developed
by Miksis et al. 1981) the asymptotic formulation we present can be extended
to other wave-structure interactions where the more general arc-length
formulation of the water-wave equations is required. In addition, there has
been significant recent interest in the study of water-wave phenomena with
dominant vorticity effects, and we reference the recent extensive survey by
Haziot et al. (2022) and references therein. The exponential-asymptotic
techniques developed in this work can also be extended to situations where
capillary ripples are forced on the surface of steep vortex-driven waves. The
leading order solution in these asymptotic regimes would then be known
analytically from the works of e.g. (Crowdy & Nelson, 2010; Crowdy & Roenby,
2014; Crowdy, 2022). These, and other exciting future directions, we shall
discuss in §6.
## 2 Mathematical formulation and outline
We consider the typical configurations shown in 1. Following Forbes (1985), in
nondimensional form, the system is formulated in terms of the arclength, $s$,
along the free surface, with unknown velocity potential $\phi=\phi(s)$, and
free-surface positions, $(x(s),y(s))$. Then, the governing equations are given
by Bernoulli’s equation, an arclength relation between $x$ and $y$, and a
boundary-integral equation.
For a single submerged point vortex at $(x,y)=(0,-1)$, the three equations are
$\displaystyle\frac{F^{2}}{2}\big{[}\phi^{\prime}(s)\big{]}^{2}+y(s)=\frac{F^{2}}{2},$
(2a)
$\displaystyle\big{[}x^{\prime}(s)\big{]}^{2}+\big{[}y^{\prime}(s)\big{]}^{2}=1,$
(2b)
$\displaystyle\phi^{\prime}(s)x^{\prime}(s)-1=\frac{\Gamma_{\text{c}}}{\pi}\frac{y(s)+1}{[x(s)]^{2}+[y(s)+1]^{2}}+\mathcal{I}[x,y,\phi^{\prime}].$
(2c)
In the above, two nondimensional parameters appear: the Froude number, $F$,
and the vortex strength, $\Gamma_{\text{c}}$, defined by
$F=\frac{c}{\sqrt{gH}}\quad\text{and}\quad\Gamma_{\text{c}}=\frac{\Gamma}{cH}.$
(3)
Here, $c$ is the speed of the fluid, $H$ is the depth of the submerged point
vortex, $g$ is the constant acceleration due to gravity, and $\Gamma$ is the
circulation of the point vortex. Furthermore, we have also introduced
$\mathcal{I}$ as the nonlinear principle-valued integral defined by
$\mathcal{I}[x,y,\phi^{\prime}]=\frac{1}{\pi}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{-\infty}^{\infty}\frac{[\phi^{\prime}(t)-x^{\prime}(t)][y(t)-y(s)]+y^{\prime}(t)[x(t)-x(s)]}{[x(t)-x(s)]^{2}+[y(t)-y(s)]^{2}}\,\mathrm{d}t.$
(4)
When the configuration with two point vortices is considered in §3.5, the
boundary-integral equation (2c) will need to be modified to (29).
### 2.1 Analytic continuation
In the exponential asymptotic procedure of §3, we study the exponentially
small terms that display the Stokes phenomenon across Stokes lines of the
problem. These Stokes lines originate from singularities of the leading order
asymptotic solution, which are located in the analytic continuation of the
domain, the arclength $s$. The analytic continuation of the governing
equations (2a)-(2c) is studied in this section.
We now analytically continue the domain $s\mapsto\sigma$, where
$\sigma\in\mathbb{C}$. Bernoulli’s equation (2a) and the arclength relation
(2b) may be analytically continued in a straightforward manner, with all
dependence on $s$ replaced by the complex valued variable $\sigma$. The
analytic continuation of the boundary integral equation (2c) is more
complicated, due to the principal value integral $\mathcal{I}$ defined in (4).
The analytic continuation of this integral is given by
$\mathcal{I}[x,y,\phi^{\prime}]=\widehat{\mathcal{I}}[x,y,\phi^{\prime}]-a\mathrm{i}\phi^{\prime}(\sigma)y^{\prime}(\sigma),$
(5)
where $a=\pm 1$ denotes the direction of analytic continuation into
$\text{Im}[\sigma]>0$ or $\text{Im}[\sigma]<0$, respectively, and
$\widehat{\mathcal{I}}$ is the complex-valued integral. Equation (5) may be
verified by taking the limit of either $\text{Im}[\sigma]\to 0^{+}$, or
$\text{Im}[\sigma]\to 0^{-}$, which yields half of a residue contribution
associated with the singular point at $t=s$ of the integrand.
Substitution of (5) into (2c) then yields the analytically continued
equations, given by
$\displaystyle\frac{F^{2}}{2}\big{[}\phi^{\prime}(\sigma)\big{]}^{2}+y(\sigma)=\frac{F^{2}}{2},$
(6a)
$\displaystyle\big{[}x^{\prime}(\sigma)\big{]}^{2}+\big{[}y^{\prime}(\sigma)\big{]}^{2}=1,$
(6b)
$\displaystyle\phi^{\prime}(\sigma)x^{\prime}(\sigma)-1+a\mathrm{i}\phi^{\prime}(\sigma)y^{\prime}(\sigma)=\frac{\Gamma_{\text{c}}}{\pi}\frac{y(\sigma)+1}{[x(\sigma)]^{2}+[y(\sigma)+1]^{2}}+\widehat{\mathcal{I}}[x,y,\phi^{\prime}].$
(6c)
The analytic continuation for situations with multiple point vortices is
similarly done, with the only difference being the inclusion of additional
point vortices in (6c).
### 2.2 Outline of paper
In this work, we will consider the following two regimes depicted in figure 1:
1. (i)
A single submerged point vortex, which is the formulation originally
considered by Forbes (1985). Imposing free stream conditions as $x\to-\infty$
results in surface waves generated by the vortex. Their amplitude is
exponentially-small as $F\to 0$. This is the limit considered by Chapman &
Vanden-Broeck (2006) in the absence of vortical effects.
2. (ii)
Two submerged point vortices of the same circulation. For certain critical
values of the Froude number, $F$, the resultant waves are confined to lie
between the two vortices. The amplitude of these is also exponentially small
as $F\to 0$.
We begin in §3 by determining these exponentially small waves using the
techniques of exponential asymptotics. This relies of the optimal truncation
of an algebraic asymptotic series for small Froude number, $F$, and deriving
the connection of this to the Stokes phenomenon that acts on the exponentially
small waves. The case for two submerged point vortices is then studied in
§3.5, where we derive the critical values of the Froude number for which the
waves are trapped. Numerical solutions are computed in §4, where comparison
occurs with the exponential asymptotic predictions for the single vortex and
double vortex cases.
## 3 Exponential asymptotics
### 3.1 Early orders of the solution
We begin by considering the following asymptotic expansions, in powers of
$F^{2}$, for the solutions, which are given by
$x(\sigma)=\sum_{n=0}^{\infty}F^{2n}x_{n}(\sigma),\quad
y(\sigma)=\sum_{n=0}^{\infty}F^{2n}y_{n}(\sigma),\quad\phi^{\prime}(\sigma)=\sum_{n=0}^{\infty}F^{2n}\phi_{n}^{\prime}(\sigma).$
(7)
Substitution of expansions (7) into equations (6a)-(6c) yields at leading
order three equations for the unknowns $x_{0}$, $y_{0}$, and
$\phi_{0}^{\prime}$. The first of these, Bernoulli’s equation (6a), yields
$y_{0}(\sigma)=0$. This may be substituted into the second equation, (6b), to
find $(x_{0}^{\prime})^{2}=1$, for which we consider $x_{0}^{\prime}=1$
without any loss of generality. This may integrated to find $x_{0}=\sigma$,
where the constant of integration has been chosen to set the origin at
$x_{0}(0)=0$. Next, $\phi_{0}^{\prime}$ is determined from equation (6c).
Since $y_{0}=0$, the integral $\widehat{\mathcal{I}}$ does not enter the
leading order equation. This yields the leading order solutions as
$y_{0}(\sigma)=0,\qquad
x_{0}(\sigma)=\sigma,\qquad\phi^{\prime}_{0}(\sigma)=1+\frac{\Gamma_{\text{c}}}{\pi}\frac{1}{(1+\sigma^{2})}.$
(8)
Note that there is a singularity in $\phi_{0}^{\prime}$ above whenever
$\sigma^{2}=-1$. This corresponds to the point vortex within the fluid at
$\sigma=-\mathrm{i}$, as well as another singularity at $\sigma=\mathrm{i}$,
which will produce a complex-conjugate contribution to the exponentially-small
solution along the free surface.
Next at order $O(F^{2})$, $y_{1}$ is found explicitly from (6a). We then find
the equation $x_{1}^{\prime}=0$ from (6b), and $\phi_{1}$ is determined
explicitly from (6c). This yields
$\left.\quad\begin{aligned}
y_{1}(\sigma)&=\frac{1}{2}\Big{(}1-\big{(}\phi_{0}^{\prime}\big{)}^{2}\Big{)},\qquad
x_{1}(\sigma)=0,\\\
\phi^{\prime}_{1}(\sigma)&=-a\mathrm{i}\phi_{0}^{\prime}y_{1}^{\prime}+\frac{\Gamma_{\text{c}}(\sigma^{2}-1)}{\pi(1+\sigma^{2})^{2}}y_{1}+\widehat{\mathcal{I}}_{1}(\sigma),\end{aligned}\quad\right\\}$
(9)
where $\widehat{\mathcal{I}}_{1}$ is the $O(F^{2})$ component of the complex-
valued integral $\widehat{\mathcal{I}}$, originally defined along the real
axis in equation (4).
### 3.2 Late-term divergence
Our derivation of the exponentially-small terms and associated Stokes
phenomenon of §3.4 requires the knowledge of the late-terms of the solution
expansion (7), $x_{n}$, $y_{n}$, and $\phi_{n}^{\prime}$, as $n\to\infty$. We
begin by determining the $O(F^{2n})$ components of equations (6a)-(6c). The
late-terms of Bernoulli’s equation are given by
$y_{n}+\phi_{0}^{\prime}\phi_{n-1}^{\prime}+\phi_{1}^{\prime}\phi_{n-2}^{\prime}+\cdots=0,$
(10a) for the arclength relation we have
$x_{0}^{\prime}x_{n}^{\prime}+x_{1}^{\prime}x_{n-1}^{\prime}+\cdots+y_{1}^{\prime}y_{n-1}^{\prime}+y_{2}^{\prime}y_{n-2}^{\prime}+\cdots=0,\\\
$ (10b) and finally the boundary integral equation yields
$x_{0}^{\prime}\phi_{n}^{\prime}+x_{1}^{\prime}\phi_{n-1}^{\prime}+\phi_{0}^{\prime}x_{n}^{\prime}+\cdots+a\mathrm{i}\big{[}\phi_{0}^{\prime}y_{n}^{\prime}+\phi_{1}^{\prime}y_{n-1}^{\prime}+y_{1}^{\prime}\phi_{n-1}^{\prime}+\cdots\big{]}\\\
+\frac{\Gamma_{\text{c}}}{\pi}\bigg{[}\frac{y_{n}}{1+x_{0}^{2}}-\frac{2y_{n}}{(1+x_{0}^{2})^{2}}+\cdots\bigg{]}-\widehat{\mathcal{I}}_{n}(\sigma)=0.$
(10c)
In (10a)-(10c) above, only the terms that will appear at the first two orders
of $n$ as $n\to\infty$ have been included.
In (10c), the $O(F^{2n})$ component of the complex-valued integral,
$\widehat{\mathcal{I}}$ has been denoted by $\widehat{\mathcal{I}}_{n}$. The
dominant components of this integral, as $n\to\infty$, require the integration
of late-term asymptotic solutions that are either a function of the real
valued integration domain, such as $y_{n}(t)$, or a function of the complex
domain, such as $y_{n}(\sigma)$. The first of these, $y_{n}(t)$, is integrated
along the real-valued free surface, away from any singular behaviour. It is
thus subdominant to the other terms appearing in equation (10c). This is
analogous to the neglection of the late terms of the complex-valued Hilbert
transform in similar free-surface problems in exponential asymptotics [c.f.
Xie & Tanveer (2002), Chapman & Vanden-Broeck (2002), Chapman & Vanden-Broeck
(2006)]. All that remains is to integrate the components of
$\widehat{\mathcal{I}}_{n}$ that involve late-term solutions evaluated in the
complex-valued domain. Of these, only that involving $y_{n}(\sigma)$ appears
in the two leading orders, as $n\to\infty$, of equation (10c). This component
is given by
$\widehat{\mathcal{I}}_{n}\sim-\frac{y_{n}(\sigma)}{\pi}\int_{-\infty}^{\infty}\frac{\phi_{0}^{\prime}(t)-1}{(t-\sigma)^{2}}\mathrm{d}t=-\frac{\Gamma_{\text{c}}}{\pi}\frac{y_{n}(\sigma)}{(\sigma+a\mathrm{i})^{2}},$
(11)
for which the integral was evaluated by substituting for $\phi_{0}^{\prime}$
from equation (8). Note that integration of $y_{n}(\sigma)$ was not required
due to the lack of any dependence on the domain of integration, $t$.
Recall that the leading order solutions were singular at $\sigma=\pm i$. For
each of the three solution expansions, this singularity first appeared in
$\phi^{\prime}_{0}$, $y_{1}$, and $x_{2}$. Since successive terms in the
asymptotic expansion involve differentiation of previous terms (for instance,
equation (10a) for $y_{n}$ involves $\phi^{\prime}_{n-1}$, whose determination
in equation (10c) requires knowledge of $y_{n-1}^{\prime}$), the strength of
this singularity will grow as we proceed into the asymptotic series.
Furthermore, this growing singular behaviour will also lead to the divergence
of the late-term solutions as $n\to\infty$, which we capture analytically with
the factorial-over-power ansatzes of
$x_{n}\sim
X(\sigma)\frac{\Gamma(n+\alpha-1)}{[\chi(\sigma)]^{n+\alpha-1}},\quad
y_{n}\sim
Y(\sigma)\frac{\Gamma(n+\alpha)}{[\chi(\sigma)]^{n+\alpha}},\quad\phi_{n}\sim\Phi(\sigma)\frac{\Gamma(n+\alpha)}{[\chi(\sigma)]^{n+\alpha}}.$
(12)
Here, $\alpha$ is a constant, $\chi$ is the singulant function that will
capture the singular behaviour of the solution at $\sigma=\pm\mathrm{i}$, and
$X$, $Y$, and $\Phi$ are functional prefactors of the divergent solutions. It
can be seen from the dominant balance as $n\to\infty$ of equations (10a) and
(10b) that $x_{n+1}=O(y_{n})$ and $y_{n}=O(\phi_{n})$, which has motivated our
precise ordering in $n$ in the ansatzes (12).
Substitution of ansatzes (12) into the $O(F^{2n})$ equations (10a)-(10c)
yields at leading order in $n$ the three equations
$Y-\phi_{0}^{\prime}\chi^{\prime}\Phi=0,\qquad\chi^{\prime}\Big{(}X+y_{1}^{\prime}Y\Big{)}=0,\qquad\chi^{\prime}\Big{(}\Phi+a\mathrm{i}\phi_{0}^{\prime}Y\Big{)}=0.$
(13)
While the last two of these equations permit the solution $\chi^{\prime}=0$,
this is unable to satisfy the first equation in (13). The remaining solutions
can be solved to give $\chi^{\prime}=a\mathrm{i}(\phi_{0}^{\prime})^{-2}$,
which we integrate to find
$\chi_{a}(\sigma)=a\mathrm{i}\int_{a\mathrm{i}}^{\sigma}\bigg{[}1+\frac{\Gamma_{\text{c}}}{\pi}\frac{1}{(1+t^{2})}\bigg{]}^{-2}\mathrm{d}t.$
(14)
Here, we have introduced the notation $\chi_{a}=\chi$, where $a=\pm 1$, to
discern between each singulant generated by the two singular points of
$\phi_{0}^{\prime}$, which are given by $\sigma=\mathrm{i}$ and
$\sigma=-\mathrm{i}$. The starting point of integration in (14) is
$\sigma=\pm\mathrm{i}$ to ensure that $\chi_{a}(a\mathrm{i})=0$. This
condition is required in order to match with an inner solution near this
singular point. Integration of (14) yields
$\displaystyle\chi_{a}(\sigma)=$ $\displaystyle
a\mathrm{i}\bigg{[}\sigma+\frac{\Gamma_{\text{c}}^{2}\sigma}{2(\Gamma_{\text{c}}+\pi)(\pi\sigma^{2}+\Gamma_{\text{c}}+\pi)}-\frac{\Gamma_{\text{c}}(3\Gamma_{\text{c}}+4\pi)}{2\sqrt{\pi}(\Gamma_{\text{c}}+\pi)^{3/2}}\tan^{-1}{\bigg{(}\frac{\sqrt{\pi}\sigma}{\sqrt{(\Gamma_{\text{c}}+\pi)}}\bigg{)}}\bigg{]}$
(15)
$\displaystyle+1+\frac{\Gamma_{\text{c}}}{2(\Gamma_{\text{c}}+\pi)}-\frac{\Gamma_{\text{c}}(3\Gamma_{\text{c}}+4\pi)}{2\sqrt{\pi}(\Gamma_{\text{c}}+\pi)^{3/2}}\tanh^{-1}{\bigg{(}\frac{\sqrt{\pi}}{\sqrt{\Gamma_{\text{c}}+\pi}}\bigg{)}}.$
### 3.3 Solution of the late-term amplitude equations
We now determine the amplitude functions, $\Phi$, $X$, and $Y$, of the late
term solutions. Note that if one of these amplitude functions is known, then
the other two may be determined by the last two equations in (13). Thus, only
one equation is required for the amplitude functions, which we find at the
next order of $n$ in the late term equation (10a). This equation is given by
$\phi_{0}^{\prime}\Phi^{\prime}=\phi_{1}^{\prime}\chi^{\prime}\Phi,$ (16)
which may be integrated to find the solution
$\Phi(\sigma)=\Lambda\exp{\bigg{(}a\mathrm{i}\int_{0}^{\sigma}\frac{\phi_{1}^{\prime}(t)}{[\phi_{0}^{\prime}(t)]^{3}}\mathrm{d}t\bigg{)}}.$
(17)
In the above, $\Lambda$ is a constant of integration, which is determined by
matching with an inner solution near the singular points $\sigma=a\mathrm{i}$.
Once $\Phi$ is known, the remaining amplitude functions are determined by the
equations $Y=a\mathrm{i}(\phi_{0}^{\prime})^{-1}\Phi$ and
$X=a\mathrm{i}\phi_{0}^{\prime\prime}\Phi$.
We now calculate the constant, $\alpha$, that appears in the factorial-over-
power ansatzes (12). This is determined by ensuring that the singular
behaviour, as $\sigma\to a\mathrm{i}$, of each ansatz is consistent with the
anticipated singular behaviours of
$x_{n}=O\Big{(}(\sigma-a\mathrm{i})^{1-3n}\Big{)},\qquad
y_{n}=O\Big{(}(\sigma-a\mathrm{i})^{1-3n}\Big{)},\qquad\phi_{n}=O\Big{(}(\sigma-a\mathrm{i})^{-3n}\Big{)}.$
(18)
In taking the inner limit of $\Phi$ from (17), we have
$\Phi=O(\sigma-a\mathrm{i})^{3/2}$. Furthermore since
$\chi=O\big{(}(\sigma-a\mathrm{i})^{3}\big{)}$, derived later in equation
(40), equating the power of the singularities for $\phi_{n}$ between the
ansatz (12) and the anticipated singular behaviour above in (18) yields the
value of $\alpha=1/2$. The constant of integration, $\Lambda$, that appears in
solution (17) for the amplitude function, $\Phi$, is derived in Appendix A by
matching the inner limit of the divergent solution, $\phi_{n}$, with an inner
solution at $\sigma=a\mathrm{i}$. This yields
$\alpha=\frac{1}{2}\qquad\text{and}\qquad\Lambda=-\frac{\Gamma_{\text{c}}(-a\mathrm{i})^{1/2}\mathrm{e}^{-\mathcal{P}(a\mathrm{i})}}{6\pi}\bigg{(}-\frac{4a\mathrm{i}\pi^{2}}{3\Gamma_{\text{c}}^{2}}\bigg{)}^{\alpha}\lim_{n\to\infty}\bigg{(}\frac{\hat{\phi}_{n}}{\Gamma(n+\alpha+1)}\bigg{)},$
(19)
where $\hat{\phi}_{n}$, determined via recurrence relation (45), is a constant
appearing in the series expansion for the outer limit of the inner solution
for $\phi$, and $\mathcal{P}(\sigma)$ is defined in equation (48).
To conclude, the late-term divergence of the asymptotic expansions (7) diverge
in a factorial-over-power manner specified by the ansatzes (12). Evaluation of
this divergence requires the constants $\alpha$ and $\Lambda$ from equation
(19), as well as the singulant function $\chi(\sigma)$ from (15) and amplitude
function $\Phi(\sigma)$ from (17). These will be required in the derivation of
the exponentially-small terms considered in the next section.
### 3.4 Stokes smoothing and Stokes lines
The exponentially-small components of the solutions are now determined. We
truncate the asymptotic expansions (7) at $n=N-1$ and considering a remainder,
yielding
$x=\underbrace{\sum_{n=0}^{N-1}F^{2n}x_{n}}_{x_{r}}+~{}\bar{x},\qquad
y=\underbrace{\sum_{n=0}^{N-1}F^{2n}y_{n}}_{y_{r}}+~{}\bar{y},\qquad\phi^{\prime}=\underbrace{\sum_{n=0}^{N-1}F^{2n}\phi_{n}^{\prime}}_{\phi^{\prime}_{r}}+~{}\bar{\phi},$
(20)
where the truncated asymptotic expansions have been denoted by $x_{r}$,
$y_{r}$, and $\phi_{r}^{\prime}$. When $N$ is chosen optimally at the point at
which the divergence expansions reorder as $n\to\infty$, given by
$N\sim\frac{\lvert\chi\rvert}{F^{2}}+\rho,$ (21)
where $0\leq\rho<1$ to ensure that $N$ is an integer, the remainders to the
asymptotic expansions (20) will be exponentially-small.
Equations for these remainders are found by substituting the truncated
expansions (20) into the analytically continued equations (6a)–(6c). These are
given by
$\displaystyle(F^{2}\phi_{0}^{\prime}+F^{4}\phi_{1}^{\prime})\bar{\phi}^{\prime}+\bar{y}$
$\displaystyle=-\xi_{\text{a}},$ (22a) $\displaystyle
2\bar{x}^{\prime}+2F^{2}y_{1}^{\prime}\bar{y}^{\prime}$
$\displaystyle=-\xi_{\text{b}},$ (22b)
$\displaystyle\bar{\phi}^{\prime}+a\mathrm{i}\phi_{0}^{\prime}\bar{y}^{\prime}$
$\displaystyle=-\xi_{\text{c}}.$ (22c)
In equations (22) above, nonlinear terms such as $\bar{x}^{2}$ were neglected
as they will be exponentially subdominant. In anticipating that
$\bar{x}=O(F^{2}\bar{y})=O(F^{2}\bar{\phi})$, terms of the first two orders of
$F^{2}$ have been retained on the left-hand side of (22a). Motivated by the
late-term analysis, in which equations for the amplitude functions were
obtained at leading order for the last two governing equations, we have only
retained the leading order terms in equations (22b) and (22c). Furthermore,
the forcing terms introduced in equations (22) are defined by
$\left.\quad\begin{aligned}
\xi_{\text{a}}&=\frac{F^{2}}{2}(\phi_{r}^{\prime})^{2}+y_{r}-\frac{F^{2}}{2},\qquad\xi_{\text{b}}=\big{(}x_{r}^{\prime}\big{)}^{2}+\big{(}y_{r}^{\prime}\big{)}^{2}-1,\\\
\xi_{\text{c}}&=\phi_{r}^{\prime}x_{r}^{\prime}-1+a\mathrm{i}\phi_{r}^{\prime}y_{r}^{\prime}-\frac{\Gamma_{\text{c}}}{\pi}\frac{y_{r}+1}{(x_{r})^{2}+(y_{r}+1)^{2}}-\widehat{\mathcal{I}}[x_{r},y_{r},\phi_{r}^{\prime}].\end{aligned}\quad\right\\}$
(23)
Since each order of these forcing terms will be identically zero up to and
including $O(F^{2(N-1)})$, they will be of $O(F^{2N})$. Only knowledge of
$\xi_{\text{a}}$ will be required in the Stokes smoothing procedure of this
section, which is given by
$\xi_{\text{a}}\sim\phi_{0}^{\prime}\phi_{N-1}^{\prime}F^{2N}.$ (24)
Homogeneous solutions to equations (22), for which the forcing terms on the
right-hand sides are omitted, are $\bar{x}\sim
F^{2}X\mathrm{e}^{-\chi/F^{2}}$, $\bar{y}\sim Y\mathrm{e}^{-\chi/F^{2}}$, and
$\bar{\phi}\sim\Phi\mathrm{e}^{-\chi/F^{2}}$, where the singulant $\chi$ and
amplitude functions $X$, $Y$, and $\Phi$ satisfy the same equations as those
found for the late-term solutions in §3.2. Next, we solve for the particular
solutions of equations (22) through variation of parameters by multiplying the
homogeneous solutions by an unknown function, $\mathcal{S}(\sigma)$, giving
$\left.\quad\begin{aligned}
\bar{x}\sim\mathcal{S}(\sigma)X(\sigma)\mathrm{e}^{-\chi(\sigma)/F^{2}},\\\
\bar{y}\sim\mathcal{S}(\sigma)Y(\sigma)\mathrm{e}^{-\chi(\sigma)/F^{2}},\\\
\bar{\phi}\sim\mathcal{S}(\sigma)\Phi(\sigma)\mathrm{e}^{-\chi(\sigma)/F^{2}},\end{aligned}\quad\right\\}$
(25)
where $Y=a\mathrm{i}\Phi/\phi_{0}^{\prime}$ and $X=-y_{1}^{\prime}Y$. The
function $\mathcal{S}$ is called the Stokes multiplier, as it will display the
Stokes phenomenon across Stokes lines of the problem, which is demonstrated
next. An equation for $\mathcal{S}$ is obtained by substituting (25) and
similar expressions for $\bar{x}$ and $\bar{y}$ into equation (22a), yielding
$F^{2}\phi_{0}^{\prime}\Phi\mathrm{e}^{-\chi/F^{2}}\mathcal{S}^{\prime}(\sigma)\sim-\xi_{\text{a}}$.
In substituting for the dominant behaviour of $\xi_{\text{a}}$ from (24) and
the factorial-over-power divergence of $\phi^{\prime}_{N-1}$ from (12), and
changing derivatives of $\mathcal{S}$ from $\sigma$ to $\chi$, we find
$\frac{\mathrm{d}\mathcal{S}}{\mathrm{d}\chi}\sim\frac{\Gamma(N+\alpha)}{\chi^{N+\alpha}}F^{2(N-1)}\mathrm{e}^{\chi/F^{2}}.$
(26)
Figure 3: The Stokes lines (bold) lie along the imaginary axis between the
two singular points of $\sigma=-\mathrm{i}$ and $\sigma=\mathrm{i}$. Branch
cuts are shown with a wavy line.
In expanding as $N\to\infty$, and substituting for
$N\sim\lvert\chi\rvert/F^{2}+\rho$ from equation (21), the right-hand side of
equation (25) is seen to be exponentially-small, except for in a boundary
layer close to contours satisfying
$\text{Im}[\chi]=0\quad\text{and}\quad\text{Re}[\chi]>0.$ (27)
These are the Stokes line conditions originally derived by Dingle (1973).
Across the Stokes lines, the solution for the Stokes multiplier $\mathcal{S}$,
$\mathcal{S}(\sigma)=S_{a}+\frac{\sqrt{2\pi}\mathrm{i}}{F^{2\alpha}}\int_{-\infty}^{\sqrt{\lvert\chi\rvert}\tfrac{\arg{(\chi)}}{F}}\exp{(-t^{2}/2)}\mathrm{d}t,$
(28)
rapidly varies from the constant $S_{a}$ to
$S_{a}+2\pi\mathrm{i}/F^{2\alpha}$. This is the Stokes phenomenon, and the
contours satisfying the Dingle conditions (27) are shown in figure 3 to lie
along the imaginary axis. For the one vortex case studied in this section, the
upstream condition as $\text{Re}[\sigma]\to-\infty$ requires that $S_{1}=0$
and $S_{-1}=-2\pi\mathrm{i}/F^{2\alpha}$.
### 3.5 Trapped waves generated by two submerged vortices
We have so far studied the case of a single submerged point vortex. When
multiple point vortices are placed within the fluid, the only change is to the
boundary integral equation, previously specified in (6c) for a single vortex.
In this section we study the formulation of two submerged point vortices of
the same nondimensional strength, $\Gamma_{\text{c}}$, located at
$z=x+\mathrm{i}y=\pm\lambda-\mathrm{i}$, for which the analytically continued
boundary integral equation is given by
$\displaystyle\phi^{\prime}(\sigma)x^{\prime}(\sigma)-1+a\mathrm{i}\phi^{\prime}(\sigma)y^{\prime}(\sigma)=\frac{\Gamma_{\text{c}}}{\pi}$
$\displaystyle\bigg{[}\frac{y(\sigma)+1}{[x(\sigma)-\lambda]^{2}+[y(\sigma)+1]^{2}}$
(29)
$\displaystyle+\frac{y(\sigma)+1}{[x(\sigma)+\lambda]^{2}+[y(\sigma)+1]^{2}}\bigg{]}+\widehat{\mathcal{I}}[x,y,\phi].$
Unlike the case for a single submerged point vortex that produces waves in the
far field for $x\to\infty$, two identical point vortices can produce solutions
for which the waves are confined to lie between the vortices,
$-\lambda<\text{Re}[\sigma]<\lambda$. This occurs for critical values of the
Froude number, which we now predict using the techniques of exponential
asymptotics developed in the previous sections.
The first two orders of the asymptotic solution for $\phi$ are now given by
$\displaystyle\phi_{0}^{\prime}(\sigma)$
$\displaystyle=1+\frac{\Gamma_{\text{c}}}{\pi}\bigg{[}\frac{1}{1+(\sigma+\lambda)^{2}}+\frac{1}{1+(\sigma-\lambda)^{2}}\bigg{]},$
(30a) $\displaystyle\phi_{1}^{\prime}(\sigma)$
$\displaystyle=-a\mathrm{i}\phi_{0}^{\prime}y_{1}^{\prime}+\frac{\Gamma_{\text{c}}y_{1}}{\pi}\bigg{[}\frac{(\sigma+\lambda)^{2}-1}{[1+(\sigma+\lambda)^{2}]^{2}}+\frac{(\sigma-\lambda)^{2}-1}{[1+(\sigma-\lambda)^{2}]^{2}}\bigg{]}+\widehat{\mathcal{I}}_{n}(\sigma),$
(30b)
which are singular at the four locations $\sigma=-\lambda+a\mathrm{i}$ (from
the vortex at $z=-\lambda-\mathrm{i}$) and $\sigma=\lambda+a\mathrm{i}$ (from
the vortex at $z=\lambda-\mathrm{i}$). Note that we have again defined $a=\pm
1$ to indicate whether $\text{Im}[\sigma]>0$ or $\text{Im}[\sigma]<0$. These
four singular points each have associated Stokes lines, shown in figure 4.
Figure 4: The Stokes lines (bold) generated by the four singular points are
shown.
In general, the waves switched on across the first Stokes lines, emanating
from the points $\sigma=-\lambda+a\mathrm{i}$, will be out of phase with the
waves switched on across the second Stokes lines, from
$\sigma=\lambda+a\mathrm{i}$. However, for certain values of $F$, the wave
switched on across the first Stokes line is then switched off by the second
Stokes line, yielding solutions with no waves for $\text{Re}[\sigma]>\lambda$.
An example of this trapped solution was shown earlier in figure 1(b).
Thus, in using the Stokes switching prediction for $\bar{\phi}$ shown in
figure 4 and writing $\bar{y}=a\mathrm{i}\bar{\phi}/\phi_{0}^{\prime}$, we
require for the two contributions of
$\left.\begin{aligned}
\bar{y}_{1}&\sim-\frac{2\pi}{F^{2\alpha}\phi^{\prime}_{0}}\Phi_{1}(\sigma)\exp{\Big{(}-\frac{\chi_{1}(\sigma)}{F^{2}}\Big{)}}+c.c.,\\\
\bar{y}_{2}&\sim-\frac{2\pi}{F^{2\alpha}\phi^{\prime}_{0}}\Phi_{2}(\sigma)\exp{\Big{(}-\frac{\chi_{2}(\sigma)}{F^{2}}\Big{)}}+c.c.,\end{aligned}\quad\right\\}$
(31)
to cancel with one another for $\text{Re}[\sigma]>\lambda$. Here, we denoted
$\chi_{1}$ and $\Phi_{1}$ as the singulant and amplitude function arising from
the $\sigma=-\lambda+a\mathrm{i}$ singularities, and $\chi_{2}$ and $\Phi_{2}$
as those arising from the $\sigma=\lambda+a\mathrm{i}$ singularities. The
first of (31), $\bar{y}_{1}$, is the contribution switched on as we pass from
left to right across the Stokes lines associated with the singular points
$\sigma=-\lambda+a\mathrm{i}$. The second, $\bar{y}_{2}$, is the contribution
switched on from left to right by the Stokes lines associated with the
$\sigma=\lambda+a\mathrm{i}$ singular point. Note that the specified
contributions in (31) are from the $a=1$ contribution, and the unspecified
complex-conjugate components are from that with $a=-1$.
We now simplify each of the expressions given in equation (31) by substituting
for the amplitude functions $\Phi_{1}$ and $\Phi_{2}$, which satisfy the same
equation as that found previously in (16). The only difference will be the
constants of integration, which we denote by $\Lambda_{1}$ and $\Lambda_{2}$.
This yields
$\left.\begin{aligned}
\bar{\phi}_{1}&\sim-\frac{4\pi\lvert\Lambda_{1}\rvert}{F^{2\alpha}\phi_{0}^{\prime}}\exp{\left(-\frac{\text{Re}[\chi_{1}]}{F^{2}}\right)}\cos{\left(\int_{0}^{\sigma}\frac{\phi_{1}^{\prime}(t)}{[\phi_{0}^{\prime}(t)]^{3}}\mathrm{d}t+\arg{[\Lambda_{1}]}-\frac{\text{Im}[\chi_{1}]}{F^{2}}\right)},\\\
\bar{\phi}_{2}&\sim-\frac{4\pi\lvert\Lambda_{2}\rvert}{F^{2\alpha}\phi_{0}^{\prime}}\exp{\left(-\frac{\text{Re}[\chi_{2}]}{F^{2}}\right)}\cos{\left(\int_{0}^{\sigma}\frac{\phi_{1}^{\prime}(t)}{[\phi_{0}^{\prime}(t)]^{3}}\mathrm{d}t+\arg{[\Lambda_{2}]}-\frac{\text{Im}[\chi_{2}]}{F^{2}}\right)}.\end{aligned}\quad\right\\}$
(32)
Since $\lvert\Lambda_{1}\rvert=\lvert\Lambda_{2}\rvert$ and
$\text{Re}[\chi_{1}]=\text{Re}[\chi_{2}]$, the prefactors multiplying each of
the cosine functions in (32) are identical, and the condition for them to
cancel, $\bar{y}_{1}+\bar{y}_{2}=0$, yields
$\displaystyle\cos{\left(\int_{0}^{\sigma}\frac{\phi_{1}^{\prime}(t)}{[\phi_{0}^{\prime}(t)]^{3}}\mathrm{d}t+\frac{\arg{[\Lambda_{1}]}+\arg{[\Lambda_{2}]}}{2}-\frac{\text{Im}[\chi_{1}+\chi_{2}]}{2F^{2}}\right)}$
(33)
$\displaystyle\qquad\qquad\qquad\qquad\times\cos{\left(\frac{\arg{[\Lambda_{1}]}-\arg{[\Lambda_{2}]}}{2}-\frac{\text{Im}[\chi_{1}-\chi_{2}]}{2F^{2}}\right)}=0.$
Note that since $\chi_{1}$ and $\chi_{2}$ satisfy the same differential
equation, $\chi^{\prime}=a\mathrm{i}(\phi_{0}^{\prime})^{-2}$, originally
derived in §3.2, the only difference between them are their constants of
integration. Therefore $\text{Im}[\chi_{1}+\chi_{2}]$ will be a function of
$\sigma$, and $\text{Im}[\chi_{1}-\chi_{2}]$ will be constant. Thus, only the
second cosine component of (33) is capable of satisfying the identity for
$\text{Re}[\sigma]>\lambda$. Since this cosine function is zero when the
argument equals $\pm\pi/2$, $\pm 3\pi/2$, and so forth, we find
$F_{k}=\sqrt{\frac{\text{Im}[\chi_{1}-\chi_{2}]}{\arg{[\Lambda_{1}]}-\arg{[\Lambda_{2}]}+\pi(2k+1)}}.$
(34)
for $k=0,1,2,\ldots$, and so forth. Equation (34) yields the discrete values
of the Froude number, $F_{k}$, for which the waves are confined to lie between
the two submerged vortices. All that remains is to evaluate
$\text{Im}[\chi_{1}-\chi_{2}]$, $\arg{[\Lambda_{1}]}$, and
$\arg{[\Lambda_{2}]}$. Each of these singulants are found by integrating
$\chi^{\prime}=\mathrm{i}(\phi_{0}^{\prime})^{-2}$, where $\phi_{0}^{\prime}$
is specified in equation (30), from the corresponding singular point. We may
decompose each singulant into a real-valued integral along the Stokes line,
and an imaginary-valued integral along the free-surface. Thus,
$\text{Im}[\chi]$ is an integral along the free-surface,
$\text{Im}[\sigma]=0$, from the intersection of the Stokes line to $\sigma$.
This yields
$\text{Im}[\chi_{1}(\sigma)-\chi_{2}(\sigma)]=\int_{-\lambda}^{\lambda}\bigg{[}1+\frac{\Gamma_{\text{c}}}{\pi}\bigg{(}\frac{1}{1+(t+\lambda)^{2}}+\frac{1}{1+(t-\lambda)^{2}}\bigg{)}\bigg{]}^{-2}\mathrm{d}t.$
(35)
In the numerical results of §4.2, the integral in (35) is evaluated with a
symbolic programming language. Note that the Stokes lines depicted in figure 4
are not truly vertical, and are slightly curved such that they intersect the
free surface at the points $-\lambda^{*}$ and $\lambda^{*}$. Thus, the range
of integration in (35) should actually lie between
$-\lambda^{*}<t<\lambda^{*}$; however since $\lambda^{*}$ is very close in
value to $\lambda$ (for $\lambda=8$ and $\Gamma_{c}=0.3$, $\lambda^{*}\approx
7.99998$), this subtlety has been ignored.
Comparisons between the analytical prediction of $F_{k}$ from (34) and
numerical results are performed in §4.2.
## 4 Numerical results
We begin in §4.1 by verifying with numerical results our analytical
predictions for the exponentially-small scaling as $F\to 0$ for the case of a
single vortex. This is given by the singulant function, $\chi$, from (15), and
comparisons are made for a range of values of the vorticity,
$\Gamma_{\text{c}}$. The analytical predictions of the Froude numbers for
trapped waves between two point vortices, given in (34), are then compared to
numerical predictions in §4.2.
A detailed description of the numerical method used is given by Forbes (1985),
which we will briefly summarise here.
1. (i)
The real-valued domain, $s$, is truncated to lie between the values of
$s_{\text{L}}$ and $s_{\text{R}}$. $N$ discretisation points are used, such
that the numerical domain is given by
$s_{k}=s_{\text{L}}+(k-1)(s_{\text{R}}-s_{\text{L}})/(N-1)$ for $1\leq k\leq
N$. The unknown solution is taken to be $y^{\prime}(s)$, which we define at
each gridpoint by $y^{\prime}_{k}=y^{\prime}(s_{k})$. The radiation conditions
are imposed by enforcing $y_{1}=0$, $y_{1}^{\prime}=0$, $x_{1}^{\prime}=1$,
$\phi_{1}^{\prime}=1$, $x_{1}=s_{\text{l}}$, and $\phi_{1}=s_{\text{L}}$, and
the initial guess for $y_{k}^{\prime}$ is either zero or a previously computed
solution.
2. (ii)
Since we assume that $y^{\prime}_{k}$ is known at the next gridpoint, the
arclength relation (2b) yields $x^{\prime}_{k}$. Trapezoidal-rule integration
then determines values for $x_{k}$ and $y_{k}$, which we use to find
$\phi_{k}^{\prime}$ from Bernoulli’s equation (2a). This process is repeated
for $k=2$ to $k=N$ to find function values at every gridpoint.
3. (iii)
The boundary-integral equation (2c) is evaluated at each gridpoint with the
known values of $x_{k}$, $y_{k}$, $\phi^{\prime}_{k}$, $x^{\prime}_{k}$, and
$y^{\prime}$. To avoid the singularity associated with the principal-valued
integral $\mathcal{I}[x,y,\phi^{\prime}]$, each unknown that is not a function
of the integration variable, $t$, is instead evaluated between gridpoints by
interpolation.
4. (iv)
This yields $N-1$ nonlinear equations from evaluating the boundary-integral
equation between each gridpoint, $(s_{k}+s_{k+1})/2$, which is closed by the
$N-1$ unknowns $y^{\prime}_{k}$ for $k=2$ to $k=N$. Solutions are found by
minimising the residual through Newton iteration. For the trapped waves
studied in §4.2, we impose an additional constraint of symmetry about $s=0$ in
the real-valued solution, $y(s)$, such that the Froude number, $F$, is
determined as an eigenvalue.
### 4.1 Waves generated by a single vortex
Figure 5: The exponentially-small dependence of the wave amplitude is shown
(dots) for numerical results for seven different values of
$\Gamma_{\text{c}}=\\{0.1,0.15,0.2,0.25,0.3,0.35,0.4\\}$. Solid lines
represent the analytical gradient found from the real part of $\chi$ in
equation (15). The behaviour of this gradient for different values of the
vortex strength $\Gamma_{\text{c}}$ is shown in figure 6.
For the numerical results presented in this section, we have used $N=2000$
grid points, and a domain specified by $s_{\text{L}}=-40$ and
$s_{\text{R}}=40$. In computing numerical solutions for a wide range of Froude
numbers, and the values of
$\Gamma_{\text{c}}=\\{0.1,0.15,0.2,0.25,0.3,0.35,0.4\\}$, the exponentially-
small scaling as $F\to 0$ of the high-frequency waves present for $s>0$ may be
measured. This is shown in the semilog plot of figure 5. We see that these
lines, each of which represents solutions with a different value of
$\Gamma_{\text{c}}$, are straight and thus the amplitude of these ripples is
exponentially small as $F\to 0$. The gradient of each of these lines is
expected to closely match the exponential scaling predicted analytically,
given by the singulant $\chi$. Along the free surface, this is given by
$\text{Re}[\chi]$ from equation (15) which takes constant values.
Figure 6: The analytical prediction for $\text{Re}[\chi]$ along the free
surface $\text{Im}[\sigma]=0$ from equation (15) is shown against the
vorticity $\Gamma_{\text{c}}$ (line). The numerical predictions, corresponding
to the slopes of the semilog plot in figure 5, are shown circled.
In figure 6, this analytical prediction is compared to the numerical values
from figure 5, and good agreement is observed. Note that there are small
instabilities present in the numerical solution which decay when the truncated
domain is extended; upon which we expect the numerical results to tend towards
the analytical prediction shown in figure 6.
Figure 7: For $F=0.45$ and $\Gamma_{\text{c}}=0.4$, a numerical solution
(dashed) is compared to an analytical solution (line) determined in §3.
Comparison between a numerical and asymptotic solution profile is shown in
figure 7 for $F=0.45$ and $\Gamma_{\text{c}}$. The numerical solution is
determined by the scheme detailed at the beginning of §4, with $N=2000$
discretisation points in the arclength, $-40\leq s\leq 40$. The asymptotic
solution plots $x(s)=x_{0}(s)+F^{2}x_{1}(s)+\bar{x}(s)$ against
$y(s)=y_{0}(s)+F^{2}y_{1}(s)+\bar{y}(s)$. These early order solutions,
$x_{0}$, $x_{1}$, $y_{0}$, and $y_{1}$ are specified in equations (8) and (9).
The exponentially-small components, $\bar{x}$ and $\bar{y}$, are implemented
from expression (25). This requires knowledge of the singulant, $\chi$, given
in (15), the amplitude functions $Y=a\mathrm{i}\Phi/\phi_{0}^{\prime}$ and
$X=-y_{1}^{\prime}Y$ determined from $\Phi$ in (17), and the Stokes
multiplier, $\mathcal{S}$, given in (28). A real-valued asymptotic solution is
obtained through evaluating the sums
$\bar{x}\rvert_{a=1}+\bar{x}\rvert_{a=-1}$ and
$\bar{y}\rvert_{a=1}+\bar{y}\rvert_{a=-1}$ on the real-valued domain,
$\sigma=s$, for $\text{Im}[\sigma]=0$. Note that in the determination of the
constant $\Lambda$, its magnitude, $\lvert\Lambda\rvert$, been fitted to equal
that found from the corresponding numerical solution, and its argument
(corresponding to a phase shift of the resultant wave) is determined from
relation (19) as $\text{arg}{[\Lambda]}=a\pi/2$.
### 4.2 Trapped gravity waves between two vortices
We considered the case of two submerged point vortices analytically in §3.5.
When each vortex had the same nondimensional circulation, $\Gamma_{\text{c}}$,
and depth equal to unity, trapped waves were seen to occur for certain
discrete values of the Froude number, $F_{k}$. In this section, we compare the
analytical prediction for $F_{k}$ from (34) with numerical results. These
trapped numerical solutions are found with the method detailed at the
beginning of §4. In imposing the additional constraint of symmetry to
eliminate waves downstream of the vortices, the special Froude number,
$F_{k}$, is determined as an eigenvalue. These results were performed for
$N=4000$ grid points, a domain between $s_{\text{L}}=-60$ and
$s_{\text{R}}=60$, and horizontal vortex placement specified as $\lambda=8$.
In figure 8, we plot the tail amplitude (for $s>\lambda$) of the asymptotic
solutions for the values of $0.3<F<0.5$, $\Gamma_{\text{c}}=0.3$, and
$\lambda=8$. This amplitude is equal to zero at the values of $F_{k}$ from
equation (34). The figure also contains additional markers denoted by (a),
where $F=0.3383$, and (b), where $F=0.4270$. This corresponds to the figure 9
where we compare numerical solutions obtained in this section, and asymptotic
solutions from §3 for those given values of $F$. The fit is excellent and the
corresponding curves are nearly visually indistinguishable at the scale of the
graphic.
Figure 8: The amplitude of oscillations present for $s>\lambda$ in the
asymptotic solutions is shown against the Froude number, $F$. Here,
$\Gamma_{\text{c}}=0.3$ and $\lambda=8$. This amplitude is equal to zero at
the locations $F_{k}$ derived in equation (34). The two points marked (a) and
(b) correspond to the profiles shown in figure 9. Figure 9: Two different
trapped wave solutions are shown for $\Gamma_{\text{c}}=0.3$ and $\lambda=8$
corresponding to (a) $F=0.3383$ and (b) $F=0.4270$. Asymptotic solutions
(solid line) are compared to numerical solutions (dashed) for (a) $k=22$ and
(b) $k=14$. In each inset, the two curves are nearly indistinguishable to
visual accuracy.
Finally, in figure 10, we compare the values of $F_{k}$ obtained analytically
and numerically. The straight lines are the analytical prediction from (34),
and dots represent the numerical values for $F_{k}$.
Figure 10: Values of the Froude number, $F_{k}$, for which the waves are
trapped between each submerged vortex are shown. The numerical results of §4.2
are shown circled, and the analytical results from equation (34) are shown
with lines. Here, $\lambda=8$, and for the numerical solutions $N=4000$,
$s_{\text{L}}=-60$, and $s_{\text{R}}=60$.
## 5 Conclusion
We have shown, through both numerical and analytical investigations, that the
waves generated by submerged point vortices are exponentially small in the
low-speed limit of $F\to 0$. Furthermore, when two submerged vortices are
considered, oscillatory waves vanish downstream for certain values of the
Froude number, $F$. Through the techniques of exponential asymptotics, we have
demonstrated how these values may be derived. Their prediction relies on the
understanding of singularities in the analytically continued domain that
generate a divergent asymptotic expansion. The remainder to this series is
exponentially small as $F\to 0$, and the study of the associated Stokes
phenomenon yields discrete values of $F$ for which the waves are trapped
between each vortex.
## 6 Discussion
The work presented here forms a basis for a number of interesting extensions
involving exponentially-small water waves with gravity, capillarity, and/or
vorticity providing singular perturbative effects.
First, it should be remarked that the classical exponential-asymptotics
theories by _e.g._ Chapman & Vanden-Broeck (2002, 2006) for capillary- and
gravity-driven surface waves produced in flows over topographies rely upon the
existence of closed-form conformal maps. In such problems, the governing
equations for the free-surface can be written in terms of a single complex-
valued unknown (_e.g._ the complex velocity), with the velocity potential
serving as the independent variable. This includes situations such as flows
past polygonal boundaries (related to the availability of the Schwarz-
Christoffel mapping). The arclength formulation we have used in this work
provides a more general setting for wave-structure interactions with arbitrary
bodies, including for instance, flows past smoothed bodies specified in
$(x,y)$-coordinates. Here, we have demonstrated that the exponential
asymptotics can be generalised to such formulations. We expect that many of
the interesting wave-structure interactions studied by _e.g._ Holmes et al.
(2013) (flow past a symmetric bottom topography) and Hocking et al. (2013)
(flow past a semi-ellipse), can be attacked using the technology we have
developed here.
Secondly, the phenomena of trapped waves is an interesting one. The
exponential asymptotics interpretation, whereby waves switched-on at one
location (the Stokes line intersection) must be switched-off at another,
provides an intuitive explanation for how trapped waves form in singularly
perturbative limits. The context, in our problem, relates to vortices fixed
within the fluid for modelling submerged obstructions, such as the submerged
cylinders studied numerically by Tuck & Scullen (1998). However, trapped waves
have been detected numerically in other geometries including submerged bumps
(Hocking et al., 2013), a semi-ellipse (Holmes et al., 2013), a trigonometric
profile (Dias & Vanden-Broeck, 2004), spikes (Binder et al., 2005), and a
rectangular bump (Lustri et al., 2012). We expect that the ‘selection
mechanism’ that produces the countably infinite set of values (34) is a kind
of universality in eigenvalue problems (cf. Chapman et al. 2022 for further
discussion and examples).
Finally, we note that in this paper, the forcing mechanism producing the waves
was via the complex-plane singularities associated with the point
vortices—then, we found that the waves were singularly perturbed due to the
inertial term in Bernoulli’s equation, thus producing exponentially small
waves, scaling as $\exp(-\text{const.}/F^{2})$. Recently, analytical solutions
have been developed for pure-vorticity-driven water waves, notably in the
works by Crowdy & Nelson (2010); Crowdy & Roenby (2014); Crowdy (2022). In
essence, we believe these solutions can serve as leading-order approximations
in the regime of small surface-tension; it might be expected that
exponentially-small parasitic ripples then exist on the surface of such
vorticity-driven profiles. This would then be similar to the work of Shelton
et al. (2021); Shelton & Trinh (2022) for parasitic capillary ripples on steep
gravity waves. Numerical and analytical work on such class of problems is
ongoing.
Acknowledgements. We are grateful for many stimulating and motivating
discussions that took place during the recent LMS-Bath symposium “New
Directions in Water Waves” held at the University of Bath in July 2022. PHT is
supported by the Engineering and Physical Sciences Research Council
[EP/V012479/1].
Declaration of interests. The authors report no conflict of interest.
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## Appendix A Inner analysis at the singularities $\sigma=\pm\mathrm{i}$
In order to determine the constant of integration of the amplitude function
$\Phi(\sigma)$ from equation (16), knowledge of the inner solutions at the
singularities $\sigma=\mathrm{i}$ and $\sigma=-\mathrm{i}$, is required. In
this section, we study the inner boundary layer at both of these locations,
for which matching with the inner limit of the outer solutions determines the
constant of integration.
First, we note that in the outer region, $\sigma=O(1)$, the asymptotic series
first reorder whenever
$\phi_{0}^{\prime}(\sigma)\sim F^{2}\phi_{1}^{\prime}(\sigma),\qquad
y_{1}(\sigma)\sim F^{2}y_{2}(\sigma),\qquad x_{2}(\sigma)\sim
F^{2}x_{3}(\sigma).$ (36)
In substituting for the early orders of the asymptotic solutions from
equations (8), (9), and (11), we see that each of (36) reorder in a boundary
layer of the same width, given by $\sigma-a\mathrm{i}=O(F^{2/3})$. We thus
introduce the inner variable, $\hat{\sigma}$, by the relation
$\sigma-a\mathrm{i}=\hat{\sigma}F^{2/3},$ (37)
for which $\hat{\sigma}=O(1)$ in the inner region. Since the asymptotic series
each reorder near the two locations of $\sigma=\mathrm{i}$ and
$\sigma=-\mathrm{i}$, we have again used the notation $a=\pm 1$ to distinguish
between these two cases.
Next, to determine the form of the inner solutions, we take the inner limit of
the outer series expansions for $\phi^{\prime}$, $x$, and $y$, by substituting
for the inner variable $\hat{\sigma}$ defined in (36) and expanding as $F\to
0$. This yields
$\phi^{\prime}\sim\frac{1}{F^{2/3}}\bigg{[}-\frac{a\mathrm{i}\Gamma_{\text{c}}}{2\pi}\frac{1}{\hat{\sigma}}+\cdots\bigg{]},\quad
y\sim
F^{2/3}\bigg{[}\frac{\Gamma_{\text{c}}^{2}}{8\pi^{2}}\frac{1}{\hat{\sigma}^{2}}+\cdots\bigg{]},\quad
x\sim a\mathrm{i}+F^{2/3}\bigg{[}\hat{\sigma}+\cdots\bigg{]},$ (38)
where the omitted terms, represented by ($\cdots$), are from the inner limit
of lower order terms of the outer asymptotic expansion. For instance, the next
term in the inner limit of $\phi^{\prime}$ is of $O(\hat{\sigma}^{-4})$. The
form of the inner limits in (38) motivates our definition of the inner
solutions, $\hat{\phi}(\hat{\sigma})$, $\hat{y}(\hat{\sigma})$, and
$\hat{x}(\hat{\sigma})$, through the equations
$\phi^{\prime}=-\frac{a\mathrm{i}\Gamma_{\text{c}}}{2\pi
F^{2/3}}\frac{\hat{\phi}(\hat{\sigma})}{\hat{\sigma}},\qquad
y=\frac{\Gamma_{\text{c}}^{2}F^{2/3}}{8\pi^{2}}\frac{\hat{y}(\hat{\sigma})}{\hat{\sigma}^{2}},\qquad
x=a\mathrm{i}+\hat{\sigma}F^{2/3}\hat{x}(\hat{\sigma}).$ (39)
The form of the inner variables introduced in (39) ensures that the first term
in the series expansion for their outer limit will be equal to unity.
Furthermore, based on the form of the inner limit of the singulant, $\chi$,
from equation (14),
$\chi\sim-\frac{4a\mathrm{i}\pi^{2}}{3\Gamma_{\text{c}}^{2}}\hat{\sigma}^{3}F^{2},$
(40)
the outer limit of the inner solutions will be a series expansion in inverse
powers of $-4a\mathrm{i}\pi^{2}\hat{\sigma}^{3}/(3\Gamma_{\text{c}})$. We thus
introduce the variable $z$, defined by
$z=-\frac{4a\mathrm{i}\pi^{2}}{3\Gamma^{2}_{\text{c}}}\hat{\sigma}^{3},$ (41)
to ensure that these series expansions are in inverse powers of $z$ alone.
### A.1 Inner equation
The leading order inner equations, as $F\to 0$, may now be derived by
substituting (39) into the outer equations (6a)-(6c), yielding
$\displaystyle\hat{y}-\hat{\phi}^{2}=0,$ (42a)
$\displaystyle\Big{(}\hat{x}+3z\hat{x}^{\prime}\Big{)}^{2}-\bigg{(}\frac{1}{3z}\hat{y}-\frac{1}{2}\hat{y}^{\prime}\bigg{)}^{2}=1,$
(42b)
$\displaystyle\hat{\phi}\bigg{(}\hat{x}-\frac{1}{6z}\hat{y}\bigg{)}\bigg{(}\hat{x}+3z\hat{x}^{\prime}-\frac{1}{3z}\hat{y}+\frac{1}{2}\hat{y}^{\prime}\bigg{)}=1.$
(42c)
The inner solutions, $\hat{\phi}(z)$, $\hat{y}(z)$, and $\hat{x}(z)$, will
satisfy equations (42a)-(42c). Rather than solve these inner equations
exactly, knowledge of the inner solutions is only required under the outer
limit of $z\to\infty$ in order to match with the inner limit of the outer
solutions to determine their divergent form. Thus, we will consider the
following series expansions for these inner unknowns,
$\hat{\phi}(z)=\sum_{n=0}^{\infty}\frac{\hat{\phi}_{n}}{z^{n}},\qquad\hat{y}(z)=\sum_{n=0}^{\infty}\frac{\hat{y}_{n}}{z^{n}},\qquad\hat{x}(z)=\sum_{n=0}^{\infty}\frac{\hat{x}_{n}}{z^{n}},$
(43)
which hold as $z\to\infty$.
At leading order as $z\to\infty$ we have, by the definition on the inner
solutions in equation (39),
$\hat{\phi}_{0}=1,\qquad\hat{y}_{0}=1,\qquad\hat{x}_{0}=1.$ (44)
Determination of $\hat{\phi}_{n}$, $\hat{y}_{n}$, and $\hat{x}_{n}$, as
$n\to\infty$, requires the evaluation of a recurrence relation, which is now
given.
Firstly, substitution of expansions (44) into the inner equation (42b) yields
$\displaystyle\hat{x}_{1}=0,\qquad 2(1-3n)\hat{x}_{n}$
$\displaystyle=\frac{1}{36}\sum_{m=0}^{n-2}(2+3m)(2n-3m-4)\hat{y}_{m}\hat{y}_{n-m-2}$
(45a)
$\displaystyle~{}~{}~{}+\sum_{m=1}^{n-1}(1-3m)(3n-3m-1)\hat{x}_{m}\hat{x}_{n-m},\qquad\text{for
$~{}n\geq 2$}.$ Next, we substitute the same expansions into the inner
equation (42c), yielding
$\displaystyle\hat{\phi}_{1}=\frac{1}{2},\qquad\hat{\phi}_{n}$
$\displaystyle=\sum_{m=2}^{n}\sum_{q=1}^{m-1}\frac{(3q-1)\hat{\phi}_{n-m}}{36}\Big{(}6\hat{x}_{q}-\hat{y}_{q-1}\Big{)}\Big{(}6\hat{x}_{m-q}-\hat{y}_{m-q-1}\Big{)}$
(45b)
$\displaystyle~{}~{}~{}-\sum_{m=1}^{n}\frac{\hat{\phi}_{n-m}}{6}\Big{(}6(2-3m)\hat{x}_{m}+(3m-2)\hat{y}_{m-1}\Big{)},\qquad\text{for
$~{}n\geq 2$}.$ Lastly, a recurrence relation for $\hat{y}_{n}$ is found from
equation (42a) to be
$\hat{y}_{1}=1,\qquad\hat{y}_{n}=\sum_{m=0}^{n}\hat{\phi}_{m}\hat{\phi}_{n-m},\qquad\text{for
$~{}n\geq 2$}.$ (45c)
Assuming that $\hat{\phi}_{n-1}$, $\hat{y}_{n-1}$, and $\hat{x}_{n-1}$ are
known, $\hat{x}_{n}$ can be determined from equation (45a), which then yields
a value for $\hat{\phi}_{n}$ from equation (45b). Lastly, $\hat{y}_{n}$ is
found by evaluating equation (45c).
### A.2 Matching and determination of the constant $\Lambda$
We now match the outer limit of the inner solution, $\hat{\phi}$, with the
inner limit of the outer solution, $\phi^{\prime}$. In writing the outer limit
of the inner solution in outer variables, we have
$\phi^{\prime}=\frac{-a\mathrm{i}\Gamma_{\text{c}}}{2\pi}\sum_{n=0}^{\infty}\frac{F^{2n}\hat{\phi}_{n}}{\Big{(}-\frac{4a\mathrm{i}\pi^{2}}{3\Gamma_{\text{c}}^{2}}\Big{)}^{n}(\sigma-a\mathrm{i})^{3n+1}},$
(46)
and for the inner limit of the outer solution,
$\displaystyle\phi^{\prime}=\sum_{n=0}^{\infty}F^{2n}\phi_{n}^{\prime}$
$\displaystyle\sim\sum_{n=0}^{\infty}-F^{2n}\chi^{\prime}\Phi\frac{\Gamma(n+\alpha+1)}{\chi^{n+\alpha+1}}$
(47)
$\displaystyle\sim\sum_{n=0}^{\infty}-\frac{4\pi^{2}\Lambda}{\Gamma_{\text{c}}^{2}(-a\mathrm{i})^{1/2}}\mathrm{e}^{\mathcal{P}(a\mathrm{i})}\frac{F^{2n}\Gamma{(n+\alpha+1)}}{\Big{(}-\frac{4a\mathrm{i}\pi^{2}}{3\Gamma_{\text{c}}^{2}}\Big{)}^{n+\alpha+1}(\sigma-a\mathrm{i})^{3n+3\alpha-1/2}}.$
In the above, the inner limit of the amplitude function $\Phi$ from equation
(17) has been taken by defining
$\mathcal{P}(\sigma)=\int_{0}^{\sigma}\bigg{[}\frac{a\mathrm{i}\phi_{1}^{\prime}(t)}{[\phi_{0}^{\prime}(t)]^{3}}-\frac{3}{2(t-a\mathrm{i})}\bigg{]}\mathrm{d}t,$
(48)
such that $\mathcal{P}(\sigma)=O(1)$ as $\sigma\to a\mathrm{i}$. Matching (46)
with (47) determines the constant, $\Lambda$, as
$\Lambda=-\frac{\Gamma_{\text{c}}(-a\mathrm{i})^{1/2}\mathrm{e}^{-\mathcal{P}(a\mathrm{i})}}{6\pi}\bigg{(}-\frac{4a\mathrm{i}\pi^{2}}{3\Gamma_{\text{c}}^{2}}\bigg{)}^{\alpha}\lim_{n\to\infty}\bigg{(}\frac{\hat{\phi}_{n}}{\Gamma(n+\alpha+1)}\bigg{)}.$
(49)
|
# LensLeech: On-Lens Interaction for Arbitrary Camera Devices
Christopher Getschmann<EMAIL_ADDRESS>0000-0002-0174-5974 Aalborg
UniversityAalborgDenmark and Florian Echtler<EMAIL_ADDRESS>Aalborg
UniversityAalborgDenmark
(2023)
###### Abstract.
Cameras provide a vast amount of information at high rates and are part of
many specialized or general-purpose devices. This versatility makes them
suitable for many interaction scenarios, yet they are constrained by geometry
and require objects to keep a minimum distance for focusing. We present the
LensLeech, a soft silicone cylinder that can be placed directly on or above
lenses. The clear body itself acts as a lens to focus a marker pattern from
its surface into the camera it sits on. This allows us to detect rotation,
translation, and deformation-based gestures such as pressing or squeezing the
soft silicone. We discuss design requirements, describe fabrication processes,
and report on the limitations of such on-lens widgets. To demonstrate the
versatility of LensLeeches, we built prototypes to show application examples
for wearable cameras, smartphones, and interchangeable-lens cameras, extending
existing devices by providing both optical input and output for new
functionality.
Mobile Interfaces, Elastomer Sensors, Optical Widgets
††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: Eighteenth International Conference on Tangible,
Embedded, and Embodied Interaction; February 11–14, 2024; Cork,
Ireland††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Human-centered
computing Interaction devices††ccs: Human-centered computing Ubiquitous and
mobile devices††ccs: Hardware Emerging interfaces Figure 1. a) The soft
silicone attachment with an integrated lens can be placed directly on and
above cameras. b) A marker point pattern is focused by the silicone lens into
the camera. c) Arbitrary devices with cameras, such as a smartphone can track
the position, rotation, and deformation of the silicone attachment on the
camera to sense input.
## 1\. Introduction
There is an increasing number of mobile devices that make use of cameras as
primary or additional sensors. At the same time, physical input has become a
scarce feature on modern, highly-integrated devices. Many of these camera
devices are limited in their input channels and the interaction techniques
they can offer due to trade-offs and design decisions to make them smaller,
more robust, or less expensive. Very small devices such as action cameras or
wearables have either very few buttons or small touch screens, requiring to
delegate even basic input tasks to paired smartphones or suffer from the fat
finger problem (Siek et al., 2005). Larger, interchangeable-lens cameras do
provide both larger touchscreens and more buttons but could benefit from
additional input options such as back-of-device interaction as well (Baudisch
and Chu, 2009). Smartphones and tablets with capacitive touchscreens offer no
physical feedback, have reachability issues (Wobbrock et al., 2008), and
require visual confirmation for input (Buxton et al., 1985). While all of
these devices could benefit from additional physical input, they have in
common that they include a powerful sensor: a camera. Yet, any camera requires
a minimal focal distance to provide focused images necessary to process rich
user input (Xiao et al., 2013; Yamada et al., 2018).
We present the LensLeech, a soft silicone attachment that can be placed
directly on or above the front element of camera lenses. The silicone is
optically clear and deformable so forces applied by fingers, hands, or
arbitrary objects can be detected visually. By using the lower surface of the
silicone body as a close-focus lens, a marker pattern on the opposite surface
is always in focus regardless of the minimum focal distance of the camera
lens.
With the LensLeech we can transform an unused or idle camera lens into a
button, a knob, or a d-pad widget (and reverse it in seconds). These widgets
provide physical feedback (when deformed), can be operated with gloves, and
are robust, versatile, and inexpensive. This allows to add knobs and d-pads to
small wearable cameras to change settings in situ, make lens caps for large
cameras touch-sensitive, or introduce novel optical attachments for
smartphones.
In summary, we contribute:
* •
a tangible deformation sensor to create buttons, knobs, and d-pads, combining
soft body, optical elements, and sensing pattern in a single object
* •
discussions on the design and fabrication of on-lens widgets
* •
an image processing pipeline for analyzing position, rotation, and deformation
* •
application examples for integration with new and existing devices
Many research approaches aim at providing novel functionality with new or
existing sensors for future devices, often built on the assumption or
requirement of a possible miniaturization and integration of this external
sensing hardware into a new device with a new form factor. However, we
explicitly aim at retrofitting existing and well-proven interaction techniques
to sensors that make them available both to legacy devices today as well as
new ones in the future. This could help to extend the lifetime of devices in
circulation by improving their usability and reducing incentives to update to
newer hardware prematurely. A user study (with regard to the input
capabilities of the widgets) is not presented as these input modalities are
well understood and can be directly applied to this new form factor.
The remainder of this paper is organized as follows: related work is discussed
with a broad overview of vision-based elastomer sensors and on-lens/around-
lens interaction techniques, then we explain our concept of soft silicone
attachments for on-lens interaction sensing. The image processing pipeline and
fabrication procedure are summarized subsequently. We built upon that by
presenting a set of scenarios and prototypes created to show real-world
applications. Finally, we discuss the limitations of using soft silicone
attachments for on-lens interaction and conclude with specific directions for
future work.
## 2\. Related Work
Relevant to the presented work are both optical deformation sensors primarily
developed for robotic applications as well as human-computer interaction
techniques and prototypes that gather input from the space on and around
camera lenses.
### 2.1. Optical Elastomer Sensors
Optical deformation sensing of soft materials is performed either by measuring
light altered by the surface or by detecting displacement of high-contrast
markers, on the surface or encapsulated in the material. Surface deformation
measurements have been proposed based on total internal reflection (Hiraishi
et al., 1988), Lambertian reflection (Johnson and Adelson, 2009; Watanabe et
al., 2014; Dong et al., 2017; Donlon et al., 2018; Taylor et al., 2021; Wang
et al., 2021) and polarization (Sato et al., 2009). The most common type of
sensor, the GelSight family, makes use of Lambertian reflection by coating the
clear elastomer with a reflective membrane. Multispectral illumination from
below allows to derive deformation depth and thus a detailed 2.5d geometry of
the reflective surface. For marker-based sensing, high-contrast points are
painted on the clear surface of the elastomer (Wang et al., 2021; Taylor et
al., 2021; Dong et al., 2017), on the interior of an opaque hull for TacTip
sensors (Winstone et al., 2012; Ward-Cherrier et al., 2018) or colored balls
are directly encapsulated in the soft material (Kamiyama et al., 2004;
Sferrazza and D’Andrea, 2019; Yamaguchi and Atkeson, 2017).
These sensors have been used extensively for tactile sensing in robotic
applications, mounting the sensor on the end effector to measure gripping
force and detect slipping. For this, the sensor assembly is designed as a
monolithic unit consisting of camera sensor, lens, and elastomer block. While
mirrors (Donlon et al., 2018; Wang et al., 2021) and fisheye lenses (Taylor et
al., 2021) have been used to shorten optical paths to create more compact
grippers these sensors are still of considerable size and rely on a tight
integration of all components, making them incompatible with arbitrary
cameras. Modular approaches offer only exchangeable elastomers while still
using a specialized camera (Lambeta et al., 2020). Additionally, all gel-based
sensors with the exception of the sensor by Obinata et al. (Obinata et al.,
2007) and Fingervision (Yamaguchi and Atkeson, 2017) block environmental light
and require white, RGB or ultraviolet illumination by integrated LEDs. For a
detailed overview refer to the reviews by Shimonomura (Shimonomura, 2019) and
Abad et al. (Abad and Ranasinghe, 2020).
In the domain of human-computer interaction, elastomer sensors have been used
for interactive surfaces (Follmer et al., 2011), clay-like projection displays
(Punpongsanon et al., 2013) and tangibles on tabletops (Weiss et al., 2009;
Hennecke et al., 2011) to support novel interaction techniques.
### 2.2. On-Lens/Around-Lens Interaction
Placing a fingertip directly on a smartphone camera lens has been proposed as
an interaction technique in LensGestures (Xiao et al., 2013). The unfocused
environmental light passing through a finger’s tissue is used to approximate
finger positions and recognize gestures. CamTrackPoint (Yamada et al., 2018)
improves on this concept by providing tactile feedback. A spring-actuated
plastic ring is integrated with a smartphone case directly over the lens for
the finger to rest on. The thin ring blocks light with a sharp transition to
black and provides a higher precision compared to tracking the blurred finger.
A proof-of-concept for more complex on-lens input techniques is presented by
Watanabe et al. (Watanabe et al., 2014): soft and optically clear toys with a
reflective surface coating are placed on the camera while a neural network is
trained to recognize deformation/gestures from internal reflections observed
through a hole in the bottom. This represents the simplest and most basic on-
lens widget: unfocused, untagged, unpowered and depending on natural
illumination, but very easy to manufacture and not obstructing the camera when
not in use.
Interaction in the space around lenses requires mirrors to both shorten the
optical path and redirect light. Clipwidgets (Visschedijk et al., 2022) makes
use of a conical mirror in a bulky smartphone case to read the state of
physical widgets such as buttons and sliders. Similar approaches have been
presented for back-of-device interaction concepts with smartphones (Wong et
al., 2016; Matsushima et al., 2017; Kitade and Yamada, 2019). Without relying
on physical input objects Handsee (Yu et al., 2019) utilizes a prism to track
hands touching and floating above a smartphone display while Surroundsee (Yang
et al., 2013) tracks objects in the whole room with a circular 360-degree
mirror above the smartphone camera.
Similar techniques have been used without mirrors or lenses in the context of
tangibles with silicone feet for pressure sensing (Weiss et al., 2009),
deformation sensing on small wearables (Weigel and Steimle, 2017), and surface
position sensing with fibers (Wimmer, 2010). Other work that is related to the
presented concept is Bokode (Mohan et al., 2009), a marker made of a lenslet
and microfilm which magnifies a grid of 2D barcodes into the defocused lens of
a camera and Sauron (Savage et al., 2013), a design tool to integrate cameras
in hollow objects that read the state of mechanical input elements.
While physical input similar to the LensLeech can be achieved on smartphones
in particular by simply redirecting electrodes of the capacitive touchscreen
(Yu et al., 2011; Schmitz et al., 2017; Matsushima et al., 2017), the
LensLeech is not limited to touchscreens and can be applied across a range of
devices, see the application examples in section 5.
Since on-lens interaction concepts such as CamTrackPoint and LensGestures make
use of unfocused light, they are limited in their expressiveness due to the
low amount of information available. While they are suitable for smartphones
and their scratch-resistant camera assemblies, these concepts translate poorly
to interchangeable lens cameras or action cams with lens front elements often
using coatings sensitive to scratches or prints from fingertips. This is one
of the fundamental issues we intend to address with our generalizable
approach.
## 3\. The design of on-lens widgets
Figure 2. Illustrative ray diagram of the combined optical system. The field
of view inside the silicone (dashed line) depends on the field of view of the
camera, the position and diameter of the entrance pupil, as well as the
distance between silicone and camera lens.
We propose that any physical attachment enabling on or around-lens interaction
with both existing and future devices should—ideally—adhere to these basic
design considerations:
* •
safe to use near or on optical components and providing credible reassurance
to the user about this. This is a prerequisite for user acceptance.
* •
non-invasive, requiring no hardware modifications of the host device or its
camera. This ensures compatibility with existing devices that benefit most
from optical attachments.
* •
passive and unpowered, requiring only ambient illumination (if possible) to
reduce size and complexity.
* •
universal; compatible with arbitrary camera/lens combinations across a wide
range of device types.
Elastomer sensors in general fulfill the first and most important of these
requirements by virtue of their nature: they are soft. However, existing
sensors fall short in most or all other points.
As discussed, these sensors combine camera and elastomer in a permanent
assembly with a fixed position and rotation, limiting the way objects can
interact with them. Additionally, they make use of known sensor and lens
combinations to allow camera calibration and optimizations of sensor geometry
(for example by backprojecting through a calibrated lens to find optimal
marker point placements). This makes these sensors more precise and reliable
but prevents them from being used with arbitrary lenses and cameras. Finally,
most sensors require constant internal illumination. Reflective membranes
(GelSight) and rubber skins (TacTip) are blocking ambient light to avoid
interference. Only sensors relying solely on point patterns (Obinata et al.,
2007; Yamaguchi and Atkeson, 2017) can tolerate ambient illumination.
We propose an elastomer sensor design suitable for interaction sensing. The
LensLeech is a tangible soft input device that resembles the gel part of an
elastomer sensor. Our all-silicone design combines a lens, compliant body, and
a colored marker pattern in a single unit (see fig. 2). This addresses all
design requirements at the cost of reduced reliability and precision compared
to elastomer sensor assemblies that are designed to measure precise gripping
forces on robot actuators.
The small form factor of the LensLeech attachment (33mm diameter, 25.5mm
height) makes it easy to grip it with two fingers and place it on a camera. By
using the lower surface of the clear silicone body as a lens, light reflected
by the deformation-sensing pattern on the surface is collimated and can be
focused on the sensor at any distance from the camera. This makes it possible
to place the silicone foot of the LensLeech directly on or slightly above the
front element of a wide range of lenses. The combined optical system of
sensor, camera lens, silicone lens, and deformation sensing pattern is limited
by the field of view of the camera, its entrance pupil, and the distance to
the silicone attachment. This is discussed in more detail in section 6
(Limitations).
Marker Pattern
When choosing a marker pattern for deformation sensing, we need to take into
account that a positive lens required to move the focal point to the surface
of the silicone body will introduce a strong magnification effect. This
amplifies any defects or irregularities in the pattern and requires the
fabrication of very small features. The most precise and reliable method is
the deposition of single droplets of silicone paint. This makes a point
pattern the preferred choice.
A point pattern is used by other optical tactile sensors such as TacTip and
GelSight as well, however, these sensors are fixed assemblies that can compute
deviations from a static reference frame. This does not apply to a silicone
sensor that can be moved and rotated freely, thus a method to align the
currently visible region of interest with the overall marker grid is required.
A common method for identifying sections of point grids are two-dimensional
DeBruijn sequences. These are sequences that contain every subsequence of a
defined size at most once. Printed as microdots on paper these have been used
for position-tracking with digital pens (Petterson and Edso, 2003) (encoding
bits as a displacement from a regular grid) and tangibles (Schüsselbauer et
al., 2021) (encoding bits as black and white). However, unlike a rigid piece
of paper which allows displacement coding of dots, the soft silicone is easily
bent or compressed and requires coding by color or contrast.
While a hexagonal arrangement of points offers the densest packing it is
incompatible with a 2D-DeBruijn sequence. Hence, we computed a DeBruijn-like
pattern with 7-point hexagons instead of 3x3 matrices using a brute-force
approach. Each overlapping hexagonal sliding window in the pattern is unique
in the given rotation (see fig. 3). An optimal pattern does contain only
hexagons that are unique in all rotations which is simplifying pattern
matching during image processing, however, this requires a minimum of three
colors at a suitable pattern size. Higher robustness to adverse lighting
conditions and a less error-prone fabrication with only two different paints
is the reason why a less-than-optimal two-color pattern is preferable.
Our hexagon patterns consist of 127 points and require 91 unique sliding
windows. This is sufficient to cover the visible region of the silicone
attachment even when placed on a wide-angle camera. While other sensors such
as TacTip and GelSight Wedge use the same or similar number of points, only a
subset of points is visible at a time for our application due to the
magnification of the silicone lens. If a unique center hexagon is enforced
during pattern generation up to 28 patterns can be discerned. This allows to
map different silicone attachments to specific input modalities.
Figure 3. Each hexagonal sliding window appears only once. Some sliding
windows are unique in all six orientations, some can be found in multiple
locations when rotated.
Image Processing
The DeBruijn-like point pattern is color-coded in blue and green to offer a
high contrast across the range of human skin tones. Coincidentally, the
fingertips have fewer variations due to smaller differences in the skin tones
of palms overall. The detection and classification of the points is the first
step of the image processing pipeline. Background removal is performed by
thresholding in the HSV color space. The diffuse top surface of the silicone
body improves this step considerably without blocking any ambient light. From
these point candidates, colors are extracted and classified by thresholding
the two classes in the hue component of the HSV colorspace using Otsu’s method
(Otsu, 1979). This is robust to errors in white balance caused by tinted
ambient illumination or fingertips and computationally less expensive than
other classification methods. Robustness is especially important since most
auto white balance algorithms overshoot for several dozen frames when a finger
is placed on the point pattern.
For pattern matching each detected point is grouped with its 6 closest
neighbors and all 6 rotation variants are checked against a lookup table.
Correct rotation is assumed when the highest number of matches is found
between neighboring sliding windows in the camera image and ground truth
pattern. Given an optimal pattern, only one rotation would result in a match
(in the absence of any errors), yet this computationally-expensive step is
necessary to limit the pattern to only two colors. This makes the pipeline’s
processing speed highly dependent on the number of detected points. On a
laptop computer (2.3 GHz 8-Core Intel i9) 34 frames per second are processed
when 39 points are visible (30% of the pattern) and 21 FPS when 69 points
(70%) are visible. Smartphone performance numbers are not reported since the
Android implementation performs segmentation on-device but outsources the
pattern-matching step of the pipeline to a server.
Figure 4. The four types of input: a) Pressing on the silicone body b) lateral
pushing in any direction c) rotation on the optical axis d) squeezing the
silicone.
The input gestures are derived directly from the matched point pattern (see
fig. 4). A press on the top is recognized by detecting locally increased
distances between neighboring points, pushing sideways by computing the
centroid of all detected points, rotation by Kabsch‘s algorithm (Kabsch,
1976), and squeeze by a global change of point distances along the squeeze
axis. The gesture detection relies on algorithm implementations in SciPy
(Virtanen et al., 2020), while processing of image data is done using OpenCV
(Bradski, 2000).
Before discussing how these input types inform examples for real-world
application, the fabrication process of both silicone body and color-coding
pattern is described briefly.
## 4\. Fabrication
Figure 5. Cross section of the mold. The curved surfaces are ground and
polished each with a precision steel ball of the required curvature. The
optical surface is polished to a 2-micron finish and the diffuse surface to 40
microns. All three sections of the mold are aligned with metal dowel pins (not
pictured). Liquid silicone is poured through a horizontal channel in the
3d-printed plastic part.
The clear silicone body is created by mixing, degassing, and pouring liquid
silicone (Trollfactory Type 19) into a mold and letting it cure. The mold
itself requires two precisely manufactured features. The lower cavity is an
optical surface (sufficiently smooth to refract light for imaging
applications) to create the spherical convex lens of 7.5mm radius for
focusing. The curved top surface of 30mm radius diffuses light. Both surfaces
are CNC-milled from acrylic before being ground and polished. For this, the
spherical surface of the acrylic part is coated with lapping paste and pressed
against a rotating steel ball of matching radius (widely available as high-
precision replacement parts for large ball bearings). After polishing the
acrylic plates are fastened to a 3d-printed center part to complete the mold
(see fig. 5). Once cured and de-molded the point pattern is applied to the
clear silicone body with two 3d-stencils milled from acrylic (one per color).
The stencil is fabricated by drilling a duplicate of the mold top part with a
circuit board drill (1.0mm) to create channels. The soft body is pressed into
the matching cavity of the stencil from below and the pigmented silicone can
be poured on the channels (see fig. 6) before removing the remaining air from
the channels in a vacuum chamber. The stencil guarantees correct placement and
uniform point size. Only silicone itself bonds reliably to cured silicone
parts, thus uncured silicone mixed with color pigments is the most suitable
paint. The main issues in this process are ensuring that the high-viscosity
silicone reliably fills the channels and avoiding oversaturation of the
silicone with pigments in a silicone oil solution, which may inhibit the
curing process. A mixture (by weight) of Smooth-On’s Psycho Paint silicone
with 15 percent dry UV-reactive pigment powder and 25 percent solvent
(toluene) to lower the viscosity worked best. After 24 hours the pigmented
silicone binds reliably to the optically-clear silicone body, creating a point
pattern on the surface that is flexible and wear-resistant.
Figure 6. Cross section of the stenciling fixture. The silicone body is
pressed upwards against the curved surface to create a seal. Once locked in
place by clamps, the liquid pigmented silicone is poured into the recess at
the top of the stencil and makes its way through the micro-drilled channels.
When the stencil is lifted a small domed blob of partially-cured silicone
paint remains on the surface. The stencil and fixture for the silicone body
are aligned with metal dowel pins to ensure precise placement for each
consecutive stencil and color.
The choice of lens curvature during mold production is a trade-off. A mold for
a lens with a stronger curvature is more demanding in fabrication but the
shorter focal length allows to reduce the height of the silicone body. At the
same time, it decreases the depth of field and the field of view, allowing to
track a lower number of points in the pattern. Additionally, interacting with
the LensLeech deforms both top surface and lens. A strong press on the top
will reduce the height of the body by several millimeters depending on the
hardness of the silicone. If the height of the silicone body does not match
the focal length the light will not exit the system collimated, resulting in a
pattern that would be out of focus. In reality, this is rarely an issue since
autofocus cameras can compensate for this, fixed-focus cameras often have a
sufficient depth of field, and the image processing pipeline is robust to low
levels of blurring.
A paraxial approximation of the focal length can be obtained using the
lensmaker’s equation. Only the refraction of the first surface is relevant for
the LensLeech geometry, so a thin, plano-convex lens in air
($d=0,R_{2}=\infty$) can be assumed:
$\frac{1}{f}=(n-1)\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}+\frac{(n-1)d}{nR_{1}R_{2}}\right)=\frac{n}{R_{1}}-\frac{1}{R_{1}}$
We have chosen a radius of $R_{1}=7.5mm$ for the lens surface and assume that
Trollfactory Type 19 has a refractive index of $n=1.41$, similar to other
platinum-cure silicones. This would result in a total focal length of 18.29mm.
To account for the deformation during interaction and the strong curvature of
the lens we increase the height of the silicone body by a factor of $1.3$ to
25mm. While the LensLeech should be able to touch the glass surface of the
camera lens, the silicone lens surface in the center requires an air gap to
refract light. Thus we extend the foot around the lens by 1.0mm to account for
deformations (cross section can be seen in fig. 2). This provided the most
reliable results during testing for high forces when pressing and squeezing
while still keeping the point pattern well within the depth of field of most
cameras when not deformed.
While fabrication is the primary challenge to making close-focus silicone lens
attachments, a thorough description would go beyond the scope of this paper.
Please refer to the companion
repository111https://github.com/volzotan/LensLeech for detailed information
about the fabrication process.
## 5\. Application Examples
Tactile on-lens input can be used in a variety of scenarios for devices with
cameras in many sizes. We present two application examples that show how on-
lens interaction can be utilized to make input on both large and small cameras
more convenient and transfer well-established tangible interaction techniques
to smartphones.
Figure 7. a) Cross section: the silicone attachment can be extended with a
3d-printed shoe matched to the specific device so it slides over the
protruding wide-angle lens of an action camera. b) The LensLeech could be used
as a rotating knob, press to confirm, squeeze to cancel.
Interactive Lens Caps for Digital Cameras
Small action cameras offer a very limited number of buttons, a tiny display
(if any), and only optionally, a touch interface on this display. While there
are techniques to facilitate touch input on very small displays (Baudisch and
Chu, 2009), it may be cumbersome. Adjusting settings is often performed via a
companion app on a smartphone which is paired with the wearable camera.
However, there are scenarios in which the phone is unavailable and direct
interaction with the device itself may be favorable. These can be casual,
everyday situations like wearing gloves or very specific use cases such as
interacting with a camera enclosed in a waterproof housing while swimming or
diving. By adopting a lens cap or protective storage case that integrates a
LensLeech (see fig. 7), we can add tangible controls such as a rotation knob
or a d-pad, and extend the number of buttons on the device for easier
navigation through nested menus. However, note that in the case of underwater
usage when the LensLeech is pressed against a waterproof housing, a different
silicone lens curvature will be required due to the refractive index of water.
Figure 8. a) The silicone attachment is placed in a 3d-printed lens cap with a
spring-loaded mechanism to allow lateral movement and rotation. b) The lens
cap on an interchangeable-lens camera.
This concept extends to larger cameras as well. Many digital consumer cameras
are infamous for convoluted menus and poor usability in general. Browsing
recorded videos and photos, changing settings in nested menus, and entering
credentials to set up wireless connections requires prolonged attention and
interaction while the sensor itself is not in use during these tasks. By
integrating a silicone attachment into a lens cap we can leverage the unused
hardware without interfering with the primary use case of the camera. Rotating
or pushing the silicone sideways could be used to traverse large lists of
settings or captured footage (see fig. 8). While the soft silicone can touch
lens coatings without damaging them, the lens cap in this case prevents direct
contact and may provide reassurance to the user for these very expensive
lenses.
Hybrid Viewfinders for Smartphones
While the LensLeech provides optical input to camera-based devices, it can be
combined with other components that offer optical output as well. This allows
to create complex passive optical add-ons to existing devices. The hybrid
viewfinder slides over the top section of a smartphone covering the front lens
and a portion of the display. By adding a beamsplitter prism to a camera
viewfinder, a section of the covered display can be reflected into the
viewfinder’s optical path (see fig. 9a) to create a hybrid optical/electronic
viewfinder for smartphone photography. By integrating the LensLeech into the
viewfinder attachment, the front camera can be used for input while the rear
camera takes images and optionally provides data for the viewfinder overlay
(see fig. 9c). Rotating the LensLeech changes the data overlay and pressing it
triggers image capture. This allows optical input and output with no hardware
modifications, transforming a smartphone into a modern rangefinder-style
camera.
Figure 9. a) Cross section of the hybrid viewfinder. The beamsplitter overlays
the light emitted by a section of the smartphone screen (blue) over the
viewfinder image of the world (yellow), while the silicone attachment rests on
the front camera (red). b) The finder can be slid over the top section of the
smartphone. The silicone attachment provides rich tangible input to control
settings and take a photo without visual confirmation as a touchscreen would
require. c) View through the finder showing an overlay of the selection menu
and a digital spirit level.
## 6\. Evaluation & Limitations
Figure 10. Mean number of identified points in relation to ambient
illumination strength. Error bars specify standard deviation. Monochromatic or
environmental light with a strong color tint will result in an undetectable
point pattern regardless of illumination strength, this is the main cause for
outliers in the plot. Note: the maximum number of points visible to the camera
varies across devices depending on pupil size and field of view.
When tested with artificially generated images (rendered images of the
deformation point pattern with a pinhole aperture instead of a lens for
focusing) the rotational error is negligible at an average of 0.03 degrees.
More relevant and considerably more challenging is the real-world performance
under low-light conditions and ambient light with color casts. As an
evaluation setup, three cameras with an attached LensLeech (Pixel 3a
smartphone, Sony A6000 + Sony 20mm 2.8 digital still camera, Raspberry Pi V1
embedded camera) were placed in complete darkness facing a display showing a
subset (201 images, three per category) of the MIT indoor scene recognition
dataset (Quattoni and Torralba, 2009). These were artificially darkened and
brightened to simulate a low-light environment (resulting in a total of 1206
images). The illuminance of each scene was measured at the surface of the
LensLeech with a TSL2591 ambient light sensor. Detection performance is
depending on the combination of sensor, lens, and environment, but in general,
it can be observed that above 150 lux reliable operation can be expected (see
fig.10). Indoor lighting conditions usually exceed 150 lux while 300-500 lux
are recommended for office work (ISO 8995-1:2002, 2002).
Figure 11. Three lenses with an identical field of view of 84° but increasing
entrance pupil diameters: a) Google Pixel 3a front camera b) Sony SEL-P1650
lens (16mm focal length) c) Sigma 16mm 1.4 DC DN (16mm focal length). For all
three images, the LensLeech is resting directly on the front element of the
camera lens.
The main limitation when using any optical attachment on lenses is the
entrance pupil diameter and its distance from the first surface of the lens.
The entrance pupil is a virtual opening within the lens barrel through which
all entering light rays pass. Size and position within lenses can vary across
lens designs, even when an image of a distant object taken with different
lenses would look identical (see fig. 11). The silicone attachment (in the
size as presented) works well on small and medium-sized lenses but requires a
different geometry on very large lenses such as professional photography or
videography lenses with large front elements and entrance pupils for better
low-light performance. As a rule of thumb: if the image of the aperture seen
through the front element of the lens is considerably larger than the silicone
lens (12mm in diameter) the number of visible points is strongly reduced. In
general, a lower bound of 19 points is required to reliably recognize input
gestures through the soft widget. Additional limiting factors on the optical
system are the field of view of the lens and the curvature of the first glass
element of the lens. A camera with a narrow field of view will reduce the
number of visible points, similar to a large entrance pupil. This makes the
presented concept more suitable for medium to wide-angle systems such as
webcams, smart home devices, smartphones, and wearable cameras. If the
LensLeech is used with hard attachments (such as the lens cap) it does not sit
directly on the glass and a strong lens curvature is not an issue.
## 7\. Discussion
Compared to other on-lens interaction concepts such as CamTrackPoint (Yamada
et al., 2018) and LensGestures (Xiao et al., 2013)) that process unfocused
light, the LensLeech is less limited in the amount of information it provides
but it requires ambient light as well. While the LensLeech performs well in
most situations, LEDs (such as smartphone flashlights and autofocus-assist
lights of still cameras) or screens of devices can be used to provide
additional artificial illumination. This can be seen in the hybrid viewfinder:
a section of the covered display illuminates the point pattern from below.
Depending on the device and application scenario this might not be a viable
option. For usage within a predefined space, near-ultraviolet flood lights can
be installed to brighten the UV-reactive pigments in the point pattern (see
fig. 12) with little interference to the brightness of the environment.
In general, an attachment solely made from silicone is simple, robust, and—to
an extent—expendable. Material cost per piece is about 2 USD/EUR when
fabricated in small quantities. This makes the LensLeech comparable to other
inexpensive attachments for mobile devices that extend I/O capabilities, such
as Google Cardboard (Cardboard, 2014) or Nintendo Labo (Labo, 2018). Similar
to the limited lifetime of corrugated cardboard, a LensLeech and its point
pattern may eventually suffer from wear and tear after extended usage.
A possible negative perception of letting an object touch the front element of
the lens is not an issue when used on smartphone lenses. The application
examples show that in other scenarios it makes sense to use device-dependent
additions such as a rigid shoe on protruding lens barrels for action cameras
or lens caps on interchangeable-lens cameras.
Figure 12. Near-UV flood illumination can considerably increase the brightness
and contrast of the UV-reactive pigments in the point pattern while it only
marginally brightens the environment. a) No ultraviolet illumination b) Single
365nm-wavelength light source.
## 8\. Future Work
A LensLeech is uniformly made from a single silicone formula with consistent
Shore hardness throughout the whole body. By making use of a two-stage mold,
the lower part of the body containing the lens could be molded separately with
a harder type of clear silicone, resulting in a lower deformation of the lens
when compressed. Also by integrating air-filled cavities and compliant
elements in multi-stage molds, tactile feedback can be provided, resulting in
a sensation when a certain amount of force is applied.
The limitation of large entrance pupil sizes can be circumvented by replacing
the single silicone lens with a grid of smaller lenses. This requires a
different fabrication technique for the mold and a point pattern that is
aligned with camera lens angle and microlens position. This limits the
silicone attachment compatibility to only a single lens, yet this may not be
an issue for applications such as model-specific lens caps.
While only a single type of LensLeech is presented, the concept is versatile.
With additional illumination, microlens arrays would allow emulating a small
touchscreen on the top surface while using an angled surface geometry makes
sensing fingerprints possible. In the near future, the emergence of cameras
under displays in smartphones would allow the use of silicone attachments as
tangible input and output devices in tabletop-like scenarios. The display can
be used both for illuminating the LensLeech to sense in dark spaces as well as
to display output in or on the body itself by refracting and redirecting the
light.
## 9\. Conclusion
We presented the LensLeech, a soft silicone attachment that allows to sense
pushing, pressing, rotating, and squeezing when placed directly on or above
lenses of arbitrary cameras. This makes it possible to add tangible input
methods to a wide range of existing and new devices, especially small action
or lifelogging cameras and smartphones. We have shown application examples
ranging from small, body-worn devices to lens caps for large cameras and
complex smartphone attachments. While the attachments are limited in their
compatibility mainly by lens geometry, the low-light performance allows them
to be used with only ambient illumination without any need for hardware
modifications on a wide range of existing devices. This simple and inexpensive
approach opens up an interaction space on lenses for rich input that was
previously inaccessible with a range of further applications in the (soft)
robotics domain.
## Reproduction Note
The example applications, source code, CAD models of molds and fixtures, a
detailed description of the fabrication process, and data/scripts for
generating plots are available publicly:
https://github.com/volzotan/LensLeech
###### Acknowledgements.
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through project EC437/1-1.
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# Operators Affiliated to Banach Lattice Properties and the Enveloping Norms
###### Abstract
Several recent papers were devoted to various modifications of limited,
Grothendieck, and Dunford–Pettis operators, etc., through involving the Banach
lattice structure. In the present paper, it is shown that many of these
operators appear as operators affiliated to well known properties of Banach
lattices, like the disjoint (dual) Schur property, the disjoint Grothendieck
property, the property (d), and the sequential w∗-continuity of the lattice
operations. It is proved that the spaces consisting of regularly versions of
the above operators are all Banach spaces. The domination problem for such
operators is investigated.
Eduard Emelyanov, Svetlana Gorokhova
Keywords: Banach lattice, affiliated operators, enveloping norm, domination
problem
MSC2020: 46B25, 46B42, 46B50, 47B60
## 1 Preliminaries
Throughout the paper, vector spaces are real; operators are linear and
bounded; letters $X$, $Y$ stands for Banach spaces; and $E$, $F$ for Banach
lattices. We denote by $B_{X}$ the closed unit ball of $X$; by $\text{\rm
L}(X,Y)$ the space of all bounded operators from $X$ to $Y$; and by $E_{+}$
the positive cone of $E$. An operator $T:E\to F$ is called regular if
$T=T_{1}-T_{2}$ for some $T_{1},T_{2}\in\text{\rm L}_{+}(E,F)$. We denote by
$\text{\rm L}_{r}(E,F)$ ($\text{\rm L}_{ob}(E,F)$, $\text{\rm L}_{oc}(E,F)$)
the space of all regular (o-bounded, o-continuous) operators from $E$ to $F$.
### 1.1
Recall that a bounded $A\subseteq X$ is said to be a limited set (resp. a DP-
set) if each w∗-null (resp. w-null) sequence in $X^{\prime}$ is uniformly null
on $A$. Similarly, a bounded $A\subseteq E$ is called an a-limited set (resp.
an a-DP-set) if each disjoint w∗-null (resp. disjoint w-null) sequence in
$E^{\prime}$ is uniformly null on $A$ (cf. [6, 7, 13, 17]). Each relatively
compact set is limited, each limited set is an a-limited DP-set, and each DP-
set is an a-DP-set.
###### Assertion 1.1.1.
(cf. [11]) Let $A\subseteq X$ be limited. Then$:$
1. (i)
Every sequence in $A$ has a w-Cauchy subsequence.
2. (ii)
If $X$ is either separable or else reflexive, then $A$ is relatively compact.
3. (iii)
If $\ell^{1}$ does not embed in $X$, then $A$ is relatively w-compact.
The following technical fact (cf. [4, Prop.1.2.1]) is useful.
###### Assertion 1.1.2.
Let $A\subseteq X$ and $B\subseteq X^{\prime}$ be nonempty. Then$:$
1. (i)
A sequence $(f_{n})$ in $X^{\prime}$ is uniformly null on $A$ iff
$f_{n}(a_{n})\to 0$ for each sequence $(a_{n})$ in $A$.
2. (ii)
A sequence $(x_{n})$ in $X$ is uniformly null on $B$ iff $b_{n}(x_{n})\to 0$
for each sequence $(b_{n})$ in $B$.
A bounded $B\subseteq X^{\prime}$ (resp. $B\subseteq E^{\prime}$) is called an
L-set (resp. an a-L-set) if each w-null sequence in $X$ (resp. each disjoint
w-null sequence in $E$) is uniformly null on $B$ (cf. [24]). The next fact
follows from Assertion 1.1.2.
###### Assertion 1.1.3.
A bounded subset $A$ of $X$ is
1. (i)
limited iff $f_{n}(a_{n})\to 0$ for all w∗-null $(f_{n})$ in $X^{\prime}$ and
all $(a_{n})$ in $A$;
2. (ii)
a DP-set iff $f_{n}(a_{n})\to 0$ for all w-null $(f_{n})$ in $X^{\prime}$ and
all $(a_{n})$ in $A$.
A bounded subset $B$ of $X^{\prime}$ is
1. (iii)
an L-set iff $b_{n}(x_{n})\to 0$ for all $(b_{n})$ in $B$ and all w-null
$(x_{n})$ in $X$.
A bounded subset $A$ of $E$ is
1. (iv)
a-limited iff $f_{n}(a_{n})\to 0$ for all disjoint w∗-null $(f_{n})$ in
$E^{\prime}$ and all $(a_{n})$ in $A$;
2. (v)
an a-DP-set iff $f_{n}(a_{n})\to 0$ for all disjoint w-null $(f_{n})$ in
$E^{\prime}$ and all $(a_{n})$ in $A$.
A bounded subset $B$ of $E^{\prime}$ is
1. (vi)
an a-L-set iff $b_{n}(x_{n})\to 0$ for all $(b_{n})$ in $B$ and all disjoint
w-null $(x_{n})$ in $E$.
### 1.2
Let us recall the following properties of Banach spaces and describe operators
affiliated to these properties.
###### Definition 1.2.1.
A Banach space $X$ is said to possess:
1. a)
the Schur property (briefly, $X\in\text{\rm(SP)}$) if each w-null sequence in
$X$ is norm null;
2. b)
the Grothendieck property (briefly, $X\in\text{\rm(GP)}$) if each w∗-null
sequence in $X^{\prime}$ is w-null;
3. c)
the Dunford–Pettis property (briefly, $X\in\text{\rm(DPP)}$) if
$f_{n}(x_{n})\to 0$ for each w-null $(f_{n})$ in $X^{\prime}$ and each w-null
$(x_{n})$ in $X$;
4. d)
the Gelfand–Phillips property (briefly, $X\in\text{\rm(GPP)}$) if each limited
subset of $X$ is relatively compact (cf. [24, p.424]).
5. e)
the Bourgain–Diestel property (briefly, $X\in(\text{\rm BDP})$) if each
limited subset of $X$ is relatively w-compact [22].
Dedekind complete AM-spaces with a strong order unit belong to (GP), for a
comprehensive rescent source on the Grothendieck property see [25]. All
separable and all reflexive Banach spaces belong to (GPP) [11]. A Dedekind
$\sigma$-complete Banach lattice $E$ belongs $(\text{\rm GPP})$ iff $E$ has
o-continuous norm [12]. In particular, $c_{0},\ell^{1}\in(\text{\rm GPP})$,
yet $\ell^{\infty}\not\in(\text{\rm GPP})$. Clearly,
$\text{\rm(GPP)}\Rightarrow\text{\rm(BDP)}$. By [11], $X\in\text{\rm(BDP)}$
whenever $X$ contains no copy of $\ell^{1}$. Applying redistribution (as in
[2]) between the domain and range to the properties of Definition 1.2.1, we
obtain the following list of the affiliated operators.
###### Definition 1.2.2.
An operator $T:X\to Y$ is called:
1. a)
an [SP]-operator if $(Tx_{n})$ is norm null for each w-null $(x_{n})$ in $X$;
2. b)
a [GP]-operator if $(T^{\prime}f_{n})$ is w-null in $X^{\prime}$ for each
w∗-null $(f_{n})$ in $Y^{\prime}$;
3. c)
a [DPP]-operator if $f_{n}(Tx_{n})\to 0$ for each w-null $(f_{n})$ in
$Y^{\prime}$ and each w-null $(x_{n})$ in $X$;
4. d)
a [GPP]-operator if $T$ carries limited sets onto relatively compact sets;
5. e)
a [BDP]-operator if $T$ carries limited sets onto relatively w-compact sets.
Note that [SP]-operators coincide with Dunford–Pettis operators,
[GP]-operators coincide with Grothendieck operators, whereas [DPP]-operators
agree with weak Dunford–Pettis operators of [1, p.349].
###### Definition 1.2.3.
Let ${\cal P}$ be a class of operators between Banach spaces. A Banach space
$X$ is said to be affiliated with ${\cal P}$ if $I_{X}\in{\cal P}$. In this
case we write $X\in({\cal P})$.
It should be clear that if $(P)$ is one of the five properties mentioned in
Definition 1.2.1, then $X\in(P)$ iff $X$ affiliated with $[P]$-operators;
symbolically $([(P)])=(P)$. It is worth noticing that the reflexivity of
Banach spaces is affiliated with w-compact operators and vice versa, whereas
the finite dimensionality is affiliated with compact operators and vice versa.
### 1.3
We recall the following classes of operators.
###### Definition 1.3.1.
An operator
1. a)
$T:X\to F$ is called almost Grothendieck (shortly, $T$ is a-G) if $T^{\prime}$
takes disjoint $\text{\rm w}^{\ast}$-null sequences of $F^{\prime}$ to w-null
sequences of $X^{\prime}$ [23, Def.3.1].
2. b)
$T:X\to F$ is called almost limited (shortly, $T$ is Lm) if $T(B_{X})$ is
a-limited; i.e., $T^{\prime}$ takes disjoint $\text{\rm w}^{\ast}$-null
sequences of $F^{\prime}$ to norm null sequences of $X^{\prime}$ [19].
3. c)
$T:E\to Y$ is called almost Dunford–Pettis (shortly, $T$ is a-DP) if $T$ takes
disjoint w-null sequences to norm null ones [35].
4. d)
$T:E\to Y$ is called almost weak Dunford–Pettis (shortly, $T$ is a-wDP) if
$f_{n}(Tx_{n})\to 0$ whenever $(f_{n})$ is w-null in $Y^{\prime}$ and
$(x_{n})$ is disjoint w-null in $E$ [4, Def.5.3.1b)].
5. e)
$T:E\to Y$ is called o-limited (shortly, $T$ is o-Lm) if $T[0,x]$ is limited
for all $x\in E_{+}$; i.e., $(T^{\prime}f_{n})$ is uniformly null on all order
intervals $[0,x]\subseteq E_{+}$ for each $\text{\rm w}^{\ast}$-null $(f_{n})$
of $Y^{\prime}$ [27].
6. f)
$T:E\to F$ is called almost o-limited (shortly, $T$ is a-o-Lm) if $T[0,x]$ is
a-limited for all $x\in E_{+}$; i.e., $(T^{\prime}f_{n})$ is uniformly null on
all order intervals $[0,x]\subseteq E_{+}$ for each disjoint $\text{\rm
w}^{\ast}$-null $(f_{n})$ of $F^{\prime}$ [28, Def.3.1].
Clearly: $\text{\rm a-Lm}(X,F)\subseteq\text{\rm a-G}(X,F)$; $\text{\rm
a-DP}(E,Y)\subseteq\text{\rm a-wDP}(E,Y)$; $\text{\rm
Lm}(E,Y)\subseteq\text{\rm o-Lm}(E,Y)$; and $\text{\rm
o-Lm}(E,F)\subseteq\text{\rm a-o-Lm}(E,F)$.
Let ${\cal P}\subseteq\text{\rm L}(E,F)$. We call elements of ${\cal P}$ by
${\cal P}$-operators and denote by ${\cal P}(E,F):={\cal P}$ the set of all
${\cal P}$-operators in $\text{\rm L}(E,F)$. The ${\cal P}$-operators satisfy
the domination property if $S\in{\cal P}$ whenever $0\leq S\leq T\in{\cal P}$.
An operator $T\in\text{\rm L}(E,F)$ is said to be ${\cal P}$-dominated if $\pm
T\leq U$ for some $U\in{\cal P}$.
### 1.4 Enveloping norms on spaces of regularly ${\cal P}$-operators.
Regularly ${\cal P}$-operators were introduced in [3, 21] and the enveloping
norms in [4, 21]. Here we recall basic results. By [34, Prop.1.3.6],
$\text{\rm L}_{r}(E,F)$ is a Banach space under the regular norm
$\|T\|_{r}:=\inf\\{\|S\|:\pm T\leq S\in\text{\rm L}(E,F)\\}$. Moreover,
$\|T\|_{r}=\inf\\{\|S\|:S\in\text{\rm L}(E,F),|Tx|\leq S|x|\ \forall x\in
E\\}\geq\|T\|$ for every $T\in\text{\rm L}_{r}(E,F)$. If $F$ is Dedekind
complete, then $(\text{\rm L}_{r}(E,F),\|\cdot\|_{r})$ is a Banach lattice and
$\|T\|_{r}=\|~{}|T|~{}\|$ for every $T\in\text{\rm L}_{r}(E,F)$. The following
definition was introduced in [21, Def.2] (cf. also [3, Def.1.5.1]).
###### Definition 1.4.1.
Let ${\cal P}\subseteq\text{\rm L}(E,F)$. An operator $T:E\to F$ is called a
regularly ${\cal P}$-operator (shortly, an r-${\cal P}$-operator), if
$T=T_{1}-T_{2}$ with $T_{1},T_{2}\in{\cal P}\cap\text{\rm L}_{+}(E,F)$. We
denote by: ${\cal P}_{r}(E,F)$ the set of all regular operators in ${\cal
P}(E,F)$; and by $\text{\rm r-}{\cal P}(E,F)$ the set of all regularly ${\cal
P}$-operators in $\text{\rm L}(E,F)$.
###### Assertion 1.4.2.
([3, Prop.1.5.2]) Let ${\cal P}\subseteq\text{\rm L}(E,F)$, ${\cal P}\pm{\cal
P}\subseteq{\cal P}\neq\emptyset$, and $T\in\text{\rm L}(E,F)$. Then the
following holds.
1. (i)
$T$ is an r-${\cal P}$-operator iff $T$ is a ${\cal P}$-dominated ${\cal
P}$-operator.
2. (ii)
Suppose ${\cal P}$-operators satisfy the domination property and the modulus
$|T|$ exists in $\text{\rm L}(E,F)$. Then $T$ is an r-${\cal P}$-operator iff
$|T|\in{\cal P}$.
The replacement of $\text{\rm L}(E,F)$ in the definition of the regular norm
by an arbitrary subspace ${\cal P}\subseteq\text{\rm L}(E,F)$:
$\|T\|_{\text{\rm r-}{\cal P}}:=\inf\\{\|S\|:\pm T\leq S\in{\cal P}\\}\ \ \ \
(T\in\text{\rm r-}{\cal P}(E,F))$ (1)
gives the so-called enveloping norm on $\text{\rm r-}{\cal P}(E,F)$ [4].
Furthermore
$\|T\|_{\text{\rm r-}{\cal P}}=\inf\\{\|S\|:S\in{\cal P}\ \&\ (\forall x\in
E)\ |Tx|\leq S|x|\\}\ \ \ (T\in\text{\rm r-}{\cal P}(E,F))$ (2)
by [4, Lm.2.2.1], and if ${\cal P}_{1}$ is a subspace of ${\cal P}$ then
$\|T\|_{\text{\rm r-}{\cal P}_{1}}\geq\|T\|_{\text{\rm r-}{\cal
P}}\geq\|T\|_{r}\geq\|T\|\ \ \ \ \ (\forall\ T\in\text{\rm r-}{\cal
P}_{1}(E,F)).$ (3)
###### Assertion 1.4.3.
([4, Thm.2.3.1]) Let ${\cal P}$ be a subspace of $\text{\rm L}(E,F)$ closed in
the operator norm. Then $\text{\rm r-}{\cal P}(E,F)$ is a Banach space under
the enveloping norm.
Let ${\cal P}\subseteq\text{\rm L}(E,F)$, and denote ${\cal
P^{\prime}}:=\\{T^{\prime}:T\in{\cal P}\\}\subseteq\text{\rm
L}(F^{\prime},E^{\prime})$. Clearly, $\text{\rm r-}{\cal
P^{\prime}}(F^{\prime},E^{\prime})=(\text{\rm r-}{\cal P}(E,F))^{\prime}$.
Since $\|S^{\prime}\|=\|S\|$, it follows from (1)
$\|T^{\prime}\|_{\text{\rm r-}{\cal P^{\prime}}}=\inf\\{\|S^{\prime}\|:\pm
T^{\prime}\leq S^{\prime}\in{\cal P^{\prime}}\\}=\inf\\{\|S\|:\pm T\leq
S\in{\cal P}\\}=\|T\|_{\text{\rm r-}{\cal P}}.$
If ${\cal P}\subseteq\text{\rm L}(E,F)$ is closed in the operator norm then
${\cal P^{\prime}}\subseteq\text{\rm L}(F^{\prime},E^{\prime})$ is also closed
in the operator norm. So, the next fact follows from Assertion 1.4.3.
###### Corollary 1.4.4.
Let ${\cal P}$ be a subspace of $\text{\rm L}(E,F)$ closed in the operator
norm. Then $\text{\rm r-}{\cal P^{\prime}}(F^{\prime},E^{\prime})$ is a Banach
space under the enveloping norm.
### 1.5
In Section 2, we introduce the main definitions and discuss basic properties
of affiliated operators, especially related to enveloping norms. Section 3 is
devoted to domination results for affiliated operators, under the
consideration, with special emphasize on the property (d) and on sequential
w-continuity of lattice operations in Banach lattices. For further unexplained
terminology and notations, we refer to [1, 2, 3, 4, 7, 14, 15, 34, 36, 37,
38].
## 2 Affiliated operators and enveloping norms
Several recent papers were devoted to various modifications of limited,
Grothendieck, L- and M-weakly compact, and Dunford–Pettis operators, through
involving the structure of Banach lattices (see, e.g. [3, 4, 6, 7, 10, 13, 17,
18, 19, 20, 23, 31, 26, 28, 29, 30, 33, 37], Definition 1.3.1). In this
section we show that many of these operators appear as operators affiliated to
well known properties of Banach lattices like the disjoint (dual) Schur
property, the disjoint Grothendieck property, the property (d), and the
sequential w∗-continuity of the lattice operations. In continuation of [4] we
shortly discuss the enveloping norms correspondent to these affiliated
operators.
### 2.1
Recall that $E$ (resp. $E^{\prime}$) has sequentially w-continuous (resp.
sequentially w∗-continuous) lattice operations if $(|x_{n}|)$ is w-null (resp.
w∗-null) for each w-null $(x_{n})$ in $E$ (resp. for each w∗-null $(x_{n})$ in
$E^{\prime}$).
###### Assertion 2.1.1.
(see [27, Prop.3.1]) The following are equivalent.
1. (i)
$E^{\prime}$ has sequentially $\text{\rm w}^{\ast}$-continuous lattice
operations.
2. (ii)
Each order interval in $E$ is limited.
In particular, the dual $E^{\prime}$ of each discrete Banach lattice $E$ with
order continuous norm has sequentially w∗-continuous lattice operations [37,
Prop.1.1], [27, Cor.3.2]. Under the disjointness assumption on a sequence in
$E$ we have the following fact.
###### Assertion 2.1.2.
(cf. [1, Thm.4.34]) For every disjoint w-null $(x_{n})$ in $E$, the sequence
$(|x_{n}|)$ is also w-null.
This is no longer true for w∗-convergence (e.g. the sequence
$f_{n}:=e_{2n}-e_{2n+1}$ is disjoint $\text{\rm w}^{\ast}$-null in
$c^{\prime}$ yet $|f_{n}|({\mathbb{1}}_{\mathbb{N}})\equiv 2\not\to 0$ [13,
Ex.2.1]). We recall the following properties of Banach lattices.
###### Definition 2.1.3.
A Banach lattice $E$ has:
1. a)
the positive Schur property (briefly, $E\in\text{\rm(PSP)}$) if each w-null
sequence in $E_{+}$ is norm null (cf. [36]);
2. b)
the positive disjoint Schur property (briefly, $E\in\text{\rm(PDSP)}$) if each
disjoint w-null sequence in $E_{+}$ is norm null;
3. c)
the disjoint Schur property (briefly, $E\in\text{\rm(DSP)}$) if each disjoint
w-null sequence in $E$ is norm null;
4. d)
the dual positive Schur property (briefly, $E\in\text{\rm(DPSP)}$) if each
w∗-null sequence in $E^{\prime}_{+}$ is norm null [7, Def.3.3];
5. e)
the dual disjoint Schur property (briefly, $E\in\text{\rm(DDSP)}$) if each
disjoint w∗-null sequence in $E^{\prime}$ is norm null [32, Def.3.2];
6. f)
the positive Grothendieck property (briefly, $E\in\text{\rm(PGP)}$) if each
w∗-null sequence in $E^{\prime}_{+}$ is w-null (cf. [37, p.760]);
7. g)
the disjoint Grothendieck property (briefly, $E\in\text{\rm(DGP)}$) if each
disjoint w∗-null sequence in $E^{\prime}$ is w-null (cf. [3, Def.2.1.3]);
8. h)
the (swl)-property (briefly, $E\in\text{\rm(swl)}$) if $(|x_{n}|)$ is w-null
for each w-null sequence $(x_{n})$ in $E$;
9. i)
the (sw∗l)-property (briefly, $E\in\text{\rm(\text{\rm sw}${}^{\ast}$l)}$) if
$(|f_{n}|)$ is w∗-null for each w∗-null sequence $(f_{n})$ in $E^{\prime}$;
10. j)
the property (d) (briefly, $E\in\text{\rm(d)}$) if $(|f_{n}|)$ is $\text{\rm
w}^{\ast}$-null for each disjoint $\text{\rm w}^{\ast}$-null sequence
$(f_{n})$ in $E^{\prime}$ [17, 37];
11. k)
the bi-sequence property (briefly, $E\in\text{\rm(bi-sP)}$) if
$f_{n}(x_{n})\to 0$ for each w∗-null $(f_{n})$ in $E^{\prime}_{+}$ and each
disjoint w-null $(x_{n})$ in $E$ [7, Def.3.1];
12. l)
the strong GP-property (briefly, $E\in\text{\rm(s-GPP)}$) if each almost
limited subset of $E$ is relatively compact;
13. m)
the strong BD-property (briefly, $E\in\text{\rm(s-BDP)}$) if each almost
limited subset of $E$ is relatively w-compact.
It is well known that $\text{\rm(PSP)}=\text{\rm(PDSP)}=\text{\rm(DSP)}$.
Indeed, $\text{\rm(PSP)}\subseteq\text{\rm(PDSP)}$ holds trivially;
$\text{\rm(PDSP)}\subseteq\text{\rm(DSP)}$ is due to Assertion 2.1.2; and, for
$\text{\rm(DSP)}\subseteq\text{\rm(PSP)}$ see [36, p.16]. We include a short
proof of the following fact.
###### Assertion 2.1.4.
([7, Thm.4.2], [37, Prop.2.4]) Let $E$ be a Banach lattice. The following are
equivalent$:$
1. (i)
$E\in\text{\rm(bi-sP)}$;
2. (ii)
$E\in\text{\rm(Pbi-sP)}$, in the sense that if $f_{n}(x_{n})\to 0$ for each
w∗-null $(f_{n})$ in $E^{\prime}_{+}$ and each disjoint w-null $(x_{n})$ in
$E_{+}$.
3. (iii)
every $\text{\rm w}^{\ast}$-null sequence $(f_{n})$ in $E^{\prime}_{+}$ is
uniformly null on each disjoint w-null $(x_{n})$ in $E_{+}$.
###### Proof.
The implication i)$\Longrightarrow$ii) is obvious, whereas
ii)$\Longrightarrow$iii) follows from Proposition 1.1.2 i).
iii)$\Longrightarrow$i) Let $(f_{n})$ be w∗-null in $E^{\prime}_{+}$ and
$(x_{n})$ be disjoint w-null in $E$. By Assertion 2.1.2, $(x_{n}^{\pm})$ are
both disjoint w-null in $E_{+}$. Then $(f_{n})$ is uniformly null on both
$(x_{n}^{\pm})$, and hence on $(x_{n})=(x_{n}^{+})-(x_{n}^{-})$. By
Proposition 1.1.2 i), $f_{n}(x_{n})\to 0$, as desired. ∎
### 2.2
Applying the redistribution between the domain and range as in Definition
1.2.2 to properties of Definition 2.1.3, we obtain the correspondent
affiliated operators.
###### Definition 2.2.1.
An operator $T:E\to Y$ is called:
1. a)
a [PSP]-operator if $\|Tx_{n}\|\to 0$ for each w-null $(x_{n})$ in $E_{+}$;
2. b)
a [PDSP]-operator if $\|Tx_{n}\|\to 0$ for each disjoint w-null $(x_{n})$ in
$E_{+}$;
3. c)
a [DSP]-operator if $\|Tx_{n}\|\to 0$ for each disjoint w-null $(x_{n})$ in
$E$;
4. d)
an [s-GPP]-operator if $T$ carries almost limited subsets of $E$ onto
relatively compact subsets of $Y$;
5. e)
an [s-BDP]-operator if $T$ carries almost limited subsets of $E$ onto
relatively w-compact subsets of $Y$.
Clearly,
$\text{\rm[s-GPP]}(E,Y)\subseteq\text{\rm[GPP]}(E,Y)\bigcap\text{\rm[s-BDP]}(E,Y)\
\ \text{\rm and}$ (4) $\text{\rm[s-BDP]}(E,Y)\subseteq\text{\rm[BDP]}(E,Y).$
(5)
[DSP]-operators coincide with the almost Dunford–Pettis operators, and hence,
by [6, Thm.2.2],
$\text{\rm[PSP]}(E,Y)=\text{\rm[PDSP]}(E,Y)=\text{\rm[DSP]}(E,Y).$ (6)
###### Definition 2.2.2.
An operator $T:X\to F$ is called:
1. a)
a [DPSP]-operator if $\|T^{\prime}f_{n}\|\to 0$ for each w∗-null $(f_{n})$ in
$F_{+}^{\prime}$;
2. b)
a [DDSP]-operator if $\|T^{\prime}f_{n}\|\to 0$ for each disjoint w∗-null
$(f_{n})$ in $F^{\prime}$;
3. c)
a [PGP]-operator if $(T^{\prime}f_{n})$ is w-null for each w∗-null $(f_{n})$
in $F_{+}^{\prime}$;
4. d)
a [DGP]-operator if $(T^{\prime}f_{n})$ is w-null for each disjoint w∗-null
$(f_{n})$ in $F^{\prime}$;
5. e)
an [swl]-operator if $(|Tx_{n}|)$ is w-null for each w-null $(x_{n})$ in $X$.
[DDSP]-operators coincide with the almost limited operators, whereas
[DGP]-operators agree with the almost Grothendieck operators.
###### Proposition 2.2.3.
$(\text{\rm[DPSP]}(X,F))^{\prime}\cup(\text{\rm[DDSP]}(X,F))^{\prime}\subseteq\text{\rm[PSP]}(F^{\prime},X^{\prime})$.
###### Proof.
Let $(f_{n})$ be disjoint w-null in $F^{\prime}_{+}$. Then $(f_{n})$ is
disjoint w∗-null in $F^{\prime}_{+}$. If $T\in\text{\rm[DPSP]}(X,F)$ or
$T\in\text{\rm[DDSP]}(X,F)$ then in both cases $\|T^{\prime}f_{n}\|\to 0$.
Thus $T^{\prime}\in\text{\rm[PDSP]}(F^{\prime},X^{\prime})$, and hence
$T^{\prime}\in\text{\rm[PSP]}(F^{\prime},X^{\prime})$ by (6). ∎
###### Definition 2.2.4.
An operator $T:E\to F$ is called:
1. a)
a [dswl]-operator if $(|Tx_{n}|)$ is w-null for each disjoint w-null
$(x_{n})$;
2. b)
an $\text{\rm[sw}^{\ast}\text{\rm l]}$-operator if $(|T^{\prime}f_{n}|)$ is
w∗-null for each w∗-null $(f_{n})$ in $F^{\prime}$;
3. c)
a [d]-operator if $(|T^{\prime}f_{n}|)$ is w∗-null for each disjoint w∗-null
$(f_{n})$ in $F^{\prime}$;
4. d)
a [bi-sP]-operator if $f_{n}(Tx_{n})\to 0$ for each w∗-null $(f_{n})$ in
$F^{\prime}_{+}$ and each disjoint w-null $(x_{n})$ in $E$;
5. e)
a [Pbi-sP]-operator if $f_{n}(Tx_{n})\to 0$ for each w∗-null $(f_{n})$ in
$F^{\prime}_{+}$ and each disjoint w-null $(x_{n})$ in $E_{+}$.
###### Proposition 2.2.5.
For a Banach lattice $F$ the following are hold.
1. i)
$F\in\text{\rm(d)}$ iff $\text{\rm r-[d]}(E,F)=\text{\rm L}_{r}(E,F)$ for
every $E$.
2. ii)
$F^{\prime}$ has sequentially $\text{\rm w}^{\ast}$-continuous lattice
operations iff
$\text{\rm[sw}^{\ast}\text{\rm l]}(E,F)=\text{\rm L}_{r}(E,F)$ for every $E$.
###### Proof.
i) For the necessity, let $E$ be a Banach lattice. It is enough to prove
$\text{\rm L}_{+}(E,F)\subseteq\text{\rm[d]}(E,F)$. So, let $0\leq T:E\to F$
and $(f_{n})$ be disjoint $\text{\rm w}^{\ast}$-null in $F^{\prime}$. Since
$F\in\text{\rm(d)}$ then $(|f_{n}|)$ is $\text{\rm w}^{\ast}$-null, and then
$(T^{\prime}|f_{n}|)$ is $\text{\rm w}^{\ast}$-null in $E^{\prime}$. It
follows from $|T^{\prime}f_{n}|\leq T^{\prime}|f_{n}|$ that
$(|T^{\prime}f_{n}|)$ is $\text{\rm w}^{\ast}$-null, and hence
$T\in\text{\rm[d]}(E,F)$. The sufficiency is immediate since
$I_{F}\in\text{\rm[d]}(F,F)$ implies $F\in\text{\rm(d)}$.
ii) Just remove the disjointness condition on $(f_{n})$ in the proof of i). ∎
The next proposition shows that [Pbi-sP]-operators agree with [bi-
sP]-opeators.
###### Proposition 2.2.6.
$\text{\rm[bi-sP]}(E,F)=\text{\rm[Pbi-sP]}(E,F)$.
###### Proof.
Clearly, $\text{\rm[bi-sP]}(E,F)\subseteq\text{\rm[Pbi-sP]}(E,F)$. Let
$T\in\text{\rm[Pbi-sP]}(E,F)$, $(f_{n})$ be w∗-null in $F^{\prime}_{+}$, and
$(x_{n})$ be disjoint w-null in $E$. By Assertion 2.1.2, $(|x_{n}|)$ is
disjoint w-null in $E$. Since $T\in\text{\rm[Pbi-sP]}(E,F)$,
$f_{n}(T|x_{n}|)\to 0$. It follows from $|f_{n}(Tx_{n})|\leq f_{n}(T|x_{n}|)$
that $f_{n}(Tx_{n})\to 0$, and hence $T\in\text{\rm[bi-sP]}(E,F)$. ∎
###### Proposition 2.2.7.
Let $T\in\text{\rm L}(E,F)$. The following holds.
1. i)
$T$ is a [d]-operator iff $T$ is almost o-limited.
2. ii)
$T$ is an $\text{\rm[sw}^{\ast}\text{\rm l]}$-operator iff $T$ is o-limited.
###### Proof.
i) For the necessity, let $T\in\text{\rm[d]}(E,F)$. Suppose $x\in E_{+}$ and
$(f_{n})$ is disjoint $\text{\rm w}^{\ast}$-null in $F^{\prime}$. By the
assumption, $(|T^{\prime}f_{n}|)$ is w∗-null, and hence $|T^{\prime}f_{n}|x\to
0$. By the Riesz–Kantorovich formula,
$|T^{\prime}f_{n}|x=\sup\\{|(T^{\prime}f_{n})y|:|y|\leq x\\}\to 0$, and hence
$(T^{\prime}f_{n})$ is uniformly null on each $[0,x]$. Thus $T\in\text{\rm
a-o-Lm}(E,F)$.
For the sufficiency, let $T\in\text{\rm a-o-Lm}(E,F)$. Suppose $(f_{n})$ is
disjoint $\text{\rm w}^{\ast}$-null in $F^{\prime}$. In order to prove
$T\in\text{[d]}(E,F)$, we need to show that
$(|T^{\prime}f_{n}|)\stackrel{{\scriptstyle\text{\rm w}^{\ast}}}{{\to}}0$. It
is enough to prove that $|T^{\prime}f_{n}|x\to 0$ for each $x\in E_{+}$. Let
$x\in E_{+}$. By the assumption, $\sup\\{|(T^{\prime}f_{n})y|:|y|\leq x\\}\to
0$. Therefore, the Riesz–Kantorovich formula implies $|T^{\prime}f_{n}|x\to
0$, and hence $T\in\text{[d]}(E,F)$.
ii) Just remove the disjointness condition on $(f_{n})$ in the proof of i). ∎
### 2.3
Affiliated operators from the previous subsection form vector spaces, which
are complete under the operator norm; the details are included in the next
lemma.
###### Lemma 2.3.1.
The following sets of affiliated operators are vector spaces which are
complete in the operator norm.
1. i)
$\text{\rm[PSP]}(E,Y)$.
2. ii)
$\text{\rm[DPSP]}(X,F)$ and $\text{\rm[DDSP]}(X,F)$.
3. iii)
$\text{\rm[PGP]}(X,F)$ and $\text{\rm[DGP]}(X,F)$.
4. iv)
$\text{\rm[swl]}(X,F)$ and $\text{\rm[dswl]}(E,F)$.
5. v)
$\text{\rm[sw}^{\ast}\text{\rm l]}(E,F)$ and $\text{\rm[d]}(E,F)$.
6. vi)
$\text{\rm[bi-sP]}(E,F)$.
7. vii)
$\text{\rm[GPP]}(X,Y)$ and $\text{\rm[s-GPP]}(E,Y)$.
8. viii)
$\text{\rm[BDP]}(X,Y)$ and $\text{\rm[s-BDP]}(E,Y)$.
###### Proof.
We skip trivial checking that all sets of affiliated operators in the lemma
are vector spaces. It remains to show that each space of affiliated operators
under the consideration is a closed in the operator norm subspace of the
correspondent space of all linear operators. As arguments here are
straightforward and standard, we present them in the basic cases.
1. i)
Let $\text{\rm[PSP]}(E,Y)\ni
T_{k}\stackrel{{\scriptstyle\|\cdot\|}}{{\to}}T\in\text{\rm L}(E,Y)$. Let
$(x_{n})$ be w-null in $E_{+}$. We need to show $\|Tx_{n}\|\to 0$. Let
$\varepsilon>0$. Pick some $k\in\mathbb{N}$ with $\|T-T_{k}\|\leq\varepsilon$.
Since $T_{k}\in\text{\rm PSP}(E,Y)$, there exists $n_{0}$ such that
$\|T_{k}x_{n}\|\leq\varepsilon$ for $n\geq n_{0}$. Take $M\in\mathbb{R}$
satisfying $\|x_{n}\|\leq M$ for all $n\in\mathbb{N}$. Since
$\|Tx_{n}\|=\|(T-T_{k})x_{n}+T_{k}x_{n}\|\leq\|T-T_{k}\|\cdot\|x_{n}\|+\|T_{k}x_{n}\|\leq\varepsilon(M+1)$
for $n\geq n_{0}$, and since $\varepsilon>0$ is arbitrary, $\|Tx_{n}\|\to 0$.
2. ii)
As the case of [DDSP](X,F) is similar, we confine ourselves to considering
[DPSP](X,F).
Let $\text{\rm[DPSP]}(X,F)\ni
T_{k}\stackrel{{\scriptstyle\|\cdot\|}}{{\to}}T\in\text{\rm L}(X,F)$, and let
$(f_{n})$ be $\text{\rm w}^{\ast}$-null in $F_{+}^{\prime}$. In order to show
$(T^{\prime}f_{n})$ is norm null, let $\varepsilon>0$ and pick $k$ with
$\|T^{\prime}-T^{\prime}_{k}\|\leq\varepsilon$. Since
$T_{k}\in\text{\rm[DPSP]}(X,F)$, there exists $n_{0}$ with
$\|T_{k}^{\prime}f_{n}\|\leq\varepsilon$ for all $n\geq n_{0}$. As $(f_{n})$
is $\text{\rm w}^{\ast}$-null, there exists $M\in\mathbb{R}$ satisfying
$\|f_{n}\|\leq M$ for all $n\in\mathbb{N}$. Since
$\|T^{\prime}f_{n}\|\leq\|T^{\prime}f_{n}-T^{\prime}_{k}f_{n}\|+\|T^{\prime}_{k}f_{n}\|\leq\|T^{\prime}_{k}-T^{\prime}\|\|f_{n}\|+\varepsilon\leq\varepsilon(M+1)$
for $n\geq n_{0}$. It follows $\|T^{\prime}f_{n}\|\to 0$, as desired.
3. iii)
As the case of [DGP](X,F) is similar, we consider [PGP](X,F) only.
Let $\text{\rm[PGP]}(X,F)\ni
T_{k}\stackrel{{\scriptstyle\|\cdot\|}}{{\to}}T\in\text{\rm L}(X,F)$, and let
$(f_{n})$ be $\text{\rm w}^{\ast}$-null in $F_{+}^{\prime}$. In order to show
that $(T^{\prime}f_{n})$ is w-null, pick a $g\in F^{\prime\prime}$, and let
$\varepsilon>0$. Fix any $k$ with
$\|T^{\prime}-T^{\prime}_{k}\|\leq\varepsilon$. Since
$T_{k}\in\text{\rm[PGP]}(X,F)$, there exists $n_{0}$ with
$|g(T_{k}^{\prime}f_{n})|\leq\varepsilon$ for all $n\geq n_{0}$. Let
$M\in\mathbb{R}$ be such $\|f_{n}\|\leq M$ for all $n\in\mathbb{N}$. Because
of
$|g(T^{\prime}f_{n})|\leq|g(T^{\prime}f_{n}-T_{k}^{\prime}f_{n})|+|g(T_{k}^{\prime}f_{n})|\leq$
$\|g\|\|T^{\prime}-T_{k}^{\prime}\|\|f_{n}\|+\varepsilon\leq(\|g\|M+1)\varepsilon$
for $n\geq n_{0}$, and since $\varepsilon>0$ is arbitrary, it follows
$g(T^{\prime}f_{n})\to 0$. Since $g\in F^{\prime\prime}$ is arbitrary,
$T\in\text{\rm[PGP]}(X,F)$.
4. iv)
We consider [swl](X,F) only. The case of [dswl](E,F) is similar.
Let $\text{\rm[swl]}(X,F)\ni
T_{k}\stackrel{{\scriptstyle\|\cdot\|}}{{\to}}T\in\text{\rm L}(X,F)$ and let
$(x_{n})$ be w-null in $X$. We need to show
$|Tx_{n}|\stackrel{{\scriptstyle\text{\rm w}}}{{\to}}0$ in $F$. Let $f\in
F^{\prime}$. There exists an $M\in{\mathbb{R}}$ with $\|x_{n}\|\leq M$ for all
$n\in{\mathbb{N}}$. Take some $\varepsilon>0$ and pick $k\in{\mathbb{N}}$ with
$\|T-T_{k}\|\leq\varepsilon$. Choose $n_{0}$ such that
$|f|(|T_{k}x_{n}|)\leq\varepsilon$ for all $n\geq n_{0}$. Then
$|f(|Tx_{n}|)|\leq|f|(|(T-T_{k})x_{n}+T_{k}x_{n}|)\leq$
$\|f\|\cdot\|T-T_{k}\|\cdot M+|f|(|T_{k}x_{n}|)\leq\varepsilon(\|f\|M+1).$
Since $\varepsilon>0$ is arbitrary, $f(|Tx_{n}|)\to 0$; and, since $f\in
F^{\prime}$ is arbitrary, $|Tx_{n}|\stackrel{{\scriptstyle\text{\rm
w}}}{{\to}}0$.
5. v)
We consider $\text{\rm[d]}(E,F)$ only. The case of
$\text{\rm[sw}^{\ast}\text{\rm l]}(E,F)$ is similar.
Let $\text{\rm[d]}(E,F)\ni
T_{k}\stackrel{{\scriptstyle\|\cdot\|}}{{\to}}T\in\text{\rm L}(E,F)$, and let
$(f_{n})$ be disjoint w∗-null in $F^{\prime}$. We need to show
$(|T^{\prime}f_{n}|)\stackrel{{\scriptstyle\text{\rm w}^{\ast}}}{{\to}}0$. It
is enough to prove that $|T^{\prime}f_{n}|x\to 0$ for each $x\in E_{+}$. Let
$x\in E_{+}$ and $\varepsilon>0$. Pick $k\in\mathbb{N}$ with
$\|T^{\prime}-T_{k}^{\prime}\|\leq\varepsilon$. By the assumption,
$|T_{k}^{\prime}f_{n}|x\to 0$. So, let $n_{0}\in\mathbb{N}$ be such that
$|T_{k}^{\prime}f_{n}|x\leq\varepsilon$ whenever $n\geq n_{0}$. As $(f_{n})$
is $\text{\rm w}^{\ast}$-null, there exists $M\in\mathbb{R}$ with
$\|f_{n}\|\leq M$ for all $n\in\mathbb{N}$. By the Riesz–Kantorovich formula,
for $n\geq n_{0}$,
$|T^{\prime}f_{n}|x=\sup\\{|(T^{\prime}f_{n})y|:|y|\leq x\\}\leq$
$\sup\\{|((T^{\prime}-T_{k}^{\prime})f_{n})y|:|y|\leq
x\\}+\sup\\{|(T_{k}^{\prime}f_{n})y|:|y|\leq x\\}\leq$
$\sup\\{\|T^{\prime}-T_{k}^{\prime}\|\cdot\|f_{n}\|\cdot\|y\|:|y|\leq
x\\}+|T_{k}^{\prime}f_{n}|x\leq\varepsilon(M\|x\|+1).$
Therefore $|T^{\prime}f_{n}|x\to 0$, and hence $T\in\text{[d]}(E,F)$.
6. vi)
Let $\text{\rm[bi-sP]}(E,F)\ni
T_{k}\stackrel{{\scriptstyle\|\cdot\|}}{{\to}}T\in\text{\rm L}(E,F)$. Let
$(f_{n})$ be w∗-null in $F^{\prime}_{+}$ and let $(x_{n})$ be disjoint w-null
in $E$. We need to show $f_{n}(Tx_{n})\to 0$. Pick $M\in{\mathbb{R}}$ such
that $\|f_{n}\|\leq M$ and $\|x_{n}\|\leq M$ for all $n\in{\mathbb{N}}$. Take
some $\varepsilon>0$. Pick $k\in{\mathbb{N}}$ with
$\|T-T_{k}\|\leq\varepsilon$. Since $T_{k}\in\text{\rm[bi-sP]}(E,F)$, there
exists $n_{0}\in{\mathbb{N}}$ such that $|f_{n}(T_{k}x_{n})|\leq\varepsilon$
for $n\geq n_{0}$. Then
$|f_{n}(Tx_{n})|\leq|f_{n}((T-T_{k})x_{n})|+|f_{n}(T_{k}x_{n})|\leq$
$\|f_{n}\|\cdot\|T-T_{k}\|\cdot\|x_{n}\|+\varepsilon\leq(M^{2}+1)\varepsilon\
\ \ (\forall n\geq n_{0}).$
Since $\varepsilon>0$ is arbitrary, $f_{n}(Tx_{n})\to 0$.
7. vii)
As the case of $\text{\rm[GPP]}(X,Y)$ is similar, we consider
$\text{\rm[s-GPP]}(E,Y)$ only.
Let $\text{\rm[s-GPP]}(E,Y)\ni
T_{k}\stackrel{{\scriptstyle\|\cdot\|}}{{\to}}T\in\text{\rm L}(E,Y)$, and let
$A\subseteq E$ be a-limited. We need to show that $T(A)$ is relatively
compact. Since a-limited sets are bounded, there exists $M\in{\mathbb{R}}$
with $\|x\|\leq M$ for all $x\in A$. Choose $\varepsilon>0$ and pick a
$k\in{\mathbb{N}}$ such that $\|T-T_{k}\|\leq\varepsilon$. Then
$Tx=T_{k}x+(T-T_{k})x\in
T_{k}(A)+\|T-T_{k}\|\cdot\|x\|B_{Y}=T_{k}(A)+\varepsilon M\cdot B_{Y}$
for all $x\in A$, and hence $T(A)\subseteq T_{k}(A)+\varepsilon M\cdot B_{Y}$.
By the assumption, $T_{k}(A)$ is relatively compact. Since $\varepsilon>0$ is
arbitrary, $T(A)$ is totally bounded and hence is relatively compact, as
desired.
8. viii)
As the case of $\text{\rm[BDP]}(X,Y)$ is similar, we consider
$\text{\rm[s-BDP]}(E,Y)$ only.
Let $\text{\rm[s-BDP]}(E,Y)\ni
T_{k}\stackrel{{\scriptstyle\|\cdot\|}}{{\to}}T\in\text{\rm L}(E,Y)$, and let
$A\subseteq E$ be a-limited. We need to show that $T(A)$ is relatively
w-compact. Since a-limited sets are bounded, there exists $M\in{\mathbb{R}}$
such that $\|x\|\leq M$ for all $x\in A$. Take $\varepsilon>0$ and pick any
$k\in{\mathbb{N}}$ with $\|T-T_{k}\|\leq\varepsilon$. Then $T(A)\subseteq
T_{k}(A)+\varepsilon M\cdot B_{Y}$, as above in vii). By the assumption,
$T_{k}(A)$ is relatively w-compact. Since $\varepsilon>0$ is arbitrary, $T(A)$
is relatively w-compact by the Grothendieck result [1, Thm.3.44].
∎
The next result follows from Theorem 1.4.3 and Lemma 2.3.1.
###### Theorem 2.3.2.
Let $E$ ad $F$ be a Banach lattices. Then $\text{\rm r-[PSP]}(E,F)$,
$\text{\rm r-[DPSP]}(E,F)$, $\text{\rm r-[DDSP]}(E,F)$, $\text{\rm
r-[PGP]}(E,F)$, $\text{\rm r-[DGP]}(E,F)$, $\text{\rm r-[swl]}(E,F)$,
$\text{\rm r-[dswl]}(E,F)$, $\text{\rm r-[sw}^{\ast}\text{\rm l]}(E,F)$,
$\text{\rm r-[d]}(E,F)$, $\text{\rm r-[bi-sP]}(E,F)$, $\text{\rm
r-[GPP]}(E,F)$,
$\text{\rm r-[s-GPP]}(E,F)$, $\text{\rm r-[BDP]}(E,F)$, and $\text{\rm
r-[s-BDP]}(E,F)$ are all Banach spaces, each under its own enveloping norm.
## 3 Domination for affiliated operators
Here we gather domination results for defined above affiliated operators. Some
of them already appeared in the literature, the others seem new.
### 3.1
The [s-GPP]-operators do not satisfy the domination property in the strong
sense that even an operator which is dominated by a rank one operator need not
to be a [GPP]-operator.
###### Example 3.1.1.
(cf. [1, Ex.5.30]) Define operators $T,S:L^{1}[0,1]\to\ell^{\infty}$ by
$T(f):=(\int_{0}^{1}f(t)dt)_{k=1}^{\infty}$, and
$S(f):=(\int_{0}^{1}f(t)r_{k}^{+}(t)dt)_{k=1}^{\infty}$, where $r_{k}$ are the
Rademacher functions on $[0,1]$. Then $T$ is a rank one operator, and hence
$T\in\text{\rm[s-GPP]}(L^{1}[0,1],\ell^{\infty})$. Moreover, $0\leq S\leq T$,
yet $S$ is not a [GPP]-operator. To see this, consider the sequence of the
Rademacher functions $(r_{n})$ in $[0,\mathbb{1}]\subseteq L^{1}[0,1]$, which
is an a-limited subset of $L^{1}[0,1]$, e.g. by Proposition 3.2.1. The
sequence $(Sr_{n})=(\frac{1}{2}e_{n})$, where $e_{n}$ are the n-th unite
vectors in $\ell^{\infty}$, has no norm convergent subsequences, and hence
$S\not\in\text{\rm[GPP]}(L^{1}[0,1],\ell^{\infty})$.
We do not know whether or not the operator $S$ in Example 3.1.1 is a
[BDP]-operator.
### 3.2
It turn out that the property (d) and the sequential w-continuity ($\text{\rm
w}^{\ast}$-continuity) of lattice operations play an important role for the
domination property. Firstly, we include some related elementary facts.
###### Proposition 3.2.1.
The following are equivalent.
1. i)
$E\in\text{\rm(d)}$.
2. ii)
Each order interval in $E$ is a-limited.
###### Proof.
i)$\Longrightarrow$ii) It suffices to show that intervals $[-a,a]$ are
a-limited. Let $a\in E_{+}$, and let $(f_{n})$ be disjoint w∗-null in
$E^{\prime}$. We need to show that $(f_{n})$ is uniformly null on $[-a,a]$. By
Assertion 1.1.2, it is enough to show that $f_{n}(a_{n})\to 0$ for each
sequence $(a_{n})$ in $[-a,a]$. So, let $(a_{n})$ be in $[-a,a]$. Since
$E\in\text{\rm(d)}$ then $(|f_{n}|)$ is w∗-null in $E^{\prime}_{+}$, and hence
$f_{n}(a)\to 0$. It follows from $-f_{n}(a)\leq f_{n}(a_{n})\leq f_{n}(a)$ for
all $n\in\mathbb{N}$ that $f_{n}(a_{n})\to 0$. By Assertion 1.1.2, $(f_{n})$
is uniformly null on $[-a,a]$, as desired.
ii)$\Longrightarrow$i) Let $(f_{n})$ be disjoint w∗-null in $E^{\prime}$. We
need to show that $(|f_{n}|)$ is w∗-null. Pick an $a\in E_{+}$. By the
assumption, $(f_{n})$ is uniformly null on $[-a,a]$, and in view of the
Riesz–Kantorovich formula, $|f_{n}|a=\sup\limits_{y\in[-a,a]}|f_{n}(y)|\to 0$.
Since $a\in E_{+}$ is arbitrary, $(|f_{n}|)$ is w∗-null, as desired. ∎
The proof of the following result of [27] consists in removing the
disjointness condition in the proof of Proposition 3.2.1.
###### Assertion 3.2.2.
The following are equivalent.
1. (i)
$E^{\prime}$ has sequentially $\text{\rm w}^{\ast}$-continuous lattice
operations.
2. (ii)
Each order interval in $E$ is limited.
Here we gather several (partially positive) domination results.
###### Theorem 3.2.3.
Let $E$ and $F$ be Banach lattices. The following spaces of operators satisfy
the domination property.
1. i)
$\text{\rm[PSP]}(E,F)$.
2. ii)
$\text{\rm[DPSP]}(E,F)$.
3. iii)
$\text{\rm[DDSP]}(E,F)$, under the assumption $F\in\text{\rm(d)}$.
4. iv)
$\text{\rm[PGP]}(E,F)$.
5. v)
$\text{\rm[DGP]}(E,F)$, under the assumption $F\in\text{\rm(d)}$.
6. vi)
$\text{\rm[dswl]}(E,F)$.
7. vii)
$\text{\rm[swl]}(E,F)$, under the assumption that $E$ has sequentially
w-continuous lattice operations.
8. viii)
$\text{\rm[sw}^{\ast}\text{\rm l]}(E,F)$, under the assumption that
$F^{\prime}$ has sequentially $\text{\rm w}^{\ast}$-continuous lattice
operations.
9. ix)
$\text{\rm[d]}(E,F)$, under the assumption $F\in\text{\rm(d)}$.
10. x)
$\text{\rm[bi-sP]}(E,F)$.
###### Proof.
As above, we restrict ourselves to basic cases.
1. i)
Let $0\leq S\leq T\in\text{\rm[PSP]}(E,F)$ and let $(x_{n})$ be w-null in
$E_{+}$. Since $T\in\text{\rm[PSP]}(E,F)$ then $\|Tx_{n}\|\to 0$. It follows
from $0\leq Sx_{n}\leq Tx_{n}$ that $\|Sx_{n}\|\to 0$, and hence
$S\in\text{\rm[PSP]}(E,F)$.
2. ii)
Let $0\leq S\leq T\in\text{\rm[DPSP]}(E,F)$ and let $(f_{n})$ be w∗-null in
$F^{\prime}_{+}$. Since $T\in\text{\rm[DPSP]}(E,F)$ then
$\|T^{\prime}f_{n}\|\to 0$. It follows from $0\leq S^{\prime}\leq T^{\prime}$
that $0\leq S^{\prime}f_{n}\leq T^{\prime}f_{n}$, and hence
$\|S^{\prime}f_{n}\|\to 0$. Thus, $S\in\text{\rm[DPSP]}(E,F)$.
3. iii)
As [DDSP]-operators agree with almost limited operators, we refer for the
proof to [17, Cor.3].
4. iv)
Let $0\leq S\leq T\in\text{\rm[PGP]}(E,F)$, and $(f_{n})$ be w∗-null in
$F_{+}^{\prime}$. In order to prove $S\in\text{\rm[PGP]}(E,F)$, it suffices to
prove $g(S^{\prime}f_{n})\to 0$ for all $g\in E^{\prime}_{+}$. Let $g\in
E^{\prime}_{+}$. Since $T\in\text{\rm[PGP]}(E,F)$, $g(T^{\prime}f_{n})\to 0$.
It follows from $0\leq g(S^{\prime}f_{n})\leq g(T^{\prime}f_{n})$ that
$g(S^{\prime}f_{n})\to 0$, as desired.
5. v)
As [DGP]-operators agree with almost Grothendieck operators, we refer for the
proof to [23, Prop.3.7].
6. vi)
Let $0\leq S\leq T\in\text{\rm[dswl]}(E,F)$, and let $(x_{n})$ be disjoint
w-null in $E$. In order to prove $S\in\text{\rm[dswl]}(E,F)$, it suffices to
prove $f(|Sx_{n}|)\to 0$ for all $f\in F^{\prime}_{+}$. So, let $f\in
F^{\prime}_{+}$. By Assertion 2.1.2, $(|x_{n}|)$ is w-null. Since
$T\in\text{\rm[dswl]}(E,F)$ then $(T|x_{n}|)=(|T(|x_{n}|)|)$ is w-null, and
hence $f(T|x_{n}|)\to 0$. It follows from $|Sx_{n}|\leq S|x_{n}|\leq T|x_{n}|$
that $f(|Sx_{n}|)\to 0$ as desired.
7. vii)
Let $0\leq S\leq T\in\text{\rm[swl]}(E,F)$, and $(x_{n})$ be w-null in $E$. It
suffices to prove $f(|Sx_{n}|)\to 0$ for all $f\in F^{\prime}_{+}$. Let $f\in
F^{\prime}_{+}$. By the assumption, $(|x_{n}|)$ is w-null. Since
$T\in\text{\rm[swl]}(E,F)$, $f(T|x_{n}|)=f(|Tx_{n}|)\to 0$. In view of
$|Sx_{n}|\leq S|x_{n}|\leq T|x_{n}|$, $f(|Sx_{n}|)\to 0$, and hence
$S\in\text{\rm[swl]}(E,F)$.
8. viii)
It follows from Proposition 2.2.5 ii).
9. ix)
It follows from Proposition 2.2.5 i).
10. x)
Let $0\leq S\leq T\in\text{\rm[bi-sP]}(E,F)$. Let $(f_{n})$ be w∗-null in
$F^{\prime}_{+}$, and let $(x_{n})$ be disjoint w-null in $E$. In order to
prove $S\in\text{\rm[bi-sP]}(E,F)$, it suffices to prove $f_{n}(Sx_{n})\to 0$.
By Assertion 2.1.2, $(|x_{n}|)$ is disjoint w-null in $E$, and, since
$T\in\text{\rm[bi-sP]}(E,F)$, then $f_{n}(T|x_{n}|)\to 0$. It follows from
$|f_{n}(Sx_{n})|\leq f_{n}(S|x_{n}|)\leq f_{n}(T|x_{n}|)$ that
$f_{n}(Sx_{n})\to 0$, and hence $S\in\text{\rm[bi-sP]}(E,F)$.
∎
In view of [1, Thm.4.31], the next fact follows form Theorem 3.2.3 vii).
###### Corollary 3.2.4.
Let $E$ be an AM-space, and let $0\leq S\leq T\in\text{\rm[swl]}(E,F)$. Then
$S\in\text{\rm[swl]}(E,F)$.
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|
# Transport Properties in Gapped Bilayer Graphene
N. Benlakhouy<EMAIL_ADDRESS>Laboratory of Theoretical Physics,
Faculty of Sciences, Chouaïb Doukkali University, PO Box 20, 24000 El Jadida,
Morocco A. El Mouhafid<EMAIL_ADDRESS>Laboratory of Theoretical
Physics, Faculty of Sciences, Chouaïb Doukkali University, PO Box 20, 24000 El
Jadida, Morocco A. Jellal<EMAIL_ADDRESS>Laboratory of Theoretical
Physics, Faculty of Sciences, Chouaïb Doukkali University, PO Box 20, 24000 El
Jadida, Morocco Canadian Quantum Research Center, 204-3002 32 Ave Vernon,
BC V1T 2L7, Canada
###### Abstract
We investigate transport properties through a rectangular potential barrier in
AB-stacked bilayer graphene (AB-BLG) gapped by dielectric layers. Using the
Dirac-like Hamiltonian with a transfer matrix approach we obtain transmission
and reflection probabilities as well as the associated conductance. For two-
band model and at normal incidence, we find extra resonances appearing in
transmission compared to biased AB-BLG, which are Fabry-Pérot resonance type.
Now by taking into account the inter-layer bias, we show that both of
transmission and anti-Klein tunneling are diminished. Regarding four band
model, we find that the gap suppresses transmission in an energy range by
showing some behaviors look like ”Mexican hats”. We examine the total
conductance and show that it is affected by the gap compared to AA-stacked
bilayer graphene. In addition, we find that the suppression in conductance is
more important than that for biased AB-BLG.
###### pacs:
73.22.Pr, 72.80.Vp, 73.63.-b
## I Introduction
The experimental realization of monolayer graphene (MLG) in 2004 by Novoselov
and Geim [1] opened up a new field in physics. Such material has attractive
electronic, optical, thermal, and mechanical properties. In particular, the
observation of Klein tunneling [3, 2], anomalous quantum Hall effect [1, 4],
and optical transparency [5]. This makes graphene a good platform for
nanoscale adaptor applications [6]. Bilayer graphene (BLG) is a system formed
by two stacked sheets of graphene. Besides that, there are two distinct kinds
of stacking: AB-BLG or AB-(Bernal) [7], and AA-BLG. AB-BLG has a parabolic
dispersion relation with four bands where two of them touch at zero energy,
whereas the other two bands split together by the interlayer hopping parameter
$\gamma_{1}\approx 0.4$ eV [8]. This structure is much more stable and its
high-quality samples are developed and studied theoretically and
experimentally [9, 10, 11, 12, 13]. AA-BLG has a linear energy gapless
spectrum with two Dirac cones switched in energy by the quantity
$\gamma_{1}\approx 0.2$ eV [14], and because of this AA-BLG attained enormous
theoretical interest [15, 16, 17, 18, 19, 20]. Such a structure is expected to
be metastable, just lately, stable samples were discovered [21, 22, 23, 24].
The AB-BLG may have clearly defined benefits than MLG, due to greater
possibilities for balancing their physical properties. For reference: quantum
Hall effect [9, 25], spin-orbit coupling and transverse electric field [26],
transmission probability in presence of electric and magnetic static fields
[27, 13], and quantum dots [28].
Experimentally, the evidence of Klein tunneling in MLG was confirmed [29, 3,
30, 31], which means that there is no electron confinement, and then a gap
must be created to overcome this issue. In fact, many methods of induction a
band gap in MLG have been elaborated such as substrates [32, 33, 34, 35, 36,
37, 38, 39] and doping with impurities [40, 41]. Regarding AB-BLG, band gap
can be realized by applying an external electric field [42, 9] or induced by
using dielectric materials like hexagonal boron nitride (h-BN) or SiC [44]. To
this end, it is theoretically showed that quantum spin Hall phase can be
identified in gapped AB-BLG even when the Rashba interaction approached zero
[44].
The introduction of an inter-layer bias to AB-BLG opens a gap in the energy
spectrum and has a major effect on electronic properties [29]. Here, we
analyze the effects of a biased AB-BLG gapped by dielectric layers to show the
impact of band gap on transport properties. In both layers of AB-BLG, band gap
is the same allowing to open a gap. Using transfer matrix method together with
current density, we calculate transmission and reflection probabilities as
well as corresponding conductance. At low-energy, $E<\gamma_{1}$, and in
presence of the band gap $\Delta_{0}$ we find that Fabry-Pérot resonances [48]
strongly appear in the transmission. Now by including also the inter-layer
bias $\delta$, we show that the total transmission and anti-Klein tunneling
significantly diminished. For energies exceeding the inter-layer coupling
$\gamma_{1}$, $E>\gamma_{1}$, we obtain a new mode of propagating giving rise
to the four transmission channels. In this case, $\Delta_{0}$ suppresses the
transmission in the energy range
$V_{0}-(\Delta_{0}+\delta)<E<V_{0}+(\Delta_{0}+\delta)$, and shows some
behaviors that look like “Mexican hats”. Finally we find that the resulting
conductance in gapped AB-BLG gets modified compared to gapped AA-BLG.
Moreover, we find that the suppression in conductance is more important than
that for biased AB-BLG [29] because the energy range for a null conductance
increases as long as $\Delta_{0}$ increase and also the number of peaks get
reduced.
The paper is organized as follows. In Sec II we construct our theoretical
model describing biased and gapped AB-BLG giving rise to four band energies.
In Sec III we explain in detail the formalism used in calculating transmission
and reflection probabilities together with conductance. In Sec IV we
numerically analyze our results and give different discussions with published
works on the topic. Finally, in Sec. V we summarize our main conclusions.
## II Theoretical model
Figure 1: The parameters of a rectangular barrier structure.
In the AB-stacked bilayer graphene the atom $B_{1}$ of the top layer is placed
directly below the atom $A_{2}$ of the bottom layer with van der Waals inter-
layer coupling parameter $\gamma_{1}$, while $A_{1}$ and $B_{2}$ do not lie
directly below or above each other. Based on [29, 44] we consider a biased and
gapped AB-BLG described by the following Hamiltonian near the point K
$\mathcal{H}=\begin{pmatrix}V_{0}+\vartheta_{1}&v_{F}\pi^{{\dagger}}&-v_{4}\pi^{{\dagger}}&v_{3}\pi\\\
v_{F}\pi&V_{0}+\vartheta_{2}&\gamma_{1}&-v_{4}\pi^{{\dagger}}\\\
-v_{4}\pi&\gamma_{1}&V_{0}-\vartheta_{2}&v_{F}\pi^{{\dagger}}\\\
v_{3}\pi^{{\dagger}}&-v_{4}\pi&v_{F}\pi&V_{0}-\vartheta_{1}\\\ \end{pmatrix}$
(1)
where $v_{F}=\frac{\gamma_{0}}{\hbar}\frac{3a}{2}\approx 10^{6}$ m/s is the
Fermi velocity of electrons in each graphene layer, $a=0.142$ nm is the
distance between adjacent carbon atoms,
$v_{3,4}=\frac{v_{F}\gamma_{3,4}}{\gamma_{0}}$ represent the coupling between
the layers, $\pi=p_{x}+ip_{y},\pi^{{\dagger}}=p_{x}-ip_{y}$ are the in-plan
momenta and its conjugate with $p_{x,y}=-i\hbar\partial_{x,y}$,
$\gamma_{1}\approx 0.4$ eV is the interlayer coupling term. The electrostatic
potential $V_{0}$ of width $d$ (Fig. 1) can be varied on the $i$-th layer
using top and back gates on the sample. $\vartheta_{1}=\delta+\Delta_{0}$,
$\vartheta_{2}=\delta-\Delta_{0}$ with $\delta$ corresponds to an externally
induced inter-layer potential difference, and $\Delta_{0}$ is the band gap.
The skew parameters, $\gamma_{3}\approx 0.315$ eV and $\gamma_{4}\approx
0.044$ eV have negligible effect on the band structure at high energy [25,
45]. Recently, it was shown that even at low energy these parameters have also
negligible effect on the transmission [29], hence we neglect them in our
calculations.
Under the above approximation and for a barrier potential configuration as
depicted in Fig. 1, the Hamiltonian (1) can be written as
$H=\left(\begin{array}[]{cccc}V_{0}+\vartheta_{1}&\nu_{F}\pi^{{\dagger}}&0&0\\\
\nu_{F}\pi&V_{0}+\vartheta_{2}&\gamma_{1}&0\\\
0&\gamma_{1}&V_{0}-\vartheta_{2}&\nu_{F}\pi^{{\dagger}}\\\
0&0&\nu_{F}\pi&V_{0}-\vartheta_{1}\\\ \end{array}\right)$ (2)
By considering the length scale $l=\hbar v_{F}/\gamma_{1}$, which represents
the inter-layer coupling length $l=1.64$ nm, we define the dimensionless
quantities: $x\equiv x/l$ and $k_{y}\equiv lk_{y}$ together with
$\delta\equiv\frac{\delta}{\gamma_{1}}$,
$\Delta_{0}\equiv\frac{\Delta_{0}}{\gamma_{1}}$,
$E\equiv\frac{E}{\gamma_{1}}$, $V_{0}\equiv\frac{V_{0}}{\gamma_{1}}$. The
eigenstates of Eq. (2) are four-components spinors
$\psi(x,y)=[{\psi}_{A_{1}},{\psi}_{B_{1}},{\psi}_{A_{2}},{\psi}_{B_{2}}]^{{\dagger}}$,
here ${\dagger}$ denotes the transpose of the row vector. As a consequence of
the transnational invariance along the $y$-direction, we have $[H,p_{y}]=0$,
and then we decompose the spinor as
$\psi(x,y)=e^{ik_{y}y}\left[\phi_{A_{1}}(x),\phi_{B_{1}}(x),\phi_{A_{2}}(x),\phi_{B_{2}}(x)\right]^{T}$
(3)
We solve the time-independent Schrödinger equation $H\psi=E\psi$ to obtain a
general solution in the region II and then require $V_{0}=\delta=\Delta_{0}=0$
to derive the solutions in the regions I and III. Indeed, by substituting Eq.
(2) and Eq. (3) we get four related differential equations
$\displaystyle-i(\partial_{x}+k_{y})\phi_{B_{1}}$ $\displaystyle=$
$\displaystyle\varepsilon_{1}\phi_{A_{1}}$ (4a)
$\displaystyle-i(\partial_{x}-k_{y})\phi_{A_{1}}$ $\displaystyle=$
$\displaystyle\varepsilon_{2}\phi_{B_{1}}-\phi_{A_{2}}$ (4b)
$\displaystyle-i(\partial_{x}+k_{y})\phi_{B_{2}}$ $\displaystyle=$
$\displaystyle\varepsilon_{3}\phi_{A_{2}}-\phi_{B_{1}}$ (4c)
$\displaystyle-i(\partial_{x}-k_{y})\phi_{A_{2}}$ $\displaystyle=$
$\displaystyle\varepsilon_{4}\phi_{B_{2}}$ (4d)
where we have set $\varepsilon_{1}=\varepsilon-\vartheta_{1}$,
$\varepsilon_{2}=\varepsilon-\vartheta_{2}$,
$\varepsilon_{3}=\varepsilon+\vartheta_{2}$,
$\varepsilon_{4}=\varepsilon+\vartheta_{1}$ and $\varepsilon=E-V_{0}$. We
solve Eq. (4a) for $\phi_{A_{1}}$, Eq. (4d) for $\phi_{B_{2}}$ and substitute
the results in Eqs. (4b,4c). This process yields
$\displaystyle(\partial_{x}^{2}-k_{y}^{2}+\varepsilon_{1}\varepsilon_{2})\phi_{B_{1}}$
$\displaystyle=$ $\displaystyle\varepsilon_{1}\phi_{A_{2}}$ (5a)
$\displaystyle(\partial_{x}^{2}-k_{y}^{2}+\varepsilon_{3}\varepsilon_{4})\phi_{A_{2}}$
$\displaystyle=$ $\displaystyle\varepsilon_{4}\phi_{B_{1}}$ (5b)
Then for constant parameters, the energy bands are solution of the following
equation
$\left[-k^{2}+\varepsilon_{1}\varepsilon_{2}\right]\left[-k^{2}+\varepsilon_{3}\varepsilon_{4}\right]-\varepsilon_{1}\varepsilon_{4}=0$
(6)
such that $k=\sqrt{k_{x}^{2}+k_{y}^{2}}$ and the four possible wave vectors
are given by
$k^{s}_{x}=\sqrt{-k_{y}^{2}+\varepsilon^{2}+\delta^{2}-\Delta_{0}^{2}\pm\sqrt{\varepsilon^{2}(1+4\delta^{2})-(\delta+\Delta_{0})^{2}}}$
(7)
where $s=\pm$ defines the modes of propagation, which will be discussed in
numerical section. Therefore, the four energy bands can be derived as
$\displaystyle\varepsilon^{s}_{\pm}=s\sqrt{k^{2}+\delta^{2}+\Delta_{0}^{2}+\frac{1}{2}\pm\sqrt{k^{2}\left(1+4\delta^{2}\right)+\left(\frac{1}{2}-2\delta\Delta_{0}\right)^{2}}}$
(8)
At this level, we have some comments in order. Indeed, firstly by taking
$\delta=0$, (8) reduces
$\varepsilon^{s}_{\pm}|_{\delta=0}=s\sqrt{k^{2}+\Delta_{0}^{2}+\frac{1}{2}\pm\sqrt{k^{2}+\frac{1}{4}}}$
(9)
Secondly for the case $\Delta_{0}=0$, we end up with Ben et al. result [29]
$\displaystyle\varepsilon^{s}_{\pm}|_{\Delta_{0}=0}=s\sqrt{k^{2}+\delta^{2}+\frac{1}{2}\pm\sqrt{k^{2}\left(1+4\delta^{2}\right)+\frac{1}{4}}}$
(10)
Now by comparing (9) and (10), we clearly notice that both quantities $\delta$
and $\Delta_{0}$ are inducing different gaps in the energy spectrum. Certainly
this difference will affect the transmission probabilities (Figs. 3, 4) as
well as conductance (Fig. 7).
Figure 2: Energy spectrum of AB-stacked graphene bilayer inside (solid curves)
and outside (dashed curves) the barrier. Here blue (brown) curves correspond
to $k^{+}(k^{-})$ propagating modes for biased and gapped $(V_{0}\neq
0,\delta\neq 0,\Delta_{0}\neq 0)$ systems. $\delta^{\prime}=\delta+\Delta_{0}$
and $\gamma_{1}^{\prime}=\sqrt{\gamma_{1}^{2}+(\delta-\Delta_{0})^{2}}$.
It is known that the perfect AB-BLG has a parabolic dispersion relation with
four bands, of which two touch each other at $k=0$. In Fig. 2 we show the
energy bands as a function of the momentum $k_{y}$, for the biased and gapped
AB-BLG. We observe that when the AB-BLG is subjected to a gap $\Delta_{0}$ and
an inter-layer bias $\delta$ the two bands are switched and placed at
$V_{0}\pm\sqrt{\gamma_{1}^{2}+(\delta-\Delta_{0})^{2}}$, and the touching
bands are shifted by $2\delta^{\prime}=2(\delta+\Delta_{0})$. One should
notice that there are two cases related to whether the wave vector
$k^{s}_{0}=\sqrt{-k_{y}^{2}+\varepsilon^{2}\pm\varepsilon}$ is real or
imaginary. Indeed for $E<\gamma_{1}$, just $k^{+}_{0}$ is real, and for that
reason, the propagation is only possible for $k^{+}_{0}$ mode. However when
$E>\gamma_{1}$, both $k^{\pm}_{0}$ are real which presenting a new propagation
mode.
As concerning the eigenspinors in regions II, we show that the solution of
Eqs. (5) is a plane wave generated by
$\phi_{B_{1}}^{2}=a_{1}e^{ik_{x}^{+}x}+a_{2}e^{-ik_{x}^{+}x}+a_{3}e^{ik_{x}^{-}x}+a_{4}e^{-ik_{x}^{-}x}$
(11)
where $a_{n}$ are coefficients of normalization, with $n=1,\cdots,4$. The
remaining components of the eigenspinors can be obtained as
$\displaystyle\phi_{A_{1}}^{2}$
$\displaystyle=a_{1}\Lambda^{+}_{+}e^{ik_{x}^{+}x}+a_{2}\Lambda^{+}_{-}e^{-ik_{x}^{+}x}+a_{3}\Lambda^{-}_{+}e^{ik_{x}^{-}x}+a_{4}\Lambda^{-}_{-}e^{-ik_{x}^{-}x}$
(12) $\displaystyle\phi_{A_{2}}^{2}$
$\displaystyle=a_{1}\rho^{+}e^{ik_{x}^{+}x}+a_{2}\rho^{+}e^{-ik_{x}^{+}x}+a_{3}\rho^{-}e^{ik_{x}^{-}x}+a_{4}\rho^{-}e^{-ik_{x}^{-}x}$
(13) $\displaystyle\phi_{B_{2}}^{2}$
$\displaystyle=a_{1}\chi^{+}_{+}\rho^{+}e^{ik_{x}^{+}x}+a_{2}\chi^{+}_{-}\rho^{+}e^{-ik_{x}^{+}x}+a_{3}\chi^{-}_{+}\rho^{-}e^{ik_{x}^{-}x}+a_{4}\chi^{-}_{-}\rho^{-}e^{-ik_{x}^{-}x}$
(14)
where we have introduced the quantities $\Lambda^{\pm}_{\pm}=\frac{-ik_{y}\pm
k_{x}^{\pm}}{\varepsilon-\vartheta_{1}}$,
$\rho^{\pm}=\frac{(\epsilon-\vartheta_{1})(\epsilon-\vartheta_{2})-k_{y}^{2}-(k_{x}^{\pm})^{2}}{\epsilon-\vartheta_{1}}$,
$\chi^{\pm}_{\pm}=\frac{ik_{y}\pm k_{x}^{\pm}}{\varepsilon+\vartheta_{1}}$. In
matrix notation, the general solution of our system in region II can be
written as
$\psi_{2}(x,y)=\mathcal{G}_{2}\cdot\mathcal{M}_{2}(x)\cdot\mathcal{C}_{2}\
e^{ik_{y}y}$ (15)
where the four-component vector ${\cal{C}}_{2}$ represents the coefficients
$a_{n}$ expressing the relative weights of the different traveling modes,
which have to be set according to the propagating region [29]. The matrices
$\mathcal{M}_{2}(x)$ and $\mathcal{G}_{2}$ are given by
$\mathcal{G}_{2}=\begin{pmatrix}1&1&1&1\\\
\Lambda^{+}_{-}&\Lambda^{+}_{+}&\Lambda^{-}_{+}&\Lambda^{-}_{-}\\\
\rho^{+}&\rho^{+}&\rho^{-}&\rho^{-}\\\
\chi^{+}_{+}\rho^{+}&\chi^{+}_{-}\rho^{+}&\chi^{-}_{+}\rho^{-}&\chi^{-}_{-}\rho^{-}\\\
\end{pmatrix},\qquad\mathcal{M}_{2}(x)=\begin{pmatrix}e^{ik_{x}^{+}x}&0&0&0\\\
0&e^{-ik_{x}^{+}x}&0&0\\\ 0&0&e^{ik_{x}^{-}x}&0\\\ 0&0&0&e^{-ik_{x}^{-}x}\\\
\end{pmatrix},\qquad\mathcal{C}_{2}=\begin{pmatrix}a_{1}\\\ a_{2}\\\ a_{3}\\\
a_{4}\\\ \end{pmatrix}$ (16)
As claimed before, to get solutions in the other regions we have to set
$V_{0}=\delta=\Delta_{0}=0$. Then the eigenspinors in region I is
$\displaystyle\phi_{A_{1}}^{1}$
$\displaystyle=\delta_{s,1}e^{ik^{+}_{0}x}+r^{s}_{+}e^{-ik^{+}_{0}x}+\delta_{s,-1}e^{ik^{-}_{0}x}+r^{s}_{-}e^{-ik^{-}_{0}x}$
(17) $\displaystyle\phi_{B_{1}}^{1}$
$\displaystyle=\delta_{s,1}\Lambda^{+}_{-}e^{ik^{+}_{0}x}+r^{s}_{+}\Lambda^{+}_{+}e^{-ik^{+}_{0}x}+\delta_{s,-1}\Lambda^{-}_{+}e^{ik^{-}_{0}x}+r^{s}_{-}\Lambda^{-}_{-}e^{-ik^{-}_{0}x}$
(18) $\displaystyle\phi_{A_{2}}^{1}$
$\displaystyle=\delta_{s,1}\rho^{+}e^{ik^{+}_{0}x}+r^{s}_{+}\rho^{+}e^{-ik^{+}_{0}x}+\delta_{s,-1}\rho^{-}e^{ik^{-}_{0}x}+r^{s}_{-}\rho^{-}e^{-ik^{-}_{0}x}$
(19) $\displaystyle\phi_{B_{2}}^{1}$
$\displaystyle=\delta_{s,1}\chi^{+}_{+}\rho^{+}e^{ik^{+}_{0}x}+r^{s}_{+}\rho^{+}\chi^{+}_{-}e^{-ik^{+}_{0}x}+\delta_{s,-1}\rho^{-}\chi^{-}_{+}e^{ik^{-}_{0}x}+r^{s}_{-}\rho^{-}\chi^{-}_{-}e^{-ik^{-}_{0}x}$
(20)
and in the region III reads as
$\displaystyle\phi_{A_{1}}^{3}$
$\displaystyle=t^{s}_{+}e^{ik^{+}_{0}x}+t^{s}_{-}e^{ik^{-}_{0}x}$ (21)
$\displaystyle\phi_{B_{1}}^{3}$
$\displaystyle=t^{s}_{+}\Lambda^{+}_{-}e^{ik^{+}_{0}x}+t^{s}_{-}\Lambda^{-}_{+}e^{ik^{-}_{0}x}$
(22) $\displaystyle\phi_{A_{2}}^{3}$
$\displaystyle=t^{s}_{+}\rho^{+}e^{ik^{+}_{0}x}+t^{s}_{-}\rho^{-}e^{ik^{-}_{0}x}$
(23) $\displaystyle\phi_{B_{2}}^{3}$
$\displaystyle=t^{s}_{+}\chi^{+}_{+}\rho^{+}e^{ik^{+}_{0}x}+t^{s}_{-}\chi^{-}_{+}\rho_{-}e^{ik^{-}_{0}x}$
(24)
Since the potential is zero in regions I and III, we have the relation
$\mathcal{G}_{1}\cdot\mathcal{M}_{1}(x)=\mathcal{G}_{3}\cdot\mathcal{M}_{3}(x)$.
We will see how the above results will be used to determine different physical
quantities. Specifically, we focus on the transmission and reflection
probabilities as well as the conductance.
## III Transmission probability and conductance
To determine the transmission and reflection probabilities, we impose the
appropriate boundary conditions in the context of the transfer matrix approach
[46, 47]. Continuity of the spinors at interfaces gives the components of the
vectors
$\mathcal{C}_{1}^{s}=\begin{pmatrix}\delta_{s,1}\\\ r_{+}^{s}\\\
\delta_{s,-1}\\\ r_{-}^{s}\\\
\end{pmatrix},\qquad\mathcal{C}_{3}^{s}=\begin{pmatrix}t_{+}^{s}\\\ 0\\\
t_{-}^{s}\\\ 0\\\ \end{pmatrix}$ (25)
where $\delta_{s,\pm}$ is the Kronecker symbol. The continuity at $x=0$ and
$x=d$ can be written in a matrix notation as
$\displaystyle\mathcal{G}_{1}\cdot\mathcal{M}_{1}(0)\cdot\mathcal{C}_{1}^{s}=\mathcal{G}_{2}\cdot\mathcal{M}_{2}(0)\cdot\mathcal{C}_{2}$
(26)
$\displaystyle\mathcal{G}_{2}\cdot\mathcal{M}_{2}(d)\cdot\mathcal{C}_{2}=\mathcal{G}_{3}\cdot\mathcal{M}_{3}(d)\cdot\mathcal{C}_{3}^{s}$
(27)
Using the transfer matrix method together with the relation
$\mathcal{G}_{1}\cdot\mathcal{M}_{1}(x)=\mathcal{G}_{3}\cdot\mathcal{M}_{3}(x)$
we can connect $\mathcal{C}_{1}^{s}$ with $\mathcal{C}_{3}^{s}$ through the
matrix $\mathcal{N}$
$\mathcal{C}_{1}^{s}=\mathcal{N}\cdot\mathcal{C}_{3}^{s}$ (28)
where
$\mathcal{N}=\mathcal{G}_{1}^{-1}\cdot\mathcal{G}_{2}\cdot\mathcal{M}_{2}^{-1}(d)\cdot\mathcal{G}_{2}^{-1}\cdot\mathcal{G}_{1}\cdot\mathcal{M}_{1}(d)$
(29)
Consequently, the transmission and reflection coefficients can be derived from
$\left(\begin{array}[]{cccc}t_{+}^{s}\\\ r_{+}^{s}\\\ t_{-}^{s}\\\
r_{-}^{s}\\\
\end{array}\right)=\left(\begin{array}[]{cccc}\mathcal{N}_{11}&0&\mathcal{N}_{13}&0\\\
\mathcal{N}_{21}&-1&\mathcal{N}_{23}&0\\\
\mathcal{N}_{31}&0&\mathcal{N}_{33}&0\\\
\mathcal{N}_{41}&0&\mathcal{N}_{43}&-1\\\
\end{array}\right)^{-1}\left(\begin{array}[]{cccc}\delta_{s,1}\\\ 0\\\
\delta_{s,-1}\\\ 0\\\ \end{array}\right)$ (30)
where $\mathcal{N}_{ij}$ are the matrix elements of $\mathcal{N}$. Then, after
some algebras, we obtain the transmission and reflection coefficients
$\displaystyle t_{+}^{s}$
$\displaystyle=\frac{\delta_{s,-1}\mathcal{N}_{13}-\delta_{s,1}\mathcal{N}_{33}}{\mathcal{N}_{13}\mathcal{N}_{31}-\mathcal{N}_{11}\mathcal{N}_{33}},\qquad
t_{-}^{s}=\frac{-\delta_{s,-1}\mathcal{N}_{11}+\delta_{s,1}\mathcal{N}_{31}}{\mathcal{N}_{13}\mathcal{N}_{31}-\mathcal{N}_{11}\mathcal{N}_{33}}$
(31) $\displaystyle r_{+}^{s}$
$\displaystyle=\mathcal{N}_{21}t_{+}^{s}+\mathcal{N}_{23}t_{-}^{s},\qquad
r_{-}^{s}=\mathcal{N}_{41}t_{+}^{s}+\mathcal{N}_{43}t_{-}^{s}$ (32)
To calculate the transmission and reflection probabilities, we have to take
into account the change in velocity of the waves when they are scattered into
a different propagation mode. For this, it is convenient to use the current
density $\bm{J}$
$\bm{J}=v_{F}\bm{\psi}^{\dagger}\begin{pmatrix}\sigma_{x}&0\\\ 0&\sigma_{x}\\\
\end{pmatrix}\bm{\psi}$ (33)
where $\sigma_{x}$ is the Pauli matrix. Then Eq. (33) gives the incident
$\bm{J}_{x}^{\text{inc}}$, reflected $\bm{J}_{x}^{\text{ref}}$ and transmitted
$\bm{J}_{x}^{\text{tra}}$ current densities. Finally the transmission $T$ and
reflection $R$ probabilities are
$T^{s}_{\pm}=\frac{k^{\pm}_{0}}{k^{s}_{0}}|t^{s}_{\pm}|^{2},\qquad
R^{s}_{\pm}=\frac{k^{\pm}_{0}}{k^{s}_{0}}|r^{s}_{\pm}|^{2}$ (34)
To preserve the probability of current, $T$ and $R$ are normalized as
$\sum_{i,j}\left(T^{j}_{i}+R^{j}_{i}\right)=1$ (35)
where the index $i=\pm$ points to the arriving mode, when the index $j=\pm$
points to the exiting mode. For example in the case of channel $k^{+}$, gives
$T^{+}_{+}+T^{-}_{+}+R^{+}_{+}+R^{-}_{+}=1$. As already mentioned, for
$E>\gamma_{1}$ we have two modes of propagation ($k^{+}_{0},k^{-}_{0}$)
leading to four transmissions $T^{s}_{\pm}$ and four reflections $R^{s}_{\pm}$
channels, through the four conduction bands. For sufficiently enough low
energy or in the two-band model, $E<\gamma_{1}$, the two modes lead to one
transmission $T$ channel and one reflection $R$ channel.
From the transmission probabilities, we can calculate the conductance $G$, at
zero temperature, using the Landauer-Büttiker formula
$G(E)=G_{0}\frac{L_{y}}{2\pi}\int_{-\infty}^{\infty}dk_{y}\sum_{i,j=\pm}T_{i}^{j}\left(E,k_{y}\right)$
(36)
with $L_{y}$ the length of the sample in the $y$-direction, and
$G_{0}=4e^{2}/h$. The factor $4$ comes from the valley and spin degeneracies
in graphene. In order to get the total conductance of the system, we need to
sum over all the transmission channels
$G_{T}=\sum_{i,j}G^{j}_{i}$ (37)
## IV NUMERICAL RESULTS AND DISCUSSION
In this section, we numerically analyze and discuss our main results. First,
we evaluate the transmission probability in the two-band model at normal
incidence (i.e. $k_{y}=0$). To understand our system more effectively in Fig.
3, we present the effect of the band gap $\Delta_{0}$ on the transmission as a
function of the incident energy $E$ and the width $d$ of the barrier. In the
(left panel), we plot the energy dependence of the transmission probability
for a barrier of width $d=10$ nm, $d=25$ nm, and $d=100$ nm for biased
$\delta=0$ and unbiased system $\delta\neq 0$ with band gap $\Delta_{0}$. For
$\Delta_{0}\neq 0$, we observe appearance of resonances in the transmission
probability for the energy range $E<V_{0}-\delta^{\prime}$,
$\delta^{\prime}=\delta+\Delta_{0}$, which can be attributed to the finite
size of the AB-BLG as well as the presence of charge carriers with different
chirality. These phenomena are known as Fabry-Pérot resonances [48]. For the
energy range $V_{0}-\delta^{\prime}<E<V_{0}+\delta^{\prime}$, there is a bowl
(window) of zero transmission for $d=100$ nm in contrary for $d=10$ nm and
$d=25$ nm the transmission is not zero. However, for
$E>V_{0}+\delta^{\prime}$, the transmission still looks like Ben et al.
results [29]. Note that the transmission of width $d=100$ nm, shows anti-Klien
tunneling, which is a direct consequence of the pseudospin conservation in the
system. In the (right panel), we plot the width dependence of the transmission
probability for the incident energies $E=\frac{1}{5}V_{0}$,
$E=\frac{2}{5}V_{0}$ and $E=\frac{8}{5}V_{0}$. It is clearly seen that for
$E=\frac{1}{5}V_{0}$ and $E=\frac{2}{5}V_{0}$ with $\delta_{0}=0$,
$\Delta_{0}=0.01\gamma_{1}$, resonance peaks show up (see upper panel), which
are absent for the case $\Delta_{0}=0$ [29]. In the middle and bottom panel,
by taking into account the effect of a finite bias $\delta=0.01\gamma_{1}$, we
observe a decrease of resonance in the transmission probability, and more
precisely when $\Delta_{0}$ is greater than $\delta$.
Figure 3: (Color online) The transmission probability at normal incidence
through a barrier of height $V_{0}=0.05\gamma_{1}$ with
$\Delta_{0}=0.01\gamma_{1}$ and $\delta=0$ (for upper panel),
$\Delta_{0}=\delta=0.01\gamma_{1}$ (for middle panel) and
$\Delta_{0}=0.03\gamma_{1}$ and $\delta=0.01\gamma_{1}$ (for bottom panel).
(Left panel): The energy dependence of the transmission probability for
barrier widths $d=10$ nm (blue), $d=25$ nm (red), and $d=100$ nm (green).
(Right panel): The width dependence of the transmission probability for
incident energies $E=\frac{1}{5}V_{0}$ (blue), $E=\frac{2}{5}V_{0}$ (red) and
$E=\frac{8}{5}V_{0}$ (green).
To investigate the effect of band gap, for energy greater than the interlayer
hopping parameter, $E>\gamma_{1}$, in Fig. 4 we show the transmission and
reflection channels as a function of the incident energy $E$ and transverse
wave vector $k_{y}$ for potential height $V_{0}=\frac{3}{2}\gamma_{1}$ and
width $d=25$ nm. The superimposed dashed curves indicate different propagating
modes inside and outside the barriers. For ungapped and unbiased AB-BLG
(pristine AB-BLG), Ben et al. [29] showed that all channels are symmetric with
respect to normal incidence, $k_{y}=0$, i.e. $T^{+}_{-}=T^{-}_{+}$ and
$R^{+}_{-}=R^{-}_{+}$. This is due to the valley equivalence, namely the
transmission probabilities of electrons moving in the opposite direction
(scattering from $k^{+}$ to $k^{-}$ in the vicinity of the first valley, and
scattering from $k^{-}$ to $k^{+}$ in the vicinity of the second valley) are
the same. Now as for our case by introducing a gap $\Delta_{0}=0.3\gamma_{1}$,
with a null inter-layer bias, $\delta=0$, we observe that the transmissions
are completely suppressed in the energy range
$V_{0}-\Delta_{0}<E<V_{0}+\Delta_{0}$ due to the absence of traveling modes.
In $T^{+}_{+}$ channel and for energies smaller than $V_{0}-\gamma_{1}$, we
find that the resonances are decreased and Klein tunneling get less
incandescent than that seen in [29]. We notice that there is asymmetric in the
transmission channels with respect to normal incidence,
$T^{+}_{-}(k_{y})=T^{-}_{+}(-k_{y})$, but reflection channels still showing
symmetric behavior, $R^{+}_{-}(k_{y})=R^{-}_{+}(k_{y})$, because the incident
electrons back again in an electron state [29]. This is not the case for
gapped AA-BLG, whereas $T^{+}_{-}$ and $T^{-}_{+}$ channels preserve the
momentum symmetry [43]. In addition, there is a significant distinction for
all reflection channels, $R^{s}_{\pm}$, between gapped AB-BLG and biased AB-
BLG. Indeed, in our case we observe that the scales of $R^{s}_{\pm}$ get
reduced inside the barrier. It is remarkably seen that our transmission
channels, $T^{s}_{\pm}$, showed some bowels in the energy spectrum instead of
“Mexican hats” as have been see in [29]. This show that $\Delta_{0}$ can be
used to control the transmission behavior in AB-BLG.
Figure 4: (Color online) Density plot of transmission and reflection
probabilities as a function of the incident energy $E$ and transverse wave
vector $k_{y}$, through a potential barrier of height $V_{0}=1.5\gamma_{1}$
and width $d=25$ nm and band gap $\Delta_{0}=0.3\gamma_{1}$ with $\delta=0$.
The dashed white and black lines represent the band inside and outside the
barrier, respectively.
In Fig. 5 we show the density plot of the transmission and reflection
channels, for biased and gapped systems, $\delta={0.3}\gamma_{1}$,
$\Delta_{0}=0.3\gamma_{1}$. The transmission is completely suppressed in the
energy range $V_{0}-\delta^{\prime}<E<V_{0}+\delta^{\prime}$,
$\delta^{\prime}=\Delta_{0}+\delta$ . We notice that the symmetric inter-layer
sublattice equivalence is also broken in this case as seen in Fig. 4. We
recall that such symmetry broken can be achieved by taking either $\delta\neq
0$ or $\Delta_{0}\neq 0$, which means that there is violation of invariance
under the exchange $k_{y}\longrightarrow-k_{y}$ as noted in [29, 49] for AB-
BLG, in contrast to the AA-BLG [50]. Therefore, the transmission and
reflection probabilities are not symmetric with respect to normal incidence as
seen in Fig. 5.
Figure 5: (Color online) The same as in Fig. 4, but now for the band gap
$\Delta_{0}=0.3\gamma_{1}$ with $\delta=0.3\gamma_{1}$. The dashed white and
black lines represent the band inside and outside the barrier, respectively.
Fig. 6 presents the same plot as in Fig. 5 except that we choose a band gap
$\Delta_{0}=0.5\gamma_{1}$ greater than inter-layer bias
$\delta=0.3\gamma_{1}$. In this situation, we notice a significant difference
in the transmission and reflection channels. Indeed, we observe that Klein
tunneling becomes less than that see for the case
$\Delta_{0}=\delta=0.3\gamma_{1}$ in Fig. 5. In addition, it is clearly seen
that some resonances disappear for the energy range $E<V_{0}-\delta^{\prime}$.
Moreover, we find that the energy bands are pushed and showed some behaviors
look like “Mexican hats”, which are more clear than those see in Fig. 5. These
results are similar to those obtained in [51], by analyzing the transmission
probabilities for a system composed of two single layer-AB bilayer-two single
layer (2SL-AB-2SL) of graphene subjected to strong gate potential. In summary,
we observe that all transmissions for $\delta\neq 0$ and $\Delta_{0}\neq 0$
are weak compared to the biased AB-BLG [29], or gapped AB-BLG (Fig. 4) cases.
Figure 6: (Color online) The same as in Fig. 4, but now for the band gap
$\Delta_{0}=0.5\gamma_{1}$ with $\delta=0.3\gamma_{1}$. The dashed white and
black lines represent the band inside and outside the barrier, respectively.
In Figs. 7 we plot the energy dependence of the corresponding conductance for
different values of the band gap and an inter-layer bias
$\delta=0.3\gamma_{1}$. The band gap $\Delta_{0}=0.3\gamma_{1}$ contributed by
opening a gap in the energy spectrum of AB-BLG at $V_{0}\pm\Delta_{0}$, and
this of course reflected on the conductance as shown in Fig. 7(a). The
resonances that are clear in the transmission probability show up as peaks,
and the total conductance $G_{\text{Tot}}$ has a convex form. For low energies
we have $G_{\text{Tot}}=G^{+}_{+}$ meaning that the propagation is only via
$k^{+}$ mode, while $k^{-}$ mode is cloaked in this regime until
$E>V_{0}+\Delta_{0}$. $G^{-}_{-}$ starts conducting by making an appearance as
a rapid increase in the total conductance. Furthermore,
$G^{+}_{-}=G^{-}_{+}=0$ since $T^{+}_{-}=T^{-}_{+}=0$ at low energy but at
$E=\gamma_{1}$ both modes are coupled and $G^{+}_{-}$, $G^{+}_{-}$ start
conducting that is why $G_{\text{Tot}}\neq G^{+}_{+}$. However the band gap
does not break the equivalence in the scattered channels of the conductance
such that $G^{-}_{+}=G^{+}_{-}$ still equivalent for all energy ranges (see
Fig. 7(a)), in contrast to the case of the double barriers [52]. By comparing
our results with those of the biased AB-BLG [29], we observe that some
shoulders of the peaks are removed and the contribution of the transmission
channels on the total conductance are not much more pronounced as a result of
the gap $\Delta_{0}$ induced by dielectric layers. This confirms that our
$\Delta_{0}$ has a significant impact on the transport properties and differs
from that induced by bias in AB-BLG [29]. Instead of contrast, the total
conductance of a gapped AA-BLG is approximately unchanged even though the band
gap has a significant impact on the intracone transport [43]. Now we involve
both of parameters by presenting Figs 7(b) and 7(c) corresponding,
respectively, to $\Delta_{0}=\delta=0.3\gamma_{1}$, and
$\Delta_{0}=0.5\gamma_{1},\delta=0.3\gamma_{1}$. As expected we observe large
suppression of the conductance in the energy range
$V_{0}-\delta^{\prime}<E<V_{0}+\delta^{\prime}$, and hence some peaks are
removed with a decrease of the total conductance $G_{\text{Tot}}$.
Figure 7: (Color online): Conductance as a function of the incident energy for
biased and gapped AB-BLG with potential height $V_{0}=1.5\ \gamma_{1}$ and
width $d=25$ nm. (a): $\Delta_{0}=0.3\gamma_{1}$, $\delta=0$. (b):
$\Delta_{0}=0.3\gamma_{1}$, $\delta=0.3\gamma_{1}$, (c):
$\Delta_{0}=0.5\gamma_{1}$, $\delta=0.3\gamma_{1}$. The solid curves
correspond to the total conductance and the dashed curves correspond to
different contributions of the four transmission channels.
## V Summary and conclusion
We have theoretically investigated the transport properties through
rectangular potential barriers of biased AB-BLG gapped by dielectric layers.
By solving Dirac equation, the four band energies are obtained to be dependent
on the band gap $\Delta_{0}$ together with the inter-layer bias $\delta$.
Subsequently, using transfer matrix method we have evaluated the corresponding
transmission, reflection probabilities, and conductance. In particular, we
have analyzed the transmission probability in the two-band model at normal
incidence, (i.e $k_{y}=0$), firstly in the presence of $\Delta_{0}$ and
secondly by taking into account $\Delta_{0}$ and $\delta$. As a result, we
have observed that the presence of $\Delta_{0}$ induces extra resonances
appearing in transmission profiles. However by adding $\delta$, we have
observed that the transmission decreased more and anti-Klein tunneling in AB-
BLG is no longer preserved.
Furthermore, we have obtained a new mode of propagation for energies exceeding
the inter-layer coupling $\gamma_{1}$. In this case, we have showed that the
band gap $\Delta_{0}$ breaks the inter-layer sublattice equivalence with
respect to $k_{y}=0$. Such asymmetry is apparent in the scattered transmission
where it depends on the incident mode. The corresponding conductance does not
incorporate this asymmetric, and the locations of their peaks are changed in
the presence of $\Delta_{0}$ compared to $\delta$ [29].
## VI Acknowledgments
The generous support provided by the Saudi Center for Theoretical Physics
(SCTP) is highly appreciated by all authors. A.J. thanks Dr. Michael Vogl for
fruitful discussion.
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|
# Thermal Casimir effect for a Dirac field on flat space with a nontrivial
circular boundary condition
Joás Venâncio<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Pernambuco, Recife, Pernambuco 50740-560, Brazil. Lameque Filho
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal da
Paraíba, João Pessoa, Caixa Postal 5008, Brazil. Herondy Mota
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal da Paraíba,
João Pessoa, Caixa Postal 5008, Brazil. Azadeh Mohammadi
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de
Pernambuco, Recife, Pernambuco 50740-560, Brazil.
###### Abstract
This work investigates the thermal Casimir effect associated with a massive
spinor field defined on a four-dimensional flat space with a circularly
compactified spatial dimension whose periodicity is oriented along a vector in
$xy$-plane. We employ the generalized zeta function method to establish a
finite definition for the vacuum free energy density. This definition
conveniently separates into the zero-temperature Casimir energy density and
additional terms accounting for temperature corrections. The structure of
existing divergences is analyzed from the asymptotic behavior of the spinor
heat kernel function and removed in the renormalization by subtracting scheme.
The only non-null heat coefficient is the one associated with the Euclidean
divergence. We also address the need for a finite renormalization to treat the
ambiguity in the zeta function regularization prescription associated with
this Euclidean heat kernel coefficient and ensure that the renormalization
procedure is unique. The high- and low-temperature asymptotic limits are also
explored. In particular, we explicitly show that free energy density lacks a
classical limit at high temperatures, and the entropy density agrees with the
Nernst heat theorem at low temperatures.
Casimir effect. Spinor field. Zeta function. Heat kernel. Boundary condition
## I Introduction
The Casimir effect is a fascinating quantum phenomenon initially proposed by
H. Casimir in 1948 Casimir . In its standard form, such an effect establishes
that two parallel, electrically neutral conducting plates in close proximity
experience an attractive force inversely proportional to the fourth power of
the distance between them. This attraction arises from alterations in vacuum
fluctuations of the electromagnetic field. Since this force between the plates
is extremely weak, the Casimir effect was initially perceived as a theoretical
curiosity. M. Sparnaay conducted the pioneering experimental attempt, however
with low precision, to detect this effect in 1958 Sparnaay . It was only
confirmed decades after by several high-accuracy experiments Bressi2002 ;
Lamoreaux ; Lamoreauxx ; Mohideen . Since then, spurred by the progress in
theories of particles and fields, the Casimir effect has been investigated in
increasingly complicated configurations, not only due to its theoretical and
mathematical aspects but also due to the countless technological applications
arising from the macroscopic manifestation of a fully quantum effect Bordag ;
Klimchitskaya ; Mostepanenko ; Ford1975 ; Dowker 1976 ; Dowker1978 ;
Appelquist1983 ; Hosotani1983 ; Brevik2002 ; Zhang2015 ; Henke2018 ;
Bradonjic2009 ; Peng2018 ; Pawlowski2013 ; Gambassi2009 ; Machta2012 ;
Milton2019 . A thorough review concerning the Casimir effect is presented in
Refs. Klimchitskaya2009 ; Milton2001 .
Although originally associated with the electromagnetic field, the Casimir
effect is not an exclusive feature of this particular field. Other fields, for
instance, scalar and spinor fields, and gauge fields (Abelian and non-
Abelian), can exhibit analogous phenomena under nontrivial boundary conditions
Stokes2015 ; Farina2006 ; Muniz2018 ; Cunha2016 ; Mobassem2014 ; Bytsenko2005
; Pereira2017 ; Photon2001 ; Chernodub2018 ; Edery2007 . However, among the
vast literature concerning the Casimir effect, the majority of the
investigations have been focused on scalar fields. The reason for this is not
conceptual but, most likely, the more significant technical complexity
involved in the formalism needed to treat spinor fields, for instance.
Spinor fields play an important role in many branches of physics since they
represent fermion fields. Additionally, they carry the fundamental
representation of the orthogonal group, making spinors the building block of
all other representations of this group. In this sense, spinors are the most
fundamental entities of a space endowed with a metric Cartan1966 ; Benn1987 ;
JoasBook2019 . In particular, studying vacuum energy associated with the
quantized version of these fields sets a scenario for which the physics
involved is quite rich.
The presence of divergencies is an inherent feature of vacuum energy when
calculated with the quantum field theory (QFT) techniques. Knowing how to deal
with them is challenging in general. This special concern has resulted in the
development of regularization and renormalization techniques in mathematical
physics, which can be applied to remove the divergences associated with the
calculations involved in the Casimir effect Oikonomou2010 ; Cheng2010 ;
Cavalcanti2004 ; Elizalde2008 . This study concentrates explicitly on a robust
and elegant regularization method employing the generalized zeta function.
This function is constructed from the eigenvalues of a differential operator,
which governs the quantum field dynamics Elizalde1995 ; Hawking1977 ;
Elizalde2012 . The divergencies are typically introduced in the partition
function in QFT by the determinant of the operator, which is an infinite
product over all eigenvalues, and encoded into the generalized zeta function
Elizalde1994 . Once we obtain the partition function, the canonical ensemble
establishes the formal connection with thermodynamics. It facilitates the
calculation of free energy, which allows for considering temperature
corrections to the vacuum energy. Basil1978 ; Plunien1986 ; Kulikov1988 ;
Maluf2020 . The structure of the existing divergences in these calculations
typically involves examining the asymptotic behavior of the two-point heat
kernel function associated with the relevant operator, as considered in M.
Kac’s seminal paper Kac1966 and further explored in Elizalde1994 ; Bordag2000
; Vassilevich2003 ; Kirstein2010 . This zeta function investigation
predominantly focuses on Laplace-type operators associated with scalar fields,
with comparatively less emphasis on Dirac-type operators associated with
spinor fields Branson1992A ; Branson1992B .
One potential explanation for this disparity is the requirement for the
considered operator, which governs the propagation of the quantum field under
specified boundary conditions, to be self-adjoint. The self-adjointness is
necessary for the construction of zeta and heat kernel functions. The most
common boundary conditions, widely used in the Casimir effect for the scalar
field, are Dirichlet and Neumann ones. However, these conditions do not
directly extend to spinor fields due to the first-order nature of Dirac
operators. Instead, the bag model boundary conditions first presented in Refs.
Chodos1974 ; Johnson1975 make the Dirac operator formally self-adjoint. This
was also investigated in Ref. Arrizabalaga2017 recently. In particular, the
Casimir effect for spinor fields under bag model boundary conditions has been
addressed in Ref. Mamayev1980 and for Majorana spinor fields with temperature
corrections in Refs. Oikonomou2010 ; Cheng2010 ; Erdas2011 ; Elizalde2012Maj .
Alternative methods to maintain the self-adjoint nature of the Dirac operator
have also been explored. For example, the Casimir effect involving spinor
fields confined by a spherical boundary has been examined in Refs. Bender1976
; Elizalde1998 using the zeta function method. This approach was recently
extended to include a spherically symmetric $\delta$-function potential in
Ref. Fucci2023 . Furthermore, Elko fields, which are spinor fields satisfying
a Klein-Gordon-like equation, allow for the imposition of boundary conditions
similar to those used for scalar fields. The finite temperature Casimir effect
for Elko spinor fields in a field theory at a Lifshitz fixed point is
discussed in Refs. Pereira2017 ; Pereira2019 ; Maluf2020 .
Boundary conditions play a pivotal role in the exploration of the Casimir
effect. Interestingly, it is possible to induce boundary conditions through
identification conditions in spaces with nontrivial topology, thereby
eliminating the need for material boundaries. Such topologies induce boundary
conditions on the quantum fields that distort the corresponding vacuum
fluctuations, such as a material boundary does, producing a Casimir-like
effect Klimchitskaya ; Milton2001 . The Casimir effect for different types of
fields and boundary conditions in spaces with nontrivial topology has been
addressed in Refs. Mostepanenko2011 ; HerondyJunior2015 ; Mohammadi2022 ;
Herondy2023 ; Farias2020 ; Xin-zhou ; Zhai ; Li ; Xin2011 .
In the present work, we have delved into the thermal Casimir effect using the
generalized zeta function approach for a massive spinor field defined on a
four-dimensional flat space with a circularly compactified spatial dimension,
whose periodicity is oriented not along a coordinate axis as usual, but along
a vector in the ${xy\text{-plane}}$, dubbed compact vector. This space
introduces a topological constraint that imposes a spatial anti-periodic
boundary condition along the compact vector on the spinor field. Up to a
coordinate origin redefinition, this condition is referred to as the anti-
helix condition in Ref. Xiang2011 , where the authors investigated the zero-
temperature Casimir effect for spinor fields induced by the helix topology.
However, to our knowledge, a study that adds thermal effects induced by this
topology in the spinor field context has not appeared in the literature. The
calculations conducted in this study not only extend the findings from Ref.
Xiang2011 to finite temperature but also revisit the results from Ref.
Bellucci2009 in a limiting case. Additionally, our study serves as a spinor
extension of the thermal Casimir effect studied in Ref. Giulia2021 , which
focused on scalar fields subjected to a helix boundary condition.
The structure of this paper is organized as follows. Section II provides a
general expression for the partition function associated with a massive Dirac
field defined on a space endowed with a flat Euclidean metric, in the path
integral representation. In Section III, we outline the mathematical framework
employed to compute the vacuum-free energy using the generalized zeta function
method. This method involves imposing an anti-periodic condition on the Dirac
field in imaginary time $t$ and analyzing existing divergences based on the
asymptotic behavior of the spinor heat kernel. In particular, we discuss the
presence of ambiguities in the zeta function regularization due to nonzero
heat kernel coefficients and the necessity of requiring vacuum energy to
renormalize to zero for large masses. In Section IV, we derive the spinor heat
kernel two-point function and the Casimir energy density, incorporating
temperature corrections induced by the anti-periodic boundary condition along
the compact vector. We also analyze the low- and high-temperature asymptotic
limits. Finally, Section V provides a summary of the paper, highlighting the
distinctions between the spinor and scalar cases. Throughout this paper, we
adopt the natural units where $c=\hbar=k_{B}=1$.
## II Path integrals
To illustrate the use of the generalized zeta function method in quantum field
theory (QFT), we revisit some known underlying facts. In the path integral
formulation, the one-loop partition function associated with a complex matter
field $\Psi$ (and its conjugate $\bar{\Psi}$) can be obtained from the
following source-free generating functional
$Z=\int\mathcal{D}\bar{\Psi}\mathcal{D}\Psi\,e^{i\mathcal{S}(\Psi,\bar{\Psi})},$
(1)
where $\mathcal{D}\bar{\Psi}\mathcal{D}\Psi$ stands for the integration
measure over the field space, whose dynamics is described by the action
${\mathcal{S}}(\Psi,\bar{\Psi})$. Such representation provides a
straightforward method for introducing temperature into QFT. This can be
achieved by defining a Euclidean action $\mathcal{S}_{E}(\Psi,\bar{\Psi})$
through a rotation in the complex plane, known as Wick rotation, with the
fields satisfying periodic (for scalar fields) or anti-periodic (for spinor
fields) conditions in imaginary time with period $\beta$. In this Euclidean
approach to QFT, $Z$ is the one-loop partition function for a canonical
ensemble at the temperature $T=\beta^{-1}$.
### II.1 Spinor fields
We can start with the path integral for spinor fields. Let
$\\{\boldsymbol{e}_{a}\\}~{}(a=1,2,3,\ldots,N)$ be an orthonormal frame field
that spans $M=(\mathbb{R}^{N},\boldsymbol{g})$, a $N$-dimensional space
endowed with a flat Euclidean metric $\boldsymbol{g}$ whose components with
respect to basis $\\{\boldsymbol{e}_{a}\\}$ are
$\boldsymbol{g}(\boldsymbol{e}_{a},\boldsymbol{e}_{b})=\delta_{ab},$ (2)
where $\delta_{ab}$ is the Kronecker delta. That is, the space can be covered
by cartesian coordinates ${\\{x^{a}\\}=\\{t,x,y,\dots,z\\}}$ such that the
line element on $M$ is given by
$\displaystyle ds^{2}=dt^{2}+dx^{2}+dy^{2}+\cdots+dz^{2}.$ (3)
The imaginary time coordinate $t$ is compactified into a finite length equal
to the inverse of temperature $\beta$, so that $M$ is closed in the
$t$-direction. This is equivalent to consider spinor fields on
$M=\mathbb{S}^{1}\times\mathbb{R}^{N-1}$ satisfying anti-periodic boundary
conditions. The associated action has the form
$\mathcal{S}_{E}(\Psi,\bar{\Psi})=\int_{0}^{\beta}dt\int
d^{N-1}x\,\sqrt{g}\,\bar{\Psi}(x)\not{D}(m)\Psi(x),$ (4)
where $g$ is the metric determinant and $\not{D}(m)$ is the standard skew-
adjoint Dirac operator, $\not{D}=\gamma^{a}\partial_{a}$, in the presence of a
mass term
$\not{D}(m)=\gamma^{a}\partial_{a}+m,\quad a=1,2,3,\ldots,N.$ (5)
The frame $\\{\boldsymbol{e}_{a}\\}$ can be faithfully represented by the
Dirac matrices $\\{\gamma_{a}\\}$ that generate the Clifford algebra over $M$
$\gamma_{a}\gamma_{b}+\gamma_{b}\gamma_{a}=2\,\boldsymbol{g}(\boldsymbol{e}_{a},\boldsymbol{e}_{b})\mathbb{1}.$
(6)
In Euclidean signature, the Dirac matrices defined above are Hermitian,
denoted by ${\gamma_{a}^{\dagger}=\gamma_{a}}$, and the conjugate spinor
$\bar{\Psi}$ is simply the Hermitian conjugate of $\Psi$, written as
${\bar{\Psi}=\Psi^{\dagger}}$. Since the dimension of the spinor space in $N$
dimensions is $[N/2]$ (the floor of the number $N/2$), $\\{\gamma_{a}\\}$ and
$\mathbb{1}$ stand for $2^{[D/2]}\times 2^{[D/2]}$ matrices. In four
dimensions ($N=4$), for instance, they are $4\times 4$ matrices. The spectral
theory of general first-order differential operator of Dirac type can be found
in Refs. Branson1992A ; Branson1992B .
Our goal is to solve the integral (1). To accomplish this, we can expand the
spinor fields $\Psi$ and $\bar{\Psi}$ in terms of four-component complete
orthonormal sets of Dirac spinors $\psi_{j}$:
$\displaystyle\Psi(x)=\sum_{j}\Psi_{j}\psi_{j}(x),$ (7)
$\displaystyle\bar{\Psi}(x)=\sum_{j}\bar{\Psi}_{j}\psi^{\dagger}_{j}(x),$ (8)
The coefficients $\Psi_{j}$ and $\bar{\Psi}_{j}$ are independent Grassmannian
variables, and the index $j$ labels the field modes. The spinors $\psi_{j}$
are eigenfunctions of $\not{D}$ with eigenvalues determined by the equation
$\not{D}\psi_{j}=i\lambda_{j}\psi_{j},\quad\forall~{}\lambda_{j}\in\mathbb{R},$
(9)
and satisfy the following orthonormality and completeness relations
$\displaystyle\int
d^{N}x\,\sqrt{g}\,\psi^{\dagger}_{j}(x)\psi_{k}(x)=\delta_{jk},$ (10)
$\displaystyle\sum_{j}\psi_{j}(x)\psi^{\dagger}_{j}(x^{\prime})=\delta(x-x^{\prime})\,\mathbb{1},$
(11)
where $\delta(x-x^{\prime})$ is Dirac delta-function in the Euclidean
coordinates $\\{x,x^{\prime}\\}$. Taking into account the orthonormality
property (10) and the field expansions (7) and (8), the action (4) can be put
into the diagonal form
$\displaystyle\mathcal{S}_{E}(\Psi,\bar{\Psi})$
$\displaystyle=\sum_{j}\lambda_{j}(m)\bar{\Psi}_{j}{\Psi}_{j},$ (12)
where $\lambda(m)$ is given by:
$\displaystyle\lambda_{j}(m)=i\lambda_{j}+m.$ (13)
Now, under the decompositions (7) and (8), the anti-periodic functional
integral over the fields can be written in terms of $\Psi_{j}$ and
$\bar{\Psi}_{j}$ as
$\displaystyle\int_{\text{anti-
periodic}}\mathcal{D}\bar{\Psi}\mathcal{D}\Psi=\int\prod_{j}\frac{1}{\mu}\,d\bar{\Psi}_{j}d\Psi_{j},$
(14)
in which an arbitrary scale parameter $\mu$ has been introduced. An
interesting discussion on the meaning of $\mu$ can be consulted in
Elizalde1990 . By using the fact that the integration rules for Grassmannian
degrees of freedom are
$\int d\Psi_{j}=0\quad\text{and}\quad\int\Psi_{j}d\Psi_{j}=0,$ (15)
which must be equally satisfied by $\bar{\Psi}_{j}$, we are eventually led to
the following result
$\displaystyle{\,}\int
d\bar{\Psi}_{j}d\Psi_{j}e^{-\lambda_{j}(m)\bar{\Psi}_{j}\Psi_{j}}$
$\displaystyle=$ $\displaystyle\int
d\bar{\Psi}_{j}d\Psi_{j}\left[1-\lambda_{j}(m)\right]\bar{\Psi}_{j}\Psi_{j}=\lambda_{j}(m).$
(16)
The exponential series’ quadratic and higher-order powers vanish identically
due to Grassmannian anticommutative properties. Assuming that Eqs. (7)-(II.1)
hold, the path integral (1) over the Grassmann-valued Dirac spinors $\Psi$ and
$\bar{\Psi}$ gives the one-loop functional determinant of the operator
$\not{D}(m)$ with a positive exponent, as follows
$\displaystyle Z$
$\displaystyle=\int\prod_{j}\frac{1}{\mu}\,d\bar{\Psi}_{j}d\Psi_{j}\,e^{-\sum_{j}\lambda_{j}(m)\bar{\Psi}_{j}\Psi_{j}}$
$\displaystyle=\prod_{j}\frac{\lambda_{j}(m)}{\mu}=\text{det}\left[\frac{\not{D}(m)}{\mu}\right].$
(17)
Note that the above functional determinant is divergent because of infinite
product over the eigenvalues. This divergence indicates a need for some
regularization procedure. In this paper, we will adopt a powerful and elegant
regularization technique that utilizes the so-called generalized zeta
function, the zeta function of an operator.
## III Generalized zeta function
Let $L$ be a positive-definite self-adjoint second-order elliptic differential
operator, i.e. the eigenvalues $\lambda_{j}$ of $L$ are real and non-negative.
The zeta function associated with the operator $L$ is defined as
$\zeta_{L}(z)=\sum_{j}\lambda_{j}^{-z},$ (18)
where the sum over $j$ means the sum over the spectrum of $L$. In $N$
dimensions, the serie (18) will converge for ${\text{Re}(z)>N/2}$ and can be
analytically continued for the other values of $z$ Seeley1967 . In particular,
it is regular at $z=0$.
Now, we can use the zeta function above to provide a regularized version of
the ill-defined product of all eigenvalues. Taking the exponential of the
derivative of the zeta function with respect to $z$, evaluated at $z=0$, the
zeta-function regularized determinant can be defined by the relation
$\text{ln~{}det}L=\sum_{j}\text{ln}\lambda_{j}:=-\zeta_{L}^{\prime}(0),$ (19)
where $\zeta_{L}^{\prime}(z)$ stands for the derivative of $\zeta_{L}(z)$ with
respect to $z$. The definition (19) is well defined because the zeta function
is regular at $z=0$, and encodes all divergences present in the sum
$\sum_{j}\text{ln}\lambda_{j}$.
Defined previously as a series over the eigenvalues of an operator, the zeta
function admits also an integral representation by making a Mellin transform,
that is
$\displaystyle\zeta_{L}(z)=\frac{1}{\Gamma(z)}\int_{0}^{\infty}d\tau\,\tau^{z-1}K_{L}(\tau),$
(20)
where $K_{L}(\tau)$ is a spectral function called global heat kernel, defined
as
$\displaystyle K_{L}(\tau)=\text{Tr}\left(e^{-\tau L}\right).$ (21)
with Tr standing for the trace operation. In the case of the operator
$\not{D}^{2}(m)$ which is a $2^{[N/2]}\times 2^{[N/2]}$ matrix in the spinor
indices, Tr should be understood with an extra factor $2^{[N/2]}$ included.
Besides that, being $\lambda_{j}$ the eigenvalues of the operator $L$, we can
rewrite ${\text{Eq.}~{}\eqref{heat kernel trace}}$ as
$\displaystyle K_{L}(\tau)=\sum_{j}e^{-\tau\lambda_{j}},$ (22)
which diverges for $\tau\rightarrow 0$. In general, the structure of the
divergences present in the zeta function can be accessed from the asymptotic
behavior of the heat kernel for small $\tau$. For $\tau\rightarrow 0$, the
heat kernel admits the following expansion
$\displaystyle
K_{L}(\tau)\sim\frac{1}{(4\pi\tau)^{N/2}}\sum_{p}c_{p}(L)\,\tau^{p},~{}p=0,\frac{1}{2},1,\frac{3}{2},\ldots,$
(23)
where $c_{p}:=c_{p}(L)$ are the heat kernel coefficients. To review many of
the basic properties of the heat kernel method in QFT, including some
historical remarks, we refer to Vassilevich2003 ; Vassilevich2011 ; Gilkey1995
.
It is now possible to obtain a link between the one-loop partition function
and the generalized zeta function. Using the cyclic property of the trace and
the fact that the hermitian chiral matrix $\gamma^{5}$ (denoted this way
independently of the dimension) anticommutes with all Dirac matrices
$\gamma^{a}$, we can show the important property
$\displaystyle\text{Tr ln}\left[\not{D}(m)\right]$ $\displaystyle=\text{Tr
ln}\left[\gamma^{5}\not{D}(m)\gamma^{5}\right]=\text{Tr
ln}\left[\not{D}^{\dagger}(m)\right]$
$\displaystyle=\frac{1}{2}\left\\{\text{Tr ln}\left[\not{D}(m)\right]+\text{Tr
ln}\left[\not{D}^{\dagger}(m)\right]\right\\}$
$\displaystyle=\frac{1}{2}\text{Tr ln}\left[\not{D}^{2}(m)\right],$ (24)
where $\not{D}^{2}(m)$ is the negative of the spinor Laplacian on $M$,
$\not{D}^{2}=\gamma^{a}\gamma^{b}\partial_{a}\partial_{b}$, in the presence of
the mass
$\displaystyle\not{D}^{2}(m)=-\not{D}^{2}+m^{2}.$ (25)
Note in particular that the spinors $\psi_{j}$ are eigenfunctions of
$\not{D}^{2}(m)$ with non-negative eigenvalues
$\displaystyle\not{D}^{2}(m)\psi_{j}=\left(\lambda_{j}^{2}+m^{2}\right)\psi_{j}.$
(26)
Employing the identity
$\text{det}L=e^{\text{Tr~{}ln}L},$ (27)
one can derive from Eq. (24) the important relation
$\text{ln}\,\text{det}\left[\not{D}(m)\right]=\frac{1}{2}\,\text{ln}\,\text{det}\left[\not{D}^{2}(m)\right],$
(28)
establishing the massive extension of the relation between the determinant of
the Dirac operator and the square root of the determinant of its associated
Laplace-type operator. From Eqs. (18), (19) and (28), the zeta-function
regularization allows us to write the one-loop partition function as follows
Bytsenko1992
$\text{ln}Z=-\frac{1}{2}\left[\zeta_{\not{D}^{2}(m)}^{\prime}(0)+\text{ln}(\mu^{2})\zeta_{\not{D}^{2}(m)}(0)\right],$
(29)
which has the same structure as the scalar case, up to a global sign
Hawking1977 ; Giulia2021 . This is expected since we are working with the zeta
function associated with operator $\not{D}^{2}(m)$, which is of Laplace type,
instead of $\not{D}(m)$.
With the expression (29), one can obtain the free energy $F$, defined as
Landau1980
$F=-\frac{1}{\beta}\,\text{ln}Z,$ (30)
which is needed for the derivation of the Casimir energy at finite
temperature. A thermodynamics quantities closely related to the free energy is
the entropy, defined as
$\displaystyle S$ $\displaystyle=-\frac{\partial F}{\partial T},$ (31)
which, as we will see later, satisfies the third law of thermodynamics (the
Nernst heat theorem).
Although the zeta-function method encodes all divergences present in the
functional determinant, the structure of these divergences, however, plays a
central role in the renormalization procedure. Let us now utilize this
mathematical machinery to discuss a generic case of the Casimir energy
associated with the spinor field in four dimensions. To achieve our purpose,
it is convenient to decompose the time dependence of the spinor field in the
Fourier basis, namely
$\psi_{j}(x)=e^{-i\omega_{n}t}\chi_{\ell}(\boldsymbol{r}),$ (32)
stemming from the fact that $\partial_{t}$ is an obvious Killing vector field
of our metric, where $\ell$ is a generic index denoting the spatial quantum
modes of the field. Imposing the anti-periodic condition in the imaginary time
$t$ on the spinor field,
$\displaystyle\psi_{j}(t,\boldsymbol{r})=-\psi_{j}(t+\beta,\boldsymbol{r}),$
(33)
one can prove that the allowed frequencies must have the form
$\displaystyle\omega_{n}=\frac{2\pi}{\beta}\left(n+\frac{1}{2}\right),\quad\forall~{}n\in\mathbb{Z}.$
(34)
The condition (33) corresponds to compacting the imaginary-time dimension $t$
into a circumference of length $\beta$. This amounts to considering spinor
fields defined over a four-dimensional space with topology of the type
$\mathbb{S}^{1}\times\mathbb{R}^{3}$, where periodicity represented by
$\mathbb{S}^{1}$ is oriented at the $t$-direction. In Refs. Kulikov1989 ;
Ahmadi2005 ; Joas2017 , spinor fields are worked out in several spaces whose
topology is formed from the direct products.
Because of the time decomposition (32), it is particularly useful to write the
operator $\not{D}^{2}(m)$ as
$\not{D}^{2}(m)=L_{1}+\not{\nabla}^{2}(m),$ (35)
where $L_{1}$ and $\not{\nabla}^{2}(m)$ are defined as follows
$\displaystyle
L_{1}=-\partial^{2}_{t}\quad\text{and}\quad\not{\nabla}^{2}(m)=-\partial^{i}\partial_{i}+m^{2}.$
(36)
$\not{\nabla}^{2}(m)$ is an elliptic, self-adjoint, second-order differential
spinor operator defined on the spatial part of $M$. The generalized zeta
function method associated with the scalar operators defined on spaces with
different conditions can be found in Refs. Hawking1977 ; Giulia2021 . The
trace of the operator $\not{D}^{2}(m)$ can also be split into temporal and
spatial parts through the trace property
$\displaystyle\text{Tr}\left[e^{-\tau\not{D}^{2}(m)}\right]=4\,\text{Tr}\left(e^{-\tau
L_{1}}\right)\text{Tr}\left[e^{-\tau\not{\nabla}^{2}(m)}\right],$ (37)
where the multiplicative factor $4$ is due to the spinor nature of
$\not{D}^{2}(m)$. The eigenvalues of $L_{1}=-\partial_{t}^{2}$ can be obtained
from Eq. (34), so we have that
$\displaystyle\text{Tr}\left(e^{-\tau
L_{1}}\right)=\sum_{n=-\infty}^{\infty}e^{-\tau\frac{4\pi^{2}}{\beta^{2}}\left(n+\frac{1}{2}\right)^{2}}.$
(38)
Defining the constant parameters ${a=4\pi^{2}\tau/\beta^{2}},b=n$ and $c=1/2$,
and using the Jacobi inversion identity Kirstein2010 ,
$\displaystyle\sum_{n=-\infty}^{\infty}e^{-a\left(b+c\right)^{2}}=\sqrt{\dfrac{\pi}{a}}\sum_{n=-\infty}^{\infty}e^{-\frac{\pi^{2}}{a}b^{2}-2\pi
ibc},$ (39)
we can rewrite Eq. (38) as follows:
$\displaystyle\text{Tr}\left(e^{-\tau
L_{1}}\right)=\dfrac{\beta}{\sqrt{4\pi\tau}}\left[1+2\sum_{n=1}^{\infty}\cos(\pi
n)e^{-\frac{\beta^{2}}{4\tau}n^{2}}\right],$ (40)
in which the first term inside the brackets represents the ${n=0}$ term in the
series. Summing up these results, one eventually obtains the integral
representation of the zeta function $\zeta_{\not{D}^{2}(m)}$ associated with
$\not{D}^{2}(m)$, which is a Laplace type operator defined in a flat space
with a metric of Euclidean signature and acts on a spinor field in thermal
equilibrium at finite temperature ${T=\beta^{-1}}$, satisfying anti-
periodicity conditions. It follows from (37), (40), and (20) that the zeta
function $\zeta_{\not{D}^{2}(m)}$ can be put in the form
$\displaystyle{~{}}\zeta_{\not{D}^{2}(m)}(z)=\frac{\beta}{\sqrt{4\pi}}\left[\frac{Z_{1}(z)}{\Gamma(z)}+Z_{2}(z,\beta)\right],$
(41)
with
$\displaystyle Z_{1}(z)$
$\displaystyle=\,\Gamma\left(z-1/2\right)\zeta_{\not{\nabla}^{2}(m)}\left(z-1/2\right),$
(42) $\displaystyle Z_{2}(z,\beta)$
$\displaystyle=\frac{2}{\Gamma(z)}\sum_{n=1}^{\infty}\cos(\pi n)\times$
$\displaystyle\times\int_{0}^{\infty}d\tau\,\tau^{z-\frac{3}{2}}e^{-\frac{\beta^{2}}{4\tau}n^{2}}K_{\not{\nabla}^{2}(m)}(\tau).$
(43)
where $\zeta_{\not{\nabla}^{2}(m)}$ and $K_{\not{\nabla}^{2}(m)}$ are the zeta
function and the global heat kernel associated with the spinor operator
$\not{\nabla}^{2}(m)$.
Once the zeta function is obtained, we should compute the vacuum-free energy,
Eq. (30), which may have divergent parts. In order to analyze such
divergences, it is convenient to perform the small-$\tau$ asymptotics
expansion of the heat kernel Bordag2000 :
$\displaystyle K_{\not{\nabla}^{2}(m)}(\tau)\sim\frac{e^{-\tau
m^{2}}}{(4\pi\tau)^{3/2}}\sum_{p=0,1/2,1,\ldots}c_{p}\,\tau^{p},$ (44)
where $c_{p}=c_{p}(\not{\nabla}^{2})$ are the heat kernel coefficients
associated with the massless operator $\not{\nabla}^{2}$. Now, using the
integral representation of $\zeta_{\not{\nabla}^{2}(m)}$, Eq. (20), we can use
this asymptotic behavior of $K_{\not{\nabla}^{2}(m)}$ to write the function
$Z_{1}(z)$ as
$\displaystyle Z_{1}(z)$
$\displaystyle=\int_{0}^{\infty}d\tau\,\tau^{z-3/2}K_{\not{\nabla}^{2}(m)}(\tau)$
$\displaystyle=\frac{1}{(4\pi)^{3/2}}\sum_{p=0,1/2,1,\ldots}\frac{c_{p}\,\Gamma\left(z+p-2\right)}{\left(m^{2}\right)^{z+p-2}},$
(45)
which has simple poles located at
$\displaystyle z+p-2=-\kappa,\quad\forall~{}\kappa\in\mathbb{N},$ (46)
since the gamma function diverges only at non-positive integers, with the
corresponding residues containing non-negative mass exponents
$\displaystyle\text{Res}\left(Z_{1}(z),-\kappa\right)=\frac{(-1)^{\kappa}\,c_{2-\kappa-z}\,m^{2\kappa}}{(4\pi)^{3/2}\kappa!}.$
(47)
As we are only interested in the limit $z\rightarrow 0$, the constraint (46)
translates into considering the series (III) up to order $p\leq 2$, to be
consistent with the poles at $\kappa=0,1,2$. In particular, this means that
the terms in the series with semi-integer $p$ have no poles, the divergent
contributions come from the dominant coefficients $c_{0},c_{1}$ and $c_{2}$,
with $c_{0}$ and $c_{1}$ multiplied by non-negative mass exponents. However,
these divergent contributions are canceled out by the pole in $\Gamma(z)$ in
the denominator of $\zeta_{\not{D}^{2}(m)}(z)$. Indeed, near $z=0$
$\displaystyle\frac{1}{\Gamma(z)}=z+\gamma_{E}z^{2}+\mathcal{O}(z^{3}),$ (48)
where $\gamma_{E}$ is the Euler constant. In particular, this implies that
$Z_{2}(0,\beta)=0$, since the remaining integral in Eq. (43) is finite at
$z=0$. Thus,
$\displaystyle{~{}}\zeta_{\not{D}^{2}(m)}(0)=\frac{\beta}{16\pi^{2}}\,c_{2}(m),$
(49)
where
$\displaystyle
c_{2}(m)=\sum_{\kappa=0}^{2}\frac{(-1)^{\kappa}}{\kappa!}c_{2-\kappa}m^{2\kappa}=\frac{m^{4}c_{0}}{2}-m^{2}c_{1}+c_{2}.$
(50)
In order to obtain the expression for $\zeta^{\prime}_{\not{D}^{2}(m)}(0)$, we
should note that while $Z_{2}(z,\beta)$ and its first derivative with respect
to $z$ are finite at $z=0$, $Z_{1}(z)$ has a pole at $z=0$ coming from the
pole of ${\zeta_{\not{\nabla}^{2}(m)}\left(z-1/2\right)}$ at this point, with
residue
$\displaystyle\text{Res}(\zeta_{\not{\nabla}^{2}(m)}\left(z-1/2\right),z=0)=-\frac{c_{2}(m)}{16\pi^{2}}.$
(51)
So, separating off this pole contribution and taking the derivative of
$\zeta_{\not{D}^{2}(m)}(z)$ with respect to $z$, after some algebra, leads to
the relation for the regularized (reg) free energy for the Dirac field as
follows
$\displaystyle
F(\beta,m,\mu)=-\frac{1}{2}\,\text{FP}[\zeta_{\not{\nabla}^{2}(m)}(-1/2)]$
$\displaystyle+\frac{c_{2}(m)}{16\pi^{2}}\left\\{\text{ln}\left(\mu^{2}\right)+2[1-\text{ln}(2)]\right\\}$
$\displaystyle+\frac{1}{\sqrt{4\pi}}\sum_{n=1}^{\infty}\cos(\pi
n)\int_{0}^{\infty}d\tau\,\tau^{-3/2}e^{-\frac{\beta^{2}}{4\tau}n^{2}}K_{\not{\nabla}^{2}(m)}(\tau),$
(52)
where $\text{FP}[\zeta_{\not{\nabla}^{2}(m)}(-1/2)]$ stands for the finite
part of $\zeta_{\not{\nabla}^{2}(m)}(-1/2)$. This result corresponds to the
massive spinor counterpart of the one obtained by Kirstein in Ref.
Kirstein2010 for the massless scalar field, in which the only nonvanishing
heat kernel coefficient is $c_{2}$. Finally, taking the limit
$\beta\rightarrow\infty$, we obtain the following expression for zero-
temperature free energy associated with the massive spinor field
$\displaystyle E(m,\mu)$
$\displaystyle=\underset{\beta\rightarrow\infty}{\text{lim}}F(\beta,m,\mu)$
$\displaystyle=-\frac{1}{2}\,\text{FP}[\zeta_{\not{\nabla}^{2}(m)}(-1/2)]+\frac{c_{2}(m)}{16\pi^{2}}\,\text{ln}(\tilde{\mu}^{2}),$
(53)
where the rescaled parameter $\tilde{\mu}=\mu e/2$ has been employed. The
$\beta$-dependent remaining term in Eq. (III) is the temperature correction to
the free energy given by
$\displaystyle\Delta F(\beta,m)$
$\displaystyle=\frac{1}{\sqrt{4\pi}}\sum_{n=1}^{\infty}\cos(\pi n)\times$
$\displaystyle\times\int_{0}^{\infty}d\tau\,\tau^{-3/2}e^{-\frac{\beta^{2}}{4\tau}n^{2}}K_{\not{\nabla}^{2}(m)}(\tau).$
(54)
At this stage, it is worth noting that there remains no singularity when
$z\rightarrow 0$. So, the spinor free energy is finite. However, when the heat
kernel coefficient $c_{2}(m)$ is nonvanishing, the zeta function
regularization prescription becomes ambiguous due to its natural dependence on
the arbitrary parameter $\mu$, which has been rescaled without loss of
generality. This scale freedom when $c_{2}(m)\neq 0$ is also responsible for
the so-called conformal anomaly Vassilevich2011 ; Vassilevich2003 . It is
worth mentioning that in the massless case ($m=0$), all information concerning
free energy ambiguity is contained in the $c_{2}$ coefficient so that such
ambiguity is present only if $c_{2}\neq 0$.
To ensure the uniqueness of the renormalization process, such ambiguity can be
removed by the subtraction of the contribution arising from the heat kernel
coefficients $c_{p}$ with $p\leq 2$. After performing this finite
renormalization, the remaining part can be expressed as the sum of the zero-
temperature Casimir energy $E_{\text{cas}}(m)$ plus the temperature correction
$\Delta F(\beta,m)$
$\displaystyle F(\beta,m)=E_{\text{cas}}(m)+\Delta F(\beta,m),$ (55)
where the Casimir energy at zero temperature $E_{\text{cas}}(m)$ is as follows
$\displaystyle
E_{\text{cas}}(m)=-\frac{1}{2}\,\text{FP}[\zeta_{\not{\nabla}^{2}(m)}(-1/2)].$
(56)
Note that as the Casimir energy exhibits a mass dependence of the type
$e^{-\tau m^{2}}$, serving as a convergence factor in the integral
representation of $\zeta_{\not{\nabla}^{2}(m)}(-1/2)$, it must vanish when the
mass tends to infinity. This is due to the fact that there can be no quantum
fluctuations at this limit.
In the same fashion, one can use the small-$\tau$ heat kernel expansion, which
can also be seen as a large-$m$ expansion, to fix the ambiguity problem
uniquely. Since the heat kernel coefficient $c_{2}(m)$ increases with non-
negative powers of the mass, one must require that $E(m,\mu)$ should be
renormalized to zero for large $m$ Bordag2000
$\displaystyle\underset{m\rightarrow\infty}{\text{lim}}E(m,\mu)\rightarrow 0,$
(57)
removing all the dependence on the scale factor $\mu$ of the Casimir energy.
It is worth pointing out that when $c_{2}(m)$ is identically null, i.e., when
the FP prescription is redundant, the finite renormalization is unnecessary
because the ambiguity is naturally removed, and hence the scale freedom is
broken.
So far, the zeta function method has been utilized to obtain a generic
expression for the vacuum free energy associated with a massive spinor field
defined on a four-dimensional flat space endowed with a Euclidean metric. In
particular, given the anti-periodic condition of spinor fields in imaginary
time, we have been able to find a constraint on the eigenvalues of $L_{1}$.
From now on, we shall consider a space with a circularly compactified
dimension that imposes an anti-periodic boundary condition along a vector on
the spinor field. By imposing this spatial condition, we will explicitly
obtain the restrictions that the eigenvalues of $\not{\nabla}^{2}(m)$ must
obey, hence evaluating the spinor vacuum free energy.
## IV Spinor field in a nontrivial compactified space
This section aims to find an analytical expression for the zero-temperature
Casimir energy and its corresponding temperature corrections induced by a
topological constraint simulating a boundary condition imposed on the spinor
field along a vector in plane. To accomplish this, we will adopt the heat
kernel approach to zeta-function regularization.
Consider the space $M=(\mathbb{R}^{4},\boldsymbol{g})$, where $\boldsymbol{g}$
is a positive-definite symmetric metric whose components are given by Eq. (2),
namely $\delta_{ab}$, so that the line element on $M$ takes the form
$\displaystyle ds^{2}=dt^{2}+dx^{2}+dy^{2}+dz^{2},$ (58)
where $\\{x^{a}\\}=\\{t,x,y,z\\}$ are cartesian coordinates. We recall that
the coordinate $t$ is compactified into a circumference length $\beta$ as
discussed in Sec. II, equivalent to equipping $M$ with the topology
$\mathbb{S}^{1}(\text{time})\times\mathbb{R}^{3}$.
Here we consider the space $\mathbb{R}^{3}$ with a circularly compactified
dimension, where the periodicity represented by the circle $\mathbb{S}^{1}$ is
oriented in the direction of a vector $\boldsymbol{L}\in\mathbb{R}^{2}$ given
by
$\displaystyle\boldsymbol{L}=a\,\boldsymbol{e}_{x}-b\,\boldsymbol{e}_{y},\quad~{}a,b\in\mathbb{R},$
(59)
referred to here as compact vector. $\boldsymbol{e}_{x}$ and
$\boldsymbol{e}_{y}$ denote the unit vectors along the directions $x$ and $y$,
respectively, and the parameters $a$ and $b$ constant displacements. In
particular, the compact dimension size is determined by the vector length
$\displaystyle L(a,b)=|\boldsymbol{L}|=\sqrt{a^{2}+b^{2}}.$ (60)
Although not along a coordinate axis as usual, the compactification in a
$\mathbb{S}^{1}$ topology along $\boldsymbol{L}$ is quite natural since $L$ is
a homogeneous function of degree $1$, that is ${L(na,nb)=n\,L(a,b)}$ for all
non-null integer $n$. Choosing a suitable frame field can recover the usual
$\mathbb{S}^{1}$ topology, as we will see later.
Along the compact vector, the spinor field is assumed to satisfy the following
boundary condition
$\displaystyle\psi_{j}(t,\boldsymbol{r})=-\psi_{j}(t,\boldsymbol{r}+\boldsymbol{L}),$
(61)
similar to the temporal anti-periodicity condition, Eq. (33). In fact, when
${b=0}$ and ${a\neq 0}$, the spinor field satisfies a spatial anti-periodicity
condition, ${\psi_{j}(t,x,y,z)=-\psi_{j}(t,x+a,y,z)}$, induced by the compact
subspace of the coordinate $x$. The condition (61) means that the spinor field
undergoes a sign change after traveling a distance $a$ in the $x$-direction
and $b$ in the $y$-direction and returns to its initial value after traveling
distances $2a$ and $2b$, namely
${\psi_{j}(t,\boldsymbol{r})=\psi_{j}(t,\boldsymbol{r}+2\boldsymbol{L})}$. In
particular, through a coordinate origin redefinition, without changing the
orientation of the axes, one can equally write the condition (61) as
${\psi_{j}(t,x+a,y,z)=-\psi_{j}(t,x,y+b,z)}$. In Ref. Xiang2011 , this latter
condition was investigated in the helix-like topology context and called the
anti-helix condition, with $a$ and $b$ labeling the circumference length and
pitch of the helix, respectively.
An ansatz for the massive spinor field in the geometry of the space $M$ was
given in Eq. (32), namely
${\psi_{j}(t,\boldsymbol{r})=e^{-i\omega_{n}t}\chi_{\ell}(\boldsymbol{r})}$,
with the spatial part $\chi_{\ell}(\boldsymbol{r})$ satisfying the eigenvalue
equation
$\displaystyle\not{\nabla}^{2}(m)\chi_{\ell}(\boldsymbol{r})=\left(\lambda_{\ell}^{2}+m^{2}\right)\chi_{\ell}(\boldsymbol{r}).$
(62)
The eigenfunctions $\chi_{\ell}$ of the above equation have the form
$\displaystyle\chi_{\ell}(\boldsymbol{r})=\mathcal{N}\,e^{i\,\boldsymbol{k}\cdot\boldsymbol{r}}\,u_{s}(\boldsymbol{k}),$
(63)
with $\mathcal{N}$ being a normalization constant and $u_{s}(\boldsymbol{k})$
being four-component spinors whose explicit form is unnecessary for our
purposes. There are four spinors for each choice of momentum $\boldsymbol{k}$,
two of which have positive energy and two with negative energy Xiang2011 .
We are interested in obtaining the finite temperature Casimir energy under the
influence of the boundary condition (61), which imposes the following non-
trivial relation for the momentum along the compact vector
$\displaystyle\boldsymbol{k}\cdot\boldsymbol{L}=k_{x}a-k_{y}b=2\pi\left(n+\frac{1}{2}\right),\quad\forall~{}n\in\mathbb{Z}.$
(64)
This means that the label $\ell$ in the spinor field (63) should be understood
as the set of quantum numbers ${\\{\ell\\}=\\{n,k_{y},k_{z},s\\}}$, since
$k_{x}$ can be eliminated employing Eq. (64). In particular, the sum over
$\ell$ becomes
$\displaystyle\sum_{\ell}\rightarrow\sum_{n}\int dk_{y}\int dk_{z}\sum_{s}.$
(65)
Thus, utilizing the identification mentioned above in the completeness
relation (11) for the spinor field $\chi_{\ell}$ obeying the boundary
condition (61), we are left with the normalization constant
$\displaystyle\mathcal{N}=\frac{1}{2\pi\sqrt{a}}.$ (66)
Given the spinor fields (63), one can determine the eigenvalues in Eq. (62),
allowing for the construction of the spinor heat kernel. Assuming that the
requirement (64) holds, the corresponding eigenvalues are found to be
$\displaystyle\lambda_{\ell}^{2}$
$\displaystyle=k_{x}^{2}+k_{y}^{2}+k_{z}^{2}$
$\displaystyle=\left[\frac{2\pi}{a}\left(n+\frac{1}{2}\right)+\frac{b}{a}\,k_{y}\right]^{2}+k_{y}^{2}+k_{z}^{2}.$
(67)
It is worth mentioning that with an appropriate choice of frame field, it is
possible to align the compactification on $\mathbb{S}^{1}$ along one of the
coordinate axes. In the momentum space, this can be achieved by defining for
instance
$\displaystyle k_{y}=\frac{a}{L}\left[k_{Y}+\frac{2\pi
b}{a\,L}\left(n+\frac{1}{2}\right)\right].$ (68)
This transformation leads to the eigenvalues (IV) to be written as
$\displaystyle\lambda_{\ell}^{2}=\left[\frac{2\pi}{L}\left(n+\frac{1}{2}\right)\right]^{2}+k_{Y}^{2}+k_{z}^{2}.$
(69)
These eigenvalues stem in particular from the spatial anti-periodic boundary
condition ${\psi_{j}(t,X,Y,z)=-\psi_{j}(t,X+L,Y,z)}$ induced by the usual
topology ${\mathbb{S}^{1}}(\text{space})\times\mathbb{R}^{2}$, whereby the
coordinate $X$ is compactified into a circumference length $L$. In fact, along
this compact dimension, the latter condition produces the discrete momentum
${k_{X}=2\pi(n+1/2)/L}$. In particular, this means that in the limiting case
${b=0}$, our results recover the ones presented in Ref. Bellucci2009 for a
specific case and include temperature corrections. In this study, the authors
investigated the Casimir effect for spinor fields in toroidally compactified
spaces, including general phases in the boundary condition along the compact
dimensions.
Building upon the previous results, we can introduce the heat kernel approach
to obtain a zeta-function analytical expression for a spinor field defined on
$M$ with the eigenvalues (IV). Instead of the global heat kernel, it is more
appropriate to utilize the local heat kernel. The reason is that the heat
kernel carries information concerning the space where the field is defined,
making it particularly valuable when focusing on the the influence of
topological constraint imposed by the boundary conditions on the thermal
vacuum fluctuations.
### IV.1 Spinor heat kernel and Casimir energy density
The spinor heat kernel
$K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r}^{\prime},\tau)$ is a
two-point function locally defined as solutions of the heat conduction
equation
$\left[\frac{\partial}{\partial\tau}+\not{\nabla}^{2}(m)\right]\,K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r}^{\prime},\tau)=0\quad\text{for}~{}\tau>0,$
(70)
supplemented with the initial condition
$\lim_{\tau\rightarrow
0}K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r}^{\prime},\tau)=\delta(\boldsymbol{r}-\boldsymbol{r}^{\prime})\,\mathbb{1}.$
(71)
The operator $\not{\nabla}^{2}(m)$ is taken to act on the first argument of
$K_{\not{\nabla}^{2}(m)}$. Similar to $\not{\nabla}^{2}(m)$,
$K_{\not{\nabla}^{2}(m)}$ is represented by a $4\times 4$ matrix.
Taking into account Eq. (26), the solutions of Eq. (70) can be expressed in
terms of the eigenvalues and eigenfunctions of $\not{\nabla}^{2}(m)$
$K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r}^{\prime},\tau)=e^{-\tau
m^{2}}\sum_{\ell}e^{-\lambda_{\ell}^{2}\tau}\chi_{\ell}(\boldsymbol{r})\chi_{\ell}^{\dagger}(\boldsymbol{r}^{\prime}).$
(72)
One can verify that the above expression provides a solution to Eq. (70), as
well as satisfying the initial condition (71) since the spinor field obeys Eq.
(11).
Inserting the spinor solution (63) along with the normalization constant (66)
and eigenvalues (IV) into spinor heat kernel (72), it follows the expression
$\displaystyle
K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r}^{\prime},\tau)$
$\displaystyle=\dfrac{e^{-\tau m^{2}}}{4\pi^{2}a}$
$\displaystyle\times\sum_{n}e^{-\frac{4\pi^{2}\tau}{a^{2}}\left(n+\frac{1}{2}\right)^{2}+\frac{2\pi
i}{a}\left(n+\frac{1}{2}\right)\Delta x}$ $\displaystyle\times\int
dk_{y}\,e^{-\tau\frac{L^{2}}{a^{2}}k_{y}^{2}+\left[i\Delta v-\frac{4\pi
b\tau}{a^{2}}\left(n+\frac{1}{2}\right)\right]k_{y}}$ $\displaystyle\times\int
dk_{z}e^{-\tau k_{z}^{2}+ik_{z}\Delta z}\,\mathbb{1},$ (73)
where
$\displaystyle\Delta v=\frac{b}{a}\,\Delta x+\Delta y.$ (74)
We can write Eq. (73) in a more compact form. To perform this, let us define
the complex parameters $w$ and $q$ as follows
$\displaystyle w=\frac{b\Delta v}{L^{2}}-\frac{\Delta
x}{a}-\frac{q}{2}\quad\text{and}\quad q=\frac{4\pi i\tau}{L^{2}},$ (75)
and introduce the following Jacobi function defined in terms of the parameters
$w$ and $q$ as Elizalde1994
$\displaystyle\theta_{3}(w,q)=\sum_{n=-\infty}^{\infty}e^{i\pi qn^{2}-2\pi
iwn}.$ (76)
Evaluating the integrals over the independent momenta $k_{y}$ and $k_{z}$ in
Eq. (73), we end up with the following relation between the spinor heat kernel
and the Jacobi function
$\displaystyle
K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r}^{\prime},\tau)=\frac{e^{-\frac{\left|\boldsymbol{r}-\boldsymbol{r}^{\prime}\right|^{2}}{4\tau}-\tau
m^{2}}}{\left(4\pi\tau\right)^{3/2}}\frac{\sqrt{-iq}}{e^{-i\pi\frac{\omega^{2}}{q}}}\,\theta_{3}(w,p)\,\mathbb{1}.$
(77)
Since we are interested in the contributions coming from the topology for the
thermal vacuum fluctuations, it is convenient to separate the Euclidean part
of the heat kernel, which should not depend on the topology parameters. This
can be done by rewriting the $\theta_{3}$ Jacobi function utilizing the
following identity
$\displaystyle\theta_{3}(w,q)=\frac{1}{\sqrt{-iq}}\,e^{-i\pi\frac{\omega^{2}}{q}}\,\theta_{3}\left(\frac{w}{p},-\frac{1}{q}\right).$
(78)
Employing this identity, leads to the following expression
$\displaystyle
K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r}^{\prime},\tau)=K_{\not{\text{E}}}(\boldsymbol{r},\boldsymbol{r}^{\prime},\tau)\sum_{n=-\infty}^{\infty}e^{-\frac{L^{2}}{4\tau}n^{2}+i\pi
n},$ (79)
with
$\displaystyle
K_{\not{\text{E}}}(\boldsymbol{r},\boldsymbol{r}^{\prime},\tau)=\frac{1}{\left(4\pi\tau\right)^{3/2}}\,e^{-\frac{\left|\boldsymbol{r}-\boldsymbol{r}^{\prime}\right|^{2}}{4\tau}-\tau
m^{2}}\,\mathbb{1}.$ (80)
where $K_{\not{\text{E}}}$ is the spinor version of the well-known Euclidean
heat kernel associated with the massive scalar Laplacian operator defined on
the flat space $\mathbb{R}^{3}$ Vassilevich2003 . Note that
$K_{\not{\text{E}}}$ is identified with the term $n=0$ in the series.
Let us now explore the heat kernel properties at the coincidence limit
$\boldsymbol{r}^{\prime}\to\boldsymbol{r}$ in Eq. (79) which results in
$\displaystyle K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r},\tau)$
$\displaystyle=K_{\not{\text{E}}}(\boldsymbol{r},\boldsymbol{r},\tau)$
$\displaystyle+\frac{e^{-\tau
m^{2}}}{(4\pi\tau)^{3/2}}\sum_{n=1}^{\infty}2\cos(\pi
n)e^{-\frac{L^{2}}{4\tau}n^{2}}\,\mathbb{1}.$ (81)
The first term on the right-hand side corresponds to the Euclidean heat
kernel, while the second term encodes information about the space topology, as
can be seen from its dependence on parameter $L$. For small $\tau$, the heat
kernel admits an expansion in powers of $\tau$, with coefficients reflecting
the space configuration. In our case, by evaluating the above series at small
$\tau$, one can see that all terms are exponentially small except for the one
associated with the Euclidean heat kernel contribution (${n=0}$). Thus, the
spinor heat kernel $K_{\not{\nabla}^{2}(m)}(\boldsymbol{r})$ on Euclidean
geometry with a circular compactification along $\boldsymbol{L}$ exhibits an
asymptotic behavior similar to the one considered in Eq. (44) with only one
non-vanishing heat kernel coefficient
$\displaystyle
K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r},\tau)\sim\frac{e^{-\tau
m^{2}}}{\left(4\pi\tau\right)^{3/2}}\sum_{p}c_{p}(\boldsymbol{r})\tau^{p}+\mathcal{O}(e^{-1/\tau}),$
(82)
where the local heat kernel coefficients $c_{p}(\boldsymbol{r})$ are given by
$\displaystyle c_{p}(\boldsymbol{r})=\delta_{0p}\,\mathbb{1}\quad\forall~{}p.$
(83)
$\mathcal{O}(e^{-1/\tau})$ stands for those terms going to zero faster than
any positive power of $\tau$ and, therefore, can be neglected. In contrast
with the global case, the local heat kernel coefficients carry spinor indices,
hence $4\times 4$ matrices. Note that $K_{\not{\nabla}^{2}(m)}(\tau)$ can be
obtained from
$K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r}^{\prime},\tau)$
performing an integral in the whole space
$\displaystyle K_{\not{\nabla}^{2}(m)}(\tau)=\int
d^{3}\boldsymbol{r}\,\sqrt{g^{(3)}}\,\text{tr}\left[K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r},\tau)\right],$
(84)
where $g^{(3)}$ is the spatial part of the metric determinant, and the trace
operation tr is taken over the spinor indices only. Thus, the global heat
kernel coefficient $c_{0}$ is found to be
$\displaystyle c_{0}$ $\displaystyle=4\,V_{3},$ (85)
where $V_{3}$ is the volume of the $3$-dimensional base space of $M$. As
discussed in Sec. III, for nonvanishing heat kernel coefficients
$c_{p}\,(p\leq 2)$, the zeta function is not finite, and the renormalization
procedure is not unique. In fact, although the vacuum energy is finite due to
the FP prescription introduced in the Casimir energy, the coefficient $c_{0}$
gives origin to the terms in the vacuum energy, which increase with non-
negative powers of the mass, besides the logarithmic dependence on the scale
factor $\mu$. To ensure a unique renormalization procedure and obtain an
unambiguous spinor vacuum free energy, all contributions associated with
$c_{0}$ should be disregarded, thereby renormalizing the energy to zero for
large masses.
After performing the finite renormalization, we can proceed with the
analytical calculation of the spinor vacuum free energy. First, we should note
that even though we are working in the local regime, the two-point function
$K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r},\tau)$ is coordinate-
independent. Therefore, considering that global quantities can be derived from
local ones by integrating over the space coordinates, the local version of the
spinor vacuum free energy differs from the global version by a volume element
and retains the same form as Eq. (55), namely
$\displaystyle\mathcal{F}(\beta,m)=\mathcal{E}_{\text{cas}}(m)+\Delta\mathcal{F}(\beta,m),$
(86)
where $\mathcal{E}_{\text{cas}}(m)$ is then the Casimir energy density at zero
temperature
$\displaystyle\mathcal{E}_{\text{cas}}(m)=-\frac{1}{2}\,\text{FP}[\zeta_{\not{\nabla}^{2}(m)}(\boldsymbol{r},-1/2)],$
(87)
and $\Delta\mathcal{F}(\beta,m)$ is the temperature correction with the form
$\displaystyle\Delta$
$\displaystyle\mathcal{F}(\beta,m)=\frac{1}{\sqrt{4\pi}}\sum_{n=1}^{\infty}\cos(\pi
n)\times$
$\displaystyle\times\int_{0}^{\infty}d\tau\,\tau^{-3/2}e^{-\frac{\beta^{2}}{4\tau}n^{2}}\text{tr}\left[K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r},\tau)\right].$
(88)
Here, the trace operation tr is taken over the spinor indices only and the
local zeta function is defined in terms of
$K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r},\tau)$ giving rise to
$\displaystyle\zeta_{\not{\nabla}^{2}(m)}$
$\displaystyle(\boldsymbol{r},z-1/2)=\frac{1}{\Gamma(z-1/2)}$
$\displaystyle\times\int_{0}^{\infty}d\tau\,\tau^{z-3/2}\,\text{tr}\left[K_{\not{\nabla}^{2}(m)}(\boldsymbol{r},\boldsymbol{r},\tau)\right].$
(89)
Then, inserting the spinor heat kernel (IV.1) into Eq. (IV.1), we conclude
from Eq. (87) that the renormalized expression for the zero temperature
Casimir energy density associated with a spinor field of mass $m$ depends on
the topology parameter $L$ according to the relation
$\displaystyle\mathcal{E}_{\text{cas}}(m,L)=\frac{2}{\pi^{2}L^{4}}\sum_{n=1}^{\infty}\frac{\cos(\pi
n)}{n^{2}}(mL)^{2}\,K_{2}(nmL),$ (90)
where $K_{2}(z)$ is the MacDonald function. Note that the FP prescription
removed the divergent contribution provided by the Euclidean heat kernel. The
above result is exactly the one shown in Ref. Xiang2011 obtained in a
different approach than the one presented here for both massive and massless
spinors. In particular, the massless one can be obtained by making use of the
following limit
$\displaystyle\underset{z\rightarrow
0}{\text{lim}}\,z^{2}K_{2}(nz)=\frac{2}{n^{2}}.$ (91)
In fact, by separating the even and odd terms in the series (90), and using
the above equation, one can promptly verify that the following massless limit
holds
$\displaystyle\mathcal{E}_{\text{cas}}(L)$
$\displaystyle=\underset{m\rightarrow
0}{\text{lim}}\mathcal{E}_{\text{cas}}(m,L)$
$\displaystyle=\frac{1}{4\pi^{2}L^{4}}\left[\zeta(4)-\zeta\left(4,\frac{1}{2}\right)\right],$
(92)
where $\zeta(z)$ is the standard Riemann zeta function and $\zeta(z,w)$ is the
Hurwitz zeta function defined for $\text{Re}(z)>1$ and $w\neq 0,-1,-2,\ldots$,
in the form Elizalde1994
$\displaystyle\zeta(z,w)=\sum_{n=0}^{\infty}(n+w)^{-z}.$ (93)
Using the relation
$\displaystyle\zeta\left(z,\frac{1}{2}\right)=(2^{z}-1)\,\zeta(z),$ (94)
along with the fact that ${\zeta(4)=\pi^{4}/90}$, we are left with the
expression for the Casimir energy density, at zero temperature, associated
with a massless spinor field
$\displaystyle\mathcal{E}_{\text{cas}}(L)=-\frac{7}{2}\,\frac{\pi^{2}}{90\,L^{4}},$
(95)
It depends only on topology parameter, in complete agreement with the massless
case obtained in Ref. Xiang2011 . In particular, its value is also $7/2$ times
the result found in the massless scalar case under periodic boundary
conditions along the compact vector $\boldsymbol{L}$ Zhai ; Giulia2021 . It is
worth mentioning that this case is unambiguous since the heat kernel
coefficient $c_{2}$ is identically zero, so the renormalization procedure is
unnecessary.
If one is interested in the limit $mL\gg 1$, then it is legitimate to consider
Mcdonald’s function behavior at ${\text{large}~{}z}$
$\displaystyle
K_{2}(z)\simeq\left(\frac{\pi}{2z}\right)^{1/2}e^{-z}\quad\text{for}\quad|\text{arg}(z)|<\pi/2.$
(96)
In this limiting case, one can see that the Casimir energy density decays
exponentially with the mass of the field
$\displaystyle\mathcal{E}_{\text{cas}}\left(m\gg\frac{1}{L}\right)=-\frac{2m^{2}}{\pi^{2}L^{2}}\sqrt{\frac{\pi}{2mL}}\,e^{-mL},$
(97)
as expected, since an infinitely heavy field should not present quantum
fluctuations and hence should not produce Casimir energy Bordag2000 . In Ref.
Maluf2020 , a similar analysis is carried out for the Casimir energy for a
real scalar field and the Elko neutral spinor field in a field theory at a
Lifshitz fixed point.
### IV.2 Finite-temperature corrections
Let us now investigate the temperature correction,
$\Delta\mathcal{F}(\beta,m)$, to the vacuum energy densities. Inserting the
heat kernel (IV.1) into $\Delta\mathcal{F}(\beta,m)$ defined in Eq. (IV.1)
leads to the following analytical expression for the temperature correction
associated with the massive spinor field, in terms of a double sum
$\displaystyle\Delta\mathcal{F}(\beta,m,L)$
$\displaystyle=\Delta\mathcal{F}_{\text{E}}(\beta,m)$
$\displaystyle+\frac{4m^{4}}{\pi^{2}}\sum_{n=1}^{\infty}\sum_{p=1}^{\infty}\cos(\pi
n)\cos(\pi p)$ $\displaystyle\times
f_{2}\left(m\beta\sqrt{p^{2}+\frac{L^{2}}{\beta^{2}}n^{2}}\right).$ (98)
For notational simplicity, we introduced the function $f_{\nu}(z)$ related to
the Mcdonald function $K_{\nu}(z)$ as follows
$\displaystyle f_{\nu}(z)=\frac{K_{\nu}(z)}{z^{\nu}}.$ (99)
The term $\Delta\mathcal{F}_{\text{E}}(\beta,m)$ is the contribution coming
from the Euclidean heat kernel and thus does not depend on the parameter $L$.
It has the following form
$\displaystyle\Delta\mathcal{F}_{\text{E}}(\beta,m)=\frac{2m^{4}}{\pi^{2}}\sum_{n=1}^{\infty}\cos(\pi
n)\,f_{2}(m\beta n).$ (100)
In particular, from Eq. (91), we conclude that the temperature correction term
in the massless limit is as follows
$\displaystyle\Delta\mathcal{F}_{\text{E}}\left(\beta\right)=-\frac{7}{2}\frac{\pi^{2}}{90}\,T^{4},$
(101)
the standard black body radiation energy density associated with the massless
spinor field. As we have seen, this contribution is directly related to the
non-null coefficient $c_{0}$. In more general spaces with nontrivial topology,
however, there may be temperature corrections to the above Stefan-Boltzmann
law, proportional to $T^{4}$, coming from heat kernel coefficients associated
with spacetime topology. These coefficients vanish in the limit of infinite
space Herondy2023 ; Basil1978 .
Since the Casimir effect is a purely quantum phenomenon, the above term should
not dominate in the high-temperature limit. Although not divergent, this
quantum term should be subtracted in the renormalization procedure to obtain a
correct classical contribution in this limit. By doing so, we end up with the
renormalized version of the free energy (86)
$\displaystyle\mathcal{F}_{\text{ren}}(\beta,m,L)=\frac{2m^{4}}{\pi^{2}}\sum_{n=1}^{\infty}\cos(\pi
n)f_{2}\left(nmL\right)$
$\displaystyle+\frac{4m^{4}}{\pi^{2}}\sum_{n=1}^{\infty}\sum_{p=1}^{\infty}\cos(\pi
n)\cos(\pi
p)f_{2}\left(m\beta\sqrt{p^{2}+\frac{L^{2}}{\beta^{2}}n^{2}}\right).$ (102)
In particular, by using the limit (91), the above expression yields the
massless contribution
$\displaystyle\mathcal{F}_{\text{ren}}(\beta,L)=\mathcal{E}_{\text{cas}}(L)+\frac{8}{\pi^{2}\beta^{4}}\sum_{n=1}^{\infty}\sum_{p=1}^{\infty}\frac{\cos(\pi
n)\cos(\pi p)}{\left(p^{2}+\frac{L^{2}}{\beta^{2}}n^{2}\right)^{2}},$ (103)
where $\mathcal{E}_{\text{cas}}(L)$ is the Casimir energy density associated
with massless spinor field at zero temperature, Eq. (95). The presence of the
double sum is convenient if one is interested in obtaining the low- and high-
temperature asymptotic limits. Although the final result is the same,
performing the sum in $p$ first is more straightforward for obtaining the
high-temperature limit, while performing the sum in $n$ first is less
complicated for obtaining the low-temperature limit. The final result for the
free energy is equivalent. Choosing which one first is a simple question of
convenience to attain our purposes.
Figure 1: Plot of the ratio
$R=\mathcal{F}_{\text{ren}}/\mathcal{E}_{\text{cas}}$ in terms of $a\,T$ for
several values of the parameter $\gamma$. $R$ decreases with $aT$ and tends to
zero when $aT$ goes to infinity.
To conduct our analysis, let us rewrite $L$ as ${L=a\sqrt{\gamma}}$ by
convenience, where $\gamma=1+(b/a)^{2}$. In Fig. 1, we have plotted the ratio
$R$ of the renormalized free energy density,
$\mathcal{F}_{\text{ren}}(\beta,L)$, to the Casimir energy density,
$\mathcal{E}_{\text{cas}}(L)$, varying with $aT$ for different values of the
parameter $\gamma$.
In each case, the plot shows the ratio
$R=\mathcal{F}_{\text{ren}}/\mathcal{E}_{\text{cas}}$ going to $1$ as $T$
approaches zero, as we should expect, and decaying to zero as $T$ approaches
infinity. In particular, this decay becomes more pronounced as the parameter
$\gamma$ increases, as illustrated by the curve for $\gamma=9$. The curves
associated with $\gamma>2$, which decay to zero faster, correspond to the case
where $b$ is greater than $a$, whereas the curve with $\gamma=2$ illustrates
the particular case $b=a$. In the limiting case when $\gamma\simeq 1$ ($b\ll
a$), the system exhibits a structure known as a quantum spring, as discussed
by Feng2010 in the context of the scalar Casimir effect.
In what follows, we will analyze the asymptotic limits of temperature
corrections to the massless free energy density above.
#### IV.2.1 High-temperature limit
Let us analyze the high-temperature limit, ${\beta\ll L}$, of the final
expression (103). In this case, it is more appropriate to perform the
summation in $p$ first, namely
$\displaystyle\frac{1}{\beta^{4}}\sum_{p=1}^{\infty}$
$\displaystyle\frac{\cos(\pi
p)}{\left(p^{2}+\frac{L^{2}}{\beta^{2}}n^{2}\right)^{2}}=-\frac{1}{2\,L^{4}}\frac{1}{n^{4}}$
$\displaystyle+\frac{T^{2}}{2L^{2}}\frac{1}{n^{2}}\,\text{csch}(n\pi L\,T)$
$\displaystyle\times\left[\coth(n\pi L\,T)+\frac{1}{n\pi L\,T}\right].$ (104)
Inserting this summation into Eq. (103), we are eventually led to the
following expression
$\displaystyle\mathcal{F}_{\text{ren}}\left(\beta,L\right)$
$\displaystyle=\frac{4\,T^{2}}{L^{2}}\sum_{n=1}^{\infty}\frac{\cos(\pi
n)}{n^{2}}\text{csch}(n\pi L\,T)$ $\displaystyle\times\left[\coth(n\pi
L\,T)+\frac{1}{n\pi L\,T}\right].$ (105)
Note that the first term on the right-hand side in Eq. (IV.2.1), after
performing the summation in $n$, gives rise to Casimir energy density
associated with massless spinor field (95) but with the opposite sign.
Therefore, the effect of this latter term is entirely compensated by the
corresponding one in the free energy density (103). Such a natural
cancellation between the massless Casimir energy density and its corresponding
temperature correction is not unusual. It is an intrinsic characteristic of
temperature corrections at the high-temperature limit Plunien1986 .
Now, our task is to evaluate the series (IV.2.1) at a high-temperature limit,
${\beta\ll L}$ (or equivalently ${L\,T\gg 1}$). Through an asymptotic
expansion up to terms of order $\mathcal{O}(e^{-\pi L\,T})$, we arrive at the
finite-temperature
$\displaystyle\ \mathcal{F}_{\text{ren}}\left(\beta\ll
L\right)\simeq-\frac{8\,T}{L^{3}}\left(1+\pi L\,T\right)\,e^{-\pi L\,T},$
(106)
which is exponentially suppressed at high $T$ and converges to zero as
${T\rightarrow\infty}$, in accordance with Fig. 1. This behavior is expected
for a spinor field since, differently from the scalar field Giulia2021 , it
lacks a temperature correction term that is linearly dependent on $T$. We
emphasize the need for the free energy density to undergo a finite
renormalization by subtracting from it the blackbody radiation contribution,
proportional to $T^{4}$, to obtain the correct classical limit, a free energy
density renormalized to zero at very high temperatures. Ref. Mostepanenko2011
found a similar result for temperature corrections associated with the spinor
field in the closed Friedmann cosmological model.
With the renormalized free energy density now available, we can obtain an
analytical expression for the renormalized entropy density. Employing the
relation (31), we have
$\displaystyle\mathcal{S}_{\text{ren}}(\beta,L)$
$\displaystyle=\frac{4\pi\,T^{2}}{L^{2}}\sum_{n=1}^{\infty}\frac{\cos(\pi
n)}{n}\text{csch}(n\pi L\,T)$ $\displaystyle\times\left[1-\frac{1}{(n\pi
L\,T)^{2}}+2\,\text{csch}^{2}(n\pi L\,T)\right.$
$\displaystyle-\left.\frac{1}{n\pi L\,T}\coth(n\pi L\,T)\right].$ (107)
The corresponding asymptotic expansion in the high-temperature regime,
${L\,T\gg 1}$, decays exponentially with the temperature $T$
$\displaystyle\mathcal{S}_{\text{ren}}\left(\beta\ll
L\right)\simeq-\frac{8}{\pi L^{3}}\left[1+\pi L\,T-(\pi
L\,T)^{2}\right]\,e^{-\pi L\,T}.$ (108)
Note that the lack of a classical term proportional to $T$ in the free energy
density results in the Casimir entropy density approaching zero at very high
temperatures, which differs from the scalar case where it is dominated by a
constant term Giulia2021 .
#### IV.2.2 Low-temperature limit
Let us now consider the asymptotic expansion of the expression (103) in the
low-temperature regime, where ${\beta\gg L}$, or equivalently ${L\,T\ll 1}$.
To accomplish this, as previously mentioned, we shall perform the summation in
$n$ first, providing
$\displaystyle\frac{1}{\beta^{4}}\sum_{n=1}^{\infty}$
$\displaystyle\frac{\cos(\pi
p)}{\left(p^{2}+\frac{L^{2}}{\beta^{2}}n^{2}\right)^{2}}=-\frac{T^{4}}{2\,\pi^{2}}\frac{1}{p^{4}}$
$\displaystyle+\frac{4\,T^{2}}{L^{2}}\frac{1}{p^{2}}\,\text{csch}\left(\frac{p\pi}{L\,T}\right)$
$\displaystyle\times\left[\frac{L\,T}{p\pi}+\coth\left(\frac{p\pi}{L\,T}\right)\right].$
(109)
Substituting it back in Eq. (103), we get
$\displaystyle\mathcal{F}_{\text{ren}}\left(\beta,L\right)$
$\displaystyle\simeq-\frac{7}{2}\,\frac{\pi^{2}}{90\,L^{4}}+\frac{7}{2}\frac{\pi^{2}}{90}\,T^{4}$
$\displaystyle+\frac{4\,T^{2}}{L^{2}}\sum_{p=1}^{\infty}\frac{\cos(\pi
p)}{p^{2}}\,\text{csch}\left(\frac{p\pi}{L\,T}\right)$
$\displaystyle\times\left[\frac{L\,T}{p\pi}+\coth\left(\frac{p\pi}{L\,T}\right)\right],$
(110)
which at the low-temperature regime up to terms of the order
$\mathcal{O}(e^{-\frac{\pi}{L\,T}})$, presents the following free energy
density asymptotic behavior
$\displaystyle\mathcal{F}_{\text{ren}}\left(\beta\gg L\right)$
$\displaystyle\simeq-\frac{7}{2}\,\frac{\pi^{2}}{90\,L^{4}}+\frac{7}{2}\frac{\pi^{2}}{90}\,T^{4}$
$\displaystyle-\frac{8\,T^{2}}{L^{2}}\left(1+\frac{L\,T}{\pi}\right)\,e^{-\frac{\pi}{L\,T}}.$
(111)
Let us make a few remarks comparing our findings and those reported in
Giulia2021 for the low-temperature behavior of the free energy associated
with the scalar field under helix topology. Apart from the additional $T^{3}$
term for temperature correction observed in the scalar case, they differ by
constant multiplicative factors that naturally arise because of the spinor
degrees of freedom. Note that at small $T$, the above asymptotic expansion is
dominated by the first term, the massless Casimir energy density at zero
temperature, Eq. (95), as expected Mostepanenko2011 ; Herondy2023 ; Basil1978
.
The entropy density can be obtained by inserting Eq. (IV.2.2) into Eq. (31),
providing
$\displaystyle\mathcal{S}_{\text{ren}}(\beta,L)$
$\displaystyle=-\frac{7\pi^{2}}{45}\,T^{3}$
$\displaystyle-\frac{4\pi}{L^{3}}\sum_{p=1}^{\infty}\frac{\cos(\pi
p)}{p}\,\text{csch}\left(\frac{p\pi}{L\,T}\right)$
$\displaystyle\times\left[1+3\left(\frac{L\,T}{p\pi}\right)^{2}+2\,\text{csch}^{2}\left(\frac{p\pi}{L\,T}\right)\right.$
$\displaystyle-\left.\frac{3L\,T}{p\pi}\coth\left(\frac{p\pi}{L\,T}\right)\right].$
(112)
Its corresponding asymptotic expansion in the low-temperature limit, where
${L\,T\ll 1}$, is found to be
$\displaystyle\mathcal{S}_{\text{ren}}\left(\beta\gg L\right)$
$\displaystyle\simeq\frac{7\pi^{2}}{45}\,T^{3}$
$\displaystyle+\frac{8\pi}{L^{3}}\left[1+\frac{3L\,T}{\pi}+3\left(\frac{L\,T}{\pi}\right)^{2}\right]\,e^{-\frac{\pi}{L\,T}}.$
(113)
As expected, the above expression tends to zero as the temperature approaches
zero. It implies that the entropy density for a massless spinor field
satisfying an anti-periodic condition along the compact vector satisfies the
third law of thermodynamics (the Nernst heat theorem) Landau1980 .
## V Conclusion
In the present work, we have investigated the thermal Casimir effect
associated with a massive spinor field defined on a four-dimensional flat
space with a circularly compactified dimension. The periodicity represented by
$\mathbb{S}^{1}$ is oriented not along a coordinate axis as usual, but along a
vector $\boldsymbol{L}$ belonging to the $xy$-plane, Eq. (59). This geometry
introduces a topological constraint inducing a spatial anti-periodic boundary
condition on the spinor field, Eq. (61), which modifies the vacuum
fluctuations, producing the Casimir effect. Imposing this boundary condition
led to the discrete eigenvalues for the momentum along vector
$\boldsymbol{L}$, Eq. (64), allowing for determining explicitly the
eigenvalues (IV). They are used to construct the generalized zeta function for
the spinor field and thus remove the formal divergences involved in the
Casimir effect.
These divergencies were introduced by the Dirac operator determinant in the
partition function originating from the infinite product over eigenvalues, Eq.
(II.1). This divergence was encoded into the generalized zeta function
employing the important relation connecting it with the partition function,
Eq. (29). It was analyzed from the asymptotic behavior of the spinor heat
kernel function, Eq. (44), and removed in the renormalization scheme by
subtraction of the divergent contribution associated with non-null heat kernel
coefficients. A rather peculiar aspect of the zeta function regularization
prescription is related to the existence of ambiguities. Such ambiguities
appear whenever the mass-dependent $c_{2}(m)$ heat kernel coefficient is
nonvanishing, Eq. (50), due to natural dependence on parameter $\mu$, Eq.
(III). For the geometry presented here, $c_{0}$ was the only non-null heat
kernel coefficient, Eq. (85), associated with the Euclidean heat kernel
contribution, Eqs. (80) and (100). In order to derive physically meaningful
expressions, all contributions associated with $c_{0}$ were dropped to ensure
that the renormalization procedure is unique and thus obtain an unambiguous
spinor vacuum free energy. Besides that, since $c_{0}$ is multiplied by mass
with a positive exponent, we adopt an additional requirement that vacuum
energy should be renormalized to zero for large masses.
We outline all the mathematical machinery required for computing the vacuum-
free energy density, starting with the construction of the partition function
for the spinor field through Euclidean path integrals. In this Euclidean
approach, we find closed and analytical expressions for the vacuum free energy
density associated with the spinor field in thermal equilibrium at finite
temperature $T=\beta^{-1}$, satisfying anti-periodic conditions in the
imaginary time $t$ and along vector $\boldsymbol{L}$. This energy density can
be expressed as a summation of the zero-temperature Casimir energy density,
Eq. (90), and temperature correction terms, Eq. (IV.2), which generalize the
results presented in Refs. Bellucci2009 ; Xiang2011 . We also analyzed the
high- and low-temperature asymptotic limits, which agree entirely with the
curves shown in Fig. 1. The ratio of the renormalized free energy density to
the Casimir energy density goes to $1$ as $T$ approaches zero and decays to
zero as $T$ approaches infinity. At high temperatures, in particular, we have
shown that the $c_{0}$ coefficient gives rise to the Stefan-Boltzmann law,
proportional to $T^{4}$. Although not divergent, this quantum term was
subtracted in the renormalization procedure to obtain a correct classical
contribution in this limit. Also, the free energy density does not possess a
classical limit at high temperatures. Except for this classical limit, all our
results for spinor fields differ from the ones for scalar fields by constant
multiplicative factors that naturally arise because of the spinor degrees of
freedom. Finally, our analysis confirms that the entropy density agrees with
the Nernst heat theorem.
## VI Acknowledgments
J. V. would like to thank Fundação de Amparo a Ciência e Tecnologia do Estado
de Pernambuco (FACEPE), for their partial financial support. A. M.
acknowledges financial support from the Brazilian agencies Conselho Nacional
de Desenvolvimento Científico e Tecnológico (CNPq), grant no. 309368/2020-0
and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). H. M.
is partially supported by CNPq under grant no. 308049/2023-3. L. F.
acknowledges support from CAPES, financial code 001.
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|
# Fruit Picker Activity Recognition with Wearable Sensors and Machine Learning
††thanks: This work is a product of a collaboration with Grow Logic, who
provided the dataset, photographs, and the context to the problem. The work
was supported by the Systems program in Agriculture & Food, CSIRO.
Joel Janek Dabrowski∗ Data61, CSIRO
Brisbane, Australia
<EMAIL_ADDRESS>Ashfaqur Rahman Data61, CSIRO
Hobart, Australia
<EMAIL_ADDRESS>
###### Abstract
In this paper we present a novel application of detecting fruit picker
activities based on time series data generated from wearable sensors. During
harvesting, fruit pickers pick fruit into wearable bags and empty these bags
into harvesting bins located in the orchard. Once full, these bins are quickly
transported to a cooled pack house to improve the shelf life of picked fruits.
For farmers and managers, the knowledge of when a picker bag is emptied is
important for managing harvesting bins more effectively to minimise the time
the picked fruit is left out in the heat (resulting in reduced shelf life). We
propose a means to detect these bag-emptying events using human activity
recognition with wearable sensors and machine learning methods. We develop a
semi-supervised approach to labelling the data. A feature-based machine
learning ensemble model and a deep recurrent convolutional neural network are
developed and tested on a real-world dataset. When compared, the neural
network achieves 86% detection accuracy.
###### Index Terms:
human activity recognition, convolutional neural network, recurrent neural
network, deep learning, agriculture, time-series
## I Introduction
It is estimated that up to 25% of all fruit and vegetable produce go to waste
before ever leaving the farm, costing Australian farmers $\$2.84$ billion
annually [1]. Waste may be caused by pests, disease, weather events, or damage
inflicted during harvesting. In harvesting, fruit are picked and placed in
harvesting bins located throughout the paddocks or orchards, and transported
to a packing house for processing prior to market distribution. The picked
fruit are spoiled when they are left in the harvesting bins exposed to heat
for extended periods of time. Farmers and managers must thus ensure that the
pickers fill the bins quickly and that the bins are transported to the cooled
packing house as soon as they are full.
In general, the machine learning paradigm provides promising approaches to
address challenges in harvesting, however examples of such applications are
scarce in the literature [2]. This study contributes to this gap in the
literature and provides a means to monitor fruit pickers and the filling of
the harvesting bins to allow for more effective management of the harvesting
process.
SensorBinsBag Figure 1: Top: Photograph of a fruit picker, a picker bag,
sensor, and harvesting bin. The bin is mounted on a trailer, which is
typically pulled by a tractor. There are typically many bins scattered across
the orchard. Bottom: Photograph of bag-emptying events where two fruit pickers
empty the bags into a bin.
As illustrated in Figure 1, fruit pickers are equipped with a fruit picker
bag. Fruit are picked and placed into these bags, and once the bag is full,
the fruit is emptied into a harvesting bin. This is referred to as a bag-
emptying event. Detecting bag-emptying events provides the means to
simultaneously monitor the fruit picker productivity and bin levels. The bag-
emptying event rate is a direct measurement of the fruit picker productivity.
Furthermore, given the known capacity of a picker bag and the knowledge of
which bin the fruit is emptied into, bag-emptying events provide a measure of
the filling rate of the bins. With this knowledge, farmers are able to more
effectively manage pickers and bins.
Given recent success in the application of deep and machine learning
approaches to human activity recognition [3], we propose a novel application
of these approaches to detect bag-emptying events using wearable sensors. A
wearable sensor is placed on a picker’s bag strap as illustrated in Figure 1
to measure the picker’s movements and their proximity to the harvest bins.
Given the sensor measurements, bag-emptying events are detected using machine
learning models. We consider two models: a feature-based machine learning
ensemble classifier and a deep Recurrent Convolutional Neural Network (RCNN).
The models are compared on a dataset collected during avocado harvesting.
In addition to bag-emptying event detection, we address a data labelling
problem where the duration of bag-emptying events are unknown. For this, the
K-means clustering algorithm is used to perform semi-supervised labelling of
the data. Approximate times of when the events occurred and several features
of the data are used to learn the duration of bag-emptying events.
To our knowledge, fruit picker activity recognition based on time series data
from wearable sensors is a novel application. The main contributions of this
study include: (1) we use a wearable accelerometer sensor for measuring fruit
picker activity, (2) we used machine learning to understand fruit picker
productivity, and (3) we develop a new methodology that uses a combination of
semi-supervised labelling and supervised learning to detect bag-emptying
events. The key advantages of our approach is it does not interfere with the
harvesting process and it is autonomous.
This article begins with a discussion on related work in Section II. The
dataset and data labelling approaches are discussed in Sections III and IV
respectively. The models are described in Section V and our methodology is
provided in Section VI. Results are presented in Section VII and the article
is concluded in Section VIII.
## II Related Work
### II-A Machine Learning in Agriculture
Machine learning has been applied to various agricultural problems and several
literature surveys have been produced [2, 4, 5]. Most applications relate to
crop management and include yield prediction, disease detection, weed
detection, crop quality prediction, and water management. Literature on the
application of machine learning approaches specifically to harvesting and the
harvesting process are scarce.
### II-B Fruit Picker Productivity
In relation to fruit picker activity recognition, several studies have
considered tracking pickers for yield mapping, e.g., see [6, 7, 8, 9, 10].
Tracking pickers can be challenging due to Global Positioning System (GPS)
signal losses through foliage. Various alternatives to GPS have been thus been
proposed [11, 12]. However, even accurate picker tracking does not necessarily
provide any direct information on the picker’s activity. In our work, we are
directly measuring the fruit picker’s activity, which to our knowledge has not
been considered in the literature before.
A more direct approach to measuring picker productivity is to have a fixed
platform located at the bin containing some form of digital scale and a device
to identify the picker (such as RFID or a bar code scanner) [13, 14, 15, 16].
The picker identifies them self, weighs the picked fruit, and releases the
weighed fruit into the bin. Drawbacks of this approach include the additional
time that is required for weighing bags and the additional supervisors
required to ensure that the weighing processes is being conducted correctly.
Our approach does not interfere with the harvesting process and does not
require additional supervisors. Furthermore, our approach considers time-
series data rather than data at a single point in time.
### II-C Human Activity Recognition and Machine Learning
Detecting fruit picker bag-emptying events can be considered as a Human
Activity Recognition (HAR) problem. Generally, HAR involves using some form of
classifier to predict an activity given data from a sensor that directly
monitors human movement, such as wearable sensors. Surveys on HAR using
wearable sensors [17] deep learning approaches to HAR [3] have been conducted.
Applications with wearable sensors in agriculture include human–robot
interaction [18] and the assessment of vibration risk with agricultural
machinery [19]. To our knowledge HAR has not been applied to monitor fruit
pickers.
A wide range of wearable sensors exist [17] including: accelerometers, global
positioning systems (GPS), radio frequency identification (RFID),
environmental sensors (such as temperature), and physiological sensors (such
as heart rate monitors). A survey has been conducted on sensor positioning on
the body [20]. Sensor positions may include the waist, arms, wrist, ankle, and
the torso. The chest is suggested to be the preferred location for medium-
level activities such as walking and house-work. The actions in such
activities are similar to that of fruit picking. The chest was thus chosen as
the location for this study.
Many HAR models begin by processing the accelerometer data using a sliding
window [17]. Various features are extracted from the data in the window as it
is slid across the dataset. Features may include mean, standard deviation,
minimum, maximum, energy, main frequency component, root mean square of the
derivative, and correlation between axes [20, 21]. The features are fed into a
classifier, which classifies the activity type. Various classifiers such as
decision trees, neural networks, Bayesian models, Markov models, and
classifier ensembles have may be considered [17]. In this study, such a
feature-based ensemble classifier is compared with a deep neural network
model.
Feature selection and design can be a tedious task that typically requires
domain knowledge. Deep learning algorithms are often designed to be end-to-end
methods that take the raw input data and output a prediction. Features are
learned within the multiple layers of network. A survey of various deep
learning architectures that have been applied to sensor-based activity
detection problems has been conducted [22]. These architectures include the
convolutional neural network (CNN) and the recurrent neural network (RNN).
The CNN is a neural network that applies a convolution operation on the data
presented to its inputs [23]. CNNs exploit local interactions in the data and
provide scale invariant features [24]. In wearable sensor HAR problems,
several CNN based models have been proposed [25, 26, 24].
The RNN is a neural network that is designed for sequential applications [23,
27]. It contains a neural network that is replicated over time where the
replications are sequentially connected. Like the CNN, the RNN has been also
applied to several HAR problems [28, 29, 30].
RNN and CNN models have been compared on HAR tasks [31]. It is found that RNNs
perform well for activities where long term dependency is required, such as
opening a door. CNNs perform well when long-term dependencies are not required
such as gait analysis. Given that RNNs and CNNs each have their own
advantages, combining the two architectures may provide a more widely
applicable and more powerful model. The combination of the architectures forms
a recurrent convolutional neural network (RCNN) and provide a promising
framework for HAR [32, 33], and is an approach we consider in this study.
## III Dataset
### III-A Sensors and Data Collection
The Haltian Nexus Prototype sensors were used in the wearable sensor. These
sensors comprise a ST LIS2DH accelerometer, a Nordic nRF52832 Wirepas radio
gateway module, and a data logger. Accelerometer readings for 3 axes were
logged at a frequency of 50Hz with a measurement range of $\pm$4 G and
sensitivity of 8 mG. The radio logged its Received Signal Strength Indicator
(RSSI) value with respect to a Haltian Thingsee POD2 Prototype node every
second. The POD2 node was located at the picker’s bin.
### III-B Avocado Farm Trial Dataset
A trial was conducted on an avocado farm. The data was acquired for two
different pickers over several hours. A sensor was attached to each pickers
bag strap and positioned on the picker’s chest as illustrated in Figure 1(a).
Bag-emptying event times were manually recorded by a human observer. The
dataset comprises 580986 samples of data with 64 bag-emptying events.
Each of the 3 data streams from the 3-axis accelerometer are filtered with a
bandpass filter. The high-cut frequency is set to reduce aliasing. The low-cut
frequency is set to remove any offsets caused by gravitational effects. The
three filtered accelerometer data streams and the RSSI data stream are
combined to form a dataset of four data streams, which are scaled to a range
of $[0,1]$.
### III-C Dataset Balancing
The dataset is imbalanced where only $28\%$ of the samples are associated with
bag-emptying events. This imbalance can create an undesirable bias in the
classifiers. Resampling is used to form a balanced dataset. Sequences of
samples are extracted to ensure the sequential nature of the data is
maintained. To introduce some form of randomness in the sampling, the length
of the extracted sequence is selected according to a normal distribution. The
mean length $\mu$ and variance $\sigma$ of the bag-emptying event sequences
are calculated from the sequence length of the manual label bag-emptying
events. For each bag-emptying event sequence, only the
$n\in\mathcal{N}(\mu,\sigma)$ preceding non-bag-emptying event samples are
preserved. All remaining non-bag-emptying event samples are removed. The
result is a dataset with sequence lengths that are similarly distributed
between classes.
## IV Bag-Emptying Event Labelling
The supervised machine learning algorithms require labels of the bag-emptying
events for training. Although the times that the bag-emptying events occurred
were recorded, the duration of the events were not recorded. Furthermore,
owing to human error, the recoded bag-emptying times are only considered to be
approximations. The bag-emptying event times are thus required to be refined
and the bag-emptying event durations are required to be determined. For this,
we consider two approaches: (1) a manual labelling approach using expert
knowledge and (2) a semi-supervised approach based on K-means clustering.
### IV-A Manual Labelling of Bag-Emptying Events
To manually determine the bag-emptying event times and durations, the
following bag-drop process is noted: (1) A pickers gait changes under the
strain of a full bag as they walk from the trees to the bin; (2) the picker
lifts the bag into the bin; (3) a flap at the bottom of the bag is opened to
release the fruit (e.g. see Figure 1); (4) the bag is shaken and tugged to
empty it; and (5) once the bag is empty, it is removed from the bin and the
bottom flap is reattached.
The scaled accelerometer and RSSI data surrounding a bag-emptying event are
illustrated in Figure 2. The plot indicates a change in the dynamics as the
picker transitions from normal picking activity to the bag-emptying event. The
RSSI increases as the picker approaches the bin and the accelerometer signal
level increases due to gait change. A spike in the accelerometer data occurs
as the picker lifts the bag into the bin (this spike is not evident in every
bag-emptying event). The accelerometer signal level remains high as the bag is
shaken, removed and reassembled. The RSSI and the accelerometer signal
decrease as the picker returns to fruit picking activities. Based on these
observations the bag-emptying times were manually refined and the duration of
the bag-emptying events were defined. The average bag-emptying event duration
is 50 seconds with the emptying of the bag typically lasting between 10 and 20
seconds.
Figure 2: Plot of the sensor data with a bag-emptying event. Sensor data is
plotted in blue. The manually defined bag-emptying event is plotted in orange
with a value of 1 indicating a bad drop event.
### IV-B Semi-supervised labelling of Bag-Emptying Events
The K-means clustering algorithm is used to perform semi-supervised learning
of the bag-emptying event labels111The K-means algorithm is selected due to
its computational efficiency, however more complex clustering algorithms could
also be considered.. The approach is to pre-define the duration of bag-
emptying events around the logged bag-emptying times. The K-means algorithm is
then used to refine the duration by clustering samples according to
statistical features of bag-emptying event and non-bag-emptying event data.
Data sample labels are initialised by defining the duration of all bag-
emptying events to be 1200 samples in length, with 500 samples before and 700
samples after the manually logged bag-drop event time. Samples within this
window are associated with bag-emptying events and samples outside this window
are associated with normal fruit picking activity.
A 256-sample sliding window is shifted sample-by-sample over the data
sequences. For each shift, a set of statistical features are extracted to
produce a feature vector associated with each sample. This feature vector
comprises the mean, standard deviation, minimum, maximum, and standard
deviation of the derivative for the associated sample. Additionally, to
capture the sequential structure of the data, an indicator is included to
specify whether the neighbouring samples labels are associated with bag-
emptying events.
K-means algorithm is initialised by grouping the feature vectors into two
clusters according to the initial bag-emptying event labels. The mean values
of each cluster form the initial values for the K-means clustering algorithm.
These clusters are refined using the K-means algorithm, which corresponds to
refining the labels of each data sample. The algorithm was typically run over
10 iterations, where the number of iterations can affect how much the clusters
change from the predefined settings.
The K-means clustering algorithm can produce false positives and false
negatives, where false positives are spurious bag-emptying events predicted
far from the logged time and false negatives are spurious non-bag-emptying
events predicted near logged bag-emptying event times.
The results of the K-means algorithm are thus filtered using the intersection
and union operators of mathematical set theory. Predicted bag-emptying event
sequences are compared with the predefined bag-emptying event sequence as
illustrated in Figure 3. To reduce false positives an intersection operator is
used. If none of the samples in the predicted sequence overlap with the
predefined sequence, the prediction is rejected. As illustrated in the bottom
plot of Figure 3, the predictions at the beginning and end of the sequence are
removed as they do not overlap with the predefined bag-emptying event
sequence. To reduce false negatives, a union operation is applied between the
predicted and predefined sequences. As illustrated in the bottom plot of
Figure 3, the false negative is removed. Note that this operator is only
applied to join predicted bag-emptying events. It is not permitted to extend
the predicted bag-emptying event duration. This operation can however result
in an unreasonably long bag-emptying event sequence if it joins two long
sequences of positive samples. If a predicted sequence is longer than twice
the predefined bag-emptying event sequence, it is rejected and the predefined
bag-emptying event sequence is used instead.
Figure 3: Plots of the process of semi-supervised bag-emptying event
labelling. The predefined labels, the K-means clustering results, and the
filtered K-means clustering results are plotted in the top, middle and bottom
figures respectively.
The overall algorithm for the semi-supervised labelling approach is presented
in Algorithm 1.
Algorithm 1 Semi-supervised Labelling of bag-emptying events using the K-means
algorithm.
0: The dataset $X$, a vector of bag-emptying event start times
$t^{\text{start}}$, a vector of bag-emptying event end times $t^{\text{end}}$,
and the dataset labels $Y$.
1: Define the window size $q=256$
2: Define an empty set of dataset labels $\hat{Y}=\varnothing$
3: for each bag-emptying event, $i$ do
4: Extract dataset sequence surrounding the $i^{\text{th}}$ event
$A=X_{t^{\text{end}}_{i-1}:t^{\text{start}}_{i+1}}$
5: Extract the labels associated with $A$
$B=Y_{t^{\text{end}}_{i-1}:t^{\text{start}}_{i+1}}$
6: for each index $j$ of sample in $A$ do
7: Extract the window of data associated with sample $j$ $W=A_{j:j+q}$
8: Compute the feature vector for sample $j$ $f_{j}=\text{features}(W)$
9: Set the class of sample $j$ according to the window
$c_{j}=\begin{cases}0&\frac{1}{q}\sum_{k=1}^{q}B_{j+k}<0.5\\\
1&\frac{1}{q}\sum_{k=1}^{q}B_{j+k}\geq 0.5\end{cases}$
10: end for
11: Update the classes using the K-means algorithm
$c\leftarrow\text{kmeans}(f,c)$
12: Filter the labels using mathematical set theory
$c\leftarrow\text{filter}(c,B)$
13: Append the filtered labels $c$ to the new label set $\hat{Y}$.
14: end for
15: return The updated labels $\hat{Y}$.
## V Models
Detecting a bag-emptying event from the wearable sensor data is a challenging
task. As described in Section IV, the events comprises various sub-activities
involved in the bag-emptying event. Furthermore, the bag-drop signals may vary
according to pickers and the environment. The models are required to handle
these variations.
### V-A Feature Based Ensemble Model
RSSIAcc. ZAcc. YAcc. XFeature vector (std., energy, RMS($dx/dt$),
RMS($d^{2}x/dt^{2}$), mean($dx/dt$), mean($d^{2}x/dt^{2}$), min, max)Naive
BayesANN (512)$\sum$$60\%$$40\%$Input dataFeature vectorEnsembleClass Figure
4: Architecture of the ensemble model. A feature vector is assembled from a
window of data. A naive Bayes and an ANN perform a classification given the
feature vector. The outputs of these classifiers are weighted and summed to
determine the class.
A traditional feature-based ensemble model is applied for detecting bag-
emptying events. The model is illustrated in Figure 4. A 256 sample sliding
window with zero overlap is applied to the data. The following features are
extracted in each window: standard deviation, energy, the RMS first and second
derivative, mean first and second derivative, minimum value, and maximum value
[21, 20].
The features are provided as inputs to an ensemble classifier comprising a
Gaussian Naive Bayes classifier and a neural network. These are two commonly
used classifiers in HAR [17] and are sufficiently different from each other to
provide diversity in the ensemble. The neural network comprises a single
hidden layer with 512 neurons with hyperbolic tangent activation functions.
The ADAM algorithm [34] is used to train the neural network. The parameters of
the Gaussian naive Bayes classifier are estimated using maximum likelihood.
Both classifiers output a prediction in the form of a probability of a bag-
emptying event. These predictions are combined in the ensemble through a
weighted summation. The Naive Bayes classifier is weighted with 60% of the
vote (which was determined through cross validation).
### V-B Recurrent Convolutional Neural Network (RCNN)
The architecture of the RCNN used in this study is illustrated in Figure 5 and
is based on the models presented in [32] and [33]. For each data stream, a set
of samples are extracted using a 256 sample sliding window with zero overlap.
The windows of samples from the 4 sensor streams are combined into a tensor,
where each stream represents channel for the input of the CNN portion of the
model. A key advantage of the RCNN over the ensemble model is that it is an
end-to-end model and does not require feature engineering.
The CNN portion of the RCNN performs feature extraction and the RNN portion of
the RCNN models temporal dynamics over the sample windows. The RCNN outputs
the class of the input window sample. The rectified linear unit (ReLU) is used
as the activation function in hidden layers, the filter and layer sizes in the
network were determined through trial and error, and the ADAM algorithm [34]
is used for training.
RSSIAcc. ZAcc. YAcc. XConv. 1 (filters = 32, kernel size = 32)Pool 1 (pool
size = 8, stride = 4)Conv. 2 (filters = 16, kernel size = 16)Pool 2 (pool size
= 4, stride = 2)Dense (512 neurons)Input tensorCNNRNNOutputs Figure 5:
Architecture of the RCNN model for a single sequence sample (based on [32] and
[33]). The input contains a tensor comprising a window of samples from each
data stream. The CNN comprises two convolutional layers, two pooling layers,
and a densely (fully) connected neural network. The CNN output is passed to an
LSTM cell of the RNN. The LSTM cell output is passed to a sigmoidal neuron
which outputs the class of the input window sample. The LSTM models temporal
dynamics over sample windows.
## VI Methodology
Both the models use a sliding window with zero overlap. The window of data is
presented to the classifier. The set of data in the window is classified to
belong either to a bag-emptying event or a non-bag-drop event. Each data
sample within the window is associated with this predicted class. During
training, a window may be slid to a position where it partially falls within a
bag-emptying event. The ground-truth class label for the window is calculated
as the average true class of all the samples in the window.
To validate models and results, a 6-fold cross validation test is performed.
The dataset is split into 6 equal data segments. Each data segment is a
continuous time series of 96831 samples. The model is trained on 5 of the data
segments and tested on the remaining segment. This is repeated such that the
model is tested on each data segment of the dataset.
Accuracy, precision, recall, and F-score are used to measure the performance
of the classifiers presented in this study. Accuracy describes the ratio of
the number of correct classifications to the total number of classified
samples. Precision describes the ratio of correct classifications to the total
number of classifications made for the particular class. It considers the
number of incorrectly predicted bag-emptying events and thus provides a
measure of the classifier quality. Recall describes the ratio of correct
classifications to the total number of items which truly belong to the
predicted class. It considers the number of bag-emptying event samples that
were missed. Recall thus provides a measure of the probability of correctly
classifying the bag-emptying event. Finally, the F-score is defined as the
harmonic mean between the precision and recall. It provides a measure to
describe both the precision and recall together.
## VII Results
### VII-A Comparison of Labelling Approaches
The median value of the accuracy, precision, recall, and F-score across the
6-fold cross validations for the predefined labels, manually defined labels,
and the learned labels are provided in Table I.
TABLE I: Feature-based ensemble model and RCNN model median value results for the predefined labels, manually defined labels, and the learned labels. Model | Measure | Predefined | Manual | Learned
---|---|---|---|---
Ensemble | Accuracy | 74% | 80% | 80%
Precision | 81% | 89% | 83%
Recall | 63% | 66% | 71%
F-score | 71% | 77% | 77%
RCNN | Accuracy | 79% | 76% | 86%
Precision | 86% | 80% | 89%
Recall | 71% | 72% | 83%
F-score | 77% | 75% | 84%
The ensemble model produces the poorest results with the predefined labels.
Improved performance is obtained with the manually and learned labels. The
performance for the manually and learned labels are similar. This is a key
result as it validates the proposed approach to learning the dataset labels.
The manual labels produce a higher precision than the learned labels. However,
the learned labels provide a higher recall than the manual labels. The recall
is considered to a more important measure as it relates to the probability of
detecting bag-emptying events.
(a) Ensemble model. (b) RCNN model.
Figure 6: Box whisker plot of the cross validation results.
The best results for the RCNN are obtained with the learned labels. This
reinforces the validation of the semi-supervised label learning approach. The
RCNN produces results that are much higher than those produced by the ensemble
model. The ensemble model’s precision for the manual labels matches the
capability of the RCNN. This however is achieved at the cost of a low recall
value of 66%. Along with the high precision value, the RCNN produces a recall
of 83%. These comparisons are reiterated by the 7% difference in F-score
results between the two models.
### VII-B Results with the Learned Labels
A box whisker plot of the results over the 6-fold cross validation test are
presented in Figure 6. The RCNN performs better than the ensemble model for
all performance measures. The precision box of the RCNN reaches high values.
However, the box is the largest of all boxes in the diagram. A larger box
indicates that there is more variability in the precision results. The quality
of the model is however still high considering that the range of the box
remains above 80%. The ensemble model’s recall box is large with whiskers that
extend to low values. This indicates a high level of uncertainty in the
ensemble models recall results. The recall box of the RCNN is narrow
indicating high certainty in the RCNN recall results. The range of the box
remains above 80% indicating good prediction ability of the RCNN model. Both
models produce narrow F-score boxes. The range of the RCNN F-score box is
higher than the ensemble model indicating superior performance overall.
A plot of the predictions and data for one of the cross validation folds is
presented in Figure 7. The learned labels correspond well with the changes in
the accelerometer and RSSI data. The predictions of the ensemble model are
sporadic resulting in several false positives and false negatives. The RCNN
model predictions are smoother over time. The RCNN however misses the third
bag-emptying event. This is possibly due to the RSSI level unexpectedly
dropping to a minimum during this bag-emptying event. Unlike the RCNN, the
ensemble model is able to detect the third bag-emptying event. This seems to
indicate that the RCNN relies more on RSSI data and the ensemble model relies
more on accelerometer data. Note that the last bag-emptying event is not
missed by the RCNN. It is detected in the following cross validation fold.
(a) Ensemble model. (b) RCNN model.
Figure 7: bag-emptying event predictions and dataset for the convolutional
recurrent neural network. The blue curves plot the dataset. The orange curve
the top figure plots the model predictions. A value of 1 indicates a bag-
emptying event.
The plots of the predictions for the remaining cross validation folds are
illustrated in Figure 8. The ensemble model results are more sporadic and the
predictions are more confident. The RCNN is less confident and the predictions
are smoother over time. This is preferred when false positives have a high
risk.
(a) (b)
(c) (d)
(e) (f)
(g) (h)
(i) (j)
Figure 8: Bag-emptying event predictions for the second to sixth cross
validation folds along the rows. Plots of the ensemble model are presented in
the left column and plots of the RCNN model are presented in the right column.
The horizontal axis is the sample number. The vertical axis is the probability
of a bag-emptying event. The blue curve plots the manual bag-emptying events
and orange curve plots the predicted bag-emptying events.
## VIII Discussion and Conclusion
In this study, we present a novel application on measuring fruit picker
productivity. The picker productivity is measured by detecting bag-emptying
events from wearable sensor data. A traditional feature-based ensemble model
and a deep convolutional recurrent neural network are applied to predict bag-
emptying events from the wearable sensor data. Furthermore, a semi-supervised
method for learning the bag-emptying event labels is presented.
Results indicate that both models are able to successfully detect the bag-
emptying events. The RCNN model is more accurate than the ensemble model but
it is less confident, which results in 3 of the 64 bag-emptying events being
missed. The ensemble model has at least one positive detection within each
bag-emptying event suggesting that all bag-emptying events. As the ensemble
model does not directly model any temporal relationship between samples, its
predictions are noisy resulting in multiple detections for a single bag-
emptying event. The RCNN models temporal dynamics to produces smooth
predictions over time and more accurate predictions of bag-emptying event
durations.
In future work, the RCNN could be improved by increasing its capacity and
training it on more data. The capacity of the RCNN can be increased by
introducing more CNN filters and by increasing the CNN depth. Such
improvements may provide the model with the capability to learn more advanced
features. The RNN can be improved by adding multiple layers and by using bi-
directional RNNs. More complex models however generally require more data.
Collecting more data is thus a priority for future work. Other than improving
models, the data captured by the sensors provide information relating to other
problems such as health and safety. For example, the accelerometer sensor can
be used to detect excessive bag weight or falls.
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|
$\displaystyle\frac{1}{1+\frac{1}{\sigma_{t-1}^{2}}\mathcal{D}_{\mathcal{F}}^{2}(X_{t-1};X_{[t-2]},\sigma_{[t-2]})}\sup_{f_{1},f_{2}\in\mathcal{F}}\frac{(f_{1}(X_{t})-f_{2}(X_{t}))^{2}}{\sum_{s=1}^{t-2}\frac{1}{\sigma_{s}^{2}}(f_{1}(X_{s})-f_{2}(X_{s}))^{2}+\lambda}$
$\displaystyle={}$
$\displaystyle\frac{1}{1+\frac{1}{\sigma_{t-1}^{2}}\mathcal{D}_{\mathcal{F}}^{2}(X_{t-1};X_{[t-2]},\sigma_{[t-2]})}\mathcal{D}_{\mathcal{F}}^{2}(X_{t};X_{[t-2]},\sigma_{[t-2]})\geq\dots$
$\displaystyle\geq{}$
$\displaystyle\frac{1}{\prod_{s=t_{0}+1}^{t-1}(1+\frac{1}{\sigma_{s}^{2}}\mathcal{D}_{\mathcal{F}}^{2}(X_{s};X_{[s-1]},\sigma_{[s-1]}))}\mathcal{D}_{\mathcal{F}}^{2}(X_{t};X_{[t_{0}]},\sigma_{[t_{0}]}),$
where the first inequality holds due to
$\displaystyle\sum_{s=1}^{t-1}\frac{1}{\sigma_{s}^{2}}(f_{1}(X_{s})-f_{2}(X_{s}))^{2}$
$\displaystyle={}$
$\displaystyle\sum_{s=1}^{t-2}\frac{1}{\sigma_{s}^{2}}(f_{1}(X_{s})-f_{2}(X_{s}))^{2}+\frac{1}{\sigma_{t-1}^{2}}(f_{1}(X_{t-1})-f_{2}(X_{t-1}))^{2}$
$\displaystyle\leq{}$
$\displaystyle\sum_{s=1}^{t-2}\frac{1}{\sigma_{s}^{2}}(f_{1}(X_{s})-f_{2}(X_{s}))^{2}+\frac{1}{\sigma_{t-1}^{2}}\mathcal{D}_{\mathcal{F}}^{2}(X_{t-1};X_{[t-2]},\sigma_{[t-2]})\sum_{s=1}^{t-2}\frac{1}{\sigma_{s}^{2}}(f_{1}(X_{s})-f_{2}(X_{s}))^{2}$
$\displaystyle={}$
$\displaystyle\left(1+\frac{1}{\sigma_{t-1}^{2}}\mathcal{D}_{\mathcal{F}}^{2}(X_{t-1};X_{[t-2]},\sigma_{[t-2]})\right)\sum_{s=1}^{t-2}\frac{1}{\sigma_{s}^{2}}(f_{1}(X_{s})-f_{2}(X_{s}))^{2}.$
Thus, we have
$\displaystyle\mathcal{D}_{\mathcal{F}}^{2}(X_{t};X_{[t_{0}]},\sigma_{[t_{0}]})$
$\displaystyle\leq\prod_{s=t_{0}+1}^{t-1}\left(1+\frac{1}{\sigma_{s}^{2}}\mathcal{D}_{\mathcal{F}}^{2}(X_{s};X_{[s-1]},\sigma_{[s-1]})\right)\mathcal{D}_{\mathcal{F}}^{2}(X_{t};X_{[t-1]},\sigma_{[t-1]})$
$\displaystyle\leq\exp\left\\{\sum_{s=t_{0}+1}^{t-1}\frac{1}{\sigma_{s}^{2}}\mathcal{D}_{\mathcal{F}}^{2}(X_{s};X_{[s-1]},\sigma_{[s-1]})\right\\}\mathcal{D}_{\mathcal{F}}^{2}(X_{t};X_{[t-1]},\sigma_{[t-1]}),$
where the second inequality holds due to the inequality $1+x\leq\exp\\{x\\}$.
∎
###### Lemma H.5.
Let $\\{\sigma_{t},\beta_{t}\\}_{t\geq 1}$ be a sequence of non-negative
numbers, $\sigma_{\mathrm{min}},\gamma,\lambda>0$, $\\{X_{t}\\}_{t\geq
1}\subset\mathcal{X}$ and $\\{\bar{\sigma}_{k}\\}_{k\geq 1}$ be recursively
defined:
$\bar{\sigma}_{t}^{2}=\max\\{\sigma_{t}^{2},\sigma_{\mathrm{min}}^{2},\gamma^{2}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})\\}.$
Then we have
$\sum_{t=1}^{T}\min\\{1,\beta_{t}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})\\}\leq\dim_{\mathcal{F}}+\max_{t\in[T]}\beta_{t}\cdot\gamma^{2}\dim_{\mathcal{F}}+\sqrt{\dim_{\mathcal{F}}}\cdot\sqrt{\sum_{t=1}^{T}\beta_{t}^{2}(\sigma_{t}^{2}+\sigma_{\mathrm{min}}^{2})},$
where $\mathcal{D}_{\mathcal{F}}$ and
$\dim_{\mathcal{F}}=\dim_{\mathcal{F}}(\sigma_{\mathrm{min}},T)$ are in
Definition 3.2.
###### Proof.
We decompose $[T]$ as the union of three disjoint sets
$\mathcal{J}_{1},\mathcal{J}_{2},\mathcal{J}_{3}$:
$\displaystyle\mathcal{J}_{1}$
$\displaystyle=\left\\{t\in[T]\Big{|}\frac{1}{\bar{\sigma}_{t}}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})>1\right\\},$
$\displaystyle\mathcal{J}_{2}$
$\displaystyle=\left\\{t\in[T]\Big{|}\frac{1}{\bar{\sigma}_{t}}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})\leq
1,\bar{\sigma}_{t}\in\\{\sigma_{t},\sigma_{\mathrm{min}}\\}\right\\},$
$\displaystyle\mathcal{J}_{3}$
$\displaystyle=\left\\{t\in[T]\Big{|}\frac{1}{\bar{\sigma}_{t}}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})\leq
1,\bar{\sigma}_{t}=\gamma\sqrt{\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})}\right\\}.$
For the summation over $\mathcal{J}_{1}$, we have
$\sum_{t\in\mathcal{J}_{1}}\min\\{1,\beta_{t}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})\\}\leq|\mathcal{J}_{1}|\leq\sum_{t\in\mathcal{J}_{1}}\min\left\\{1,\frac{1}{\bar{\sigma}_{t}^{2}}\mathcal{D}_{\mathcal{F}}^{2}(X_{t};X_{[t-1]},\sigma_{[t-1]})\right\\}\leq\dim_{\mathcal{F}}.$
Next, for the summation over $\mathcal{J}_{2}$, we have
$\displaystyle\sum_{t\in\mathcal{J}_{2}}\min\\{1,\beta_{t}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})\\}$
$\displaystyle\leq{}$
$\displaystyle\sum_{t\in\mathcal{J}_{2}}\beta_{t}\bar{\sigma}_{t}\cdot\frac{1}{\bar{\sigma}_{t}}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})\\}$
$\displaystyle\leq{}$
$\displaystyle\sum_{t=1}^{T}\beta_{t}\max\\{\sigma_{t},\sigma_{\mathrm{min}}\\}\frac{1}{\bar{\sigma}_{t}}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})\\}$
$\displaystyle\overset{(a)}{\leq}{}$
$\displaystyle\sqrt{\sum_{t=1}^{T}\beta_{t}^{2}(\sigma_{t}^{2}+\sigma_{\mathrm{min}}^{2})}\cdot\sqrt{\sum_{t=1}^{T}\min\left\\{1,\frac{1}{\bar{\sigma}_{t}^{2}}\mathcal{D}_{\mathcal{F}}^{2}(X_{t};X_{[t-1]},\sigma_{[t-1]})\right\\}}$
$\displaystyle\leq{}$
$\displaystyle\sqrt{\dim_{\mathcal{F}}}\cdot\sqrt{\sum_{t=1}^{T}\beta_{t}^{2}(\sigma_{t}^{2}+\sigma_{\mathrm{min}}^{2})},$
where $(a)$ holds due to Cauchy-Schwartz inequality and
$\frac{1}{\bar{\sigma}_{t}}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})\leq
1$. Then, for the summation over $\mathcal{J}_{3}$, we have
$\displaystyle\sum_{t\in\mathcal{J}_{3}}\min\\{1,\beta_{t}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})\\}$
$\displaystyle\leq{}$
$\displaystyle\sum_{t\in\mathcal{J}_{3}}\beta_{t}\bar{\sigma}_{t}\cdot\frac{1}{\bar{\sigma}_{t}}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})$
$\displaystyle\overset{(a)}{\leq}{}$
$\displaystyle\max_{t\in[T]}\beta_{t}\cdot\gamma^{2}\sum_{t=1}^{T}\min\left\\{1,\frac{1}{\bar{\sigma}_{t}^{2}}\mathcal{D}_{\mathcal{F}}^{2}(X_{t};X_{[t-1]},\sigma_{[t-1]})\right\\}$
$\displaystyle\leq{}$
$\displaystyle\max_{t\in[T]}\beta_{t}\cdot\gamma^{2}\dim_{\mathcal{F}},$
where $(a)$ holds due to
$\bar{\sigma}_{t}=\gamma^{2}\frac{1}{\bar{\sigma}_{t}}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})$
and
$\frac{1}{\bar{\sigma}_{t}}\mathcal{D}_{\mathcal{F}}(X_{t};X_{[t-1]},\sigma_{[t-1]})\leq
1$. Finally, putting pieces together finishes the proof. ∎
###### Lemma H.6 (Modified from Lemma 2 in Zhang et al. 2021a).
Let $\lambda_{1},\lambda_{2},\lambda_{4}>0$, $\lambda_{3}\geq 1$ and
$i^{\prime}=\lceil\log_{2}\lambda_{1}\rceil$. Let
$a_{0},a_{1},a_{2},\dots,a_{i^{\prime}}$ be non-negative reals such that
$a_{i}\leq\lambda_{1}$ for any $0\leq i\leq i^{\prime}$, and
$a_{i}\leq\lambda_{2}\sqrt{a_{i+1}+2^{i+1}\cdot\lambda_{3}}+\lambda_{4}$ for
any $0\leq i<i^{\prime}$. Then we have
$\displaystyle a_{0}$
$\displaystyle\leq\max\left\\{\left(\lambda_{2}+\sqrt{\lambda_{2}^{2}+\lambda_{4}}\right)^{2},\lambda_{2}\sqrt{4\lambda_{3}}+\lambda_{4}\right\\}\leq\lambda_{2}\sqrt{4\lambda_{3}}+4\lambda_{2}^{2}+3\lambda_{4},$
$\displaystyle a_{1}$
$\displaystyle\leq\max\left\\{\left(\lambda_{2}+\sqrt{\lambda_{2}^{2}+\lambda_{4}}\right)^{2},\lambda_{2}\sqrt{8\lambda_{3}}+\lambda_{4}\right\\}\leq\lambda_{2}\sqrt{8\lambda_{3}}+4\lambda_{2}^{2}+3\lambda_{4}.$
|
# (N)$\text{NLO}+\text{NLL}^{\prime}$ accurate predictions for plain and
groomed 1-jettiness in neutral current DIS
Max Knobbe111Email<EMAIL_ADDRESS>Institut für Theoretische
Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077
Göttingen, Germany Daniel Reichelt222Email<EMAIL_ADDRESS>Institute for Particle Physics Phenomenology, Department of Physics, Durham
University, Durham DH1 3LE, United Kingdom Steffen Schumann555Email:
<EMAIL_ADDRESS>Institut für Theoretische Physik,
Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen,
Germany
###### Abstract
The possibility to reanalyse data taken by the HERA experiments offers the
chance to study modern QCD jet and event-shape observables in deep-inelastic
scattering. To address this, we compute resummed and matched predictions for
the 1-jettiness distribution in neutral current DIS with and without grooming
the hadronic final state using the soft-drop technique. Our theoretical
predictions also account for non-perturbative corrections from hadronisation
through parton-to-hadron level transfer matrices extracted from dedicated
Monte Carlo simulations with S HERPA . To estimate parameter uncertainties in
particular for the beam-fragmentation modelling we derive a family of replica
tunes to data from the HERA experiments. While NNLO QCD normalisation
corrections to the NLO+NLL’ prediction are numerically small, hadronisation
corrections turn out to be quite sizeable. However, soft-drop grooming
significantly reduces the impact of non-perturbative contributions. We
supplement our study with hadron-level predictions from S HERPA based on the
matching of NLO QCD matrix elements with the parton shower. Good agreement
between the predictions from the two calculational methods is observed.
###### Contents
1. 1 Introduction
2. 2 Phase space and observable definition
3. 3 DIS Monte Carlo simulations with S HERPA
1. 3.1 MEPS@NLO predictions for DIS
2. 3.2 Tuning the beam fragmentation model against HERA data
4. 4 (N)$\text{NLO}+\text{NLL}^{\prime}$ resummation for 1-jettiness in DIS
1. 4.1 NLL resummation in the C AESAR approach
2. 4.2 Grooming in DIS
3. 4.3 Calculational tools and setup
5. 5 Results for (groomed) 1-jettiness in DIS
6. 6 Conclusions
7. A Tuning details
## 1 Introduction
Event shape observables offer great potential for detailed studies of the
intriguing dynamics of Quantum Chromodynamics (QCD), thereby providing insight
into various strong interaction phenomena. For example, they offer sensitivity
to the strong coupling constant $\alpha_{S}$, the colour charges of the QCD
quanta, and parton density functions, when considering hadronic initial state
particles. Predictions for event shape distributions can be obtained from
fixed-order perturbation theory, all-orders resummation of logarithmically
enhanced contributions, as well as detailed particle-level simulations as
provided by Monte Carlo event generators. Accordingly, they form a rather
unique testbed for a variety of theoretical approaches, ranging from cutting-
edge multi-loop calculations to detailed aspects in the modelling of the non-
perturbative parton-to-hadron transition.
Event shapes have played a central role in the QCD measurement program of past
$e^{+}e^{-}$ collider experiments, see for instance [1, 2, 3, 4, 5]. Also at
hadron–hadron machines they are considered in studies of hadronic final
states. Possibly even more prominently, closely related jet-substructure
observables have attracted enormous attention and sparked the development of
modern grooming and tagging techniques, see Ref. [6] for a recent review. Also
in deep-inelastic lepton–nucleon scattering experiments several event shape
variables have been measured [7, 8, 9, 10, 11, 12]. However, the LEP and HERA
experiments phased out in the years 2000 and 2007, respectively, such that
later breakthroughs in calculational methods and modern observable definitions
have not yet been fully exploited.
Their complementarity and partially reduced complexity when compared to
present day LHC measurements, make the LEP and HERA data a real treasure for
additional tests of our theoretical understanding and simulation capabilities.
In the past years a small number of re-analyses of the LEP data have been
published, see for instance [13, 14, 15, 16]. Furthermore, there are efforts
to provide open data sets that can directly be used by the entire community
[17, 18].
To open the treasure chest of their large data set for modern QCD studies the
HERA H1 collaboration has recently started to publish a series of new,
fascinating measurements that allow one to confront contemporary state-of-the-
art predictions with precise DIS data. Besides their relevance for
benchmarking our present day tools, such analyses build an important stepping
stone towards future electron–hadron colliders like the EIC at BNL [19, 20] or
the LHeC at CERN [21, 22].
We here compile predictions for the 1-jettiness event shape in the Breit frame
[23], that is equivalent to the well known thrust variable [24], for the HERA
kinematics, _i.e._ lepton–proton collisions at $\sqrt{s}=319\;\text{GeV}$.
Furthermore, we consider grooming of the hadronic final states based on the
soft-drop method prior to the observable evaluation. We derive differential
distributions for groomed and ungroomed $\tau^{b}_{1}$ differential in the
photon virtuality $Q^{2}\in[150,20000]\;\text{GeV}^{2}$, and the events
inelasticity $y\in[0.05,0.94]$. We perform Monte Carlo simulations with the S
HERPA generator based on next-to-leading-order (NLO) matrix elements for the
one- and two-jet final states matched to the parton shower and hadronised
using S HERPA ’s new cluster fragmentation model [25]. To estimate the
hadronisation modelling uncertainties in particular related to the beam
remnant fragmentation we derive a set of replica tunes [26] to a selection of
DIS measurements from the H1 and ZEUS experiments.
Furthermore, we compute resummed predictions at next-to-leading-logarithmic
(NLL) accuracy in the observable value based on the implementation of the C
AESAR resummation formalism [27] in the S HERPA framework [28]. These get
matched to the NNLO QCD result for the inclusive DIS process and the NLO
matrix elements for the two-jet channel. For the NNLO QCD corrections we rely
on an implementation in S HERPA presented in [29]. This results in
predictions of $\text{NLO}+\text{NLL}^{\prime}$ accuracy for the actual event-
shape distributions, while we achieve NNLO precision for the total event rate.
In consequence, we refer to our predictions as being
(N)$\text{NLO}+\text{NLL}^{\prime}$ accurate. To account for non-perturbative
corrections we derive parton-to-hadron level transfer matrices differential in
the event shape variables that we extract from particle level simulations with
S HERPA [30], thereby also accounting for the cluster-model parameter
uncertainties through the set of replica tunes to HERA data.
Our calculations are targeted on an upcoming measurement by the H1 experiment,
for that preliminary results have recently been presented [31, 32]. Results
based on simulations with S HERPA in a similar fiducial phase space have been
compared to data from jet-substructure observables in neutral current DIS in
[33]. Our study extends earlier work on the simulation of DIS events with S
HERPA [34]. Furthermore, this is the first time we include NNLO QCD correction
in resummation calculations with S HERPA .
The article is organised as follows: in Sec. 2 we introduce the considered
observables and define the fiducial phase space used in our study of the
hadronic final states produced in $ep$ collisions at HERA. In Sec. 3 we
describe the setup used to simulate DIS events with S HERPA as well as the
tuning of its beam-fragmentation parameters. In Sec. 4 we present our
framework to compile (N)$\text{NLO}+\text{NLL}^{\prime}$ predictions, based on
the implementation of the C AESAR formalism in S HERPA . Here, we also
present our approach to treat non-perturbative corrections based on transfer
matrices extracted from MC simulations, see Sec. 4.1. We present our final
(N)$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ results in Sec. 5, alongside
with MC predictions from S HERPA . We compile our conclusions and give an
outlook in Sec. 6.
## 2 Phase space and observable definition
We consider deep-inelastic scattering (DIS) of leptons with momentum $p$ of
off protons with momentum $P$ at HERA energies, _i.e._
$E_{l}=27.6\;\text{GeV}$ and $E_{p}=920\;\text{GeV}$, resulting in a centre-
of-mass energy of $\sqrt{s}=319\;\text{GeV}$. Denoting the outgoing lepton
momentum as $p^{\prime}$, we define the momentum difference, at LO carried by
the virtual photon, as
$q=p-p^{\prime}\equiv(0,0,0,-Q)~{},$ (1)
where the last equivalence defines the Breit frame, which we will assume
whenever frame-specific formulae are given. We also introduce the usual
Bjorken variable $x_{B}$ and inelasticity $y$
$\displaystyle x_{B}$ $\displaystyle=\frac{Q^{2}}{2P\cdot q}\,,$ (2)
$\displaystyle y$ $\displaystyle=\frac{P\cdot q}{P\cdot p}~{}.$ (3)
We consider events with $150<Q^{2}/\text{GeV}^{2}<2\cdot 10^{4}$ and
$0.05<y<0.94$. No other cuts are applied, but we have studied 1-jettiness in
smaller bins of $Q^{2}$ and $y$, and will only discuss a selection of results
here***Results over the full range of $Q^{2},y$ in several bins of both
variables are available upon request..
We take into account all final state particles apart from the outgoing lepton
for the calculation of event-shape variables. We study a well known
observable, referred to as thrust $\tau_{Q}$ [24] or alternatively 1-jettiness
$\tau^{b}_{1}$ [23]. Several equivalent definitions exist in the literature.
For concreteness we define it by dividing the event into a current hemisphere
$\mathcal{H}_{C}$ and a beam hemisphere $\mathcal{H}_{B}$. Working in the
Breit frame, we can introduce two reference vectors
$n_{\pm}=(1,0,0,\pm 1)$ (4)
and denote the hemispheres according to the final state particles momentum
fractions along those,
$\displaystyle\mathcal{H}_{C}=\\{p_{i}:p_{i}\cdot n_{+}>p_{i}\cdot
n_{-}\\}\quad\text{and}\quad\mathcal{H}_{B}=\\{p_{i}:p_{i}\cdot
n_{+}<p_{i}\cdot n_{-}\\}~{}.$ (5)
We can now define thrust as the sum of the longitudinal momentum components of
all particles in the current hemisphere. As we prefer to work with an
observable that vanishes in the soft limit, _i.e._ the limit where all final
state partons apart from the struck quark have arbitrarily small momenta, we
ultimately use
$\tau=1-\frac{2}{Q}\sum_{p_{i}\in\mathcal{H}_{C}}p_{i}^{z}\,.$ (6)
Despite this definition only summing over one of the hemispheres, thrust,
_i.e._ 1-jettiness, is actually sensitive to emissions anywhere in the event,
and indeed is a global event shape in the sense of _e.g._ [27]. Note this
statement depends on the precise definition, including the normalisation
factor here given by $Q/2$, that differs in the thrust variant we use for
tuning in the following.
In addition we study 1-jettiness calculated based on events that have been
groomed of soft wide-angle radiation. Soft-drop grooming was first introduced
in [35] as a jet substructure technique, including as a special case the
modified Mass Drop Tagger [36, 37]. It has since been generalised and applied
also to jets at lepton colliders [38, 18] and event shapes at both lepton [38,
39] and hadron [40] colliders. A version applicable to DIS was proposed in
[41], based on the C ENTAURO jet algorithm [42], that accounts for the
forward-backward asymmetry when considering the Breit frame. This sequential
cluster algorithm is based on the distance measure between particles with
momenta $p_{i},p_{j}$
$\displaystyle d_{ij}$
$\displaystyle=(\Delta\bar{z}_{ij})^{2}+2\bar{z}_{i}\bar{z}_{j}(1-\cos\Delta\phi_{ij})\,,$
(7) $\displaystyle\text{with}\;\;\bar{z}_{i}$
$\displaystyle=2\sqrt{1+\frac{q\cdot p_{i}}{x_{B}P\cdot
p_{i}}}\quad\text{and}\quad~{}\Delta\bar{z}_{ij}=\bar{z}_{i}-\bar{z}_{j}\,.$
(8)
Note that [42] discusses more general functional forms of the distance
measure, while we concentrate here on the definition given in [41]. As in all
other soft-drop grooming methods the objects of interest, in this case the
full event, are first clustered according to this sequential algorithm, and
then the reverse clustering history is considered. The last cluster step is
undone, and the softness of the softer of the two branches is evaluated. For
the DIS case, [41] suggests to use
$z_{i}=\frac{P\cdot p_{i}}{P\cdot q}$ (9)
as a measure for softness. The formal soft-drop criterion then reads
$\frac{\min[z_{i},z_{j}]}{z_{i}+z_{j}}>z_{\text{cut}}~{},$ (10)
with $z_{\text{cut}}$ the grooming parameter. If this is satisfied, _i.e._
both branches are classified as hard, the algorithm terminates. Otherwise the
softer branch (with smaller $z$) is dropped, and the procedure is repeated
with the harder branch. This iteration stops when either Eq. (10) is
satisfied, or there is only one particle left in the hard branch such that no
further unclustering is possible.
We finally recalculate 1-jettiness, using Eq. (6) but restricting the sum to
particles in the current hemisphere that have not been dropped during
grooming, thereby considering variable values for $z_{\text{cut}}$.
## 3 DIS Monte Carlo simulations with S HERPA
We derive hadron-level predictions for the DIS event shapes using a pre-
release version of S HERPA -3.0 [43], that will supersede the current S
HERPA -2.2 series [44]. This major release features extended physics-modelling
capabilities, including, for example, the automated evaluation of electroweak
(EW) corrections at the one-loop order [45, 46, 47] or in the Sudakov
approximation [48, 49], a complete reimplementation of the cluster
hadronisation model [25], as well as an improved user interface based on Yaml
[50]. To analyse our simulated event samples we employ the R IVET analysis
package [51]. For jet clustering we use the C ENTAURO plugin [42] within the
F AST J ET framework [52].
### 3.1 MEPS@NLO predictions for DIS
The basics of simulating DIS processes by merging parton-shower evolved
higher-multiplicity tree-level matrix elements within the S HERPA framework
have been presented in [34]. We here lift this to next-to-leading order (NLO)
accurate QCD matrix elements. To this end, we consider the massless single and
dijet production channels in neutral current DIS at NLO, and three- and four-
jets at leading order (LO), _i.e._
$e^{-}p\to e^{-}+1,2\,j\,@\,\text{NLO}+3,4\,j\,@\,\text{LO},$ (11)
where we consider $u,d,s$ quarks to be massless and add additional LO
processes for the remaining massive quarks. The massless and massive channels
get matched to the S HERPA Catani–Seymour dipole shower [53] and merged
according to the MEPS@NLO [54] and MEPS@LO [55] truncated shower formalism,
respectively. The contributing one-loop amplitudes are obtained from O PEN L
OOPS [56], that employs the C OLLIER library [57] for the evaluation of
tensor and scalar integrals. All tree-level matrix elements are provided by C
OMIX [58], and PDFs are obtained from LHAPDF [59].
To determine the perturbative scales entering the calculation, the final
states of the multi-parton final states get clustered to a two-to-two core
process [55]. For the reconstructed core the factorisation, renormalisation,
and parton shower starting scale are set to
$\displaystyle\mu_{\text{F}}=\mu_{\text{R}}=\mu_{\text{Q}}:=\mu_{\text{core}}\,.$
(12)
For jet-associated DIS three configurations need to be distinguished [34]:
1. (i)
virtual photon exchange, _i.e._ $ej\to ej$, where
$\mu_{\text{core}}^{2}=Q^{2}$,
2. (ii)
interaction of the virtual photon with a QCD parton, _i.e._ $\gamma^{*}j\to
j_{1}j_{2}$, with $\mu^{2}_{\text{core}}=m_{\perp,1}m_{\perp,2}$ defined as
the product of the two jet transverse masses
$m_{\perp,i}=\sqrt{m^{2}_{i}+p_{\perp,i}^{2}}$ relative to the beam axis,
3. (iii)
and pure QCD channels, _i.e._ $jj\to jj$, where
$\mu^{2}_{\text{core}}=-\frac{1}{\sqrt{2}}\left(s^{-1}+t^{-1}+u^{-1}\right)^{-1}$
is a scaled harmonic mean of the Mandelstam variables $s,t,u$.
Beyond the core process, the arguments of the strong-coupling factors are
determined by the clustering algorithm [55]. The merging-scale parameter,
separating the different jet-multiplicity contributions, is dynamically set to
$Q_{\text{cut}}=\frac{\bar{Q}_{\text{cut}}}{\sqrt{1+\bar{Q}^{2}_{\text{cut}}/Q^{2}}}\,,\quad\text{using}\quad\bar{Q}_{\text{cut}}=5\,\text{GeV}\,.$
(13)
As parton density functions we use the NNLO PDF4LHC21_40_pdfas set [60] with
$\alpha_{S}(M^{2}_{Z})$=0.118.
To estimate perturbative uncertainties, we consider 7-point variations of the
factorisation ($\mu_{F}$) and renormalisation ($\mu_{R}$) scales in the matrix
element and the parton shower that get evaluated on-the-fly [61], _i.e._
$\\{(\tfrac{1}{2}\mu_{\text{R}},\tfrac{1}{2}\mu_{\text{F}}),(\tfrac{1}{2}\mu_{\text{R}},\mu_{\text{F}}),(\mu_{\text{R}},\tfrac{1}{2}\mu_{\text{F}}),(\mu_{\text{R}},\mu_{\text{F}}),(\mu_{\text{R}},2\mu_{\text{F}}),(2\mu_{\text{R}},\mu_{\text{F}}),(2\mu_{\text{R}},2\mu_{\text{F}})\\}\,.$
(14)
The resummation scale $\mu_{Q}$ we keep fixed. The final uncertainty estimate
is derived by forming an envelope of all variations.
### 3.2 Tuning the beam fragmentation model against HERA data
Ref. [25] presented a new cluster fragmentation model for S HERPA that will
be used in S HERPA -3, superseding the old cluster model described in [62],
that was used in the S HERPA -1.X [63] and S HERPA -2.X [44] released. A
particular feature of the new implementation is a specific treatment of the
fragmentation of hadronic clusters that contain beam remnant particles. To
calibrate the corresponding model parameters we performed dedicated tunes
using HERA data for hadronic final state observables in neutral current DIS.
Broadly speaking, a cluster hadronisation simulation features two basic
components, a cluster-formation and a cluster-decay model [64, 65]. Based on
the pre-confinement property of QCD [66], finite mass colour neutral mesonic
and baryonic clusters can be formed from the final state of a parton shower
evolution of a hard scattering event. These primary clusters are then subject
to an iterative fission process that ultimately results in the transition to
known hadronic resonances, whose decays can be treated by a dedicated package.
Both elements of the hadronisation model introduce sets of parameters that
need to be carefully adjusted by comparing model predictions and measurements
for suitable observables, a process commonly known as tuning.
In Ref. [26] the free model parameters were calibrated against hadronic
observables measured in electron–positron annihilation experiments. However,
in leptonic collisions the beam fragmentation modelling is not probed and the
corresponding parameters remained unconstrained. This affects in particular
the parametrisation of the decay of clusters that contain a remnant particle
of an incident hadron, _e.g._ a (anti-)quark and (anti-)diquark from the
break-up of the incoming proton in DIS. We consider the two-body decay of a
beam cluster with flavours $f_{1}$ and $\bar{f}_{2}$, where a (di)quark-
flavour pair $f\bar{f}$ is drawn from the vacuum, resulting in
${\cal{C}}[f_{1}\bar{f}_{2}]\to{\cal{C}}_{1}[f_{1}\bar{f}]\;{\cal{C}}_{2}[f\bar{f}_{2}]\,.$
(15)
To fix the kinematics of the two-body decay in the rest frame of ${\cal{C}}$,
the absolute value of the transverse momentum of the decay products
${\cal{C}}_{1}$ and ${\cal{C}}_{2}$ is selected according to a Gaussian
distribution ${\cal{N}}(0,k^{2}_{T,0}/2)$ that is truncated at the parton-
shower cut-off $p_{T,\text{min}}$, _i.e._
${\cal{P}}(k_{T})\propto\exp\left(-k_{T}^{2}/k_{T,0}^{2}\right)\Theta(p^{2}_{T,\text{min}}-k_{T}^{2})\,.$
(16)
The parameter $k_{T,0}$ is thereby considered as independent of the incident
cluster type. The direction of the two-component $\vec{k}_{T}$ is picked
uniformly in the transverse plane, with $f_{1}$ and $\bar{f}_{2}$ pointing
along the positive and negative $z$-axis, respectively. This leaves one to fix
the longitudinal momentum fractions $z^{(1),(2)}$ with respect to the light-
like vectors $n^{\mu}_{\pm}=(1,0,0,\pm 1)$. For the case of a beam-remnant
cluster, still working in its rest frame, these are distributed according to
${\cal{P}}(z)\propto
z^{\alpha_{B}}(1-z)^{\beta_{B}}\cdot\exp\left\\{-\gamma_{B}\frac{1}{z}\left(\frac{k^{2}_{T}+(m_{f_{1}}+m_{\bar{f}_{2}})^{2}}{k^{2}_{T,0}}\right)\right\\}\,.$
(17)
Note the similarity to the symmetric Lund string fragmentation function [67].
This results in the four-momenta of the decay products being given by
$\displaystyle p^{\mu}_{{\cal{C}}_{1}}$ $\displaystyle=$
$\displaystyle\frac{m_{{\cal{C}}}}{2}\left(z^{(1)}n^{\mu}_{+}+(1-z^{(2)})n^{\mu}_{-}\right)+k^{\mu}_{T}\,,$
(18) $\displaystyle p^{\mu}_{{\cal{C}}_{2}}$ $\displaystyle=$
$\displaystyle\frac{m_{{\cal{C}}}}{2}\left((1-z^{(1)})n^{\mu}_{+}+z^{(2)}n^{\mu}_{-}\right)-k^{\mu}_{T}\,.$
(19)
According to Eq. (17) the relevant free parameters specifically steering the
decays of beam clusters are $\alpha_{B}$, $\beta_{B}$, and $\gamma_{B}$. To
calibrate those we performed dedicated tunes based on a variety of hadronic
observables measured by the HERA experiments H1 and ZEUS. The remaining
hadronisation parameters are set according to the LEP data tune described in
Ref. [26].
We employ the A PPRENTICE tuning tool [68], with reference data for DIS
analyses at centre of mass energies of $\sqrt{s}=$300\text{\,}\mathrm{GeV}$$,
_i.e_. lepton energies of $27.5\text{\,}\mathrm{GeV}$ and proton energies of
$820\text{\,}\mathrm{GeV}$. The tuning requires an initial set of Monte Carlo
runs, that are then used to generate a polynomial, bin-wise approximation of
the Monte Carlo response with respect to changes in the hadronisation-model
parameters. The predictions for the grid points are generated using the
calculational setup described in Sec. 3.1.
The selection of observables considered for the tuning includes classic
variables sensitive to hadronisation. In particular, we use event-shape
distributions like thrust and jet broadening [9], energy flows and charged
particle spectra [69, 70] and multiplicities [71, 72], as well as quark
fragmentation functions [73, 74]. Further details on the used analyses and
observables are provided in App. A.
Given we consider model parameters newly introduced that have not been tuned
before, we have little prior knowledge about their preferred values and thus
need to start out with rather wide parameter ranges. To narrow these down, we
make an initial pass to get a rough idea of the relevant regions. The
corresponding ranges are outlined in Tab. 1. For a second run we restrict the
tuning ranges using the results of the exploration run, resulting in an
iterative procedure to further narrow down the considered parameter intervals.
The initial run, with largely unconstrained parameter values also serves the
purpose of filtering out the most sensitive observables from the considered
analyses. Observables or observable regions that remain unchanged under the
variation of the tuning parameters are not suited for the following tunes and
therefore dropped.
Similar to the procedure described in Ref. [26], we generate a set of
equivalent tunes that only differ by the Monte Carlo runs used to construct
the polynomial approximations as described above. The tunes are thus fully
equivalent and can be used to estimate the non-perturbative model-parameter
uncertainties as illustrated in Fig. 1 for a selection of data from the HERA
experiments. We call these alternative parameter sets replica tunes. To
reflect the uncertainty associated with the three beam-fragmentation
parameters we here consider seven such replicas, _cf._ Tab. 1 for the
resulting uncertainty variations.
parameter | parameter tag | tuning range | central tune | uncertainty variation
---|---|---|---|---
$\alpha_{B}$ | ALPHA_B | [-1, 20] | 14.2 | [13.9, 14.8]
$\beta_{B}$ | BETA_B | [0.5, 4] | 1.59 | [1.14, 1.60]
$\gamma_{B}$ | GAMMA_B | [1, 20] | 8.11 | [8.06, 9.47]
Table 1: A HADIC++ model parameters considered in the tuning. Quoted are the
initial parameter interval, the obtained central-tune value, and uncertainty
ranges extracted from 7 replica tunes.
Figure 1: S HERPA predictions for the hadronisation tune, for observables
measured by the H1 and ZEUS experiments at
$\sqrt{s}=$296\text{\,}\mathrm{GeV}$$. Shown is the transverse energy flow
(left) [69], thrust $\tau^{\prime}$ (center) [9] and the charged particle
multiplicity $n_{\mathrm{ch}}$ (right) [71]. Note, the statistical
uncertainties of the simulated data is small compared to the non-perturbative
tuning uncertainties indicated by the blue band.
## 4 (N)$\text{NLO}+\text{NLL}^{\prime}$ resummation for 1-jettiness in DIS
The 1-jettiness observable considered here is equivalent to thrust in DIS,
which has originally been resummed at NLL accuracy in [24, 75]. The more
general $n$-jettiness [76, 77] was suggested for lepton–hadron collisions in
[78], and has been resummed to NNLL accuracy [79]. For 1-jettiness, analytic
fixed order results at LO have been presented in [80], and the NLL calculation
has been matched to fixed order at NLO accuracy in [81]. The resummed
calculations in this formalism for event shapes in DIS were extended to N3LL
in [82]. Grooming for DIS has first been suggested in [41] based on jets
defined with the C ENTAURO jet algorithm [42]. The same Ref. [41] also
provided NNLL results for both 1-jettiness and jet mass after soft drop
grooming. Non-perturbative corrections have there been modelled through a two-
parameter shape function [83, 84]. To our knowledge there are no published
results studying these observables including matching to fixed order or using
a fixed order calculation alone.
### 4.1 NLL resummation in the C AESAR approach
To perform the NLL resummation of logarithms $L$ of event shapes in DIS we use
the implementation of the C AESAR formalism [27] available in the S HERPA
framework [28, 85]. For a recursive infrared and collinear (rIRC) safe
observable, the cumulative cross section for observable values up to
$v=\exp(-L)$ can be expressed to all orders, in general as a sum over partonic
channels $\delta$, as follows:
$\begin{split}\Sigma_{\mathrm{res}}(v)&=\sum_{\delta}\Sigma_{\mathrm{res}}^{\delta}(v)\,,\,\,\text{with}\\\
\Sigma_{\mathrm{res}}^{\delta}(v)&=\int
d\mathcal{B_{\delta}}\frac{\mathop{d\sigma_{\delta}}}{\mathop{d\mathcal{B_{\delta}}}}\exp\left[-\sum_{l\in\delta}R_{l}^{\mathcal{B_{\delta}}}(L)\right]\mathcal{P}^{\mathcal{B}_{\delta}}(L)\mathcal{S}^{\mathcal{B_{\delta}}}(L)\mathcal{F}^{\mathcal{B_{\delta}}}(L)\mathcal{H}^{\delta}(\mathcal{B_{\delta}})\,,\end{split}$
(20)
where $\frac{\mathop{d\sigma_{\delta}}}{\mathop{d\mathcal{B_{\delta}}}}$ is
the fully differential Born cross section for channel $\delta$ and
$\mathcal{H}$ implements the kinematic cuts applied to the Born phase space
$\mathcal{B}$. For a 2-jet observable like thrust in DIS, there is only one
relevant partonic Born channel, corresponding to an incoming and an outgoing
quark. This also implies that the soft function $\mathcal{S}$, which
implements colour evolution, is trivial in our case. Further, since we are
dealing with an additive observable, the multiple emission function
$\mathcal{F}$ is simply given by
$\mathcal{F}(L)=e^{-\gamma_{E}R^{\prime}}/\Gamma(1+R^{\prime})$, with
$R^{\prime}(L)=\partial R/\partial L$ and $R(L)=\sum_{l\in\delta}R_{l}(L)$.
The collinear radiators $R_{l}$ for the hard legs $l$ were computed in [27]
for a general observable $V$ scaling for the emission of a soft-gluon of
relative transverse momentum $k_{t}^{(l)}$ and relative rapidity $\eta^{(l)}$
with respect to leg $l$ as
$V(k)=\left(\frac{k_{t}^{\left(l\right)}}{\mu_{Q}}\right)^{a}e^{-b_{l}\eta^{\left(l\right)}}d_{l}\left(\mu_{Q}\right)g_{l}\left(\phi\right)\,.$
(21)
For the case of 1-jettiness we are focusing on in this publication, we have
$a=b_{l}=1$, and fixing $\mu_{Q}^{2}=Q^{2}$ also $d_{l}g_{l}=1$ since there is
no dependence on the azimuthal angle $\phi$. The precise form of the logarithm
can be varied according to
$L\to\ln\left[\frac{x_{L}}{v}-x_{L}+1\right]\to\ln\frac{x_{L}}{v}\quad\text{as}\quad
v\to 0\,,$ (22)
to estimated the impact of sub-leading logarithms, while leaving the
distribution at the kinematic endpoint $v\sim 1$ unchanged. Note this implies
an additional contribution to $R_{l}(L)$ to restore NLL accuracy.
The PDF factor $\mathcal{P}$, in our study applicable only to the hadronic
beam, is here given by
$\mathcal{P}=\frac{f_{q}(x,e^{-2L/(a+b)}\mu_{F}^{2})}{f_{q}(x,\mu_{F}^{2})}\,,$
(23)
corrects for the true initial-state collinear scale. We thereby account for
the full DGLAP evolution by calculating a simple ratio. For the purpose of
matching to a fixed order calculation, we also need the expansion of the ratio
to a given order in $\alpha_{\text{s}}$. We generally follow the approach of
[27] to implement the expansion of a leading order approximation. This of
course introduces additional effects beyond our considered logarithmic
accuracy. We argue it is safe to ignore those, given the generally small
numerical size of these contributions as seen for example in [28]. We here for
the first time apply the C AESAR implementation in S HERPA to an observable
that is sensitive to the PDF ratio (note this only applies to the ungroomed
version of thrust) and at the same time match to the NLO calculation for the
differential distribution and the NNLO result for the inclusive DIS process.
We hence need to take care of the expansion to one order higher. Following
[27], the numerator of Eq. (23) can to NLL accuracy be written and expanded in
powers of $\alpha_{\text{s}}$ as
$\displaystyle\mathbf{f}(x,e^{-2L/(a+b)}\mu_{F}^{2})$
$\displaystyle=\exp\left[-T\left(\frac{L}{a+b}\right)\mathbf{P}\otimes\right]\mathbf{f}(x,\mu_{F}^{2})$
$\displaystyle\sim
1-\left(T^{(1)}\left(\frac{L}{a+b}\right)+T^{(2)}\left(\frac{L}{a+b}\right)\right)\mathbf{P}\otimes\mathbf{f}(x,\mu_{F}^{2})$
$\displaystyle\phantom{=1}+\frac{1}{2}\left(T^{(1)}\left(\frac{L}{a+b}\right)\right)^{2}\mathbf{P}\otimes\mathbf{P}\otimes\mathbf{f}(x,\mu_{F}^{2})+\mathcal{O}\left(\alpha_{\text{s}}^{3}\right),$
(24)
where $T^{(i)}$ denotes the $i$th term obtained by expanding the integrated
strong coupling
$T(L)=-\frac{1}{\pi\beta_{0}}\ln(1-2\alpha_{\text{s}}\beta_{0}L)$ (25)
in powers of $\alpha_{\text{s}}$. The bold-faced symbols represent matrices
(of splitting functions, $\mathbf{P}$) and vectors
($\mathbf{f}=(f_{u},f_{d},f_{s},\dots)$) in flavour space, and the convolution
is given by
$\mathbf{P}\otimes\mathbf{f}(x,\mu_{F}^{2})=\int_{x}^{1}\frac{dz}{z}\mathbf{P}\left(\frac{x}{z}\right)\mathbf{f}(z,\mu_{F}^{2})\,.$
(26)
New terms at $\mathcal{O}(\alpha_{\text{s}}^{2})$ hence originate from the
higher order expansion of $T$, mixed terms with other parts of the resummation
multiplying the leading order expansion, and the convolution of two splitting
functions with the PDF in the last line of Eq. (24). The last one is the only
one that requires a non-trivial implementation. We use the expressions from
[86] for convoluted splitting functions, and solve the final integral for the
convolution with the PDF through Monte Carlo integration, as done at leading
order.
We match our resummed calculation in the multiplicative matching scheme along
the lines of [85], which we briefly recap here. The matching to fixed order is
done at the level of cumulative distributions $\Sigma(v)$. Note that we have
dropped the label for the partonic channel since in our case there is a single
one only. We expand the inclusive cross section $\sigma_{\text{fo}}$ as well
as the fixed-order and resummed cumulative distributions, $\Sigma_{\text{fo}}$
and $\Sigma_{\text{res}}$ in series of $\alpha_{\text{s}}$:
$\displaystyle\sigma_{\text{fo}}$
$\displaystyle=\sigma^{(0)}+\sigma^{(1)}_{\text{fo}}+\sigma^{(2)}_{\text{fo}}+\dots\,,$
(27) $\displaystyle\Sigma_{\text{fo}}(v)$
$\displaystyle=\sigma^{(0)}+\Sigma^{(1)}_{\text{fo}}(v)+\Sigma^{(2)}_{\text{fo}}(v)+\dots\,,$
(28) $\displaystyle\Sigma_{\text{res}}(v)$
$\displaystyle=\sigma^{(0)}+\Sigma^{(1)}_{\text{res}}(v)+\Sigma^{(2)}_{\text{res}}(v)+\dots\,,$
(29)
where the number in parentheses indicates the respective order in
$\alpha_{\text{s}}$, and $\sigma^{(0)}$ denotes the Born-level cross section.
Our final matched expression for the cumulative distribution, with the
dependencies on the observable value suppressed, reads:
$\Sigma_{\text{matched}}=\Sigma_{\text{res}}\left(1+\frac{\Sigma^{(1)}_{\text{fo}}-\Sigma^{(1)}_{\text{res}}}{\sigma^{(0)}}+\frac{\Sigma^{(2)}_{\text{fo}}-\Sigma^{(2)}_{\text{res}}}{\sigma^{(0)}}-\frac{\Sigma^{(1)}_{\text{res}}}{\sigma^{(0)}}\frac{\Sigma^{(1)}_{\text{fo}}-\Sigma^{(1)}_{\text{res}}}{\sigma^{(0)}}\right)\,.$
(30)
Note that, compared to our earlier works, we use $\Sigma^{(2)}$ directly, thus
reproducing the inclusive cross section to one order higher, _i.e._ NNLO, what
requires the calculation of $\sigma^{(2)}_{\text{fo}}$. Importantly, the
resummed NLL result $\Sigma_{\text{res}}$ is multiplied by
$\frac{\Sigma^{(1)}_{\text{fo}}-\Sigma^{(1)}_{\text{res}}}{\sigma^{(0)}}\to\frac{\alpha_{\text{s}}}{2\pi}C_{1}\quad\text{as}\quad
v\to 0\,.$ (31)
We refer to the distribution that includes all NLL terms in
$\Sigma_{\text{res}}$ and additionally the coefficient $C_{1}$ as
$\text{NLL}^{\prime}$ accurate. Through this matching procedure we achieve a
formal accuracy of $\text{NLO}+\text{NLL}^{\prime}$ for the differential
distribution and NNLO for the inclusive event rate, referred to as
(N)$\text{NLO}+\text{NLL}^{\prime}$ in what follows.
In addition to the perturbative contribution described above, there is a
significant non-perturbative component to the distribution of event shapes,
that we necessarily need to take into account in order to accurately describe
actual collider data. While it has been shown in various circumstances that
soft-drop grooming reduces the impact of hadronisation corrections, see for
example [83, 38, 39, 40, 87, 30], it is typically still necessary to account
for a remaining small non-perturbative contribution. We here adopt the
approach of [30] to extract transfer matrices from Monte Carlo simulations.
Transfer matrices are defined as
$\mathcal{T}_{hp}=\frac{\int\mathop{dP}\frac{\mathop{d\sigma}}{\mathop{dP}}\Theta_{p}\left(P\right)\Theta_{h}\left(H(P)\right)}{\int\mathop{dP}\frac{\mathop{d\sigma}}{\mathop{dP}}\Theta_{p}\left(P\right)}\,,$
(32)
with
$\displaystyle\Theta_{p}\left(P\right)$
$\displaystyle=\prod_{i=1}^{m}\theta(V_{i}(P)-v^{\text{min}}_{p,i})\theta(v^{\text{max}}_{p,i}-V_{i}(P))\,,$
(33) $\displaystyle\Theta_{h}\left(H(P)\right)$
$\displaystyle=\prod_{i=1}^{m}\theta\left(V_{i}\left(H(P)\right)-v^{\text{min}}_{h,i}\right)\theta\left(v^{\text{max}}_{h,i}-V_{i}\left(H(P)\right)\right)\,,$
(34)
for a transition between the parton level phase space $P$ and the
corresponding hadron level configuration $H(P)$, characterised by a set of
observables $V_{i}$ that can be calculated on both of them. For our purpose,
we assume that the requirements on the DIS kinematics, _cf._ Sec. 2,
sufficiently fix the remaining degrees of freedom other than 1-jettiness
$\tau$. This first of all means that we do not allow non-perturbative
corrections to change the underlying Born kinematics, _i.e._ the $Q^{2},\,y$
bin, in contrast to, for example, measurements performed on jets with a
potentially affected transverse momentum spectrum. On the other hand this
implicitly assumes that it is valid to average the corrections for all
configurations with a common 1-jettiness value. Hence, we are only concerned
with events migrating between different bins in $\tau$ within a given
$Q^{2},\,y$ bin. The transfer matrices as defined above can readily be
extracted from the S HERPA event generator by analysing the different stages
of the events evolution, _i.e._ after parton showering but before
hadronisation and thereafter. For practical details of our event generation
setup see Sec. 3. Our final results are then calculated from the resummed and
matched parton level bins $\Delta\sigma_{p}^{\text{PL}}$ as
$\mathop{\Delta\sigma_{h}^{\text{HL}}}=\sum_{p}\mathcal{T}_{hp}\mathop{\Delta\sigma_{p}^{\text{PL}}}\,.$
(35)
### 4.2 Grooming in DIS
The framework described above has already been employed to obtain resummed
predictions for soft-drop thrust in lepton–lepton collisions at
$\text{NLO}+\text{NLL}^{\prime}$ precision [39], for soft-drop groomed
hadronic event shapes [40] and groomed jet substructure observables at the LHC
[87, 88, 30]. The extensions made in [40] to accommodate the phase space
constraints implied by soft-drop grooming, with general parameters
$z_{\text{cut}}$ and $\beta$, are directly applicable here. Note that [41]
does not define a $\beta\neq 0$ version of grooming in DIS, and we make no
attempt here to extend it.
The applicability of the results from [40] to DIS event shapes relies on two
statements. First, within the current hemisphere the phase space constraints
to radiation in the soft and collinear limits correspond to the case of final
state radiation in general hadronic collisions. Second, in the beam hemisphere
any soft and collinear radiation is groomed away. Accordingly, we can treat
radiation in $\mathcal{H}_{B}$ equivalent to the initial state radiation case
in [40], even if the precise shape of the phase space boundary is different,
but such difference does not enter at NLL accuracy. We analyse the behaviour
of the C ENTAURO algorithm and the associated soft-drop grooming variant in
the language of the C AESAR framework in the following to illustrate this.
Recall that we are working in the Breit frame. At NLL accuracy, we have to
take into account ensembles of soft particles, well separated in rapidity,
around a Born configuration consisting of the proton momentum
$P^{\mu}=\frac{Q}{2x_{B}}n^{\mu}_{+}$ (36)
and the outgoing struck quark in $n_{-}$ direction. The virtual photon carries
momentum
$q=\frac{Q}{2}(n_{-}-n_{+})~{}.$ (37)
We parameterise the momenta of additional soft gluons as
$k_{i}^{\mu}=k_{t}^{i}\left(\frac{e^{\eta_{i}}}{2}n_{-}^{\mu}+\frac{e^{-\eta_{i}}}{2}n_{+}^{\mu}+n_{\perp}^{\mu}\right)\,,$
(38)
where $n_{\perp}$ is a transverse unit vector perpendicular to $n_{+}$ and
$n_{-}$. The variable introduced in the C ENTAURO algorithm, _cf._ Eq. (8),
can be written using the phase space variables $\eta_{i}$, $k_{t}^{i}$ as
$\bar{z}_{i}=2e^{-\eta_{i}}~{},$ (39)
such that the expression for the distance measure, _cf._ Eq. (7), becomes
$d_{ij}=4\left(e^{-2\eta_{i}}+e^{-2\eta_{j}}+2e^{-(\eta_{i}+\eta_{j})}\cos\Delta\phi_{ij}\right)\sim
4e^{-2\eta_{i}}\,,$ (40)
where we have identified the behaviour for strong ordering in $\eta$,
$\eta_{i}\ll\eta_{j}$. In this limit, the algorithm builds up a single jet
containing the hard quark by adding the next remaining gluon that is most
collinear to this jet. The last clustering will add the gluon most collinear
to the beam direction to the jet. If all gluons are separated in rapidity well
enough, there are no other clusters to be taken care of.
From this discussion it is clear that all comparisons of scales during soft
drop will be between a soft gluon and a jet containing the hard quark. At Born
level, the four-momentum of the jet will be approximately that of the quark,
and the gluon will be the softer of the two. With this in mind the hardness
measure for soft drop for soft momentum $k_{i}$ can be written as
$z_{i}\sim\frac{k_{t}^{i}}{Q}e^{\eta_{i}}~{}.$ (41)
Within the current hemisphere, the phase space restriction, on an emission
that passes the soft-drop criterion, is given by
$\frac{k_{t}e^{\eta}}{Q}>z_{\text{cut}}~{},$ (42)
which precisely matches the one given in [40] for $\beta=0$ (see Sec. 3.4
point (iv), and note that the hard quark has energy $Q/2$ in the Breit frame).
Note that particles outside of the current hemisphere will enter in Eq. (42)
with negative rapidity $\eta$. They will hence be groomed away unless they are
at very high $k_{t}$, only causing logarithms of $z_{\text{cut}}$. We note
again that the precise shape of the phase space boundary is different from
what is given in [40] for initial states. The main point is however that only
logarithms of $z_{\text{cut}}$ are produced, which we ignore noting again that
we work in the limit $v\ll z_{\text{cut}}$.
### 4.3 Calculational tools and setup
As already stated, the resummation calculation for 1-jettiness is accomplished
with the C AESAR plugin to S HERPA that hooks into the event generation
framework†††Note, during the course of this work the plugin has been ported to
the S HERPA -3.0 release series.. S HERPA thereby provides all the process
management, and gives access to the C OMIX matrix element generator [58], as
well as phase-space integration and event-analysis functionalities. We make
use of S HERPA ’s interface to L HA P DF [59] and use the PDF4LHC_40_pdfas
PDF set, as we do for the parton-shower simulations outlined in the previous
section. The value of the strong coupling is set accordingly, _i.e._
$\alpha_{S}(M^{2}_{Z})=0.118$. The S HERPA framework is also used to compile
all the required higher-order tree-level and one-loop calculations. For the
NLO QCD computations we use the S HERPA implementation of the Catani–Seymour
dipole subtraction [89] and the interfaces to the R ECOLA [90, 91] and O PEN
L OOPS [92] one-loop amplitude generators. The calculation of NNLO accurate
predictions for DIS has been automated in S HERPA in [29], and we use it to
compute cross sections $\sigma^{(2)}_{\text{fo}}$ at order
$\alpha_{\text{s}}^{2}$ differential in $Q^{2}$ and $y$ to achieve overall
NNLO accuracy for inclusive cross sections. This corresponds to an accuracy of
the distribution differential in thrust at NLO, and we refer to the combined
accuracy of our fixed order predictions including cross sections as (N)NLO.
The plugin implements the building blocks of the C AESAR master formula Eq.
(20), along with the necessary expansion in $\alpha_{s}$ used in the matching
with fixed-order calculations. The building blocks are evaluated fully
differentially for each Born-level configuration $\mathcal{B}_{\delta}$ of a
given momentum configuration. Jet clustering and grooming functionalities are
accessed through the interface of S HERPA to F AST J ET [52]. Non-
perturbative corrections are extracted from dedicated runs of the S HERPA
generator using the identical setup described in Sec. 3, thereby employing the
functionality of the R IVET analysis tool to provide access to intermediate
evolution stages through the H EP MC event record [93].
## 5 Results for (groomed) 1-jettiness in DIS
Having outlined our calculational techniques for describing hadronic final
state observables in neutral current DIS, we can finally present our numerical
results for the 1-jettiness event shape. We begin by discussing selected
results for the ungroomed case. We have compiled predictions for a wide range
of $Q^{2}$ values, _i.e._ $Q^{2}\in[150,20000]\;\text{GeV}^{2}$. Furthermore,
we consider the production cross section differential in the events
inelasticity, thereby covering the region $y\in[0.05,0.94]$. For brevity, we
here focus on three kinematic regions corresponding to medium values of
$y\in[0.4,0.7]$ and rather low ($Q^{2}\in[150,200]\,\text{GeV}^{2}$), medium
($Q^{2}\in[440,700]\,\text{GeV}^{2}$) and high
($Q^{2}\in[3500,8000]\,\text{GeV}^{2}$) photon virtuality.
Along with the central predictions we show error bands indicating the
perturbative uncertainty obtained from 7-point variations of
$\mu_{R},\mu_{F}$, in both the shower and the semi-analytic calculation, and
in addition a variation of $x_{L}=0.5,2$ in the latter, _cf._ Eq. (22).
Furthermore, we include an uncertainty estimate related to the tuning of beam-
fragmentation parameters based on replica tunes, see Sec. 3.2. Generally, this
contribution is found to be rather small compared to the perturbative
uncertainties. We observe the overall uncertainties for the NLO QCD matrix
element plus parton-shower simulations and the resummation predictions to be
of similar sizes.
We first analyse the behaviour of the $\text{NLO}+\text{NLL}^{\prime}$
resummation calculation upon inclusion of the NNLO normalisation correction
and non-perturbative effects. To this end we compile in Fig. 2 corresponding
predictions for the three kinematic regions specified before. From the lower
panels, showing the ratio to the respective $\text{NLO}+\text{NLL}^{\prime}$
result, it can be read off, that correcting the normalisation to NNLO accuracy
has a rather small impact. The differential cross section receives a small
negative correction, of at most a few percent at small $\tau$ in the lower
$Q^{2}$ region. Note, however, that even the smallest $Q^{2}$ values in this
analysis remain sizeable compared to the overall range accessible for the HERA
experiments. Somewhat more significant is the reduction in the perturbative
uncertainties when going from NLO to NNLO, in particular for the bulk of the
distributions, _i.e._ low values of 1-jettiness.
Next, we consider the inclusion of non-perturbative corrections based on the
transfer-matrix approach described in Sec. 4.1. As clearly visible in Fig. 2
these significantly alter the shape of the distributions, introducing a
sizeable shift towards larger 1-jettiness values. In particular for the low
and medium $Q^{2}$ region the first bin gets almost entirely depopulated. In
contrast, for values of $\tau\approx 0.1\dots 0.2$ corrections can reach up to
$+100\%$. The effect of hadronisation corrections is less pronounced at higher
$Q^{2}$. We furthermore note, that the non-perturbative corrections through
the bin migration via transfer matrices partially compensate the dependence of
the perturbative calculation on scale variations and in particular of
$\mu_{R}$.
Figure 2: Distributions of ungroomed 1-jettiness in selected $y-Q^{2}$ bins,
at different stages of the calculation, at $\text{NLO}+\text{NLL}^{\prime}$
accuracy, including the normalisation at NNLO
((N)$\text{NLO}+\text{NLL}^{\prime}$) accuracy, and including non-perturbative
corrections. All results correspond to DIS kinematics with $y\in[0.4,0.7]$ and
the plots represent from left to right regions of
$Q^{2}/\text{GeV}^{2}\in[150,200]$, $[440,700]$, and $[3500,8000]$,
respectively. The lower panels present the ratio to the plain
$\text{NLO}+\text{NLL}^{\prime}$ result.
We close this first discussion of the resummed predictions for ungroomed
1-jettiness by pointing to the distinct peak at $\tau\approx 1$ for the low
and medium $Q^{2}$ distributions, emerging after a significant decline of the
differential cross section from lower to larger observable values. For the
given observable definition the configuration $\tau=1$ can be attributed to
events with an empty current hemisphere ${\mathcal{H}}_{C}$ [80]. Such
configurations first appear when considering the NLO real-emission correction
to the DIS process, when both final state partons feature negative
longitudinal momenta in the Breit frame, such that 1-jettiness defaults to 1,
see Eq. (6). We here account for these configurations through matching to the
exact NLO QCD result for $\tau$, _i.e._ including the full
${\cal{O}}(\alpha_{S})$ corrections to the two-parton channel. It can be
observed, that hadronisation corrections reduce the amount of $\tau\approx 1$
events, what can be expected, as the fragmentation of partons originally in
the beam hemisphere might spill over hadrons in the current hemisphere.
We now turn to the presentation of the hadron level results from MEPS@NLO
simulations with S HERPA as outlined in Sec. 3 and compare those to the
(N)$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ predictions. In Fig. 3 we
compare the respective results for the three considered kinematic regions. We
observe an overall fair agreement between the matrix element improved shower
simulations at hadron level obtained from S HERPA and the resummed and
matched calculation at (N)$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ accuracy,
corrected for non-perturbative effects. In general the merged prediction
features a somewhat harder spectrum, _i.e._ favours somewhat larger observable
values. This might also be attributed to the inclusion of the exact tree-level
three- and four-jet matrix elements, see Eq. (11). These contributions feature
LO scale dependence and are thus the source for the somewhat enlarged
theoretical uncertainties in the shower simulation towards larger values of
$\tau$. However, the regions of small 1-jettiness agree within uncertainties
for all three kinematic regions, up until the peak of the respective
distribution. Towards the kinematic endpoint, the two approaches tend to agree
again, with both calculations predicting very similar cross sections for
events with $\tau\sim 1$.
Figure 3: Distributions of 1-jettiness in selected $y-Q^{2}$ bins, _i.e._
$y\in[0.4,0.7]$ and, from left to right, $Q^{2}/\text{GeV}^{2}\in[150,200]$,
$[440,700]$, and $[3500,8000]$, respectively. Shown are hadron level MEPS@NLO
predictions from S HERPA and results at
(N)$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ accuracy. The lower panels
present the ratio to the MEPS@NLO result.
Besides the plain 1-jettiness event shape we here also consider the effect of
soft-drop grooming the hadronic final state. In Fig. 4 we show resummed
predictions for groomed 1-jettiness, referred to as $\tau^{\text{SD}}$ in what
follows, integrated over the full $Q^{2}$ range, _i.e._
$Q^{2}\in[150,20000]\;\text{GeV}^{2}$, and the inelasticity region
$y\in[0.2,0.7]$. We compiled predictions for three commonly considered values
of $z_{\text{cut}}$, namely $z_{\text{cut}}=0.05,0.1,0.2$, thereby always
assuming the angular grooming parameter $\beta=0$. As seen for the ungroomed
case, we note rather small effects of the NNLO normalisation corrections
compared to the $\text{NLO}+\text{NLL}^{\prime}$ calculation. Also the
systematic uncertainties hardly change from NLO to NNLO. However, the size of
the non-perturbative corrections is significantly reduced relative to the
ungroomed case, staying below $50\%$ and being largely flat over a wide range
of $\tau^{\text{SD}}$, apart from very low values of 1-jettiness and at the
endpoint $\tau^{\text{SD}}\sim 1$. This confirms the potential of soft-drop
grooming to mitigate hadronisation effects for event shape observables also in
DIS, seen before in $e^{+}e^{-}$ [38, 39] and $pp$ collisions [40].
Figure 4: Distributions of groomed 1-jettiness, at different stages of the
calculation, at $\text{NLO}+\text{NLL}^{\prime}$ accuracy, including the
normalisation at NNLO ((N)$\text{NLO}+\text{NLL}^{\prime}$) accuracy, and
including non-perturbative corrections. From left to right the plots represent
predictions for the grooming parameter $z_{\text{cut}}=0.05,0.1,0.2$,
respectively. The lower panels present the ratio to the plain
$\text{NLO}+\text{NLL}^{\prime}$ result.
The comparison of the (N)$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ results
with hadron level simulations at MEPS@NLO accuracy is presented in Fig. 5. For
all the $z_{\text{cut}}$ values, we observe good agreement between our S
HERPA simulation and the resummation calculation somewhat better than for the
ungroomed case. In all three cases, the
(N)$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ calculation predicts a larger
cross section in the $\tau\sim 1$ bin, although still compatible within the
uncertainty of the event generator for $z_{\text{cut}}=0.05$ and the combined
uncertainty for both calculations for $z_{\text{cut}}=0.1$. Apart from this
last bin, for these two $z_{\text{cut}}$ values the resummation calculation is
consistently below the S HERPA simulation. In the case of
$z_{\text{cut}}=0.05$, this happens flat over the full spectrum
$\tau^{\mathrm{SD}}<1$, while for increasing $z_{\text{cut}}$ a slight shape
develops, with the (N)$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ cross section
decreasing faster for $\tau^{\mathrm{SD}}<z_{\text{cut}}$ than what is seen in
the Monte Carlo simulation.
Figure 5: Distributions of groomed 1-jettiness. Shown are hadron level
MEPS@NLO predictions from S HERPA and results at
(N)$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ accuracy. From left to right the
plots represent predictions for the grooming parameter
$z_{\text{cut}}=0.05,0.1,0.2$, respectively. The lower panels present the
ratio to the MEPS@NLO result.
It will be interesting to compare the
(N)$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ predictions and the S HERPA
MEPS@NLO simulations with the data of upcoming measurements by the H1
experiment. This will shed light on the found deviations between the two sets
of predictions and possibly guide the development of yet improved theoretical
predictions, _e.g._ through the inclusion of next-to-next-to-leading
logarithmic corrections.
## 6 Conclusions
We presented the calculation of theoretical predictions for the 1-jettiness
event shape in neutral current DIS at HERA energies. The here considered
1-jettiness observable, evaluated in the Breit frame, is equivalent to the
well-known thrust variable that has been widely studied at lepton and hadron
colliders. Besides plain 1-jettiness we also considered its variant after
soft-drop grooming the hadronic final state using different values of the
grooming parameter $z_{\text{cut}}$. We consider the triple-differential cross
section in the observable, momentum transfer $Q^{2}$, and the events
inelasticity $y$.
Based on the C AESAR formalism we derive NLL accurate results matched to the
exact NLO QCD matrix element for the two-jet DIS matrix element. Furthermore,
we include the exact NNLO QCD corrections to the inclusive DIS process,
thereby achieving full NNLO accuracy for the integrated observable
distribution. We furthermore correct our results of
(N)$\text{NLO}+\text{NLL}^{\prime}$ accuracy for non-perturbative
hadronisation effects through a transfer matrix that takes into account
migration in the observable value when going from parton to hadron level. The
corresponding corrections have been extracted from Monte Carlo simulations at
MEPS@NLO accuracy with the S HERPA generator. To this end, we have performed
tunes of the beam-fragmentation parameters of S HERPA ’s new cluster
fragmentation model against data from the H1 and ZEUS experiments. We thereby
also derived replica tunes that account for the parametric uncertainties.
For plain 1-jettiness we have shown results for three kinematic regions,
corresponding to medium inelasticity $y$ and ranges of rather low, medium, and
high $Q^{2}$ values. While the impact of the NNLO contributions is found to be
very small, hadronisation corrections significantly sculpt the differential
distributions, pushing events from lower to larger 1-jettiness values. When
comparing the hadronisation corrected (N)$\text{NLO}+\text{NLL}^{\prime}$
predictions with hadron level predictions from S HERPA good agreement is
found, with larger deviations dominantly in the region $0.2<\tau<0.6$. Quite
good agreement is found regarding events at the endpoint of the distribution,
_i.e._ $\tau\simeq 1$. For the low and medium $Q^{2}$ regions the distribution
here develops a significant peak, that can be attributed to events with an
empty current hemisphere.
For the soft-drop groomed variant of 1-jettiness we have shown predictions for
three values of $z_{\text{cut}}$, integrated over a wide range of $Q^{2}$,
_i.e._ $Q^{2}\in[150,20000]\;\text{GeV}^{2}$, and $y\in[0.2,0.7]$. For all
values of $z_{\text{cut}}$ non-perturbative corrections to the resummed
predictions get significantly reduced, when comparing to the ungroomed case.
Furthermore, an improved agreement with the hadron level predictions from S
HERPA is found.
It will be exciting to confront the two types of predictions with actual data
from the HERA collider that are currently being analysed by the H1 experiment.
We can expect that in particular for the ungroomed 1-jettiness observable data
should be able to discriminate between the two predictions. This will motivate
and guide the development and advancement of the theoretical predictions. For
DIS parton shower simulations there are recent developments towards the
inclusion of NNLO QCD corrections [29] and to achieve formal NLL accuracy [94,
95, 96, 97]. This would allow to match the precision of the analytic
predictions we presented in this study. Improving the analytic calculation
might require the inclusion of higher-logarithmic corrections or improved
means to account for non-perturbative corrections. Furthermore, a detailed
analysis of systematic differences between analytic NLL resummation and shower
algorithms implementing unitarity and momentum conservation along the lines of
[98] might help to pin down the origin of the observed differences.
## Acknowledgements
We would like to thank Daniel Britzger and Henry Klest for triggering us to
dive into DIS event shapes and a very fruitful communication. We furthermore
thank Johannes Hessler and Vinicius Mikuni for discussions. We are indebted to
Stefan Höche for assistance with the NNLO corrections and we are grateful to
Frank Krauss for help with S HERPA ’s new beam fragmentation model.
MK and SS acknowledge support from BMBF (05H21MGCAB) and funding by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project
number 456104544 and 510810461. DR is supported by the STFC IPPP grant
(ST/T001011/1).
## Appendix A Tuning details
We here collate more detailed information on the tuning of the A HADIC++
beam-fragmentation parameters. The R IVET analyses and considered observable
measurements by the H1 and ZEUS HERA experiments used for the tuning are
summarised in Tab. 2.
R IVET Analysis name [reference] | Observables | Virtuality range [$\text{GeV}^{2}$]
---|---|---
H1_2006_I699835 [9] | thrust, jet broadening | $Q^{2}\in[256,400]$
H1_1994_S2919893 [70] | transverse energy flow | $Q^{2}\in[10,100]$
| energy-energy correlation | $Q^{2}\in[10,100]$
H1_1995_I394793 [73] | quark fragmentation functions | $Q^{2}>100$
H1_1996_I422230 [72] | charged multiplicity distributions | $Q^{2}\in[10,1000]$
H1_1996_I424463 [99] | charged particle spectra | $Q^{2}\in[5,50]$
H1_1997_I445116 [74] | quark fragmentation functions | $Q^{2}\in[100,8000]$
| charged hadron energy spectra | $Q^{2}\in[100,8000]$
H1_2000_S4129130 [69] | transverse energy flow | $Q^{2}\in[10,2200]$
Table 2: R IVET analysis tags, observables and corresponding photon
virtuality ranges used for the tuning.
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|
School of Physics and Astronomy, University of Minnesota Twin Cities,
Minneapolis, Minnesota 55455, USA<EMAIL_ADDRESS>
# Evaluation of the mean excitation energies of gaseous and liquid argon
M. Strait
###### Abstract
Current and future experiments need to know the stopping power of liquid
argon. It is used directly in calibration, where commonly the minimum-ionizing
portion of muon tracks is used as a standard candle. Similarly, muon range is
used as a measure of muon energy. More broadly, the stopping power figures
into the simulation of all charged particles, and so uncertainty propagates
widely throughout data analysis of all sorts. The main parameter that controls
stopping power is the mean excitation energy, or I-value. Direct experimental
information for argon’s I-value come primarily from measurements of gaseous
argon, with a very limited amount of information from solid argon, and none
from liquid argon. A powerful source of indirect information is also available
from oscillator strength distribution calculations. We perform a new
calculation and find that from oscillator strength information alone, the
I-value of gaseous argon is $(187\pm 5)$ eV. In combination with the direct
measurements and other calculations, we recommend $(187\pm 4)$ eV for gaseous
argon. For liquid argon, we evaluate the difference in central value and
uncertainty incurred by the difference of phase and recommend $(197\pm 7)$ eV.
All uncertainties are given to 68% C.L.
Interaction of radiation with matter, Neutrino detectors 2212.06286
Evaluation of the mean excitation energy of liquid argon
M. Strait<EMAIL_ADDRESS>School of Physics and Astronomy, University of
Minnesota Twin Cities, Minneapolis, Minnesota 55455, USA
## 1 Introduction
The International Commission on Radiation Units and Measurements (ICRU)
recommended values and uncertainties for the mean excitation energy of gaseous
argon twice, once in 1984 [1] and again in 2016 [2]. The first of these was in
Report 37 which gives an evaluated value of $(188\pm 10$) eV. The uncertainty
is not at 68% C.L., but is a “[figure] of merit, arrived at by subjective
judgments, and with a meaning that is not easily defined.” The report further
explains that errors are given at roughly 90% C.L., and that one “could
convert them to ‘standard deviations’ by multiplying them by a factor of about
one half.” These two statements indicate difference confidence levels.
However, ICRU-37 derives from NBSIR 82-2550 [3], which gives only the first of
these uncertainty interpretations, so this note will consider ICRU-37
uncertainties to be at 90% C.L.; this also agrees with ICRU-90’s comments on
ICRU-37. The 68% C.L. uncertainty is therefore 6 eV.
ICRU-37 uses four experimental results for stopping power and range [4, 5, 6,
7] in their evaluation. Other methods of estimating I-values are cited, such
as the semi-empirical oscillator strength distribution calculated in Ref. [8],
but they are not used in the evaluation.
ICRU Report 90 [2] updates the I-value of gaseous argon to $(187\pm 3)$ eV,
where the uncertainty now means roughly the usual one standard deviation (“an
interval having a confidence of approximately 67%”). The only new experimental
result used is Ref. [9], which does not add much. The decrease in the
recommended uncertainty is almost entirely from inclusion of oscillator
strength distribution calculations.
This note re-evaluates the sources of information on gaseous argon’s I-value.
First, direct experimental evidence from stopping power and range measurements
are reviewed in section 2. Second, the state of oscillator strength
distribution calculations is reviewed, and a new calculation performed, in
section 3. Other indirect methods of estimating the I-value are also reviewed
in this section. In section 4 all of the information on the I-value of gaseous
argon is combined into a new recommended value and uncertainty. In the
following section 5, information on phase effects are reviewed and we evaluate
an I-value for liquid argon. Finally, the major implications of the
recommended I-value for liquid argon for an experiment such as DUNE [10] are
given in section 6.
## 2 Gaseous argon experiments
### 2.1 Brolley & Ribe 1955
Brolley & Ribe [4] measured the stopping power of deuterons in argon gas
relative to air. Deuterons with initial energy 10.05 MeV were sent through a
cell filled with argon, and the pressure in the cell was adjusted until a
downstream NaI(Tl) crystal registered a drop of 25% of the initial energy.
This is a stopping power measurement that samples the $dE/dx$ only for fast
projectiles. It therefore suffers less than a range measurement from
uncertainties related to slow particles, primarily in the difficulty of
evaluating shell corrections.
The authors do not directly report an I-value for argon, but rather quote an
“absolute stopping cross section $dE/dx$” of $(3.72\pm 0.08)\times
10^{-15}\,\mathrm{eV\,cm}^{2}$ for argon gas at 219.2 mmHg, i.e.
$56.1\,\mathrm{MeV\,g^{-1}\,cm^{2}}$. From this, the ICRU inferred an I-value
of $(190\pm 15)$ eV. The uncertainty in the cross section appears to have been
evaluated from a combination of the degree of consistency between runs at the
same argon pressure, 1.4%, and the uncertainty of the initial beam energy,
1.3%.
The authors are not clear about the confidence level of their uncertainties. A
reanalysis shows that their $\pm 0.08\times 10^{-15}\,\mathrm{eV\,cm}^{2}$
directly corresponds to ICRU’s $\pm 15$ eV. Given that the three uncertainties
quoted by the ICRU for Refs. [4, 5, 6] are $\pm 15$ eV, $\pm 7$ eV and $\pm
10$ eV, that these measurements have consistent central values, and that the
final ICRU recommendation is $10$ eV at 90% C.L. (“subjective”), it appears
that the ICRU has assumed that Brolley & Ribe’s $\pm 0.08\times
10^{-15}\,\mathrm{eV\,cm}^{2}$ is at roughly 68% C.L. This is the most
conservative likely interpretation (i.e. the other sensible choices are 90% or
95% confidence, and these would indicate smaller uncertainties), so the same
will be assumed in this note’s evaluation.
### 2.2 Martin & Northcliffe 1962
Martin & Northcliffe [5] measured energy loss of few-MeV alpha particles in
gaseous argon and report an I-value of $(190\pm 17)$ eV. The ICRU-37 table
shows this as $(190\pm 7)$ eV instead, as does predecessor document NBSIR
82-2550. As Ref. [5] unambiguously says “$\pm 17\,$eV” twice, this appears to
be a simple mistake, not a re-evaluation. With this misreading, Martin &
Northcliffe would be the best experimental result, while the correct
uncertainty makes it subdominant.
The authors give a detailed list of sources of uncertainty as well as a
discussion of theoretical difficulties in calculating corrections at low
energy needed to obtain the I-value for argon. The confidence level of their
result would seem to be subjective — “obtained by estimating the maximum and
minimum slopes consistent with the errors displayed.” This note’s evaluation
treats the uncertainty as being at 68% C.L., but this may be very
conservative. Like the previous experiment, this is a stopping power
measurement and is relatively less vulnerable to uncertainties related to very
slow particles.
### 2.3 Hanke & Bichsel 1970
In another stopping power measurement, Hanke & Bichsel [6] used alpha
particles from radioactive decay to measure the I-value of gaseous argon. The
authors quote 182 eV and 167 eV as their results, for two evaluations of shell
corrections. ICRU-37 uses a re-evaluation of $(188\pm 10)$ eV. Hanke & Bichsel
provides the most information on gaseous argon’s I-value (given that we have
used the correct uncertainty of 17 eV from Ref. [5]). From context, the ICRU
quoted uncertainty is probably meant to be at 68% C.L., but unfortunately it
is not clear how it was obtained.
### 2.4 Besenbacher et al 1979
Ref. [7] reports on a measurement of stopping power for protons in the range
40 keV to 1 MeV and alpha particles in the range 100 keV to 2.4 MeV. The
I-value of 194 eV is quoted with no error in their Table II. ICRU has not
evaluated an uncertainty either, nor is it clear how it would be done. At such
low energies, shell corrections and other complications are very important.
For the purpose of this note’s evaluation, it is assumed that this experiment
is a factor of several less precise than those that report uncertainties, and
an error of $\pm 30$ eV has been assigned. The final result below is
insensitive to the precise value of this error; so long as it is several times
larger than the uncertainty of the more reliable inputs, this note’s
evaluation of the central value and uncertainty are both unchanged to within
the precision displayed.
### 2.5 Baumgart et al 1983
Ref. [9] (not used in ICRU-37) reports on a measurement of the stopping power
of argon to protons of between 60 and 800 keV. Similarly to the previous
experiment, a value of 190 eV is given, but no uncertainty is quoted, and it
is not clear how one would be extracted. Shell corrections are, again, a major
concern. An error of $\pm 30$ eV has again been assigned for this note’s
evaluation.
## 3 Calculations for gaseous argon
### 3.1 Oscillator strength distribution
ICRU-37 says that the most reliable calculations of I-value come from semi-
empirical dipole oscillator strength distributions (OSD), i.e. the
photoabsorption cross section as a function of energy. ICRU-37 does not use
OSD calculations as part of its evaluation of the recommended I-value for
argon, but ICRU-90 does, with much of the reason for its small recommended
uncertainty of 3 eV being its adopted uncertainty of 2 eV for the calculation
of Kamakura et al 2006 [11], who reported 191 eV. However, neither this
reference, nor the older calculations from Kumar & Meath 1985 [12] and
Eggarter 1975 [8], report an uncertainty themselves. ICRU-90 explains its own
by saying that it is “based on those quoted for similar results.”
This is on very shaky ground. Similar results would only have similar
uncertainties if the uncertainties in the underlying photon cross section data
were similar between the various materials. But the underlying data for
Kamakura come from the review of Berkowitz 2002 [13], who cautions that
“Information on the oscillation strengths [of argon] is still rather limited”,
a warning that does not appear for similar cases (e.g. O, $\mathrm{O_{2}}$, N,
$\mathrm{N}_{2}$, Ne). Moreover, ICRU-90 averages several OSD calculations,
then expands all the errors such that the reduced $\chi^{2}$ is unity. But the
calculations are not independent, being based on mostly the same underlying
data, so they cannot be validly combined in this way.
The calculation of Kamakura is based on the recommended oscillator strengths
from Berkowitz. Here we will repeat the calculation, adding more recent
experimental data, and estimating uncertainties. The result is $(187\pm 5)$
eV, and is found by
$\log
I=\left.\left(\sum_{n}f_{n}\log(E_{n})+\int_{\mathrm{IP}}^{\infty}\frac{df}{dE}\log(E)dE\right)\middle/S(0)\right.,$
where
$S(0)=\sum_{n}f_{n}+\int_{\mathrm{IP}}^{\infty}\frac{df}{dE}dE,$
and in each case the sum is over discrete states and the integral is over the
continuum from the ionization potential to infinity. $E$ is the incoming
photon energy and $f$ is the oscillator strength, i.e.
$f=\frac{2\epsilon_{0}m_{e}c}{\pi e^{2}\hbar}\sigma,$
where $\sigma$ is the photoabsorption cross section, $\epsilon_{0}$ is the
permittivity of free space, $m_{e}$ is the mass of the electron, $c$ is the
speed of light, and $e$ is the elementary charge.
By the Thomas-Reiche-Kuhn sum rule, $S(0)=Z$, i.e. 18 for argon, up to
relativistic and multipole effects that are expected to be negligible for
low-$Z$ elements. We find $S(0)$ to be $17.8\pm 0.8$, in good agreement with
the sum rule. No explicit correction to the oscillator strengths to make
$S(0)$ equal to 18 is made, as the I-value is invariant under a uniform
scaling of oscillator strengths, and we do not impose additional constraints
that would motivate any non-uniform scaling.
The following discussion examines energy ranges from lowest to highest. Our
method for adopting central values for each part of the spectrum will closely
follow the choices and methods of Berkowitz’s review, updated with newer data.
Table 1: Oscillator strengths for discrete transitions. The rightmost column indicates the primary source of information. Uncertainties are judgments made here based on consistency between several measurements or as stated by the original experimental group [14][15]. Level | Energy (eV) | $f$ | Group
---|---|---|---
4s | 11.62 | $0.0580\pm 0.0034$ | Gibson and Risley
4s′ | 11.83 | $0.2214\pm 0.0068$ | Gibson and Risley
5s | 14.09 | $0.026\pm 0.003$ | Chan
3$\mathrm{\bar{d}}$ | 14.15 | $0.090\pm 0.009$ | Chan
5s′ | 14.26 | $0.012\pm 0.001$ | Chan
3d′ | 14.30 | $0.106\pm 0.011$ | Chan
3d | 13.86 | $0.00110\pm 0.00011$ | Chan
4d | 14.71 | $0.0019\pm 0.0002$ | Chan
6s | 14.85 | $0.0144\pm 0.0014$ | Chan
4$\mathrm{\bar{d}}$ | 14.86 | $0.0484\pm 0.0048$ | Chan
4d′ | 15.00 | $0.0209\pm 0.0021$ | Chan
6s′ | 15.02 | $0.0221\pm 0.0022$ | Chan
5d | 15.12 | $0.0041\pm 0.0008$ | Chan
7s | 15.19 | $0.0139\pm 0.0014$ | Chan
5$\mathrm{\bar{d}}$ | 15.19 | $0.043\pm 0.010$ | Chan
Others | 15.5 | $0.18\pm 0.02$ | Berkowitz
Table 2: Contributions to the uncertainty on the I-value in the OSD calculation, by energy range. Groups of ranges treated as fully correlated are shown without separating horizontal lines and with subtotals. The total uncertainty is found by adding each group in quadrature. Energy | Uncertainty (eV)
---|---
4s, 4s′ (Gibson and Risley) | 0.3
Other discrete below the IP (Chan) | 1.5
Unresolved discrete below the IP (Berkowitz) | 0.2
Window resonances 26–29 eV (Madden, Berrah) | 0.1
1s$\rightarrow$4pm, 3202.3 eV (Deslattes) | 00.01
15.7596–15.9371 (Berkowitz) | 0.2
15.9371–29.3295 (Samson, Carlson) | 2.7
29.3295–48.0 (Samson, Carlson) | 0.8
Subtotal | 3.6
48.0–79.3 (Watson, Samson, Suzuki) | 0.1
79.3–243 (Watson, Samson, Suzuki, Henke) | 0.1
Subtotal | 0.2
243–250 (Suzuki) | 00.02
243–336 (Suzuki, Chan) | 0.6
336–500 (Suzuki) | 1.1
500–929 (Suzuki) | 0.9
Subtotal | 2.5
929–3202 (Wuilleumier, Zheng, Henke, Suzuki) | 0.8
3206–10k (Wuilleumier, Zheng, Millar, McCrary) | 1.4
Subtotal | 2.2
10k–100k (Chantler) | 0.3
First, the discrete spectrum. The values listed in table 1 were chosen. The
uncertainties adopted are primarily those stated by the experimental groups.
As an exception, the uncertainties for the 5d and 5$\mathrm{\bar{d}}$ levels
are expanded by a factor of two because Berkowitz finds the values
suspiciously large. Within each of the two experimental groups, Chan et al
[15] and Gibson & Risley [14], we conservatively take the uncertainties to be
fully correlated. The final entry in the table is Berkowitz’s estimate for all
unresolved discrete transitions close to the ionization potential. Two
uncertainties are assigned to Berkowitz’s estimate, a 10% uncertainty fully
correlated to the discrete transitions attributed to Chan, since Berkowitz’s
number is an extrapolation based primarily on Chan’s measurements, and an
uncorrelated 10% uncertainty to cover the extrapolation procedure. Despite the
conservatism used in these decisions, the overall uncertainty to the I-value
from transitions below the ionization potential, shown in table 2, is only
$\pm 1.5$ eV, which is a minor contributor to the total.
[width=0.69]osd-figure3.pdf
Figure 1: Oscillator strength distribution for gaseous argon, 15–48 eV,
showing data of Samson 1966 [16], Carlson et al 1973 [17], Samson et al 1991
[18], and Samson & Stolte 2002 [19]. This note’s evaluation is shown as the
solid red line and Berkowitz 2002’s evaluation [13] in dot-dashed red. The
shaded energy range from 26.6 to 29.2 eV is handled specially; see the text.
The vertical line at 29.3295 eV is the boundary between polynomial fits used
in the evaluations. The bottom pane, on this plot and the following plots,
shows the fractional differences between the present evaluation and the
various data and other evaluations.
For the narrow energy range between the $\mathrm{{}^{2}P_{3/2}}$ and
$\mathrm{{}^{2}P_{1/2}}$ ionization potentials, Berkowitz assumes a constant
cross section of 20.75 Mb. We accept this and assign a 20% uncertainty to this
cross section to cover the data of Samson 1966 [16] as shown in figure 1. This
section does not contribute significantly to the I-value or its uncertainty.
Following Berkowitz, the continuum is partitioned into several energy ranges,
and the oscillator strength data fit to polynomials within these ranges. These
are of the form
$\mathrm{\frac{df}{dE}=\frac{eV}{Ry}}\sum_{i=2}^{7}a_{i}y^{i},$
where $\mathrm{eV/Ry}=1/13.606$, $y=15.9371\,\mathrm{eV}/E$. In Berkowitz’s
evaluation, a 4-term polynomial is used, i.e. $a_{6}=a_{7}=0$. Here, we use a
6-term polynomial for the first range and a 4-term polynomial for all the
rest.
Table 3: Coefficients for the piecewise polynomial fit to various energy ranges, used in the OSD calculation. Energy (eV) | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | $a_{6}$ | $a_{7}$
---|---|---|---|---|---|---
15.9371–29.3295 | $-74.283\,0$ | $\phantom{+}386.182$ | $-494.182$ | $-29.413\,5$ | $402.690$ | $-187.177$
29.3295–48.0 | $\phantom{+}122.781$ | $-890.881$ | $2\,080.83$ | $-1\,516.569$ | — | —
48.0–79.3 | $\phantom{+}16.428\,0$ | $-66.036\,0$ | $-10.122\,0$ | $\phantom{+}203.586$ | — | —
79.3–243 | $\phantom{+}10.457\,5$ | $\phantom{+}27.350\,8$ | $-512.203$ | $1\,122.41$ | — | —
336–500 | $\phantom{+}27.194\,2$ | $7\,158.62$ | $-134\,219$ | $\phantom{+}729\,590$ | — | —
500–929 | $-36.618\,3$ | $\phantom{+}13\,081.3$ | $-296\,263$ | $1\,967\,090$ | — | —
929–3202 | $\phantom{+}31.255\,2$ | $\phantom{+}280.033$ | $\phantom{+}732\,348$ | $-28\,474\,100$ | — | —
3206–10k | $-43.245\,8$ | $\phantom{+}207\,562$ | $-39\,331\,400$ | $3\,787\,040\,000$ | — | —
For the first range, 15.9371 to 29.3295 eV, the data of Samson & Stolte 2002
[19] has become available since Berkowitz’s review. See figure 1. This
motivates refitting, and we have found that a substantially better fit results
from adding two additional terms to Berkowitz’s polynomial. The coefficients
are shown in table 3. The region of window resonances from 26.6 to 29.2 eV is
excluded from the fit; these are treated as separate discrete transitions
below. Samson 1966 and Samson 2002 are used in the fit, but the Samson 1966
data from 29 to 30 eV is excluded from the fit for both this energy range and
the following range, as it is in substantial conflict with other measurements.
The data of Carlson et al 1973 [17] is not directly used, but is displayed to
illustrate the consistency and to guide the choice of adopted uncertainty.
Samson 1966 does not state an uncertainty except to suppose that the overall
error on the sum of all continuous oscillator strengths is about 5%. Samson
2002 says that uncertainties range from 1 to 3% but unfortunately does not
specify which energies have the lower or higher uncertainties. Carlson states
no uncertainties. The older two measurements are largely within 3% of Samson
2002 and within 3% of the fit, with the largest difference between the fit and
Samson 1966 being 6% around 26 eV and the largest difference for Carlson being
4% around 19 eV. We adopt 3% as the uncertainty for this range. Here and in
the following, the number given indicates the uncertainty on the overall
normalization of the range.
The window resonances evident in figure 1 are included as discrete transitions
with total oscillator strength of $-0.055$, following Berkowitz. Given that
two experimental groups, Madden et al 1969 [20] and Berrah et al 1996 [21]
have results within 10% of each other, we assign a 10% uncertainty. This is a
tiny contributor to the overall uncertainty on the I-value, and so it is not
examined more closely.
In the following range, 29.3295–48.0 eV, we performed the fit on the data of
Samson 1966, excluding 29–30 eV, Carlson 1973, excluding $>$37 eV where
evidently background dominates, Samson 1991, and Samson 2002. We note that the
very low point for 43.8 eV from Samson 1966 is a probable typo — the original
data table gives 23 cm-1, but 32 cm-1 would be much more consistent with both
the surrounding points and later experimental results. Since a 6-term
polynomial gave very similar results to a 4-term polynomial, the latter was
used; this reasoning holds for each of the following energy ranges as well.
Nearly the same result is obtained if Samson 2002 is fit alone. Given the
less-good agreement between experiments in this region, a 4% uncertainty is
adopted. We note the rather large fractional difference between our evaluation
and that of Berkowitz towards the end of this range. Our evaluation is clearly
a better fit to all of the data, but in any case the _absolute_ difference is
very small since there is little oscillator strength between 45 and 48 eV.
Since both the 15.9371–29.3295 eV and 29.3295–48.0 eV ranges are dominated by
the work of Samson, their uncertainties are treated as fully correlated. The
15.9371–48 eV range is the dominant contributor of uncertainty to the I-value,
giving $\pm 3.6$ eV.
[width=0.69]osd-figure4.pdf
Figure 2: Oscillator strength distribution for gaseous argon, 48–230 eV,
showing the data of Samson 1966 [16], Henke et al 1967 [22], Watson 1972 [23],
Samson et al 1991 [18], Samson & Stolte 2002 [19], and Suzuki & Saito 2005
[24]. The evaluation of Berkowitz 2002 [13] is also shown. The conventions are
the same as for figure 1. Note suppressed zero.
For 48.0 eV to 79.3 eV, we fit a 4-term polynomial to the data of Watson 1972
[23], Samson 2002 and Suzuki & Saito 2005 [24]. See figure 2. Watson states 3%
uncertainty. Suzuki says that uncertainties are between 0.05% and 2%, and
within 1% for most energies, but gives no information as to which energies
have which uncertainties. Accordingly, we assign 2% uncertainties in the fit.
The three experiments have good agreement, with only one point of Watson lying
more than 2% from the fit. With this robust confirmation from different
groups, we adopt a 2% uncertainty for this range.
From 79.3 to 243 eV, the data of the same three groups is fit, plus that of
Henke et al 1967 [22]. This latter data is treated as having 5% uncertainties,
despite the very small errors reported by the authors in this unpublished
report. There is somewhat more disagreement between the various results in
this range, and so we adopt a 3% uncertainty. As nearly all the information
comes from the same groups as the previous energy range, the two are treated
as fully correlated.
[width=0.69]osd-figure5.pdf
Figure 3: Oscillator strength distribution for gaseous argon, 230–500 eV,
showing the data of Henke et al 1967 [22], Chan et al 1992 [15], and Suzuki &
Saito 2005 [24], as well as the evaluations of Henke et al 1993 [25] and
Berkowitz 2002 [13]. Between 230 and 336 eV, Berkowitz uses the data of Chan
1992 without a functional form.
From 243 to 250 eV, the L-edge, we directly use the data of Suzuki 2005 shown
in figure 3. From 250 to 336 eV, we directly average Suzuki 2005 with Chan
1992, without any functional form, scaled by their stated uncertainties.
Although Chan’s data covers the entire range from the ionization potential to
500 eV, we do not display it for the L-edge because with a resolution of 1 eV,
it lacks the ability to resolve the structure of the edge. (We have not
displayed it for lower energies because it would add little information.)
From 336 to 500 eV, we display Chan’s data, but do not use it. Above 336 eV,
it rises steadily away from the work of other groups, and, as Berkowitz
states, its use in this region would give too much contribution to the sum
rule. We fit the data of Suzuki to a polynomial, as above. From 243 to 500 eV,
we assign an uncertainty of 5%, given that the evaluation rests heavily on a
single group’s data, Suzuki’s, and the main second source, Chan, does not
agree very well, even in the 250–336 eV range. Two points from Henke 1967 do
agree well with Suzuki, but this is quite sparse. We also display on the plot
the Henke et al 1993 [25] evaluation for comparison only.
[width=0.69]osd-figure6.pdf
Figure 4: Oscillator strength distribution for gaseous argon, 500–10,000 eV,
with data of Wuilleumier 1965 [26], Henke et al 1967 [22], McCrary et al 1970
[27], Millar & Greening 1974 [28], Suzuki & Saito 2005 [24], and Zheng et al
2006 [29], and the evaluations of Henke et al 1993 [25], Chantler 1995 [30],
and Berkowitz 2002 [13]. Chantler 1993 is divided into the total cross
section, which is what attenuation experiments measure, and the
photoabsorption cross section, which is of interest for the OSD calculation.
The difference is used to correct the attenuation measurements of McCrary and
Millar.
From 500 to 929 eV, we again fit a polynomial to Suzuki’s data. See figure 4.
In this range there are four points by Henke 1967 which agree very well, and
we choose an uncertainty of 3%, fully correlated with the 243–500 eV range.
For this range and the following, The evaluation of Chantler et al 1995 [30]
is displayed along with Henke’s evaluation, for comparison.
From 929 to 3202 eV, just before the K-edge, we fit the data of Wuilleumier
1965 [26], Henke 1967, Suzuki 2005, and Zheng et al 2006 [29] to a polynomial.
Given the level of agreement between these several groups, we assign a 3%
uncertainty, uncorrelated with the previous regions.
Between 3202 and 3206 eV, we treat the oscillator strength as consisting of
the discrete 1s$\rightarrow$4pm resonance, measured by Deslattes et al 1983
[31], with a strength of 0.0022 and a 10% uncertainty. This is for
completeness only, as it has a negligible impact on the I-value or its
uncertainty.
From 3206 eV to 10 keV, we fit the data of Wuilleumier 1965, Zheng 2005,
Millar & Greening 1974 [28] and McCrary et al 1970 [27]. (Here we depart from
direct use of Berkowitz’s energy divisions by using a single fit for this
range, which Berkowitz divides into two sections.) For the higher energy data
of Millar and McCrary, we use the evaluations of Chantler to subtract the non-
photoabsorption portion of the cross section. Again an uncertainty of 3% is
assigned based on the agreement between groups. Given the overlap of the
experimental groups between the 3206 eV to 10 keV range and the 929 to 3202 eV
range, we treat these two ranges as fully correlated. This choice is not
crucial; a less conservative choice to consider them uncorrelated would lower
the uncertainty on the I-value by only 0.2 eV.
[width=0.69]osd-figure7.pdf
Figure 5: Oscillator strength distribution for gaseous argon from $10^{4}$ to
$10^{5}$ eV. The data of McCrary et al 1970 [27], and Millar & Greening 1974
[28] are shown, along with the evaluations of Henke 1993 [25] and Chantler
[30]. Subtraction to obtain oscillator strength from attenuation measurements
is done as in figure 4. The Chantler evaluation is used directly in the
present evaluation.
From 10 to 100 keV, we follow Berkowitz and directly use Chantler’s
evaluation. We assign a 3% uncertainty based on agreement with the data of
Millar and McCrary, being somewhat conservative because of the subtraction
procedure necessary to isolate the photoabsorption component of attenuation at
high energies. See figure 5.
Above 100 keV, we are unaware of any photoabsorption data, and we follow
Berkowitz, using the formula of Bethe & Salpeter [32], directly evaluating it
up to 1 GeV and then integrating the asymptotic form from there to infinity.
Because the range from 100 keV to infinity gives almost no contribution to the
I-value, changing it by only 0.1 eV if it is neglected entirely, no
uncertainty is assigned.
As stated above, the result for the I-value from combining all of these energy
ranges is $(187\pm 5)$ eV. There are several energy ranges each contributing
significantly to the uncertainty on the I-value. From the ionization potential
to 48 eV has the biggest contribution because of the large amount of
oscillator strength present. The range from the L-edge to 929 eV contributes
near the second most because of its fairly large oscillator strength and
larger fractional uncertainties. Similarly, the range just above the K-edge is
a significant contributor to the uncertainty. Discrete transitions below the
IP are next most important given their substantial contribution to the total
oscillator strength and 10% uncertainties. In contrast, the range 48–243 eV
has a very small contribution because of the fairly small oscillator strengths
coupled with low uncertainties, and above 10 keV, the contribution to the
uncertainty is small because the oscillator strength is small.
As a byproduct of this calculation, we can estimate the quantities $I(-1)$ and
$I(1)$, where $I(p)$ is defined as:
$\log
I(p)=\left.\left(\sum_{n}f_{n}E_{n}^{p}\log(E_{n})+\int_{\mathrm{IP}}^{\infty}\frac{df}{dE}E^{p}\log(E)dE\right)\middle/S(p)\right.,$
and
$S(p)=\sum_{n}f_{n}E_{n}^{p}+\int_{\mathrm{IP}}^{\infty}E^{p}\frac{df}{dE}dE,$
such that the I-value that is the focus of this note is $I(0)$.
We find $I(-1)=26.51\pm 0.28$. For $I(-1)$, the lower energies are more
important. The uncertainty comes almost entirely from discrete transitions
below the IP and from the region just above the L-edge, in equal proportions.
The result for $I(1)$ is $(3600\pm 80)\,\mathrm{eV}^{2}$. For $I(1)$, higher
energies are more important. Nearly all of the uncertainty comes from the
region above the L-edge ($\pm 60\,\mathrm{eV}^{2}$) and from the 10-100 keV
range where we use the evaluation of Chantler ($\pm 40\,\mathrm{eV}^{2}$).
This assumes an uncertainty below $\sim$10% above 100 keV where we use the
formula of Bethe and Salpeter. Unlike for $I(0)$, the region above 100 keV is
not negligible for $I(1)$; neglecting it reduces the result by
$200\,\mathrm{eV}^{2}$. For this reason, we do not venture an estimate for
$I(2)$ since it is even more sensitive to high energies and we are unaware of
any photoabsorption data above 100 keV where the majority of the uncertainty
is most likely to lie.
#### 3.1.1 Comparison with other results
In 2010, Kumar and Thakkar gave another estimate of argon’s I-value, 186.3 eV
with an estimated $\pm 2$% ($\pm 3.7\,$eV) uncertainty [33]. There are three
major differences between their evaluation and ours. First, their method uses
the sum rule and molar refractivity data as constraints. We choose to use
oscillator strengths alone, without constraints from molar refractivity data.
Second, for any given energy interval, they choose a single data set or
evaluated set of oscillator strengths for their fit. This has the effect of
discarding modern oscillator strength data in many energy ranges. For instance
from 319.9 eV to 100 keV, the 1973 evaluation of Veigele [34] is used even
though data from Millar 1974, Suzuki 2005 and Zheng 2006 all exist in this
range. In contrast, we include all modern data, and in each energy range use a
fit to all data judged as reliable.
Third, we display the underlying uncertainties assigned to each part of the
oscillator strength distribution and the effect that each has towards the
final uncertainty on the I-value. It is not clear how to trace Kumar and
Thakkar estimated uncertainty on the I-value back to the underlying data.
Despite all these differences, the present evaluation arrives at a very
similar result for the I-value. For $I(1)$, Kumar and Thakkar find
$3620\,\mathrm{eV}^{2}$ $\pm 3$% ($\pm 110\,\mathrm{eV}^{2}$), also very
similar to our result. For $I(-1)$, they find 26.53 $\pm 1$% ($\pm 0.27$),
essentially identical to our result.
### 3.2 Periodic trends
As an independent method of estimating the I-value, an interpolation can be
done from nearby elements. First, using the I-values for aluminum, silicon and
calcium, a simple interpolation on a plot of Z vs. I/Z can be made. The result
is 189 eV. I-values certainly do not lie on a smooth curve, so this result
cannot be taken too literally. Nevertheless, it would be surprising, given the
clear periodic trends, if argon’s I-value lay much above 200 eV or much below
170 eV, even in the abscense of other methods of evaluation.
[width=0.64]calci.pdf
Figure 6: Reproduction of ICRU-37 Fig 3.2 for $Z=1$–38. Note suppressed zero.
The vertical dashed line indicates argon’s atomic number.
A more sophisticated treatment is given in eq. 4.1 of ICRU-37,
$I_{\mathrm{int}}(Z)=\frac{I_{c}(Z)}{Z_{2}-Z_{1}}\left[\frac{I(Z_{1})}{I_{c}(Z_{1})}(Z_{2}-Z)+\frac{I(Z_{2})}{I_{c}(Z_{2})}(Z-Z_{1})\right],$
where $Z$ is the atomic number of the element whose I-value is to be
interpolated, $Z_{1}$ and $Z_{2}$ are the atomic numbers of the next lower and
next higher element with experimentally determined I-values, $I_{c}$ indicates
a calculated I-value and a bare $I$ indicates an experimental I-value. ICRU’s
choice of calculated I-values for gasses are the set from Chu & Powers 1972
[35], and for solids those from Ziegler 1980 [36], displayed in ICRU-37 Fig
3.2 and reproduced here in figure 6. Another set of calculated I-values are
those from Dehmer et al 1975 [37] and Inokuti et al 1981 [38], where the
former covers $Z=1$–18 and the latter 19–38; these are also shown in figure 6.
Although ICRU does not calculate an interpolated value for argon, as it has
experimental data, the result is 194 eV, given that $Z_{1}=14$ (silicon),
$I(Z_{1})=173$ eV, $I_{c}(Z_{1})=123.5$ eV, $Z_{2}=20$ (calcium),
$I(Z_{2})=191$ eV, $I_{c}(Z_{2})=147.9$ eV, and $I_{c}(Z)=146.3$ eV. This
procedure accounts for the phase of the substance. If argon is treated as a
solid, then $I_{c}(Z)=147.9$ eV instead, and the result is 196 eV.
The uncertainties on the experimental data of silicon and calcium (the nearest
elements with experimental data on each side of argon) contribute 3 eV to the
uncertainty of this interpolation. The choice of theoretical input is much
more important. If, instead of the ICRU recommendation, Dehmer and Inokuti are
used (the unused curve of their Fig. 3.2), the result shifts 20 eV upwards to
214 eV. These inputs are assumed to be worse, but still represent a reasonable
calculation, and so we may qualitatively take the theory uncertainty to be
somewhat smaller than the difference. Therefore, this evaluation includes the
I-value interpolated from periodic trends as $(194\pm 12)$ eV.
### 3.3 Hartree-Fock wave functions
Only two calculations for argon listed in ICRU-90 come with uncertainties
stated by the original authors. One has such large uncertainties as to be
irrelevant. The other is Bell et al 1972 [39], which uses a method involving
Hartree-Fock wave functions. Since it does not share underlying data with the
OSD calculations, it can be included separately in this note’s evaluation.
Bell’s result, as given by the ICRU table, is $(174\pm 3.5)$ eV. What Bell
actually says is 12.8 Ry, that “the predictions of different representations
[agreed] to within 2% in all cases,” and that a “full error analysis […] is
beyond the scope of the present paper.” Bell goes on to compare the
calculation for helium with a more sophisticated treatment, finding that the
difference is 6% and “Errors in $I$ arising from the Hartree-Fock
approximation are probably similar for the other atoms considered here”. Since
both the 2% and 6% errors are relevant, an uncertainty of 11 eV is more
correct, and is used in the evaluation in this note.
## 4 Evaluation for gaseous argon
[width=0.64]argonIpdf.pdf
Figure 7: Evaluation of the mean excitation energy of gaseous argon, using
experimental results [4, 5, 6, 9, 7], interpolation from periodic trends, and
calculations [39, 11]. The ICRU-90 recommendation is shown for comparison,
interpreting their central value and uncertainty as a Gaussian PDF. All curves
except for the one for ICRU-90 share a normalization.
The five direct experimental results are combined with the three indirect
methods to produce this note’s evaluation (see figure 7). Each underlying
result is represented as a PDF which is the sum of a Gaussian plus a uniform
distribution. The Gaussian’s mean and standard deviation are the quoted
central value and uncertainty of the given result. The uniform distribution
has the range 130–240 eV and a normalization representing a subjective
judgment about how likely the result is to be incorrect (through any means,
e.g., unaccounted for systematic error, incorrect calculation, typographic
error, etc.), typically 5–10%. This reflects the tendency of older papers (and
sometimes newer papers) to have results that are incompatible with each other
at many times the stated errors. The Gaussian is assigned the remainder of the
normalization.
The underlying PDFs are multiplied together to produce the evaluated PDF,
which is integrated from the mean to find the evaluated uncertainty. The
present evaluation for gaseous argon is $(187\pm 4)$ eV.
Since the method just described unfortunately must include subjective
estimates of the correctness of past experiments and calculations, a second
method was used to check how robust the result is. In this method, it is
assumed that exactly one of the input experiments or calculations is
incorrect. The average is taken with Gaussian distributions only, but with
each input experiment or calculation dropped in turn, producing several
results. A weighted average is then produced from these results with the
weights set by the uncertainty of the experiment or calculation that was
excluded. The result of this alternate procedure is $(187\pm 5)$ eV, which
does not differ significantly from the main procedure. If our own OSD
calculation is exempted from this procedure, the result is $(187\pm 4)$ eV,
identical to the main result.
Our answer is ultimately very similar to ICRU-90’s, with the same central
value and an uncertainty just 1 eV larger. It has, however, been arrived at
through a substantially different process. Our uncertainty is larger than
ICRU-90’s for several reasons: First, the ICRU misread the Martin &
Northcliffe’s error as 7 eV rather than 17 eV. Second, the present note’s OSD
evaluation is $(187\pm 5)$ eV, which has a larger uncertainty than ICRU
assigned to the several OSD calculations used in their evaluation. Finally,
ICRU-90 misunderstood the uncertainty of the Bell calculation, understating it
by a factor of three.
This note’s evaluation for gaseous argon is dominated by the OSD calculation
presented above. If it is excluded from the average, the result is instead
$(187\pm 6)$ eV. If only direct experimental evidence from range and stopping
power measurements is used, the value $(189\pm 8)$ eV. It can be seen that
there is good agreement among the several methods used to estimate the
I-value.
## 5 Evaluation for liquid argon
[width=0.64]phaseinterp.pdf
Figure 8: Effect of phase on I-value. The points with error bars are each
discussed in the text. For compounds water and the hydrocarbons, $Z$ is a
weighted average of the constituent elements. The atomic number of argon is
shown as a vertical dotted line. A smooth curve is drawn to give an estimate
of the effect for argon.
Liquid argon is not just a very dense gaseous argon; there is binding energy
associated with the phase change. Naively, if electrons are bound more
strongly, the stopping power should decrease. Data on this effect is limited,
but instructive. This effect is predicted to be large for strongly bound
systems such as metals, and smaller for molecular substances. Examples of the
latter:
* •
Hydrogen: ICRU-37 (Table 5.7) recommends 19.2 eV for gaseous $\mathrm{H_{2}}$
and 21.8 eV for liquid hydrogen (14% higher).
* •
Water: The ICRU-90 recommended value for liquid water is $(78\pm 2)$ eV.
ICRU-37 gives $(71.6\pm 2)$ eV for water vapor and $(75\pm 3)$ eV for liquid
water. The addendum for ICRU-73 [40] gives 69.1 eV for water vapor and 78 eV
for liquid water. ICRU-90 cites this without recommending a value for water
vapor. The phase effect on the I-value ranges from 5% to 13% depending on the
numbers chosen. For the purposes of this evaluation, the phase effect is
considered to be $(9\pm 4)$%.
* •
Nitrogen: ICRU-37 recommends $(82\pm 2)$ eV for gaseous $\mathrm{N}_{2}$ and
$(90.5\pm 2.6)$ eV for liquid nitrogen. After converting their 90% C.L. errors
to standard ones, this gives an increase of $(10\pm 3)$%.
* •
Oxygen: ICRU-37 recommends $(95\pm 2)$ eV for gaseous $\mathrm{O}_{2}$ and
$(104.3\pm 2.6)$ eV for liquid oxygen ($(10\pm 3)$%).
* •
n-propane, n-pentane, n-hexane, n-heptane: ICRU-37 recommends I-values
differing by 10% or 11% between the liquid and gas phases. All of the gaseous
measurements were performed by the same group, as were all of the liquid
measurements, so the uncertainties are considered fully correlated. Averaging
them gives $(11\pm 5)$%.
* •
Bromine: ICRU-37 recommends 343 eV for gaseous bromine and 357 eV for
condensed bromine, although both are interpolations from adjacent elements.
Taking into account the uncertainties of this procedure as was done for argon
in section 3.2, this is considered to be a change of $(4\pm 6)$%.
* •
Iodine: ICRU-37 recommends 474 eV for gaseous iodine and 491 eV for condensed
iodine. Using the same procedure as for bromine, an evaluation of $(4\pm 6)$%
is obtained.
None of these are noble gasses like argon, but several have similar boiling
points and are likewise non-polar, which implies that the relevant binding
energies are similar. The effect decreases with atomic number, as expected,
since a smaller fraction of the electrons participate in chemical binding. By
fitting a smooth curve to the changes as a function of $Z$ (see figure 8), we
would expect around a 7% increase in the I-value of argon. An uncertainty of
3% is estimated, based on the mean of the uncertainties for water, nitrogen
and oxygen, the closest three experimental data points.
There is a limited amount of experimental evidence for argon itself. As
summarized in ICRU-49 [41], two groups have studied the difference of stopping
power between solid and gaseous argon, but unfortunately only with alpha
particles below 3 MeV. Chu et al 1978 [42] found a 5–10% decrease in stopping
power for solid argon below 1.0 MeV, and none for 1–2 MeV. Besenbacher et al
1981 [43] found no phase effect to within their 3% uncertainty for 0.5–3.0 MeV
alphas, implying less than a 5% change in the mean excitation energy.
Moreover, Besenbacher 1981 points out that their measurements of solid argon
are compatible with Chu 1978; the apparent difference comes from the use of
different values for the gaseous stopping power. The situation remains a
little confused, but the result from Besenbacher alone still allows for a
change of the I-value of up to 9 eV. This is surprisingly small compared to
what’s expected from the results from the non-polar molecules listed above. It
may be that results from slow alpha particles do not reliably translate to
results for fast protons.
Solid argon likely has a very similar stopping power to liquid argon, as both
are condensed phases. Unfortunately, to the author’s knowledge, there are no
substances for which the I-value has been measured in both solid and liquid
phases. This lack of experimental evidence motivates conservatism in the error
assignments.
Since data from other molecules suggests an increase in the I-value of $(7\pm
3)$%, while data from solid argon suggests $(0\pm 5)\%$, it seems reasonable
to take the weighted average of these and arrive at an estimated I-value
increase of $(5\pm 3)$% for argon in a condensed phase. This gives $(197\pm
7)$ eV as the present evaluation for the mean excitation energy of liquid
argon.
## 6 Implications
Geant4 [44], by default, uses the (older) ICRU-37 central value for gaseous
argon, 188 eV, regardless of phase. In a liquid argon neutrino detector, the
two most important quantities that the I-value feed into are muon $dE/dx$ at
minimum ionization and muon range. The former is used for calorimetric
calibration and the latter to measure muon energy.
At muon minimum ionizing, around 270 MeV, changing the I-value from 188 eV to
197 eV decreases $dE/dx$ by 0.3%. This shift increases towards lower energy
and is 0.5% at 100 MeV. The effect on calibration is dependent on the details
of the procedure, but is perhaps 0.4%. Muon range at 1 GeV is increased by
0.3%. As with $dE/dx$, this shift increases as muon energy decreases, and is
0.6% at 100 MeV.
As examples of two other quantities that affect simulation and reconstruction,
the effect on proton $dE/dx$ and range is also calculated. Proton $dE/dx$ is
decreased by 0.7% at 50 MeV and 0.6% at 400 MeV. Across these energies, range
increases by between 0.6% and 0.7% with the largest change around 100 MeV.
Since the change in the I-value is 9 eV and the recommended uncertainty is 7
eV, the uncertainties on all these ranges and stopping powers are nearly as
large as the shifts quoted above. Notably, the effect of a change in I-value
on calorimetric energy calibration with muon $dE/dx$ and the muon energy
reconstruction with range have the same sign: the whole neutrino event energy
is overestimated by using gaseous argon’s I-value. Nor is there any
cancellation of this kind of uncertainty from use of a liquid argon near
detector.
Energy reconstruction has many uncertainties besides those from the I-value,
but the author’s estimate is that the I-value dominates muon energy estimation
uncertainty below 1 GeV. Most of the information about $\nu_{\mu}$ energy
comes from the muon, and $\Delta m^{2}_{32}$ is directly proportional to the
measured energies of oscillation minima and maxima. All other oscillation
parameters rest, if not as directly, on energy reconstruction as well. The
I-value is, therefore, a critical parameter for DUNE and other liquid argon
detectors.
To conclude, this note recommends a new value and uncertainty for the mean
excitation energies of gaseous and liquid argon, $(187\pm 4)$ eV and $(197\pm
7)$ eV, respectively. The central value for liquid argon is significantly
higher than that most recently recommended by the ICRU for gaseous argon, and
the uncertainty is substantially larger. While this recommendation is believed
to be a useful improvement, it is notable that it rests strongly on an
indirect calculation based on oscillator strength distributions. Direct
experimental evidence is also used, but none of the inputs are clean. Three of
the five experiments lack any original statement of uncertainty, while the
remaining two give numerical uncertainties, but without confidence levels. Of
these latter two, only one directly states an I-value. With so much freedom of
interpretation, another evaluation could easily arrive at a different result.
This fact should further motivate experimental work to improve the
uncertainty.
## 7 Acknowledgments
This work was supported by the U.S. Department of Energy. Thanks also to
Andrew Furmanski, Tom Junk, and Abigail Waldron for helpful discussions.
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|
§ CONCLUSION
Within this paper, we proposed an online communication-efficient distributed changepoint detection method, and it can achieve similar performance as an idealistic setting but save many transmission costs. Numerically, we show that the local threshold and window size have an impact on the performance of our algorithm, and there is a trade-off in choosing a local threshold and window size. In application, we recommend choosing a large local threshold in general cases. But when the change is extremely small, the choice of the local threshold depends on the communication and storage budgets. If the communication budget is much more limited, choosing a large threshold with a large window size is sensible. If the storage cost is much more expensive, choosing a small threshold with small window size will approximately achieve the idealistic performance.
The violation of independent assumptions will negatively affect the power of our proposed method. We tried to solve this problem by inflating thresholds or estimating the long-run variance. Both ways can, to some extent, improve our algorithm when the auto-correlation problem is not severe. However, both approaches fail to detect changes in highly auto-correlated data. Therefore, one of the future research directions is how to detect change within highly auto-correlated data in real-time.
Yang gratefully acknowledges the financial support of the EPSRC via the STOR-i Centre for Doctoral Training (EP/S022252/1). This research was also supported by EPSRC grant EP/R004935/1 (Eckley) together with financial support from BT Research (Eckley, Fearnhead, Yang). The authors are also grateful to Lawrence Bardwell who played a key role in inspiring this work, and Dave Yearling (BT) for several helpful conversations that helped shape this research.
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# Fast Methods for Computing Photometric Variability of Eccentric Binaries:
Boosting, Lensing, and Variable Accretion
Daniel J. D’Orazio Niels Bohr International Academy, Niels Bohr Institute,
Blegdamsvej 17, 2100 Copenhagen, Denmark<EMAIL_ADDRESS>Paul C.
Duffell Department of Physics and Astronomy, Purdue University, 525
Northwestern Avenue, West Lafayette, IN 47907-2036, USA<EMAIL_ADDRESS>Christopher Tiede Niels Bohr International Academy, Niels Bohr Institute,
Blegdamsvej 17, 2100 Copenhagen, Denmark<EMAIL_ADDRESS>
###### Abstract
We analyze accretion-rate time series for equal-mass binaries in co-planar
gaseous disks spanning a continuous range of orbital eccentricities up to
$0.8$, for both prograde and retrograde systems. The dominant variability
timescales match that of previous investigations; the binary orbital period is
dominant for prograde binaries with $e\gtrsim 0.1$, with a $5\times$ longer
“lump” period taking over for $e\lesssim 0.1$. This lump period fades and
drops from $5\times$ to $4.5\times$ the binary period as $e$ approaches $0.1$,
where it vanishes. For retrograde orbits, the binary orbital period dominates
at $e\lesssim 0.55$ and is accompanied by a $2\times$ longer-timescale
periodicity at higher eccentricities. The shape of the accretion-rate time
series varies with binary eccentricity. For prograde systems, the orientation
of an eccentric disk causes periodic trading of accretion between the binary
components in a ratio that we report as a function of binary eccentricity. We
present a publicly available tool, binlite, that can rapidly ($\lesssim 0.01$
sec) generate templates for the accretion-rate time series, onto either binary
component, for choice of binary eccentricity below $0.8$. As an example use-
case, we build lightcurve models where the accretion rate through the
circumbinary disk and onto each binary component sets contributions to the
emitted specific flux. We combine these rest-frame, accretion-variability
lightcurves with observer-dependent Doppler boosting and binary self-lensing.
This allows a flexible approach to generating lightcurves over a wide range of
binary and observer parameter space. We envision binlite as the access point
to a living database that will be updated with state-of-the-art hydrodynamical
calculations as they advance.
keywords
## 1 Introduction
The binary-gas-disk interaction arises in a multitude of astrophysical
environments. It is important for binary orbital evolution (see, e.g., Lai &
Muñoz, 2022): from sculpting planetary system architectures (e.g., Ward, 1997;
Nelson, 2018), to impacting stellar binary demographics (e.g., Valli et al.,
2024), to facilitating mergers of stellar-mass (e.g., Stone et al., 2017) and
supermassive (e.g., Begelman et al., 1980) black hole binaries (SMBHBs).
Gas disks also offer a way to observe such systems in the electromagnetic (EM)
spectrum through accretion. In the realm of stellar binary+disk systems, radio
and millimeter-wavelength observations have revealed a variety of disks
feeding and forming young stellar binaries as well as planetary systems (e.g.,
Tobin et al., 2016; Alves et al., 2019; Czekala et al., 2021; Zurlo et al.,
2023). In the time domain, photometric variability associated with periodic
accretion onto stellar binaries has been observed in multiple systems (e.g.,
Tofflemire et al., 2017a, b), providing further data on the disk+binary
interaction.
Resolving disks around SMBHBs is more difficult than in the stellar case due
to a lack of known sub-parsec-separation SMBHBs, uncertainty in the emission
structure surrounding the accreting SMBHB at the relevant wavelengths, and the
high spatial resolution needed for imaging of these distant sources (see
D’Orazio & Loeb, 2018, 2019). Similar to the stellar analogue, however,
accretion onto SMBHBs could be observable via bright, periodically modulated
EM emission. While there is no definitive evidence for the sub-parsec-
separation SMBHBs that will merge within the age of the Universe, accretion
rates onto these systems can be as high as for single SMBHs (e.g., Farris et
al., 2012; D’Orazio et al., 2013), suggesting that they could be a sub-
population of the quasars. In addition to being bright, such a binary-quasar
population could be identified by its periodic imprint on quasar lightcurves,
on $\sim$year or shorter timescales (Haiman et al., 2009; Kelley et al., 2021;
Xin & Haiman, 2021; Haiman et al., 2023). Such periodicity can arise from the
binary’s modulation of the accretion rate (e.g., Hayasaki & Mineshige, 2008;
MacFadyen & Milosavljević, 2008; D’Orazio et al., 2013; Farris et al., 2014),
or observer-dependent relativistic effects due to the binary orbit (D’Orazio
et al., 2015; D’Orazio & Di Stefano, 2018; Hu et al., 2020). Both offer a way
to identify such systems in photometric time-domain data, with multiple
searches having identified $\sim 250$ candidates to date (see D’Orazio &
Charisi, 2023, and references therein).
Time-domain searches require predictions for periodic signatures of binary
accretion and also characterization of the intrinsic variability noise (e.g.,
Vaughan et al., 2016; Zhu & Thrane, 2020, for the SMBHB case). Here we make a
step towards the former, by characterising the variable accretion rates of
eccentric binaries embedded in circumbinary disks (CBDs). We present an
analysis of accretion variability measured from 2D isothermal numerical
hydrodynamical calculations of gas disks accreting onto equal mass binaries,
for a continuous range of binary eccentricities $e\leq 0.8$ and for both
prograde D’Orazio & Duffell (2021, hereafter DD21) and retrograde
configurations of the binary and disk angular momentum Tiede & D’Orazio (2024,
hereafter TD23). We use this data to build a publicly available tool named
binlite that can rapidly generate accretion-rate time series data, for any
binary eccentricity in the simulated range via Fourier decompositions of the
simulation data.
Section 2 describes our methods while Sections 3.1, 3.2, and 3.3 present the
results of our periodicity analysis, accretion-rate time series
reconstruction, and calculation of preferential accretion rates. As an example
use-case and to demonstrate the wide range of periodic lightcurves that can
arise from accreting eccentric binaries, Section 4 presents a method for
generating light-curves of accreting black hole binaries, in a chosen
observing band, while including the observer-dependent relativistic effects of
Doppler boosting and gravitational self-lensing for multiple observer viewing
angles.
We envision this tool and the data it is built from as a starting point from
which further sophistication in numerical models and post-processing can be
added with the goal of generating a publicly available, living-lightcurve
database for modelling, interpreting, and searching for emission from
accreting binary systems. We discuss these future prospects and current
limitations in Section 5.
## 2 Methods
Throughout we consider a binary of total mass $M$, with equal mass components
(described by mass ratio $q\equiv M_{2}/M_{1}=1$), orbital eccentricity $e$,
semi-major axis $a$, and orbital angular frequency $\Omega_{b}$. A locally
isothermal, circumbinary disk accretes onto the binary and is modelled with
viscous hydrodynamics in the two dimensions in the plane of the binary orbit.
In this case the disk is characterized by the disk aspect ratio in vertical
hydrostatic equilibrium, $h$, which describes the relative importance of
pressure forces, and the kinematic coefficient of viscosity $\nu$. Because of
the simplified physics, one can scale results to any value of $M$ or $a$,
which amounts to choosing an orbital timescale via $\Omega_{b}$. Throughout we
scale the accretion rate by the equivalent steady-state value for a single
mass, $\dot{M}_{0}=3\pi\Sigma_{0}\nu$, for arbitrary surface-density scale
$\Sigma_{0}$. We consider both prograde and retrograde configurations of the
binary orbit with respect to the CBD. In both cases the equations of
hydrodynamics are solved using the moving-mesh code DISCO (Duffell, 2016).
#### Accretion From a Prograde Circumbinary Disk
The accretion-rate time-series data for prograde disks around eccentric
binaries is taken directly from the output of 2D numerical viscous
hydrodynamical calculations described in DD21. These calculations assume a
locally isothermal equation of state, which keeps the aspect-ratio a constant
value of $h=0.1$ in the circumbinary disk, and a spatially constant
coefficient of kinematic viscosity $\nu=10^{-3}a^{2}\Omega_{b}$. Specifically,
we utilize the main calculations in DD21, which evolve the binary and disk for
25,000 binary orbits, with the first 500 orbits relaxing the disk around a
binary on a circular orbit, and the following 20,000 binary orbits sweeping
the binary eccentricity linearly from $e=0$ to $e=0.9$. Here we utilize the
accretion rates measured via a sink prescription (Eq. 3 of DD21) onto each
component of the binary, as a function of time.
#### Accretion From a Retrograde Circumbinary Disk
The accretion-rate time-series data for retrograde disks around eccentric
binaries is taken directly from the output of 2D numerical viscous
hydrodynamical calculations described in TD23. These calculations assume the
same disk and binary parameters as the prograde case except that the binary
eccentricity is swept linearly from $e=0.0-0.8$ over a timescale of 10,500
orbits, with 500 orbits to relax the disk around a binary with a circular
orbit. We note that the accretion sinks in TD23 have half the characteristic
sink size and are implemented to be “torque free”, compared to the standard
sink implementation in DD21 (see Dempsey et al., 2020; Dittmann & Ryan, 2021,
for further clarificaiton on these sink types).
### 2.1 Accretion-Rate Variability Timescales
We first compute the dominant variability timescales as a function of binary
orbital eccentricity. We follow Duffell et al. (2020); TD23 and compute a 2D
periodogram of the accretion-rate time series by taking the norm of the
quantity,
$\mathcal{P}(e,\omega)=\frac{1}{\sqrt{2\pi\sigma^{2}_{\mathcal{P}}}}\int^{t(e_{f})}_{t(e_{0})}{\mathrm{e}^{-\frac{1}{2}\frac{\left(t(e)-\tau\right)^{2}}{\sigma^{2}_{\mathcal{P}}}}\dot{M}(\tau)\mathrm{e}^{-i\omega\tau}d\tau},$
(1)
which picks out the power in Fourier components with frequency $\omega$ in a
window of the accretion-rate time series centered on time $t(e)$ and of
characteristic width $2\sqrt{2\log 2}\sigma_{\mathcal{P}}$. For our choice of
$\sigma_{\mathcal{P}}=30(2\pi\Omega^{-1}_{b})$, this corresponds to a small
window of $\sim 70$ orbits in eccentricity centered around any $e$ in the time
series.
$\begin{array}[]{cc}\includegraphics[scale={0.6}]{fin_fig/edot2D_DoDuf21sweep_tround_adjst_Nouts600xNPs600}&\includegraphics[scale={0.6}]{fin_fig/edot2D_RetroDISCO_tround_adjst_Nouts300xNPs300_nmn0p1_nmx5p5}\end{array}$
Figure 1: 2D periodograms of the total accretion rate from prograde (left)
and retrograde (right) circumbinary disks onto binaries with orbital
eccentricity $e$ ranging from 0.0 to 0.8. The y-axis indicates the timescale
in units of orbital periods. Black indicates regions where there is no power,
while purple and yellow regions have increasingly more power. Both panels are
normalized to the same color scale.
### 2.2 Fourier Reconstruction
Our primary goal is to generate accretion-rate time series, which are periodic
over the binary orbital period, for any chosen value of the orbital
eccentricity. This is possible given the continuous sweep of our solutions
through binary parameter space. We start with the total accretion rate onto
the binary $\dot{M}(t)$ and its two components $\dot{M}_{1}(t)$ and
$\dot{M}_{2}(t)$ computed over the entire 25,000 (10,500) orbit sweep from the
numerical calculation of DD21 (TD23). Because these calculations carried out a
linear sweep in eccentricity with time, we also have a linear relation between
the time $t$ and the orbital eccentricity, $t(e)$.
The accretion-rate time series at a given orbital eccentricity, onto either
component of the binary, is reconstructed with a Fourier series,
$\displaystyle\dot{M}_{\rm{Rec}}(t,e)=\alpha_{0}$ $\displaystyle+$
$\displaystyle\sum_{n}\alpha_{n}(e)\cos{\left(n\Omega t\right)}$ (2)
$\displaystyle+$ $\displaystyle\sum_{n}\beta_{n}(e)\sin{\left(n\Omega
t\right)},$
for chosen fundamental frequency $\Omega$ and its integer multiples. That is,
a simple, yet accurate reconstruction is possible when the primary power is
concentrated at integer multiples of one frequency. A natural choice is the
binary frequency $\Omega_{b}$, and we show in Section 3.1 that this is indeed
the best choice except for a few regions of parameter space where a lower
frequency dominates, but still at approximate integer multiples of
$\Omega_{b}$.
The Fourier amplitudes are computed for the nearly continuous range of orbital
eccentricities by convolving the accretion-rate time series with a Gaussian
centered around a chosen eccentricity $e$ that arises at time $t(e)$ in the
eccentricity sweep,
$\displaystyle\alpha_{0}(e)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi\sigma^{2}}}\int^{t(e_{f})}_{t(e_{0})}\dot{M}(t^{\prime})e^{-\frac{1}{2}\left[\frac{\left(t^{\prime}-t(e)\right)^{2}}{\sigma^{2}}\right]}\
dt^{\prime}$ (3) $\displaystyle\alpha_{n}(e)$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{2\pi\sigma^{2}}}\int^{t(e_{f})}_{t(e_{0})}\dot{M}(t^{\prime})e^{-\frac{1}{2}\left[\frac{\left(t^{\prime}-t(e)\right)^{2}}{\sigma^{2}}\right]}\cos\left({n\Omega
t^{\prime}}\right)\ dt^{\prime}$ $\displaystyle\beta_{n}(e)$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{2\pi\sigma^{2}}}\int^{t(e_{f})}_{t(e_{0})}\dot{M}(t^{\prime})e^{-\frac{1}{2}\left[\frac{\left(t^{\prime}-t(e)\right)^{2}}{\sigma^{2}}\right]}\sin\left({n\Omega
t^{\prime}}\right)\ dt^{\prime},$
where $\sigma$ chooses the window width of $\dot{M}$ times-series data over
which to construct the Fourier series. In practice, we calculate
reconstructions up to $e=0.81$ for prograde systems and $e=0.791$ for
retrograde systems. Throughout we choose $\sigma=10(2\pi\Omega^{-1}_{b})$,
which corresponds to a Gaussian full width at half maximum of $\approx 23.5$
orbits.
Tracking accretion rates onto both components of an equal-mass-ratio binary is
necessary because, for prograde disks, some binary eccentricities excite disk
eccentricities that allow the accretion-rate to periodically favor one binary
component over the other (Dunhill et al., 2015; Muñoz & Lai, 2016; Siwek et
al., 2023b). The simulations of DD21 find that for $0.0\leq e\leq 0.18$ and
$e\geq 0.38$ the cavity is eccentric and precesses on super-orbital timescales
($\mathcal{O}(10^{2})$ orbital periods). This causes the accretion to favor
one binary component for approximately one half of the precession period of
the eccentric cavity. When binary and cavity eccentricity vectors pass through
a perpendicular configuration, the accretion-rate ratio quickly swaps to favor
the other component for the other half of the cavity precession period. Hence,
when the circumbinary cavity is eccentric and precessing, for prograde disks
around binaries with eccentricities in the range $0.0\leq e\leq 0.18$ and
$e\geq 0.38$, there are three possible accretion states: one where the primary
dominates accretion, one where the secondary dominates accretion, and one
shorter-lived stated where the two share the accretion rate as they swap
between the first two states. In these cases the accretion-rate ratio averages
to unity when taken over a disk precession period.
DD21 finds that prograde disks around binaries with eccentricities in the
range $0.18\leq e\leq 0.38$ are much more symmetric around the origin and
either do not precess (due to lack of disk eccentricity or to locking with the
binary eccentricity vector) or have much longer precession periods than for
higher or lower binary eccentricities. Similar observations were made from the
numerical calculations of Miranda et al. (2017) and Siwek et al. (2023b).
Siwek et al. (2023b) classify regions of parameter space where the disk
eccentricity is “locked” to the binary eccentricity, finding such a state for
$e=0.2$, $q=1$, and otherwise finding precessing states for
$e=0.0,0.4,0.6,0.8$, in agreement with DD21. Here we find that even in the
locked regime, the small asymmetry of the disk can still cause the accretion
rate to be spilt unequally between the binary components, but with a different
nature than for the precessing solutions. While the eccentric binary-disk
dynamics are worth understanding further in this regime, for the purposes of
this study, we note that even in this “symmetric" non-precessing disk state,
asymmetries arise that cause unequal accretion rates onto the binary
components (See Section 4).
For retrograde systems, where persistent disk eccentricities are not excited,
the accretion rates are always split evenly between the binary components.
$\begin{array}[]{cc}\includegraphics[scale={0.35}]{fin_fig/Progr_All_Mdot_recon_ecc0.0999619_Nfourier29}&\includegraphics[scale={0.35}]{fin_fig/Progr_All_Mdot_recon_ecc0.199592_Nfourier29}\vspace{-23pt}\\\
\includegraphics[scale={0.35}]{fin_fig/Progr_All_Mdot_recon_ecc0.299222_Nfourier29}&\includegraphics[scale={0.35}]{fin_fig/Progr_All_Mdot_recon_ecc0.399682_Nfourier29}\vspace{-23pt}\\\
\includegraphics[scale={0.35}]{fin_fig/Progr_All_Mdot_recon_ecc0.499312_Nfourier29}&\includegraphics[scale={0.35}]{fin_fig/Progr_All_Mdot_recon_ecc0.599773_Nfourier29}\\\
\includegraphics[scale={0.35}]{fin_fig/Progr_All_Mdot_recon_ecc0.699403_Nfourier29}&\includegraphics[scale={0.35}]{fin_fig/Progr_All_Mdot_recon_ecc0.799863_Nfourier29}\end{array}$
Figure 2: Match of accretion-rate templates constructed with our code binlite
(dashed curves), compared to the simulation output (solid curves). The total
accretion rate onto the binary is plotted in black, while the breakdown of
accretion onto each binary component is represented by the coral and teal
curves. Accretion rates are plotted in units of the steady-state value
$\dot{M}_{0}$. The time series is displayed over four binary orbits comprising
the center of a $\sim 20$ orbit segment from which the Fourier reconstruction
is built. Deviations between the reconstruction and simulation output for
$e=0.5,0.6,0.7$ derive from inter-orbital variability (peak-to-peak
variations) whose presence is apparent from the noisy region in the top-right
portion of the corresponding periodogram in Figure 1. The binary components
have equal masses but the eccentric cavity causes preferential accretion that
could equivalently be favoring either black hole. Hence, labeling of coral and
teal curves can be interchanged.
## 3 Results
### 3.1 Periodicity Analysis
#### Prograde Periodogram
The left panel of Figure 1 shows $|\mathcal{P}(e,\omega)|$ for prograde
accretion computed via Eq. (1) over a grid of $600\times 600$ values of
$\omega$ and $e$. Bright colors denote significant power while dark colors
denote lack of power at that corresponding point in parameter space – the
y-axis indicates the periodicity timescale in units of the binary orbital
period, and the x-axis indicates binary eccentricity.
For $e\lesssim 0.1$ we find that, in agreement with previous works (e.g.,
Muñoz & Lai, 2016; Miranda et al., 2017; Zrake et al., 2021), the power is
concentrated at the orbital period and its harmonics, but is dominant in a
small range of timescales centered around five binary orbital periods. This
corresponds to the timescale for the cavity “lump” to circulate and
periodically alter the feeding rate to the binary (MacFadyen & Milosavljević,
2008; D’Orazio et al., 2013).
For $0.05\lesssim e\lesssim 0.1$, power in the “lump-timescale" splits into
branches centered on the five-binary-orbit feature. This branching can be seen
most easily in the higher frequency ($1/2.5P^{-1}_{b}$) harmonic of the lump
timescale on the left side of the left panel of Figure 1. The dominant lump-
branch drops in frequency from $1/5P^{-1}_{b}$ to $1/4.5P^{-1}_{b}$ as $e$
increases from $0.05\rightarrow 0.1$. At $e=0.1$ the lump branches vanish and
are replaced by power that is almost entirely focused at the orbital timescale
and its higher harmonics. Orbital period timescales then dominate for
$e\gtrsim 0.1$. At higher orbital eccentricities, particularly for
$0.5\lesssim e\lesssim 0.8$, the power is noisy at lower frequencies. This
arises due to less stability in the accretion-rate time series from orbit to
orbit. We demonstrate this further in the next section.
Our primary takeaway for the purpose of generating reconstructions of the
accretion-rate time series is that the orbital period is a dominant feature
for all eccentricities $e\geq 0.1$, hence we choose $\Omega_{b}$ as our
fundamental frequency for the Fourier reconstruction, Eqs. (2) and (3). For
$e<0.1$, we choose $\Omega_{b}/5$ as the fundamental frequency, but with more
terms in the reconstruction.
#### Retrograde Periodogram
The right panel of Figure 1 shows $|\mathcal{P}(e,\omega)|$ for retrograde
accretion computed via Eq. (1) over a grid of $300\times 300$ values of
$\omega$ and $e$. A different version of this is also published in TD23 (over
a different range of timescales). Our main purpose for showing it here is to
emphasize that the orbital timescale periodicity is strong for all
eccentricities. However, in the retrograde case, a strong, two-times-orbital
periodicity arises for $e\sim 0.55$. Hence, for retrograde systems we choose
the fundamental Fourier reconstruction frequency to be $\Omega_{b}/2$. Note
that the retrograde periodogram is much less noisy than its prograde
counterpart, indicating steadier accretion rate-times series, even for high
eccentricities.
### 3.2 Accretion-Rate Time Series
#### Prograde Time Series
Figure 2 presents example accretion-rate time series for prograde binaries
with eight different values of orbital eccentricity. The solid lines show the
accretion rates measured directly from the numerical calculations while the
dashed lines are the accretion templates built from our Fourier reconstruction
(Eqs. (2) and (3)) using $\Omega=\Omega_{b}$ and a total of 30 Fourier
components. Accretion rates onto each component are denoted in coral and teal,
while the total is plotted in black. Vertical dotted lines denote the time of
pericenter while vertical dot-dashed lines denote apocenter. Note that even
though we have included enough Fourier components to capture sharp features in
the time series (e.g., the bottom right panel of Figure 2), small deviations
between the reconstructions (dashed) and the simulation (solid) are apparent
for $e\gtrsim 0.5$. This is due to the inter-orbit variability which manifests
as the noisy upper-right region in the prograde periodogram (left panel) of
Figure 1, i.e., at some eccentricities the accretion rate is less steady from
one orbit to the next, affecting our reconstructions which are built from an
average over $\sim 20$ orbits.
Figure 3 demonstrates a reconstruction for a prograde binary with $e=0.01$,
where the $\omega\sim\Omega_{b}/5$ periodicity of the cavity lump dominates.
In this case the Fourier reconstruction uses $\Omega=\Omega_{b}/5$ and $60$
Fourier components. Here, the more complex nature of the variability is
apparent in the less exact match of reconstruction and direct simulation (see
again the more complex structure in the $e<0.01$ portion of the left
periodogram in Figure 1). Despite this, the reconstruction captures the main
qualitative features of the time series, including crucially, the periodicity
at both $\Omega_{b}$ and $\Omega_{b}/5$, and reliable reconstruction of the
contribution of each component accretion rate to the total.
$\begin{array}[]{c}\includegraphics[scale={0.35}]{fin_fig/Progr_All_Mdot_recon_ecc0.00946463_Nfourier59}\end{array}$
Figure 3: The same as Figure 2, but for a prograde, $e=0.01$ system, where
variability at $\Omega_{b}$ and $\Omega_{b}/5$ co-exist.
#### Retrograde Time Series
Figure 4 shows the reconstructed total, primary, and secondary accretion rates
for retrograde binary-disk systems using $\Omega=\Omega_{b}/2$ and 30 terms in
the Fourier reconstruction. The reconstructed and simulated cases capture both
orbital and twice-orbital periodicity very well and result in nearly identical
reconstructed vs. simulated curves. This is due to the much more steady nature
of retrograde disk solutions across the parameter space. The total retrograde
accretion rates are described further in TD23, while here we additionally show
the component accretion rates. These are nearly identical to each other, as is
expected for equal-mass binaries when no other asymmetry arises, i.e., the
eccentric disk of the prograde case.
$\begin{array}[]{cc}\includegraphics[scale={0.35}]{fin_fig/Retro_All_Mdot_recon_ecc0.00948017_Nfourier29}&\includegraphics[scale={0.35}]{fin_fig/Retro_All_Mdot_recon_ecc0.099293_Nfourier29}\\\
\includegraphics[scale={0.35}]{fin_fig/Retro_All_Mdot_recon_ecc0.199916_Nfourier29}\vspace{-23pt}&\includegraphics[scale={0.35}]{fin_fig/Retro_All_Mdot_recon_ecc0.299708_Nfourier29}\\\
\includegraphics[scale={0.35}]{fin_fig/Retro_All_Mdot_recon_ecc0.3995_Nfourier29}\vspace{-23pt}&\includegraphics[scale={0.35}]{fin_fig/Retro_All_Mdot_recon_ecc0.499292_Nfourier29}\\\
\includegraphics[scale={0.35}]{fin_fig/Retro_All_Mdot_recon_ecc0.599916_Nfourier29}&\includegraphics[scale={0.35}]{fin_fig/Retro_All_Mdot_recon_ecc0.699708_Nfourier29}\end{array}$
Figure 4: The same as Figure 2 but for retrograde, eccentric binaries. Here
the accretion rates onto the primary (coral) and secondary (teal) are nearly
identical because there is no precessing, circumbinary cavity.
### 3.3 Accretion-Rate Ratio $Q$
Over long enough timescales ($\mathcal{O}(10^{2}-10^{3})$ binary orbits), the
accretion rates onto equal-mass binaries will average to unity (e.g., Siwek et
al., 2023a)111Though this may not be the case when binary eccentricity and
disk eccentricity vectors are locked relative to each other, as could be the
case for our $e\sim 0.2$ results – this requires further investigation..
However, when computing lightcurves, the relevant quantity is the accretion-
rate ratio between the binary components over timescales spanning orbital
periods. To quantify this, we define the ratio
$Q=\left<\dot{M}_{2}/\dot{M}_{1}\right>$, averaged over an integer number of
orbits for which the eccentric accretion-rate imbalance operates (much shorter
than a disk precession timescale).
The ratio of component accretion rates, $Q$ is always unity for retrograde,
equal-mass binaries (see Figure 4) but is a function of eccentricity for
prograde binaries. In Figure 5 we present $Q$ measured from the prograde
simulations as a function of binary eccentricity. The grey x’s represent 1000
values drawn from the reconstructed accretion rates averaged over 5 orbits,
sampled evenly in $e$. Because preferential accretion trades between primary
and secondary, we present the minimum of $Q$ and $Q^{-1}$, with both values
being valid choices when modelling equal-mass binaries considered here. When
plotted this way, the extreme values of the ratio of accretion rates can be
approximated by a simple function of eccentricity, inspired by the ratio of
pericenter and apocenter distances of the binary components,
$\displaystyle Q_{\rm{min}}$ $\displaystyle\approx$
$\displaystyle\frac{1-\mathcal{P}(e)e}{1+\mathcal{A}(e)e}$ (4)
$\displaystyle\mathcal{P}(e)$ $\displaystyle=$ $\displaystyle 2-e^{2}-2e^{3}$
$\displaystyle\mathcal{A}(e)$ $\displaystyle=$ $\displaystyle 2+e^{2},$
which is drawn as the solid blue line in Figure 5.
For most binary eccentricities the accretion-rate periodically switches back
and forth between favoring each of the binary components. Hence, the ratio of
accretion rates used to compute lightcurves can take values between
$Q_{\rm{min}}$ and $Q^{-1}_{\rm{min}}$. As can be seen from the density of
grey x’s in Figure 5, the binary spends more time accreting at some ratios
than others: $Q=1$ is sparsely sampled because this value is encountered
during the relatively rapid stage where the accretion rate switches from
favoring one component to favoring the other. The black points and associated
error bars show the average accretion ratio from a set of constant-
eccentricity verification runs (detailed in DD21) and the black triangles
illustrate the smallest value over the run. Those values of $Q$ that lie
within the error bars from these constant-eccentricity runs provide a good
indicator for where the binary spends most of its time accreting.
For intermediate values of eccentricity, the disk is more symmetric around the
origin and is either slowly precessing or not precessing at all. This results
in regions with narrow spreads of accretion-rate ratio $Q$, seen as the
clustering of grey x’s between $0.018\lesssim e\lesssim 0.38$ in Figure 5.
While the spread of $Q$ values is much smaller in this range than outside of
it, $Q$ is not continuous in $e$ here and jumps through four different states
of monotonically varying $Q(e)$, while also experiencing islands of
oscillating disk solutions near the transitions between these four states.
This behavior is likely due to locking of the angle between disk and binary
eccentricity and would be worth studying further for elucidating the
binary+disk dynamics in this regime and its relation to observability via,
e.g., accretion variability.
$\begin{array}[]{c}\includegraphics[scale={0.5}]{fin_fig/Qofecc_csteccs_Ne1000}\end{array}$
Figure 5: The ratio $Q\equiv\left<\dot{M}_{2}/\dot{M}_{1}\right>$ of
accretion rates onto each binary component, averaged over a duration shorter
than the disk precession frequency. The grey x’s are measured from our
reconstruction while the blue line is an analytic approximation for the
extreme values. As a check of the values derived from the eccentricity sweep,
the black points show results measured from simulations with fixed
eccentricities – black points with error bars represent the average and
standard deviation of $\rm{min}\left(Q,Q^{-1}\right)$ over a precession period
of the disk , and the triangles denote the smallest values over the same
range.
## 4 Application: Construction of Boosted and Lensed Lightcurves
As an example-use case and to demonstrate the wide range of lightcurve shapes
that can arise from accreting, eccentric binaries, we develop a simple
procedure for converting the accretion-rate time series of the previous
section to a rest-frame flux. Primarily for application to accreting black
hole binaries, we then convert this to an observer-dependent flux by including
relativistic orbital Doppler boosting of emission emanating from the minidisks
around each binary component (D’Orazio et al., 2015), as well as binary self-
lensing in the point mass, point-source limit (D’Orazio & Di Stefano, 2018; Hu
et al., 2020).
Doppler boosting and binary lensing have been put forth as mechanisms for
causing unique periodic variability in accreting SMBHB systems (D’Orazio et
al., 2015; D’Orazio & Di Stefano, 2018; Hu et al., 2020; Davelaar & Haiman,
2022; Ingram et al., 2021; D’Orazio & Charisi, 2023; Major Krauth et al.,
2023). Modeling these signatures consistently requires knowledge of the
fraction of light coming from each of the minidisks and the CBD. It should
also be combined with the intrinsic variability of the source in order to
understand systems where both hydrodynamical variability as well as observer
dependent effects are jointly operating.222A combination of Doppler boosting
and hydrodynamic variability near merger is simulated in Tang et al. (2018).
Hence, the reconstructed accretion-rate time series of the previous section
allow us to significantly build upon toy models for the variability of
accreting SMBHBs.
We model the emission from the accreting binary in a given frequency band as a
constant specific flux from the circumbinary disk, $F^{0}_{\nu,\mathrm{CBD}}$,
plus time-dependent emission from the primary and secondary minidisks. We
model the minidisk emission as a constant, average flux times a time-dependent
function $F^{0}_{\nu,1}p(t)$ and $F^{0}_{\nu,2}s(t)$, where
$p(t)\equiv\dot{M}_{1}/\left<\dot{M}_{1}+\dot{M}_{2}\right>$ and
$s(t)\equiv\dot{M}_{2}/\left<\dot{M}_{1}+\dot{M}_{2}\right>$ are the
reconstructed time-variable accretion rates computed with binlite (Section
3.2) and normalized by the average total accretion rate at that eccentricity.
#### Lightcurve Generation
With boosting and lensing taken into account the total observed flux is
$\displaystyle
F_{\nu}=F^{0}_{\nu,1}\mathcal{D}_{1}\mathcal{M}_{1}p(t)+F^{0}_{\nu,2}\mathcal{D}_{2}\mathcal{M}_{2}s(t)+F^{0}_{\nu,\mathrm{CBD}},$
(5)
where $\mathcal{D}_{i}\equiv D^{3-\alpha}_{i}$ is the time-dependent Doppler-
boost magnification for Doppler factor $D_{i}$ and frequency-dependent log-
spectral slope $\alpha$, in the observing band (assumed here to be the same
for all disk components), while $\mathcal{M}_{i}$ is the time-dependent
lensing magnification for the specified binary component (see D’Orazio & Di
Stefano, 2018; Hu et al., 2020). Defining
$\displaystyle F^{0}_{\nu,\mathrm{Tot}}$ $\displaystyle\equiv$ $\displaystyle
F^{0}_{\nu,1}+F^{0}_{\nu,2}+F^{0}_{\nu,\mathrm{CBD}},$
$\displaystyle\chi_{\nu,1}$ $\displaystyle\equiv$ $\displaystyle
F^{0}_{\nu,1}/F^{0}_{\nu,\mathrm{Tot}},$ $\displaystyle\chi_{\nu,2}$
$\displaystyle\equiv$ $\displaystyle F^{0}_{\nu,2}/F^{0}_{\nu,\mathrm{Tot}},$
(6)
we can write the observed in-band flux, normalized to the total average (rest
frame) flux as,
$\displaystyle\frac{F_{\nu}}{\left<F^{0}_{\nu,\mathrm{Tot}}\right>}$
$\displaystyle=$
$\displaystyle\chi_{\nu,1}\mathcal{D}_{1}\mathcal{M}_{1}p(t)+\chi_{\nu,2}\mathcal{D}_{2}\mathcal{M}_{2}s(t)$
(7) $\displaystyle+$ $\displaystyle(1-\chi_{\nu,1}-\chi_{\nu,2}),$
which reduces to
$\displaystyle\frac{F_{\nu}}{\left<F^{0}_{\nu,\mathrm{Tot}}\right>}=(1-\chi_{\nu,2})\mathcal{D}_{1}\mathcal{M}_{1}p(t)+\chi_{\nu,2}\mathcal{D}_{2}\mathcal{M}_{2}s(t),$
(8)
for a simplified case where the minidisks are assumed to outshine the CBD.
Hence, we need to know the relative fluxes, in a specified frequency band,
from each minidisk and the circumbinary disk. This requires knowing the disk
spectra, which depends on physics of radiative energy balance in the accretion
flow not captured by our isothermal simulations. For simplicity, we treat the
circumbinary disk and both minidisks as separate components and approximate
the spectra of each component with composite black-body spectra of the
optically thick alpha-disk solutions. In this case, the spectrum is set by
each disk temperature profile, $T(r)\propto(M\dot{M}/r^{3})^{1/4}$, which, for
equal-mass binary components differs between minidisks only through the
accretion rate, and by an extra mass factor $2^{1/4}$ for the circumbinary
disk surrounding the total binary mass.
This allows us to compute spectra of each of the disk components by choosing a
total accretion rate through the CBD, a binary mass ratio, and using the split
in accretion rates onto the binary components measured from the reconstructed
accretion rates of Section 3. Over long enough timescales, the average
accretion rate through the circumbinary disk is split evenly for equal mass
binary components. However, as discussed in Sections 3.2 and 3.3, for prograde
eccentric binaries, this balance can be shifted back and forth between
components over periods of $\mathcal{O}(100)$ binary orbits as the eccentric
disk precesses with respect to the binary argument of pericenter. To quantify
this, we use the accretion-rate ratio explored in Section 3.3 and Figure 5,
$Q\equiv\left<\dot{M}_{2}/\dot{M}_{1}\right>$. We require that the minidisks
are fed by a circumbinary disk with total mass-accretion rate $\dot{M}_{\rm
CBD}$. Then the measured $Q$ and choice of $\dot{M}_{\rm CBD}$ specify the
system:
$\displaystyle\dot{M}_{1}$ $\displaystyle=$ $\displaystyle\dot{M}_{\rm
CBD}\left(1+Q\right)^{-1};\quad\dot{M}_{2}=Q\dot{M}_{1},$ (9)
so that, evaluated at the same radius,
$\displaystyle T_{\rm CBD}=2^{1/4}(1+Q)^{1/4}T_{1};\quad T_{2}=Q^{1/4}T_{1},$
(10)
where we have assumed that the binary components have equal masses in the
second line. The average fluxes from each disk component are,
$\displaystyle F^{0}_{\nu,1}$ $\displaystyle=$
$\displaystyle\frac{2\pi\cos{I}}{d^{2}}\int^{r_{o,1}}_{r_{i,1}}B_{\nu}\left[T_{i,1}\left(\frac{r}{r_{i,1}}\right)^{-3/4}\right]\
rdr$ (11) $\displaystyle F^{0}_{\nu,2}$ $\displaystyle=$
$\displaystyle\frac{2\pi\cos{I}}{d^{2}}\int^{r_{o,2}}_{r_{i,2}}B_{\nu}\left[Q^{1/4}T_{i,1}\left(\frac{r}{r_{i,1}}\right)^{-3/4}\right]\
rdr$ $\displaystyle F^{0}_{\nu,\mathrm{CBD}}$ $\displaystyle=$
$\displaystyle\frac{2\pi\cos{I}}{d^{2}}\int^{r_{o,\mathrm{CBD}}}_{r_{i,\mathrm{CBD}}}B_{\nu}\left[2^{1/4}(1+Q)^{1/4}T_{i,1}\left(\frac{r}{r_{i,1}}\right)^{-3/4}\right]\
rdr,$
for a source at distance $d$ and a common disk-inclination angle $I$. For the
examples here, we assume that the CBD extends from $r_{i,\mathrm{CBD}}=2a$ to
$r_{o,\mathrm{CBD}}=100a$ and that the minidisks extend from the Schwarzschild
inner-most stable circular orbit (ISCO) of the black hole, e.g.,
$r_{i,1}=6GM_{1}/c^{2}$, to the tidal truncation radius, $r_{0}\approx
0.27a=0.27(\Omega_{b})^{-2/3}(GM)^{1/3}$ (Roedig et al., 2014). The quantity
$T_{i,1}$ is the temperature in the primary minidisk at $r_{i,1}$. We
emphasise that in this model, Eqs. (11) set the average flux scale for each
disk component while time variability comes from the binlite accretion-rate
time series of Section 3.2.
#### Example Lightcurves
We compute example lightcurves in the V-band (optical) for different binary
viewing angles and eccentricities in Figures 6 and 7. For these examples we
choose binary parameters $M=2\times 10^{9}{M_{\odot}}$, $P=1$yr, place the
source at a luminosity distance of $1.5$ Gpc ($z\approx 0.29$), and prescribe
a total accretion rate onto the binary of $10\%$ of the Eddington rate, with
$10\%$ accretion efficiency. To set the amplitude of Doppler-boost variability
we choose a spectral index in the observing band of $\alpha=-1$ (see, e.g.,
Charisi et al., 2018) and keep $\cos{I}=1$ fixed for easy comparison
throughout. For the accretion-rate ratio we use Eq. (4). For an eccentricity
of $e=0.4$, this results in a value of $Q=0.169$, and V-band flux ratios of
$\chi_{V,1}=0.308$ and $\chi_{V,2}=0.160$. Hence, in this example, the
minidisks are contributing $\approx 53\%$ of the total V-band flux. Note that
this relative contribution can be a strong function of observing band, binary
masses, and accretion rate, ranging from 0 to 1. We compute the flux in the
V-band, $F_{V}(t)$ by multiplying Eq. (7) by the sum of Eqs. (11) evaluated at
$\nu=5.5\times 10^{14}$ Hz, and then computing an approximate V-band apparent
magnitude $m_{V}=-2.5\log_{10}\left[F_{V}(t)/F_{V,0}\right]$, using the V-band
zero-point flux of $F_{V,0}=3630.22$ Jy.
$\begin{array}[]{cc}\includegraphics[scale={0.35}]{fin_fig/LCmagV_vs_t_e0.40_omega0.00_chi10.307801_chi20.160371_Mbin9.30103_Pbin1yr_fEddCBD0.1_t00.00P}&\includegraphics[scale={0.35}]{fin_fig/LCmagV_vs_t_e0.40_omega90.00_chi10.307801_chi20.160371_Mbin9.30103_Pbin1yr_fEddCBD0.1_t00.00P}\\\
\includegraphics[scale={0.35}]{fin_fig/LCmagV_vs_t_e0.40_omega180.00_chi10.307801_chi20.160371_Mbin9.30103_Pbin1yr_fEddCBD0.1_t00.00P}&\includegraphics[scale={0.35}]{fin_fig/LCmagV_vs_t_e0.40_omega270.00_chi10.307801_chi20.160371_Mbin9.30103_Pbin1yr_fEddCBD0.1_t00.00P}\end{array}$
Figure 6: V-band apparent magnitude lightcurves for prograde systems, (see
Section 4). Each panel is drawn for a different azimuthal viewing angle of the
binary orbit relative to the argument of pericenter, $\varpi$, and each
differently colored line is for a different binary inclination to the line of
sight. The dashed line represents a face-on binary and so exhibits purely
accretion-rate-induced flux variability (compare to Fig. 2). The strongest
Doppler and lensing effects arise for the red line, drawn for a binary
inclined close to the line of sight, $I=85^{\circ}$. Each panel assumes a
binary with $M=2\times 10^{9}{M_{\odot}}$ and $P=1$yr, at a distance of $1.5$
Gpc, and accreting at $10\%$ of the Eddington rate.
$\begin{array}[]{cc}\includegraphics[scale={0.35}]{fin_fig/LCmagV_vs_t_e0.20_omega0.00_chi10.278139_chi20.206308_Mbin9.30103_Pbin1yr_fEddCBD0.1_t00.00P}&\includegraphics[scale={0.35}]{fin_fig/LCmagV_vs_t_e0.40_omega0.00_chi10.307801_chi20.160371_Mbin9.30103_Pbin1yr_fEddCBD0.1_t00.00P}\\\
\includegraphics[scale={0.35}]{fin_fig/LCmagV_vs_t_e0.60_omega0.00_chi10.317951_chi20.141664_Mbin9.30103_Pbin1yr_fEddCBD0.1_t00.00P}&\includegraphics[scale={0.35}]{fin_fig/LCmagV_vs_t_e0.80_omega0.00_chi10.29837_chi20.176344_Mbin9.30103_Pbin1yr_fEddCBD0.1_t00.00P}\end{array}$
Figure 7: The same as Figure 6 except each panel is for a different binary
orbital eccentricity and the same viewing azimuth. Note that, as in Figure 6,
viewing angles $\varpi$ and $I$ affect only the Doppler boost and lensing
signatures while the binary eccentricity affects both accretion variability
and Doppler+lensing magnifications.
Figures 6 and 7 show that a wide range of lightcurve morphologies arise when
allowing viewing angle and eccentricity to vary. It is significant that each
lightcurve is uniquely fixed by binary and observer parameters. Specifically,
the Doppler+lensing-induced and accretion-rate-induced features cannot be
shifted in phase independently of one another. This is because both accretion-
rate and the Doppler+lensing variability have features fixed to specific
values of the binary phase. The peaks and troughs of the Doppler boost
modulation (as well as its shape) and the lensing flares occur at unique
values of the binary phase for a given eccentricity and observing angle, while
accretion-rate variability for eccentric orbits encodes the binary phase via
accretion-rate peaks that occur near pericenter for prograde orbits and
between pericenter and apocenter for retrograde orbits (see Figure 2).
In contrast, for a near-circular-orbit binary, the peak of orbital timescale
variability is related to the passage of a binary component by the near-side
of the lopsided circumbinary cavity (e.g., D’Orazio et al., 2013) and so
depends on the relative orientation of the cavity and not the binary phase
with respect to the observer’s line of sight. Put another way, a lightcurve
exhibiting hydrodynamic and Doppler+lensing variability for a near-circular
orbit binary would only allow identification up to an undetermined orientation
of the circumbinary disk cavity on the sky. For an eccentric binary this free
parameter is eliminated, and the lightcurve model is fully specified by binary
parameters and the observer’s viewing angle.
That the lightcurves in Figures 6 and 7 are unique to the chosen binary
parameters and observer angles offers considerable constraining power compared
to the circular-orbit case. It also makes this signature more difficult to
duplicate via non-SMBHB drivers of variability.
Finally, we note that for equal-mass binaries it is often assumed that the
orbital Doppler effect is nullified or greatly diminished since both black-
hole minidisks are assumed to be emitting at the same luminosity, and via the
two-body problem, will have opposite line-of-sight velocities (e.g., D’Orazio
et al., 2015). However, because the accretion rate can be split unequally
between the components of eccentric binaries (Fig. 5), the Doppler boost can
still cause significant modulations for equal mass binaries when the orbital
eccentricity is non-zero.
## 5 Discussion and Conclusion
We have analysed the variability (Fig. 1) and relative magnitudes (Fig. 5) of
accretion rates measured from viscous hydrodynamical simulations of disks
accreting onto equal-mass, eccentric binaries in both prograde and retrograde
configurations. With the goal of generating lightcurve models to facilitate
searches for accreting binaries, we developed a tool, named binlite, which can
rapidly generate accretion-rate time series at any eccentricity in our
continuous sweep of simulations ($e\leq 0.8$). We then post-processed these
accretion-rate time series to generate simple models for lightcurves at
optical wavelengths, including also observer-dependent effects of orbital
Doppler boosting and gravitational self-lensing.
It is important to note that the details of the accretion rates presented here
will likely vary with different included physics, sink prescriptions, and
numerical methods for solving the equations of hydrodynamics. However, while
these simulations correspond to a simplest non-trivial inclusion of 2D,
viscous, isothermal hydrodynamics, they capture some robust features that lead
to accretion-rate periodicities observed over a wide range of calculations
that include different physics: 3D (e.g., Moody et al., 2019), self-gravity
(Franchini et al., 2024b), magneto-hydrodynamics (MHD, e.g., Shi et al., 2012;
Shi & Krolik, 2015), General Relativity (e.g., Noble et al., 2021), non-
isothermal equations of state (e.g., Westernacher-Schneider et al., 2022; Wang
et al., 2023), for fixed and live binaries (e.g., Franchini et al., 2023), and
are robust over a wide range of numerical techniques (Duffell et al., 2024).
Hence, while exact shapes of accretion-rate times series will depend on the
physical parameters and numerical methods employed, the accretion-rate times
series available through binlite can give insight into the types of accretion
variability expected and aid in building templates with which to search for
such signatures.
Furthermore, the simple lightcurve models presented in Section 4, could be
expanded and adapted to numerous situations. More complex spectra that take
into account different accretion flow properties could be added to this
picture, e.g., emission characteristics of radiatively inefficient accretion
flows, (see the Methods Section of D’Orazio et al., 2015), or those tailored
to proto-planetary disks (e.g., Zhu, 2015). Timescales for the disk spectra to
respond to the changing accretion rate could also be taken into account. For
example, the lightcurve generation procedure presented here could be modified
by smoothing the reconstructed accretion rates in time with a smoothing kernel
set by a buffering timescale due to, e.g., photon diffusion. Beyond this, the
fluid properties of the disk (via post-processing or inclusion of radiative
cooling terms in the energy equation) can be used to generate mock spectra, as
has been done for a much smaller parameter space in a number of works using
viscous hydrodynamics (e.g., Farris et al., 2015; Tang et al., 2018;
Westernacher-Schneider et al., 2022; Krauth et al., 2023; Franchini et al.,
2024a; Cocchiararo et al., 2024), as well as general relativistic MHD (e.g.
d’Ascoli et al., 2018; Combi et al., 2021; Gutiérrez et al., 2022; Avara et
al., 2023).
In addition to advancing lightcurve models with the accretion-rate time series
investigated here, the accretion rates accessible with this tool should also
be updated with the newest, and a wider range of, simulation results.
Utilising both simple and fast simulations, which will expand available data
to a wider range of parameter space (e.g., a wider range of binary and disk
parameters for the types of simulations analysed here (D’Orazio et al., 2016;
Tiede et al., 2020; Dittmann & Ryan, 2022, 2023)), and also simulations
including more physics that can improve accuracy in smaller portions of
parameter space. We plan to add such improvements over time from our own
calculations and also from the wider community.
## 6 Public availability : binlite
We have developed a simple Python package for rapidly generating periodic
accretion rate time series and associated flux series at any eccentricity in
our continuous sweep simulations.
binlite is available in the Python Package Index, and it can be installed with
⬇
python -m pip install binlite
and imported locally as
⬇
import binlite as blt
It contains two main modules
⬇
blt.accretion
blt.flux
for generating variability series of the mass accretion onto the binary and
the flux at a given frequency (under the assumptions detailed in Section 4)
respectively. The source code and more detailed documentation are also
available at github.com/nbia-gwastro/binlite.
D.J.D. received funding from the European Union’s Horizon 2020 research and
innovation programme under Marie Sklodowska-Curie grant agreement No.
101029157. D.J.D. and C.T. acknowledge support from the Danish Independent
Research Fund through Sapere Aude Starting Grant No. 121587. P.D. acknowledges
support from the National Science Foundation under grant AAG-2206299.
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# Periods of the Long-Term Variability of the Blazar 0716+714 and
Their Inter-Correlations in a Helical Jet Model
Marina S. Butuzova Crimean Astrophysical Observatory of RAS, 298409, Nauchny,
Russia
###### Abstract
Various quasi-periods for the long-term variability of the radio emission,
optical emission, and structural position angle of the inner part of the
parsec-scale jet in the blazar 0716+714 have been detected. The relationships
between these quasi-periods are interpreted assuming that the variability
arises due to helical structure of the jet, which is preserved from regions
near the jet base to at least 1 milliarcsecond from the core observed in radio
interferometric observations. The radiating jet components should display
radial motions with Lorentz factors of $\approx 3$, and decelerate with
distance from the jet base. The best agreement with the data is given in the
case of non-radial motions of these components with a constant physical speed.
It is also shown that the helical shape of the jet strongly influences
correlations both between fluxes observed in different spectral ranges and
between the flux and position angle of the inner part of the parsec-scale jet.
blazar, S5 0716+714, helical jet
Astronomy Reports, Volume 62, Issue 10, pp.654-663, 2018, DOI:
10.1134/S1063772918100037
## 1 Introduction
Blazars are a class of Active Galactic Nuclei (AGN) whose relativistic parsec-
scale jets are oriented close to the line of sight. Therefore, the flux
density emitted by the jet is enhanced by relativistic effects in the
observer’s frame, so that it dominates the emission of other parts of the AGN.
This may be able to explain the fact that no lines are observed in the
spectrum of the blazar 0716+714. Indirect estimates of its redshift have
yielded $z\approx 0.31$ (Bychkova et al., 2006; Nilsson et al., 2008),
$0.2315<z<0.3407$ (Danforth et al., 2013), and $z\geq 0.52$ (Sbarufatti et
al., 2005). This object is variable over the entire electromagnetic spectrum,
on both short and long time scales (see, e.g., Wagner et al., 1996; Raiteri et
al., 2003; Wu et al., 2007; Poon et al., 2009; Gorshkov et al., 2011; Volvach
et al., 2012; Rani et al., 2013; Liao et al., 2014; Bychkova et al., 2015).
0716+714 has also been observed with Very Long Baseline Interferometry (VLBI)
in the framework of a 2 cm survey using the Very Long Baseline Array (VLBA),
and monitored as part of the MOJAVE project (see, e.g., Bach et al., 2005;
Lister et al., 2013; Pushkarev et al., 2017). The bright compact VLBI core
visible in these maps is the region of the jet where the medium becomes
optical thin to radiation at the given wavelength. For more than 150 sources
(including 0716+714), the position of the VLBI core has been found to shift
closer to the jet base with increasing frequency (Pushkarev et al., 2012),
interpreted as an effect of synchrotron self-absorption in the jet (Marcaide &
Shapiro, 1984; Lobanov, 1998; Kovalev et al., 2008). That is, the magnetic-
field strength and density of the radiating particles decrease with distance
from the jet base, so that the medium becomes optically thin to radiation at
increasingly lower frequencies. Synchrotron self-absorption could also be
responsible for the observed delays between flares observed at different radio
frequencies on single dishes (see, e.g., Kudryavtseva et al., 2011; Agarwal et
al., 2017). Without interpretation by synchrotron selfabsorption a number of
authors have found time delays between variability at different radio
frequencies and in different spectral ranges.
For example, Raiteri et al. (2003) found that the delay between the
variability of 0716+714 at 22$-$23 GHz and 15 GHz is 6$-$9 days, between
22$-$23 and 8 GHz is 22$\pm$2${}^{\text{d}}$, and between 15 and 5 GHz is
53$\pm$2${}^{\text{d}}$. No reliable correlation between the optical and 15
GHz fluxes has been found. However, Rani et al. (2013) found a correlation
between the V and 230 GHz fluxes with a time delay of $\approx 65^{\text{d}}$.
The delay between the optical and gamma-ray ranges is $\approx 1.4^{\text{d}}$
(Larionov et al., 2013). The millimeter wave length variability lags the
gamma-ray variability by $82\pm 32^{\text{d}}$ (Rani et al., 2014). Wu et al.
(2007) and Poon et al. (2009) attempted to determine the delays between the
variability of 0716+714 in different optical bands. If there are such delays,
they are less than the time resolution of those observations.
Thus, the flux variability at low frequencies is delayed compared to the
variability at higher frequencies. Analyses of long-term series of
observations reveal different quasi-periods111Here and below, quasi-periods
refer to variability periods detected in certain time intervals at a specified
confidence level using specialized methods such as discrete correlation
functions (Edelson & Krolik, 1988), structure functions (Simonetti et al.,
1985), and others. for the long-term optical ($\approx 3.3$ yrs (Raiteri et
al., 2003)) and radio (5.5$-$6 yrs (Raiteri et al., 2003; Bychkova et al.,
2015; Liu et al., 2012) variability, and also for variations in the position
angle of the inner jet PA${}_{\text{in}}$ ($\approx 11$ yrs Lister et al.
(2013)).
The variations of PA${}_{\text{in}}$ can be explained in a natural way if the
jet has a helical shape (Bach et al., 2005; Lister et al., 2013), which also
often provides the simplest interpretation of a variety of observed properties
of AGNs. For example, in the case of 0716+714, variations in the spectral
energy distribution (Ostorero et al., 2001) and the kinematics of features in
the parsec-scale jet (Butuzova, 2018) have both been explained in this way.
The helical jet shape also gives rise to periodic variations in the viewing
angle of the radiating regions, leading to corresponding variations of the
Doppler factor, which should be manifest as longterm periodicity of the flux
variations. The differences in the long-term periods for the radio
variability, optical variability, and PA${}_{\text{in}}$ variations can be
explained by either an overall deceleration of the speed of the jet or non-
radial motions of the radiating regions in the jet. The latter is most
probable for 0716+714 (see Section 2). Section 3 presents interpretations of
the following results. The first is the opposite results obtained in searches
for correlations between the radio and optical flux variations using the same
statistical method (Raiteri et al., 2003; Rani et al., 2013) The second is the
fact that time intervals when there is a strong positive correlation between
PA${}_{\text{in}}$ and the gamma-ray flux alternate with intervals in which
there is a strong negative correlation between these quantities (Rani et al.,
2014). A discussion of the obtained results and our conclusions are presented
in Section 4.
## 2 Relationship between the long-term variability periods for various
observed quantities
Analysis of many-year radio and optical light curves of the blazar 0716+714
indicate the presence of quasi-periods in the long-term variability. Optical
data for 1994$-$2001 indicated a period of about 3.3 yrs (Raiteri et al.,
2003). This period was never confirmed, possibly because other studies
considered data obtained over shorter time intervals. For example, Rani et al.
(2013) did not detect any reliable periods for the long-term variability in
their analysis of data for 2007$-$2010\. It is difficult to detect long-term
periodicity in the optical due to the superposition of short-term flares on
the longer-term trend. In the radio, where the flare component is weaker,
variability periods of 5.6$-$6 yrs at 14.5 and 15 GHz for 1978$-$2001 (Raiteri
et al., 2003), 5.5$-$6 yrs at 22 GHz for 1992.7$-$2001.2 (Bach et al., 2005),
and $5.8\pm 0.4$ yrs at 15 GHz for data after 2001 (Liu et al., 2012) have
been found.
Data from more than 30 years of observations carried out on telescopes of the
Crimean Astrophysical Observatory (CrAO), the Metsahovi Radio Observatory, and
the University of Michigan Radio Astronomy Observatory at frequencies from 4.8
to 36.8 GHz also indicate the presence of a period of $\approx 8$ yrs,
together with shorter periods (Bychkova et al., 2015). Quasi-periodicity is
also present in the variations of the position angle of the inner parsec-scale
jet of 0716+714. Analysis of data for 26 epochs of observations (from 1992.7)
at 2.9, 8.4, 15.3, and 22.2 GHz led to the detection of variations in the
position angle of the jet lying within 1 mas from the VLBI core with a period
of $7.4\pm 1.5$ yrs and an amplitude of $3.5^{\circ}$ (Bach et al., 2005).
Observations at 15 GHz obtained from 1994.5$-$2011.5 displayed a period for
the PA${}_{\text{in}}$ variations of 10.9 yrs and an amplitude of $11^{\circ}$
(in this case, for the mean position angle of all jet features at distances
$0.15-1$ mas from the core, weighted according to their flux densities)
(Lister et al., 2013).
Therefore, the lack of agreement between the periods for the radio and optical
variability seems to suggest that the brightness variations of the blazar
0716+714 in these spectral ranges are associated with physical processes
occurring in its jet. On the other hand, the periodic variations of the
position angle of the inner jet testify to periodic variations in the jet
direction, leading to variations in the spectral flux density, since
$F_{\nu}\propto\nu^{-\alpha}\delta^{s+\alpha},$ (1)
where $\nu$ is the observing frequency, $\alpha$ the spectral index, $s=3$ if
the depth of the radiating region can be neglected ($s=2$ otherwise), and the
Doppler factor is
$\delta=\delta\left(\theta,\beta\right)=\sqrt{1-\beta^{2}}\left(1-\beta\cos\theta\right)^{-1}.$
(2)
Here, $\theta$ is the angle between the velocity vector for a jet component
and the line of sight at the given time, and $\beta$ is the physical speed of
the radiating feature in units of the speed of light $c$. If $\beta$ is
constant, periodic variations of $\theta$ will give rise to long-term
variability of the radio and optical flux densities and of PA${}_{\text{in}}$
with the same period. However, the periods for these three types of variations
are different. We will elucidate the origins of this contradiction under the
hypotehsis that the jet is helical in shape. We will explore this using the
schematic representation of a jet comprised of individual components forming a
helical line on the surface of an notional cone (Butuzova, 2018). This
corresponds to results of recent studies based on stacked VLBI images for
individual sources (Pushkarev et al., 2017), which indicate that, for many
sources, including 0716+714, the jet features on scales from hundreds to
thousands of parsecs are located inside a cone. We take jet components to be
individual radiating regions of the jet that become observable when they reach
distances from the VLBI core of $\lesssim 0.1$ mas. For our subsequent
arguments, it is not important whether these components are regions of
enhanced particle density or shocks where electrons are accelerated and
subsequently injected into the surrounding space. The position of a component
on the surface of this cone can be described by an azimuth angle $\varphi$
measured along a circular arc formed by a planar cross section of the cone
perpendicular to its axis. (A detailed schematic of the jet and its
geometrical parameters are described by Butuzova (2018)). The coordinate
origin for $\varphi$ was taken to be the point located in the plane of the
line of sight and the cone axis, on the far side of the cone relative to the
observer.
Taking into account synchrotron self-absorption, we assumed that the medium
becomes transparent to the optical radiation of a jet component when it
reaches circle 1 (Fig. 1) formed by the cross section of the cone by a plane
orthogonal to the cone axis at a distance $r_{1}$ from its apex. Continuing
from the active nucleus, the jet component reaches the analogously formed
circle 2 at the distance $r_{2}$. In this region, the medium becomes
transparent to the radio emission of the jet formed in its VLBI core (at the
given frequency). Moving farther, the component reaches circle 3 at a distance
$r_{3}$ from the cone apex, where it is manifest on VLBI maps as the closest
component to the core. We took this to be the distance at which
PA${}_{\text{in}}$ is measured. For each of these circles, we introduced a
notional point moving such that it coincides with the position of the jet
component intersecting the corresponding circle at a given time. As follows
from formulas (1), (2), the periods of the optical and radio variability will
then be equal to the period of rotation of notional points around circles 1
and 2, respectively. Figure 1 shows that the period for variations of
PA${}_{\text{in}}$ will similarly be equal to the period of rotation of a
notional point around circle 3 (see also Butuzova, 2018, Fig. 2). In order for
the helical structure of the jet to be preserved over a long time, we assumed
that the speed of the components was constant, or at least that this speed
varies with distance from the active nucleus in the same way for all
components.
Figure 1: Schematic of the arrangement of regions in the jet in which the
observed optical and radio emission arise and the region in which
PA${}_{\text{in}}$ becomes measureable (the regions where a jet component
crosses the circles 1, 2, and 3, respectively). The helical jet (without
details of its components) is presented at a specified time by the thick bold
curve. The thin curve shows the position of part of the jet in the vicinity of
circle 3 at some subsequent time. We can see that the point where the jet
crosses circle 3 shifts with time (similarly for circles 1 and 2). Over the
variability periods for the optical emission, radio emission or
PA${}_{\text{in}}$, these points undergo a full revolution about circles 1, 2,
and 3, respectively.
### 2.1 Radial Motion of the Jet Components
We will first consider the case when the jet components move outward along the
generating cone (so-called radial, or ballistic motion, see Fig. 2). Without
loss of generality, we can assume that the difference in the azimuth angles of
each of two successive components is some value $\varphi_{d}$ in radians. We
denote $\Delta t^{\prime}_{1}$ to be the time interval in the comoving frame
between the times when any two successive components cross circle 1. Over this
time, a notional point moving along circle 1 describes an arc with length
$\varphi_{d}$. For simplicity, we assumed that $2\pi$ is a multiple of
$\varphi_{d}$. In this case, some number $n$ of components cross circle 1 over
the rotation period of the notional point. Since the interval between two
events in the observer’s frame is smaller than the interval in the source rest
frame by a factor $\delta$,
$\Delta t^{\prime}=\delta\Delta t,$ (3)
the variability period for the optical emission in the observer’s frame will
be
$P_{1}=\sum_{j=1}^{n}\frac{\Delta
t^{\prime}_{1}}{\delta\left(\theta\left(\varphi\right)\right)}\approx\frac{n\Delta
t^{\prime}_{1}}{\delta\left(\theta_{0},\beta\right)}.$ (4)
The right-hand side of (4) was obtained as follows. The angle $\theta$ of the
components crossing circle 1 over the period varies from $\theta_{0}-\xi$ to
$\theta_{0}+\xi$, where $\theta_{0}$ is the angle between the cone axis and
the line of sight. According to formula (2), the Doppler factor varies
cyclically in some interval. Deviations of the Doppler factor from its mean
value can be neglected, since their magnitudes are not large, due to the
smallness of the angle $\xi$, and their sum over the period is zero. For our
further estimates, we took the mean Doppler factor to be
$\delta\left(\theta_{0},\beta\right)$.
Figure 2: Schematic for interpreting the variability periods in the case of
ballistic motion of the jet features. Part of the helical jet in the vicinity
of the circle 1 is shown by the bold curve. The filled squares represent
several jet components. The trajectories of each of the components lie along
the generator of the cone (dashed curves). The hollow squares show the
positions where these components will intersect the circles 1 and 2. The arrow
indicates the direction of motion of a notional point along the circle 2
(similarly for the circle 1).
Moving farther, the component crosses circle 2. The variability period for the
radio emission $P_{2}$ can be written similarly to (4), but with the subscript
“1” replaced with “2”. Due to the character of the motion (Fig. 2), the
azimuthal angle of each jet component does not vary with time. The number of
components crossing circles 1 and 2 during the variation period $\varphi$ is
also constant. The time intervals between the moments of intersection by two
successive components of circles 1 and 2 are equal $\left(\Delta
t^{\prime}_{1}=\Delta t^{\prime}_{2}\right)$. In the absence of deceleration
of the components, we should have $P_{2}=P_{1}$. Since this is not observed,
we supposed that the speed of the components crossing circle 2 was
$\beta_{2}=a\beta_{1}$. Here, $0<a<1$ and $a>\beta_{2}$ (otherwise,
$\beta_{1}>1$). It follows from (4) that the ratio of the variability periods
in the optical and radio will be
$\frac{P_{1}}{P_{2}}=\frac{\delta\left(\theta_{0},\beta_{2}\right)}{\delta\left(\theta_{0},\beta_{1}\right)}.$
(5)
We used (2) and (5) to write the equation
$\frac{1-\beta_{1}\cos\theta_{0}}{1-a\beta_{1}\cos\theta_{0}}\sqrt{\frac{1-a^{2}\beta_{1}^{2}}{1-\beta_{1}^{2}}}=\frac{P_{1}}{P_{2}},$
(6)
which was solved numerically for $a$ for values $\beta_{1}=0.3-0.9999$
(corresponding to Lorentz factors $\Gamma=1-70$) with $P_{1}=3.3$ yrs and
$P_{2}=5.8$ yrs.
We found that one root is always greater than one. The other root is negative
when $\beta_{1}<0.52$ and satisfies our conditions when $\beta_{1}\geqslant
0.52$. When $a\approx 0.962$ and $\beta_{1}=0.9948$, the maximum value
$\beta_{2}\approx 0.957$ is reached, corresponding to $\Gamma=3.5$. Solving
(6) for $P_{2}$ and the variation period for the position angle of the inner
jet for this interval of $\beta_{1}$ values yields the maximum value
$\beta_{2}=0.949$ (for $a\approx 0.954$ and $\beta_{1}=0.995$), which
corresponds to $\Gamma=3.1$. Thus, agreement of the variability periods
observed at difference distances from the jet base can be achieved when
$\Gamma\sim 3-4$. This does not agree with the values $\Gamma\sim 10-20$
inferred from observations of superluminal motions of jet components in
0716+714 (see, e.g., Bach et al., 2005; Nesci et al., 2005; Pushkarev et al.,
2009).
### 2.2 Non-Radial Motion of the Jet Components
Let us now consider a helical jet whose components move non-ballistically,
i.e., at some angle to a radial trajectory. We denote $p$ to be the pitch
angle (angle between the generating cone and the velocity vector of a jet
component), and $\psi$ to be the angle between the tangent to the helix of the
jet and the generating cone at a given point (Fig. 3). If $p=\psi$, the jet
will appear stationary in space, and will always cross circles 1, 2, and 3 at
the same points. In this case, there should be no periodic variability, since
the angle $\theta$, and consequently the Doppler factor, do not change in the
regions responsible for the observed quantities. If $p\neq\psi$, we will
observe the jet helix rotating about its axis. Due to the conical geometry of
the jet, the variations of the azimuth angle $\varphi$ decrease with
increasing $r$. However, we are interested in variations of $\varphi$ at the
constant distances $r_{i}$ from the cone apex to the circles $i$ corresponding
to the regions making the main contributions to the optical ($i=1$) and radio
($i=2$) emission, and to the region where PA${}_{\text{in}}$ can be measured
($i=3$) (Fig. 1).
Figure 3: Schematic of a helical jet with non-ballistic motion of its
components (shown by the filled squares). Part of the helical path of the jet
located near circle 1 is shown (bold curve). The thin curves show the position
of this part of the jet and the components at some later time, with $\psi\neq
p$. The triangles on circle 1 show the places where the jet components cross
circle 1. The arrows show the direction of motion of a notional point along
circle 1.
Variations of the azimuthal angle of the part of the jet reaching a given
distance $r_{i}$ from the cone apex can be found from the schematic presented
in Fig. 4, under the condition that $\beta c\,dtr_{i}$:
$d\varphi\approx\frac{\beta c\,dt\,\sin(\psi-p)}{r_{i}\sin\xi\cos\psi},$ (7)
where $\xi$ is the opening angle of the cone ($\xi=1^{\circ}$, Butuzova,
2018). Since the angular frequency of a notional point moving along circle $i$
is $\omega_{i}=d\varphi/dt$, we find that the ratio of the periods of two
observable quantities $i$ and $k$ is equal to the ratio of the distances from
the cone apex to the region of the jet where these quantities are measured:
$\frac{P_{i}}{P_{k}}=\frac{r_{i}}{r_{k}}.$ (8)
Figure 4: Schematic of part of the jet (bold curves) at a distance $r_{i}$
from the cone apex at the initial time (1) and after a time dt (2).
Substituting various pairs of the known variability periods for 0716+714 into
(8) yields the three independent relations
$\begin{split}r_{2}=&1.76\,r_{1},\\\ r_{3}=&3.30\,r_{1},\\\
r_{3}=&1.88\,r_{2}.\end{split}$ (9)
It follows from the last two equations of (9) that $r_{2}/r_{1}\approx 1.76$,
which is equal to the directly inferred ratio $r_{2}/r_{1}$ in the first
equation of (9). Thus, the observed periods for the long-term variability in
the ratio, optical and inner-jet position angle PA${}_{\text{in}}$ show good
consistency. This supports a picture with non-radial motion of the jet
features with $p\neq\psi$ and an absence of deceleration at the distances from
the active nucleus considered here.
Let us suppose that PA${}_{\text{in}}$ is measured at a specified distance
from the VLBI core at 15 GHz, equal to 0.15 mas. Then,
$r_{3}-r_{2}=0.15$(mas)$/\sin\theta_{0}$. Using the third equation of (9) and
$\theta_{0}=5.3^{\circ}$ (Butuzova, 2018), we obtain the distance $r_{2}=1.84$
mas. In a $\Lambda$CDM model with $H_{0}=71$ km s-1Mpc-1, $\Omega_{m}=0.27$,
and $\Omega_{\Lambda}=0.73$ (Komatsu et al., 2009) and adopting a redshift of
$z=0.3$ for 0716+714, the physical distance from the cone apex to the position
of the VLBI core is 8.1 pc. This is consistent with the distance of the VLBI
core from the black hole of 6.68 pc at 15.4 GHz determined by Pushkarev et al.
(2012) using these same cosmological parameters. This provides additional
support for our picture of the jet. Continuing our reasoning using (9), we
find that the distance between the jet apex and the region where the optical
emission becomes observable is 4.6 pc. Due to the small delay in the
variability (at the limit of the time resolution of high frequency data of
Rani et al., 2013; Larionov et al., 2013), we infer that the gamma-ray and
optical emission is formed in the same region, or at least in closely spaced
regions. Rani et al. (2014) estimated that the gamma-ray emission arises from
a region located $3.8\pm 1.9$ pc closer to the black hole relative to the VLBI
core (observed at 43 and 86 GHz), also consistent with our results.
## 3 Correlation between flux and inner-jet position angle
Assuming that the long-term variability of the blazar 0716+714 is due to
periodic variations in the direction of motion of the jet components, we
expect there should be a relationship between the measured spectral flux
density $F_{\nu}$ and the inner-jet position angle. Such relationships between
the gamma-ray flux $F_{\gamma}$ and both the flux from the VlBI core at 43 and
86 GHz and PA${}_{\text{in}}$ have been investigated for 0716+714 by Rani et
al. (2014). They found that time intervals with a strong positive correlation
between $F_{\gamma}$ and PA${}_{\text{in}}$ alternate with intervals in which
there is a strong negative correlation between these two quantities.
This result was explained by Rani et al. (2014) by the fact that, in a curved
(possibly helical) jet, the regions responsible for the observed quantities
are located at different distances from the active nucleus, and therefore have
different viewing angles $\theta$. If the $\theta$ values for two regions are
roughly the same, there will be a strong positive correlation between the
corresponding observed quantities. If the $\theta$ values are different, there
may be a strong negative correlation. However, for the helical jet we are
considering here, there is no direct relationship between $F_{\nu}$ and
PA${}_{\text{in}}$, which can be explained as follows. According to Butuzova
(2018) (Eq. (1)), the position angle is
$\text{PA}_{\text{in}}=\text{PA}_{0}+\Delta\text{PA}=\text{PA}_{0}+\frac{\sin\xi\sin\varphi}{\cos\xi\sin\theta_{0}+\sin\xi\cos\theta_{0}\cos\varphi}$
(10)
(PA0 is the mean value of PA${}_{\text{in}}$) and the angle between the
component velocity and the line of sight depend on the geometrical parameters
of the cone and the position of the component relative to the cone axis and
the line of sight (i.e., on the angle $\varphi$). We can use these equations,
where the periodicity appears only due to variations in the azimuthal angle,
to model the observed correlation between the flux and PA${}_{\text{in}}$.
Let us first consider the case of radialmotion of the components. We find from
(1) with the substitution of (2), in which (see Butuzova, 2018, Eq. (5))
$\theta=\theta_{b}=\arccos\left(\cos\xi\cos\theta_{0}-\sin\xi\sin\theta_{0}\cos\varphi\right),$
that the extrema of the function $F_{\nu}$ occur at values $\varphi=n\pi$
(maxima for $n=1$, 3, 5, etc. and minima for $n=0$, 2, 4, etc.). That is, the
qualitative variations of $F_{\nu}$ do not depend on the choice of $\nu$,
$\alpha$, $s$, and $\beta$. The upper panel of Fig. 5 presents the variations
of $F_{\nu}$ calculated using (1) and (10) (for $\alpha=0.5$, $s=2$, and
$\beta=0.995$, which corresponds to $\Gamma=10$) and deviations of the inner-
jet position angle from its mean value $\Delta$PA (for $\theta_{0}/\xi=5.3$)
as functions of the azimuthal angle $\varphi$. The flux is normalized so as to
enable a visual comparison of its variations with the variations of
$\Delta$PA. The resulting curves were divided into several sections in
$\varphi$, such that qualitative variations in the behavior of both quantities
did not arise within each section.
Figure 5: Deviations of the position angle of the inner jet from its mean
value $\Delta$PA (solid curve) and the observed flux $F_{\nu}$ (dashed curve)
over the variation period for $\varphi$ (upper), and the correlation
coefficient $r_{p}$ between these quantities (lower) for ballistic motion of
features of a helical jet.
The corresponding formulae were used to compose datasets of $\Delta$PA and
$F_{\nu}$ values for each section, for variations of $\varphi$ in steps of
$0.5^{\circ}$, and the Pearson correlation coefficient ($r_{p}$) between these
datasets was calculated. The resulting $r_{p}$ value will have its maximum
possible value, since, in contrast to the observational data, there is no
measurement error. An alternation of intervals of strong positive and negative
correlations can be seen (Fig. 5, lower panel). Intermediate values of the
correlation coefficient are present only in short intervals. This theoretical
result is in qualitative agreement with the observations of the behavior of
the gamma-ray flux and the inner-jet position angle considered by Rani et al.
(2014). Thus, we found that, for the same flux value (corresponding to the
same viewing angle $\theta_{b}$), either a positive or negative correlation
with PA${}_{\text{in}}$ could be present in the observations.
In the case of non-radial component motions, the variations of the viewing
angle are given by Eqs. (11)-(13) from (Butuzova, 2018), which we do not
present here due to their unwieldiness. The extrema of the function $F_{\nu}$
occur at the values
$\varphi=-\arcsin\left(\frac{\sin
p}{\sqrt{\cos^{2}p\sin^{2}\xi+\sin^{2}p}}\right)+\pi n,$ (11)
where even $n$ correspond to maxima and odd $n$ to minima of the function
$F_{\nu}$. According to (11), for the pitch angle found by Butuzova (2018) of
$p=5.5^{\circ}$, the flux reaches a maximum when $\varphi\approx 100^{\circ}$.
As $p$ is decreased, the peak $F_{\nu}$ shifts toward 180∘, and the maximum
$F_{\nu}$ occurs for $\varphi\approx 90^{\circ}$ when $p>20^{\circ}$. The
upper panel of Fig. 6 plots the functions $F_{\nu}$ and $\Delta$PA for the
same parameters as in the previous case. The correlation coefficient $r_{p}$
was constructed using an analogous procedure (Fig. 6, middle panel). This
figure shows that a strong positive correlation between the inner jet position
angle and the observed flux should always be present. However, since these
quantities arise in regions located at different distances from the jet base
in our model (see Section 2), we can conclude with confidence that the
azimuthal angles of the components for which $F_{\nu}$ and $\Delta$PA are
observed at a given time differ by some amount $\Delta\varphi$.
Figure 6: Upper: variations of the inner-jet position angle relative to the
mean value $\Delta$PA (dashed curve) and of the flux $F_{\nu}$ for the case of
zero difference in the azimuthal angles of the regions responsible for the
observed quantities (curve 1) and for $\Delta\varphi\approx 79^{\circ}$ (curve
2). The central and lower panels show the correlation coefficients $r_{p}$ for
the former and latter cases, respectively.
Agreement with the results of Rani et al. (2014) requires that this difference
be $\Delta\varphi\approx 79^{\circ}$ (curve 2 for $F_{\nu}$ in the upper panel
and curve for $r_{p}$ in the lower panel in Fig. 6). On the other hand, strong
positive or negative correlations between $F_{\nu}$ and $\Delta$PA are also
possible for radial component motions if the difference in the azimuthal
angles is $\Delta\varphi\approx\pi/2$ or $\approx\pi$ (see Fig. 5).
Consequently, the correlation between the observed quantities may be
insignificant when analyzing data over long time intervals, as was found by
Raiteri et al. (2003), for example, in their analysis of the radio and optical
fluxes during 1994$-$2001\. In contrast, the data for the shorter interval
2007$-$2010 analyzed by Rani et al. (2013) revealed a correlation between the
indicated quantities at a significance level of more than 99$\%$. Further, the
difference in the azimuthal angles of the regions responsible for the observed
quantities can appreciably affect both the correlation coefficient between
PA${}_{\text{in}}$in and $F_{\nu}$ and the duration of the time interval when
a given correlation coefficient is observed. Finally, the character of the
motions of individual components of helical jet cannot be determined by
analyzing the correlation between PA${}_{\text{in}}$ and $F_{\nu}$. It is
important to note that, when investigating correlations between fluxes
observed in different spectral ranges, distinct correlation coefficients in
the different time intervals will also be present.
## 4 Discussion and conclusion
The hypothesis that AGN jets may be helical has been widely applied for
several decades to interpret various observed properties such as their
microvariability (Camenzind & Krockenberger, 1992), the shape and variations
of the spectral energy distributions for the blazars Mrk 501 (Villata &
Raiteri, 1999) and 0716+714 Ostorero et al. (2001), the long-term brightness
variability of OJ 287 (Sillanpaa et al., 1988) and 0716+714 (Nesci et al.,
2005), variations in the speeds and non-radial component motions (Rastorgueva
et al., 2009), and quasi-periodicity of variations of the inner-jet position
angle for 0716+714 (Bach et al., 2005; Lister et al., 2013). A helical jet
shape could form due to precession of the jet nozzle or the development of
(magneto)hydrodynamical instabilities. Kelvin–Helmholtz instability (Hardee,
1982) has been widely studied as a means of estimating the physical parameters
of jets and the ambient medium (e.g., for 3C 120 (Hardee, 2003) and 0836+710
(Perucho et al., 2012)). Alternatively, wavelike perturbations at the
boundaries of the observed isophotes of jets could be related to the
development of a magnetohydrodynamical analog of wind instability (Gestrin &
Kontorovich, 1986).
In this study, we have supposed that the radiating jet components form a
helical curve, without considering their physical nature. For example, the jet
components could be individual radiating parts of the jet (plasmoids or
regions of shocks) or volume elements (in the case of a spatially continuous
radiating jet). A helical shape suggests the presence of periodic variations
of the angle between the jet velocity and the line of sight at some constant
distance from the core, which should be manifest as long-term quasi-periodic
variability of the radiation flux over the entire observed range of the
electromagnetic spectrum of the blazar 0716+714. The differences in the quasi-
periods for the variations of PA${}_{\text{in}}$ (Bach et al., 2005; Lister et
al., 2013) and of the radio- (Raiteri et al., 2003; Bychkova et al., 2015;
Bach et al., 2005; Liu et al., 2012) and optical (Raiteri et al., 2003) fluxes
can most simply be explained in the jet geometry considered if the radiation
in different spectral ranges is emitted at different distances from the jet
apex. This spatial separation of regions radiating at different frequencies
can arise due to synchrotron self-absorption in the jet or the energy losses
of the radiating electrons. In both cases, the higher the frequency of the
observed emission, the closer to the jet base the region in which it is
generated. The delays in the flux variability observed at different
frequencies also testify to the action of this effect (see, e.g., Raiteri et
al., 2003; Larionov et al., 2013; Rani et al., 2014).
In this study, we have brought the variability periods in different spectral
ranges into agreement in the case of radial and non-radial motions of the
radiating components. The former case requires a low Lorentz factor for the
components (no more than 4) with overall deceleration, at least from the
region where the optical emission is formed to 0.15-0.5 mas from the core,
where the position angle of the inner jet is measured. This does not agree
with observations of features in the parsec-scale jet of 0716+714 (Bach et
al., 2005; Nesci et al., 2005; Pushkarev et al., 2009). Moreover, it was shown
by Butuzova (2018) that differences between the relative speeds of components
in the inner and outer jet observed in different years can be explained in the
framework of a helical-jet model with non-ballistic component motions, such as
are observed in the jet (Bach et al., 2005; Rastorgueva et al., 2009). It was
shown that period ratio is equal to ratio of the physical distances from the
jet apex of the regions responsible for the measured quantities. This enables
us to introduce another absolute distance scale, which will subsequently
facilitate deeper studies of the jet properties. In our picture of the jet and
the appearance of long-term variability due to geometric effects, the radio
periods found by Bychkova et al. (2015) cannot carry information about the
properties of the central engine without taking into account the non-radial
motions of the jet components.
The helical shape of a jet with spatially separated regions responsible for
the observed emission at different frequencies and region where the inner jet
position angle is measured complicates studies of correlations between the
observed quantities. It has been shown that there cannot be a constant
correlation coefficient between quantities formed in regions at different
fixed distances from the jet apex. A strong positive correlation observed in
one time interval will be replaced with a negative correlation in another.
This is due to both the different azimuthal angles of these regions and the
fact that $\varphi$ varies irregularly in the observer’s rest frame, due to
variations of $\delta$, especially for non-radial component motions (Butuzova,
2018). This agrees with certain observational facts.
For example, Rani et al. (2013) noted an alternation of intervals when
positive and negative correlations between PA${}_{\text{in}}$ and the gamma-
ray flux of the blazar 0716+714 were observed. Rani et al. (2013) also
indicate that a strong correlation between the gamma-ray and optical fluxes
was observed over roughly 500 days (Pearson correlation coefficient
$r_{p}=0.66$), while $r_{p}=0.36$ over the following $\approx$400 days. A
correlation was also found between the radio flux and PA${}_{\text{in}}$
during 1994$-$2014 ($r_{p}=0.44$) (Liu et al., 2012), while no correlation
between these quantities was found for data obtained from August 2008 through
September 2013 (Rani et al., 2014).
We can introduce some clarity into this picture only if we know the
geometrical parameters of the helical jet, the character of the motion of the
jet components,and the arrangement of the studied regions responsible for
various observed quantities relative to the plane containing the axis of the
helical jet and the line of sight, and not only relative to the line of sight,
as was supposed in the simplest case (Rani et al., 2014). We also showed in
Section 3 that it is not possible to determine the character of the component
motions in a helical jet based on the correlation between the radio flux and
the inner-jet position angle, as was done by Liu et al. (2012).
Our hypothesis that the jet of the blazar 0716+714 has the form of a helical
curve located on the surface of a cone may seem somewhat idealized. However,
this is consistent with the results of many years of VLBI observations (Lister
et al., 2013; Pushkarev et al., 2017). In addition, a helical jet with non-
radial component motions makes it possible to find agreement between estimates
of the component velocities in the VLBI jet and the viewing angle obtained in
different studies, which often differ appreciably (Butuzova, 2018). As we have
shown, such a jet can provide a simple explanation for the differences in the
observed long-term quasi-periods for different quantities, as well as the
differences in the correlations between the observed quantities in different
time intervals.
The author thanks A.B. Pushkarev for useful comments.
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|
∎
11institutetext: R. Machleidt 22institutetext: Department of Physics,
University of Idaho, Moscow, ID 83844, USA
22email<EMAIL_ADDRESS>
# What is ab initio?
R. Machleidt
(Received: date / Accepted: date)
###### Abstract
Microscopic nuclear theory is based on the tenet that atomic nuclei can be
accurately described as collections of point-like nucleons interacting via
two- and many-body forces obeying nonrelativistic quantum mechanics—and the
concept of the ab initio approach is to calculate nuclei accordingly. The
forces are fixed in free-space scattering and must be accurate. We will
critically review the history of this approach from the early beginnings until
today. An analysis of current ab initio calculations reveals that some
mistakes of history are being repeated today. The ultimate goal of nuclear
theory are high-precision ab initio calculations which, as it turns out, may
be possible only at the fifths order of the chiral expansion. Thus, for its
fulfillment, nuclear theory is still facing an enormous task.
## 1 Introduction
The tenet of microscopic nuclear theory is that atomic nuclei can be
accurately described as collections of point-like nucleons interacting via
two- and many-body forces obeying nonrelativistic quantum mechanics—the forces
being fixed in free-space scattering.
The microscopic or ab initio approach to nuclear structure and reactions is
then defined as calculating the properties of nuclei in accordance with the
tenet.
It is the purpose of this note to discuss how consistent or inconsistent the
fundamental model of nuclear theory has been pursued through the history of
nuclear physics and to provide an outlook for the future.
## 2 Early history of the microscopic approach
The microscopic approach to nuclear structure is almost as old as nuclear
physics itself. Brueckner and co-workers introduced Brueckner theory as early
as 1954 [1] and performed the first semi-realistic microscopic nuclear matter
calculation in 1958 [2]. Already that same year, Brueckner discussed finite
nuclei proposing the local density approximation [3].
In the second half of the 1960’s, one of the hottest topics in nuclear
structure physics was calculating the properties of finite nuclei without
recourse through nuclear matter using Brueckner-Hartree-Fock (BHF) theory. The
Oak Ridge National Laboratory (ORNL) with its computer power played a leading
role in this effort that was guided by Thomas Davies and Michel Baranger [4,
5]. BHF (and coupled cluster) calculations of finite nuclei continued into the
early 1970s with work by the Bochum [6] and the Bonn-Jülich groups [7].
In parallel to the above developments, research on the microscopic derivation
of the shell-model effective interaction was conducted (again, applying
Brueckner theory) that had been kicked off by Kuo and Brown in 1966 [8].
Applying the nucleon-nucleon ($NN$) potentials available at the time, the BHF
approach reproduced about one half of the binding energies of closed-shell
nuclei which, in the early phase, was seen as a great success [4], but in the
long run did not satisfy demands for more quantitative predictions. Therefore,
a departure from the microscopic approach happened around 1973 as reflected
most notably in a lead-talk by Michel Baranger at the International Conference
on Nuclear Physics in Munich in 1973 [9].
The shell-model effective interaction suffered a similar fate at the
International Conference on Effective Interactions and Operators in Nuclei in
Tucson, Arizona, in 1975, organized by Bruce Barrett [10].
And so it happened that in the early 1970s, the microscopic approach was
abandoned and replaced by phenomenological effective interactions (also know
as mean-field models): the Skyme interaction [11] as revived by Vautherin and
co-workers [12, 13], the Gogny force [14, 15], and the relativistic mean-field
model of Walecka [16, 17].
Ironically, the calculations with those effective interactions continued to be
called “microscopic”, for which John Negele had provided the (debatable)
justification in his Ph.D. thesis of 1970 [18]. Before calculating finite
nuclei in the local density approximation, Negele had adjusted the
insufficient binding of nuclear matter provided by the Reid soft-core
potential [19] (11 MeV per nucleon) by hand to the presumed empirical value of
15.68 MeV making “the assumption that when higher-order corrections have been
evaluated carefully, nuclear-matter theory will indeed produce the correct
binding” [18]. Negele had many followers [20, 21, 22].
However, the true “deeper reason” for those effective interactions was much
simpler: “To get better results!” [23]. Clearly, the trends that won
popularity in the early 1970s were a setback for the fundamental research in
nuclear structure.
Nuclear structure theory at its basic level is not about fitting data to get
“good” results. Fundamental nuclear structure theory is about answering the
question:
> Do the same nuclear forces that explain free-space scattering experiments
> also explain the properties of finite nuclei and nuclear matter when applied
> in nuclear many-body theory?
One can think of many reasons why the basic tenet should be wrong. According
to the EMC effect, nucleons swell when inserted into nuclei which might affect
the force between nucleons [24]. Meson exchange in the nuclear medium may be
different than in free-space for various reasons [25, 26, 27]. The excitation
of resonances, e. g. $\Delta(1232)$ isobars, within the nucleon-nucleon
interaction process is subject to changes when happening in a nuclear medium
[28, 29, 30, 31]. And many more ideas have been advanced, like e. g., Brown-
Rho scaling [32]. In fact, in the 1970s, a popular belief was that medium
effects on the $NN$ interaction may be the solution to the problem of lacking
saturation [33].
Thus, it is a good question to ask whether medium modifications of nuclear
forces show up in a noticeable way and/or are even needed for quantitative
nuclear structure predictions. But when we re-adjust the free-space forces
arbitrarily to get “good” results, then we will never find out. Note also that
at some (high) energy and high density, the picture of point-like nucleons is
bound to break down [34]. So, the issue behind the nuclear theory tenet is:
Are the energies typically involved in conventional nuclear structure physics
low enough to treat nucleons as structure-less objects?
To come back to history: the renunciation of the truly microscopic approach
lasted about two decades (essentially the 1970s and 80s). Then, in the early
1990s, the microscopic theory was revived by the Argonne-Urbana group [35,
36]. The crucial element in those new microscopic calculations was the
inclusion of a three-nucleon force (3NF). The idea of a nuclear 3NF was not
new. In fact, it is almost as old as meson theory itself [37]. But for years
it had been considered just an academic topic, too difficult to incorporate
into actual calculations, anyhow. But the persistent failure to saturate
nuclear matter at reasonable energies and densities, as well as the the
underbinding of nuclei, finally compelled nuclear structure physicists to take
a serious look at the 3NF issue, as explained in the exemplary Comment by Ben
Day [38] based upon first test calculations by the Urbana group [39]. The 3NF
definitely improved nuclear saturation and the properties of light nuclei,
even though nothing was perfect [36].
## 3 Recent history
After the year of 2000, two changes occurred. First, the term ‘microscopic’
was increasingly replaced by the term ‘ab initio’ [40]—for reasons nobody
knows (but nothing to worry about because both mean the same). Second and more
importantly, nuclear forces based upon chiral effective field theory (EFT)
entered the picture [41, 42]. This development was of great advantage. Note
that for a microscopic approach to be truly microscopic, the free-space forces
need to be accurate. But with phenomenological or meson-theoretic forces it
was difficult to define what sufficiently accurate means, since the errors in
those theories are unknown. However, in the framework of an EFT, the
theoretical uncertainty can be determined and, thus, related with the accuracy
of the predictions. Hence, in the framework of an EFT:
> Accurate free-space forces are forces that predict experiment within the
> theoretical uncertainty of the EFT at the given order.
After 2000, it also became well established that predictive nuclear structure
must include 3NFs, besides the usual two-nucleon force (2NF) contribution.
Another advantage of chiral EFT is then that it generates 2NFs and multi-
nucleon forces simultaneously and on an equal footing. In the $\Delta$-less
theory [43, 44], 3NFs occur for the first time at next-to-next-to-leading
order (NNLO) and continue to have additional contributions in higher orders.
If an explicit $\Delta$-isobar is included in chiral EFT ($\Delta$-full theory
[45, 46, 47, 48]), then 3NF contributions start already at next-to-leading
order (NLO).
In the initial phase, the 3NFs were typically adjusted in $A=3$ and/or the
$A=4$ systems and the ab initio calculations were driven up to the oxygen
region [49]. It turned out that for $A\raisebox{-1.29167pt}{\small$\
\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ }16$ the ground-state energies
and radii are predicted about right, no matter what type of chiral or
phenomenological potentials were applied (local, nonlocal, soft, hard, etc.)
and what the details of the 3NF adjustments to few-body systems were [49, 50,
51, 52, 53, 54].
However, around the year of 2015, the picture changed, when the many-body
practitioners were able to move up to medium-mass nuclei (e. g., the calcium
or even the tin regions). Large variations of the predictions now occurred
depending on what forces were used, and cases of severe underbinding [55] as
well as of substantial overbinding [56] were observed. Ever since, the nuclear
structure community understands that accurate ab initio explanations of
intermediate and heavy nuclei is an outstanding problem.
There have been several attempts to predict the properties of medium-mass
nuclei with more accuracy. Of the various efforts, we will now list four
cases, which are representative for the status, and will denote each case with
a short label for ease of communication. We restrict ourselves to cases, where
the properties of medium-mass nuclei and nuclear matter have been calculated,
because the simultaneous description of both systems is part of the
problem.111Other interesting cases are the models by Soma et al. [57] and
Maris et al. [54] for which, however, presently no nuclear matter results are
available.
Figure 1: Upper panel: Ground-state energies per nucleon, $E/A$, of selected
closed-shell oxygen, calcium, and nickel isotopes as obtained in the “Hoppe”
case [58]. Results are shown for various chiral interactions as denoted. The
blue and orange bands give the NNLO and N3LO uncertainty estimates,
respectively. $\Lambda=450$ MeV in all cases except the green curve. Black
bars indicate experimental data. Lower panel: Same as upper panel, but for
charge radii. (Reproduced from Ref. [58] with permission.)
Figure 2: Ground-state energies per nucleon (top panel) and point-proton rms
radii (bottom panel) for selected medium-mass isotopes as obtained in the
“Hüther” case [59]. The light blue and pink bands represent the theoretical
uncertainties at NNLO and N3LO, respectively. $\Lambda=450$ MeV. Black bars
indicate the experimental data. (Figure courtesy of R. Roth)
* •
“Magic” [60, 61]: A seemingly successful interaction for the intermediate mass
region commonly denoted by “1.8/2.0(EM)” (sometimes dubbed “the Magic force”).
It is a similarity renormalization group (SRG) evolved version of the N3LO 2NF
of Ref. [42] complemented by a NNLO 3NF adjusted to the triton binding energy
and the point charge radius of 4He. With this force, the ground-state energies
all the way up to the tin isotopes are reproduced perfectly—but with charge
radii being on the smaller side [62, 63]. Nuclear matter saturation is also
reproduced reasonably well, but at a slightly too high saturation density
[60].
* •
“GO” [64, 65]: A family of $\Delta$-full NNLO potentials constructed by the
Göteborg/Oak Ridge (GO) group. The authors claim to obtain “accurate binding
energies and radii for a range of nuclei from $A=16$ to $A=132$, and provide
accurate equations of state for nuclear matter” [65].
* •
“Hoppe” [66, 58]: Recently developed soft chiral 2NFs [67] at NNLO and N3LO
complemented with 3NFs at NNLO and N3LO, respectively, to fit the triton
binding energy and nuclear matter saturation. These forces applied in in-
medium similarity renormalization group (IM-SRG [68]) calculations of finite
nuclei up to 68Ni predict underbinding and slightly too large radii [58], see
Fig. 1.
* •
“Hüther” [59]: The same 2NFs used in “Hoppe”, but with the 3NFs adjusted to
the triton and 16O ground-state energies. The interactions so obtained
reproduce accurately experimental energies and point-proton radii of nuclei up
to 78Ni [59], see Fig. 2. However, when the 2NF plus 3NF combinations of
“Hüther” are utilized in nuclear matter, then overbinding and no saturation at
realistic densities is obtained [69], see Fig. 3.
Figure 3: Energy per nucleon, $E/A$, as a function of density, $\rho$, of
symmetric nuclear matter as obtained in calculations with the 2NFs and 3NFs
consistently at NNLO [69]. In the two cases shown, the 2NF is the same, while
the 3NFs are the ones used in the calculations of finite nuclei in the “Hoppe”
and “Huether” cases as denoted. $\Lambda=450$ MeV in both cases. The error
bars show the theoretical uncertainties around saturation, which is expected
to occur in the area of the gray box.
Obviously, in some cases, there appears to be a problem with achieving
simultaneously accurate results for nuclear matter and medium-mass nuclei: In
the “Hoppe” case, nuclear matter is saturated correctly, but nuclei are
underbound; while in the “Hüther” case, nuclei are bound accurately, but
nuclear matter is overbound. Other cases seem to have solved this problem. But
are they all truly ab initio? Our assessment:
* •
“Magic”: The construction of this force includes some inconsistencies. The 2NF
is SRG evolved, while the 3NF is not. Moreover, the SRG evolved 2NF is used
like an original force with the induced 3NFs omitted. Note that ab inito also
implies that the forces are based upon some sort of theory in a consistent
way. This is here not true and, thus, this case is not ab initio.
* •
“GO”: In Ref. [70] it has been shown that the predictions by the $\Delta$-full
$NN$ potentials at NNLO constructed by the Gőteborg-Oak Ridge (GO) group [65]
are up to 40 times outside the theoretical error of chiral EFT at NNLO. So,
they fail on accuracy. The reason for their favorable reproduction of the
energies (and radii) of intermediate-mass nuclei, can be traced to incorrect
$P$-wave and $\epsilon_{1}$ mixing parameters [70]. Thus, this case is
especially far from being ab initio. It is just a repetition of the mistakes
of the early 1970s.
* •
“Hoppe”: In this case, the 2NF and 3NF forces are consistently chiral EFT
based. Moreover, the 2NFs are accurate. However, there is another accuracy
aspect that is, in general, quietly ignored [71, 72]: Are the 3NFs accurate?
The accuracy of the chiral 3NF at NNLO was thoroughly investigated in Ref.
[73] for a variety of cutoffs ranging from 400-550 MeV and large variations of
the NNLO 3NF parameters, $c_{D}$ and $c_{E}$. A typical result is shown in
Fig. 4. It is seen that the 3$N$ data are reproduced within the truncation
errors at NNLO (green bands). On the other hand, it is also clearly seen that
the theoretical uncertainties are very large. Moreover, it was found in Ref.
[73] that the cutoff dependence is weak and that the variations of the 3NF
LECs $c_{D}$ and $c_{E}$ make only small differences relative to the large
uncertainties. Thus, we can assume that the NNLO 3NFs used in “Hoppe” will
yield results that lie within the NNLO uncertainties shown in Fig. 4 by the
green bands and, consequently, the “Hoppe” 3NF is accurate. Hence, “Hoppe”
passes on all accounts and is, therefore, truly ab initio.
* •
“Hüther”: An assessment similar to “Hoppe” applies. Thus, this case is also
truly ab initio.
Figure 4: Predictions for the differential cross section, nucleon and
deuteron analyzing powers $A^{n}_{y}$ and $A^{d}_{y}$ as well as deuteron
tensor analyzing powers $A_{yy}$, $A_{xz}$, and $A_{xx}$ in elastic
nucleon–deuteron scattering at a laboratory energy of 135 MeV at NLO (yellow
bands) and NNLO (green bands). The light- (dark-) shaded bands indicate 95%
(68%) confidence levels. The dotted (dashed) lines show the results based on
the CD-Bonn $NN$ potential [74] (CD-Bonn $NN$ potential in combination with
the Tucson-Melbourne 3NF [75]). Black symbols represent the data together with
their experimental errors. (Reproduced from Ref. [73].)
The bottom line is that not all calculations, which have been published in the
literature under the label of ab initio, are really ab initio. Indeed, of the
cases we considered here, only 50% pass the test. But we need to point out
that even in the two cases we declared ab initio, there are concerns. The NNLO
predictions by Hoppe and Hüther for finite nuclei barely overlap within their
theoretical uncertainties and, for nuclear matter, they do not overlap at all.
Obviously, there are problems with the error estimates and the uncertainties
are much larger than the shown ones. The true NNLO truncation errors of the
Hoppe and Hüther calculations are probably as large as the differences between
the two predictions. In this way, the two predictions are actually consistent
with each other, in spite of their seeming discrepancy. Chiral EFT is a model-
independent theory and, thus, different calculations at the same order should
agree within truncation errors.
At N3LO the predictions differ even more. However, for current N3LO
calculations, a strong caveat is in place. As pointed out in Ref. [76], there
is a problem with the regularized 3NF at N3LO (and higher orders) in all
present nuclear structure calculations. The N3LO 3NFs currently in use are all
regularized by a multiplicative regulator applied to the 3NF expressions that
are derived from dimensional regularization. This approach leads to a
violation of chiral symmetry at N3LO and destroys the consistency between two-
and three-nucleon forces [76]. Consequently, all current calculations that
include a N3LO 3NF contain an uncontrolled error and are, therefore,
unreliable. When a consistent regularization scheme has been found, the
calculations have to be repeated. At the present time, reliable predictions
exist only at NNLO, NLO, and LO.
## 4 The future: ab initio plus precision
Figure 5: Latest ab initio predictions by the LENPIC collaboration [54]:
Ground-state energies and point-proton radii for doubly magic oxygen and
calcium isotopes obtained from the $NN$ potential of Ref. [77] complemented by
NNLO 3NFs using a cutoff of 450 MeV (left-hand panel) and of 500 MeV (right-
hand panel). The blue squares represent the predictions by complete NNLO
calculations with the blue error bands showing the chiral NNLO truncation
uncertainties at the 95% confidence level. The green and purple points and
pink error bands are based upon incomplete calculations and are to be ignored.
Black bars indicate the experimental data. (Reproduced from Ref. [54] with
permission.)
It is comforting to know that at least a few correct ab initio calculations do
exist. But these cases show that the precision at NNLO is very poor. The same
is true for the latest LENPIC calculations [54], see Fig. 5 (which we did not
include in our case study, because nuclear matter results are lacking). At
N3LO (if one day correct such calculations become available) the precision
will most likely not be substantially better.
As stated at the outset, the purpose of the ab initio approach is to test if
the tenet of nuclear theory is correct or not. Within huge errors as, e. g. in
Fig. 4, any approach may come out right. So, that is not a good basis for a
reliable test. We need more precision! This is in particular true for the 3NF
and the reproduction of the 3$N$ data, which has been thoroughly investigated
in Refs [73, 78] with the conclusion that, at N4LO, there is a chance to
achieve the desirable precision—for several reasons. The long- and
intermediate-range topologies of the 3NF at N4LO are expected to be much
larger than the corresponding ones at N3LO because, at N4LO, the subleading
$\pi NN$ seagull vertex is involved with LECs $c_{i}$, which are large [79,
80]. This will provide the 3NF at N4LO with more leverage as compared to N3LO.
Moreover, at N4LO, 13 new 3$N$ contact terms occur [81] with essentially free
parameters introducing considerable flexibility [82, 78] (see also Ref. [83]).
Worth mentioning is also that, at N4LO, the 3NF includes all 20 operators of
the most general 3NF [84]. Furthermore, the plentiful N4LO 3NF terms may also
provide what is needed to improve the status of the medium-mass nuclei and
nuclear matter.
Thus, the future of truly microscopic nuclear structure is to go for complete
N4LO calculations—a gigantic task.
## 5 Summary and outlook
To summarize, let me just reiterate the main statements.
The tenet of microscopic nuclear theory is:
> Atomic nuclei can be accurately described as collections of point-like
> nucleons interacting via two- and many-body forces obeying nonrelativistic
> quantum mechanics—the forces being fixed in free-space scattering.
And in the ab initio approach, nuclei are calculated accordingly.
We need to critically investigate if the tenet is true. To that end, we have
to answer the question:
> Do the same nuclear forces that explain free-space scattering experiments
> also explain the properties of finite nuclei and nuclear matter when applied
> in nuclear many-body theory?
Either way, the answer is of fundamental relevance. The correct answer can
only be obtained if the free-space forces are accurate, where accurate is
defined by:
> Accurate free-space forces are forces that predict experiment within the
> theoretical uncertainty of the applied EFT at the given order.
Moreover, one would also require that the applied nuclear forces are based
upon some sort of theory in a consistent way.
Without strictly adhering to these principles, the true answer to the
fundamental question will not be found. Once again, the goal is not to obtain
“good” results, but to understand whether there are non-negligible medium
effects on nuclear forces when inserted into the nuclear many-body problem.
In our community, the term ab initio is often used in a way that is too lose
and many calculations that are presented as ab initio do not pass muster. Such
calculations repeat the mistakes of history and, thus, do not move us forward.
The ultimate goal of nuclear theory should be to conduct calculations that
test the tenet with high precision. There is strong evidence that this
precision can only be achieved at N4LO of the chiral EFT expansion.
Calculations of this kind, which must also include all many-body forces at
that order, are very challenging, and the current status of ab initio
calculations is far from meeting that goal.
In this context, it should be mentioned that the uncertainties of the many-
body calculations must also be included in the error analysis. With
calculations now moving up to heavy nuclei, current many-body techniques need
to be tested critically for which bechmark calculations would be the right
tool.
The work that is left to do in microscopic nuclear theory is monumental.
###### Acknowledgements.
This work was supported in part by the U.S. Department of Energy under Grant
No. DE-FG02-03ER41270.
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# Challenges to observation of many-body localization
Piotr Sierant The Abdus Salam International Center for Theoretical Physics,
Strada Costiera 11, 34151, Trieste, Italy Institute of Theoretical Physics,
Jagiellonian University in Kraków, Łojasiewicza 11, 30-348 Kraków, Poland
Jakub Zakrzewski<EMAIL_ADDRESS>Institute of Theoretical Physics,
Jagiellonian University in Kraków, Łojasiewicza 11, 30-348 Kraków, Poland
Mark Kac Complex Systems Research Center, Jagiellonian University in Krakow,
Kraków, Poland.
###### Abstract
We study time dynamics of 1D disordered Heisenberg spin-1/2 chain focusing on
a regime of large system sizes and a long time evolution. This regime is
relevant for observation of many-body localization (MBL), a phenomenon that is
expected to freeze the dynamics of the system and prevent it from reaching
thermal equilibrium. Performing extensive numerical simulations of the
imbalance, a quantity often employed in the experimental studies of MBL, we
show that the regime of a slow power-law decay of imbalance persists to
disorder strengths exceeding by at least a factor of 2 the current estimates
of the critical disorder strength for MBL. Even though we investigate time
evolution up to few thousands tunneling times, we observe no signs of the
saturation of imbalance that would suggest freezing of system dynamics and
provide a smoking gun evidence of MBL. We demonstrate that the situation is
qualitatively different when the disorder is replaced by a quasiperiodic
potential. In this case, we observe an emergence of a pattern of oscillations
of the imbalance that is stable with respect to changes in the system size.
This suggests that the dynamics of quasiperiodic systems remain fully local at
the longest time scales we reach provided that the quasiperiodic potential is
sufficiently strong. Our study identifies challenges in an unequivocal
experimental observation of the phenomenon of MBL.
## I Introduction
Generic isolated quantum many-body systems initialized in an out-of-
equilibrium state are expected to approach featureless thermal states
described by the eigenstate thermalization hypothesis [1, 2, 3]. Many-body
localization (MBL) [4, 5] has been put forward as a mechanism that prevents
the approach to equilibrium due to an interplay of interactions and strong
disorder.
The phenomenon of MBL has received a lot of attention over the last decade [6,
7, 8]. The MBL phase is characterized by presence of local integrals of motion
[9, 10, 11, 12, 13, 14, 15] that inhibit the transport [6, 16], and slow down
the spreading of the quantum entanglement [17, 18]. MBL has been investigated
numerically in disordered spin chains [19, 20, 21, 22] that map onto spinless
fermionic chains, in systems of spinful fermions [23, 24, 25, 26] or bosons
[27, 28, 29] and found in systems with random interactions [30, 31, 32] or in
various types of quasiperiodic systems [33, 34, 35]. All those investigations
were confirming the belief that MBL is a robust mechanism of ergodicity
breaking, that can be expected to occur in a wide class of local, one-
dimensional quantum many-body systems provided that a sufficiently strong
quenched disorder is present.
This belief was challenged in [36] where it was argued that MBL might not be
stable in the asymptotic sense, i.e. in the limit of an infinite time and
system size, and the observations of earlier works indicate only a presence of
an MBL regime found at a finite system size and finite times. This lead to an
intense debate about the stability of MBL [37, 38, 39] and its dynamical
properties [40, 41, 42, 43, 44]. Despite these works, it is presently unclear
whether a stable MBL phase exists much deeper in the MBL regime than it was
previously estimated [45] or whether there is no stable MBL phase at all [46].
An example of the latter scenario is provided by disordered constrained spin
chains which, despite hosting a wide non-ergodic regime at finite system sizes
[47] become ergodic in the thermodynamic limit [48].
Figure 1: Interactions induce a slow decay of the imbalance $I(t)$ that
persists to long times. This is visualized comparing results for non-
interacting ($\Delta=0$) and interacting ($\Delta=1$) systems. Data for
disordered XXZ model (1) at disorder strength $W=4$. The squares denote the
Heisenberg time $t_{H}$ that scales exponentially with system size $L$.
The double limit of infinite time and system size is the source of
difficulties in establishing the status of MBL. On one hand, one may
investigate properties of eigenstates of many-body systems, that encode the
properties of the system at infinite time. However, the eigenstates can be
found in an unbiased fashion only for relatively small system sizes $L$ (for
instance, for the usually studied spin-1/2 chains, $L\leq 24$ [49, 50]), which
does not allow for a fully controlled extrapolation of the results to the
thermodynamic limit $L\rightarrow\infty$. On the other hand, tensor network
algorithms [51, 52] such as Time Evolving Block Decimation (TEBD) [53, 54] or
Time-Dependent Variational Principle (TDVP) [55, 56, 57, 58] allow one to
study time evolution of systems comprised of hundreds or even thousands of
sites. Unfortunately, the time evolution of many-body systems can be traced
faithfully with such algorithms only up to times restricted by the growth of
the entanglement in the system. Since, in strongly disordered systems, the
entanglement entropy grows only logarithmically in time, maximal times of
several hundred tunneling times were achieved in [59, 25, 60, 61].
Nevertheless, there is no straightforward way of extrapolating these results
to the infinite time limit.
Figure 1 illustrates the difficulties in assessing whether the system is
ergodic or MBL in a quench experiment. It shows the time evolution of the so-
called imbalance $I(t)$ for a disordered XXZ spin-1/2 chain (precise
definitions are given in the following section). An ergodic system has no
memory of its initial state and the imbalance vanishes in the long-time limit:
$I(t)\stackrel{{\scriptstyle t\rightarrow\infty}}{{\rightarrow}}0$. In
contrast, the information about the initial density profile persists
indefinitely in the MBL phase in which $I(t)\stackrel{{\scriptstyle
t\rightarrow\infty}}{{\rightarrow}}I_{0}>0$. For the non-interacting system
($\Delta=0$) one clearly sees that after initial oscillations, the imbalance
saturates to a constant value. Such a behavior allows for a straightforward
experimental observation of Anderson localization in the absence of
interactions [62, 63]. The main effect of interactions is that the imbalance
decays to much longer times, as exhibited by data for $\Delta=1$. The time
scale at which $I(t)$ ceases to decay is of the order of Heisenberg time
$t_{H}$ [64] that is proportional to an inverse of the mean level spacing of
the system and hence it is exponentially large in the system size $L$. In a
consequence, the data presented in Fig. 1 allow us only to conclude that at
the considered disorder strength $W=4$, the system is in a finite time MBL
regime [45]. The value of the imbalance in the $t\rightarrow\infty$ limit is
clearly decreasing with the system size $L$ and it is impossible to determine
from the data in Fig. 1 whether in the limit $L\rightarrow\infty$,
$t\rightarrow\infty$ the system remains MBL at $W=4$ or whether the ergodicity
is restored.
The presence of MBL regime has been demonstrated in a number of numerical
works as well as in experiments with cold atoms and ions [65, 66, 67, 68, 69,
70, 71]. The aim of this work is to determine whether we can observe
unambiguous signatures of the MBL phase in the time evolution of disordered
many-body systems. To that end we perform extensive numerical simulations of
disordered XXZ spin-1/2 chain and concentrate on the time evolution of density
correlation functions.
Let us note that we, on purpose, limit our discussion to short-ranged
interactions although MBL has been addressed also for long-range (e.g. dipolar
[72, 73, 74, 75, 76], Ising-type [77, 78, 79, 80, 81] or cavity-mediated [82,
83]) interactions. Similarly we do not address the existence and properties of
localization in disorder-free potentials (such as e.g. tilted lattices) - the
subject of intensive recent studies [84, 85, 86, 87, 88, 89, 90, 91, 92, 93,
94, 95, 96, 97]. We want to concentrate on the “pure”, traditional MBL case.
The paper is structured as follows. In Sec. II we introduce the XXZ spin
chain. We provide results for small system sizes and formulate tentative
criteria for observation of MBL phase in Sec. III. Then, we verify whether
those criteria are fulfilled by dynamics of the XXZ spin chain in the regime
of large disorder strengths and system sizes in Sec. V. Subsequently, we
investigate time evolution of entanglement entropy in that regime in Sec. VI.
Finally, instead of random disorder we consider time dynamics of the system
with a quasiperiodic potential in Sec. VII. We draw our conclusions in Sec.
VIII.
## II Model and observables
In this work we concentrate on 1D XXZ spin chain with Hamiltonian given by
$H=J\sum_{i=1}^{{L-1}}\left(S^{x}_{i}S^{x}_{i+1}+S^{y}_{i}S^{y}_{i+1}+\Delta
S^{z}_{i}S^{z}_{i+1}\right)+\sum_{i=1}^{L}h_{i}S^{z}_{i}$ (1)
where $\vec{S}_{i}$ are spin-1/2 matrices, $J=1$ is fixed as the energy unit,
open boundary conditions are assumed and $h_{i}\in[-W,W]$ are independent,
uniformly distributed random variables. The Jordan-Wigner transformation
allows to map XXZ spin chain (1), to a system of interacting spinless
fermions, with the tunneling matrix element equal to $J$ and nearest-neighbor
interaction strength $\Delta$. This allows to make connection between
disordered XXZ model and optical lattice experiments (as e.g. in [65]). The
random-field XXZ spin chain has been widely studied in the MBL context, see
e.g. [98, 22, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108]. Various
estimates of the critical disorder strength $W_{C}$ for the transition to MBL
phase include: $W_{C}\approx 3.7$ [22], $W_{C}\approx 3.8$ [109],
$W_{C}\approx 4.2$ [110, 60], $W_{C}\gtrapprox 5$ [59, 111], $W_{C}\approx
5.4$ [50].
Besides the random disorder $h_{i}\in[-W,W]$, we also consider the case of
quasiperiodic (QP) potential, for which $h_{j}=W^{\mathrm{QP}}\cos(2\pi
kj+\phi)$, where $k=(\sqrt{5}-1)/2$ and $\phi$ is a random phase taken from
the uniform distribution between $[0,2\pi]$. The QP potential breaks the
translation invariance of the system playing a role similar to disorder and
leading to MBL at a critical strong amplitude of the QP potential
$W^{\mathrm{QP}}_{C}$, with various estimates ranging from
$W^{\mathrm{QP}}_{C}\approx 1.5$ [33, 112, 113, 114, 115, 116] through
$W^{\mathrm{QP}}_{C}\approx 2.4$ [117, 118], up to $W^{\mathrm{QP}}_{C}\approx
4$ [119]. Its important to note that the properties of the transition to MBL
phase in QP systems are distinct from the transition in system with random
disorder [34, 120, 121, 122].
We analyze dynamics of imbalance
$I(t)=D\sum_{i=1+l_{0}}^{L-l_{0}}\langle\psi(t)|S^{z}_{i}|\psi(t)\rangle\langle\psi|S^{z}_{i}|\psi\rangle,$
(2)
where $|\psi(t)\rangle=e^{-iHt}|\psi\rangle$, $\ket{\psi}$ is the initial
state, the constant $D$ assures that $I(0)=1$, $l_{0}>0$ diminishes the
influence of boundaries (in our calculations we take $l_{0}=2$). The results
are averaged over $n_{\mathrm{real}}$ disorder realizations. As the initial
state we take the Néel state with every second spin pointing up and every
second spin down
$|\psi\rangle=|\uparrow\downarrow\ldots\uparrow\downarrow\rangle$. In the
following Section we also take $\ket{\psi}$ as a product state of eigenstates
of $S^{z}_{i}$ operators with average energy $\bra{\psi}H\ket{\psi}$ being in
the middle $10\%$ of the spectrum of $H$ – we refer to such a choice as to a
density correlation function $C(t)$.
We note that other observables, see e.g. [102, 123], suffer from finite size
and finite time limitations similar as (2). Hence, it seems that their
behavior is always governed by the broad distributions of relaxation time
scales [124], and that is why we concentrate on the very simple observable
given by (2), which has another advantage of being directly accessible in
experiments with cold atoms [65].
To find the time evolved state $\ket{\psi(t)}$ we employ Chebyshev expansion
of the evolution operator $e^{-iHt}$ [125], which allows us to investigate
time evolution of systems of $L\leq 20$ sites up to the Heisenberg time
$t_{H}=2\pi/\overline{s}\sim e^{cL}$ (where $\overline{s}$ is the average
level spacing in the middle of the spectrum and $c$ determines the scaling of
Hilbert space dimension with system size: for spin-1/s chains $c=\ln 2$). For
larger system sizes $L=50,100,200$ we use a TDVP algorithm, with bond
dimension $\chi$, specified later in the text for each $W$ and $L$ considered.
In the latter case we focus on relatively large disorder strengths $W\geq 8$
which allows us to investigate time evolution up to a few thousand tunneling
times $J^{-1}$.
## III How to observe an MBL phase?
Numerical [126] as well as experimental [67] investigations of the imbalance
$I(t)$ indicate a presence of a wide regime of disorder strength $W$ in which
the imbalance decays according to a power-law $I(t)\sim
t^{-\overline{\beta}}$. As a criterion for a transition to MBL, the work [59]
introduced the condition that $\overline{\beta}$ vanishing within error bars
implies the onset of MBL. The problem with such a criterion is that the error
bars on $\overline{\beta}$ can be significantly reduced with increasing time
of evolution and number of disorder samples, pushing the tentative boundary of
MBL to larger and larger disorder strengths. An alternative was put forward in
[61], which used a cut-off $\overline{\beta}_{\mathrm{cut}}$ such that
$\overline{\beta}<\overline{\beta}_{\mathrm{cut}}$ implies MBL behavior. The
cut-off value of $\overline{\beta}_{\mathrm{cut}}$ was taken from a comparison
of critical disorder strength estimated from gap ratio statistics as
$W_{C}\approx 4$ for system size $L\approx 20$ and the decay rate of imbalance
at that system size.
The latter criterion also runs into problems. If we assume a simplified model
of the decay of the imbalance, in which $I(t)\sim t^{-\overline{\beta}}$ for
$t<t_{H}$, and then $I(t)=\mathrm{const}$ for $t>t_{H}$ (which is mildly
consistent with data shown in Fig. 1), then the value of the imbalance at
infinite time is: $I(\infty)=I(t_{H})=e^{-cL\overline{\beta}}$. Hence, in
order to have a finite value of imbalance in the $t\rightarrow\infty$ the
exponent governing decay of imbalance should vanish at least as
$\overline{\beta}\sim L^{-1}$. Keeping this in mind we now examine the
dynamics of the density correlation function $C(t)$ in a system of moderate
size $L\leq 20$.
Figure 2: Time evolution of density correlation function $C(t)$ in disordered
XXZ model. Panels a) and b) – $C(t)$ for various system sizes $L=10,...,L=20$
at disorder strengths $W=3,5$, data averaged over $n_{\mathrm{real}}>10^{4}$
disorder realizations. Panels c) and d) – time evolution of the flowing beta
function $\beta(t)$ that locally describes the exponent of decay of $C(t)$.
The red squares denote the Heisenberg time $t_{H}\sim e^{cL}$.
Figure 2a) shows $C(t)$ for disorder $W=3$ for which the XXZ spin chain is in
the ergodic phase. The density correlation function, as well as the imbalance
are characterized by oscillations at small times due to the coupling between
neighboring spins. Those oscillations are gradually damped with time $t$, and
the slow decay becomes the main feature of the dynamics of $I(t)$ and $C(t)$.
With an increasing system size, the power-law decay of $C(t)$ persists to
longer and longer times, not changing much beyond the Heisenberg time $t_{H}$.
The interaction induced decay of $C(t)$ is evidently getting more abrupt with
increasing $L$. The situation is, in fact quite similar for $W=5$ (see Fig.
2b) ), which, according to the majority of estimates (e.g. [22, 35, 110]) is
already in the MBL phase. While the decay of $C(t)$ is much slower than for
$W=3$, it persist to long-times and the saturation value of $C(t)$ is
decreasing with $L$.
To investigate the slow decay of $C(t)$ in more quantitative fashion, we
consider a time-dependent $\beta(t)$ function [117], that is obtained from the
fit $C(t_{1})=at_{1}^{-\beta(t)}$ in the interval $t\in[t_{1},1.5t_{1}]$. The
resulting $\beta(t)$ functions are shown in Fig. 2 c),d). For $W=3$ we observe
that at first, the decay of $C(t)$ is well described by a power-law
($\beta(t)$ is constant) and then the decay gradually slows down, stopping at
the time scale approximately order of magnitude larger than $t_{H}$. For
$W=5$, the slow down of the decay of $C(t)$ occurs at smaller times, however,
a non-vanishing $\beta(t)$ up to Heisenberg time signals further, non-
negligible decay of the density correlation function.
Results presented in this section show that the correlation functions decay up
to Heisenberg time or even longer. Moreover, comparison of results for $W=3$
and $W=5$ indicates that it is hard to propose an accurate phenomenological
model for the decay of $C(t)$. Nevertheless, building on intuitions obtained
in this section, we conclude that an unambiguous observation of MBL phase
should include at least one of the two conditions:
1. (A)
the value of the exponent $\overline{\beta}$ that is decreasing with system
size as $L^{-1}$ \- in such a case even if the power-law decay persists up to
the Heisenberg time, the imbalance is non-vanishing in the limit $t\to\infty$;
2. (B)
a decrease of value of $\beta(t)$ with time $t$ that occurs in a system size
independent fashion indicating the saturation of the imbalance at all
experimentally accessible times beyond a certain time scale.
The results for small system sizes indicate that if the dynamics of the
imbalance satisfies either the criterion A or B, the system is in an
asymptotic MBL phase. In that sense, the conditions A and B can be thought of
as conditions sufficient for the observation of MBL phase. The conditions A
and B must be verified with care and their fulfillment is not in a strict
sense a proof for a stable MBL phase: a system that satisfies either of them
could still be ergodic. For instance, one may imagine a decrease of $\beta(t)$
in time in a system size independent fashion below a certain (large from the
experimental perspective) time scale combined with an onset of a fast decay of
imbalance beyond a certain larger time scale. Nevertheless, such scenarios
seem to be ruled out by the results for small system sizes and for that reason
we treat the conditions A and B as sufficient for observation of MBL phase. At
the same time, we would like to note that neither of the conditions is a
necessary criterion for an observation of MBL phase. Other scenarios in which
the system breaks ergodicity can be envisioned. For instance, the imbalance
may behave in a non-monotonous in time manner with a non-zero infinite time
average in the large system size limit, disallowing the analysis of $I(t)$
with a power-law decay.
With those remarks in mind, we now turn to an analysis of time dynamics of
large systems in the strong-disorder, long-time regime, which seems to be the
most suitable one to find signatures of the MBL phase. The criteria A and B
will be the guiding principles of our analysis. First, however, let us briefly
consider a non-interacting system.
Figure 3: Comparison of the time evolution for noninteracting system between exact propagation and TDVP approximate algorithm ($L/2$ fermions for the system size $L=50$ at disorder strength $W=10$). Top: the imbalance, $I(t)$ (left) and the entanglement entropy in the middle of the chains, $S(t)$ (right) obtained in exact propagation (blue curves extending to larger times) and via TDVP (lighter, orange line). Bottom shows the difference between exact and TDVP results for imbalance (left) and entropy (right). Table 1: Details of numerical simulations for $W=10$ and $\Delta=0$: system size $L=50$, maximal time reached in time evolution $t_{\mathrm{max}}$, the bond dimension $\chi$ (not displayed for the exact numerical calculation), number of disorder realizations $n_{\mathrm{real}}$, and the exponent $\overline{\beta}$ obtained from the fit $I(t)\sim t^{-\overline{\beta}}$ in interval $t\in[100,t_{\mathrm{max}}]$. The error of $\overline{\beta}$ is estimated by resampling over the disorder realizations (here, as well in the rest of this manuscript). | $t_{\mathrm{max}}$ | $\chi$ | $n_{\mathrm{real}}$ | $\overline{\beta}$
---|---|---|---|---
$L$=50 | 1500 | 128 | 1000 | $(0.97\pm 1.12)\cdot 10^{-4}$
$L$=50 | 1500 | - | 1000 | $(0.96\pm 1.12)\cdot 10^{-4}$
$L$=50 | 5000 | - | 1000 | $(-0.23\pm 0.43)\cdot 10^{-4}$
## IV Non-interacting test case
We consider now the Hamiltonian (1) and set $\Delta=0$ which via Jordan-Wigner
transformation maps to a set of non-interacting spinless fermions in a random
on-site potential $h_{i}$. This model is known to be Anderson localized [127]
for an arbitrary amplitude of the disorder $W$. Since the model is non-
interacting, we calculate the time evolution of an initial state $\ket{\psi}$
in numerically exact fashion in time polynomial in system size (see Appendix
A.3). As the initial state we take the Néel state
$|\psi\rangle=|\uparrow\downarrow\ldots\uparrow\downarrow\rangle$. The
obtained time evolved imbalance $I(t)$ provides a reference for our
approximate time propagation using TDVP. Note that while in the non-
interacting case obtaining the exact solutions for arbitrary disorder
realization is a straightforward task, this is not so for TDVP – in the latter
case the algorithm keeps track of a matrix product state that belongs to the
full many-body Hilbert space in a manner similar to the interacting case.
The TDVP algorithm used is described in detail in the Appendix A. The
convergence of TDVP crucially relies on a value of the bond dimension $\chi$.
The time evolved state $\ket{\psi_{\chi}(t)}$ obtained with TDVP becomes a
better and better approximation of the exact time evolved state
$\ket{\psi(t)}$ as $\chi$ increases. However, the simulation cost increases
with the value of the bond dimension as $\chi^{3}$. Hence, one has to choose
the value of $\chi$ such that the observables of interest are converged with
the bond dimension, i.e. do not change with increase of $\chi$ so that one can
safely assume that their value approximates well the value in the exact time
evolved state $\ket{\psi(t)}$. For the interacting model (1) we present
details on the convergence of results with the bond dimension $\chi$ in
Appendix A. In the remainder of this section we compare the exact solution
$\ket{\psi(t)}$ for the non-interacting case ($\Delta=0$) with the time
evolved state obtained with TDVP.
For our test we take disorder amplitude $W=10$ and propagate the Néel state up
to time $t_{\mathrm{max}}=1500$ for 1000 disorder realizations using TDVP with
bond dimension $\chi=128$. Fig. 3a) compares the obtained imbalance $I(t)$
with the result of exact numerical solution for non-interacting model. The
exact solution and TDVP result agree very well up to $t_{\mathrm{max}}=1500$
reach in TDVP simulation. The exact imbalance typically exceeds the TDVP
result, the difference, shown in Fig. 3c) , grows in time and saturates around
$t=800$ at $2\cdot 10^{-6}$. The TDVP slightly underestimates the imbalance in
agreement with the findings of [60]. Nevertheless, both TDVP as well as the
exact results show that the exponent $\overline{\beta}$ governing the decay of
the imbalance is vanishing within the estimated error bars as shown in Tab. 1.
The vanishing $\overline{\beta}$ fulfills trivially the criterion A for
observation of localization.
We also calculated entanglement entropy $S(t)$ for a bipartition of the
lattice into subsystems $A$ and $B$ of length $L/2$:
$S(t)=-\mathrm{Tr}_{A}[\rho_{A}\ln\rho_{A}],$ (3)
where $\rho_{A}=\mathrm{Tr}_{B}\ket{\psi(t)}\bra{\psi(t)}$, $\mathrm{Tr}_{C}$
denotes trace with respect to degrees of freedom of subsystem $C$ and
$\ket{\psi(t)}$ is the state of the system. The entanglement entropy $S(t)$ is
shown in Fig. 3 b). We observe that after an initial increase, the entropy
oscillates around a constant value - similarly to the imbalance $I(t)$. As
Fig. 3 d)shows, TDVP slightly overestimates the entanglement entropy. The
ratio between the error of TDVP simulation and the value of the observable is
roughly two orders of magnitude larger than for the imbalance. Nevertheless,
the results from TDVP and the numerically exact simulation practically overlap
showing that TDVP provides a reliable information about the entanglement
entropy growth.
Encouraged by this comparison we shift towards the interacting case for which
a comparison with the exact dynamics is not possible. There, we necessarily
rely on self-consistency tests of our simulations described in Appendix A.
## V Time evolution of imbalance at strong disorder in large systems
Figure 4: Time evolution of imbalance $I(t)$ for systems of size
$L=50,100,200$ at disorder strength $W=8$, details of the simulations and fits
are given in Tab. 2. Top: the shaded lines denote $I(t)$ whereas the solid
lines denote a running overage of $I(t)$ over window $(t-25,t+25)$, dashed
lines denote power-law fits $I(t)\sim t^{-\overline{\beta}}$ in time interval
$t\in[100,1500]$. Bottom: the running beta function $\beta(t)$, dashed lines
show the value of $\overline{\beta}$, the error of $\beta(t)$ is estimated by
resampling over the disorder realizations (here, as well in the rest of this
manuscript).
Taking into account the various estimates of the critical disorder strength
$W_{C}$ for transition to MBL phase, discussed in Sec. II, we fix the disorder
amplitude at $W=8$ and $W=10$. Such disorder strengths, according to the
aforementioned estimates of $W_{C}$, are expected to lay significantly above
the transition to the MBL phase.
Table 2: Details of numerical simulations for $W=8$: system size $L$, maximal time reached in time evolution $t_{\mathrm{max}}$, the bond dimension $\chi$, number of disorder realizations $n_{\mathrm{real}}$, and the exponent $\overline{\beta}$ obtained from the fit $I(t)\sim t^{-\overline{\beta}}$ in interval $t\in[100,1500]$. The error of $\overline{\beta}$ is estimated by resampling over the disorder realizations (here, as well in the rest of this manuscript). | $t_{\mathrm{max}}$ | $\chi$ | $n_{\mathrm{real}}$ | $\overline{\beta}$
---|---|---|---|---
$L$=50 | 1500 | 128 | 4000 | $(10.03\pm 1.23)\cdot 10^{-4}$
$L$=100 | 1500 | 128 | 2000 | $(11.07\pm 0.97)\cdot 10^{-4}$
$L$=200 | 1500 | 160 | 1000 | $(11.03\pm 0.81)\cdot 10^{-4}$
The evolution of imbalance $I(t)$ for $W=8$ is shown in Fig. 4 whereas the
details of numerical simulations are shown in Tab. 2. After an initial
transient decay and oscillations that last up to $t\approx 100$, we observe a
slow but steady monotonic decrease of $I(t)$ that persists up to the largest
time $t_{\mathrm{max}}=1500$ reached in the simulation. The value of
$t_{\mathrm{max}}$ is not sufficiently large to unambiguously pin-point the
functional form of the decay of $I(t)$. Nevertheless, we observe that the
imbalance is well fitted by a power-law decay $I(t)\sim t^{-\overline{\beta}}$
in the interval $t\in[100,1500]$. The values of the exponent
$\overline{\beta}$, shown in Tab. 2, are positive confirming that the slow
decay of $I(t)$ is present (for a discussion of the stability of the value of
$\overline{\beta}$ with respect to the choice of the fitting interval see
Appendix. B). Moreover, within the estimated error bars, the values of
$\overline{\beta}$ are the same for system sizes $L=50,100,200$, indicating
clearly that the condition A for the observation of MBL phase is not met at
$W=8$.
To check whether the condition B is fulfilled, we consider the flowing beta
function $\beta(t)$ obtained from fitting $I(t_{1})=at_{1}^{-\beta(t)}$ in the
interval $t\in[t_{1},1.5t_{1}]$. The result, shown in the bottom panel of Fig.
4, indicates that the decay of the imbalance slows down considerably for
$t{\approx}150$. However, beyond that time the value of the $\beta(t)$
oscillates around the exponent $\overline{\beta}$. Therefore, we see no traces
of slowing-down of the decay of imbalance at $W=8$.
In conclusion, for $W=8$, neither the criterion A nor B is fulfilled. Hence,
we proceed to repeat our analysis for larger disorder strength $W=10$.
Figure 5: Time evolution of imbalance for $W=10$, denotation the same as in
Fig. 4. Details of the simulations and fits given in Tab. 3.
Time evolution of the imbalance $I(t)$, as well as the flowing $\beta(t)$
function are shown in Fig. 5.
Table 3: Details of numerical simulations for $W=10$, denotations the same as in Tab. 2. | $t_{\mathrm{max}}$ | $\chi$ | $n_{\mathrm{real}}$ | $\overline{\beta}$
---|---|---|---|---
$L$=50 | 1500 | 128 | 4000 | $(3.93\pm 0.82)\cdot 10^{-4}$
$L$=100 | 1500 | 128 | 2000 | $(3.60\pm 0.53)\cdot 10^{-4}$
$L$=200 | 1200 | 160 | 1000 | $(3.50\pm 0.87)\cdot 10^{-4}$
$L$=50 | 5000 | 192 | 2000 | $(3.08\pm 0.51)\cdot 10^{-4}$
While the decay of imbalance clearly slowed down considerably, as reflected by
the values of the exponent $\overline{\beta}$ shown in Tab. 3, upon the
increase of disorder strength from $W=8$ to $W=10$, the system size dependence
of $\overline{\beta}$ remains the same: the values of $\overline{\beta}$ are,
within the estimated error bars, similar for $L=50,100,200$, clearly not
satisfying the criterion A. The flowing $\beta(t)$ function, shown in the
bottom panel of Fig. 5 indicates that the decay of imbalance is relatively
fast around $t\approx 200$ and then slows down considerably at $t\approx 500$
for which the value of $\beta(t)$ is vanishing. However, around $t\approx 800$
the flowing $\beta(t)$ function acquires again the value similar to
$\overline{\beta}$ and the decay of imbalance persists and the criterion B is
not met.
To make sure that our conclusions for $W=10$ are valid, we increased the
maximal time reached in our simulations to $t_{\mathrm{max}}=5000$ for system
size $L=50$, the results are presented in Fig. 6. We indeed observe that the
slow decay of imbalance $I(t)$ persists up to the longest time achieved in our
simulation. This is exemplified by the power-law fit $I(t)\sim
t^{-\overline{\beta}}$ that accurately matches the decay of imbalance in the
whole interval $t\in[100,5000]$, with the exponent $\overline{\beta}$ close to
the values obtained for the shorter time intervals, see Tab. 3. Moreover, the
flowing $\beta(t)$ function oscillates around the value $\overline{\beta}$ in
the whole interval of available times. We see no signs of the slow down of
decay of $I(t)$, which leads us to conclude that the criterion B is not
fulfilled for $W=10$.
Figure 6: Time evolution of imbalance for $W=10$ in extended time interval,
denotation the same as in Fig. 4. Details of the simulations and fits given in
Tab. 3.
In conclusion, we found no clear signatures of the MBL phase in results
presented in this sections, even though we considered significantly larger
times and disorder strengths than in earlier studies [59, 60]. One immediate
question is whether we can go even further in the attempts to observe MBL
phase and consider larger disorder strength $W$ and bigger maximal time
$t_{\mathrm{max}}$. The factor that limits such a continuation most severely
is the slow-down of decay of $I(t)$ with $W$. In order to observe in a
statistically significant way a decay of $I(t)$ at larger $W$ the increase of
$t_{\mathrm{max}}$ should be coupled with an increase of the number of
disorder realizations $n_{\mathrm{real}}$. This considerably increases the
resources needed for such numerical simulations. The same considerations apply
to experiments with quantum many-body systems which are limited by a finite
coherence time (typically limited to at most 1000 tunneling times [96] thus
shorter than the times considered by us) as well as resources needed to
perform disorder averages.
## VI Time evolution of entanglement entropy
The time dependence of the entanglement entropy is one of the tools that may
be used to identify the existence of MBL phase. While typically in the
deconfined systems the entanglement entropy grows linearly in time when the
evolution is started from the low entanglement, e.g. separable state, in MBL
one expects a logarithmic entanglement entropy growth [128, 129]. It is,
therefore, instructive to study the entropy growth in our case in the regime
of large disorder strengths and long times probed in our numerical
simulations. Since the Hamiltonian (1) conserves the total magnetization
$\sum_{i=1}^{L}S^{z}_{i}$, the entanglement entropy $S$ of subsystem $A$
consisting of lattice sites $1,\ldots L/2$ can be written as a sum
$S(t)=S_{n}(t)+S_{c}(t)$, where $S_{n}(t)$ is the number entropy and $S_{c}$
denotes the configurational entropy [130, 131, 132, 133, 69, 82, 97]. The
number entropy is given by
$S_{n}(t)=-\sum_{n}p(n)\ln p(n),$ (4)
where $p(n)$ is the probability that $\sum_{i=1}^{L/2}S^{z}_{i}$ is equal to
$n$. (We note that $\sum_{i=1}^{L/2}S^{z}_{i}$ is proportional to the total
number of spinless fermions in subsystem $A$ after Jordan-Wigner
transformation of (1), explaining the term “number entropy”.) The
configurational entropy is given by
$S_{c}(t)=-\sum_{n}p(n)\mathrm{Tr}[\rho(n)\ln\rho(n)],$ (5)
where $\rho(n)$ is the block of the reduced density matrix in sector with
$\sum_{i=1}^{L/2}S^{z}_{i}=n$.
Figure 7: Time evolution of entanglement entropy for $L=50$ and $W=10$. Top:
configuration entanglement entropy $S_{c}(t)$ denoted by solid line, dashed
lines denote power-law and logarithmic fits $f(t)$. The inset shows the
residual $f(t)-S_{c}(t)$. Bottom: The number entanglement entropy $S_{n}(t)$
is denoted by solid line, dashed line denotes a double-logarithmic fit
$f_{2}(t)$. The inset shows the residual $f_{2}(t)-S_{n}(t)$.
Our results for the entanglement entropies $S_{n}(t)$ and $S_{c}(t)$ are shown
in Fig. 7. The configurational entropy $S_{c}(t)$ is expected to grow
logarithmically in time [128, 129] in the MBL regime. We observe that after an
initial transient at times $t\lessapprox 10$, the growth of $S_{c}(t)$ is well
described by a power-law $S_{c}(t)\propto t^{\gamma}$ with $\gamma=0.250(2)$
in the interval $t\in[10,600]$. This behavior resembles the time dynamics of
entanglement entropy observed in the ergodic regime at moderate values of
disorder $W\approx 2.5$ [126]. However, at longer times, the increase of
$S_{c}(t)$ slows down and is well fitted by $S_{c}(t)=a+b\ln t$ with
$a=-0.04437(7)$ and $b=0.02001(9)$ for $t\in[400,5000]$ in agreement with
expectations for the MBL regime. The growth of the number entropy is
significantly slower, and is very well fitted by a double logarithmic formula
$S_{n}(t)=a+b\ln\ln t$ with $a=0.1496(6)$ and $b=0.0120(3)$ in a wide regime
of times $t\in[20,5000]$. This confirms the prediction of [40, 44] for the
significantly larger system size and disorder strength than tested before.
In conclusion, the slow decay of imbalance observed in Sec. V is accompanied
by a logarithmic increase of the configurational entanglement entropy
$S_{c}(t)$ and a double logarithmic growth of the number entropy $S_{n}(t)$.
Those quantities provide a complementary to the imbalance insight into the
dynamics of the slow delocalization of the system. At the same time, they do
not allow for an observation of the MBL phase in the fashion similar to the
imbalance. For a localized system, one expects a saturation of $S_{n}(t)$
[41]. The upper limit, $S_{n}=\ln(3){\approx 1.01}$, predicted in [41] is much
higher than the values reached by a very slow double logarithmic growth of
$S_{n}(t)$ observed in Fig. 7. Note also that a very recent study, [134],
instead of such a a slow double logarithmic growth predicts a power-law
approach of ${S}_{n}$ to its asymptotic value at $t\to\infty$. This cannot be
tested for the large system sizes $(L\geq 50)$ considered by us since we are
unable to determine the asymptotic value of $\lim_{t\to\infty}S_{n}(t)$.
## VII Quasiperiodic systems
In this section we attempt at observation of MBL phase in dynamics of the
system with QP potential, defined in Sec. II. To that end we investigate the
impact of the amplitude of QP potential $W^{\mathrm{QP}}$ on time evolution of
imbalance $I(t)$.
Figure 8: Time evolution of imbalance $I(t)$ for QP potential. Top: results
for the amplitude of QP potential $W^{\mathrm{QP}}=2$ the shaded lines denote
$I(t)$ whereas the solid lines denote a running overage of $I(t)$ over window
$(t-25,t+25)$, dashed lines denote power-law fits $I(t)\sim
t^{-\overline{\beta}}$ in time interval $t\in[500,5000]$. Bottom: the same for
$W^{\mathrm{QP}}=3$. Details of simulations are given in Tab. 4
The results for $W^{\mathrm{QP}}=2,3$ are shown in Fig. 8. The behavior of
$I(t)$ is qualitatively similar to the systems with random disorder: after an
initial transient, the decay of imbalance is well fitted by a power-law
$I(t)\sim t^{-\overline{\beta}}$. The exponent $\overline{\beta}$ is clearly
increasing with system size both for $W^{\mathrm{QP}}=2$ and
$W^{\mathrm{QP}}=3$, as shown in Tab. 4, suggesting that the system
delocalizes in the thermodynamic limit at those values of $W^{\mathrm{QP}}$
and neither the criterion A nor B for observation of MBL phase is met.
Table 4: Details of numerical simulations QP potential, denotations the same as in Tab. 2. The bond dimension $\chi$ is not displayed for calculation performed with the Chebyshev expansion of the evolution operator. | $t_{\mathrm{max}}$ | $\chi$ | $n_{\mathrm{real}}$ | $\overline{\beta}$
---|---|---|---|---
$L$=12, $W^{\mathrm{QP}}$=2 | 5000 | - | $10^{6}$ | $(1.8\pm 0.2)\cdot 10^{-3}$
$L$=16, $W^{\mathrm{QP}}$=2 | 5000 | - | $10^{5}$ | $(9.5\pm 0.2)\cdot 10^{-3}$
$L$=20, $W^{\mathrm{QP}}$=2 | 5000 | - | $5\cdot 10^{4}$ | $(19.0\pm 0.1)\cdot 10^{-3}$
$L$=12, $W^{\mathrm{QP}}$=3 | 5000 | - | $10^{6}$ | $(3.3\pm 0.4)\cdot 10^{-4}$
$L$=16, $W^{\mathrm{QP}}$=3 | 5000 | - | $10^{5}$ | $(8.8\pm 0.3)\cdot 10^{-4}$
$L$=20, $W^{\mathrm{QP}}$=3 | 5000 | - | $5\cdot 10^{4}$ | $(8.9\pm 0.6)\cdot 10^{-4}$
$L$=12, $W^{\mathrm{QP}}$=4 | 5000 | - | $10^{6}$ | $(2.1\pm 0.4)\cdot 10^{-4}$
$L$=16, $W^{\mathrm{QP}}$=4 | 5000 | - | $10^{5}$ | $(2.8\pm 0.3)\cdot 10^{-4}$
$L$=50, $W^{\mathrm{QP}}$=4 | 4000 | 128 | | $(3.0\pm 1.3)\cdot 10^{-4}$
$L$=12, $W^{\mathrm{QP}}$=5 | 10000 | - | $10^{6}$ | $(0.3\pm 0.7)\cdot 10^{-4}$
$L$=16, $W^{\mathrm{QP}}$=5 | 10000 | - | $10^{5}$ | $(1.1\pm 0.8)\cdot 10^{-4}$
$L$=50, $W^{\mathrm{QP}}$=5 | 4500 | 128 | 2000 | -
$L$=100, $W^{\mathrm{QP}}$=5 | 3000 | 128 | 1000 | -
$L$=200, $W^{\mathrm{QP}}$=5 | 2500 | 128 | 600 | -
Figure 9: Time evolution of imbalance $I(t)$ for QP potential for
$W^{\mathrm{QP}}=4,5$, denotation the same as in Fig. 8. Details of the
simulations and fits given in Tab. 4.
The decay of imbalance $I(t)$ slows down considerably when the amplitude of
the QP potential is increased to $W^{\mathrm{QP}}=4$ as shown in Fig. 9. The
exponents $\overline{\beta}$ governing the power-law decay of imbalance for
$W^{\mathrm{QP}}=4$ are comparable to the exponents obtained for $W=10$ for
the random disorder. However, the behavior of the running averages of $I(t)$
(shown by the solid lines in Fig. 9) is different: we observe significant
oscillations around the fitted power-law decay. The pattern of those
oscillations is not stable with increasing the system size, $L$.
This behavior changes qualitatively for $W^{\mathrm{QP}}=5$. For this
amplitude of the QP potential we observe an emergence of a pattern of
oscillations of $I(t)$ at times $t\gtrapprox 200$ that remains the same when
the system size is increased from $L=12$ to $L=200$. This is the first case
for which we observe that the increase of the system size does not enhance its
delocalization. Instead, this result shows that the dynamics of a small system
comprised of $L=12$ sites is reproduced in the bulk of the large system of
$L=200$ sites. Such a behavior suggests that the system remains MBL in the
thermodynamic limit at $W^{\mathrm{QP}}=5$, although our approach is
inherently limited to dynamics at finite times and cannot give a definite
answer about the fate of the system at $t\rightarrow\infty$.
Two remarks are in order. Firstly, the values of the running average of $I(t)$
are not changing monotonically with $L$: the curve for $L=16$ is on the top
whereas that for $L=50$ on the bottom. This is caused by the statistical
fluctuations associated with the finite number of disorder realizations
$n_{\mathrm{real}}$ as well as by the erratic changes of $2\pi kL$ modulo
$2\pi$ with $L$ that determine the number of full periods of the QP potential
in the whole chain. Secondly, the emergent pattern of oscillations of $I(t)$
prevents us from determining whether the imbalance $I(t)$ slowly decays in
time. Performing a power-law fit in the interval $t\in[1000,10000]$ we have
found non-vanishing values of $\overline{\beta}$ as shown in Tab. 4. However,
$\overline{\beta}$ changes significantly when the interval in which the fit is
performed changes. This shows that the criteria A and B are effectively
inapplicable to the dynamics of imbalance in QP potential.
We refer the reader to Appendix C for further numerical studies of the
imbalance in QP potential where we show that the persistent oscillations tend
to decay for even larger values of the amplitude $W^{QP}$. We also show there
that the character of the oscillations depends on the parameter $k$ which
determines the quasiperiodicity of the potential by simulating the dynamics
also for $k=\sqrt{2}/2$.
Figure 10: Time evolution of entanglement entropy for QP potential with
$W^{QP}=5$. Top: configuration entanglement entropy $S_{c}(t)$ for system
sizes $L=16,20,50,200$. The inset shows the same but on log-log scale. The
dashed lines divide the time into intervals I, II, III (see text). Bottom: The
corresponding number entanglement entropy $S_{n}(t)$.
To explore the dynamics in the QP potential from a different perspective, we
calculate the number, $S_{n}(t)$, and the configurational, $S_{c}(t)$,
entropies for $W^{QP}=5$. The results are shown in Fig. 10. Rather than
observing an anticipated monotonic increase of the entanglement entropies, we
may distinguish three time intervals (A,B,C) in the time dependence of
$S_{c,n}(t)$. In the interval $I$, for $t\lesssim 4500$, we observe an
algebraic in time increase of $S_{c}(t)$ (compare the inset in the top panel
in Fig. 10 drawn in the log-log scale). This behavior is accompanied by a slow
increase of the number entropy $S_{n}(t)$ which initially follows a
logarithmic growth, saturates around $t\approx 300$ and then again seems to
follow a logarithmic growth. In the interval I, the results for the small
$L=16,20$ and large system sizes $L=50,200$ practically overlap both for
$S_{c}(t)$ and $S_{n}(t)$. This is another property suggesting the locality of
the dynamics at $W^{QP}=5$. We observe for $L=16,20$ that the behavior of
$S_{c}(t)$ changes qualitatively at larger times: $S_{c}(t)$ is approximately
constant in the region II ($4500<t<20000$) and grows logarithmically in time
in the region III ($t>20000$). Both for $S_{c}(t)$ as well as for $S_{n}(t)$
the results for $L=16$ and $L=20$ are practically overlapping in the time
intervals II and III. Unfortunately, the regimes II and III are inaccessible
in TDVP calculations for large system sizes. This prevents us from deciding
whether the initial power-law growth of $S_{c}(t)$ is continued in the large
time limit for large system sizes (leading to a slow approach towards
ergodicity) or whether the features of the entanglement growth at $L=20$ are
consistent with the behavior system for $L\rightarrow\infty$ (leading to a
stable MBL phase).
## VIII Conclusions
In this work we have addressed the problem of a possible experimental
observation of MBL. The presence of interactions gives rise to a slow dynamics
towards equilibrium in strongly disordered systems. This leads us to argue
that an observation of even a very slow decay of correlation functions in a
finite interval of time is insufficient to claim an unambiguous observation of
MBL.
For relatively small systems comprising of less than $L=20$ lattice sites, we
calculated time dynamics beyond the Heisenberg time which allowed us to
extrapolate the results to the infinite time limit. Building on intuitions
obtained in that way, we formulated the criteria A and B for an observation of
the MBL phase. The criterion A requires a slowdown of the decay of density
correlation functions as $L^{-1}$ when the system size $L$ is increased. The
criterion B demands a saturation of correlation functions beyond a certain
time scale in a system size independent manner. We would like to emphasize
that these criteria are neither sufficient nor necessary conditions to prove
that a system is MBL. Rather, we perceive the criteria A and B as hints of
whether the dynamics of a given system breaks the ergodicity or not.
Performing large scale tensor network simulations of time evolution of
disordered XXZ spin chain of up to $L=200$ sites we did not find a regime of
parameters in which the criterion A or B for observation of MBL would be
satisfied. For considered disorder strengths we always encountered the slow
but persistent decay of imbalance hinting at a slow approach of the system
towards the eventually delocalized future. This conclusion was obtained even
though we focused on the regime of disorder strengths lying significantly
above the current estimates of the critical disorder strength for transition
to MBL phase and pushed the maximal time reached in our simulations to few
thousands tunneling times. In that respect, our results are consistent with
the nonexistence of MBL phase in the thermodynamic limit, see also [36, 46].
We also revisited the dynamics of the entanglement entropy confirming the
logarithmic growth of its configurational part and the double logarithmic
increase of the number entropy in the regime of long times and large system
sizes confirming predictions of [40, 44].
Finally, we investigated the time evolution of QP systems. The dynamics of
quasiperiodic system is very much similar to random system at intermediate
values of the amplitude $W^{\mathrm{QP}}$, with a slow, power-law like decay
of imbalance. However, for a stronger QP potential, at $W^{\mathrm{QP}}=5$, we
demonstrated an emergence of a pattern of oscillations in the imbalance
$I(t)$. This pattern remains stable with the increase of the system size. This
qualitatively different behavior of the imbalance in a striking fashion shows
that the dynamics of QP systems at sufficiently large potential strengths
becomes local. While we were eventually not able to fully exclude the decay of
the imbalance in the infinite time limit, the result for QP systems appears to
be not far from being sufficient to claim an observation of MBL phase. In any
case, our results show that the asymptotic properties of transition to MBL
phase may be probed more easily in QP systems (see [122] for the analysis of
QP system from the spectral perspective).
We would like to stress that our results, especially for disordered systems,
do not exclude the existence of a stable MBL phase. Rather, they provide lower
bounds on time scales and disorder strengths required to observe the freezing
of system dynamics in the long time limit that defines the MBL phase. Those
lower bounds are relevant both for future numerical simulations of disordered
systems as well as for experiments with quantum simulators.
###### Acknowledgements.
This work would not be possible without the help of Titas Chanda who provided
us with his tensor network codes and generously helped with their
implementation. Thank you, Titas! We are also indebted to Anatoly Polkovnikov
and Dries Sels for discussions as well as to Elmer V. H. Doggen for questions
regarding our error analysis. The numerical computations have been possible
thanks to the support of PL-Grid Infrastructure. The TDVP simulations have
been performed using ITensor library (https://itensor.org). This research has
been supported by National Science Centre (Poland) under project
2019/35/B/ST2/00034 (J.Z.)
## Appendix A Tests on the numerical accuracy of the presented results
The results presented in the main text are obtained using different numerical
techniques that will be described in detail below. We also provide details of
the numerical method for used for the non-interacting system.
There are two types of errors in our results. The first is the statistical
error which arises due to fluctuation of results from one disorder realization
to another at fixed parameters of the system. The resulting errors in the
exponent $\overline{\beta}$ governing the decay of imbalance $I(t)$, as well
as in the running $\beta(t)$ function are estimated by the bootstrap
technique, i.e. by resampling over the disorder realizations. In the figures
we plot the imbalance $I(t)$ as well as a running average of the imbalance.
Importantly, however, in the fits that determine $\beta(t)$ and
$\overline{\beta}$ we always use the full data for the imbalance $I(t)$. The
second type of uncertainties are the systematic errors that might occur when
the numerical simulations are not fully converged. Those systematic errors are
particularly relevant for TDVP results. Below, we describe numerical tests
that confirm that the values $\chi$ used by us in the main text are sufficient
for the results to be converged, i.e. independent of the value of the bond
dimension $\chi$.
### A.1 Chebyshev time propagation
For small system sizes ($L\leq 20$) we use Chebyshev propagation scheme as
described in detail in [125]. In a nutshell, this approach approximates the
time evolution operator $U(\Delta t)=\exp(-iH\Delta t)$ over time period
$\Delta t$ as
$U(\Delta t)\approx\mathrm{e}^{-\mathrm{i}b\Delta t}\left(J_{0}(a\Delta
t)+2\sum_{k=1}^{N}(-i)^{k}J_{k}(a\Delta
t)T_{k}\left(\mathcal{H}\right)\right),$ (6)
where $a=(E_{\rm max}-E_{\rm min})/2$, $b=(E_{\rm max}+E_{\rm min})/2$ and
$E_{\rm min}$ ($E_{\rm max}$) is the energy of the ground state (the highest
excited eigenstate) of the Hamiltonian $H$. The Hamiltonian is rescaled to
$\mathcal{H}=\frac{1}{a}(H-b)$ so that spectrum of $\mathcal{H}$ belongs to
the $[-1,1]$ interval, $J_{k}(t)$ is the Bessel function of the order $k$ and
$T_{k}(x)$ is the Chebyshev polynomial of order $k$. The order of expansion
$N$ needed to assure convergence of the expansion (6) for a given time step
$\Delta t$ is computed in the following way. We take a random normalized state
$\ket{\psi_{R}}$, calculate the state $U(\Delta t)\ket{\psi_{R}}$ with a
certain trial order of expansion $N_{tr}$ and compute its norm. If the norm of
$U(\Delta t)\ket{\psi_{R}}$ deviates from unity by more than $10^{-13}$, we
know that $N_{tr}$ needs to be increased; otherwise $N_{tr}$ is decreased.
This allows us to perform a binary search for $N_{tr}$ in the interval
$N_{tr}\in[5,5000]$ (the upper boundary is determined by the maximal time step
$\Delta t$ and parameters of the model). The result of this binary search,
$N^{0}_{tr}$, is then incremented by $20\%$, yielding the desired order
$N=1.2N^{0}_{tr}$. We calculate the order of expansion whenever the time step
$\Delta t$ changes in our algorithm. To calculate the time evolution of an
initial state $\ket{\psi(0)}$ we repeatedly apply (6) to obtain
$\ket{\psi(\Delta t)}$, $\ket{\psi(2\Delta
t)}$,$\ldots$,$\ket{\psi(t_{\mathrm{max}})}$. We have tested this procedure
for system sizes $L\leq 16$ comparing $\ket{\psi(t_{\mathrm{max}})}$ with
state $\ket{\psi_{ED}(t_{\mathrm{max}})}=U(t_{\mathrm{max}})\ket{\psi(0)}$
evolved using time evolution operator $U(t_{\mathrm{max}})$ determined by
means of the full exact diagonalization of the Hamiltonian $H$. For
$t_{\mathrm{max}}=10^{5}$ we checked that the norm
$||\ket{\psi(t_{\mathrm{max}})}-\ket{\psi_{ED}(t_{\mathrm{max}})}||$ is
smaller than $10^{-10}$ in the whole parameter range considered in this work.
The deviation from unity of the norm of the state propagated with the
Chebyshev expansion: $1-||\ket{\psi(t_{\mathrm{max}})}||$ was smaller than
$10^{-12}$ for all system sizes considered in this work. For $L\leq 16$ the
corresponding deviations in the value of $C(t)$ function (as compared to
$\ket{\psi_{ED}(t_{\mathrm{max}})}$) were smaller than $10^{-13}$. We note
that the main advantage of the Chebyshev expansion is that it efficiently
utilizes the sparse matrix structure of the Hamiltonian of the system. This is
due to the fact that a single time propagation step $U(\Delta t)\ket{\psi(t)}$
reduces to $\mathcal{O}(N)$ matrix-vector products and a calculation of linear
combinations of vectors.
### A.2 Tensor network approaches
Chebyshev propagation scheme is not effective for larger system sizes since it
operates on the quantum states expressed as vectors in the full Hilbert space
that is exponentially large in system size. In contrast, tensor network
techniques parameterize only a fraction of the full Hilbert space, encoding
the state of the system in a matrix product state (MPS). This allows us to
investigate time evolution of systems larger than $L>25$. The tensor network
techniques were developed over the years starting from seminal works of Vidal
[53, 54] and White [135]. The link between the two approaches was illuminated
in [136]. Those schemes are known as time-dependent density matrix
renormalization group techniques (tDMRG) or time evolving block decimation
TEBD techniques. The important modification came with the variational approach
leading to algorithms based on Time Dependent Variational Principle (TDVP)
optimal for an assumed limitation of the Hilbert space [55, 56, 57, 58]. The
time evolution can be calculated effectively with tensor network approaches
only when the bond dimension $\chi$ of the MPS is sufficiently large to encode
the state of the system. This gives rise to an upper limit on the entanglement
entropy in the state of the system for a given $\chi$. This, in turn,
translates into maximal time $t_{\mathrm{max}}$ to which time evolution of the
system can be accurately simulated with TDVP/tDMRG for a given bond dimension
$\chi$. Calculations in our work rely on the fact that for disorder strengths
$W=8-10$ the spreading of entanglement in the system is very slow which allows
us to probe the time evolution at times equal to few thousand tunneling times.
The TDVP algorithm for time evolution consists of two stages. In the first
stage we use the so called 2-site TDVP which allows for an accurate estimation
of the errors. They appear mainly due to the truncation of the Hilbert space
via Schmidt decomposition between the sites. When a disregarded Schmidt weight
exceeds $10^{-12}$ the Hilbert space is enlarged so in this stage the
algorithm is practically exact until the bond dimension reaches the prescribed
value $\chi$ at a given bond. At this stage we switch (at this bond) to 1-site
TDVP algorithm, from this moment errors due to the Hilbert space truncation
start to accumulate. This is a standard, well developed strategy [60, 61]
which we follow in our work.
The detailed comparison of the performance of TEBD and TDVP algorithms for the
random-field XXZ chain, but for lower disorder amplitudes than in the present
work, was performed in in our previous work [60]. It was shown, in particular,
that the TEBD algorithm that is unconverged, i.e. the bond dimension is not
sufficiently large to follow the time evolution of the state up to the
requested time $t_{\mathrm{max}}$, spuriously indicates a stabilization of the
imbalance $I(t)$ suggesting a localization in the system. In contrast,
unconverged TDVP has a tendency to show a delocalization in the system by
overestimating the degree of decay of the imbalance $I(t)$. An analogous
behavior of TDVP was also observed in a different disorder-free models in
[58].
Figure 11: Comparison of the imbalance $I(t)$ (averaged over times
$[t-10,t+10]$) for system size $L=200$ and disorder strength $W=10$ obtained
with TEBD and TDVP algorithms. The bond dimension is fixed as $\chi=128$ and
the results are averaged over $24$ disorder realizations. The inset shows the
difference between the imbalances $I(t)$ for TDVP and TEBD propagation
schemes. Figure 12: Comparison of the imbalance $I(t)$ (averaged over times
$[t-10,t+10]$) for system size $L=50$ and disorder strength $W=8$ obtained
with TEBD and TDVP algorithms. The bond dimension is fixed as $\chi=128$ and
the results are averaged over $1000$ disorder realizations. The dashed line
shows the fitted power-law decay of $I(t)$. The inset shows the difference
between the imbalances $I(t)$ for TDVP and TEBD propagation schemes.
This motivates us to compare results for the imbalance $I(t)$ obtained with
TEBD and TDVP algorithms as shown in Fig. 11. The results are averaged over 24
disorder realizations for $L=200$ and $W=10$. We observe that the agreement
between TDVP and TEBD results is excellent indicating the convergence for
individual disorder realizations. The difference between the curves at late
times oscillates around ${4}\cdot 10^{-5}$. This small discrepancy can be
compared with the total change of the value of imbalance $\Delta I=7\cdot
10^{-4}$ in the interval $t\in[100,1200]$ for $L=200$ (cf. Fig. 5). The latter
value is more than an order of magnitude larger than the discrepancy between
TDVP and TEBD results. This suggests that the exponent of power-law decay
$\overline{\beta}=(3.50\pm 0.87)\cdot 10^{-4}$ for $L=200$ (see Tab. 3) is
accurately estimated.
While TEBD is faster “per time step” for such a large disorder amplitude
($W=10$) we must take very small time step $\Delta t=0.001$ for TEBD to obtain
converged results. The error of the approximate unitary evolution may be
estimated by a relative energy change in TEBD algorithm
$[(\braket{\psi_{TEBD}(t_{\mathrm{max}})}{H}{\psi_{TEBD}(t_{\mathrm{max}})}-E_{0})/E_{0}$
where $\ket{\psi_{TEBD}(t)}$ is the state obtained in TEBD time evolution and
$E_{0}=\braket{\psi(0)}{H}{\psi(0)}$]. It remains below $10^{-4}$ for even the
most unfavorable disorder realisation (for $\chi=128$). At the same time, the
total accumulated error, equal to sum of squares of Schmidt coefficients
disregarded in all time steps, associated with necessary truncations inherent
to TEBD is below $10^{-5}$. As shown in the following, the resulting error is
sufficiently small to obtain an accurate estimate of the exponent
$\overline{\beta}$. The required small step makes, however, the application of
TEBD scheme not practical. For large disorder amplitudes and large time
scales, it is more efficient to use TDVP. It allows us to keep the time step
at a reasonable value, $\Delta t=0.1$ (the time scale is fixed by $J=1$ in
(1)). We have checked by decreasing the time step that the chosen value leads
to accurate results. The agreement of TDVP results with the numerically exact
results for the non-interacting case, shown in Fig. 3 provides another test of
the convergence of our results with the time step.
While the comparison for $L=200$ is carried out for 24 disorder realizations
only, we supplement it with comparison for $W=8$ and $L=50$ carried out for
over a 1000 disorder realizations in Fig. 12. Again the discrepancy between
curves is small (of the order of ${4}\cdot 10^{-5}$ as for $L=200$ data in
Fig. 11. Larger number of disorder realizations allows us to extract reliably
$\beta$ values from both simulations. They agree very well with each other
indicating the agreement between both aogorithms used.
Figure 13: The imbalance $I(t)$ (averaged over times $[t-10,t+10]$) for system
size $L$ and disorder strength $W$ obtained with TDVP algorithm with bond
dimension $\chi$, dashed lines show power-law fits $I(t)\sim
t^{-\overline{\beta}}$ in time interval $t\in[100,t_{\mathrm{max}}]$ . Top
panel: $L=200$, $W=8$, results averaged over 250 disorder realizations,
$t_{\mathrm{max}}=1500$. Center panel: $L=100$, $W=10$, results averaged over
1000 disorder realizations,$t_{\mathrm{max}}=1500$. Bottom panel: $L=200$,
$W=10$, results averaged over 984 disorder realizations,
$t_{\mathrm{max}}=1200$. The insets show the difference between the imbalances
for the larger and the smaller value of $\chi$. Figure 14: Top panel:
Comparison of the imbalance $I(t)$ (averaged over times $[t-10,t+10]$) for
system size $L=50$ and disorder strength $W=10$ obtained with TDVP algorithm
with bond dimension $\chi=[50,90,128,192]$. The results are averaged over
$1000$ disorder realizations. The inset shows the difference $\Delta
I(t)=I_{\chi}(t)-I_{\chi=192}(t)$ between the imbalance obtained with TDVP
with the largest bond dimension $\chi=192$ and the imbalances obtained with
$\chi=128,90,50$. Bottom panel: the same, but data for the smaller bond
dimensions: $\chi=48,64,96,128$ obtained with TEBD algorithm. Note the
difference in the range of the horizontal axes of the two panels. Figure 15:
Top panel: Comparison of the values of the exponent $\overline{\beta}$
governing the decay of the imbalance $I(t)$ for disorder strength $W=10$ and
system size $L=50$ obtained with TDVP propagation scheme. The fitting was
performed in the interval $t\in[100,t_{\mathrm{max}}]$ and results are shown
as a function of $1/\chi$, solid lines denote second order polynomial fits in
$1/\chi$, the red point at $1/\chi=0$ shows the result
$\overline{\beta}=(3.08\pm 0.51)\cdot 10^{-4}$ from Tab. 3. Bottom panel: The
exponent $\overline{\beta}$ as a function of $1/\chi$ for TDVP and TEBD
algorithms (for $L=50$, $W=10$). The values of $\overline{\beta}$ presented in
both are extracted from data shown in Fig. 14.
The above comparison of TEBD and TDVP algorithms suggests a good convergence
of our TDVP results. To further investigate the accuracy of the TDVP scheme,
we compare results obtained for a varying bond dimension $\chi$. Fig. 13
summarizes our results. In each of the investigated cases we observe that the
curves showing the imbalance $I(t)$ practically overlap for the both bond
dimensions considered (cf. the insets in Fig. 13). We observe that the
exponents $\overline{\beta}$ govering the power-law decay of imbalance for
smaller and larger $\chi$ are consistent with each other indicating a good
convergence of the data with the bond dimension. At the same time, we observe
that the $\overline{\beta}$ slightly decreases with the increase of the bond
dimension $\chi$ in each of the analyzed cases.
This dependence is further analyzed in the top panel of Fig. 14 in which we
have supplemented the data for $\chi=128$ and $\chi=192$ with results for
smaller bond dimensions $\chi=50,90$. Interestingly, the agreement of results
for $\chi\geq 50$ up to time $t\approx 1000$ shows that already the results
for $\chi=50$ are a good estimate of the imbalance $I(t)$ in that time
interval at $W=10$. At larger times, the results for $\chi=50$ are unconverged
and show a spurious signatures of delocalization in the system consistently
with our expectations based on [60]. The bottom panel of Fig. 14 compares the
TDVP results for $\chi=192$ with the imbalance obtained with TEBD and bond
dimensions $\chi=48,64,96,128$. Contrary to the expectations from [60], we see
that TEBD also indicates weaker and weaker decay of the imbalance $I(t)$ as
the bond dimension $\chi$ is increased. The difference between the TDVP and
TEBD results for the largest $\chi$ presented is no bigger than $4\cdot
10^{-5}$ indicating that both algorithms yield consistent estimates of
$\overline{\beta}$, $\beta(t)$.
To clarify the dependence of the results on the value of $\chi$, we plot the
values of the exponent $\overline{\beta}$ as function of $1/\chi$ in the top
panel of Fig. 15. The value of the exponent $\overline{\beta}$ decreases
monotonously with the bond dimension $\chi$. The change in the value of
$\beta$ when $\chi$ increases from $50$ to $192$ is the smallest for
$t_{\mathrm{max}}=1500$ (indicating that smaller bond dimensions are needed to
get converged results for $t<1500$), and increases with the increase of
$t_{\mathrm{max}}$. Nevertheless, the extrapolations of $\overline{\beta}$
with a second order polynomial in $1/\chi$ give consistent results for all
considered values of $t_{\mathrm{max}}$. Importantly, those extrapolations are
in agreement with the result $\overline{\beta}=(3.08\pm 0.51)\cdot 10^{-4}$
from Tab. 3, confirming the convergence of our simulations with the bond
dimension. The bottom panel of Fig. 15 shows a comparison of
$\overline{\beta}$ for $t_{\mathrm{max}}=1500$ for TEBD and TDVP results. The
values of $\overline{\beta}$ are nearly independent of $\chi$ for
$\chi\gtrapprox 90$ confirming that both algorithms are very close to being
converged at those bond dimensions for $t<t_{\mathrm{max}}=1500$. Finally, the
extrapolation of those results to large $\chi$ limit yields the consistent
values of $\overline{\beta}$ for both TEBD and TDVP in line with our message
about the persistence of a slow decay of the imbalance even at the large
disorder strength $W=10$.
### A.3 Time evolution for free fermions
Here, for completeness, we provide details of the standard (see [137] and
references therein) approach to time evolution of a system of non-interacting
fermions used by us in Sec. IV. The Hamiltonian (1), upon Jordan-Wigner
transformation, becomes
$\hat{H}=2J\sum_{i=1}^{L-1}\left(\hat{c}^{{\dagger}}_{i}\hat{c}_{i+1}+\hat{c}^{{\dagger}}_{i+1}\hat{c}_{i}+\frac{\Delta}{2}\hat{n}_{i}\hat{n}_{i+1}\right)+\sum_{i=1}^{L}h_{i}\hat{n}_{i},$
(7)
where $\hat{c}^{{\dagger}}_{i}$ ($\hat{c}_{i}$) is creation (anihilation)
operator of spinless fermion at site $i$, canonical anti-commutation relation
$\\{\hat{c}_{i},\hat{c}^{{\dagger}}_{j}\\}=\delta_{ij}$ is fulfilled, and the
number operator is given as $\hat{n}_{i}=\hat{c}^{{\dagger}}_{i}\hat{c}_{i}$.
For $\Delta=0$, the model (7) becomes non-interacting. Then, it can be written
as a quadratic form of the fermionic operators
$\hat{H}=\sum_{i,j=1}^{L}h_{ij}\hat{c}^{{\dagger}}_{i}\hat{c}_{j},$ (8)
where we have introduced a $L\times L$ matrix $\mathbf{h}=(h_{ij})$. Time
dependence of the fermion anihilation operator is given by
$\hat{c}_{i}(t)=e^{i\hat{H}t}\,\hat{c}_{i}e^{-i\hat{H}t}=\sum_{j=1}^{L}(e^{-i\mathbf{h}t})_{ij}\hat{c}_{j},$
(9)
where the second equality can be obtained from the Baker–Campbell–Hausdorff
formula. Defining a $L\times L$ correlation matrix
$\mathbf{C}(t)=(\mathbf{C}(t))_{i,j}=\bra{\psi}\hat{c}^{{\dagger}}_{i}(t)\hat{c}_{i}(t)\ket{\psi},$
(10)
and using (9), we find that
$\mathbf{C}(t)=e^{i\mathbf{h}t}\mathbf{C}(0)\,e^{-i\mathbf{h}t}.$ (11)
The correlation matrix $\mathbf{C}(0)$ at $t=0$ is determined by the initial
state, and for the Néel state the only non-vanishing coefficients are
$\mathbf{C}(0)_{2k,2k}=1$ where $k=1,\ldots,L/2$. The imbalance is given by
$I(t)=D\sum_{i=1+l_{0}}^{L-l_{0}}(-1)^{i}(\mathbf{C}(t))_{ii},$ (12)
where the constant $D$ assures that $I(0)=0$. Finally, to calculate
entanglement entropy for a bipartition of the system into subsystems
consisting of sites $1,\ldots,l_{A}$ and $l_{A}+1,\ldots L$, we calculate
eigenvalues $\lambda_{i}$ of the submatrix $(\mathbf{C}(t))_{i,j=1}^{l_{A}}$
and compute the entanglement entropy as [138, 139]
$S(t)=-\sum_{i=1}^{l_{A}}[\lambda_{i}\ln(\lambda_{i})+(1-\lambda_{i})\ln(1-\lambda_{i})].$
(13)
The formulas (12) and (13) allow us to calculate the imbalance and
entanglement entropy for the XXZ spin chain with $\Delta=0$ with numerical
cost scaling as $L^{3}$.
Figure 16: The exponent $\overline{\beta}$ obtained from the fit $I(t)\sim
t^{-\overline{\beta}}$ in interval $t\in[t_{\mathrm{min}},t_{\mathrm{max}}]$
as a function of $t_{\mathrm{min}}$ for $W=8,10$ and system size $L=50,200$.
## Appendix B Stability of the power-law fits to choice of time interval
In the main text, the imbalance was fitted by a power-law decay: $I(t)\sim
t^{-\overline{\beta}}$ in interval $t\in[t_{\mathrm{min}},t_{\mathrm{max}}]$,
where $t_{\mathrm{min}}=100$ and the value of $t_{\mathrm{max}}$ was equal to
the maximal time reached in time evolution
($t_{\mathrm{max}}=1200,1500,5000$). In this appendix, we discuss the impact
of changes of $t_{\mathrm{min}}$ on the value of exponent $\overline{\beta}$.
The result is shown in Fig. 16. For $W=8$, we observe that the value
$\overline{\beta}$ remains, within the estimated error bars, constant in the
interval $t_{\mathrm{min}}=[80,200]$, justifying the choice
$t_{\mathrm{min}}=100$ made in the main text. Inclusion of times $t\lessapprox
80$ leads to an increase of $\overline{\beta}$ – consistently with the small
$t$ behavior of the imbalance shown in Fig. 4. When
$t_{\mathrm{min}}\gtrapprox 200$, the precision of estimation of
$\overline{\beta}$ decreases as the fitting interval
$[t_{\mathrm{min}},t_{\mathrm{max}}]$ gets narrower. Similar trends are
observed for $W=10$ for data with $t_{\mathrm{max}}=1500$. However, the
stability of the fit is greatly improved when $t_{\mathrm{max}}=5000$: then,
the choices of $t_{\mathrm{min}}$ from interval $[80,1000]$ lead to the values
of $\overline{\beta}$ that agree within the estimated error bars.
The value of $\overline{\beta}$ decreases approximately $3$ times when $W$ is
increased from $8$ to $10$. However, $\overline{\beta}$ clearly remains
positive within the estimated error bars, showing that the imbalance $I(t)$
indeed decays in time. Extrapolating the trend of changes in
$\overline{\beta}$, we may expect that $\overline{\beta}\approx 10^{-4}$ at
$W=12$. Assuming a similar scaling of the statistical error of
$\overline{\beta}$, already at $W=12$ we would need to either increase the
number of disorder realizations or increase $t_{\mathrm{max}}$ as compared to
their respective values at $W=8,10$ to be certain that the value of
$\overline{\beta}$ at $W=12$ is positive.
Figure 17: Persistent oscillations for QP potential. The imbalance $I(t)$
(averaged over times $[t-25,t+25]$) is shown by solid lines for various
amplitudes of QP potential $W^{QP}$, shades show the imbalance without time
averaging. The results are averaged over more than $5000$ realizations of QP
potential and the system size is fixed as $L=16$. Top panel shows results for
$k=\frac{\sqrt{5}-1}{2}$ whereas the bottom panel for $k=\frac{\sqrt{2}}{2}$.
The range of the vertical axis is the same for all subplots.
## Appendix C Oscillations of the imbalance for quasiperiodic systems
In Sec. VII of the main text, we have demonstrated an emergence of persistent
oscillations of the imbalance $I(t)$ for sufficiently strong QP potential. In
this Appendix we provide further details on this phenomenon.
Since the oscillations do not depend on the system size (at least for $L\geq
12$), we fix the system size as $L=16$ and investigate the time evolution of
the imbalance $I(t)$ varying the amplitude $W^{QP}$ of the QP potential as
well as the wave vector $k$ that determines the shape of the QP potential
(recall that $h_{j}=W^{\mathrm{QP}}\cos(2\pi kj+\phi)$). The results are shown
in Fig. 17.
By comparing the results for fixed $k=\frac{\sqrt{5}-1}{2}$, we note that the
amplitude of oscillations diminishes when $W^{QP}=5$ is increased to
$W^{QP}=8$. This could be expected as in the limit of $W^{QP}\to\infty$, the
initial Néel state becomes an eigenstate of the XXZ model. In that limit, the
oscillations are absent and the imbalance remains trivially equal to unity
throughout the time evolution. Thus, the imbalance oscillations occur only in
a limited range of amplitudes of the QP potential: $W^{QP}$ must be
sufficiently large to give rise to a very slow dynamics (unlike in Fig. 8) but
not large enough to give rise to a trivial dynamics. A similar decrease of the
oscillations of $I(t)$ upon the increase of $W^{QP}$ is visible in Fig. 17 for
$k=\frac{\sqrt{2}}{2}$.
The differences in the pattern of oscillations for $k=\frac{\sqrt{5}-1}{2}$
and $k=\frac{\sqrt{2}}{2}$ (well pronounced for the smaller values of $W^{QP}$
in Fig. 17) demonstrate that the oscillations of $I(t)$ depend in a non-
trivial fashion on the value of the constant $k$. By investigating time
dynamics of the density correlation function $C(t)$ for initial states that
are eigenstates of the $S^{z}_{i}$ operator but are different than the Néel
state, we noted that the pattern of the oscillations of $I(t)$ depends
strongly also on the initial state. In particular, the density correlation
function $C(t)$ averaged over such initial states shows no long-time
oscillations. This shows that the emergence of the pattern of oscillations is
determined by the interplay of a spatial structure of the initial state and
the constant $k$ of the QP potential.
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|
# VT-CLIP: Enhancing Vision-Language Models with Visual-guided Texts
###### Abstract
Contrastive Language-Image Pre-training (CLIP) has drawn increasing attention
recently for its transferable visual representation learning. However, due to
the semantic gap within datasets, CLIP’s pre-trained image-text alignment
becomes sub-optimal on downstream tasks, which severely harms its transferring
performance. To better adapt the cross-modality embedding space, we propose to
enhance CLIP via Visual-guided Texts, named VT-CLIP. Specifically, we guide
textual features of different categories to adaptively explore informative
regions on the image and aggregate visual features by attention mechanisms. In
this way, the texts become visual-guided, namely, more semantically correlated
with downstream images, which greatly benefits the category-wise matching
process. In few-shot settings, we evaluate our VT-CLIP on 11 well-known
classification datasets to demonstrate its effectiveness.
Index Terms— Contrastive language-image pre-training, Few-shot learning,
Image classification
## 1 Introduction
With the exploring of better network architectures, traditional deep learning
has achieved extraordinary performances over a wide range of vision tasks,
such as image classification [1], object detection [2] and so on [3]. Although
these methods are expert at specific scenarios, they lack the ability of
general vision representation and are hard to transfer to open-set
applications. Considering the wide coverage of languages, CLIP [4] proposes to
pre-train the visual representation learning contrastively with natural
language signals. Supervised by large-scale image-text pairs, CLIP extracts
both features of input images and texts by separate encoders and matches the
paired ones in the same embedding space. Such pre-trained cross-modality
alignment endows CLIP the capability of recognizing new visual concepts on
downstream tasks. Specifically, given a new dataset with images to be
recognized, one could construct the textual inputs of CLIP by the category
names, termed as prompts, and convert the classification task into a matching
problem. By this, CLIP is able to conduct zero-shot recognition in open-
vocabulary settings.
To further improve the downstream performance of CLIP, existing works
introduce different fine-tuning methods under the few-shot settings. Inspired
by prompt tuning in natural language processing [5], Context Optimization
(CoOp) [6] freezes CLIP’s pre-trained weights and adopts learnable prompts to
learn the best-fitted textual inputs other than the original hand-crafted
ones. From another perspective, CLIP-Adapter [7] appends a lightweight adapter
module [8] over the image or text encoders and is also fine-tuned with frozen
CLIP’s weights. However, they are constrained by the following limitations.
The learnable prompts in CoOp are set before the large-scale text encoder,
which is much time-consuming to back-propagate the training gradients through
such 12-layer transformer [9] for every iteration. CLIP-Adapter only conducts
feature adaption independently for each branch, and lacks cross-modal
interactions for image and language. Also, both of their learned parameters
are fixed for all images during inference, losing the adaption flexibility for
varying visual inputs.
Fig. 1: Visualization of attention map and prediction score. In our VT-CLIP,
the informative regions on the image gain more attention weights from the
texts, which improves the prediction score on ground-truth category.
In this paper, we propose VT-CLIP, which equips CLIP with Visual-guided Texts
to exert thorough adaptions of both modalities under few-shot settings.
Specifically, we introduce a visual-guided attention module to conduct feature
communication after both encoders, which views texts as the queries, and
images as keys and values. For textual branch, we input the hand-crafted
prompts with explicit semantics and utilize the encoded features for
interaction with images. For visual branch, we extract the intermediate
spatial features of images as keys and values instead of the final global
ones, which could provide more fine-grained contextual information. By
calculating the per-pixel similarities, different categories in texts could
explore informative visual regions and gather related features weighted by
their attention scores. After this, the texts are visual-guided and better
fitted for the later matching stage. As visualized in Figure 1, the textual
feature of ”airline” category focuses on the corresponding visual regions, but
other unmatched categories do not, as expected. Importantly, our visual-guided
texts are adaptive for different samples, since the attention map is
dynamically produced by the input images. We conduct extensive experiments on
11 well-known classification datasets, which fully demonstrates the
outstanding enhancement ability of visual-guided texts over CLIP.
## 2 RELATED WORK
Recently, vision-Language Models shows great potential in learning generic
visual representation with nature language supervision, which allowing zero-
shot transfer ability for various downstream classification tasks. Inspired by
the success of pre-train models [10], [11] [12] and SimVLM [13] use attention
architecture improve the performance of vision-language tasks. At present, the
recent breakthrough in vision-language learning, particularly CLIP [4] and
ALIGN [14] are driven by the noisy large-scale datasets available in the
Internet, which is 400 million image-text pair for CLIP and 1.8 billion noisy
image-text pairs for ALIGN.To fine-tune vision-Language Models on downstream
tasks like few-shot classification task, CoOp [15] propose to learn soft
prompts represented by continuous context vectors as alternative for hand-
craft prompt while CLIP-Adapter propose to adopts an additional bottleneck
layer to learn new features and performs residual style feature blending with
the original pre-trained features. Though CoOp and CLIP-Adapter achieve
significant performance in the perspective of prompt learning and feature
adapters, our VT-CLIP explores the impact of instance-level image visual
feature on refining text feature with a cross-attention module.
Prompt Learning are designed to better mine the knowledge from pre-trained
models without fine-tuning the entire model, which generate a prompting
template or function to bridge gap between the pre-training objective and
downstream tasks [5, 16, 17, 18]. Prompt engineering is an important topic in
prompt learning. Early research focus on designing hand-crafted prompts, which
generate cloze-style prompts like “fill-in-the-blank” cloze tests and benefits
a number of downstream tasks, such as sentiment analysis [16]. Recently, [17,
18] introduce gradient-based approaches which optimize continuous vectors in
the word embedding space. The limitation of prompt engineering is that hand-
crafted prompt template requires specific domain knowledge and the prompt
content learned by optimization lacks interpretability. In this paper, we
demonstrate guiding text feature with instance-level image feature through
cross-attention module is an alternative for prompt learning on large-scale
vision-language models, which is more interpretable and simpler in
architecture.
Fig. 2: Structures of VT-CLIP. Class-level text feature from CLIP text encoder
is updated by spatial visual image feature with cross attention module
## 3 METHODS
We first revisit the zero-shot CLIP in Section 3.1, and then introduce the
details of our proposed VT-CLIP in Section 3.2.
### 3.1 Zero-shot CLIP
CLIP is pre-trained to align image and text pair information through
contrastive training. CLIP contains two independent encoders for visual and
textual feature encoding. Specifically, the image encoder consists of a visual
backbone, which is ResNet [1] or ViT [19], and an attention pooling layer,
while the text encoder is a conventional Transformer Encoder . With large-
scale data traning, informative representation for both modalities are deeply
learned by CLIP’s image encoder and text encoder, thus obtaining zero-shot
classification ability. In details, for a dataset contains $K$ categories,
denoted as {$C_{1},\dots,C_{K}$}, CLIP first places all category names into
the hand-crafted template $H$ proposed by [4] to get the textual inputs, which
then are fed into the tokenizer $T$ and text encoder to obtain text features,
$T_{c}\in R^{1\times C}$. Meanwhile, the input image $I\in
R^{H\times{W\times{3}}}$, where $H$ and $W$ are the height and width of the
image respectively, is first encoded by visual backbone, termed as $VB$,
getting contextual-level spatial features $V_{s}$. After that, an attention
pooling operation is adopted to get global visual features $V_{c}$, i.e.,
$\displaystyle V_{s}=\operatorname{VB}(I),V_{s}\in R^{H_{s}\times W_{s}\times
C_{s}}$ (1) $\displaystyle V_{c}=\operatorname{Pooling}(V_{s}),V_{c}\in
R^{1\times C}$ (2) $\displaystyle
T_{c}=\operatorname{TextEn}\big{(}T([H;C_{i}])\big{)},i\in\\{1,\dots,K\\},$
(3)
where $TextEn$ denotes text encoder, and $C$ is the class-level feature
dimension. The $H_{s},W_{s},C_{s}$ are the height , width and channel
dimension for spatial feature. Via attention pooling, CLIP generates the
global visual features from spatial features which contain more local
contextual-level information. Finally, the similarity scores are calculated
as,
$\displaystyle P=\operatorname{Softmax}(V_{c}T_{c}^{T}/\tau),$ (4)
where $SoftMax(\cdot)$ and $P$ denote the softmax function and the similarity
scores for $K$ categories, and $\tau$ is a temperature parameter learned by
CLIP.
### 3.2 VT-CLIP
Different from the perspective of prompt learning , we present a new approach
to enhance vision-language model. We suppose that generic soft prompt which is
invariant to images with various content is kind of insufficient. Hence, we
propose VT-CLIP, which dynamically refine the text features using visual
spatial features. Specifically, through a visual-guided cross-attention
module, we leverage the contextual-level spatial feature, which is obtained
before pooling, to guide the text feature to adaptively explore informative
regions on the image. The learned refinement is fused with original text
features a by residual connection to preserve the robustness and
effectiveness. Following the standard architecture of the transformer decoder
blocks [9], our proposed visual-guided cross-attention module includes a self-
attention layer, a co-attention layer and a feed forward network, where
$T_{c}$ and $V_{s}$ are fed into the cross attention module, with $T_{c}$
serving as query, and $V_{s}$ as key and value, i.e.,
$\displaystyle VT_{c}=\operatorname{CrossAttn}(V_{s},V_{s},T_{c})+T_{c},$ (5)
$\displaystyle VT_{c}\in R^{1\times C}$
where $CrossAttn$ is the visual-guided cross-attention module, and $VT_{c}$
denotes the adapted text features.
Through the interaction in cross-attention, the adapted text features $VT_{c}$
become more semantically correlated with the paired image. Then, the
similarity scores are predicted via obtained $VT_{c}$, that is,
$\displaystyle P=\operatorname{Softmax}(V_{c}VT_{c}^{T}/\tau).$ (6)
During training, we freeze both visual baskbone and textual encoder, and only
optimize weights in visual-guided cross-attention module via cross-entropy
loss.
Fig. 3: Experiment Results––Main results of few-shot learning on 11 datasets.
VT-CLIP shows overall better performance over baselines across different
training shots.
## 4 EXPERIMENTS
### 4.1 Training Settings
We evaluate the performance of VT-CLIP on 11 widely-adopted image
classification datasets and follow the few-shot evaluation protocol of CLIP
[4], that is, training on $1$, $2$, $4$, $8$ and $16$ shots and testing on
full test set. During training, we adopt pre-trained visual backbone ResNet-50
from [4] with all the weights frozen, and follow the data preprocessing
protocol of CLIP, which consists of random cropping, resizing, and random
horizontal flip. Following CoOp [6], VT-CLIP is trained with batch size $32$
and learning rate $2\times 10^{-3}$ for all 11 datasets. Instead of using
learnable continuous prompts in CoOp, we adopt the same hand-crafted prompt as
CLIP. We compare VT-CLIP with three baseline works, i.e., Zero-shot CLIP [4],
CoOp [6] and CLIP-Adapter [7]. Also, in order to thoroughly demonstrate the
superiority of proposed VT-CLIP, we use the best baseline variants.
### 4.2 Performance Comparison & Analysis
The main results are presented in Figure 3. As shown in the top-left chart of
Figure 3, VT-CLIP shows outstanding average performance over three baselines,
and the accuracy gain increases as the training shots get more, which
indicates VT-CLIP serves as an effective and reliable enhancer under few-shot
settings. Also, as shown in all the twelve charts in Figure 3, our VT-CLIP
outperforms other works significantly under each shot setting on 11 datasets.
What’s more, unlike CoOp’s poor performance with little training samples, VT-
CLIP achieves more stable scores, which indicates our VT-CLIP is not sensitive
to data scale. Additionally, it is clear that VT-CLIP obtains consistently
prominent results on all the 11 datasers, which demonstrates more considerable
generalization ability than CoOp, as seen in charts of OxfordPets [20] and
Food101 [21], where CoOp falls behind even zero-shot CLIP under 16-shot
setting. As for CLIP-Adapter, VT-CLIP not only surpasses it on different
datasets, but also contains better interpretability of leveraging contextual
visual features to guide text features to be more semantically correlated to
the certain downstream task. The consistent superiority of VT-CLIP over 11
datasets fully demonstrates the effectiveness and generality of our proposed
method.
## 5 Ablation Study
In this section, we conduct several ablation studies for VT-CLIP. All
experiments below adopt the 16-shot setting.
Heads | 4 | 8 | 16 | 32
---|---|---|---|---
Caltech101 (%) | 93.06 | 93.10 | 92.37 | 92.62
DTD (%) | 64.42 | 65.72 | 64.78 | 65.43
Table 1: Head Number. Performance with different number of heads in cross
attention.
We explore the number of heads in cross attention on Caltech-
101 [22] and DTD [23]. The heads number in attention mechanism equals to the
number of heterogeneous scores computed for values indicating fitting ability
of model. As presented in Table 1, the best performance is achieved with two
heads for cross attention module. To further demonstrate the design of our
method, we conduct experiments on increase the number of cascaded attention
layers. The results are in Table 2. We observe that performance degrades as
the cascaded layers increase which indicate the complex model with multiple
cross attention layer tend to overfit the insufficient training data under few
shot scenario.
Layers | 1 | 2 | 3 | 4
---|---|---|---|---
Caltech101(%) | 93.10 | 93.06 | 92.58 | 92.29
DTD (%) | 65.72 | 64.60 | 65.60 | 64.78
Table 2: Layer Number. Performance with different number of cascaded cross
attention layers.
## 6 CONCLUSION
We propose VT-CLIP, a novel enhancement of CLIP for few-shot classification
which focuses on leveraging the contextual visual features to guide the text
features to highlight the important regions via a visual-guided cross-
attention module. In this way, the deep interaction between the image and text
branches in vision-language model is of great potential in enhancing the
model’s ability. Also, extensive experiments demonstrate that VT-CLIP
outperforms all the competitive baselines in few-shot settings on 11 widely-
used datastes. Ablation studies are conducted to further prove our design and
give a view of the extensive performance of VT-CLIP. In the future, we hope to
combine VT-CLIP with prompt learning based approaches to push the boundary
further for fine-tuning vision-language models. We will also explore the
potential of VT-CLIP on other vision or textual tasks.
## References
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|
# EvoGrad: A Dynamic Take on the Winograd Schema Challenge with Human
Adversaries
###### Abstract
While Large Language Models (LLMs) excel at the Winograd Schema Challenge
(WSC), a coreference resolution task testing common-sense reasoning through
pronoun disambiguation, they struggle with instances that feature minor
alterations or rewording. To address this, we introduce EvoGrad, an open-
source platform that harnesses a human-in-the-loop approach to create a
dynamic dataset tailored to such altered WSC instances. Leveraging ChatGPT’s
capabilities, we expand our task instances from 182 to 3,691, setting a new
benchmark for diverse common-sense reasoning datasets. Additionally, we
introduce the error depth metric, assessing model stability in dynamic tasks.
Our results emphasize the challenge posed by EvoGrad: Even the best performing
LLM, GPT-3.5, achieves an accuracy of 65.0% with an average error depth of
7.2, a stark contrast to human performance of 92. 8% accuracy without
perturbation errors. This highlights ongoing model limitations and the value
of dynamic datasets in uncovering them.
Keywords: Winograd Schema Challenge, Common-sense Reasoning, Large Language
Models
EvoGrad: A Dynamic Take on the Winograd Schema Challenge with Human
Adversaries
Jing Han Sun, Ali Emami
---
University of Montreal/Mila, Brock University
Montreal, Canada, Saint Catharines, Canada
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract content
## 1\. Introduction
The Winograd Schema Challenge (WSC), a co-reference resolution task, was
developed to gauge the common-sense reasoning of automated systems Winograd
(1972); Levesque et al. (2011). Given subtly varying sentence pairs, the task
is to correctly associate a pronoun with a noun, as illustrated below:
Tom told Ralph, “Check,” as he moved his bishop. (Answer: Tom) Tom told Ralph,
“Check,” as he took his bishop. (Answer: Ralph)
In these examples, chess knowledge informs our interpretation of the pronoun
his—either referring to Tom or Ralph—based on the action performed, either a
move or take. While humans find such tasks intuitive, they pose a challenge
for statistical models, especially when lacking exposure to basic rules or
common knowledge. Yet, recent developments of extensive common-sense reasoning
datasets and benchmarks have allowed LLMs to achieve near-human performance on
WSC variants Brown et al. (2020); Sakaguchi et al. (2020). This impressive
accomplishment raises the question: has the WSC, seen as a definitive
alternative to the Turing Test, been definitively “defeated” Kocijan et al.
(2022)?
Figure 1: Interface of EvoGrad at https://evograd.com
At the same time, evidence suggests that even slight alterations to a WSC task
can significantly undermine a model’s performance Jia and Liang (2017);
Trichelair et al. (2018, 2019); Balasubramanian et al. (2020); Lin et al.
(2020a); Elazar et al. (2021a). This instability may reflect a discrepancy
between current supervision paradigms and the dynamic nature of common sense
acquisition. It suggests the potential value of exploring various approaches,
including the human-and-model-in-the-loop concept, as part of a broader
strategy to address these challenges Nie et al. (2020); Kiela et al. (2021);
Lu et al. (2022).
Existing datasets, often curated by select scientific communities or
crowdsourcing platforms, may also unintentionally bias models toward certain
knowledge instances or values, which may not be universally shared. This
consideration underscores the need for diverse, dynamic, and inclusive
benchmarks in the journey towards systems equipped with generalized common
sense.
Consider the chess example mentioned earlier. While the original WSC sentences
test the model’s understanding of the game’s basic rules, perturbations can
further probe deeper nuances and potential biases:
Maria told Jane, “Your move,” as she adjusted her queen. (Answer: Maria) Maria
told Jane, “Your move,” as she glanced at her clock. (Answer: Jane)
In these variations, the emphasis shifts from the action performed on a chess
piece to the broader context of a timed chess match. Slight word changes can
dramatically alter the correct answer, exposing potential model biases or gaps
in understanding. Such perturbations, especially when generated by diverse
human contributors, ensure a broader and more comprehensive test of a model’s
common-sense reasoning capabilities.
In this paper, we propose a revisit to the WSC within the framework of human-
and-model-in-the-loop. We introduce EvoGrad, an open-source, user-centric
platform dedicated to the active generation and expansion of nuanced examples
for the WSC through human-in-the-loop interactions. Our work contributes three
primary advancements:
A novel data construction mechanism: We enhance the WSC with our unique
approach to human-adversarial perturbations, combining human creativity with
the efficiency of ChatGPT. This innovative union, along with our use of
Wordnet for synonym-based variation, led to a dataset expansion from 182 to
3691 instances, setting a new standard for dynamic, diverse, and high-quality
common-sense reasoning datasets. Notably, our evaluations highlight the
challenging nature of EvoGrad, revealing significant gaps in model abilities
when compared to human benchmarks.
A new metric for model stability: In response to the instability of
transformer-based models on WSC-like tasks Abdou et al. (2020), we introduce a
metric termed error depth. This measure, derived from our data construction
process, offers a quantifiable assessment of model stability. We advocate for
its inclusion in evaluation reports alongside accuracy-based metrics, which
could discourage the development of models that achieve high scores due to
incorrect reasoning.
Online platform for user contributions: Available at https://evograd.com111All
aspects of the website remain anonymous during the submission and review
process to maintain the integrity of the user-contributed data and ensure
unbiased evaluation., our platform encourages public participation in the
continuous expansion of the dataset. Users can modify existing task instances
and observe the predictions of a chosen LLM, fostering a more participatory
and immersive data construction process (Figure 1).
$s_{0}$$per_{2}(s_{0},\text{screamed})$$s_{1(2)}$$per_{4}(s_{1(2)},\text{Melissa})$$s_{2(2,4)}$$per_{5}(s_{1(2)},\text{although})$$s_{2(2,5)}$$per_{9}(s_{0},\text{annoying})$$s_{1(9)}$$per_{7}(s_{1(9)},\text{became})$$s_{2(7,9)}$
Figure 2: Evolution figure of the sentence “Kevin yelled at Jim because he was
so upset.” up to depth level 2.
## 2\. Related Work
### 2.1. WSC-based Datasets
The Winograd Schema Challenge (WSC) Levesque et al. (2011) inspired various
datasets for pronominal coreference resolution, each tackling specific
challenges in the WSC or model evaluations. Datasets like Winogrande Sakaguchi
et al. (2020) and KnowRef Emami et al. (2019) address the WSC’s size
constraints. WinoGender Rudinger et al. (2018), WinoBias Zhao et al. (2018),
and KnowRef-60k Emami et al. (2020) focus on model biases, while WinoWhy Zhang
et al. (2020) and WinoLogic He et al. (2021) target common sense deficiencies
in models. Some research efforts enhanced the original WSC task Wang et al.
(2018); Trichelair et al. (2018); Kocijan et al. (2019); Elazar et al.
(2021a); Zahraei and Emami (2024) and utilized crowd-sourcing for task
development Isaak and Michael (2019); Sakaguchi et al. (2020). While these
static datasets each offer distinct strengths, they often introduce challenges
that necessitate prolonged research and iterations. EvoGrad, on the other
hand, adopts a dynamic framework, allowing for swift adjustments and
refinements in response to emerging challenges.
### 2.2. Dynamic Datasets
Dynamic datasets, updated over time to present new challenges, have been
developed for various tasks Zellers et al. (2019); Lin et al. (2020b).
Adversarial frameworks, as seen in Adversarial SQuAD, SWAG, HellaSWAG, CODAH
and ANLI, exemplify this approach Jia and Liang (2017); Zellers et al. (2018,
2019); Chen et al. (2019); Nie et al. (2020). Techniques such as AFLite
address biases through adversarial filtering Le Bras et al. (2020), while
other methods use continuous learning or a human-model collaborative process
Lan et al. (2017); Yang et al. (2018); Wallace et al. (2019); Dinan et al.
(2019); Nie et al. (2020); Xu et al. (2021); Kiela et al. (2021). ANLI and
Dynabench are notable for their multi-round adversarial data collection Nie et
al. (2020); Kiela et al. (2021). EvoGrad, while aligning with the dynamic
dataset philosophy, specifically targets WSC-based tasks. It merges human-and-
model collaboration, continuous learning, and domain-specific insights for
evolutionary data creation, amplifying the depth and relevance of WSC
challenges to shed light on common-sense reasoning.
### 2.3. Data Augmentation Methods in NLP
Data augmentation techniques in NLP create new examples from existing ones,
obviating the need for novel data collection Shi et al. (2021); Feng et al.
(2021). These methods include token-level manipulation, text generation
restricted, soft data enhancement, and structure-aware data augmentation Wang
and Yang (2015); Bergmanis et al. (2017); Zhang et al. (2018); Xu et al.
(2016). Our approach, mainly a token-level manipulation technique, extends
beyond the substitution of words to include the addition and removal of
tokens, allowing more significant sentence transformations Zmigrod et al.
(2019); Lu et al. (2020); Shi et al. (2018). We also measure the depth of
changes (Section 3.6) relative to the original sentence, providing insights
into model stability as a function of perturbations.
### 2.4. Large Language Models in Data Augmentation and Annotation
Large language models have emerged as effective tools for NLP data
augmentation and annotation, often exceeding the performance of crowd-workers
in terms of efficiency and cost Gilardi et al. (2023). These models have been
shown to be effective in tasks such as zero-shot gender identification and
providing explanations for implicit hate speech Kuzman et al. (2023); Huang et
al. (2023). AugGPT, for instance, outperforms traditional text augmentation
methods in few-shot learning scenarios by rephrasing sentences Dai et al.
(2023). Similarly, ChatGPT has shown potential to simplify social computing
tasks by replicating human-like annotations Zhu et al. (2023). Building on
these insights, we introduce an enhanced data augmentation method that
encompasses token substitutions, additions, and removals, aiming to address
common-sense reasoning deficiencies in the WSC and related tasks.
## 3\. EvoGrad
### 3.1. Dataset Evolution by Perturbation
We adopt an evolutionary approach to dataset expansion, initiating the process
with randomly selected instances from the original Winograd Schema Challenge
(WSC273) Levesque et al. (2011) and Winogrande Sakaguchi et al. (2020), which
are correctly resolved by all evaluated models.
Our method introduces a one-word perturbation to each sentence, effectively
mutating it via substitution. We define a perturbation function $per_{j}(s,w)$
that replaces the token at index $j$ in sentence $s$ with the token $w$.
Though primarily substitution-based, this function can also facilitate the
addition or removal of words, denoted as $per_{j}(s,w_{j}+w)$ and
$per_{j}(s,\epsilon)$ respectively, with $\epsilon$ symbolizing an empty
string.
The function is generalized as follows:
$per_{j}(s_{k(i_{1},...,i_{k})},w)=s_{(k+1)(i_{1},...,i_{j},...,i_{k+1})}\\\
j\not\in\\{i_{1},...,i_{k}\\}\text{ \& }i_{1}<...<i_{k+1}$ (1)
In this equation, $s_{k(i_{1},...,i_{k})}$ signifies the $k$th perturbation on
the base sentence $s_{0}$, wherein tokens at indices $i_{1},...,i_{k}$ have
been modified from $s_{0}$ (Equation 1). The term $k$ denotes the ‘depth’ or
generation of the sentence.
The conditions set for $j$ and indices $i_{1},...,i_{k+1}$ ensure that a depth
increment corresponds solely to the perturbation of a token distinct from
those previously perturbed (i.e.,${i_{1},...,i_{k}}$). Although repeated
modifications at the same token position are not prohibited, such sentences
maintain their original depths. This approach follows our depth
interpretation, emphasizing model stability against sentences that are
increasingly divergent from the original. This methodological choice
facilitates the systematic generation of progressively varied sentences,
thereby enriching the dataset.
The perturbation function is applied iteratively, generating a cascade of
output instances from each input instance. This process is illustrated in
Figure 2 by the sentence ‘Kevin yelled at Jim because he was so upset.’
Through several iterations of the perturbation function, we generate a wide
spectrum of sentences, each incrementally divergent from the original.
### 3.2. Scaling with ChatGPT
Beyond user contributions, we strategically employed
ChatGPT222https://chat.openai.com/ to vastly expand our dataset. We
initialized the process with 14 seed sentences (7 from WSC273 and 7 from
Winogrande-valid) and designed an elaborate prompt that enabled ChatGPT to act
as an ‘expert human annotator’. The prompts were meticulously crafted to guide
the model generation process via demonstrative examples and called for
frequent self-reflection to ensure the quality of the output. One unique
aspect of these prompts was the incorporation of a segmented generation
process, interspersed with feedback to ensure quality control and continuous
self-assessment. For each instance, we verified semantic coherence and
implemented a validation step to ensure pronouns and co-references matched
commonly accepted or typical human readings. An illustrative dialogue sample
can be found in the Appendix in section A.1.
This rigorous approach to prompt engineering culminated in the generation of
approximately 100 new instances per seed sentence. We further diversified
these generated sentences by modifying words, altering the correct antecedent,
and varying the total perturbation depth from the original sentences. This
strategy effectively harnessed the power of human creativity and the
scalability of the model to significantly expand our dataset. As a result, we
managed to augment our initial 182-instance dataset to a much more extensive
collection of 1,414 sentences, thereby facilitating a more comprehensive
evaluation of model performance on dynamic WSC tasks.
### 3.3. Scaling with Wordnet
To increase the diversity of our dataset, we utilized Wordnet Fellbaum (2010),
a lexical database, to augment the 1,414 sentences obtained from our ChatGPT
Scaling stage. This process enabled us to nearly triple our dataset size to a
final count of 3,691 sentences.
Our strategy was to introduce variability while preserving the context of the
sentence and grammatical accuracy. We achieved this by iterating over each
sentence and randomly selecting a word—excluding stop words and named
entities—for replacement. Once a word was selected, a random synonym from
Wordnet was chosen as its substitute. In cases where the chosen word was a
verb, we ensured that the replacement synonym matched the tense of the
original verb.
This approach allowed us to maintain the integrity of our original dataset
while significantly enhancing its size and complexity. The resulting sentences
provided a rich basis for model testing, aiding in the generation of a more
diverse and nuanced set of pronoun disambiguation scenarios.
Dataset | Sub | Size | Method
---|---|---|---
EvoGrad-S | - | 182 | Human (14 orig.)
EvoGrad-M | Train | 1010 | ChatGPT (1-10)
| Val | 202 | ChatGPT (11-12)
| Test | 202 | ChatGPT (13-14)
EvoGrad-L | Train | 2963 | WordNet (M Train)
| Val | 526 | WordNet (M Val)
| Test | 202 | ChatGPT (13-14)
Table 1: Summary of EvoGrad Allocation Source | Sentence | Answer | Depth
---|---|---|---
Original (WSC) | I poured water from the bottle into the cup until _ was full. | cup | 0
Human-perturbed | I poured water from the bottle into the cup because _ was empty. | cup | 2
ChatGPT-scaled | I poured water from the bottle, filling the cup until _ was empty. | bottle | 4
Wordnet-scaled | I decanted water from the feeding bottle into the cup until _ was empty. | feeding bottle | 4
Table 2: Sample instances of EvoGrad derived from an original WSC sentence,
showcasing the different methods of sentence generation and perturbation.
### 3.4. The Dataset
Table 1 outlines the construction and allocation process for our datasets,
specifically EvoGrad-small (S), EvoGrad-medium (M) and EvoGrad-large (L). The
initial dataset, EvoGrad-S, comprised 182 instances, all of which were
adaptations induced by humans from an original set of 14 sentences.
Subsequently, we generated the EvoGrad-M dataset, which was divided into three
distinct subsets: ‘train’, ‘val’, and ‘test’. These subsets were created by
perturbing the original sentences using ChatGPT, resulting in a total of 1,414
instances.
Finally, our most extensive dataset, EvoGrad-L, was constructed by augmenting
both the ‘train’ and ‘val’ subsets of EvoGrad-M using Wordnet, leading to an
overall count of 3,691 instances. The ‘test’ subset was retained from the
EvoGrad-M ‘test’ dataset and was generated through further perturbation of
EvoGrad-S sentences via ChatGPT. To illustrate the range of perturbations and
their sources, we provide sample instances in Table 2 derived from an original
WSC sentence.
### 3.5. The Platform
To foster collaborative development of EvoGrad, we have developed an
interactive platform, accessible at https://evograd.com. Here, global users
can actively contribute to the dataset’s evolution by modifying existing
sentences.
In the Build dataset page, users can select an original or perturbed sentence
from a drop-down menu labeled Original Sentence. They are then guided to input
a modified version of this sentence, replacing the target pronoun with an
underscore, in the New Sentence field. Following the Winogrande format
Sakaguchi et al. (2020), users also provide the two potential noun antecedents
in the Option 1 and Option 2 fields, specifying the correct answer.
To enhance user engagement, our platform offers immediate feedback. Users can
choose an LLM from a list - including BERT Devlin et al. (2019), RoBERTa Liu
et al. (2019), and Albert Lan et al. (2020)—and observe the model’s live
prediction. By clicking Submit, this prediction is generated, and the newly
provided data is incorporated into the dataset.
We prioritize transparency by allowing the dataset, stored as a CSV file, to
be downloaded and inspected directly from the platform. To ensure the quality
and appropriateness of the submissions, we manually validate all entries.
Users are further supported with examples and guidelines. A glimpse of the
platform’s interface is depicted in Figure 1.
Original sentence: Although she was being prosecuted, Monica was welcomed into
the sanctuary of the church by Samantha because _was a sinful criminal.
---
Perturbed Sentence | Prediction | True Label | Depth
Although she was being prosecuted, Monica was welcomed into the sanctuary of the church by Samantha because _was a guilty criminal. | Monica | Monica | 1 ✓
Although she was being prosecuted, Monica was welcomed into the sanctuary of the church by Samantha because _was a compassionate person. | Samantha | Samantha | 2 ✓
Even though she was being prosecuted, Monica was guided into the safe haven of the church by Samantha because _was a virtuous person. | Monica | Samantha | 5 ✗
While under prosecution, Monica was brought into the spiritual refuge of the church by Samantha because _was a good-natured woman. | Monica | Samantha | 6 ✗
While being prosecuted, Monica was welcomed into the church’s refuge by Samantha because _was a law-abiding person. | Monica | Samantha | 5 ✗
Table 3: Sample of perturbations constructed from Eq.1 on a Winogrande
example, with predictions corresponding to RoBERTa fine-tuned on Winogrande-
XL. The model’s incorrect predictions occur at depths 5,6 and 5, respectively,
corresponding to the number of modified tokens from the original. Therefore,
this sample of 5 perturbed instances has an average error depth (ED) of 5.333.
### 3.6. Error Depth
Given our dataset construction methodology, we propose the error depth (ED)
metric to evaluate model stability. While accuracy is a widely used metric to
gauge model performance on prediction tasks such as the WSC, it might not
effectively capture a model’s resilience against instances that progressively
deviate from the original.
There are scenarios where models predict correctly but possibly for the wrong
reasons. Sole reliance on accuracy can obscure these nuances. Ideally, a model
should demonstrate stability against token substitutions. Although, in the
context of the WSC, a token change can alter the answer label, a truly robust
model should not be overly sensitive to such modifications.
The error depth metric quantifies a model’s performance on sentences that
increasingly diverge from a correctly understood original. Specifically, the
error depth denotes the number of perturbations made to the original sentence
before the model produces its first incorrect prediction.
For clarity, let’s define the symbols:
* •
$s_{0}$: The original seed sentence.
* •
$\text{label}(s)$: The true label of sentence $s$.
* •
$\text{pred}(s)$: The model’s predicted label for sentence $s$.
* •
$n_{wrong}$: The number of incorrect predictions made by the model on
perturbed versions of the original sentence.
With these definitions, the error depth (ED) is formulated as:
$\displaystyle\overline{ED}$
$\displaystyle\overset{def}{=}\frac{1}{n_{wrong}}\sum_{k}^{n_{wrong}}k$ (2)
$\displaystyle\text{if }\text{label}(s_{0})=\text{pred}(s_{0})\text{ and }$
$\displaystyle\text{label}(s_{k(i_{1},...,i_{k})})\neq\text{pred}(s_{k(i_{1},...,i_{k})})$
Refer to Table 3 for an application of the metric to perturbations of a
sentence. In this demonstration, the model mispredicts three sentences: two
after five perturbations and one after six. Thus,
$\overline{ED}=(5+5+6)/3=5.333$. The error depth functions as an instance-
level metric, assessing a model’s stability for individual sentences.
Averaging over all instances yields $\overline{ED}$, which, when paired with
accuracy, offers a comprehensive assessment of a model’s performance on tasks
like the WSC.
### 3.7. Human Performance
Three English-proficient annotators reviewed EvoGrad-M Val and EvoGrad-L Val,
achieving mean accuracies of 95.2% and 92.8%, respectively. Importantly, they
did not exhibit an average error depth, effectively handling perturbations to
the full depth of the dataset. A high inter-annotator agreement was recorded
with a Fleiss’ Kappa of $\kappa=0.914$.
Model | Tuning | Wino-valid | EvoGrad-M-val | EvoGrad-L-val
---|---|---|---|---
BERT | EvoGrad-M | - | 60.4 (6.913) | -
EvoGrad-L | - | - | 54.9 (6.867)
Wino | 62.75 | —– (7.302) | —– (7.258)
Wino + EvoGrad-M | 63.06 | —– (7.308) | -
Wino + EvoGrad-L | 62.98 | - | —– (7.232)
RoBERTa | EvoGrad-M | - | 58.4 (6.762) | -
EvoGrad-L | - | - | 60.3 (6.727)
Wino | 76.09 | —– (6.286) | 6.393
Wino + EvoGrad-M | 76.09 | —– (6.286) | -
Wino + EvoGrad-L | 76.64 | - | 6.652
ALBERT | EvoGrad-M | - | 55.4 (6.989) | -
EvoGrad-L | - | - | 57.2 (6.853)
Wino | 64.64 | —– (7.971) | —– (7.670)
Wino + EvoGrad-M | 64.48 | —– (8.000) | -
Wino + EvoGrad-L | 64.64 | - | —– (7.694)
GPT-3* | EvoGrad-M | - | 59.41 (7.122) | -
EvoGrad-L | - | - | 56.08 (6.753)
GPT-3.5* | EvoGrad-M | - | 67.33 (7.061) | -
EvoGrad-L | - | - | 65.02 (7.245)
Table 4: Accuracy (and error depth) results of models on Winogrande-valid and
EvoGrad-val sets after training on Winogrande-XL and/or EvoGrad-train. Bold
values represent the highest accuracy and underlined values represent the
highest error depth for each model in each dataset. A single dash (-) denotes
that the model was not tuned on that specific dataset variant, hence was not
tested. Dashed (—–) values indicate that accuracy was not tested due to
potential contamination from EvoGrad’s seed examples being taken from
Winogrande, though error depth was still evaluated. Models marked with an
asterisk (*) were evaluated using few-shot learning rather than fine-tuning.
## 4\. Experiments and Results
### 4.1. Model Setup
We evaluated three primary transformer-based models that are masked language
models: BERT Devlin et al. (2019), RoBERTa Liu et al. (2019), and ALBERT Lan
et al. (2020). These models have been recognized for their strong performance
on the WSC and have led the benchmark results. Each of the models were fine-
tuned on the Winogrande-XL dataset Sakaguchi et al. (2020), which contains
approximately 40,000 task instances and is designed to reduce potential
annotation biases.
Additionally, we evaluated two left-to-right language models, specifically
GPT-3 (text-davinci-003) Brown et al. (2020) and GPT-3.5 (gpt-3.5-turbo-0613),
on the Winogrande-XL and EvoGrad datasets.
For BERT and RoBERTa, we first aimed to replicate top-performing models from
existing literature. Using Huggingface’s package Wolf et al. (2020), we
achieved validation accuracies of 62.75% for BERT-large-uncased and 76.09% for
RoBERTa-large. Although slightly below the reported accuracies in Sakaguchi et
al. (2020), variations in hyperparameter tuning may account for the
differences. A similar approach was taken for ALBERT-large-v2, with a
resulting accuracy of 64.64%.
Hyperparameters for BERT, RoBERTa, and ALBERT were selected from:
* •
Learning rates: $1e-5$, $3e-5$, $5e-5$
* •
Epochs: 3, 4, 5, 8
* •
Batch sizes: 8, 16
For training on EvoGrad-train (both medium and large versions), given its
resemblance but smaller size to Winogrande, we experimented with:
* •
Learning rates: $1e-5$, $3e-5$, $5e-5$
* •
Epochs: 1, 2, 4, 8
* •
Batch sizes: 8, 16, 32, 64
For evaluations using GPT-based models, we adopted a few-shot learning
approach. Each instance was evaluated using an instruction-based prompt
consisting of 30 random instances from the respective training set.
Figure 3: Distribution of the top three perturbations for models trained on
Winogrande + Evograd-L. From left to right: BERT, RoBERTa, and ALBERT.The
segments represent the relative frequency of each perturbation type: ‘–NN’
(noun removal), ‘+NN’ (noun addition), and either ‘–JJ’ (adjective removal) or
‘–IN’ (preposition removal).
Model | Trained on | EvoGrad-M-val | EvoGrad-L-val
---|---|---|---
BERT | EvoGrad-M | +NN (150), –NN (148), –JJ (105) | -
| EvoGrad-L | - | –NN (578), +NN (471), –JJ (342)
| Wino | –NN (92), –JJ (62), +NN (61) | –NN (365), +NN (294), –JJ (228)
| Wino + EvoGrad-M | –NN (108), +NN (90), –JJ (78) | -
| Wino + EvoGrad-L | - | –NN (373), +NN (303), –JJ (233)
RoBERTa | EvoGrad-M | –NN (170), +NN (146), –JJ (120) | -
| EvoGrad-L | - | –NN (494), +NN (416), –JJ (283)
| Wino | –NN (17), +NN (12), –JJ (11) | –NN (76), +NN (54), –JJ (41)
| Wino + EvoGrad-M | –NN (17), +NN (12), –JJ (11) | -
| Wino + EvoGrad-L | - | –NN (61), +NN (43), –IN (32)
ALBERT | EvoGrad-M | –NN (189), +NN (161), –JJ (131) | -
| EvoGrad-L | - | –NN (542), +NN (479), –JJ (316)
| Wino | –NN (92), –JJ (62), +NN (61) | –NN (272), +NN (208), –JJ (169)
| Wino + EvoGrad-M | –NN (92), –JJ (62), +NN (61) | -
| Wino + EvoGrad-L | - | –NN (294), +NN (220), –JJ (183)
GPT-3 | EvoGrad-M | –NN (173), +NN (144), –JJ (118) | -
| EvoGrad-L | - | –NN (505), +NN (448), –JJ (306)
GPT-3.5 | EvoGrad-M | –NN (161), –JJ (115), +NN(111) | -
| EvoGrad-L | - | –NN (464), +NN (364), –JJ (290)
Table 5: Top 3 perturbations and their count on incorrect predictions on
EvoGrad-val sets after fine-tuning on Winogrande-XL and EvoGrad-train.
### 4.2. Results
Our evaluation results, as shown in Tables 4 and Figure 3, offer insight into
model performance under different training conditions. We trained models
exclusively on EvoGrad-train, on Winogrande-XL (denoted as Wino), or
sequentially on both Winogrande and EvoGrad-train (denoted as Wino + EvoGrad).
This approach allowed us to understand how different training datasets
influence model robustness and stability.
Table 4 displays the models’ accuracies on the Winogrande-valid dataset
alongside their average error depth on the EvoGrad datasets. The error depth
indicates the perturbative distance at which a model starts to fail, providing
insights into model stability. While accuracy is the main metric, error depth
(shown in parentheses) gives a complementary view of model performance. Due to
the potential overlap between EvoGrad and Winogrande, we have omitted the
accuracy scores for Winogrande-trained models in EvoGrad. GPT-based models
were only evaluated on EvoGrad instances as they are evaluated through few-
shot learning.
Figure 3 visualizes the three most frequent perturbation types that lead to
incorrect predictions by the models. Each perturbation is categorized by its
effect on parts of speech. For instance, “+NN (150)” indicates a noun was
added in 150 of the incorrect predictions. A comprehensive breakdown of the
perturbation counts and their types, spanning all parts of speech observed, is
provided in Table 5.
## 5\. Discussion
#### Influence of EvoGrad on Language Model Performance
Table 4 illustrates the varied impacts of EvoGrad on Transformer models,
leading to several key insights:
* •
BERT’s improved performance post-EvoGrad training underscores its ability to
integrate the dataset’s specific perturbations effectively. This adaptability
implies that BERT may be particularly effective for tasks requiring deeper
linguistic insight or sensitivity to subtle contextual changes.
* •
RoBERTa consistently performs well both before and after training EvoGrad,
showcasing its robustness. However, its lower error depth compared to its
accuracy points to a potential trade-off between performance and stability.
This observation underscores the need to balance generalization with stability
to perturbations.
* •
The negligible change in ALBERT’s performance across various training regimes
raises questions regarding the model’s saturation point and its alignment with
the dataset. This warrants further investigation of the limits of adaptability
for certain models.
* •
While GPT-based models, especially GPT-3.5, demonstrate competitive
performance, their error depths highlight challenges related to stability.
This trend suggests that some of the newer models might prioritize
adaptability at the expense of robustness.
Figure 3 sheds light on the areas where language models are most vulnerable,
particularly in handling noun and adjective modifications. Addressing these
specific challenges is imperative for the enhancement of common-sense
reasoning in future model iterations.
#### Robustness and Adaptability to New Tasks
One of the challenges in deep learning is ensuring that the models remain
adaptable and robust when exposed to new tasks or datasets. Whether through
fine-tuning or few-shot learning, a model’s ability to incorporate new
information without significant detriment to its original capabilities is
vital. In our experiments, the transformer models exhibited this adaptability,
particularly when introduced to EvoGrad. For instance, when models were fine-
tuned on EvoGrad, their performance on the Winogrande validation set generally
improved or remained consistent (Table 4), indicating that they did not lose
their grasp of previously acquired knowledge. However, GPT-based models,
through few-shot learning, demonstrated their versatility in quickly adapting
to new tasks without the need for extensive retraining. These observations
underscore the potential of current architectures in handling evolving
datasets and tasks, highlighting their robustness in diverse learning
scenarios.
#### Evolution and Community Involvement with EvoGrad
The current rendition of EvoGrad represents only the first phase in a series
of envisioned enhancements. As the platform matures, our goal is to achieve
multiple cycles of data augmentation, model training, and fine-tuning,
striving to foster a greater social impact in the AI domain. In making EvoGrad
accessible to a diverse audience, including those new to WSC-style challenges,
we have incorporated clear prompts and guidelines, drawing inspiration from
our initial work with the 182 instances in EvoGrad-small.
Looking ahead, we are also planning to expand the platform to incorporate
other foundational NLP tasks by integrating datasets such as OntoNotes 5.0 for
Named Entity Recognition (NER) Weischedel et al. (2012), Natural Questions
(NQ) Kwiatkowski et al. (2019) for Question Answering (QA), and the SemEval
tasks for Sentiment Analysis, thereby broadening the scope and utility of
EvoGrad.
Recognizing the scale at which EvoGrad could grow, we understand the crucial
role of user-driven validation. While our dedicated team of in-house
researchers currently curates the dataset to ensure its quality, we’re eager
to transition this role to our users in the near future. This strategy not
only offloads the validation responsibility but also promises a more dynamic,
participatory, and community-centric approach to refining LLMs.
## 6\. Conclusion
In this work, we introduced EvoGrad, a dynamic platform that extends the
Winograd Schema Challenge with a human-and-model-in-the-loop methodology. The
dataset, enriched through our platform, incorporates contributions from human
experts, language model expansions, and lexical resource utilization. We also
introduced the “error depth" metric as a novel means to assess model stability
in evolving tasks. While our evaluations showed potential benefits of using
the augmented data from EvoGrad across different training regimes, the
disparity between human and machine performance on this task underlines its
complexity and the ongoing challenges in enhancing common-sense reasoning in
LLMs.
## Ethics Statement
We are presenting our publicly-accessible platform to those outside the
scientific and crowd-sourcing communities. However, our platform is still
limited to those in society who have access to a mobile device/personal
computer and internet access; a large but underrepresented group of people in
the world do not. We therefore use our platform as only a first step towards
more inclusiveness, which we open to people outside the small community of
science and crowd-sourcing, but wish to be involved in efforts that will
include the underrepresented mentioned.
We also cannot assume that everyone’s foremost priorities involve contributing
towards such endeavours as ours – many members of society are currently in
turmoils of war, famine, or even indifference or aversion towards AI, which
all may amount to their non-involvement in projects related to ours.
Accordingly, the direction towards progress is best achieved outside the
laboratory; after all, if diversity and community involvement in the
development of tasks such as ours is as correlated to positive results in AI,
our efforts as researchers should also extend towards the education, well
being, and thriving of members in society, without which our goal of a global
task is never truly realized.
## Acknowledgements
This work was supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC) and by the New Frontiers in Research Fund (NFRF).
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## Appendix A Appendix
### A.1. ChatGPT Dialogue During Scaling Process
Author:
You are an expert linguist, human annotator and are familiar with the Winograd
Schema Challenge. Consider the sentence: Although they ran at about the same
speed, Sue beat Sally because _had such a good start. This sentence is in the
format of a Winograd schema, where two antecedents, in this case "Sue" and
"Sally" are competing for reference in the position, _. Common sense allows us
to conclude that one of the two antecedents is most likely, in this case, Sue.
I am intending on building a dataset composed of modifications to the above
sentence. Based on this original sentence, I need you to generate 100 new
sentences that are modified by either adding, replacing, or substituting words
in the original sentence. They should be organized in of the following format,
copied from a csv file :
index sentence option1 option2 answer distance
where the original sentence has the following form: 0 Although they ran at
about the same speed, Sue beat Sally because _had such a good start. Sue Sally
1 0
Here, index refers to the index number for the sentence, sentence refers to
the sentence itself (notice the necessity of the _as the position for one of
the two antecedents), option1 refers to one antecedent, option2 refers to the
other, answer refers to the correct antecedent that belongs in _– "1"
corresponds to option1, and "2" corresponds to option2. Notice, for this
original sentence, the answer is "1" because having a good start is a likely
explanation for someone (i.e., Sue) beating another in a race. Finally,
"distance" corresponds to the number of modifications made on the original
sentence (whose distance is, by definition, 0). It is equivalent to the "token
edit distance".
I want you to create a diverse range of new entries, that vary in terms of the
modifications, the total distance, as well as the correct answer.
For example, the sentence "Although they sprinted at about the same speed, Sue
beat Sally because _had such a good start." would have an answer of 1 and
distance of 1 (because ran was changed to sprinted, but the correct answer
still remains option1, which in this case is Sue), and the sentence "Although
they sprinted at about the same speed, Sue beat Sally although _had such a
good start." would have the answer of 2 and distance of 2, because the correct
answer is now "Sally", and there were two word modifications – the second one
here was "although" instead of "because", which flips the answer. Notice that
with the right change in words, the answer can flip – I want to encourage you
to find such minimal number of changes when you’re creating new entries. You
are also allowed to change the words in the options too (e.g., Sue and Sally).
I would like the dataset to have a similar number of answer=1 entries than
answer=2 entries.
Please generate the first 10, formatted as a table here. First, you may ask
any clarification questions to make sure you understand the task.
ChatGPT:
Thank you for the instructions. Here are the first 10 entries: […]
Author:
Excellent work. Please generate 30 more, but try to keep the distance lower.
Don’t forget you can change the names too!
ChatGPT:
Sure! Here are the next 30 entries with smaller distances and changes in
names: […]
Author:
These sentences didn’t actually have the distances you claimed they had. These
should correspond to token edit distances. Please be careful when generating
sentences to actually correspond to distances – it’s okay if the distance is a
bit higher. I just want you to be accurate. You also forgot to change the
names (as possible modifications) at one point. Please generate 10 for me to
see if it’s okay.
ChatGPT:
I apologize for the inaccuracies in the distances and missing name changes in
the previous examples. […] Here are the next 10 sentences: […]
Author:
These were much better. Please generate 30 more. Pay attention to not
miscalculate the distance.
ChatGPT:
I appreciate your feedback. Here are the next 30 sentences, with the correct
token edit distances and a careful attention to detail: […]
Author:
Great job. Please generate 30 more. Again, the token distance should be
correctly calculated. For example, in your sentence:
Even though they raced at the same speed, Sue beat Sally although _had a
powerful start.
You said the distance is 3. It’s actually 5 here, because Although was dropped
and replaced with Even (1), then "though" was added (2), then raced was
replaced by ran (3), then about was dropped (4), then good was replaced by
powerful (5).
ChatGPT:
I understand your concern and I apologize for the mistake. I’ll ensure more
accuracy in calculating the token edit distance. Here are the next 30
sentences: […]
Author:
Great job. Please generate 20 more. Remember that changing the names is valid
but counts as a modification (increments distance).
ChatGPT:
Thank you for your feedback. Here are the next 20 sentences, this time
including changes in names: […]
##
* (1)
|
EXO-17-028
$HeadURL:svn+ssh://svn.cern.ch/reps/tdr2/papers/EXO-17-028/trunk/EXO-17-028.tex$
$Id:EXO-17-028.tex4665202018-06-2812:01:53Zalverson$
EXO-17-028
# Search for heavy Majorana neutrinos in same-sign dilepton channels in
proton-proton collisions at $\sqrt{s}=13\TeV$
###### Abstract
A search is performed for a heavy Majorana neutrino ($\mathrm{N}$), produced
by leptonic decay of a $\PW$ boson propagator and decaying into a $\PW$ boson
and a lepton, with the CMS detector at the LHC. The signature used in this
search consists of two same-sign leptons, in any flavor combination of
electrons and muons, and at least one jet. The data were collected during 2016
in proton-proton collisions at a center-of-mass energy of 13, corresponding to
an integrated luminosity of 35.9. The results are found to be consistent with
the expected standard model backgrounds. Upper limits are set in the mass
range between 20 and 1600in the context of a Type-I seesaw mechanism, on
$\abs{V_{\Pe\mathrm{N}}}^{2}$, $\abs{V_{\mu\mathrm{N}}}^{2}$, and
$\abs{V_{\Pe\mathrm{N}}V^{*}_{\mu\mathrm{N}}}^{2}/(\abs{V_{\Pe\mathrm{N}}}^{2}+\abs{V_{\mu\mathrm{N}}}^{2})$,
where $V_{\ell\mathrm{N}}$ is the matrix element describing the mixing of
$\mathrm{N}$ with the standard model neutrino of flavor $\ell=\Pe,\mu$. For
$\mathrm{N}$ masses between 20 and 1600, the upper limits on
$\abs{V_{\ell\mathrm{N}}}^{2}$ range between $2.3\times 10^{-5}$ and unity.
These are the most restrictive direct limits for heavy Majorana neutrino
masses above 430.
## 0.1 Introduction
The observation of neutrino oscillations [1], a mixing between several
neutrino flavors, established that at least two of the standard model (SM)
neutrinos have nonzero masses and that individual lepton number is violated.
The nonzero masses of the neutrinos are arguably the first evidence for
physics beyond the SM. Upper limits on the neutrino masses have been
established from cosmological observations [1], as well as direct
measurements, including those of tritium decays [2, 3]. The extremely small
values of these masses are difficult to explain in models that assume
neutrinos to be Dirac particles [4, 5].
The leading theoretical candidate to explain neutrino masses is the so-called
“seesaw” mechanism [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], in
which a new heavy Majorana neutrino $\mathrm{N}$ is postulated. In the seesaw
mechanism, the observed small neutrino masses, $m_{\nu}$, result from the
large mass of $\mathrm{N}$, with $m_{\nu}\sim
y^{2}_{\nu}v^{2}/m_{\mathrm{N}}$. Here $y_{\nu}$ is a Yukawa coupling, $v$ is
the Higgs vacuum expectation value in the SM, and $m_{\mathrm{N}}$ is the mass
of the heavy-neutrino state. One model that incorporates the seesaw mechanism,
and can be probed at the LHC, is the neutrino minimal standard model
($\nu$MSM) [20, 21, 22, 23]. In this model, the existence of new heavy
neutrinos could not only explain the very small masses of the SM neutrinos,
but also provide solutions to other problems in cosmology, such as the origin
of dark matter or the matter-antimatter asymmetry of the early universe [22,
23].
In this paper, we present the results of a search for a heavy Majorana
neutrino in the $\nu$MSM, which incorporates new heavy-neutrino states without
additional vector bosons. Searches for heavy Majorana neutrinos at hadron
colliders have been proposed by many theoretical models [24, 25, 26, 27, 28].
Numerous experiments have looked for heavy neutrinos in the mass range from
several keV to some hundred GeV, with no evidence seen, and a summary of the
limits on $\abs{V_{\ell\mathrm{N}}}^{2}$ versus $m_{\mathrm{N}}$ for these
experiments is given in Ref. [29], where $V_{\ell\mathrm{N}}$ is a matrix
element describing the mixing between the heavy neutrino and the SM neutrino
of flavor $\ell={\Pe},\;\mu$, or $\tau$. Direct searches for heavy neutrinos
have been performed at the CERN LEP collider [30, 31, 32] and, more recently,
at the CERN LHC [33, 34, 35, 36, 37]. These searches use a model-independent
phenomenological approach, assuming that $m_{\mathrm{N}}$ and
$V_{\ell\mathrm{N}}$ are free parameters.
The searches performed by the DELPHI [30] and L3 [31, 32] Collaborations at
LEP looked for the $\Pe^{+}\Pe^{-}\to\mathrm{N}\nu_{\ell}$ process, where
$\nu_{\ell}$ is any SM neutrino. For $\ell=\mu,\tau$ the limits on
$\abs{V_{\ell\mathrm{N}}}^{2}$ were set for $m_{\mathrm{N}}<90\GeV$, while for
$\ell=\Pe$ the limits extend to $m_{\mathrm{N}}<200\GeV$. Several experiments
obtained limits for low neutrino masses ($m_{\mathrm{N}}<5\GeV$), including
the LHCb Collaboration [33] at the LHC, which set limits on the mixing of a
heavy neutrino with an SM muon neutrino. The searches by L3, DELPHI, and LHCb
include the possibility of a finite heavy-neutrino lifetime, such that
$\mathrm{N}$ decays with a vertex displaced from the interaction point. In the
search reported here, however, it is assumed that $\mathrm{N}$ decays close to
the point of production, since in the mass range of this search
($m_{\mathrm{N}}>20\GeV$) the decay length is expected to be less than
$10^{-10}$m [38].
This search probes the decay of a $\PW$ boson, in which an SM neutrino
oscillates into a new state $\mathrm{N}$. In this analysis, only $\ell=\Pe$ or
$\mu$ processes are considered. In the previous CMS analyses [34, 35], only
the Drell–Yan (DY) production of $\mathrm{N}$
($\cPq\cPaq^{\prime}\to\PW^{*}\to\mathrm{N}\ell^{\pm}\to\ell^{\pm}\ell^{{}^{\prime}\pm}\cPq^{\prime}\cPaq$),
shown in Fig. 1 (left) was considered, while in this study the photon-
initiated production of $\mathrm{N}$
($\cPq\gamma\to\PW\cPq^{\prime\prime}\to\mathrm{N}\ell^{\pm}\cPq^{\prime\prime}\to\ell^{\pm}\ell^{\pm}\cPq^{\prime\prime}\cPq^{\prime}\cPaq$),
as shown in Fig. 1 (right), is also taken into account. The diagram in Fig. 1
(right) shows a possible production of $\mathrm{N}$ via $\PW\gamma$ fusion,
which we refer to by the generic term vector boson fusion (VBF). The inclusion
of the VBF channel enhances the sensitivity of this analysis for $\mathrm{N}$
masses above several hundred GeV [39], where the $t$-channel photon-initiated
processes become the dominant production mechanism for
$\PW^{*}\to\mathrm{N}\ell$ [40, 39].
Figure 1: Feynman diagram representing a resonant production of a Majorana
neutrino ($\mathrm{N}$), via the $s$-channel Drell–Yan process (left) and its
decay into a lepton and two quarks, resulting in a final state with two same-
sign leptons and two quarks from a $\PW$ boson decay. Feynman diagram for the
photon-initiated process (right).
Since $\mathrm{N}$ is a Majorana particle and can decay to a lepton of equal
or opposite charge to that of its parent $\PW$ boson, both opposite- and same-
sign (SS) lepton pairs can be produced. This search targets same-sign dilepton
(SS2$\ell$) signatures since these final states have very low SM backgrounds.
We search for events where the $\mathrm{N}$ decays to a lepton and a $\PW$
boson, and the $\PW$ boson decays hadronically, as this allows the
reconstruction of the mass of the $\mathrm{N}$ without the ambiguity
associated with the longitudinal momentum of an SM neutrino. For the DY
channel production, the final state is
$\ell^{+}\ell^{{}^{\prime}+}\cPq^{\prime}\cPaq$. The charge-conjugate decay
chain also contributes and results in an
$\ell^{-}\ell^{{}^{\prime}-}\cPaq^{\prime}\cPq$ final state. In the VBF
channel, production of an additional forward jet is produced in the event.
An observation of the
$\ell^{\pm}\ell^{{}^{\prime}\pm}\cPq^{\prime}\cPaq(\cPq^{\prime\prime})$
process would constitute direct evidence of lepton number violation. The study
of this process in different dilepton channels improves the likelihood for the
discovery of $\mathrm{N}$, and constrains the mixing matrix elements. The
dielectron ($\Pe\Pe$), dimuon ($\mu\mu$), and electron-muon ($\Pe\mu$)
channels are searched for and allow constraints to be set on
$\abs{V_{\Pe\mathrm{N}}}^{2}$, $\abs{V_{\mu\mathrm{N}}}^{2}$, and
$\abs{V_{\Pe\mathrm{N}}V^{*}_{\mu\mathrm{N}}}^{2}/(\abs{V_{\Pe\mathrm{N}}}^{2}+\abs{V_{\mu\mathrm{N}}}^{2})$,
respectively [38]. In the $\Pe\mu$ channel, the leptons from the $\PW$ boson
and the $\mathrm{N}$ decay can be either $\Pe$ and $\mu$, or $\mu$ and $\Pe$,
respectively, so the branching fraction for this channel is twice as large as
that for the $\Pe\Pe$ or $\mu\mu$ channels.
The most recent CMS search for heavy Majorana neutrinos in events with two
leptons and jets was performed for the mass range $m_{\mathrm{N}}=40$–500in
the $\Pe\Pe$, $\mu\mu$, and $\Pe\mu$ channels at $\sqrt{s}=8\TeV$ [34, 35]. A
similar search was also performed by the ATLAS Collaboration in the $\Pe\Pe$
and $\mu\mu$ channels [36]. The CMS Collaboration performed a search for heavy
Majorana neutrinos in final states with three leptons using the 2016 data set
[37], setting limits on $\abs{V_{\Pe\mathrm{N}}}^{2}$ and
$\abs{V_{\mu\mathrm{N}}}^{2}$, for the mass range $m_{\mathrm{N}}=1$–1200. In
the case of trilepton channels, events that contain both an electron and a
muon ($\Pe\Pe\mu,\mu\mu\Pe$) present an ambiguity about which of the leptons
mixes with $\mathrm{N}$, and it is thus impossible to probe
$\abs{V_{\Pe\mathrm{N}}V^{*}_{\mu\mathrm{N}}}^{2}/(\abs{V_{\Pe\mathrm{N}}}^{2}+\abs{V_{\mu\mathrm{N}}}^{2})$.
This ambiguity is not present in the current analysis with dilepton channels,
allowing limits to be set on
$\abs{V_{\Pe\mathrm{N}}V^{*}_{\mu\mathrm{N}}}^{2}/(\abs{V_{\Pe\mathrm{N}}}^{2}+\abs{V_{\mu\mathrm{N}}}^{2})$.
The CMS analysis at $\sqrt{s}=8\TeV$ showed that the efficiency for signal
events drops for masses above 400, as a consequence of the Lorentz-boosted
topology of the decay products of $\mathrm{N}$, which causes the signal jets
to overlap and be reconstructed as a single jet. The signal efficiency can be
recovered by including events containing a wide jet that is consistent with
the process $\PW\to\cPq\cPaq^{\prime}$, where the decay products of the $\PW$
boson are merged into a single jet [41]. It was also observed that the signal
efficiency dropped significantly when the mass of $\mathrm{N}$ was below the
$\PW$ boson mass ($m_{\mathrm{\PW}}$). For the $\mu\mu$ channel, the signal
acceptance was 0.65 (10.9)% for $m_{\mathrm{N}}=60\,(125)\GeV$. For
$m_{\mathrm{N}}<m_{\mathrm{\PW}}$ the final-state leptons and jets are very
soft and fail the momentum requirements applied in the 8analysis. In the
present analysis, cases where one of the signal jets fails the selection
criteria are recovered by including events with only one jet.
In this paper, a new search for $\mathrm{N}$ in the $\Pe\Pe$, $\mu\mu$, and
$\Pe\mu$ channels is presented using CMS data collected in 2016 at
$\sqrt{s}=13\TeV$. We search for events with two isolated leptons with the
same electric charge, with the presence of either a) two or more jets, b)
exactly one jet, or c) at least one wide jet. We look for an excess of events
above the expected SM background prediction by applying selection criteria to
the data to optimize the signal significance for each mass hypothesis. Heavy
Majorana neutrinos with a mass in the range of 20 to 1700are considered. There
are three potential sources of SS2$\ell$ backgrounds: SM sources in which two
prompt SS leptons are produced (a prompt lepton is defined as an electron or
muon originating from a $\PW$/$\PZ$/$\gamma^{*}$ boson or $\tau$ lepton
decay), events resulting from misidentified leptons, and opposite-sign
dilepton events (, from $\PZ\to\ell^{+}\ell^{-}$,
$\PW^{\pm}\PW^{\mp}\to\,\ell^{+}\nu\ell^{-}\overline{\nu}$) in which the sign
of one of the leptons is mismeasured. The last source is negligible for the
$\mu\mu$ and $\Pe\mu$ channels.
## 0.2 The CMS detector
The central feature of the CMS apparatus is a superconducting solenoid of 6m
internal diameter. The solenoid provides a magnetic field of 3.8T along the
direction of the counterclockwise rotating beam as viewed from above the plane
of the accelerator, taken as the $z$ axis of the detector coordinate system,
with the center of the detector defined to be at $z=0$. The azimuthal angle
$\phi$ is measured in radians in the plane perpendicular to the $z$ axis,
while the polar angle $\theta$ is measured with respect to this axis. Within
the solenoid volume are a silicon pixel and strip tracker, a lead tungstate
crystal electromagnetic calorimeter (ECAL), and a brass and scintillator
hadron calorimeter (HCAL), each composed of a barrel and two endcap sections.
The ECAL provides a coverage in pseudorapidity $\abs{\eta}<1.48$ in the barrel
region and $1.48<\abs{\eta}<3.00$ in the two endcap regions, where
pseudorapidity is defined as $\eta=-\ln[\tan(\theta/2)]$. Forward calorimetry
extends the pseudorapidity coverage provided by the barrel and endcap
detectors. Muons are detected in gas-ionization detectors, providing a
coverage of $\abs{\eta}<2.4$, and are embedded in the steel flux-return yoke
outside the solenoid. The first level of the CMS trigger system [42], composed
of custom hardware processors, uses information from the calorimeters and muon
detectors to select up to 100kHz of the most interesting events. The high-
level trigger (HLT) processor farm uses information from all CMS subdetectors
to further decrease the event rate to roughly 1kHz before data storage. A more
detailed description of the CMS detector can be found in Ref. [43].
## 0.3 Simulated samples
Samples of simulated events are used to estimate the background from SM
processes containing prompt SS leptons originating from hard-scattering
processes and to determine the heavy Majorana neutrino signal acceptance and
selection efficiency. The backgrounds from SM sources are produced using the
2.2.2 or 2.3.3 Monte Carlo (MC) generator [44] at leading order (LO) or next-
to-leading order (NLO) in perturbative quantum chromodynamics (QCD), with the
exception of $\Pg\Pg\to\PZ\PZ$ which is simulated at LO with 7.0 [45], and the
diboson production processes ($\PW\PZ$ and $\PZ\PZ$) that are generated at NLO
with the v2 [46, 47, 48, 49] generator.
The NNPDF3.0 [50] LO (NLO) parton distribution function (PDF) sets are used
for the simulated samples generated at LO (NLO). For all signal and background
samples, showering and hadronization are described using the 8.212 [51]
generator, with the CUETP8M1 [52] underlying event tune. The response of the
CMS detector is modeled using [53]. Double counting of the partons generated
with and is removed using the MLM [54] and FxFx [55] matching schemes in the
LO and NLO samples, respectively.
The $\mathrm{N}$ signals are generated using 2.6.0 at NLO precision, where the
decay of $\mathrm{N}$ is simulated with MadSpin [56], following the
implementation of Refs. [57, 58]. This includes the production of $\mathrm{N}$
via the charged-current DY and VBF processes. For the charged-current DY
production mechanism, we employ the NNPDF31_nnlo_hessian_pdfas PDF set [50],
while to include the photon PDF in the VBF ($\PW\gamma$ fusion) mechanism we
use the LUXqed17_plus_PDF4LHC15_nnlo_100 PDF set [59]. The NLO cross section,
obtained using the generator at $\sqrt{s}=13\TeV$, for the DY (VBF) process
has a value of 58.3 (0.050)pb for $m_{\mathrm{N}}=40\GeV$, dropping to 0.155
($9.65\times 10^{-4}$)pb for $m_{\mathrm{N}}=100\GeV$, and to $9.92\times
10^{-6}$ ($1.69\times 10^{-5}$)pb for $m_{\mathrm{N}}=1000\GeV$, assuming
$\abs{V_{\ell\mathrm{N}}}^{2}=0.01$. The VBF process becomes the dominant
production mode for scenarios where the mass of $\mathrm{N}$ is greater than
${\approx}700\GeV$. Only the final states with two leptons (electrons or
muons) and jets are generated.
Additional $\Pp\Pp$ collisions in the same or adjacent bunch crossings
(pileup) are taken into account by superimposing minimum bias interactions
simulated with on the hard-scattering process. The simulated events are
weighted such that the distribution of the number of additional pileup
interactions, estimated from the measured instantaneous luminosity for each
bunch crossing, matches that in data. The simulated events are processed with
the same reconstruction software as used for the data.
## 0.4 Event reconstruction and object identification
The reconstructed vertex with the largest value of summed physics-object
$\pt^{2}$ is taken to be the primary $\Pp\Pp$ interaction vertex, where is the
transverse momentum of the physics-objects. Here the physics objects are the
jets, clustered using the jet finding algorithm [60, 61] with the tracks
assigned to the vertex as inputs, and the associated missing transverse
momentum, , which is defined as the magnitude of the vector sum of the momenta
of all reconstructed particles in an event.
The global event reconstruction, based on the particle-flow algorithm [62],
aims to reconstruct and identify each individual particle in an event, with an
optimized combination of all subdetector information. In this process, the
identification of the particle type (photon, electron, muon, charged hadron,
neutral hadron) plays an important role in the determination of the particle
direction and energy. Photons are identified as ECAL energy clusters not
linked to the extrapolation of any charged-particle trajectory to the ECAL.
Electrons are identified as primary charged-particle tracks and potentially
many ECAL energy clusters corresponding to this track extrapolation to the
ECAL and to possible bremsstrahlung photons emitted along the way through the
tracker material. Muons are identified as tracks in the central tracker
consistent with either a track or several hits in the muon system, associated
with no significant associated energy deposits in the calorimeters. Charged
hadrons are identified as charged-particle tracks neither identified as
electrons, nor as muons. Finally, neutral hadrons are identified as HCAL
energy clusters not linked to any charged-hadron trajectory, or as ECAL and
HCAL energy excesses with respect to the expected charged-hadron energy
deposit.
The energy of photons is directly obtained from the ECAL measurement,
corrected for zero-suppression effects. The energy of electrons is determined
from a combination of the track momentum at the primary interaction vertex,
the corresponding ECAL cluster energy, and the energy sum of all
bremsstrahlung photons attached to the track. The energy of muons is obtained
from the corresponding track momentum. The energy of charged hadrons is
determined from a combination of the track momentum and the corresponding ECAL
and HCAL energy, corrected for zero-suppression effects and for the response
function of the calorimeters to hadronic showers. Finally, the energy of
neutral hadrons is obtained from the corresponding corrected ECAL and HCAL
energies.
### 0.4.1 Lepton selection
Electron candidates are selected in the region $\abs{\eta}<2.5$, excluding
$1.44<\abs{\eta}<1.57$. Their identification that is based on a multivariate
discriminant built from variables that characterize the shower shape and track
quality. To reject electrons originating from photon conversions in the
detector material, electrons must have no measurements missing in the
innermost layers of the tracking system and must not be matched to any
secondary vertex containing another electron [63]. To reduce the rate of the
electron sign mismeasurement, charges measured from independent techniques are
required to be the same, using the “selective method” for the charge
definition as explained in Ref. [63], which we refer to as “tight charge”.
Requiring the electrons to have tight charge reduces the signal efficiency by
1–20%, depending on $m_{\mathrm{N}}$, while the background from mismeasured
sign is reduced by a factor of 10. To ensure that electron candidates are
consistent with originating from the primary vertex, the transverse
(longitudinal) impact parameter of the leptons with respect to this vertex
must not exceed 0.1 (0.4). These electrons must also satisfy
$\abs{\mathrm{d}_{xy}}/\sigma(\mathrm{d}_{xy})<4$, where $\mathrm{d}_{xy}$ is
the transverse impact parameter relative to the primary vertex, estimated from
the track fit, and $\sigma(\mathrm{d}_{xy})$ is its uncertainty.
Muons are selected in the range $\abs{\eta}<2.4$. The muon trajectory is
required to be compatible with the primary vertex, and to have a sufficient
number of hits in the tracker and muon systems. The transverse (longitudinal)
impact parameter of the muons with respect to this vertex must not exceed 0.05
(0.40). These muons must also satisfy
$\abs{\mathrm{d}_{xy}}/\sigma(\mathrm{d}_{xy})<3$.
To distinguish between prompt leptons (a prompt lepton is defined as an
electron or muon originating in a $\PW$/$\PZ$/$\gamma^{*}$ boson or $\tau$
lepton decay) originating from decays of heavy particles, such as electroweak
(EW) bosons or heavy neutrinos, and those produced in hadron decays or hadrons
misidentified as leptons, a relative isolation variable
($I^{\ell}_{\text{rel}}$) is used. It is defined for electrons (muons) as the
pileup-corrected [63, 64] scalar $\pt$ sum of the reconstructed charged
hadrons originating from the primary vertex, the neutral hadrons, and the
photons, within a cone of $\Delta
R=\sqrt{\smash[b]{(\Delta\eta)^{2}+(\Delta\phi)^{2}}}=0.3\,(0.4)$ around the
lepton candidate’s direction at the vertex, divided by the lepton candidate’s
.
Electrons that pass all the aforementioned requirements and satisfy
$I^{\Pe}_{\text{rel}}<0.08$ are referred to as “tight electrons”. Electrons
that satisfy $I^{\Pe}_{\text{rel}}<0.4$, and pass less stringent requirements
on the multivariate discriminant and impact parameter are referred to as
“loose electrons”. Muons that pass all the aforementioned requirements and
satisfy $I^{\mu}_{\text{rel}}<0.07$ are referred to as “tight muons”. Muons
that satisfy $I^{\mu}_{\text{rel}}<0.6$, and pass a less stringent requirement
on the impact parameter and track quality requirements are referred to as
“loose muons”. Electrons within $\Delta R<0.05$ of a muon are removed, as
these particles are likely a photon radiated from the muon.
### 0.4.2 Identification of jets and missing transverse momentum
For each event, hadronic jets are clustered from the reconstructed particle-
flow objects with the infrared and collinear safe anti-jet clustering
algorithm [60], implemented in the package [65]. Two different distance
parameters, $\mathrm{R}=0.4$ and 0.8, are used with this algorithm, producing
objects referred to as AK4 and AK8 jets, respectively. The jet momentum is
determined as the vector sum of all particle momenta in the jet, and is found
from simulation to be within 5 to 10% of the true parton momentum over the
entire spectrum and detector acceptance. Additional $\Pp\Pp$ interactions
within the same or nearby bunch crossings can contribute additional tracks and
calorimetric energy depositions to the jet momentum. To mitigate this effect,
tracks identified to be originating from pileup vertices are discarded and an
offset correction is applied to correct for remaining contributions. Jet
energy corrections are derived from simulation to bring the measured response
of jets to that of particle level jets on average. In situ measurements of the
momentum balance in dijet, photon+jet, $\PZ\text{+}\text{jet}$, and multijet
events are used to estimate residual differences in jet energy scale in data
and simulation, and appropriate corrections are applied [66]. The jet energy
resolution is typically 15% at 10, 8% at 100, and 4% at 1. Additional
selection criteria are applied to remove jets potentially dominated by
anomalous contributions from various subdetector components or reconstruction
failures. The AK4 (AK8) jets must have $>20$ (200)and $\abs{\eta}<2.7$ to be
considered in the subsequent steps of the analysis. To suppress jets matched
to pileup vertices, AK4 jets must pass a selection based on the jet shape and
the number of associated tracks that point to non-primary vertices [67].
The AK8 jets are groomed using a jet pruning algorithm [68, 69]: subsequent to
the clustering of AK8 jets, their constituents are reclustered with the
Cambridge–Aachen algorithm [70, 71], where the reclustering sequence is
modified to remove soft and wide-angle particles or groups of particles. This
reclustering is controlled by a soft threshold parameter $z_{\text{cut}}$,
which is set to 0.1, and an angular separation threshold $\Delta
R>m_{\text{jet}}/p_{\text{T},\text{jet}}$. The jet pruning algorithm computes
the mass of the AK8 jet after removing the soft radiation to provide a better
mass resolution for jets, thus improving the signal sensitivity. The pruned
jet mass is defined as the invariant mass associated with the four-momentum of
the pruned jet.
In addition to the jet grooming algorithm, the “$N$-subjettiness” of jets [72]
is used to identify boosted vector bosons that decay hadronically. This
observable measures the distribution of jet constituents relative to candidate
subjet axes in order to quantify how well the jet can be divided into $N$
subjets. Subjet axes are determined by a one-pass optimization procedure that
minimizes $N$-subjettiness [72]. The separation in the phi-azimuth plane
between all of the jet constituents and their closest subjet axes are then
used to compute the $N$-subjettiness as
$\tau_{N}=1/d_{0}\Sigma_{k}p_{\mathrm{T},k}\text{min}(\Delta R_{1,k},\Delta
R_{2,k},...,\Delta R_{N,k})$ with the normalization factor
$d_{0}=\Sigma_{k}p_{\mathrm{T},k}R_{0}$ where $R_{0}$ is the clustering
parameter of the original jet, $p_{\mathrm{T},k}$ is the transverse momentum
of the $k$-constituent of the jet and $\Delta
R_{N,k}=\sqrt{\smash[b]{(\Delta\eta_{N,k})^{2}+(\Delta\phi_{N,k})^{2}}}$ is
its distance to the $N$-th subjet. In particular, the ratio between $\tau_{2}$
and $\tau_{1}$, known as $\tau_{21}$, has excellent capability for separating
jets originating from boosted vector bosons from jets originating from quarks
and gluons [72]. To select a high-purity sample of jets originating from a
hadronically decaying $\PW$ bosons, the AK8 jets are required to have
$\tau_{21}<0.6$ and a pruned jet mass between 40 and 130. We refer to these
selected jets as $\PW$-tagged jets. The efficiency of the $\tau_{21}$
selection for AK8 jets is measured in a -enriched sample in data and
simulation. To correct for observed differences between the estimated and
measured efficiencies a scale factor of $1.11\pm 0.08$ is applied to the event
for each AK8 jet that passes the $\tau_{21}$ requirement in the simulation
[67].
Identifying jets originating from a bottom quark can help suppress backgrounds
from production. To identify such jets the combined secondary vertex algorithm
[73] is used. This algorithm assigns to each jet a likelihood that it contains
a bottom hadron, using many discriminating variables, such as track impact
parameters, the properties of reconstructed decay vertices, and the presence
or absence of low-leptons. The average tagging efficiency for jets above 20is
63%, with an average misidentification probability for light-parton jets of
about 1%.
To avoid double counting due to jets matched geometrically with a lepton, any
AK8 jet that is within $\Delta R<1.0$ of a loose lepton is removed from the
event. Moreover, if an AK4 jet is reconstructed within $\Delta R<0.4$ of a
loose lepton or within $\Delta R<0.8$ of an AK8 jet, it is not used in the
analysis.
The is adjusted to account for the jet energy corrections applied to the event
[66]. The scalar sum of all activity in the event ($S_{\mathrm{T}}$) is used
in the selection of our signal region selection and is defined as the sum of
all AK4 and AK8 jets, leptons, and . The transverse mass, $m_{\text{T}}$,
which is used as a requirement to suppress backgrounds from leptonic $\PW$
boson decays, is defined as follows:
$m_{\text{T}}(\ell,\ptmiss)=\sqrt{\smash[b]{2\pt^{\ell}\ptmiss[1-\cos(\Delta\phi_{\ell,\ptvecmiss})]}},$
(1)
where $\pt^{\ell}$ is the transverse momentum of the lepton and
$\Delta\phi_{\ell,\ptvecmiss}$ is the azimuthal angle difference between the
lepton momentum and vector.
## 0.5 Event selection
Events used in this search are selected using several triggers, requiring the
presence of two charged leptons ($\Pe$ or $\mu$). All triggers require two
loosely isolated leptons, where the leading- (trailing-)lepton must have
$\pt>23\,(12)\GeV$ for the $\Pe\Pe$, $\pt>17\,(8)\GeV$ for the $\mu\mu$, and
$\pt>23\,(8)\GeV$ for the $\Pe\mu$ trigger at the HLT stage. The offline
requirements on the leading (trailing) lepton $\pt$ are governed by the
trigger thresholds, and are $\pt>25\,(15)\GeV$ for the $\Pe\Pe$,
$\pt>20\,(10)\GeV$ for the $\mu\mu$, and $\pt>25\,(10)\GeV$ for the $\Pe\mu$
channels. The efficiency for signal events to satisfy the trigger in the
$\Pe\Pe$, $\mu\mu$, and $\Pe\mu$ channels is above 0.88, 0.94, and 0.88,
respectively.
### 0.5.1 Preselection criteria
At a preselection stage, events are required to contain a pair of SS leptons.
To remove backgrounds with soft misidentified leptons, the invariant mass of
the dilepton pair is required to be above 10. Dielectron events with an
invariant mass within 20of the $\PZ$ boson mass [1] are excluded to reject
background from $\PZ$ boson decays in which one electron sign is mismeasured.
In order to suppress backgrounds from diboson production, such as $\PW\PZ$,
events with a third lepton identified using a looser set of requirements and
with $\pt>10\GeV$ are removed. Preselected events are required to have at
least one AK4 or one AK8 jet passing the full jet selection. The same
preselection is applied in all three channels ($\Pe\Pe$, $\mu\mu$, $\Pe\mu$).
### 0.5.2 Selection criteria for signals
The kinematic properties of signal events from heavy-neutrino decays depend on
its mass. To distinguish between the two $\PW$ bosons involved in the
production and decay sequence, we refer to the $\PW$ boson that produces
$\mathrm{N}$ in Fig. 1 (left) as the $\PW$ boson propagator and the $\PW$
boson that decays to a quark and anti-quark pair as the hadronically decaying
$\PW$ boson. Two search regions (SRs) are defined. In the low-mass SR
($m_{\mathrm{N}}\lesssim 80\GeV$), the $\PW$ boson propagator is on-shell and
the final-state system of dileptons and two jets should have an invariant mass
close to the $\PW$ boson mass. In the high-mass SR ($m_{\mathrm{N}}\gtrsim
80\GeV$), the $\PW$ boson propagator is off-shell but the hadronically
decaying $\PW$ boson is on-shell, so the invariant mass of the jets from the
hadronically decaying $\PW$ will be consistent with the $\PW$ boson mass.
Since the kinematic properties of the signal depend on $m_{\mathrm{N}}$, we
define four event categories to maximize the discovery potential over the full
mass range. The low- and high-mass SRs are further split based on the jet
configuration. The four signal categories used in the analysis are defined as:
* •
low- and high-mass SR1: number of AK4 jets $\geq$ 2 and number of AK8 jets =
0,
* •
low-mass SR2: number of AK4 jets = 1 and number of AK8 jets = 0,
* •
high-mass SR2: number of AK8 jets $\geq$ 1.
Taking the three flavor channels into account, the analysis has 12 separate
SRs.
In each SR, the technique of selecting jets associated with the hadronic $\PW$
boson decay is different. If there are any $\PW$-tagged AK8 jets in the event,
the AK8 jet with pruned jet mass closest to $m_{\mathrm{\PW}}$ is assumed to
be from the hadronic $\PW$ boson decay. For the high-mass SRs, if there are
two or more AK4 jets in the event and no AK8 jets, the two AK4 jets with the
invariant mass closest to $m_{\mathrm{\PW}}$ are assigned to the hadronically
decaying $\PW$ boson. In the low-mass SRs, the $\PW$ boson propagator is
reconstructed from $\mathrm{N}$ (one lepton + jet(s)) and the additional
lepton, and if there are more than two jets, the jets are selected such that
the mass is closest to $m_{\mathrm{\PW}}$. If only one jet is reconstructed in
the low-mass SR then this is assigned as being from the hadronic $\PW$ boson
decay. The jet(s) assigned to the hadronic $\PW$ boson decay are referred to
by the symbol $\PW_{\text{jet}}$ to simplify notation in the rest of the
paper.
Before optimizing the signal significance for each mass hypothesis we apply a
set of loose selections to the preselection events to select the low- and
high-mass SRs. These requirements are chosen to remove a large fraction of the
backgrounds while keeping the signal efficiency high. In the low-mass SRs, the
invariant mass of the two leptons and $\PW_{\text{jet}}$ is required to be
less than 300. To remove backgrounds from leptonic $\PW$ boson decays, events
must have less than 80. To remove backgrounds from top quark decays, events
are vetoed if they contain a -tagged AK4 jet. In the high-mass SRs, the
following selections are used. For SR1 the events are required to have
$30<m(\PW_{\text{jet}})<150\GeV$ for the invariant mass of the
$\PW_{\text{jet}}$ and $\pt>25\GeV$ for the leading AK4 jet. For SR2 the
pruned jet mass must satisfy $40<m(\PW_{\text{jet}})<130\GeV$. Since the is
correlated with the energy of the final-state objects, this requirement is not
used in high-mass SRs. Instead, we use $(\ptmiss)^{2}/S_{\mathrm{T}}$, which
has a stronger discriminating power between high-mass signals and backgrounds.
The $(\ptmiss)^{2}/S_{\mathrm{T}}$ must be less than 15. These selections are
summarized in Table 0.5.2.
Selection requirements, after applying the preselection criteria, for the low-
and high-mass signal regions. A dash indicates that the variable is not used
in the selection. Region $(\ptmiss)^{2}/S_{\mathrm{T}}$
$m(\ell^{\pm}\ell^{\pm}\PW_{\text{jet}})$ $m(\PW_{\text{jet}})$
$p_{\mathrm{T}}^{\mathrm{j}}$ () () () () () Low-mass SR1+SR2 $<$80 — $<$300 —
$>$20 High-mass SR1 — $<$15 — 30–150 $>$25 High-mass SR2 — $<$15 — 40–130
$>$200
#### Optimization of signal selection
After applying the selection criteria in Table 0.5.2, the signal significance
is optimized by combining several different variables using a modified Punzi
figure of merit [74]. The Punzi figure of merit is defined as
$\epsilon_{\mathrm{S}}/(a/2+\delta B)$ where $a$ is the number of standard
deviation, and is set equal to 2 to be consistent with the previous CMS
analysis, $\epsilon_{\mathrm{S}}$ is the signal selection efficiency, and
$\delta B$ is the uncertainty in the estimated background. The signal regions
are optimized separately for each mass hypothesis and for each of the three
flavor channels.
The variables used to optimize the signal selection, which are all optimized
simultaneously, are: the transverse momentum of the leading lepton
$\pt^{\ell_{1}}$, and of the trailing lepton $\pt^{\ell_{2}}$; the invariant
mass of the two leptons and the selected jet(s)
$m(\ell^{\pm}\ell^{\pm}\PW_{\text{jet}})$; the angular separation between the
$\PW_{\text{jet}}$ and the trailing lepton $\Delta
R(\ell_{2},\PW_{\text{jet}})$; minimum and maximum requirements on the
invariant mass of the lepton (leading or trailing) and the selected jet(s)
$m(\ell_{i}\PW_{\text{jet}})$, where $i$=1,2; and the invariant mass of the
two leptons $m(\ell^{\pm}\ell^{\pm})$. We consider the variable
$m(\ell_{i}\PW_{\text{jet}})$, as this should peak at $m_{\mathrm{N}}$ for the
signal. Since it is not known which lepton comes from the $\mathrm{N}$ decay,
the event is accepted if either $m(\ell_{i}\PW_{\text{jet}})$ satisfies the
requirements. The optimized window requirements for some SRs are enlarged to
give complete coverage of the signal parameter space at negligible loss of
sensitivity. The selection requirements for each mass hypothesis are
summarized later in Section 0.8, in Tables 0.8–0.8 for both low- and high-mass
SRs. The overall signal acceptance ranges between 0.10–0.27% and 17–33% for
$m_{\mathrm{N}}=20$–1500, respectively. Here, the lower acceptance at low
$m_{\mathrm{N}}$ is due to the selection requirements on the of the leptons
and jets in a signal with very soft jets and leptons. The overall signal
acceptance includes trigger efficiency, geometrical acceptance, and
efficiencies of all selection criteria.
## 0.6 Background estimate
The SM backgrounds leading to a final state with two SS leptons and jets are
divided into the following categories:
* •
SM processes with multiple prompt leptons: These backgrounds are mainly from
events with two vector bosons ($\PW^{\pm}\PW^{\pm}$, $\PW\PZ$, $\PZ\PZ$). We
also consider as background a $\PW$ or $\PZ$ boson decaying leptonically and
is accompanied by radiation of an initial- or final-state photon that
subsequently undergoes an asymmetric conversion. These processes produce a
final state that can have three or four leptons. If one or more of the charged
leptons fail the reconstruction or selection criteria these processes can
appear to have only two SS leptons.
* •
Misidentified leptons: These are processes that contain one or more leptons
that are either misidentified hadrons, are from heavy-flavor jets, from light
meson decays, or from a photon in a jet. These leptons are generally less
isolated than a prompt lepton from a $\PW/\PZ$ boson decay and tend to have
larger impact parameters. The main processes with a misidentified lepton in
the SRs include $\PW$+jets events and events, but multijet and DY events also
contribute.
* •
Sign mismeasurement: If the signs of leptons are mismeasured in events with
jets and two opposite-sign leptons (OS2$\ell$), these events could contaminate
a search region. When the sign of a lepton is mismeasured the lepton will on
average have a larger impact parameter in comparison to a lepton from a prompt
EW boson decay. Although the rate of mismeasuring the sign of an electron is
small, the abundance of OS2$\ell$ events from DY dilepton production means
that this background is significant. It is suppressed by tight requirements on
the impact parameter and on the charge of the electron. The muon sign
mismeasurement rate is known to be negligible, based on studies in simulation
and with cosmic ray muons [75], and is not considered in this analysis.
### 0.6.1 Background from prompt SS leptons
Background events that contain two prompt SS leptons are referred to as the
prompt-lepton background. These backgrounds are estimated using simulation. To
remove any double counting from the misidentified-lepton background estimate
based on control samples in data, the leptons have to originate in the decay
of either a $\PW/\PZ/\gamma^{*}$ boson, or a $\tau$ lepton. The largest
contribution comes from $\PW\PZ$, $\PZ\PZ$, and asymmetric photon conversions,
including those in $\PW\gamma$ and $\PZ\gamma$ events. The background from
$\PW\PZ$ and $\PW\gamma^{*}$ production, with $\PW\to\ell\nu$ and
$\PZ(\gamma^{(*)})\to\ell\ell$, can yield the same signature as $\mathrm{N}$
production: two SS isolated leptons and jets, when one of the opposite-sign
same-flavor (OSSF) leptons is not identified and QCD/pileup jets are
reconstructed in the event. This is the largest prompt contribution in both
the low- and high-mass SRs. This background is estimated from simulation, with
the simulated yield normalized to the data in a control region (CR) formed by
selecting three tight leptons with $\pt>25,15,10\GeV$ and requiring an OSSF
lepton pair with invariant mass $m(\ell^{\pm}\ell^{\mp})$ consistent with the
$\PZ$ boson mass: $\abs{m(\ell^{\pm}\ell^{\mp})-m_{\mathrm{\PZ}}}<15\GeV$. In
addition, events are required to have $\ptmiss>50\GeV$ and
$m_{\text{T}}(\ell_{\PW},\ptmiss)>20\GeV$, where the $\ell_{\PW}$ is the
lepton not used in the OSSF pair that is consistent with the $\PZ$ boson. The
ratio of the predicted to observed $\PW\PZ$ background yield in this CR is
found to be $1.051\pm 0.065$. This factor and its associated uncertainty (both
statistical and systematic) is used to normalize the corresponding simulated
sample. The systematic uncertainty on this factor is determined by varying, in
the simulation, the properties that are listed in Section 0.7.2, by $\pm 1$
standard deviation from its central value.
Production of $\PZ\PZ$ events with both $\PZ$ bosons decaying leptonically,
with two leptons not identified, results in a possible SS2$\ell$ signature.
This process is estimated from simulation, and the simulated yield is
normalized in a CR containing four leptons that form two OSSF lepton pairs
with invariant masses consistent with that of the $\PZ$ boson. The ratio of
data to simulation from the CR is found to be $0.979\pm 0.079$, and is used to
normalize the simulated $\PZ\PZ$ sample. A $\PZ$ boson -dependent EW
correction to the cross section [76, 77, 78] is not included in the simulated
samples. It would correct the cross section by at most 25%, given the range of
$\PZ$ boson probed in this analysis. Since this correction is larger than the
uncertainty on the ratio of data to simulation in the CR, we increase the
uncertainty on the normalization to 25%.
External and internal photon conversions can produce an SS2$\ell$ final state
when a photon is produced with a $\PW$ or $\PZ$ boson, and this photon
undergoes an asymmetric external or internal conversion
($\gamma^{(*)}\to\ell^{+}\ell^{-}$) in which one of the leptons has very low
and fails the lepton selection criteria. This background mostly contributes to
events in the $\Pe\Pe$ and $\Pe\mu$ channels. It is obtained from simulation
and verified in a data CR enriched in both external and internal conversions
from the $\PZ\text{+}\text{jets}$ process, with $\PZ\to\ell\ell\gamma^{(*)}$
and $\gamma^{(*)}\to\ell\ell$, where one of the leptons is outside the
detector acceptance. The CR is defined by
$\abs{m(\ell^{\pm}\ell^{\mp})-m_{\mathrm{\PZ}}}>15\GeV$ and
$\abs{m(\ell^{\pm}\ell^{\mp}\ell^{\pm})-m_{\mathrm{\PZ}}}<15\GeV$. The ratio
of data to expected background in the CR is $1.093\pm 0.075$, and this ratio
is used to normalize the MC simulation.
Other rare SM processes that can yield two SS leptons include events from EW
production of SS $\PW$ pairs, and double parton scattering, while any SM
process that yields three or more prompt leptons produces SS2$\ell$ final
states if one or more of the leptons fails to pass the selection. Processes in
the SM that can yield three or more prompt leptons include triboson processes
and production associated with a boson ($\ttbar\PW$, $\ttbar\PZ$, and
$\ttbar\PH$). Such processes generally have very small production rates (less
than 10% of total background after the preselection) and in some cases are
further suppressed by the veto on -tagged jets and requirements on . They are
estimated from simulation and assigned a conservative uncertainty of 50%,
which accounts for the uncertainties due to experimental effects, event
simulation, and theoretical calculations of the cross sections.
### 0.6.2 Background from misidentified leptons
The most important background source for low-mass signals originates from
events containing objects misidentified as prompt leptons. These originate
from hadron decays, light-quark or gluon jets, and are typically not well
isolated. Examples of these backgrounds include: multijet production, in which
one or more jets are misidentified as leptons; $\PW(\to\ell\nu)$+jets events,
in which one of the jets is misidentified as a lepton; and decays, in which
one of the top quark decays yields a prompt isolated lepton
$(\cPqt\to\PW\cPqb\to\ell\nu\cPqb)$ and the other lepton of same sign arises
from a bottom quark decay or a jet misidentified as an isolated prompt lepton.
The simulation is not reliable in estimating the misidentified-lepton
background for several reasons, including the lack of statistically large
samples (because of the small probability of a jet to be misidentified as a
lepton) and inadequate Modeling of the parton showering process. Therefore,
these backgrounds are estimated using control samples of collision data.
An independent data sample enriched in multijet events (the “measurement”
sample) is used to calculate the probability misidentifying a jet that passes
minimal lepton selection requirements (“loose leptons”) to also pass the more
stringent requirements used to define leptons after the full selection (“tight
leptons”). The misidentification probability is applied as an event-by-event
weight to the application sample. The application sample contains events in
which one lepton passes the tight selection, while the other lepton fails the
tight selection but passes the loose selection ($N_{n\overline{n}}$), as well
as events in which both leptons fail the tight selection, but pass the loose
criteria ($N_{\overline{n}\,\overline{n}}$). The total contribution to the
signal regions (, the number of events with both leptons passing the tight
selection, $N_{nn}$), is then obtained for each mass hypothesis by weighting
events of type $n\overline{n}$ and $\overline{n}\,\overline{n}$ by the
appropriate misidentification probability factors and applying the signal
selection requirements to the application sample. To account for the double
counting we correct for $\overline{n}\,\overline{n}$ events that can also be
$n\overline{n}$.
The measurement sample is selected by requiring a loose lepton and a jet,
resulting in events that are mostly dijet events, with one jet containing a
lepton. Only one lepton is allowed and requirements of $\ptmiss<80\GeV$, and
$m_{\text{T}}(\ell,\ptmiss)<25\GeV$ are applied. The loose lepton and jet are
required to be separated in azimuth by $\Delta\phi>2.5$ and the momentum of
the jet is required to be greater than the momentum of the lepton. These
requirements suppress contamination from $\PW$ and $\PZ$ boson decays.
Contamination of prompt leptons in the measurement sample from EW processes is
estimated and subtracted using simulation. The normalization of the prompt
lepton simulation is validated in a data sample enriched in $\PW$+jets events
by requiring events with a single lepton, $\ptmiss>40\GeV$, and
$60<m_{\text{T}}(\ell,\ptmiss)<100\GeV$. The minimum uncertainty that covers
the discrepancy between the data and simulation in single-lepton $\PW$+jets
events (across all $\eta$ and $\pt$ bins considered in the analysis) is 30
(13)% for electrons (muons) and is assigned as the uncertainty in the prompt
lepton normalization. The larger uncertainty for prompt electron events is to
allow for the disagreement between data and simulation in single-electron
$\PW$+jets events for high-electrons.
The method is validated using a sample of simulated , $\PW$+jets, and DY
events. The misidentification probabilities used in this validation are
obtained from simulated events comprised of jets produced via the strong
interaction, referred to as QCD multijet events. The predicted and observed
numbers of events in the $\Pe\Pe$, $\mu\mu$, and $\Pe\mu$ channels agree
within 10% for the $\PW$+jets and DY samples, and within 25% for samples. The
latter figure is reduced to 18% after rejecting events with a -tagged jet.
### 0.6.3 Background from opposite-sign leptons
To estimate backgrounds due to sign mismeasurement, the probability of
mismeasuring the lepton sign is studied. Only mismeasurement of the electron
sign is considered, and this background is estimated only in the $\Pe\Pe$
channel. The probability of mismeasuring the sign of a prompt electron is
obtained from simulated $\PZ\to\Pe^{\pm}\Pe^{\mp}$ events and is parametrized
as a function of separately for electrons in the barrel and endcap
calorimeters. The average value and statistical uncertainty for the sign
mismeasurement probabilities are found to be $(1.65\pm 0.12)\times 10^{-5}$ in
the inner ECAL barrel region ($\abs{\eta}<0.8$), $(1.07\pm 0.03)\times
10^{-4}$ in the outer ECAL barrel region ($0.8<\abs{\eta}<1.5$), and $(0.63\pm
0.01)\times 10^{-3}$ in the endcap region. The sign mismeasurement
probabilities are then validated with data separately for the barrel and
endcap regions.
To estimate the background due to sign mismeasurement in the $\Pe\Pe$ channel,
a weight $W_{p}$ is applied to data events with all the SR selections applied,
except that here the leptons are required to be oppositely signed (OS2$\ell$
events). $W_{p}$ is given by $W_{p}=p_{1}/(1-p_{1})+p_{2}/(1-p_{2})$, where
$p_{1(2)}$ is the probability for the leading (trailing) electron sign to be
mismeasured and is determined from simulated events. The of leptons with a
mismeasured sign will be misreconstructed. To correct for the misreconstructed
measurement in the OS2$\ell$ events the lepton is shifted up by 1.8%, which is
determined from simulation.
To validate the sign mismeasurement probability for the barrel (endcap)
region, a control sample of $\PZ\to\Pe^{\pm}\Pe^{\mp}$ events in the data is
selected, requiring both electrons to pass through the barrel (endcap) region
and demanding the invariant mass of the electron pair to be between 76 and
106. The difference between the observed and predicted numbers of
$\Pe^{\pm}\Pe^{\pm}$ events is used as a scale factor to account for the
modeling in the simulation. The observed number of events in the data is
determined by fitting the $\PZ$ boson mass peak. The predicted number of
events is determined by weighting the OS2$\ell$ events with the value $W_{p}$.
The scale factors and their associated statistical uncertainties in the barrel
and endcap regions are found to be $0.80\pm 0.03$ and $0.87\pm 0.03$,
respectively.
To validate the combined sign mismeasurement probability and scale factors in
the data, a control sample of $\PZ\to\Pe^{\pm}\Pe^{\mp}$ events is again
selected, as described above, but here requiring that one electron is found in
the endcap and the other, in the barrel region. The difference in the
predicted and observed numbers of $\Pe^{\pm}\Pe^{\pm}$ events in this sample
is 12%. The same procedure was performed using $\PZ\to\Pe^{\pm}\Pe^{\mp}$
events in the data but requiring no $\eta$ restrictions on the electrons and
requiring that the event has only one jet, yielding an agreement within 10%
between the predicted background and the data.
Prompt leptons and backgrounds from sign mismeasurement can contaminate the
application sample of the misidentified-lepton background, resulting in an
overprediction of this background. This contamination is removed using
simulation. The contamination from the prompt-lepton background is generally
less than 1%. However, for the backgrounds from leptons with sign
mismeasurement or leptons from photon conversions, the contamination can be as
large as 2% in the signal region and up to 30% in CR2, that is enriched in
backgrounds with mismeasured lepton sign.
### 0.6.4 Validation of background estimates
To test the validity of the background estimation methods, several signal-free
data CRs are defined. The background estimation method is applied in these
regions and the results are compared with the observed yields. These CRs are
used to validate the backgrounds separately in each of the three flavor
channels and are defined as follows:
* •
CR1: (SS2$\ell$), at least one -tagged AK4 jet,
* •
CR2: (SS2$\ell$), $\Delta R(\ell_{1},\ell_{2})>2.5$ and no -tagged AK4 jet,
* •
CR3: (SS2$\ell$), low-mass SR1 and either $\geq$ 1 -tagged jet or
$\ptmiss>100\GeV$,
* •
CR4: (SS2$\ell$), low-mass SR2 and either $\geq$ 1 -tagged jet or
$\ptmiss>100\GeV$,
* •
CR5: (SS2$\ell$), high-mass SR1 and either $\geq$ 1 -tagged jet or
$(\ptmiss)^{2}/S_{\mathrm{T}}>20\GeV$,
* •
CR6: (SS2$\ell$), high-mass SR2 and either $\geq$ 1 -tagged jet or
$(\ptmiss)^{2}/S_{\mathrm{T}}>20\GeV$.
The numbers of predicted and observed background events in each CR are shown
in Table 0.6.4. In the control regions CR1 and CR2, the backgrounds estimated
from data are dominant and validated in events both with and without -tagged
jets, while in the remaining CRs all backgrounds are validated in regions that
are close to the SRs (the misidentified-lepton background accounts for about
90% of the total background in CR1 and CR2 and about 50% across the remaining
CRs). The contribution from signal events is found to be negligible in all
control regions, with signal accounting for less than 1% of the yields in most
CRs and at most 5%, when assuming a coupling consistent with the upper limits
from previous results. In all regions the predictions are in agreement with
the observations within the statistical and systematic uncertainties described
in Section 0.7, which is dominated by the 30% uncertainty in the
misidentified-lepton background. Within each region, the observed
distributions of all relevant observables also agree with the predictions,
within the uncertainties
Observed event yields and estimated backgrounds in the control regions. The
uncertainties in the background yields are the sums in quadrature of the
statistical and systematic components. Channel Control region Estimated
background Observed $\Pe\Pe$ CR1 $\phantom{0.0}366\pm 73\phantom{0.0}$ 0378
CR2 $\phantom{0.0}690\pm 100\phantom{.0}$ 0671 CR3 $\phantom{0.0}222\pm
42\phantom{0.0}$ 0242 CR4 $\phantom{00.0}48\pm 11\phantom{0.0}$ 0038 CR5
$\phantom{0.0}334\pm 56\phantom{0.0}$ 0347 CR6 $\phantom{00}25.7\pm
4.3\phantom{00}$ 0028 $\mu\mu$ CR1 $\phantom{0.0}880\pm 230\phantom{.0}$ 0925
CR2 $\phantom{0.0}890\pm 200\phantom{.0}$ 1013 CR3 $\phantom{0.0}420\pm
100\phantom{.0}$ 0439 CR4 $\phantom{0.0}156\pm 42\phantom{0.0}$ 0174 CR5
$\phantom{0.0}560\pm 120\phantom{.0}$ 0568 CR6 $\phantom{00}35.1\pm
7.0\phantom{00}$ 0038 $\Pe\mu$ CR1 $\phantom{.0}1010\pm 240\phantom{.0}$ 1106
CR2 $\phantom{.0}1350\pm 230\phantom{.0}$ 1403 CR3 $\phantom{0.0}650\pm
140\phantom{.0}$ 0706 CR4 $\phantom{0.0}143\pm 32\phantom{0.0}$ 0150 CR5
$\phantom{0.0}920\pm 180\phantom{.0}$ 0988 CR6 $\phantom{00.0}62\pm
11\phantom{0.0}$ 0064
## 0.7 Systematic uncertainties
The estimate of backgrounds and signal efficiencies are subject to a number of
systematic uncertainties. The relative sizes of these uncertainties for each
type of background and signal, in each SR, are listed in Table 0.7. Table 0.7
shows the contributions from the uncertainty in the signal and backgrounds
(for two mass hypotheses, $m_{\mathrm{N}}=50$ and 500), expressed as a
percentage of the total uncertainty.
Summary of the relative systematic uncertainties in heavy Majorana neutrino
signal yields and in the background from prompt SS leptons, both estimated
from simulation. The relative systematic uncertainties assigned to the
misidentified-lepton and mismeasured-sign backgrounds estimated from control
regions in data and simulation are also shown. The uncertainties are given for
the low- (high-)mass selections. The range given for each systematic
uncertainty source covers the variation across the mass range. Upper limits
are presented for the uncertainty related to the PDF choice in the background
estimates, however this source of uncertainty is considered to be accounted
for via the normalization uncertainty and was not applied explicitly as an
uncertainty in the background. Source / Channel $\Pe\Pe$ signal $\Pe\Pe$
bkgd. $\mu\mu$ signal $\mu\mu$ bkgd. $\Pe\mu$ signal $\Pe\mu$ bkgd. (%) (%)
(%) (%) (%) (%) Simulation: SM cross section — .12–14 (15–27). — .13–18
(22–41). — .12–14 (16–30). Jet energy scale .02–5 (0–1).0 .002–6 (5–6).00
.02–8 (0–1).0 .003–5 (4–7).00 .01–6 (0–1).0 .001–4 (3)–.000 Jet energy
resolution .01–2 (0–0.3) .001–2 (2–6).00 .01–2 (0–0.3) 00–0.8 (1–3).00 –00.8
(0–0.3) 00–0.8 (0–3).00 Jet mass scale 0–0.3 (0–0.1) .000–1 (1–3).00 0–0.2
(0–0.1) 00–0.3 (0.7)–00 0–0.1 (0–0.1) 00–0.2 (0–5).00 Jet mass resolution
0–0.4 (0–0.3) .000–1 (0–2).00 0–0.1 (0–0.2) 00–0.1 (0–0.5)0 0–0.4 (0–0.3)
00–0.4 (0–3).00 Subjettiness .00–1 (0–8).0 00–1.0 (1–7).00 0–0.3 (0–8).0
00–0.1 (0–8).00 0–0.2 (0–8).0 00–0.4 (0–8).00 Pileup .02–3 (1)–.00 –.0002
(0–2).00 .00–1 (0–1).0 .000–1 (0–3).00 –00.7 (0.8)–0 –.0002 (2–4).00
Unclustered energy 0–0.7 (0–0.1) –.0001 (2–5).00 .00–1 (0–0.1) .000–1 (3–4).00
0–0.5 (0–0.1) –000.9 (1–2).00 Integrated luminosity –02.5 (2.5)–0 –002.5
(2.5)–00 –02.5 (2.5)–0 –002.5 (2.5)–00 –02.5 (2.5)–0 –002.5 (2.5)–00 Lepton
selection .02–4 (4)–.00 .002–4 (2–6).00 –.003 (3–4).0 –.0003 (3–5).00 –.002
(3)–.00 –.0002 (2–6).00 Trigger selection .03–4 (1)–.00 –.0003 (3–5).00 0–0.9
(0–0.4) .000–1 (0–0.8)0 –.003 (0–0.2) –.0003 (2)–.000 tagging 0–0.8 (0–1).0
–000.7 (1)–.000 0–0.5 (0–0.6) .000–1 (1–3).00 0–0.7 (0–0.7) .000–1 (1–4).00
Theory: PDF variation 0–0.7 (0–0.2) $<15$ ($<20$) 0–0.7 (0–0.1) $<15$ ($<20$)
0–0.7 (0–0.2) $<15$ ($<20$) Scale variation .01–5 (0–0.1) — .01–4 (0–0.3) —
.01–5 (0–0.2) — Estimated from data: Misidentified leptons — –.0030 (30)–.00 —
–.0030 (30)–.00 — –.0030 (30)–.00 Mismeasured sign — .29–41 (53–88). — — — —
Fractional contributions to the total background systematic uncertainties
related to the uncertainties in the prompt SS lepton, misidentified-lepton,
and mismeasured-sign backgrounds. The numbers are for the SR1 (SR2) in the
case of $m_{\mathrm{N}}$ = 50 and 500. Channel $m_{\mathrm{N}}$ Prompt-lepton
Misidentified-lepton Mismeasured-sign () (%) (%) (%) $\Pe\Pe$ 50 53 (49) 43
(46) 4.5 (4.9) 500 60 (75) 3.6 (4.6) 37 (21) $\mu\mu$ 50 38 (42) 62 (58) — 500
100 (100) 0.0 (0.0) — $\Pe\mu$ 50 52 (45) 48 (55) — 500 99 (100) 1.3 (0.0) —
### 0.7.1 Background uncertainties
The main sources of systematic uncertainties are associated with the
background estimates. The largest uncertainty is that related to the
misidentified-lepton background. The systematic uncertainty in this background
is determined by observing the change in the background estimate with respect
to variations in isolation requirement (and several other selection criteria)
for the loose leptons, modifying the requirement for the away-side jet (the
jet that is required to be back-to-back with the lepton in the measurement
region). In addition, uncertainties in the jet flavor dependence of the
misidentification probability, and in the prompt-lepton contamination in the
measurement region are taken into account By combining these sources, a
systematic uncertainty of 8.9–20% is assigned. This uncertainty depends on the
lepton flavor and the SR. The validity of the prediction of the misidentified
lepton background was checked by estimating this background using simulated
events alone. The results disagreed with those obtained from the various CRs
by up to 30%, and this value is assigned as the systematic uncertainty in this
background estimate.
The systematic uncertainties in the mismeasured electron sign background are
determined by combining weighted average of the uncertainties in barrel/endcap
scale factors from background fits, and the uncertainty on the parameterized
sign mismeasurement probabilities. To evaluate the uncertainties in the sign
mismeasurement probability scale factors, we vary the range and the number of
bins used in the fitting of the data, as well as the requirement on the
subleading lepton , and, when combining all these sources, we assign a
systematic uncertainty in the scale factors of 9%. The uncertainty in the sign
mismeasurement probability arising from the choice of parameterization
variables was estimated by considering alternative variables such as
$(\ptmiss)^{2}/S_{\mathrm{T}}$ and . A variation of up to 11% was observed.
The background estimate method was tested using only simulation, in which
OS2$\ell$ events were weighted using the sign mismeasurement probabilities
with no scale factors applied. The predicted and observed number of events in
simulation disagree by up to 7%, and this value is assigned as another source
of systematic uncertainty in estimating the sign mismeasurement background.
The three sources discussed above are combined to give a systematic
uncertainty of 16% on this background. This uncertainty covers the difference
between the predicted and observed numbers of events in both data samples
enriched in backgrounds with mismeasured electrons as discussed in Section
0.6.3.
The simulated sample used to measure the sign mismeasurement probabilities has
low statistics for events with electron above 100. When combined with the
uncertainty related to the low statistics of simulated electrons in bins with
high electron , for backgrounds from mismeasured electron sign, an overall
systematic uncertainty of 29–88% is assigned, depending on electron $\eta$ and
. The large uncertainty in this background applies only to the cases where the
SR has two high-electrons. The effect on the total systematic uncertainty in
the background is at most 5%.
### 0.7.2 Simulation uncertainties
The systematic uncertainties in the normalization of the irreducible SM
diboson backgrounds are taken from the data CR used to normalize the
backgrounds. The assigned uncertainties are 6% for $\PW\PZ$, 25% for $\PZ\PZ$
and 8% for $\PZ\gamma$ and $\PW\gamma$ backgrounds. Since other SM processes
that can yield two SS leptons, including triboson, $\ttbar\mathrm{V}$, and
$\PW^{\pm}\PW^{\pm}$, have small background yields in the SR, we assign a
conservative uncertainty of 50%, which includes the uncertainties due to
experimental effects, event simulation, and theoretical calculations of the
cross sections. The overall systematic uncertainty in the prompt-lepton
background, including the contributions discussed below, is 12–18% for the
low-mass selection and 16–43% for the high-mass selection, depending on the
lepton channel. To evaluate the uncertainty due to imperfect knowledge of the
integrated luminosity [79], jet energy/mass scale, jet energy/mass resolution
[66], tagging [73], lepton trigger and selection efficiency, as well as the
uncertainty in the total inelastic cross section used in the pileup
reweighting procedure in simulation, the input value of each parameter is
changed by $\pm 1$ standard deviation from its central value. Energy not
clustered in the detector affects the overall scale, resulting in an
uncertainty in the event yield due to the upper threshold on . The theory
uncertainties in the acceptance of the signal events are determined by varying
the renormalization and factorization scales up and down by a factor of two
relative to their nominal values, and following the PDF4LHC recommendations
[80] to estimate the uncertainty due to the choice of the PDF set. The
uncertainty related to the PDF choice in the background estimates was
evaluated, and an upper limit on the uncertainty was added to Table 0.7,
although this uncertainty was not applied explicitly in the results but
considered to be accounted for via the normalization uncertainty taken from
the normalization control regions.
## 0.8 Results and discussion
The data yields and background estimates after the application of the low- and
high-mass SR selections are shown in Table 0.8. The predicted backgrounds
contributed by events with prompt SS leptons, leptons with mismeasured sign,
and misidentified leptons are shown along with the total background estimate
and the number of events observed in data. The uncertainties shown are the
statistical and systematic components, respectively. The data yields are in
good agreement with the estimated backgrounds. Kinematic distributions also
show good agreement between data and SM expectations. Figures 2–3 show for
illustration: the invariant mass of the two leptons (of the leading lepton and
the selected jets); the invariant mass of the trailing lepton and the selected
jets; and the invariant mass of the two leptons and the selected jets for low-
(high-)mass SRs. In Fig. 2, the $m(\ell^{\pm}\ell^{\pm}\mathrm{jj})$ signal
distribution peaks somewhat below $m_{\mathrm{\PW}}$, because of the selection
requirements imposed.
Observed event yields and estimated backgrounds for the signal region
selections. The background predictions from prompt SS leptons, misidentified
leptons, leptons with mismeasured sign, and the total background are shown
together with the number of events observed in data. The uncertainties shown
are the statistical and systematic components, respectively. A dash indicates
that the background is considered negligible. SR Prompt-lepton Misidentified-
lepton Mismeasured-sign Total bkgd. $\mathrm{N}_{\text{obs}}$ $\Pe\Pe$ channel
Low-mass SR1 $206\pm 10\pm 21\phantom{0}$ $128\pm 5\pm 38\phantom{0}$ $29.8\pm
0.2\pm 12.3$ $364\pm 11\pm 45\phantom{0}$ 324 Low-mass SR2 $281\pm 12\pm
28\phantom{0}$ $143\pm 7\pm 43\phantom{0}$ $36.4\pm 0.2\pm 10.7$ $461\pm 14\pm
53\phantom{0}$ 460 High-mass SR1 $236\pm 10\pm 25\phantom{0}$ $141\pm 6\pm
42\phantom{0}$ $45.2\pm 0.3\pm 24.0$ $422\pm 12\pm 55\phantom{0}$ 382 High-
mass SR2 $8.0\pm 1.3\pm 1.6$ $2.0\pm 0.6\pm 0.6$ $0.91\pm 0.05\pm 0.80$
$10.9\pm 1.5\pm 1.9\phantom{0}$ 10 $\mu\mu$ channel Low-mass SR1 $151\pm 6\pm
16\phantom{0}$ $276\pm 7\pm 83\phantom{0}$ — $426\pm 9\pm 84\phantom{0}$ 487
Low-mass SR2 $209\pm 8\pm 19\phantom{0}$ $393\pm 9\pm 118$ — $602\pm 12\pm
120$ 663 High-mass SR1 $166\pm 6\pm 20\phantom{0}$ $244\pm 6\pm 73\phantom{0}$
— $410\pm 9\pm 76\phantom{0}$ 502 High-mass SR2 $7.1\pm 0.8\pm 1.9$ $4.4\pm
0.8\pm 1.3$ — $11.5\pm 1.1\pm 2.3\phantom{0}$ 13 $\Pe\mu$ channel Low-mass SR1
$418\pm 13\pm 37\phantom{0}$ $432\pm 10\pm 130$ — $850\pm 17\pm 135$ 907 Low-
mass SR2 $566\pm 17\pm 47\phantom{0}$ $464\pm 12\pm 139$ — $1031\pm 21\pm
147\phantom{0}$ 1042 High-mass SR1 $463\pm 14\pm 42\phantom{0}$ $409\pm 10\pm
123$ — $871\pm 17\pm 129$ 901 High-mass SR2 $16.8\pm 1.9\pm 3.6\phantom{0}$
$7.4\pm 1.3\pm 2.2$ — $24.2\pm 2.3\pm 4.2\phantom{0}$ 31
Figure 2: Observed distributions of the invariant mass of the two leptons
(upper), invariant mass of the subleading lepton and jets (middle), and the
invariant mass of the reconstructed $\PW$ propagator (lower), compared to the
expected SM background contributions, for the low-mass SR1 (left) and SR2
(right), after combining the events in the $\Pe\Pe$, $\mu\mu$, and $\Pe\mu$
channels. The hatched bands represent the sums in quadrature of the
statistical and systematic uncertainties. The solid and dashed lines show the
kinematic distributions of two possible signal hypothesis. The lower panels
show the ratio between the observed and expected events in each bin, including
the uncertainty bands that represent the statistical (cyan) and total
uncertainties (orange).
Figure 3: Observed distributions of the invariant mass of the leading lepton
and jets (upper), invariant mass of the subleading lepton and jets (middle),
and the invariant mass of the reconstructed $\PW$ propagator (lower), compared
to the expected SM background contributions, for the high-mass SR1 (left) and
SR2 (right), after combining the events in the $\Pe\Pe$, $\mu\mu$, and
$\Pe\mu$ channels. The hatched bands represent the sums in quadrature of the
statistical and systematic uncertainties. The solid and dashed lines show the
kinematic distributions of two possible signal hypothesis. The lower panels
show the ratio between the observed and expected events in each bin, including
the uncertainty bands that represent the statistical (cyan) and total
uncertainties (orange).
The expected signal depends on both $m_{\mathrm{N}}$ and the mixing matrix
elements $\abs{V_{\Pe\mathrm{N}}}^{2}$, $\abs{V_{\mu\mathrm{N}}}^{2}$, or
$\abs{V_{\Pe\mathrm{N}}V^{*}_{\mu\mathrm{N}}}^{2}/(\abs{V_{\Pe\mathrm{N}}}^{2}+\abs{V_{\mu\mathrm{N}}}^{2})$,
and the values are summarized in Table 0.8 for a few mass points. Tables
0.8–0.8 show the optimized selections applied on top of the low- and high-mass
SRs requirements for each mass hypothesis. These tables also present the
observed event counts in data and the expected background for each signal mass
hypothesis. The data are generally consistent with the predicted backgrounds
in all three flavor channels. The largest deviation observed is in the
$\mu\mu$ channel of SR1, at a signal mass of 600, and has a local significance
of 2.3 standard deviations. The corresponding point of SR2 does not show a
matching fluctuation.
Numbers of expected signal events after all the selections are applied. The
matrix element squared is equal to $1\times 10^{-4}$, $1\times 10^{-2}$, and 1
for $m_{\mathrm{N}}=50$, 200, and 1000, respectively. $m_{\mathrm{N}}$
$\Pe\Pe$ channel $\mu\mu$ channel $\Pe\mu$ channel () SR1 SR2 SR1 SR2 SR1 SR2
50 15 21 28 76 14 26 200 5.5 0.74 9.7 1.9 7.0 1.1 1000 0.43 4.0 0.80 7.5 0.57
4.5
Exclusion limits at 95% confidence level (CL) are set on the heavy Majorana
neutrino mixing matrix elements as a function of $m_{\mathrm{N}}$. The limits
are obtained using criterion [81, 82] based on the event yields in Tables
0.8–0.8. Log-normal distributions are used for both the signal and nuisance
parameters. The combined limits from SR1 and SR2, on the absolute values of
the matrix elements $\abs{V_{\Pe\mathrm{N}}}^{2}$,
$\abs{V_{\mu\mathrm{N}}}^{2}$, and
$\abs{V_{\Pe\mathrm{N}}V^{*}_{\mu\mathrm{N}}}^{2}/(\abs{V_{\Pe\mathrm{N}}}^{2}+\abs{V_{\mu\mathrm{N}}}^{2})$
are shown in Figs. 4–5, also as a function of $m_{\mathrm{N}}$. We assume the
systematic uncertainties in SR1 and SR2 to be fully correlated when
calculating these limits. The limits are calculated separately for each of the
three channels. For an $\mathrm{N}$ mass of 40the observed (expected) limits
are $\abs{V_{\Pe\mathrm{N}}}^{2}<9.5\,(8.0)\times 10^{-5}$,
$\abs{V_{\mu\mathrm{N}}}^{2}<2.3\,(1.9)\times 10^{-5}$, and
$\abs{V_{\Pe\mathrm{N}}V^{*}_{\mu\mathrm{N}}}^{2}/(\abs{V_{\Pe\mathrm{N}}}^{2}+\abs{V_{\mu\mathrm{N}}}^{2})<2.7\,(2.7)\times
10^{-5}$, and for an $\mathrm{N}$ mass of 1000the limits are
$\abs{V_{\Pe\mathrm{N}}}^{2}<0.42\,(0.32)$,
$\abs{V_{\mu\mathrm{N}}}^{2}<0.27\,(0.16)$, and
$\abs{V_{\Pe\mathrm{N}}V^{*}_{\mu\mathrm{N}}}^{2}/(\abs{V_{\Pe\mathrm{N}}}^{2}+\abs{V_{\mu\mathrm{N}}}^{2})<0.14\,(0.14)$.
The mass range below $m_{\mathrm{N}}=20\GeV$ is not considered because of the
very low selection efficiency in this region. Furthermore, since the
$\mathrm{N}$ lifetime is inversely proportional to
$m_{\mathrm{N}}^{5}\abs{V_{\ell\mathrm{N}}}^{2}$, for $m_{\mathrm{N}}<20\GeV$
it becomes significant and results in displaced decays. Thus the prompt lepton
requirement is not satisfied. The behavior of the limits around
$m_{\mathrm{N}}=80\GeV$ is caused by the fact that as the mass of the heavy
Majorana neutrino approaches the $\PW$ boson mass, the lepton produced
together with the $\mathrm{N}$ or the lepton from the $\mathrm{N}$ decay has
very low .
The present search at 13extends the previous CMS SS2$\ell$ plus jets searches
at 8 [34, 35] to both higher $\mathrm{N}$ masses as well as lower masses. In
those earlier searches, two AK4 jets were required in the low- and high-mass
SRs, while in the present analysis at $\sqrt{s}=13\TeV$, the search has been
extended in the low-mass SR to include events with exactly one AK4 jet, and in
the high-mass SR to include events with at least one AK8 jet. As seen in Figs.
4–5, the exclusion limits for the mixing matrix elements are extended both for
low and high $\mathrm{N}$ mass, and now cover $\mathrm{N}$ masses from 20 to
1600. In the range previously studied, the present limits significantly
improve over the previous results except in the region from 60–80, where they
are equivalent. The 13data were taken at higher collision rates and thus with
higher trigger thresholds and pileup rates, which impacted the sensitivity of
the search in the low-mass region. This region is covered with high efficiency
by a recent search in trilepton channels [37].
Figs. 4 shows the exclusion limits for $\abs{V_{\Pe\mathrm{N}}}^{2}$ and
$\abs{V_{\mu\mathrm{N}}}^{2}$ overlaid with the 13CMS limits from the
trilepton channel [37]. For low-mass signals the trilepton analysis is more
sensitive, since it has both fewer backgrounds from misidentified leptons and
higher signal efficiency. However for high-mass signals the signal
efficiencies are compatible, and with the inclusion of the signal region using
AK8 jets, and the larger signal cross section in the dilepton channel this
analysis has more stringent limits for masses of $\mathrm{N}$ above 100.
Selection requirements on discriminating variables determined by the
optimization for each Majorana neutrino mass point in the low-mass signal
regions. The last columns show the overall signal acceptance for the DY
channel. The quoted uncertainties include both the statistical and systematic
contributions. $m_{\mathrm{N}}$ $p^{\ell_{1}}_{\mathrm{T}}$ $\pt^{\ell_{2}}$
$m(\ell^{\pm}\ell^{\pm}\PW_{\text{jet}})$ $m(\ell_{1}\PW_{\text{jet}})$
$m(\ell_{2}\PW_{\text{jet}})$ $m(\ell^{\pm}\ell^{\pm})$ Total bkgd.
$\mathrm{N}_{\text{obs}}$ DY $A\epsilon$ () () () () () () () (%) $\Pe\Pe$
channel SR1 20 25–70 60 $<$190 $<$160 $<$160 10–60 $\phantom{}48.9\pm
9.5\phantom{0}$ 45 $0.12\pm 0.02$ 30 25–70 60 $<$190 $<$160 $<$160 10–60
$\phantom{}48.9\pm 9.5\phantom{0}$ 45 $0.13\pm 0.02$ 40 25–70 60 $<$190 $<$160
$<$160 10–60 $\phantom{}48.9\pm 9.5\phantom{0}$ 45 $0.21\pm 0.03$ 50 25–70 60
$<$190 $<$160 $<$160 10–60 $\phantom{}48.9\pm 9.5\phantom{0}$ 45 $0.24\pm
0.03$ 60 25–70 60 $<$190 $<$160 $<$160 10–60 $\phantom{}48.9\pm
9.5\phantom{0}$ 45 $0.18\pm 0.02$ 70 25–70 60 $<$190 $<$160 $<$160 10–75
$\phantom{.0}64\pm 12\phantom{.0}$ 58 $0.10\pm 0.01$ 75 25–70 60 $<$190 $<$160
$<$160 10–100 $\phantom{.0}68\pm 12\phantom{.0}$ 67 $0.13\pm 0.02$ $\Pe\Pe$
channel SR2 20 25–70 60 $<$100 $<$70 $<$70 10–60 $\phantom{}50.3\pm
8.5\phantom{0}$ 55 $0.26\pm 0.03$ 30 25–70 60 $<$100 $<$70 $<$70 10–60
$\phantom{}50.3\pm 8.5\phantom{0}$ 55 $0.30\pm 0.04$ 40 25–70 60 $<$100 $<$70
$<$70 10–60 $\phantom{}50.3\pm 8.5\phantom{0}$ 55 $0.35\pm 0.04$ 50 25–70 60
$<$100 $<$70 $<$70 10–60 $\phantom{}50.3\pm 8.5\phantom{0}$ 55 $0.32\pm 0.03$
60 25–70 60 $<$100 $<$70 $<$70 10–60 $\phantom{}50.3\pm 8.5\phantom{0}$ 55
$0.24\pm 0.03$ 70 25–70 60 $<$100 $<$70 $<$70 10–75 $\phantom{.0}65\pm
10\phantom{.0}$ 70 $0.06\pm 0.01$ 75 25–70 60 $<$100 $<$70 $<$70 10–80
$\phantom{.0}67\pm 10\phantom{.0}$ 70 $0.11\pm 0.02$ $\mu\mu$ channel SR1 20
20–80 15–50 $<$160 $<$150 $<$150 20–60 $\phantom{}15.3\pm 3.4\phantom{0}$ 18
$0.10\pm 0.02$ 30 20–80 15–50 $<$160 $<$150 $<$150 20–60 $\phantom{}15.3\pm
3.4\phantom{0}$ 18 $0.18\pm 0.03$ 40 20–80 15–50 $<$160 $<$150 $<$150 20–60
$\phantom{}15.3\pm 3.4\phantom{0}$ 18 $0.34\pm 0.05$ 50 20–80 15–50 $<$160
$<$150 $<$150 20–60 $\phantom{}15.3\pm 3.4\phantom{0}$ 18 $0.40\pm 0.04$ 60
20–80 15–50 $<$160 $<$150 $<$150 20–60 $\phantom{}15.3\pm 3.4\phantom{0}$ 18
$0.33\pm 0.04$ 70 20–80 15–50 $<$160 $<$150 $<$150 10–75 $\phantom{}20.3\pm
4.4\phantom{0}$ 21 $0.17\pm 0.02$ 75 20–80 15–50 $<$160 $<$150 $<$150 20–100
$\phantom{}18.9\pm 4.0\phantom{0}$ 19 $0.19\pm 0.03$ $\mu\mu$ channel SR2 20
20–80 15–50 $<$100 $<$70 $<$70 20–60 $\phantom{}25.9\pm 5.9\phantom{0}$ 29
$0.28\pm 0.03$ 30 20–80 15–50 $<$100 $<$70 $<$70 20–60 $\phantom{}25.9\pm
5.9\phantom{0}$ 29 $0.51\pm 0.05$ 40 20–80 15–50 $<$100 $<$70 $<$70 20–60
$\phantom{}25.9\pm 5.9\phantom{0}$ 29 $0.8\pm 0.1$ 50 20–80 15–50 $<$100 $<$70
$<$70 20–60 $\phantom{}25.9\pm 5.9\phantom{0}$ 29 $1.1\pm 0.1$ 60 20–80 15–50
$<$100 $<$70 $<$70 20–60 $\phantom{}25.9\pm 5.9\phantom{0}$ 29 $0.73\pm 0.07$
70 20–80 15–50 $<$100 $<$70 $<$70 10–75 $\phantom{}37.5\pm 7.1\phantom{0}$ 41
$0.20\pm 0.03$ 75 20–80 15–50 $<$100 $<$70 $<$70 20–80 $\phantom{}29.7\pm
6.7\phantom{0}$ 34 $0.24\pm 0.03$ $\Pe\mu$ channel SR1 20 25–60 15–40 $<$185
$<$135 $<$135 20–60 $\phantom{}34.0\pm 6.4\phantom{0}$ 34 $0.08\pm 0.02$ 30
25–60 15–40 $<$185 $<$135 $<$135 20–60 $\phantom{}34.0\pm 6.4\phantom{0}$ 34
$0.12\pm 0.02$ 40 25–60 15–40 $<$185 $<$135 $<$135 20–60 $\phantom{}34.0\pm
6.4\phantom{0}$ 34 $0.21\pm 0.02$ 50 25–60 15–40 $<$185 $<$135 $<$135 20–60
$\phantom{}34.0\pm 6.4\phantom{0}$ 34 $0.20\pm 0.03$ 60 25–60 15–40 $<$185
$<$135 $<$135 20–60 $\phantom{}34.0\pm 6.4\phantom{0}$ 34 $0.17\pm 0.02$ 70
25–60 15–40 $<$185 $<$135 $<$135 10–75 $\phantom{.0}51\pm 10\phantom{.0}$ 49
$0.09\pm 0.01$ 75 25–60 15–40 $<$185 $<$135 $<$135 20–100 $\phantom{}46.5\pm
8.7\phantom{0}$ 49 $0.17\pm 0.03$ $\Pe\mu$ channel SR2 20 25–60 15–40 $<$100
$<$65 $<$65 20–60 $\phantom{}51.7\pm 9.2\phantom{0}$ 50 $0.21\pm 0.02$ 30
25–60 15–40 $<$100 $<$65 $<$65 20–60 $\phantom{}51.7\pm 9.2\phantom{0}$ 50
$0.27\pm 0.03$ 40 25–60 15–40 $<$100 $<$65 $<$65 20–60 $\phantom{}51.7\pm
9.2\phantom{0}$ 50 $0.45\pm 0.04$ 50 25–60 15–40 $<$100 $<$65 $<$65 20–60
$\phantom{}51.7\pm 9.2\phantom{0}$ 50 $0.40\pm 0.03$ 60 25–60 15–40 $<$100
$<$65 $<$65 20–60 $\phantom{}51.7\pm 9.2\phantom{0}$ 50 $0.24\pm 0.03$ 70
25–60 15–40 $<$100 $<$65 $<$65 10–75 $\phantom{}75.8\pm 12.4\phantom{}$ 65
$0.09\pm 0.01$ 75 25–60 15–40 $<$100 $<$65 $<$65 20–80 $\phantom{}62.8\pm
10.9\phantom{}$ 57 $0.12\pm 0.03$
Selection requirements on discriminating variables determined by the
optimization for each Majorana neutrino mass point in the $\Pe\Pe$ channel
high-mass SRs. The last column shows the overall signal acceptance for the DY
and VBF channels. The quoted uncertainties include both the statistical and
systematic contributions. The dash indicates that no selection requirement is
made. $m_{\mathrm{N}}$ $p^{\ell_{1}}_{\mathrm{T}}$ $\pt^{\ell_{2}}$
$m(\ell^{\pm}\ell^{\pm}\PW_{\text{jet}})$ $m(\ell\PW_{\text{jet}})$
$(\ptmiss)^{2}/S_{\mathrm{T}}$ Total bkgd. $\mathrm{N}_{\text{obs}}$ DY
$A\epsilon$ VBF $A\epsilon$ () () () () () () (%) (%) $\Pe\Pe$ channel SR1 85
$>$25 $>$15 $>$110 45–95 $<$6 $\phantom{0}9.5\pm 2.8\phantom{0}$ 9
$\phantom{0.}0.11\pm 0.02\phantom{.}$ — 90 $>$25 $>$15 $>$110 50–100 $<$6
$\phantom{}12.5\pm 3.5\phantom{0}$ 10 $\phantom{0.}0.23\pm 0.05\phantom{.}$ —
100 $>$25 $>$15 $>$120 50–110 $<$6 $\phantom{}20.3\pm 5.0\phantom{0}$ 15
$\phantom{0.0}1.1\pm 0.1\phantom{.0}$ — 125 $>$30 $>$25 $>$120 90–140 $<$6
$\phantom{}17.7\pm 4.5\phantom{0}$ 17 $\phantom{0.0}2.6\pm 0.2\phantom{.0}$ —
150 $>$40 $>$25 $>$180 130–160 $<$6 $\phantom{}14.7\pm 3.8\phantom{0}$ 9
$\phantom{0.0}3.1\pm 0.2\phantom{.0}$ — 200 $>$55 $>$40 $>$220 160–225 $<$6
$\phantom{}12.4\pm 2.7\phantom{0}$ 10 $\phantom{0.0}4.9\pm 0.4\phantom{.0}$ —
250 $>$70 $>$60 $>$310 220–270 $<$6 $\phantom{0}6.0\pm 1.7\phantom{0}$ 4
$\phantom{0.0}5.9\pm 0.4\phantom{.0}$ — 300 $>$80 $>$60 $>$370 235–335 $<$6
$\phantom{0}8.2\pm 2.1\phantom{0}$ 6 $\phantom{0.0}7.6\pm 0.5\phantom{.0}$
$\phantom{.0}3.0\pm 0.3\phantom{.0}$ 400 $>$100 $>$65 $>$450 335–450 $<$6
$\phantom{0}2.5\pm 1.4\phantom{0}$ 4 $\phantom{0.0}6.6\pm 0.5\phantom{.0}$
$\phantom{.0}3.0\pm 0.2\phantom{.0}$ 500 $>$125 $>$65 $>$560 400–555 $<$6
$\phantom{0}1.5\pm 0.8\phantom{0}$ 5 $\phantom{0.0}5.5\pm 0.4\phantom{.0}$
$\phantom{.0}2.7\pm 0.2\phantom{.0}$ 600 $>$125 — $>$760 400–690 $<$6
$\phantom{0}0.9\pm 0.6\phantom{0}$ 1 $\phantom{0.0}3.8\pm 0.3\phantom{.0}$
$\phantom{.0}1.7\pm 0.2\phantom{.0}$ 700 $>$125 — $>$760 400–955 $<$6
$\phantom{0}1.7\pm 0.7\phantom{0}$ 1 $\phantom{0.0}4.0\pm 0.3\phantom{.0}$
$\phantom{.0}2.8\pm 0.2\phantom{.0}$ 800 $>$125 — $>$760 400–1130 $<$6
$\phantom{0}1.7\pm 0.7\phantom{0}$ 1 $\phantom{0.0}3.6\pm 0.3\phantom{.0}$
$\phantom{.0}3.0\pm 0.3\phantom{.0}$ 900 $>$125 — $>$760 400–1300 $<$6
$\phantom{0}1.7\pm 0.7\phantom{0}$ 1 $\phantom{0.0}3.2\pm 0.2\phantom{.0}$
$\phantom{.0}2.9\pm 0.2\phantom{.0}$ 1000 $>$125 — $>$760 400–1490 $<$6
$\phantom{0}1.7\pm 0.7\phantom{0}$ 1 $\phantom{0.0}2.6\pm 0.2\phantom{.0}$
$\phantom{.0}2.4\pm 0.2\phantom{.0}$ 1100 $>$125 — $>$760 400–1490 $<$6
$\phantom{0}1.7\pm 0.7\phantom{0}$ 1 $\phantom{0.0}2.2\pm 0.2\phantom{.0}$
$\phantom{.0}2.0\pm 0.2\phantom{.0}$ 1200 $>$125 — $>$760 400–1600 $<$6
$\phantom{0}1.7\pm 0.7\phantom{0}$ 1 $\phantom{0.0}2.0\pm 0.2\phantom{.0}$
$\phantom{.0}1.8\pm 0.2\phantom{.0}$ 1300 $>$125 — $>$760 400–1930 $<$6
$\phantom{0}1.7\pm 0.7\phantom{0}$ 1 $\phantom{0.0}1.8\pm 0.1\phantom{.0}$
$\phantom{.0}1.6\pm 0.2\phantom{.0}$ 1400 $>$125 — $>$760 400–1930 $<$6
$\phantom{0}1.7\pm 0.7\phantom{0}$ 1 $\phantom{0.0}1.5\pm 0.1\phantom{.0}$
$\phantom{.0}1.3\pm 0.1\phantom{.0}$ 1500 $>$125 — $>$760 400–1930 $<$6
$\phantom{0}1.7\pm 0.7\phantom{0}$ 1 $\phantom{0.0}1.3\pm 0.1\phantom{.0}$
$\phantom{.0}1.2\pm 0.2\phantom{.0}$ $\Pe\Pe$ channel SR2 85 $>$25 $>$15 — —
$<$15 $\phantom{}10.9\pm 2.9\phantom{0}$ 10 $\phantom{0.}0.001\pm
0.001\phantom{.}$ — 90 $>$25 $>$15 — 90–220 $<$15 $\phantom{0}3.4\pm
1.0\phantom{0}$ 2 $\phantom{0.}0.003\pm 0.002\phantom{.}$ — 100 $>$25 $>$15 —
100–220 $<$15 $\phantom{0}3.4\pm 1.0\phantom{0}$ 2 $\phantom{0.}0.005\pm
0.003\phantom{.}$ — 125 $>$60 $>$15 — 123–145 $<$15 $\phantom{0}0.2\pm
0.1\phantom{0}$ 0 $\phantom{0.}0.04\pm 0.01\phantom{.}$ — 150 $>$90 $>$15 —
125–185 $<$15 $\phantom{0}1.3\pm 0.5\phantom{0}$ 0 $\phantom{0.}0.19\pm
0.03\phantom{.}$ — 200 $>$100 $>$20 — 173–220 $<$15 $\phantom{0}0.8\pm
0.3\phantom{0}$ 1 $\phantom{0.}0.60\pm 0.07\phantom{.}$ — 250 $>$100 $>$25 —
220–305 $<$15 $\phantom{0}2.1\pm 1.2\phantom{0}$ 3 $\phantom{0.0}2.2\pm
0.2\phantom{.0}$ — 300 $>$100 $>$30 — 270–330 $<$15 $\phantom{0}1.3\pm
0.6\phantom{0}$ 1 $\phantom{0.0}3.5\pm 0.4\phantom{.0}$ $\phantom{0.0}0.6\pm
0.1\phantom{.0}$ 400 $>$100 $>$35 — 330–440 $<$15 $\phantom{0}3.1\pm
1.3\phantom{0}$ 3 $\phantom{0.0}9.1\pm 0.9\phantom{.0}$ $\phantom{0.0}2.9\pm
0.3\phantom{.0}$ 500 $>$120 $>$35 — 440–565 $<$15 $\phantom{0}2.8\pm
1.0\phantom{0}$ 1 $\phantom{.0}14.3\pm 1.4\phantom{.0}$ $\phantom{0.0}6.1\pm
0.6\phantom{.0}$ 600 $>$120 — — 565–675 $<$15 $\phantom{0}0.8\pm
0.3\phantom{0}$ 1 $\phantom{.0}17.4\pm 1.8\phantom{.0}$ $\phantom{.0}11.0\pm
1.0\phantom{.0}$ 700 $>$140 — — 635–775 $<$15 $\phantom{0}0.8\pm
0.3\phantom{0}$ 2 $\phantom{.0}19.4\pm 2.0\phantom{.0}$ $\phantom{.0}13.1\pm
1.3\phantom{.0}$ 800 $>$140 — — 740–1005 $<$15 $\phantom{0}0.9\pm
0.4\phantom{0}$ 0 $\phantom{.0}20.8\pm 2.1\phantom{.0}$ $\phantom{.0}14.0\pm
1.3\phantom{.0}$ 900 $>$140 — — 865–1030 $<$15 $\phantom{0}0.2\pm
0.1\phantom{0}$ 0 $\phantom{.0}19.2\pm 2.0\phantom{.0}$ $\phantom{.0}13.2\pm
1.3\phantom{.0}$ 1000 $>$140 — — 890–1185 $<$15 $\phantom{0}0.3\pm
0.1\phantom{0}$ 1 $\phantom{.0}21.5\pm 2.2\phantom{.0}$ $\phantom{.0}15.3\pm
1.5\phantom{.0}$ 1100 $>$140 — — 1035–1395 $<$15 $\phantom{0}0.1\pm
0.1\phantom{0}$ 1 $\phantom{.0}20.3\pm 2.1\phantom{.0}$ $\phantom{.0}14.7\pm
1.4\phantom{.0}$ 1200 $>$140 — — 1085–1460 $<$15 $\phantom{0}0.1\pm
0.0\phantom{0}$ 1 $\phantom{.0}20.8\pm 2.2\phantom{.0}$ $\phantom{.0}15.3\pm
1.5\phantom{.0}$ 1300 $>$140 — — 1140–1590 $<$15 $\phantom{0}0.1\pm
0.0\phantom{0}$ 1 $\phantom{.0}20.5\pm 2.2\phantom{.0}$ $\phantom{.0}15.5\pm
1.6\phantom{.0}$ 1400 $>$140 — — 1245–1700 $<$15 $\phantom{0}0.1\pm
0.0\phantom{0}$ 0 $\phantom{.0}19.6\pm 2.1\phantom{.0}$ $\phantom{.0}15.1\pm
1.6\phantom{.0}$ 1500 $>$140 — — 1300–1800 $<$15 $\phantom{0}0.04\pm
0.02\phantom{0}$ 0 $\phantom{.0}19.5\pm 2.1\phantom{.0}$ $\phantom{.0}15.2\pm
1.6\phantom{.0}$
Selection requirements on discriminating variables determined by the
optimization for each Majorana neutrino mass point in the $\mu\mu$ channel
high-mass SRs. The last column shows the overall signal acceptance for the DY
and VBF channels. The quoted uncertainties include both the statistical and
systematic contributions. The dash indicates that no selection requirement is
made. $m_{\mathrm{N}}$ $p^{\ell_{1}}_{\mathrm{T}}$ $\pt^{\ell_{2}}$
$m(\ell^{\pm}\ell^{\pm}\PW_{\text{jet}})$ $m(\ell\PW_{\text{jet}})$
$(\ptmiss)^{2}/S_{\mathrm{T}}$ Total bkgd. $\mathrm{N}_{\text{obs}}$ DY
$A\epsilon$ VBF $A\epsilon$ () () () () () () (%) (%) $\mu\mu$ channel SR1 85
$>$25 $>$10 $>$90 40–100 $<$9 $\phantom{}26.0\pm 6.3\phantom{0}$ 30
$\phantom{0.}0.50\pm 0.05\phantom{.}$ — 90 $>$25 $>$10 $>$90 45–105 $<$9
$\phantom{}34.5\pm 7.5\phantom{0}$ 35 $\phantom{0.0}1.2\pm 0.1\phantom{.0}$ —
100 $>$25 $>$15 $>$110 55–115 $<$9 $\phantom{}18.6\pm 4.2\phantom{0}$ 20
$\phantom{0.0}2.6\pm 0.2\phantom{.0}$ — 125 $>$25 $>$25 $>$140 85–140 $<$7
$\phantom{}11.7\pm 2.7\phantom{0}$ 12 $\phantom{0.0}5.1\pm 0.4\phantom{.0}$ —
150 $>$35 $>$35 $>$150 110–170 $<$7 $\phantom{0}8.9\pm 1.9\phantom{0}$ 11
$\phantom{0.0}6.6\pm 0.5\phantom{.0}$ — 200 $>$50 $>$40 $>$250 160–215 $<$7
$\phantom{0}4.6\pm 1.2\phantom{0}$ 4 $\phantom{0.0}8.1\pm 0.6\phantom{.0}$ —
250 $>$85 $>$45 $>$310 215–270 $<$7 $\phantom{0}3.0\pm 0.9\phantom{0}$ 2
$\phantom{.0}11.0\pm 0.8\phantom{.0}$ — 300 $>$100 $>$50 $>$370 225–340 $<$7
$\phantom{0}2.6\pm 1.0\phantom{0}$ 2 $\phantom{.0}13.2\pm 0.9\phantom{.0}$
$\phantom{.0}5.2\pm 0.4\phantom{.0}$ 400 $>$110 $>$60 $>$490 295–490 $<$7
$\phantom{0}0.9\pm 0.4\phantom{0}$ 3 $\phantom{.0}11.7\pm 0.8\phantom{.0}$
$\phantom{.0}5.1\pm 0.4\phantom{.0}$ 500 $>$110 $>$60 $>$610 370–550 $<$7
$\phantom{0}0.4^{\phantom{0}+\phantom{0}0.6\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.4\phantom{0}\phantom{0}}$
3 $\phantom{0.0}8.6\pm 0.6\phantom{.0}$ $\phantom{.0}4.1\pm 0.3\phantom{.0}$
600 $>$110 — $>$680 370–630 $<$7
$\phantom{0}0.3^{\phantom{0}+\phantom{0}0.3\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.3\phantom{0}\phantom{0}}$
3 $\phantom{0.0}7.4\pm 0.5\phantom{.0}$ $\phantom{.0}4.1\pm 0.3\phantom{.0}$
700 $>$110 — $>$800 370–885 $<$7
$\phantom{0}0.2^{\phantom{0}+\phantom{0}0.4\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.2\phantom{0}\phantom{0}}$
2 $\phantom{0.0}6.7\pm 0.4\phantom{.0}$ $\phantom{.0}3.9\pm 0.3\phantom{.0}$
800 $>$110 — $>$800 370–890 $<$7
$\phantom{0}0.2^{\phantom{0}+\phantom{0}0.4\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.2\phantom{0}\phantom{0}}$
2 $\phantom{0.0}6.0\pm 0.4\phantom{.0}$ $\phantom{.0}5.4\pm 0.3\phantom{.0}$
900 $>$110 — $>$800 370–1225 $<$7
$\phantom{0}0.3^{\phantom{0}+\phantom{0}0.4\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.3\phantom{0}\phantom{0}}$
2 $\phantom{0.0}5.4\pm 0.4\phantom{.0}$ $\phantom{.0}5.0\pm 0.3\phantom{.0}$
1000 $>$110 — $>$800 370–1230 $<$7
$\phantom{0}0.3^{\phantom{0}+\phantom{0}0.4\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.3\phantom{0}\phantom{0}}$
2 $\phantom{0.0}4.6\pm 0.3\phantom{.0}$ $\phantom{.0}4.2\pm 0.3\phantom{.0}$
1100 $>$110 — $>$800 370–1245 $<$7
$\phantom{0}0.3^{\phantom{0}+\phantom{0}0.4\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.3\phantom{0}\phantom{0}}$
2 $\phantom{0.0}4.1\pm 0.3\phantom{.0}$ $\phantom{.0}3.8\pm 0.3\phantom{.0}$
1200 $>$110 — $>$800 370–1690 $<$7
$\phantom{0}0.3^{\phantom{0}+\phantom{0}0.4\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.3\phantom{0}\phantom{0}}$
2 $\phantom{0.0}3.6\pm 0.2\phantom{.0}$ $\phantom{.0}3.4\pm 0.3\phantom{.0}$
1300 $>$110 — $>$800 370–1890 $<$7
$\phantom{0}0.3^{\phantom{0}+\phantom{0}0.4\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.3\phantom{0}\phantom{0}}$
2 $\phantom{0.0}3.2\pm 0.2\phantom{.0}$ $\phantom{.0}3.0\pm 0.2\phantom{.0}$
1400 $>$110 — $>$800 370–1940 $<$7
$\phantom{0}0.3^{\phantom{0}+\phantom{0}0.4\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.3\phantom{0}\phantom{0}}$
2 $\phantom{0.0}2.7\pm 0.2\phantom{.0}$ $\phantom{.0}2.7\pm 0.2\phantom{.0}$
1500 $>$110 — $>$800 370–2220 $<$7
$\phantom{0}0.3^{\phantom{0}+\phantom{0}0.4\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.3\phantom{0}\phantom{0}}$
2 $\phantom{0.0}2.5\pm 0.2\phantom{.0}$ $\phantom{.0}2.3\pm 0.2\phantom{.0}$
$\mu\mu$ channel SR2 85 $>$25 $>$10 — — $<$15 $\phantom{}11.4\pm
3.5\phantom{0}$ 13 $\phantom{0.}0.001\pm 0.001\phantom{.}$ — 90 $>$25 $>$10 —
90–170 $<$15 $\phantom{0}4.1\pm 1.3\phantom{0}$ 4 $\phantom{0.}0.003\pm
0.003\phantom{.}$ — 100 $>$25 $>$15 — 98–145 $<$15 $\phantom{0}1.0\pm
0.3\phantom{0}$ 0 $\phantom{0.}0.006\pm 0.003\phantom{.}$ — 125 $>$60 $>$15 —
110–150 $<$15 $\phantom{0}0.8\pm 0.3\phantom{0}$ 0 $\phantom{0.}0.08\pm
0.01\phantom{.}$ — 150 $>$70 $>$15 — 145–175 $<$15 $\phantom{0}1.0\pm
0.4\phantom{0}$ 2 $\phantom{0.}0.28\pm 0.04\phantom{.}$ — 200 $>$100 $>$20 —
175–235 $<$15 $\phantom{0}1.3\pm 0.8\phantom{0}$ 0 $\phantom{0.0}1.4\pm
0.1\phantom{.0}$ — 250 $>$140 $>$25 — 226–280 $<$15 $\phantom{0}0.3\pm
0.2\phantom{0}$ 0 $\phantom{0.0}3.0\pm 0.3\phantom{.0}$ — 300 $>$140 $>$40 —
280–340 $<$15 $\phantom{0}0.4\pm 0.3\phantom{0}$ 0 $\phantom{0.0}5.4\pm
0.5\phantom{.0}$ $\phantom{0.0}0.7\pm 0.1\phantom{.0}$ 400 $>$140 $>$65 —
340–445 $<$15 $\phantom{0}0.5\pm 0.3\phantom{0}$ 2 $\phantom{.0}13.3\pm
1.3\phantom{.0}$ $\phantom{0.0}2.7\pm 0.3\phantom{.0}$ 500 $>$140 $>$65 —
445–560 $<$15 $\phantom{0}0.8\pm 0.5\phantom{0}$ 0 $\phantom{.0}22.4\pm
2.2\phantom{.0}$ $\phantom{0.0}6.8\pm 0.7\phantom{.0}$ 600 $>$140 — — 560–685
$<$15 $\phantom{0}0.7\pm 0.4\phantom{0}$ 0 $\phantom{.0}30.2\pm
2.9\phantom{.0}$ $\phantom{.0}20.4\pm 1.8\phantom{.0}$ 700 $>$140 — — 635–825
$<$15 $\phantom{0}0.8\pm 0.4\phantom{0}$ 2 $\phantom{.0}34.6\pm
3.4\phantom{.0}$ $\phantom{.0}24.7\pm 2.2\phantom{.0}$ 800 $>$140 — — 755–960
$<$15 $\phantom{0}0.4\pm 0.3\phantom{0}$ 0 $\phantom{.0}34.8\pm
3.5\phantom{.0}$ $\phantom{.0}24.9\pm 2.3\phantom{.0}$ 900 $>$140 — — 840–1055
$<$15
$\phantom{0}0.2^{\phantom{0}+\phantom{0}0.2\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.2\phantom{0}\phantom{0}}$
1 $\phantom{.0}35.8\pm 3.6\phantom{.0}$ $\phantom{.0}26.9\pm 2.5\phantom{.0}$
1000 $>$140 — — 900–1205 $<$15
$\phantom{0}0.1^{\phantom{0}+\phantom{0}0.2\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.1\phantom{0}\phantom{0}}$
1 $\phantom{.0}38.4\pm 3.9\phantom{.0}$ $\phantom{.0}28.9\pm 2.7\phantom{.0}$
1100 $>$140 — — 990–1250 $<$15
$\phantom{0}0.1^{\phantom{0}+\phantom{0}0.2\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.1\phantom{0}\phantom{0}}$
1 $\phantom{.0}36.7\pm 3.7\phantom{.0}$ $\phantom{.0}29.2\pm 2.7\phantom{.0}$
1200 $>$140 — — 1035–1430 $<$15
$\phantom{0}0.2^{\phantom{0}+\phantom{0}0.3\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.2\phantom{0}\phantom{0}}$
1 $\phantom{.0}38.5\pm 4.0\phantom{.0}$ $\phantom{.0}30.1\pm 2.8\phantom{.0}$
1300 $>$140 — — 1100–1595 $<$15 $\phantom{0}0.3\pm 0.3\phantom{0}$ 1
$\phantom{.0}38.5\pm 4.0\phantom{.0}$ $\phantom{.0}30.7\pm 3.0\phantom{.0}$
1400 $>$140 — — 1285–1700 $<$15
$\phantom{0}0.1^{\phantom{0}+\phantom{0}0.2\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.1\phantom{0}\phantom{0}}$
1 $\phantom{.0}35.9\pm 3.8\phantom{.0}$ $\phantom{.0}29.4\pm 2.8\phantom{.0}$
1500 $>$140 — — 1330–1800 $<$15
$\phantom{0}0.1^{\phantom{0}+\phantom{0}0.2\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.1\phantom{0}\phantom{0}}$
1 $\phantom{.0}36.4\pm 3.9\phantom{.0}$ $\phantom{.0}30.0\pm 2.9\phantom{.0}$
Selection requirements on discriminating variables determined by the
optimization for each Majorana neutrino mass point in the $\Pe\mu$ channel
high-mass SRs. The last column shows the overall signal acceptance for the DY
and VBF channels. The quoted uncertainties include both the statistical and
systematic contributions. The dash indicates that no selection requirement is
made. $m_{\mathrm{N}}$ $p^{\ell_{1}}_{\mathrm{T}}$ $\pt^{\ell_{2}}$
$m(\ell^{\pm}\ell^{\pm}\PW_{\text{jet}})$ $m(\ell\PW_{\text{jet}})$
$(\ptmiss)^{2}/S_{\mathrm{T}}$ Total bkgd. $\mathrm{N}_{\text{obs}}$ DY
$A\epsilon$ VBF $A\epsilon$ () () () () () () (%) (%) $\Pe\mu$ channel SR1 85
$>$30 $>$10 $>$120 55–95 $<$7 $\phantom{}26.1\pm 6.2\phantom{0}$ 25
$\phantom{0.}0.21\pm 0.03\phantom{.}$ — 90 $>$30 $>$10 $>$120 60–100 $<$7
$\phantom{}37.4\pm 8.4\phantom{0}$ 32 $\phantom{0.}0.59\pm 0.07\phantom{.}$ —
100 $>$25 $>$20 $>$110 60–115 $<$7 $\phantom{}23.6\pm 4.8\phantom{0}$ 21
$\phantom{0.0}1.3\pm 0.1\phantom{.0}$ — 125 $>$30 $>$30 $>$140 90–140 $<$7
$\phantom{}25.5\pm 5.9\phantom{0}$ 16 $\phantom{0.0}3.1\pm 0.2\phantom{.0}$ —
150 $>$45 $>$35 $>$150 100–170 $<$7 $\phantom{}34.1\pm 6.0\phantom{0}$ 26
$\phantom{0.0}5.1\pm 0.3\phantom{.0}$ — 200 $>$65 $>$35 $>$270 170–230 $<$7
$\phantom{}11.1\pm 2.8\phantom{0}$ 14 $\phantom{0.0}6.1\pm 0.4\phantom{.0}$ —
250 $>$75 $>$60 $>$300 200–280 $<$7 $\phantom{}11.1\pm 2.3\phantom{0}$ 9
$\phantom{0.0}8.9\pm 0.5\phantom{.0}$ — 300 $>$95 $>$60 $>$340 255–325 $<$7
$\phantom{0}5.8\pm 1.7\phantom{0}$ 8 $\phantom{0.0}9.0\pm 0.6\phantom{.0}$
$\phantom{.0}3.4\pm 0.3\phantom{.0}$ 400 $>$120 $>$60 $>$530 325–450 $<$7
$\phantom{0}2.2\pm 1.0\phantom{0}$ 7 $\phantom{0.0}7.4\pm 0.4\phantom{.0}$
$\phantom{.0}3.0\pm 0.3\phantom{.0}$ 500 $>$150 $>$60 $>$580 315–530 $<$7
$\phantom{0}1.8\pm 1.1\phantom{0}$ 6 $\phantom{0.0}6.6\pm 0.5\phantom{.0}$
$\phantom{.0}3.0\pm 0.2\phantom{.0}$ 600 $>$175 — $>$670 315–740 $<$7
$\phantom{0}1.2\pm 0.9\phantom{0}$ 4 $\phantom{0.0}5.9\pm 0.4\phantom{.0}$
$\phantom{.0}3.5\pm 0.3\phantom{.0}$ 700 $>$180 — $>$720 350–1030 $<$7
$\phantom{0}1.6\pm 1.1\phantom{0}$ 3 $\phantom{0.0}5.2\pm 0.3\phantom{.0}$
$\phantom{.0}3.8\pm 0.2\phantom{.0}$ 800 $>$180 — $>$720 400–1030 $<$7
$\phantom{0}1.6\pm 1.1\phantom{0}$ 3 $\phantom{0.0}4.5\pm 0.3\phantom{.0}$
$\phantom{.0}3.7\pm 0.2\phantom{.0}$ 900 $>$185 — $>$720 450–1040 $<$7
$\phantom{0}1.0\pm 0.7\phantom{0}$ 2 $\phantom{0.0}3.8\pm 0.2\phantom{.0}$
$\phantom{.0}3.3\pm 0.2\phantom{.0}$ 1000 $>$185 — $>$720 500–1415 $<$7
$\phantom{0}1.0\pm 0.7\phantom{0}$ 2 $\phantom{0.0}3.4\pm 0.2\phantom{.0}$
$\phantom{.0}3.0\pm 0.2\phantom{.0}$ 1100 $>$185 — $>$720 550–1640 $<$7
$\phantom{0}1.0\pm 0.7\phantom{0}$ 1 $\phantom{0.0}2.8\pm 0.2\phantom{.0}$
$\phantom{.0}2.6\pm 0.2\phantom{.0}$ 1200 $>$185 — $>$720 600–1780 $<$7
$\phantom{0}1.0\pm 0.7\phantom{0}$ 1 $\phantom{0.0}2.4\pm 0.2\phantom{.0}$
$\phantom{.0}2.3\pm 0.2\phantom{.0}$ 1300 $>$185 — $>$720 650–1880 $<$7
$\phantom{0}0.8\pm 0.7\phantom{0}$ 1 $\phantom{0.0}2.1\pm 0.1\phantom{.0}$
$\phantom{.0}1.9\pm 0.2\phantom{.0}$ 1400 $>$185 — $>$720 650–1885 $<$7
$\phantom{0}0.8\pm 0.7\phantom{0}$ 1 $\phantom{0.0}1.8\pm 0.1\phantom{.0}$
$\phantom{.0}1.7\pm 0.2\phantom{.0}$ 1500 $>$185 — $>$720 650–1885 $<$7
$\phantom{0}0.8\pm 0.7\phantom{0}$ 1 $\phantom{0.0}1.5\pm 0.1\phantom{.0}$
$\phantom{.0}1.5\pm 0.1\phantom{.0}$ 1700 $>$185 — $>$720 650–2085 $<$7
$\phantom{0}0.8\pm 0.7\phantom{0}$ 1 $\phantom{0.0}1.2\pm 0.1\phantom{.0}$
$\phantom{.0}1.3\pm 0.1\phantom{.0}$ $\Pe\mu$ channel SR2 85 $>$25 $>$10 — —
$<$15 $\phantom{}24.2\pm 6.4\phantom{0}$ 31 $\phantom{0.}0.001\pm
0.002\phantom{.}$ — 90 $>$25 $>$10 — 90–240 $<$15 $\phantom{}13.4\pm
3.7\phantom{0}$ 22 $\phantom{0.}0.003\pm 0.002\phantom{.}$ — 100 $>$30 $>$15 —
100–335 $<$15 $\phantom{}14.1\pm 4.1\phantom{0}$ 21 $\phantom{0.}0.009\pm
0.003\phantom{.}$ — 125 $>$35 $>$25 — 115–150 $<$15 $\phantom{0}0.6\pm
0.4\phantom{0}$ 2 $\phantom{0.}0.03\pm 0.01\phantom{.}$ — 150 $>$45 $>$30 —
132–180 $<$15 $\phantom{0}1.4\pm 0.5\phantom{0}$ 2 $\phantom{0.}0.14\pm
0.02\phantom{.}$ — 200 $>$70 $>$30 — 180–225 $<$15 $\phantom{0}1.5\pm
0.5\phantom{0}$ 3 $\phantom{0.}0.86\pm 0.09\phantom{.}$ — 250 $>$75 $>$55 —
225–280 $<$15 $\phantom{0}1.2\pm 0.4\phantom{0}$ 2 $\phantom{0.0}1.7\pm
0.2\phantom{.0}$ — 300 $>$95 $>$55 — 280–340 $<$15 $\phantom{0}1.2\pm
0.7\phantom{0}$ 1 $\phantom{0.0}4.4\pm 0.4\phantom{.0}$ $\phantom{0.0}0.8\pm
0.1\phantom{.0}$ 400 $>$125 $>$55 — 340–475 $<$15 $\phantom{0}2.0\pm
1.2\phantom{0}$ 1 $\phantom{.0}11.8\pm 1.1\phantom{.0}$ $\phantom{0.0}2.7\pm
0.3\phantom{.0}$ 500 $>$145 $>$60 — 460–555 $<$15 $\phantom{0}0.7\pm
0.3\phantom{0}$ 0 $\phantom{.0}16.7\pm 1.6\phantom{.0}$ $\phantom{0.0}5.2\pm
0.5\phantom{.0}$ 600 $>$160 — — 555–645 $<$15 $\phantom{0}1.4\pm
0.9\phantom{0}$ 1 $\phantom{.0}20.2\pm 1.9\phantom{.0}$ $\phantom{.0}13.2\pm
1.2\phantom{.0}$ 700 $>$170 — — 610–780 $<$15 $\phantom{0}2.0\pm
0.9\phantom{0}$ 2 $\phantom{.0}25.0\pm 2.4\phantom{.0}$ $\phantom{.0}17.6\pm
1.6\phantom{.0}$ 800 $>$170 — — 730–895 $<$15 $\phantom{0}0.8\pm
0.4\phantom{0}$ 2 $\phantom{.0}26.1\pm 2.5\phantom{.0}$ $\phantom{.0}18.3\pm
1.6\phantom{.0}$ 900 $>$180 — — 845–1015 $<$15 $\phantom{0}0.5\pm
0.2\phantom{0}$ 0 $\phantom{.0}25.6\pm 2.5\phantom{.0}$ $\phantom{.0}18.5\pm
1.7\phantom{.0}$ 1000 $>$180 — — 930–1075 $<$15 $\phantom{0}0.2\pm
0.2\phantom{0}$ 0 $\phantom{.0}23.5\pm 2.3\phantom{.0}$ $\phantom{.0}17.6\pm
1.6\phantom{.0}$ 1100 $>$180 — — 1020–1340 $<$15 $\phantom{0}0.3\pm
0.3\phantom{0}$ 0 $\phantom{.0}26.9\pm 2.7\phantom{.0}$ $\phantom{.0}19.6\pm
1.7\phantom{.0}$ 1200 $>$180 — — 1080–1340 $<$15
$\phantom{0}0.1^{\phantom{0}+\phantom{0}0.2\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.1\phantom{0}\phantom{0}}$
0 $\phantom{.0}25.9\pm 2.6\phantom{.0}$ $\phantom{.0}19.9\pm 1.8\phantom{.0}$
1300 $>$180 — — 1155–1595 $<$15
$\phantom{0}0.2^{\phantom{0}+\phantom{0}0.2\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.2\phantom{0}\phantom{0}}$
0 $\phantom{.0}27.1\pm 2.7\phantom{.0}$ $\phantom{.0}20.7\pm 1.9\phantom{.0}$
1400 $>$180 — — 1155–1615 $<$15
$\phantom{0}0.2^{\phantom{0}+\phantom{0}0.3\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.2\phantom{0}\phantom{0}}$
0 $\phantom{.0}26.7\pm 2.7\phantom{.0}$ $\phantom{.0}20.8\pm 2.0\phantom{.0}$
1500 $>$180 — — 1345–1615 $<$15
$\phantom{0}0.0^{\phantom{0}+\phantom{0}0.1\phantom{0}\phantom{0}}_{\phantom{0}-\phantom{0}0.0\phantom{0}\phantom{0}}$
0 $\phantom{.0}21.6\pm 2.2\phantom{.0}$ $\phantom{.0}18.0\pm 1.7\phantom{.0}$
1700 $>$180 — — 1400–1800 $<$15 $\phantom{0}0.7\pm 0.6\phantom{0}$ 0
$\phantom{.0}19.8\pm 2.1\phantom{.0}$ $\phantom{.0}17.0\pm 1.7\phantom{.0}$
Figure 4: Exclusion region at 95% CL in the $\abs{V_{\Pe\mathrm{N}}}^{2}$
(upper) and $\abs{V_{\mu\mathrm{N}}}^{2}$ (lower) vs. $m_{\mathrm{N}}$ plane.
The dashed black curve is the expected upper limit, with one and two standard-
deviation bands shown in green and yellow, respectively. The solid black curve
is the observed upper limit. The brown line shows constraints from EWPD [83].
Also shown are the upper limits from other direct searches: DELPHI [30], L3
[31, 32], ATLAS [36], and the upper limits from the CMS $\sqrt{s}=8\TeV$ 2012
data [35] and the trilepton analysis [37] based on the same 2016 data set as
used in this analysis. Figure 5: Exclusion region at 95% CL in the
$\abs{V_{\Pe\mathrm{N}}V^{*}_{\mu\mathrm{N}}}^{2}/(\abs{V_{\Pe\mathrm{N}}}^{2}+\abs{V_{\mu\mathrm{N}}}^{2})$
vs. $m_{\mathrm{N}}$ plane. The dashed black curve is the expected upper
limit, with one and two standard-deviation bands shown in green and yellow,
respectively. The solid black curve is the observed upper limit. Also shown
are the upper limits from the CMS $\sqrt{s}=8\TeV$ 2012 data [35].
## 0.9 Summary
A search for heavy Majorana neutrinos, $\mathrm{N}$, in final states with
same-sign dileptons and jets has been performed in proton-proton collisions at
a center-of-mass energy of 13, using a data set corresponding to an integrated
luminosity of 35.9. No significant excess of events compared to the expected
standard model background prediction is observed. Upper limits at 95%
confidence level are set on the mixing matrix element between standard model
neutrinos and $\mathrm{N}$ ($\abs{V_{\ell\mathrm{N}}}$) in the context of a
Type-I seesaw model, as a function of $\mathrm{N}$ mass. The analysis improves
on previous 8searches by including single-jet events into the signal region,
which increases sensitivities. For an $\mathrm{N}$ mass of 40the observed
(expected) limits are $\abs{V_{\Pe\mathrm{N}}}^{2}<9.5\,(8.0)\times 10^{-5}$,
$\abs{V_{\mu\mathrm{N}}}^{2}<2.3\,(1.9)\times 10^{-5}$, and
$\abs{V_{\Pe\mathrm{N}}V^{*}_{\mu\mathrm{N}}}^{2}/(\abs{V_{\Pe\mathrm{N}}}^{2}+\abs{V_{\mu\mathrm{N}}}^{2})<2.7\,(2.7)\times
10^{-5}$, and for an $\mathrm{N}$ mass of 1000the limits are
$\abs{V_{\Pe\mathrm{N}}}^{2}<0.42\,(0.32)$,
$\abs{V_{\mu\mathrm{N}}}^{2}<0.27\,(0.16)$, and
$\abs{V_{\Pe\mathrm{N}}V^{*}_{\mu\mathrm{N}}}^{2}/(\abs{V_{\Pe\mathrm{N}}}^{2}+\abs{V_{\mu\mathrm{N}}}^{2})<0.14\,(0.14)$.
The search is sensitive to masses of $\mathrm{N}$ from 20 to 1600. The limits
on the mixing matrix elements are placed up to 1240for
$\abs{V_{\Pe\mathrm{N}}}^{2}$, 1430for the $\abs{V_{\mu\mathrm{N}}}^{2}$, and
1600for
$\abs{V_{\Pe\mathrm{N}}V^{*}_{\mu\mathrm{N}}}^{2}/(\abs{V_{\Pe\mathrm{N}}}^{2}+\abs{V_{\mu\mathrm{N}}}^{2})$.
These are the most restrictive direct limits on the $\mathrm{N}$ mixing
parameters for heavy Majorana neutrino masses greater than 430, and are the
first for masses greater than 1200.
###### Acknowledgements.
We congratulate our colleagues in the CERN accelerator departments for the
excellent performance of the LHC and thank the technical and administrative
staffs at CERN and at other CMS institutes for their contributions to the
success of the CMS effort. In addition, we gratefully acknowledge the
computing centers and personnel of the Worldwide LHC Computing Grid for
delivering so effectively the computing infrastructure essential to our
analyses. Finally, we acknowledge the enduring support for the construction
and operation of the LHC and the CMS detector provided by the following
funding agencies: the Austrian Federal Ministry of Science, Research and
Economy and the Austrian Science Fund; the Belgian Fonds de la Recherche
Scientifique, and Fonds voor Wetenschappelijk Onderzoek; the Brazilian Funding
Agencies (CNPq, CAPES, FAPERJ, FAPERGS, and FAPESP); the Bulgarian Ministry of
Education and Science; CERN; the Chinese Academy of Sciences, Ministry of
Science and Technology, and National Natural Science Foundation of China; the
Colombian Funding Agency (COLCIENCIAS); the Croatian Ministry of Science,
Education and Sport, and the Croatian Science Foundation; the Research
Promotion Foundation, Cyprus; the Secretariat for Higher Education, Science,
Technology and Innovation, Ecuador; the Ministry of Education and Research,
Estonian Research Council via IUT23-4 and IUT23-6 and European Regional
Development Fund, Estonia; the Academy of Finland, Finnish Ministry of
Education and Culture, and Helsinki Institute of Physics; the Institut
National de Physique Nucléaire et de Physique des Particules / CNRS, and
Commissariat à l’Énergie Atomique et aux Énergies Alternatives / CEA, France;
the Bundesministerium für Bildung und Forschung, Deutsche
Forschungsgemeinschaft, and Helmholtz-Gemeinschaft Deutscher
Forschungszentren, Germany; the General Secretariat for Research and
Technology, Greece; the National Research, Development and Innovation Fund,
Hungary; the Department of Atomic Energy and the Department of Science and
Technology, India; the Institute for Studies in Theoretical Physics and
Mathematics, Iran; the Science Foundation, Ireland; the Istituto Nazionale di
Fisica Nucleare, Italy; the Ministry of Science, ICT and Future Planning, and
National Research Foundation (NRF), Republic of Korea; the Lithuanian Academy
of Sciences; the Ministry of Education, and University of Malaya (Malaysia);
the Ministry of Science of Montenegro; the Mexican Funding Agencies (BUAP,
CINVESTAV, CONACYT, LNS, SEP, and UASLP-FAI); the Ministry of Business,
Innovation and Employment, New Zealand; the Pakistan Atomic Energy Commission;
the Ministry of Science and Higher Education and the National Science Center,
Poland; the Fundação para a Ciência e a Tecnologia, Portugal; JINR, Dubna; the
Ministry of Education and Science of the Russian Federation, the Federal
Agency of Atomic Energy of the Russian Federation, Russian Academy of
Sciences, the Russian Foundation for Basic Research, and the National Research
Center “Kurchatov Institute”; the Ministry of Education, Science and
Technological Development of Serbia; the Secretaría de Estado de
Investigación, Desarrollo e Innovación, Programa Consolider-Ingenio 2010, Plan
Estatal de Investigación Científica y Técnica y de Innovación 2013-2016, Plan
de Ciencia, Tecnología e Innovación 2013-2017 del Principado de Asturias, and
Fondo Europeo de Desarrollo Regional, Spain; the Ministry of Science,
Technology and Research, Sri Lanka; the Swiss Funding Agencies (ETH Board, ETH
Zurich, PSI, SNF, UniZH, Canton Zurich, and SER); the Ministry of Science and
Technology, Taipei; the Thailand Center of Excellence in Physics, the
Institute for the Promotion of Teaching Science and Technology of Thailand,
Special Task Force for Activating Research and the National Science and
Technology Development Agency of Thailand; the Scientific and Technical
Research Council of Turkey, and Turkish Atomic Energy Authority; the National
Academy of Sciences of Ukraine, and State Fund for Fundamental Researches,
Ukraine; the Science and Technology Facilities Council, UK; the US Department
of Energy, and the US National Science Foundation. Individuals have received
support from the Marie-Curie program and the European Research Council and
Horizon 2020 Grant, contract No. 675440 (European Union); the Leventis
Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation;
the Belgian Federal Science Policy Office; the Fonds pour la Formation à la
Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium); the
Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the
F.R.S.-FNRS and FWO (Belgium) under the “Excellence of Science - EOS” - be.h
project n. 30820817; the Ministry of Education, Youth and Sports (MEYS) of the
Czech Republic; the Lendület (“Momentum”) Program and the János Bolyai
Research Scholarship of the Hungarian Academy of Sciences, the New National
Excellence Program ÚNKP, the NKFIA research grants 123842, 123959, 124845,
124850 and 125105 (Hungary); the Council of Scientific and Industrial
Research, India; the HOMING PLUS program of the Foundation for Polish Science,
cofinanced from European Union, Regional Development Fund, the Mobility Plus
program of the Ministry of Science and Higher Education, the National Science
Center (Poland), contracts Harmonia 2014/14/M/ST2/00428, Opus
2014/13/B/ST2/02543, 2014/15/B/ST2/03998, and 2015/19/B/ST2/02861, Sonata-bis
2012/07/E/ST2/01406; the National Priorities Research Program by Qatar
National Research Fund; the Programa de Excelencia María de Maeztu, and the
Programa Severo Ochoa del Principado de Asturias; the Thalis and Aristeia
programs cofinanced by EU-ESF, and the Greek NSRF; the Rachadapisek Sompot
Fund for Postdoctoral Fellowship, Chulalongkorn University, and the
Chulalongkorn Academic into Its 2nd Century Project Advancement Project
(Thailand); the Welch Foundation, contract C-1845; and the Weston Havens
Foundation (USA).
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## .10 The CMS Collaboration
Yerevan Physics Institute, Yerevan, Armenia
A.M. Sirunyan, A. Tumasyan Institut für Hochenergiephysik, Wien, Austria
W. Adam, F. Ambrogi, E. Asilar, T. Bergauer, J. Brandstetter, M. Dragicevic,
J. Erö, A. Escalante Del Valle, M. Flechl, R. Frühwirth1, V.M. Ghete, J.
Hrubec, M. Jeitler1, N. Krammer, I. Krätschmer, D. Liko, T. Madlener, I.
Mikulec, N. Rad, H. Rohringer, J. Schieck1, R. Schöfbeck, M. Spanring, D.
Spitzbart, A. Taurok, W. Waltenberger, J. Wittmann, C.-E. Wulz1, M. Zarucki
Institute for Nuclear Problems, Minsk, Belarus
V. Chekhovsky, V. Mossolov, J. Suarez Gonzalez Universiteit Antwerpen,
Antwerpen, Belgium
E.A. De Wolf, D. Di Croce, X. Janssen, J. Lauwers, M. Pieters, M. Van De
Klundert, H. Van Haevermaet, P. Van Mechelen, N. Van Remortel Vrije
Universiteit Brussel, Brussel, Belgium
S. Abu Zeid, F. Blekman, J. D’Hondt, I. De Bruyn, J. De Clercq, K. Deroover,
G. Flouris, D. Lontkovskyi, S. Lowette, I. Marchesini, S. Moortgat, L.
Moreels, Q. Python, K. Skovpen, S. Tavernier, W. Van Doninck, P. Van Mulders,
I. Van Parijs Université Libre de Bruxelles, Bruxelles, Belgium
D. Beghin, B. Bilin, H. Brun, B. Clerbaux, G. De Lentdecker, H. Delannoy, B.
Dorney, G. Fasanella, L. Favart, R. Goldouzian, A. Grebenyuk, A.K. Kalsi, T.
Lenzi, J. Luetic, N. Postiau, E. Starling, L. Thomas, C. Vander Velde, P.
Vanlaer, D. Vannerom, Q. Wang Ghent University, Ghent, Belgium
T. Cornelis, D. Dobur, A. Fagot, M. Gul, I. Khvastunov2, D. Poyraz, C. Roskas,
D. Trocino, M. Tytgat, W. Verbeke, B. Vermassen, M. Vit, N. Zaganidis
Université Catholique de Louvain, Louvain-la-Neuve, Belgium
H. Bakhshiansohi, O. Bondu, S. Brochet, G. Bruno, C. Caputo, P. David, C.
Delaere, M. Delcourt, B. Francois, A. Giammanco, G. Krintiras, V. Lemaitre, A.
Magitteri, A. Mertens, M. Musich, K. Piotrzkowski, A. Saggio, M. Vidal Marono,
S. Wertz, J. Zobec Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro,
Brazil
F.L. Alves, G.A. Alves, L. Brito, M. Correa Martins Junior, G. Correia Silva,
C. Hensel, A. Moraes, M.E. Pol, P. Rebello Teles Universidade do Estado do Rio
de Janeiro, Rio de Janeiro, Brazil
E. Belchior Batista Das Chagas, W. Carvalho, J. Chinellato3, E. Coelho, E.M.
Da Costa, G.G. Da Silveira4, D. De Jesus Damiao, C. De Oliveira Martins, S.
Fonseca De Souza, H. Malbouisson, D. Matos Figueiredo, M. Melo De Almeida, C.
Mora Herrera, L. Mundim, H. Nogima, W.L. Prado Da Silva, L.J. Sanchez Rosas,
A. Santoro, A. Sznajder, M. Thiel, E.J. Tonelli Manganote3, F. Torres Da Silva
De Araujo, A. Vilela Pereira Universidade Estadual Paulista a, Universidade
Federal do ABC b, São Paulo, Brazil
S. Ahujaa, C.A. Bernardesa, L. Calligarisa, T.R. Fernandez Perez Tomeia, E.M.
Gregoresb, P.G. Mercadanteb, S.F. Novaesa, SandraS. Padulaa, D. Romero Abadb
Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of
Sciences, Sofia, Bulgaria
A. Aleksandrov, R. Hadjiiska, P. Iaydjiev, A. Marinov, M. Misheva, M. Rodozov,
M. Shopova, G. Sultanov University of Sofia, Sofia, Bulgaria
A. Dimitrov, L. Litov, B. Pavlov, P. Petkov Beihang University, Beijing, China
W. Fang5, X. Gao5, L. Yuan Institute of High Energy Physics, Beijing, China
M. Ahmad, J.G. Bian, G.M. Chen, H.S. Chen, M. Chen, Y. Chen, C.H. Jiang, D.
Leggat, H. Liao, Z. Liu, F. Romeo, S.M. Shaheen6, A. Spiezia, J. Tao, C. Wang,
Z. Wang, E. Yazgan, H. Zhang, J. Zhao State Key Laboratory of Nuclear Physics
and Technology, Peking University, Beijing, China
Y. Ban, G. Chen, A. Levin, J. Li, L. Li, Q. Li, Y. Mao, S.J. Qian, D. Wang, Z.
Xu Tsinghua University, Beijing, China
Y. Wang Universidad de Los Andes, Bogota, Colombia
C. Avila, A. Cabrera, C.A. Carrillo Montoya, L.F. Chaparro Sierra, C. Florez,
C.F. González Hernández, M.A. Segura Delgado University of Split, Faculty of
Electrical Engineering, Mechanical Engineering and Naval Architecture, Split,
Croatia
B. Courbon, N. Godinovic, D. Lelas, I. Puljak, T. Sculac University of Split,
Faculty of Science, Split, Croatia
Z. Antunovic, M. Kovac Institute Rudjer Boskovic, Zagreb, Croatia
V. Brigljevic, D. Ferencek, K. Kadija, B. Mesic, A. Starodumov7, T. Susa
University of Cyprus, Nicosia, Cyprus
M.W. Ather, A. Attikis, M. Kolosova, G. Mavromanolakis, J. Mousa, C. Nicolaou,
F. Ptochos, P.A. Razis, H. Rykaczewski Charles University, Prague, Czech
Republic
M. Finger8, M. Finger Jr.8 Escuela Politecnica Nacional, Quito, Ecuador
E. Ayala Universidad San Francisco de Quito, Quito, Ecuador
E. Carrera Jarrin Academy of Scientific Research and Technology of the Arab
Republic of Egypt, Egyptian Network of High Energy Physics, Cairo, Egypt
Y. Assran9,10, S. Elgammal10, S. Khalil11 National Institute of Chemical
Physics and Biophysics, Tallinn, Estonia
S. Bhowmik, A. Carvalho Antunes De Oliveira, R.K. Dewanjee, K. Ehataht, M.
Kadastik, M. Raidal, C. Veelken Department of Physics, University of Helsinki,
Helsinki, Finland
P. Eerola, H. Kirschenmann, J. Pekkanen, M. Voutilainen Helsinki Institute of
Physics, Helsinki, Finland
J. Havukainen, J.K. Heikkilä, T. Järvinen, V. Karimäki, R. Kinnunen, T.
Lampén, K. Lassila-Perini, S. Laurila, S. Lehti, T. Lindén, P. Luukka, T.
Mäenpää, H. Siikonen, E. Tuominen, J. Tuominiemi Lappeenranta University of
Technology, Lappeenranta, Finland
T. Tuuva IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
M. Besancon, F. Couderc, M. Dejardin, D. Denegri, J.L. Faure, F. Ferri, S.
Ganjour, A. Givernaud, P. Gras, G. Hamel de Monchenault, P. Jarry, C. Leloup,
E. Locci, J. Malcles, G. Negro, J. Rander, A. Rosowsky, M.Ö. Sahin, M. Titov
Laboratoire Leprince-Ringuet, Ecole polytechnique, CNRS/IN2P3, Université
Paris-Saclay, Palaiseau, France
A. Abdulsalam12, C. Amendola, I. Antropov, F. Beaudette, P. Busson, C.
Charlot, R. Granier de Cassagnac, I. Kucher, S. Lisniak, A. Lobanov, J. Martin
Blanco, M. Nguyen, C. Ochando, G. Ortona, P. Paganini, P. Pigard, R. Salerno,
J.B. Sauvan, Y. Sirois, A.G. Stahl Leiton, A. Zabi, A. Zghiche Université de
Strasbourg, CNRS, IPHC UMR 7178, Strasbourg, France
J.-L. Agram13, J. Andrea, D. Bloch, J.-M. Brom, E.C. Chabert, V. Cherepanov,
C. Collard, E. Conte13, J.-C. Fontaine13, D. Gelé, U. Goerlach, M. Jansová,
A.-C. Le Bihan, N. Tonon, P. Van Hove Centre de Calcul de l’Institut National
de Physique Nucleaire et de Physique des Particules, CNRS/IN2P3, Villeurbanne,
France
S. Gadrat Université de Lyon, Université Claude Bernard Lyon 1, CNRS-IN2P3,
Institut de Physique Nucléaire de Lyon, Villeurbanne, France
S. Beauceron, C. Bernet, G. Boudoul, N. Chanon, R. Chierici, D. Contardo, P.
Depasse, H. El Mamouni, J. Fay, L. Finco, S. Gascon, M. Gouzevitch, G.
Grenier, B. Ille, F. Lagarde, I.B. Laktineh, H. Lattaud, M. Lethuillier, L.
Mirabito, A.L. Pequegnot, S. Perries, A. Popov14, V. Sordini, M. Vander
Donckt, S. Viret, S. Zhang Georgian Technical University, Tbilisi, Georgia
|
# Molecular machines for quantum error correction
Thiago Guerreiro<EMAIL_ADDRESS>Department of Physics, Pontifícia
Universidade Católica, Rio de Janeiro, Brazil
###### Abstract
Inspired by biological molecular machines we explore the idea of an active
quantum robot whose purpose is delaying decoherence. A conceptual model
capable of partially protecting arbitrary logical qubit states against single
physical qubit errors is presented. Implementation of an instance of that
model - the entanglement qubot - is proposed using laser-dressed Rydberg
atoms. Dynamics of the system is studied using stochastic wavefunction
methods.
## I Introduction
The living cell can be seen as a Brownian computer [1]. At its core, machines
of molecular dimensions store, correct and process information in the presence
of noise, with the goal of keeping the state of the living creature away from
thermodynamical equilibrium. The machinery of life [2] is responsible for gene
expression, matter transport across the cell and energy harvesting, among a
vast number of other tasks [3]. An example of such molecular devices is RNA
polymerase (RNAP): an enzyme with $\sim 40.000$ atoms, roughly $\SI{10}{nm}$
of linear size, capable of synthesising a strand of RNA from a DNA template in
the presence of Brownian noise, at error rates as low as $10^{-7}$ [4].
Molecular devices such as RNAP have inspired nanotechnology [5, 6] and various
artificial molecular machines were built, such as molecular ratchets [7],
pumps [8], motors [9], and gene editing tools [10].
Detailed unified understanding of biological molecular machines according to
the tradition of theoretical physics is yet to be achieved [11], but there is
little doubt that experimental [12] and computational methods [13] in physics
play a key role in that endeavour. It is also expected that the coming age of
quantum information processing will illuminate biological systems through
simulation of quantum chemistry [14] and quantum enhanced learning [15, 16].
Conversely [17], one could ask whether biological molecular machines will
inspire new ideas for engineering autonomous molecular-sized quantum
information processing devices with the goal of keeping quantum states away
from thermodynamical equilibrium. It is the purpose of this work to explore
this idea.
Figure 1: (a) Schematics for a conceptual qubot model capable of partially
protecting an arbitrary logical qubit state against decoherence. (b) Example
of a possible potential landscape describing the interaction between the
nucleus atoms; for this plot the radial dependence of (2) is considered
$J_{\alpha}(R)=(d^{2}/R^{3})j_{\alpha}$, with $j_{y}=-3j_{x},j_{z}=6j_{x}$.
A quantum molecular machine would be a device composed of at most a few
thousand atoms capable of autonomously storing, protecting and/or processing
quantum states in the presence of external decoherence and thermalization. We
refer to these bio-inspired devices as quantum robots, or qubots [18].
Devising qubots is a problem in coherent quantum chemistry [19, 20] much like
engineering artificial molecular machines is a problem in synthetic chemistry
[21]. Hence, the ultracold atom [22] and molecular toolbox [23, 24] is
expected to play a key role in the conception of these active quantum devices.
As we will see, qubots exploit open system dynamics to achieve their purpose
and thus have a close connection to the idea of engineered environments
constructed to produce desired quantum states [25, 26, 27, 28, 29, 30, 31,
32]. Their nature, however, is much closer to that of artificial molecular
ratchets and pumps that respond to the environment and consume resources to
maintain nonequilibrium states [33].
In what follows, we explore various aspects around the idea of qubots. We
begin by introducing a conceptual model for a quantum robot capable of
partially protecting a logical qubit state against single physical qubit
errors. It is interesting that the model can handle almost all combinations of
phase and bit-flip errors since, as pointed out by Kitaev, it is generally
easy to get rid of one kind of errors, but not both [34]. The construction is
somewhat inspired by the surface code [35], only here syndrome detection and
correction are part of the system’s dynamics rather than a consequence of
measurement followed by external conditional action. Next, a specific physical
implementation of instances of the model based on laser-dressed Rydberg atoms
is discussed. More specifically, we exhibit potential landscapes implementing
an entanglement qubot, a device that stabilizes a Bell state against single
qubit errors. The stabilized Bell state is only one possible state of the
logical qubit, but in this case we can view the qubot as preserving a
maximally entangled state. An ensemble of entanglement qubots could therefore
preserve vast amounts of entanglement, a useful resource. Simulation of the
entanglement qubot dynamics is performed with the help of stochastic
wavefunction methods, and we evaluate the effects of coupling the motional
degrees of freedom of the robot to an external heat bath. We conclude with a
discussion on potential future developments regarding active quantum matter.
## II Conceptual model
We would like to introduce the conceptual model of a quantum robot capable of
protecting an arbitrary logical qubit state against errors. Our quantum robot
consists of two parts, called the nucleus and the correctors. See Figure 1(a)
for a schematic representation. A pair of particles denoted $a$ and $b$
constitute the nucleus. Quantum information is stored in the particles’
internal spin degrees of freedom taken to be two spin 1/2 systems with Hilbert
space $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ and basis states denoted
$\\{|0\rangle|0\rangle,|0\rangle|1\rangle,|1\rangle|0\rangle,|1\rangle|1\rangle\\}$.
Particle $a$ is held fixed at the origin by an optical tweezer while $b$ is
subject to the potential
$\displaystyle V(R)=V_{t}(R)+V_{I}(R)\ ,$ (1)
where $R$ is the relative distance between $a$ and $b$, $V_{t}(R)$ is a trap
potential for particle $b$ and
$\displaystyle V_{I}(R)=J_{z}Z_{a}Z_{b}+J_{x}X_{a}X_{b}+J_{y}Y_{a}Y_{b}\ ,$
(2)
is the interaction energy between the particles, where
$X_{\lambda},Y_{\lambda},Z_{\lambda}$ are the Pauli operators for particle
$\lambda$ ($=a,b$) and the coefficients $J_{\alpha}=J_{\alpha}(R)$ form a
spatial-dependent spin-spin interaction pattern. We assume for simplicity that
particle $b$ can only move along the direction $\hat{R}$.
As an example of trap potential one may consider an optical tweezer,
$\displaystyle V_{t}(R)$ $\displaystyle=$ $\displaystyle
V_{0}\left(R-\delta\right)^{2}\ .$ (3)
where $V_{0}$ and $\delta$ are constants. Tunneling outside the confining
potential is considered negligible. Note also that dipole-dipole interactions
among atoms and polar molecules is of the form (2), and typically for
molecules [36, 37] and spin impurities in diamond [38],
$\displaystyle J_{\alpha}=(d^{2}/R^{3})j_{\alpha}\ ,$ (4)
where $d$ is the dipole moment [39] and $j_{\alpha}$ a proportionality
constant with $\alpha=x,y,z$. Through the remaining of this section we will
consider this radial dependence as an illustration of the qubot functioning.
Note however that effective spin interactions of the so-called $XYZ$ form with
more general radial dependencies can be engineered within a number of
different systems, including trapped ions [40, 41], atoms in dressed Rydberg
states [42, 43] and microwave-excited polar molecules in optical lattices [45,
46]. In the next section an implementation using laser dressed Rydberg atoms
will be discussed.
Bell states of the particles’ spins are eigenstates of $V_{I}$ with
eigenvalues given by
$\displaystyle V_{I}|\psi^{-}\rangle$ $\displaystyle=$
$\displaystyle\left(-J_{x}-J_{y}-J_{z}\right)|\psi^{-}\rangle\ ,$ (5)
$\displaystyle V_{I}|\phi^{-}\rangle$ $\displaystyle=$
$\displaystyle\left(-J_{x}+J_{y}+J_{z}\right)|\phi^{-}\rangle\ ,$ (6)
$\displaystyle V_{I}|\psi^{+}\rangle$ $\displaystyle=$
$\displaystyle\left(J_{x}+J_{y}-J_{z}\right)|\psi^{+}\rangle\ ,$ (7)
$\displaystyle V_{I}|\phi^{+}\rangle$ $\displaystyle=$
$\displaystyle\left(J_{x}-J_{y}+J_{z}\right)|\phi^{+}\rangle\ .$ (8)
This implies that the total potential $V(R)$ exhibits collective spin-
dependent landscapes.
As an example consider the trap potential (3) and the spin pattern (4). If
local equilibrium positions $R_{0}(|\psi\rangle)$ exist, they satisfy the
condition
$\displaystyle
R_{0}^{4}(R_{0}-\delta)=\dfrac{3d^{2}\langle\psi|W|\psi\rangle}{2V_{0}}\ ,$
(9)
where
$\langle\psi|W|\psi\rangle=\langle\psi|\left(j_{z}Z_{a}Z_{b}+j_{x}X_{a}X_{b}+j_{y}Y_{a}Y_{b}\right)|\psi\rangle$
are possible expectation values with respect to each of the four Bell states.
Figure 1(b) shows the total potential landscape seen by particle $b$ for each
of the spin Bell states, displaying the spin-dependent potentials. Note that
the state $|\psi^{+}\rangle$ does not exhibit a minimum; this is not a problem
provided the protected logical qubit states do not involve $|\psi^{+}\rangle$.
In between equilibrium points of the potential landscapes in Figure 1(b) there
are corrective sites, where devices we call correctors are present. Correctors
are represented in Figure 1(a) as loops. The function of the corrective
devices is executing a unitary operation on the spin subspace once the
particle approaches their site. There are two correctors, denoted $L1$ and
$L2$. For illustration of the device functioning, in the remaining of this
section we treat the correctors $L1$ and $L2$ as qubits. Note however that
there are a number of ways of implementing such devices and alternatives to
the qubit model will be discussed in the following implementation section.
Consider the $L1$ device has basis states
$\\{|\mu_{0}^{1}\rangle,|\mu_{1}^{1}\rangle\\}$. Whenever the particle enters
one of the $L1$ loops, the unitary operation $Z_{b}X_{L1}$ is executed, where
$X_{L1}=|\mu_{0}^{1}\rangle\langle\mu_{1}^{1}|+|\mu_{1}^{1}\rangle\langle\mu_{0}^{1}|$.
It is important that $L1$ is insensitive to whether particle $b$ entered the
innermost or outermost loop, since obtaining that information would collapse
the spin state of the system as it is correlated to motion. The $L2$ system,
or middle corrector, has basis states
$\\{|\mu_{0}^{2}\rangle,|\mu_{1}^{2}\rangle\\}$ and whenever particle $b$
enters $L2$, the unitary $X_{b}X_{L2}$ is executed, where $X_{L2}$ is once
again the bit-flip operator on the corresponding basis states of $L2$.
We have the following operations:
$\displaystyle L1:\ Z_{b}X_{L1}\ ,\ \ L2:\ X_{b}X_{L2}\ .$ (10)
Note these unitaries act on the spins conditional on the particle’s position.
Hence, when tracing out the position degree of freedom, action of the
corrective sites manifests as dissipative maps on the spin subspace.
Logical basis states of the nucleus are defined as
$\displaystyle|\bar{0}\rangle$ $\displaystyle=$
$\displaystyle|\psi^{-}\rangle$ (11) $\displaystyle|\bar{1}\rangle$
$\displaystyle=$ $\displaystyle|\phi^{-}\rangle$ (12)
and an arbitrary logical qubit state is
$\displaystyle|\Psi\rangle=\alpha|\bar{0}\rangle+\beta|\bar{1}\rangle$ (13)
Note that a superposition of the $|\bar{0}\rangle,|\bar{1}\rangle$ states
implies particle $b$ is in a superposition of singlet and triplet spin states,
implying a superposition of different spatial equilibrium points.
To understand how the qubot delays decoherence and partially protects the
logical qubit, one must follow carefully what happens to the particles when a
physical error occurs in one of the spins. Single physical qubit errors are
assumed to be much more likely than multi-qubit errors [35] and the
depolarizing channel is considered as decoherence model. A summary of possible
errors and how they act on logical basis states is shown in Table 1.
Error | $\ \ |\psi^{-}\rangle$ | $\ \ |\phi^{-}\rangle$ | Corrected state
---|---|---|---
$X_{a}$ | $-|\phi^{-}\rangle$ | $-|\psi^{-}\rangle$ | $-\alpha|\bar{0}\rangle-\beta|\bar{1}\rangle$
$X_{b}$ | $+|\phi^{-}\rangle$ | $+|\psi^{-}\rangle$ | $\alpha|\bar{0}\rangle+\beta|\bar{1}\rangle$
$Z_{a}$ | $+|\psi^{+}\rangle$ | $+|\phi^{+}\rangle$ | $\alpha|\bar{0}\rangle-\beta|\bar{1}\rangle$
$Z_{b}$ | $-|\psi^{+}\rangle$ | $+|\phi^{+}\rangle$ | $-\alpha|\bar{0}\rangle-\beta|\bar{1}\rangle$
$Z_{a}X_{a}$ | $-|\phi^{+}\rangle$ | $-|\psi^{+}\rangle$ | $\alpha|\bar{0}\rangle-\beta|\bar{1}\rangle$
$Z_{b}X_{b}$ | $+|\phi^{+}\rangle$ | $-|\psi^{+}\rangle$ | $-\alpha|\bar{0}\rangle-\beta|\bar{1}\rangle$
Table 1: Effect of physical errors on logical basis states and the final
corrected state after action of the qubot. Figure 2: Example of a qubot
cycle: (a) particle $b$ rests in its equilibrium position, while the spin
states form a singlet $|\bar{0}\rangle=|\psi^{-}\rangle$; (b) an error occurs,
changing the potential landscape seen by $b$; (c) the particle is forced into
loop $L2$, which restores the original spin state and (d) the particle goes
back to the original equilibrium position.
As an illustration, consider the example of a bit-flip in the first spin
described by the $X_{a}$ operator. Initially, an arbitrary logical qubit state
$|\Psi\rangle=\alpha|\psi^{-}\rangle+\beta|\phi^{-}\rangle$ is in a
superposition of equilibrium positions $R_{0}(|\psi^{-}\rangle)$ and
$R_{0}(|\phi^{-}\rangle)$ given by solutions of (9). The $X_{a}$ error changes
the spin state of the particles according to
$\displaystyle\alpha|\psi^{-}\rangle+\beta|\phi^{-}\rangle\rightarrow-\alpha|\phi^{-}\rangle-\beta|\psi^{-}\rangle$
(14)
and hence the particles’ interaction potential is changed accordingly. After
the error, the possible positions of particle $b$ are no longer equilibrium
points of the potential landscapes. For the case in which $b$ was initially at
$R_{0}(|\psi^{-}\rangle)$, the particles repel, forcing $b$ into $L2$.
Similarly, for $R_{0}(|\phi^{-}\rangle)$, occurrence of the error causes an
attractive force which pulls $b$ into $L2$. Once $b$ reaches the loop, the
operator $X_{b}X_{L2}$ is applied, restoring the logical qubit to the original
state and driving the system back to the initial superposition of equilibrium
points. Naturally this process introduces kinetic energy in the form of
phonons, which must be removed if particle $b$ is to settle back in the
original state. This implies the need for a dissipative force acting on $b$
which could be provided by state-independent cooling of the atom motion. For
now, we will assume that such cooling is present, and this phonon issue will
be discussed further in the implementation section.
Similar processes occur for $X_{b}$ and $Z_{b}$ errors: a combination of spin-
motion dynamics and subsequent application of the loop operators corrects
errors and restores the system to the initial arbitrary logical state. The
qubot is also able to correct a concatenation of phase and bit-flip errors,
given by $Y_{b}$. Note that this requires a passage through two correctors.
The present qubot model is not able to correct all errors. As can be seen in
Table 1, logical basis states transform under $Z_{a}$ with opposite parity,
thus inducing a phase error in the logical qubit. This imparts on the $Y_{a}$
error since $iY_{a}=Z_{a}X_{a}$. This imperfection can be traced back to the
fact that the qubot uses two physical qubits to encode a logical state. The
quantum Hamming bound [47] implies that for single qubit errors, a minimum of
five qubits are required to achieve complete fault tolerance for one logical
qubit. Despite this partial fault tolerance the qubot can delay decoherence of
arbitrary logical qubit states, and for some specific states it is even able
to preserve it regardless of the error, as for example the singlet
$|\psi\rangle=|\psi^{-}\rangle$. More general models implementing perfect
quantum error correcting codes [48] can nevertheless be devised at the expense
of more particles or higher spin states. Note that to protect arbitrary
logical qubit states, the qubot potential landscapes must distinguish between
all the four elements of the Bell basis, as in Figure 1(b). If the landscape
for two or more Bell states is indistinguishable, certain errors will cause no
effect upon the atom preventing the action of the correctors. Note also that
the order of the potential minima for each Bell state defines the choice of
position and action for the corrective sites, as well as the choice of logical
basis states.
It is instructive to consider the qubot operation under a depolarizing channel
acting on particle $b$ alone. Denote environment states as $|e_{j}\rangle$.
Decoherence causes the joint particle-environment-corrector state to evolve
according to,
$\displaystyle|\Psi\rangle|e_{0}\rangle|\mu_{0}^{1}\mu_{0}^{2}\rangle\rightarrow\sqrt{1-p}\left(\alpha|\psi^{-}\rangle+\beta|\phi^{-}\rangle\right)|e_{0}\rangle|\mu_{0}^{1}\mu_{0}^{2}\rangle$
$\displaystyle+\sqrt{\dfrac{p}{3}}\left(\alpha|\phi^{-}\rangle+\beta|\psi^{-}\rangle\right)|e_{1}\rangle|\mu_{0}^{1}\mu_{0}^{2}\rangle$
$\displaystyle+\sqrt{\dfrac{p}{3}}\left(-\alpha|\psi^{+}\rangle+\beta|\phi^{+}\rangle\right)|e_{2}\rangle|\mu_{0}^{1}\mu_{0}^{2}\rangle$
$\displaystyle+\sqrt{\dfrac{p}{3}}\left(\alpha|\phi^{+}\rangle-\beta|\psi^{+}\rangle\right)|e_{3}\rangle|\mu_{0}^{1}\mu_{0}^{2}\rangle$
(15)
where $p$ denotes the error probability. Equation (15) describes the
depolarizing dynamics suffered by the logical qubit, with the first term
proportional to $\sqrt{1-p}$ corresponding to no decoherence and the
subsequent terms proportional to $\sqrt{p/3}$ corresponding to errors on the
logical qubit. Note that at this stage, the corrective devices remain
unaffected while the system undergoes errors and the environment learns when
an error has occurred. Tracing out the environment, the above evolution
induces a dissipative map on the spin system increasing its entropy and
causing decoherence of the original state.
With the occurrence of errors the potential landscapes acting on $b$ undergo a
change forcing the action of the correctors upon the spin state of the
nucleus. Purity of the logical qubit is restored at the expense of an increase
in entropy for the correctors; after a correction event, (15) evolves to
$\displaystyle|\Psi\rangle\left(\sqrt{1-p}|e_{0}\rangle|\mu_{0}^{1}\mu_{0}^{2}\rangle+\sqrt{\dfrac{p}{3}}|e_{1}\rangle|\mu_{0}^{1}\mu_{1}^{2}\rangle\right.$
$\displaystyle\left.-\sqrt{\dfrac{p}{3}}|e_{2}\rangle|\mu_{1}^{1}\mu_{0}^{2}\rangle-\sqrt{\dfrac{p}{3}}|e_{3}\rangle|\mu_{1}^{1}\mu_{1}^{2}\rangle\right)$
(16)
where we can see that the original logical qubit state is restored and the
environment gets correlated to the correctors’ state. After a single error
correction, the correctors’ states must be reset to the pure initial state
$|\mu_{0}^{1}\mu_{0}^{2}\rangle$. This is a non-unitary operation which
requires energy expenditure, similar to erasing a quantum state [49, 50] and
can be implemented as a non-equilibrium stochastic process. This corresponds
to a consumption of resources by the qubot analogous to the consumption of
resources by biological molecular machines and living organisms. Irrespective
of the physical implementation of the corrective sites, such consumption of
resources is a mandatory part of the qubot operation in accordance to the laws
of thermodynamics.
## III Implementation
Potential engineering. Spin-spin interactions of the form (2) suitable for
implementing quantum robots could be engineered in a number of different
atomic and molecular systems. In this section a physical implementation using
laser-dressed Rydberg atoms [42, 43, 44] is discussed. As will be shown,
instances of the qubot described in the previous section can be realized for
realistic experimental parameters, provided one chooses the correct logical
basis elements and position of corrective sites. We will focus on a qubot that
stabilizes an effective entangled spin state against a depolarizing
environment similar to the one outlined in [18]. We shall refer to this device
as an entanglement qubot.
Figure 3: Level schematics for the entanglement qubot. Figure 4: (a) Spin
pattern, corresponding to the coefficients of Eq. (20) for the parameters
$n=60$, $\Delta_{-}=-\Delta_{+}=2\pi\times\SI{50}{MHz}$ and
$\Omega_{-}=\Omega_{+}/3=\SI{2\pi\times 3}{MHz}$. (b) Collective spin-
dependent potential landscapes. Each trace corresponds to a Bell state of the
qubot nucleus. Corrective devices $L1$ and $L2$ are positioned outside the
potential minima, for example as the dashed vertical lines indicate.
A pair of 87Rb atoms labelled $a$ and $b$ constitute the qubot nucleus.
Effective spin states are provided by hyperfine levels of $b$, specifically
$\displaystyle|0\rangle$ $\displaystyle=$
$\displaystyle|5^{2}S_{1/2},F=1,m_{F}=1\rangle\ ,$ (17)
$\displaystyle|1\rangle$ $\displaystyle=$
$\displaystyle|5^{2}S_{1/2},F=2,m_{F}=2\rangle\ ,$ (18)
with energy difference $\omega_{01}$. The atom-atom interaction potential is
induced by dressing the $|0\rangle,|1\rangle$ states with two strongly
interacting Rydberg Zeeman sublevels in the $n^{2}P_{1/2}$ manifold via Rabi
oscillations with detunings $\Delta_{\pm}$ and frequencies $\Omega_{\pm}$
using $\sigma^{\pm}$ polarized light. The interaction between Rydberg states
arises from a van der Waals potential of the form $C_{6}R^{-6}$, and a fixed
orientation of the two particles is considered, with the atoms polarized
perpendicular to the plane. Large detunings guarantee that only a small
fraction of the Rydberg states is admixed to the $|0\rangle,|1\rangle$ levels
while maintaining a long lifetime. Following [42], the Rydberg states are
$\displaystyle|r_{\pm}\rangle=|n^{2}P_{1/2},m_{j}=\pm
1/2\rangle|m_{I}=3/2\rangle\ ,$ (19)
with an energy difference $\Delta E_{r}$. Detunings are chosen such that the
energy conservation condition $\Delta E_{r}=(\Delta_{+}-\Delta_{-})$ is
satisfied. A level diagram is shown in Figure 3. The atoms are trapped in one
dimensional potentials, insensitive to their internal states. State-
independent trapping of Rydberg dressed atoms can be achieved in so-called
magic [51, 52] and magnetic traps [53]. While atom $a$ is fixed at the origin,
$b$ is able to move under the influence of a force resulting from the
combination of an external tweezer and the atom-atom interaction potential.
As in quantum chemistry [19, 20], the time scale associated to electronic
dynamics is much shorter than the time scale of nuclei motion. An effective
spin dependent Born-Oppenheimer potential can therefore be derived at fixed
atomic separations $R$. In the limit of large detunings
$\Omega_{\pm}\ll\Delta_{\pm}$ and for
$\Delta_{+}/\Delta_{-}<0,\Delta_{+}+\Delta_{-}<0$, adiabatic elimination [66]
can be used in the rotating frame to obtain an effective interaction acting on
the subspace generated by the $|0\rangle,|1\rangle$ states to fourth order in
$\Omega_{\pm}/\Delta_{\pm}$,
$\displaystyle
V_{I}(R)=J_{z}Z_{a}Z_{b}+J_{x}X_{a}X_{b}+J_{y}Y_{a}Y_{b}+J_{\parallel}\left(Z_{a}+Z_{b}\right)\
,$ (20)
where $J_{\alpha}(R)$ ($\alpha=x,y,z$) are radial steplike coefficients
depending on the Rabi frequencies $\Omega_{\pm}$, detunings $\Delta_{\pm}$ and
van der Waals $C_{6}$ coefficients for the $n^{2}P_{1/2}$ manifold.
$J_{\parallel}$ is an effective magnetic field, which we assume can be
cancelled by an additional weak non-homogeneous field on the order of 2G. See
Appendices A and B for explicit definitions, formulas and details on the
potential and effective magnetic field, respectively.
A plot of the $J_{\alpha}$ spin pattern for $n=60$, detunings
$\Delta_{-}=-\Delta_{+}=2\pi\times\SI{50}{MHz}$ and Rabi frequencies
$\Omega_{-}=\Omega_{+}/3=\SI{2\pi\times 3}{MHz}$ can be seen in Figure 4(a).
Note these are in the same parameter region as used for realizing the quantum
spin ice Hamiltonian on a kagome lattice in [42, 54]. The parameters defining
a qubot potential are not unique, allowing some freedom in the construction;
for an example of a different set of numbers and the resulting spin pattern
see the Appendix C.
From the spin pattern coefficients together with Eqs.(5)-(8) and a trap
potential $V_{t}(R)$ we can derive the collective spin-dependent potentials
acting on particle $b$. Consider a trap potential provided by two neighboring
optical tweezers,
$\displaystyle
V_{t}(R)=V_{0}\left[\left(R-\delta_{1}\right)^{2}+\left(R-\delta_{2}\right)^{2}\right]$
(21)
where $V_{0}=\SI{15}{kHz/\mu m^{2}}$, $\delta_{1}=\SI{1.6}{\mu m}$ and
$\delta_{2}=\SI{2.0}{\mu m}$. The resulting spin-dependent potential
landscapes $V(R)$ can be seen in Figure 4(b), where each trace corresponds to
a different Bell state of the two atoms. Note equilibrium positions are
separated by approximately $\SI{0.3}{\mu m}$. Trap frequencies are
approximately $\omega_{t}/2\pi\approx\SI{1}{kHz}$. Possible positions for the
corrective sites $L1$ and $L2$, corresponding to the transformations (10), are
represented by dashed vertical lines. Note the potential landscapes for the
Bell states $|\psi^{-}\rangle$ and $|\psi^{+}\rangle$ overlap. This implies
that one cannot choose either $|\psi^{-}\rangle$ or $|\psi^{+}\rangle$ as
protected states, as in this case, phase errors could not be corrected. The
protected logical state is chosen to be $|\phi^{+}\rangle$.
Corretive sites. Correctors $L1$ and $L2$ were previously considered to be
qubits acting as an entropy sink for maintaining the purity of the protected
logical qubit state carried by the nucleus. The interaction between
superconducting quantum electronics and atomic [55], molecular [56] and
mesoscopic particles [57] has been extensively studied in the context of
hybrid quantum systems and the coupling between NV centers and superconductors
has been observed [58]. A number of different implementations involving
superconducting qubit systems is therefore expected.
Beyond qubits, one may consider additional atoms as candidates for
implementing corrective devices. Controlled atomic collisions [59] would
provide the mechanism for position-dependent unitary operations. One could
envision a lattice with arrays of data particles interpolated with corrective
particles, analogous to the surface code [35]; occurrence of errors would
alter the interaction between data particles, enabling or inhibiting motion
and tunneling - and consequently interactions - with neighboring corrective
sites. It would be as a surface code in motion, where errors induce controlled
motion leading to correction feedbacks. It is important to stress that in the
course of the qubot action, entropy of the corrective atoms would increase and
a dissipative map for restarting the correctors in their original state would
have to be continuously enforced, for example through an amplitude damping
channel [18].
Corrective devices could also be implemented using Rabi oscillations between
the $|0\rangle,|1\rangle$ levels. By carefully tuning the Rabi frequency of
the transition and the profile of the spin-dependent potentials in Figure 4(b)
it is in principle possible to engineer the transit time of atom $b$ through
$L1$ and $L2$ such that $Z_{b}$ and $X_{b}$ operations are applied, analogous
to the transit time stimulated decay in ammonia masers [60] and Ramsey
interferometry in atomic fountain clocks [61]. In this implementation -
probably the most practical from an experimental point-of-view - the
electromagnetic field assumes the role of entropy sink since conditional $X$
and $Z$ operations on the atom would introduce uncertainties in the intensity
and phase of the field, respectively. A schematics of this implementation is
shown in Figure 5.
Figure 5: Corrective sites as Rabi oscillations.
Operation, cooling and lifetime. Operation of the qubot proceeds as described
in the previous section: occurrence of an error induces a change in the
potential landscape seen by atom $b$ thus forcing it into one of the
corrective sites $L1$ or $L2$. Note that errors can occur due to external
environmental influence or intrinsically due to thermal and quantum
fluctuations of the atomic motion. Consider atom $b$ in a thermal state. For
temperatures on the order of 10nK, reachable for atomic ensembles [62], the
occupation number of atomic motion is $\bar{n}\approx 0.1$ pointing out that
the atom is effectively in the trap ground state. Zero point motion of the
atom is approximately
$R_{\mathrm{zpm}}\simeq\sqrt{\hbar/2m\omega_{t}}\approx\SI{0.23}{\mu m}$,
indicating that at 10nK quantum fluctuations can cause the atom to reach the
corrective sites even when no environmental error took place, inducing change
in the qubot state. Hence, intrinsic fluctuation errors are expected to
constitute a portion of total errors. In the next section, a model of the
qubot operation taking into account intrinsic and external errors will be
discussed.
Errors can be effectively corrected provided the qubot nucleus undergoes
constant cooling of its motional degrees of freedom to dissipate the kinetic
energy gained by mechanical forces due to potential changes. Such cooling
mechanism needs to preserve the quantum information stored in the nucleus, so
it must be insensitive to the quantum state stored in the spins. State-
insensitive cooling of neutral atoms can be achieved via superfluid immersion
[63], cavity cooling [64] or sympathetic cooling through spin-independent
Rydberg interactions with neighboring atoms [65].
What is the order of magnitude of the expected lifetime for the protected
entangled state? The $60P_{1/2}$ Rydberg state has a lifetime on the order of
$\tau_{r}\approx\SI{133}{\mu s}$ [70]. This implies a bare lifetime for the
effective spin state of
$\tau_{s}\approx(2\Delta_{-}/\Omega_{-})^{2}\tau_{r}\approx\SI{9}{ms}$ [42],
corresponding to a spin decoherence rate $\Gamma\approx\SI{111}{Hz}$. A decay
process to the ground state $|0\rangle$ is defined by the following
transformations,
$\displaystyle|0\rangle|e_{0}\rangle$ $\displaystyle\rightarrow$
$\displaystyle|0\rangle|e_{0}\rangle$ (22)
$\displaystyle|1\rangle|e_{0}\rangle$ $\displaystyle\rightarrow$
$\displaystyle\sqrt{1-\tau_{s}^{-1}dt}|1\rangle|e_{0}\rangle+\sqrt{\tau_{s}^{-1}dt}|0\rangle|e_{1}\rangle$
(23)
where the first ket corresponds to the spin of the particle while the second
ket represents the environment state. Action of this quantum channel upon the
elements of the Bell basis can be written in terms of strings of Pauli errors
[71]. It is thus expected that the qubot is able to extend the lifetime of
Rydberg dressed entangled states.
## IV Dynamics Simulation
Exploration of the qubot requires simulation of its error-correction dynamics.
Any such simulation must take into account the effects of quantum fluctuations
of atomic motion, as these fluctuations are in themselves a source of errors
that can disturb the protected Bell state. A first principles description of
the spin and motion degrees of freedom is intricate as the spin state is
subject to transformations conditional on the motion state, which in itself is
conditioned on the spin through the spin-dependent potential. As Wheeler would
say [72]: spin tells matter how to move, matter tells spin how to turn.
To capture the essential features of the qubot we propose an open quantum
system model in which the motion and spin degrees of freedom follow a set of
discrete-time coupled stochastic Schrodinger equations. Each realization of
the evolution is described in terms of sequences of quantum state pairs,
denoted $|\psi\rangle$ for the spin and $|\phi\rangle$ for the motion degree
of freedom. Averaging over many realizations of the stochastic process results
in the mean behavior of the system.
The spin and motion degrees of freedom act as environments for each other.
This idea can be used to motivate the model as follows. For simplicity,
discretize (1D) space into a set of points $R_{k}$. The position state reads
$\displaystyle|\phi\rangle=\sum_{k}\phi(R_{k})|R_{k}\rangle$ (24)
where $|\phi(R_{k})|^{2}$ gives the probability of finding the particle at
position $R_{k}$. The initial state evolves in a small time increment $\delta
t$ according to
$\displaystyle|\psi\rangle|\phi\rangle\xrightarrow{\delta
t}\sum_{i}\phi(R_{i})(T(R_{i})|\psi\rangle)(W(|\psi\rangle)|R_{i}\rangle)$
$\displaystyle=|\Psi(t+\delta t)\rangle$ (25)
where $T(R_{i})$ is the identity operator unless $R_{i}=R_{L1}$ or
$R_{i}=R_{L2}$, for which
$\displaystyle T(R_{L1})=Z_{b}$ (26) $\displaystyle T(R_{L2})=X_{b}$ (27)
The operator $W(|\psi\rangle)$ contains information on the spin-dependent
potential and is responsible for the evolution of the motion state. Expanding
$|\Psi(t+dt)\rangle$,
$\displaystyle|\Psi(t+\delta t)\rangle$ $\displaystyle=$
$\displaystyle\sum_{i\neq
L1,L2}\phi(R_{i})|\psi\rangle(W(|\psi\rangle)|R_{i}\rangle)$ (28)
$\displaystyle+$
$\displaystyle\phi(R_{L1})(Z_{b}|\psi\rangle)(W(|\psi\rangle)|R_{L1}\rangle)$
$\displaystyle+$
$\displaystyle\phi(R_{L2})(X_{b}|\psi\rangle)(W(|\psi\rangle)|R_{L2}\rangle)$
Assuming the spin state is continuously monitored in the Bell basis, the above
state continuously collapses to a random separable state allowing the phase
information and correlations of the global state to be ignored. Note that
under this monitoring assumption one can describe the dynamics of the system
within a simpler scenario and yet verify the error correction capability of
the proposed qubot. Moreover, monitoring of the joint spin state in the Bell
basis can be achieved by continuous measurement of the force acting on
particle $a$, since the interaction between the particles is given by their
joint spin state. The motion state then acts as an environment for the spin,
inducing corrective jump operators,
$\displaystyle L_{1}=\sqrt{\gamma_{L1}}Z_{b}$ (29) $\displaystyle
L_{2}=\sqrt{\gamma_{L2}}X_{b}$ (30)
where we define correction rates as
$\displaystyle\gamma_{L1}dt$ $\displaystyle=$
$\displaystyle|\phi(R_{L1})|^{2}$ (31) $\displaystyle\gamma_{L2}dt$
$\displaystyle=$ $\displaystyle|\phi(R_{L2})|^{2}$ (32)
Note that the probability of a given corrective jump occuring is also the
probability of finding the particle in the corresponding corrective site. In
addition to corrective jumps the spin state is also under the effect of a
depolarizing channel due to an external decoherence environment, defined in
terms of the collapse operators
$\displaystyle L_{3}=\sqrt{\dfrac{\Gamma}{3}}X_{b}\ ,\
L_{4}=\sqrt{\dfrac{\Gamma}{3}}Y_{b}\ ,L_{5}=\sqrt{\dfrac{\Gamma}{3}}Z_{b}\ ,\
$ (33)
where $\Gamma$ is the decoherence rate.
Conversely spin acts as an environment to the motion state. If no spin
corrective jump occurs the motion state is left almost unperturbed, according
to (28), and evolves through the unitary predicted by the spin state
$|\psi\rangle$ plus the effects of a damping collapse operator provided by an
additional spin-insensitive cooling environment with damping rate $\kappa$
acting as a drain of kinetic energy, as discussed previously. On the other
hand, if a corrective jump $L_{1}$ or $L_{2}$ happens the motion state
collapses to $|R_{L1}\rangle$ or $|R_{L2}\rangle$, respectively. The collapsed
state subsequently evolves according to the unitary predicted by the spin
state $|\psi\rangle$ plus the additional damping collapse operator. When spin
jumps happen, the motion Hamiltonian must be updated accordingly for the next
time iteration.
This evolution can be implemented via a coupled Monte-Carlo method. First,
define the Motion Monte-Carlo procedure (MMC) for a damped harmonic oscillator
as following:
* (1)
Define motion state $|\phi\rangle$ and Hamiltonian $H$;
* (2)
Compute $\delta v=\kappa\delta t\langle\phi|a^{\dagger}a|\phi\rangle$;
* (3)
Choose uniformly distributed random number $q\in[0,1]$;
* (4)
If $q<\delta v$, update $|\phi\rangle\leftarrow a|\phi\rangle/\sqrt{\delta
v/\delta t}$;
* (5)
If $q>\delta v$, update $|\phi\rangle\leftarrow e^{-i\hat{H}\delta
t}|\phi\rangle/\sqrt{1-\delta v}$, where $\hat{H}=H-\frac{i}{2}a^{\dagger}a$;
We denote by MMC$(|\phi\rangle,H,\delta t)$ the output of the above procedure
for input state $|\phi\rangle$, Hamiltonian $H$, over a time step $\delta t$.
This output consists of the updated motion state after one time step.
The following algorithm, dubbed Spin-Motion Monte Carlo (SMMC), summarizes one
time iteration of the qubot dynamics:
* (1)
Define (update) motion and spin states $|\phi\rangle$ and $|\psi\rangle$ and
motion Hamiltonian $H=H(|\psi\rangle)$;
* (2)
Define correction rates $\gamma_{L1}\delta t=|\langle
R_{L1}|\phi\rangle|^{2},\gamma_{L2}\delta t=|\langle R_{L2}|\phi\rangle|^{2}$,
where $\delta t$ is the discrete time increment;
* (3)
Compute $\delta p_{k}=\delta t\langle\psi|L_{k}^{\dagger}L_{k}|\psi\rangle$
and $\delta p=\sum_{k}\delta p_{k}$;
* (4)
Choose uniformly distributed random number $r\in[0,1]$;
* (5)
If $r<\delta p$, update $|\psi\rangle\leftarrow
L_{k}|\psi\rangle/\sqrt{dp_{k}/\delta t}$ with probability $\delta
p_{k}/\delta p$;
(5.1) If jumps $L_{k}$ with $k=1$ or $2$ occurred, update
$|\phi\rangle\leftarrow|R_{L_{k}}\rangle$ and run
MMC$(|R_{L_{k}}\rangle,H,\delta t)$. After MMC update the motion state and the
motion Hamiltonian to $H=H(L_{k}|\psi\rangle)$;
(5.2) If jumps $L_{k}$ with $k=3,4$ or $5$ occurred, run
MMC$(|\phi\rangle,H,\delta t)$. After MMC update the motion state and the
motion Hamiltonian to $H=H(L_{k}|\psi\rangle)$;
* (6)
If $r>\delta p$, update $|\psi\rangle\leftarrow e^{-iH_{s}\delta
t}|\psi\rangle/\sqrt{1-\delta p}$, where
$H_{s}=-i\sum_{k}L_{k}^{\dagger}L_{k}$;
(6.1) Run MMC$(|\phi\rangle,H,\delta t)$. After MMC update the motion state
and the motion Hamiltonian to $H=H(|\psi\rangle)$;
* (7)
Go to (1) for next iteration.
A time series of quantum states $\\{|\psi(t)\rangle,|\phi(t)\rangle\\}$ is
called a quantum trajectory of the system, and can be obtained by iterating
SMMC. Mean behavior of the qubot can be obtained by averaging quantities of
interest over many quantum trajectories. For example, we can define the
overlap between the qubot spin state and the protected Bell state as
$F=\mathbb{E}\left[|\langle\psi(t)|\phi^{+}\rangle|^{2}\right]$, where
$\mathbb{E}\left[...\right]$ denotes the ensemble average over all quantum
trajectories. The quantity $F$ then measures how close the qubot spin state is
on average to the protected state and hence quantifies how well the qubot
functions.
To simplify the dynamics simulation, spin-dependent potentials are taken to be
harmonic traps of equal resonance frequency. This removes any issues due to
anharmonicity in the potentials and allows for the definition of fixed phonon
creation and annihilation operators. The potentials shown in Figure 4(b) are
approximated as
$\displaystyle
V(|\psi\rangle,R)=\dfrac{m\omega_{t}^{2}}{2}\left[R-R_{0}(|\psi\rangle)\right]^{2}$
(34)
where $\omega_{t}/2\pi=\SI{1}{kHz}$ and the trap position
$R_{0}(|\psi\rangle)$ is given by
Figure 6: Coupled spin-motion Monte-Carlo simulation of the qubot, $10^{3}$
quantum trajectories. Top: average fidelity to the $|\phi^{+}\rangle$ Bell
state as a function of time for the qubot plus a depolarizing channel (thick
green line) compared to the action of a depolarizing channel alone (thin
purple line). Middle: average position of the atom with corresponding quantum
uncertainty (light blue shade). Bottom: average correction rates $\gamma_{L1}$
(light yellow line) and $\gamma_{L2}$ (thick green line). The parameters used
in the plot are: decoherence rate $\Gamma=\SI{100}{Hz}$, trap frequency
$\omega_{t}=\SI{1}{kHz}$, damping rate
$\kappa=\SI{0.1}{ms}\times\omega_{t}^{2}$, initial wavepacket uncertainty
$\Delta R=\SI{0.22}{\mu m}$, $R_{L2}=-R_{L1}=\SI{0.63}{\mu m}$.
$\displaystyle
R_{0}(|\psi\rangle)=\left\\{\begin{array}[]{ll}R_{01},&\mathrm{if}\
|\phi^{+}\rangle\\\ R_{10},&\mathrm{if}\ |\phi^{-}\rangle\\\
R_{00},&\mathrm{if}\ |\psi^{\pm}\rangle\\\ \end{array}\right.$ (38)
The positions $R_{\alpha\beta}$ are dependent on the details of the
experimental implementation. Inspired by Figure 4(b) we consider
$R_{01}=\SI{1.90}{\mu m}$, $R_{10}=\SI{2.20}{\mu m}$ and $R_{00}=\SI{1.64}{\mu
m}$. Since the Hamiltonian always appears inside a commutator, constant terms
can be neglected without affecting the dynamics. Defining the origin of our
reference frame at the minimum of the potential $V(|\phi^{+}\rangle)$ and
neglecting constant shifts, the Hamiltonian reads
$\displaystyle H(|\psi\rangle)=\omega_{t}a^{\dagger}a-m\omega_{t}^{2}\Delta
R_{0}(|\psi\rangle)R_{\mathrm{zpm}}\left(a^{\dagger}+a\right)$ (39)
with $a^{\dagger},a$ the creation and annihilation operators for the
$|\phi^{+}\rangle$ potential, given by,
$\displaystyle
a=\sqrt{\dfrac{m\omega_{t}}{2}}\left(R+\dfrac{i}{m\omega_{t}}P\right)$ (40)
$\displaystyle
a^{\dagger}=\sqrt{\dfrac{m\omega_{t}}{2}}\left(R-\dfrac{i}{m\omega_{t}}P\right)$
(41)
with $R,P$ the atom position and momentum operators of particle $b$,
respectively, $R_{\mathrm{zpm}}$ the corresponding zero-point motion and
$\Delta R_{0}(|\psi\rangle)=R_{0}(|\psi\rangle)-R_{0}(|\phi^{+}\rangle)$. The
effect of a change in the spin state can be interpreted as the appearance of
an additional force acting on particle $b$.
Figure 6 shows the result of iterating SMMC averaged over $10^{3}$ quantum
trajectories, implemented using QuTiP [73], for the initial Bell-position
state $|\phi^{+}\rangle|\chi\rangle$, where $|\chi\rangle$ is a Gaussian
wavepacket in position with uncertainty $\Delta R$. See the Figure caption for
details on the parameters used in the simulation. The top graph shows the mean
overlap $F=\mathbb{E}\left[|\langle\psi(t)|\phi^{+}\rangle|^{2}\right]$ as a
function of time for the qubot (thick green line) compared to the depolarizing
channel alone (thin purple line). We can see that initially the qubot overlap
drops faster than the free spins, but it stabilizes at about $70\%$, while
free decohering spins decrease significantly below. The middle plot shows the
atom position and its quantum uncertainty as a function of time: action of the
qubot stabilizes the location of the atom. Note that motion of the atom
towards one corrective site is expected to increase correction rates of that
site and decrease correction rates of the other. This behavior can be seen in
the bottom graph, where rates are shown as a function of time. As expected,
$\gamma_{L1}$ (light yellow line) displays significant anti-correlation with
$\gamma_{L2}$ (thick green line).
The effect of finite temperature can be evaluated by adapting SMMC to include
motion collapse operators $\sqrt{\kappa(\bar{n}+1)}a$ and
$\sqrt{\kappa\bar{n}}a^{\dagger}$ representing contact with a thermal bath of
phonons at temperature $T$ with coupling $\kappa$ and thermal occupation
number $\bar{n}$, where $\bar{n}=1/(e^{\hbar\omega_{t}/k_{B}T}-1)$. When in
contact with a thermal bath, the particle initially in the ground state
evolves to a thermal state with mean number of phonons $\bar{n}$, increasing
the position spread and consequently the intrinsic qubot error rate. The spin
overlap is thus expected to decrease with temperature.
Figure 7: Effect of contact with a thermal bath at temperature $T$ upon the
steady state time-averaged overlap $\langle F\rangle_{s}$. Time average is
considered starting at 10ms, when the overlap has already achieved its steady
value. Error bars correspond to one standard deviation. Each point is
evaluated from $10^{2}$ quantum trajectories. Coupling to the heat bath is
$\kappa=\SI{0.1}{ms}\times\omega_{t}^{2}$ and all remaining parameters are the
same as in Figure 6.
The time-averaged steady state overlap $\langle F\rangle_{s}$ as a function of
temperature is plotted in Figure 7. Each point is the result of time-averaging
$10^{2}$ quantum trajectories with error bars corresponding to one standard
deviation. As expected, the effect of contact with a heat bath is to decrease
the overlap.
Figure 8: Influence of corrector positioning. Top: steady state overlap.
Bottom: mean correction rates. Averages are considered from $\SI{10}{ms}$
onward, when the device is well settled in the steady state. Error bars
correspond to one standard deviation. Each point is evaluated from $10^{2}$
quantum trajectories. All remaining parameters are the same as in Figure 6.
Quantum fluctuations of the atomic motion can induce internal errors if the
atom interacts with the correctors when no external (decoherence) error has
taken place. To quantify that effect, the steady state overlap $\langle
F\rangle_{s}$ and correction rates $\langle\gamma\rangle_{s}$ are numerically
calculated for different values of the $L1$ position $|R_{L1}|$, shown in
Figure 8; $R_{L1}=-R_{L2}$ is assumed. Note that if the correctors are too
close to the equilibrium position of $|\phi^{+}\rangle($
$|R_{L1}|<\SI{0.40}{\mu m}$), the steady state overlap $\langle F\rangle_{s}$
falls below 50%, while the mean rate for ‘correction’ events are on the order
of 1kHz, due to the atom fluctuating towards $L1$ or $L2$ even in the absence
of an error. As $|R_{L1}|$ is increased, the steady state overlap increases,
reaching a maximum value $\langle F\rangle_{s}\approx 0.7$ for
$|R_{L1}|\approx 0.63$, and then decreases again as the correctors are placed
further apart from the atom. The mean correction rates can be seen to decrease
as the position $|R_{L1}|$ is further increased, which is intuitive since
larger distances imply longer correction times. The optimal operation point
$|R_{L1}|\approx 0.63$ is such that the mean correction rates
$\langle\gamma\rangle$ are of the same order of the decoherence rate
$\Gamma=\SI{100}{Hz}$. See Appendix D for more details.
## V Discussion
Throughout this work we discussed quantum robots, devices as the one
conceptualized in [18], capable of harnessing interactions between its
constituent parts and the surrounding environment to achieve targeted tasks
such as state protection against decoherence. We have introduced for the first
time a model of a qubot capable of partially protecting an arbitrary logical
qubit state against general single physical qubit errors. The first physical
implementation of an instance of such device, capable of protecting a Bell
state against the detrimental action of a depolarizing environment has been
described, as well as Monte-Carlo simulations of the qubot dynamics and the
inclusion of effects due to contact of the device with a thermal bath.
From where we stand, several directions for future exploration can be sighted.
For instance, a more thorough investigation of the capabilities of the
proposed entanglement qubot remains to be done: by tuning the relevant
parameters such as the Rydberg level detunings $\Delta_{\pm}$ and trap
potential $V_{t}(R)$ can we engineer a qubot capable of protecting entangled
states other than the $|\phi^{+}\rangle$ state? What about implementing a
system analogous to the conceptual model, capable of protecting an arbitrary
logical qubit? Could we extend the device to handle multiple qubits? Would the
protection work against general physical errors? We have focused on the
implementation using Rydberd-dressed atoms, but that is certainly not the only
possibility. What other opportunities are offered by considering different
physical setups for qubots? Polar molecules provide a promising platform [36,
45, 46, 19] with the possibility of coupling to superconducting quantum
electronics [56].
Synthetic molecular machines are one of the frontiers of nanotechnology [6, 8,
9, 21]. Enabled by the idea of a quantum robot we can envision extensions of
the molecular machinery toolbox where the quantum states of the nanomachines
play a fundamental role in their dynamics. These devices would combine
resources from the environment, stochasticity and non-equilibrium to execute
coupled quantum motion and processing of quantum information entering the
realm of quantum nanomechanics. For example, in the entanglement qubot one
could set the correction sites to perform the operation $L1=L2=X_{b}$, and
initiate the spins in the state $|\psi^{+}\rangle$. This would cause a
periodic spin-driven motion of the atom. It would be interesting to
investigate the possibility of building quantum time crystals [38, 74, 75]
using this scheme.
Quantum robots with no moving parts are also a hitherto unexplored direction.
In such devices an error in one degree of freedom would unleash a chain of
reactions in other internal non-mechanical parts of the system, which would
act back on the affected degree of freedom and steer it to a desired state.
This touches upon the theoretical issue of quantum feedback [76, 77], in a
situation where the feedback itself is carried by quantum mechanical
information, rather than the usual classical information scheme in which a
measurement result is used to counter-act on the system.
Finally, a very intriguing thought is the combination of a large number of
quantum robots interacting with each other. Large numbers of interacting
classical active agents display fascinating emergent behavior [78, 79].
Ensembles of active quantum agents on the other hand remain unexplored. Qubots
offer a concrete path towards experimentally uncovering the physics of quantum
active matter.
## Acknowledgements
I thank Bruno Melo, Lucianno Defaveri, Bruno Suassuna, Igor Brandão and George
Svetlichny for useful discussions and feedback on the manuscript. This work
was financed in part by the Coordenacão de Aperfeiçoamento de Pessoal de Nível
Superior - Brasil (CAPES) - Finance Code 001, Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq) and the FAPERJ Scholarship No.
E-26/202.830/2019.
## VI Appendix A: effective potentials
As described in the main text, admixing strongly interacting Rydberg states
from the $n^{2}P_{1/2}$ manifold to the low-lying $5^{2}S_{1/2}$ Zeeman
sublevels induces spatial dependent spin-spin interactions of the form (20).
For completeness we reproduce the main results of [42] outlining the toolbox
for engineering a wide range of effective spin interactions.
The interaction coefficients $J_{\alpha}$ are calculated by adiabatic
elimination of the Rydberg levels $|r_{\pm}\rangle$ up to fourth order in
$\Delta/\Omega$, and are given by
$\displaystyle
J_{z}(R)=\dfrac{1}{4}\left(\tilde{V}_{--}(R)-2\tilde{V}_{+-}(R)+\tilde{V}_{++}(R)\right)\
,$ (42) $\displaystyle
J_{x}(R)=2\left(\tilde{W}_{+-}(R)+\tilde{W}_{++}(R)\right)\ ,$ (43)
$\displaystyle J_{y}(R)=2\left(\tilde{W}_{+-}(R)-\tilde{W}_{++}(R)\right)\ ,$
(44) $\displaystyle
J_{\parallel}(R)=\dfrac{1}{4}\left(\tilde{V}_{--}(R)-\tilde{V}_{++}(R)\right)\
,$ (45)
where the functions $\tilde{W}_{\alpha\beta},\tilde{V}_{\alpha\beta}$ are
effective radial dependent steplike potentials,
$\displaystyle\tilde{V}_{\alpha\alpha}(R)=\dfrac{\Omega^{2}_{\bar{\sigma}}}{2\Delta_{\bar{\sigma}}}-\dfrac{\Omega^{4}_{\bar{\sigma}}}{4\Delta_{\bar{\sigma}}^{3}}+\dfrac{\Omega_{\bar{\alpha}}^{4}}{4\Delta_{\bar{\alpha}}^{2}}\dfrac{V_{++}-2\Delta_{\alpha}}{W_{++}^{2}-(V_{++}-2\Delta_{+})(V_{++}-2\Delta_{-})}$
(46)
$\displaystyle\tilde{V}_{+-}(R)=\dfrac{\Omega^{2}_{-}}{4\Delta_{-}}+\dfrac{\Omega^{2}_{+}}{4\Delta_{+}}-\dfrac{\Omega_{+}^{2}\Omega_{-}^{2}}{16\Delta_{+}^{2}\Delta_{-}}-\dfrac{\Omega_{+}^{2}\Omega_{-}^{2}}{16\Delta_{-}^{2}\Delta_{+}}-\dfrac{\Omega^{4}_{-}}{16\Delta_{-}^{3}}-\dfrac{\Omega^{4}_{+}}{16\Delta_{+}^{3}}+\dfrac{\Delta_{\pm}^{2}\Omega_{+}^{2}\Omega_{-}^{2}}{16\Delta_{+}^{2}\Delta_{-}^{2}}\dfrac{(\Delta_{\pm}-V_{+-})}{(\Delta_{\pm}-V_{+-})^{2}-W_{+-}^{2}}$
(47) $\displaystyle\tilde{W}_{+-}(R)$ $\displaystyle=$
$\displaystyle\dfrac{\Omega_{+}^{2}\Omega_{-}^{2}}{16\Delta_{+}^{2}\Delta_{-}^{2}}\dfrac{\Delta_{\pm}^{2}W_{+-}}{(\Delta_{\pm}-V_{+-})^{2}-W_{+-}^{2}}$
(48) $\displaystyle\tilde{W}_{++}(R)$ $\displaystyle=$
$\displaystyle\dfrac{\Omega_{+}^{2}\Omega_{-}^{2}}{4\Delta_{+}\Delta_{-}}\dfrac{W_{++}}{W_{++}^{2}-(V_{++}-2\Delta_{+})(V_{++}-2\Delta_{-})}$
(49)
written in terms of the $n^{2}P_{1/2}$ van der Waals potentials
$V_{\alpha\beta},W_{\alpha\beta}$. Note the single particle light-shifts have
been included in the above expressions. Moreoever,
$\tilde{V}_{+-}=\tilde{V}_{-+}$, and we have defined
$\Delta_{\pm}=\Delta_{+}+\Delta_{-}$ and $\bar{\alpha}=-\alpha$. In the
parameter region $\Delta_{+-}<0$, $\Delta_{+}/\Delta_{-}<0$ resonant Rydberg
excitations are avoided for all values of $R$. For atomic orientation
$\theta=\pi/2$ (polar), $\phi=0$ (azimuthal) the van der Waals potentials are
$\displaystyle V_{\alpha\beta}=\dfrac{c_{\alpha\beta}}{R^{6}}\ ,\
W_{+-}=\dfrac{w}{R^{6}}=-\dfrac{1}{3}W_{++}\ .$ (50)
where the so-called $C_{6}$ coefficients $c_{\alpha\beta}$ and $w$ are
obtained from second order perturbation theory, and are given by
$\displaystyle c_{++}$ $\displaystyle=$
$\displaystyle\dfrac{2}{81}\left(5C_{6}^{(a)}+14C_{6}^{(b)}+8C_{6}^{(c)}\dfrac{}{}\right)$
(51) $\displaystyle c_{+-}$ $\displaystyle=$
$\displaystyle\dfrac{2}{81}\left(C_{6}^{(a)}+10C_{6}^{(b)}+16C_{6}^{(c)}\dfrac{}{}\right)$
(52) $\displaystyle w$ $\displaystyle=$
$\displaystyle\dfrac{2}{81}\left(C_{6}^{(a)}+C_{6}^{(b)}-2C_{6}^{(c)}\dfrac{}{}\right)$
(53)
The indivitual channel coefficients $C_{6}^{(\nu)}$, $\nu=a,b,c$ are not
dependent of magnetic quantum numbers and characterize the interaction
strengh. There is one channel for each non-vanishing matrix element of the
dipole-dipole interaction potential [42],
$\displaystyle a$ $\displaystyle:$ $\displaystyle\ P_{1/2}+P_{1/2}\rightarrow
S_{1/2}+S_{1/2}$ (54) $\displaystyle b$ $\displaystyle:$ $\displaystyle\
P_{1/2}+P_{1/2}\rightarrow D_{3/2}+D_{3/2}$ (55) $\displaystyle c$
$\displaystyle:$ $\displaystyle\ P_{1/2}+P_{1/2}\rightarrow D_{3/2}+S_{1/2}$
(56)
and each $C_{6}^{(\nu)}$ is calculated from the radial part of the dipole-
dipole matrix element [68],
$\displaystyle
C^{(\nu)}_{6}=\sum_{n_{\alpha}n_{\beta}}\dfrac{e^{4}}{\delta_{\alpha\beta}}\left(R_{nl}^{n_{\alpha}l_{\alpha}}R_{nl}^{n_{\beta}l_{\beta}}\right)^{2}$
(57)
where
$\displaystyle R_{nl}^{n_{i}l_{i}}=\int
drr^{2}\psi_{n,l,j}(r)^{*}r\psi_{n_{i},l_{i},j_{i}}(r)\ ,$ (58)
and $\delta_{\alpha\beta}$is the energy defect between levels $n_{\alpha}$ and
$n_{\beta}$.
Figure 9: $C_{6}^{(\nu)}$ coefficients as a function of principal quantum
number for the $n^{2}P_{1/2}$ manifold.
To numerically obtain the coefficients (57), and consequently the step-like
potentials (47) and (49), we use the ARC python library for alkali Rydberg
atoms [69]. Numerical calculation results are shown in Figure 9 as a function
of the principal quantum number for the $n^{2}P_{1/2}$ manifold. For $n=60$,
as used in the main text, we find
$\displaystyle-C_{6}^{(a)}\approx 2\pi\times\SI{2.7E5}{MHz\cdot\mu m^{6}}$
(59) $\displaystyle C_{6}^{(b)}\approx 2\pi\times\SI{1.1E3}{MHz\cdot\mu
m^{6}}$ (60) $\displaystyle C_{6}^{(c)}\approx
2\pi\times\SI{4.9E4}{MHz\cdot\mu m^{6}}$ (61)
## VII Appendix B: Magnetic field $J_{\parallel}$
Besides the $J_{\alpha}(R)$ coefficients, the Rydberg dressing generates an
effective magnetic field term $J_{\parallel}(Z_{a}+Z_{b})$ in the interaction
energy. Under the influence of this term, Bell states of the $ab$ pair are no
longer eigenstates of the interaction. To obtain the spin dependent potential
landscapes given by the eigenvalues in Eqs.(5)-(8), we need to cancel
$J_{\parallel}$ by applying an external spatial dependent static field. How
large such a field needs to be? A plot of $J_{\parallel}$ can be seen in
Figure 10.
Figure 10: $J_{\parallel}$ profile.
Note that $\langle J_{\parallel}\rangle\approx\SI{1401}{kHz}$. Considering the
Landé factor $|g_{F}|\approx\SI{0.70}{MHz/G}$ for the $5^{2}S_{1/2}$ states
[67] this effective magnetic field can be cancelled by an additional weak non-
homogeneous field of order of magnitude $|B_{c}|\approx\SI{2}{G}$.
## VIII Appendix C: Alternative spin pattern
Alternative spin dependent potentials, defined by parameters different from
the ones employed in the main text are shown in Figure 11. Here, we consider
detunings $\Delta_{+}=-2\pi\times\SI{70}{MHz}$,
$\Delta_{-}=2\pi\times\SI{30}{MHz}$, Rabi frequencies
$\Omega_{+}=\Omega_{-}=-2\pi\times\SI{7}{MHz}$ and the trap potential
$\displaystyle V_{t}(R)=V_{0}\left(R-\delta\right)^{2}$ (62)
where $V_{0}=\SI{15}{kHz/\mu m^{2}}$ and $\delta=\SI{2.30}{\mu m}$.
Figure 11: (a) Alternative spin pattern profile. (b) Resulting spin dependent
potentials. (c) Resulting spin dependent potentials after adding the spin-
independent harmonic potential.
Note the resulting landscapes in Figure 11(c) suggest $|\phi^{-}\rangle$ as
protected state, while corrective loops $L1$ and $L2$ should be reversed with
respect to the choice discussed in the main text. The effective magnetic field
has a mean value $\langle J_{\parallel}\rangle\approx\SI{1803}{kHz}$, which
requires a slightly higher compensating magnetic field, but still on the order
of a few Gauss. The spatial profile $J_{\parallel}(R)$ is shown in Figure 12.
Figure 12: Alternative effective magnetic field.
## IX Appendix D: Optimal operation
To evaluate the effect of positioning of the correctors $L1$ and $L2$, we ran
SMMC, as described in the main text, for different values of the positions
$R_{L1}=-R_{L2}$.
Figure 13: Overlap $F$ for different values of the corrector position
$|R_{L1}|=|R_{L2}|$ obtained from simulating $10^{2}$ quantum trajectories.
For each trace position values are $(0.47,0.52,0.58,0.63,0.69,0.74,0.80)\
\SI{}{\mu m}$ from top to bottom, respectively. Grey dashed lines represent
the overlap of free decohering spins, for comparison. All remaining parameters
are the same as in Figure 6 in the main text.
Figure 13 shows traces of the overlap $F$ as a function of time. Each trace
corresponds to a different corrector position (see caption), and the overlap
of free spins under the action of the depolarizing channel is shown as the
grey dashed line for comparison. The points in Figure 8 (see main text) are
obtained by time-averaging the overlap above $\SI{10}{ms}$ for each of the
traces in Figure 13.
We can see that if the corrector’s positions are too close to the atom’s
equilibrium position, the overlap quickly decays due to internal errors,
occurring when a quantum fluctuation in the atomic position places it near the
corrective site. This fast drop in overlap can be mitigated by positioning the
correctors further apart from the $|\phi^{+}\rangle$ equilibrium point. There
is, however, a trade-off: the maximum steady state overlap $\approx 70\%$ is
reached for a position $|R_{L1}|\approx\SI{0.63}{\mu m}$, while placing the
correctors further than that reduce the correction rates below the decoherence
rate and consequently the steady state overlap.
Decoherence causes the overlap to decrease exponentially according to
$e^{-\Gamma t}=e^{-t/\tau_{D}}$, where
$\tau_{D}=\Gamma^{-1}=(\SI{100}{Hz})^{-1}=\SI{10}{ms}$ is the characteristic
decay time of the system. Decoherence effectivelly freezes when the system
reaches its steady state, which happens after a stabilization time $t_{s}$
elapses. From Figure 6 in the main text, we see that $t_{s}\approx\SI{4}{ms}$,
yielding an expected overlap of $F\approx e^{-t_{s}/\tau_{D}}\approx 0.67$, in
accordance to the simulation results.
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|
# What drives the European carbon market? Macroeconomic factors and
forecasts††thanks: The authors gratefully acknowledge the participants at the
4th IWEEE in Bolzano for their useful feedback. Luca Rossini acknowledges
financial support from the Italian Ministry MIUR under the PRIN-PNRR project
Mapping and Pricing of Methane Emissions from the European Electricity Sector
(MAP-of-MeLEES) (grant P2022H483A). This research used the Computational
resources provided by the Core Facility INDACO, which is a project of High-
Performance Computing at the University of Milan.
Andrea Bastianin University of Milan, Italy and Fondazione Eni Enrico Mattei
(FEEM<EMAIL_ADDRESS>Elisabetta Mirto University of Milan, Italy.
<EMAIL_ADDRESS>Yan Qin London Stock Exchange Group, Norway.
<EMAIL_ADDRESS>Luca Rossini University of Milan, Italy and Fondazione Eni
Enrico Mattei (FEEM<EMAIL_ADDRESS>
Abstract: Putting a price on carbon – with taxes or developing carbon markets
– is a widely used policy measure to achieve the target of net-zero emissions
by 2050. This paper tackles the issue of producing point, direction-of-change,
and density forecasts for the monthly real price of carbon within the EU
Emissions Trading Scheme (EU ETS). We aim to uncover supply- and demand-side
forces that can contribute to improving the prediction accuracy of models at
short- and medium-term horizons. We show that a simple Bayesian Vector
Autoregressive (BVAR) model, augmented with either one or two factors
capturing a set of predictors affecting the price of carbon, provides
substantial accuracy gains over a wide set of benchmark forecasts, including
survey expectations and forecasts made available by data providers. We extend
the study to verified emissions and demonstrate that, in this case, adding
stochastic volatility can further improve the forecasting performance of a
single-factor BVAR model. We rely on emissions and price forecasts to build
market monitoring tools that track demand and price pressure in the EU ETS
market. Our results are relevant for policymakers and market practitioners
interested in monitoring the carbon market dynamics.
Key Words: Bayesian inference; Carbon prices; Climate Changes; EU ETS;
Forecasting.
JEL Codes: C11; C32; C53; Q02; Q50.
## 1 Introduction
Climate change is one of the greatest challenges currently addressed by
governments, central banks, and other national and supranational regulators.
The size and complexity of the call to fight climate change are evident, even
when – setting aside its environmental consequences – we focus solely on its
direct socio-economic impacts (Carleton and Hsiang,, 2016). The literature has
highlighted a multitude of effects and transmission channels that connect the
environmental influences of climate change to various economic aggregates
(Dell et al.,, 2014; Ciccarelli and Marotta,, 2024). Moreover, when designing
mitigation and adaptation policies, a trade-off emerges between the stringency
of measures and the undesired outcomes that these might induce (Fullerton and
Muehlegger,, 2019; Hsiang et al.,, 2019).
Among the policy measures implemented to achieve the target of net-zero
emissions by 2050 – as laid down in the 2015 Paris Agreement – carbon taxes
and carbon pricing have lately received a lot of attention from academics and
policymakers interested in quantifying the desired and unintended
macroeconomic effects of putting a price on carbon (see e.g. European Central
Bank, 2021a, ; Parry et al.,, 2021; Pan et al.,, 2023). Given this literature
and the need to incorporate climate and carbon market modules into
macroeconomic models used by central banks, having access to reliable short-
and medium-term forecasts of the price of carbon is becoming increasingly
important (European Central Bank, 2021b, ; NGFS,, 2022).
This paper focuses on producing point, direction-of-change, and density
forecasts of the real price of carbon at a monthly sampling frequency in the
world’s largest carbon market: the EU Emission Trading Scheme (EU ETS). Our
research pursues two intertwined targets. First, we aim to uncover supply- and
demand-side forces that can contribute to improving the prediction accuracy of
models at short- and medium-term horizons. The second main objective of the
paper is to highlight which methodological choices have the potential to
improve the forecast accuracy of the estimated models.
Since the inception of the EU ETS in 2005, the research on carbon pricing has
addressed a variety of issues (see Chevallier,, 2012; Pan et al.,, 2023, for a
survey). In particular, Bjørnland et al., (2023) and Känzig, (2023) use
macroeconomic and financial data, along with microdata to evaluate the impact
of carbon price shocks. Recently Känzig and Konradt, (2023) and Moessner,
(2022) focus on European countries and the inflationary effects in a set of
OECD countries, respectively.
Our paper directly contributes to the literature dealing with forecasting the
price of carbon and identifying its determinants such as weather, energy
prices, macroeconomic and financial conditions (see e.g. Chevallier, 2011a, ;
Chevallier, 2011b, ; Koop and Tole,, 2013; Lei et al.,, 2022; Mansanet-
Bataller et al.,, 2007; Tan et al.,, 2022). Whether the results that emerge
from this strand of the literature on carbon price forecasting – mostly based
on daily and weekly data – can be directly applied to building monthly or
quarterly econometric models through appropriate time aggregation of variables
has to be empirically determined.
Differently from the literature, we focus on the real monthly price of carbon,
which enables us to provide results that are relevant to the dialogue about
the macroeconomic consequences of carbon pricing. Indeed, our analysis can be
closely related to Chevallier, 2011b , which evaluate the role of energy
prices and business cycle movements in driving the price of carbon, and to
Bjørnland et al., (2023), which use a Bayesian structural vector
autoregressive (VAR) model with endogenous variables (i.e. real price of
carbon, verified emissions, and industrial production) to identify demand and
supply shocks driving the emissions and the real price of carbon. Moreover,
while most of the papers dealing with carbon price forecasting rely on
frequentist methods, we use Bayesian techniques to readily incorporate
Stochastic Volatility (SV) dynamics into the models and to generate density
forecasts.111One notable exception is Koop and Tole, (2013) that also rely on
Bayesian methods but consider daily data. Reliance on SV models is another
novelty of our paper; in fact, several previous analyses have focused either
on conditionally homoskedastic models or have modeled conditional volatility
dynamics using Generalized AutoRegressive Conditional Heteroskedasticity
models.
After having assessed a suite of benchmark univariate time series models, we
consider small-scale Vector Autoregressive (VAR) models with endogenous
variables capturing forces affecting the real price of carbon. First, we focus
on the model put forth by Bjørnland et al., (2023) as a starting point. Next,
we extend their VAR model to include factors capturing the influence of
multiple predictors that affect the EU ETS real price.
While we initially compare models based on their ability to deliver accurate
point forecasts, we subsequently extend these results in several directions.
The first extension evaluates different specifications for their ability to
yield accurate sign forecasts and prediction densities. Directional accuracy
is mostly relevant at short-term forecast horizons, while accurate prediction
densities are useful for assessing the uncertainty surrounding point forecasts
and for quantifying the probability of extreme price movements both in the
short- and medium-run. Next, we examine the role of time-varying volatility in
improving forecast accuracy (Clark and Ravazzolo,, 2015; Chan,, 2023) and
assess the presence of forecast instabilities (Rossi,, 2021).
We show that a simple VAR model, augmented with either one or two factors
capturing key predictors of the price of carbon, provides substantial accuracy
gains over a wide set of benchmark forecasts, including survey expectations
and forecasts made available by data providers. We extend the forecasting
study to verified emissions and demonstrate that, in this case, SV can further
improve the performance of a single-factor VAR model. Lastly, we show how
model-based forecasts can be used to build market monitoring tools that track
demand and price pressure in the EU ETS market.
The rest of the paper is organized as follows. Section 2 offers an overview of
the EU ETS; data and details of the forecasting exercise are illustrated in
Section 3. Results for real price and verified emissions are presented in
Sections 4 and 5, respectively. Section 6 concludes.
## 2 The EU Emissions Trading System
The EU ETS is a cap and trade system that started in 2005 intending to reduce
carbon emissions. In this system, the maximum quantity of emissions (the cap)
is set through unit permits (European Unit Allowances, EUA), which allow the
owner of the permit to produce 1 ton of CO2 or an equivalent quantity of other
greenhouse gases. The European Commission sets a yearly cap on the total
greenhouse gas emissions that can be produced by actors participating in the
system. Since the aim is to decrease emissions over time, every year the cap
is lower than the year before, and consequently, the maximum allocation of EUA
is reduced.
The EU ETS system is a financial market where actors can acquire EUA on the
primary market through an auctioning system, and trade derivatives on the
secondary market. A certain amount of permits is originally granted for free
each year according to the needs of specific sector emissions, although the
remaining amount of available allowances is allocated on the primary market
through uniform price auctions with single rounds and sealed bids, conducted
daily by the European Energy Exchange (EEX). Since EUA have been classified as
financial instruments, the associated derivatives - such as spot, futures,
options, and forward contracts - can be traded on secondary markets, both on
exchange and over the counter. While auctions take place on the EEX, trading
takes place also on the Intercontinental Exchange (ICE).
EUA are handed out to the market through a system of benchmark-based
allocation or auctions. If emissions at the end of the year result to be lower
than the cap set for each installation participating in the market, permits
can be traded among actors for an economic value to be determined on the
secondary market. In case emissions exceed the threshold, sanctions are
applied to economic agents participating in the market.
Actors participating in the EU ETS market entail industries belonging to high
emissions sectors: electricity and heat generation, energy-intensive industry
sectors (including oil refineries, steel works, and production of iron,
aluminum, metals, paper, etc.), aviation within the European Economic Area
and, starting from 2024, maritime transport. Participation in the EU ETS is
mandatory for companies in the covered sectors, however, for some of the
sectors only production plants bigger than a threshold size are included.
Historically, the EU ETS evolution went through four phases. To meet the
objectives set by the European Commission in terms of emissions reduction,
each phase aims to reduce the number of EUA granted to each participating
sector. The cap can be lowered by setting a decreasing number of allowances to
be allocated each year or by establishing a yearly linear reduction factor,
e.g. a linear reduction of 1.74% and 2.2% of the baseline 2008-2012 emissions
have been set respectively from 2013 and 2021 onwards, with no end date,
resulting in a year-on-year reduction by up to 43 million allowances
(International Carbon Action Partnership,, 2023).
The pilot phase (2005-2007) aimed to verify rules, regulations, emission
detection systems, as well as the regulatory framework. In this phase, the
allocation system was grandfathering: all EUA were allocated freely to
industries, up to the cap set for each regulated sector. The second phase,
which lasted from 2008 to 2012, was characterized by the introduction of the
allocation of permits employing an auctioning system. In this phase, roughly 2
up to 5% of the total permits were allocated through auctions. This share
increased to reach 54$\%$ in the third phase, which lasted from 2013 to 2020,
and it includes more sectors and gases. Lastly, the fourth phase (2021-2030)
has the aim of reducing net emissions by at least 55% by 2030 compared to
1990, as set in the European Climate Law, by further lowering the cap and
targeting the carbon leakage phenomenon.
In July 2021, the European Commission adopted a series of legislative
proposals regarding EU ETS aimed at increasing the pace of emissions cuts.
These include, among others, covering more sectors and gases, gradually
lowering the number of emission allowances each year, and reinforcing the
Market Stability Reserve (MSR), which aims at reducing the surplus of
allowances in the market. An excessive allowances surplus would lead to lower
carbon prices, rendering, therefore, the ETS system less effective, by
decreasing incentives of the economic actors participating in the market to
lower emissions. The MSR is automatically applied when the total number of
allowances on the market exceeds a certain threshold. In Phase IV, the free
EUA allocation system was granted a ten-year extension, and specific measures
have been taken for sectors exposed to a higher risk of carbon leaking.
Figure 1: Allocated and verified emissions for all stationary installations in
EU-27 countries (excluding the aviation sector). Notes: authors’ elaboration
of data from https://www.eea.europa.eu.
The way allowance allocation has evolved throughout the years is described in
Figure 1. In Phase II, auctions were introduced, but between 95% and 97% of
allowances were still distributed for free. This changed completely during
Phase III, where auctions became the predominant allocation method for most
sectors, covering between 40% and 55% of the total allocated permits. Phase
III also needed to deal with the aftermath of an excessive surplus of unused
EUA in Phase II. The number of allocated EUA decreased in 2014, increasing the
price of allowances and a reduction in the number of verified emissions and
unused EUA (Bjørnland et al.,, 2023). In Phase IV, auctions remain the main
allocation method on the primary market.
## 3 Data and methods
Following Bjørnland et al., (2023), we focus on forecasting the end-of-month
price of the one-month ahead futures contract traded on ICE that represents
the most closely watched series by practitioners. Moreover, we deflate the
nominal futures price by using the Euro area harmonized index of consumer
prices. The left panel of Figure 2 shows the real price series jointly with
shaded grey areas representing the recessions in the Euro Area.
As for the predictors, we follow the approach of Boivin and Ng, (2006) and
Baumeister et al., (2022). Therefore, instead of collecting a large number of
series, we carefully select 21 predictors that capture demand and supply-side
forces driving the price of carbon. More precisely, we concentrate on
variables within the following categories:
* •
Economic activity (8 series): we collect data on aggregate industrial
production (IP) for the EU-19 area, as well as indices for sectors covered by
the EU ETS (i.e. electricity, gas, steam, and air conditioning supply, basic
metals, manufacture of paper and its products, coke and refined petroleum
products, chemical products, non-metallic mineral products) from Eurostat.
Moreover, we consider the Euro Stoxx 50 stock price index, sourced from
Refinitiv Eikon.
* •
Energy prices (7 series): we consider the prices of Brent crude oil, TTF
natural gas (front-month and front-year), ARA API-2 coal (front-year), German
power price (front-year), clean dark, and clean spark spreads (front month).
These variables are sourced from Refinitiv Eikon.
* •
Technical indicators (3 series): we select some of the variables that
practitioners use to track the functioning of the EU ETS market (Marcu et
al.,, 2023). The auction coverage ratio, defined as the total number of bids
in an auction divided by the number of available EUA, is a proxy for the
actual auction demand relative to supply on the primary market. As a rule of
thumb, a value greater (lower) than two indicates a high (low) auction demand
relative to supply. The auction clearing price, and a volatility proxy based
on the monthly auction price range. These variables are sourced from the EUA
Primary Market Auction Report maintained by EEX.
* •
Weather conditions (2 series): temperature and precipitation anomalies for
EU-19 countries are constructed as differences from long-term moving averages
using data sourced from the Weather for Energy Tracker maintained by the
International Energy Agency (IEA).
Figure 2: Real EU ETS price (left), actual and interpolated verified emissions
for EU-19 countries (right) from March 2005 to September 2023.
Notes: Interpolated emissions are obtained using IP indices of six sectors
covered by the EU ETS. Shaded areas represent recessions in the Euro Area, as
determined by the CEPR-EABCN (https://eabcn.org).
Moreover, we also collect information on verified emissions for all stationary
installations in six sectors covered by the EU ETS222We do not include the
aviation sector because it is covered by the EU ETS since 2012, while we
interpolate data on verified emissions using IP indices starting from 2006. in
the EU-19 countries. These data, available on an annual basis, refer to the
actual amount of greenhouse gas emissions produced by a company or entity, as
reported in its emissions report and verified by an accredited verifier by 31
March of the following year.333Based on verified emissions, companies must
surrender a corresponding number of emission allowances by the end of April of
that year. If a company’s verified emissions exceed the number of allowances
it holds, it may need to purchase additional allowances from the market to
cover the excess emissions. Following Bjørnland et al., (2023) and Känzig,
(2023), we temporally disaggregate annual data with the Chow and Lin, (1971)
method. Annual and interpolated monthly verified emissions are shown in the
right panel of Figure 2.444More precisely, we consider the IP indices and
emissions for all sectors covered by the EU ETS (except aviation) and
construct an emission-weighted IP index to interpolate annual verified
emissions. The MATLAB library provided by Quilis, (2013) is used to implement
the Chow-Lin method. Since verified emissions for 2023 were not available, we
used the 2022 value to interpolate data for 2023.
The forecasting exercise is based on data spanning from June 2012 to September
2023, comprising a total of 136 monthly observations. The start date of the
sample is dictated by the availability of data on auctions, which are
particularly relevant for explaining the allocation mechanism of emission
allowances during Phases III and IV of the EU ETS (see Figure 1). Forecasts
are generated using an expanding window approach: each time a new forecast is
produced, the estimation sample is updated by adding a new observation. The
first estimation sample ends in December 2017, and the last forecast is issued
in September 2022. We consider forecast horizons of one month up to one year
ahead. The forecast evaluation sample is the same for all forecast horizons
and consists of 58 observations spanning from December 2018 to September 2023.
We denote the level of the real price of carbon in month $t$ as $R_{t}$ and
the log price as $r_{t}=\log R_{t}$. Models are estimated using the first
difference of the logarithm of the real price of carbon, $\Delta r_{t}$, and
forecasts are constructed iteratively from the estimated models and converted
into levels as follows:
$\displaystyle\hat{R}_{t+h|t}$
$\displaystyle=\exp\left(r_{t}+\sum_{\ell=1}^{h}\Delta\hat{r}_{t+\ell|t}\right),$
where $\Delta\hat{r}_{t+\ell|t}$ is the $\ell$-step ahead forecasted value.
The evaluation of point forecasts relies on the relative Root Mean Squared
Forecast Error (RMSFE) that represents the ratio of the RMSFE of a model to
the RMSFE of the benchmark, such as the Random Walk (RW). Therefore, a
relative RMSFE lower than unity is taken as evidence that a certain model is
more accurate than the benchmark.
Sign, or direction-of-change forecasts, are defined as:
$\text{sign}\left(\hat{R}_{t+h|t}-R_{t}\right)$, where $\mathrm{sign}(x)$
equals -1 if $x<0$, 0 if $x=0$, and 1 if $x>0$. We use the Success Ratio (SR),
defined as the proportion of correctly predicted signs, to gauge the
directional accuracy. A SR greater than 0.5 indicates a gain in accuracy
relative to the RW model that implies a no-change forecast.
As a further measure of forecasting, we rely on the quantile-based continuous
ranked probability score (qCRPS) of Gneiting and Ranjan, (2011), which is a
density forecasting measure denoted by
$\widehat{QS}_{t}=\frac{1}{J-1}\sum_{j=1}^{J-1}\widehat{QS}_{t}^{\alpha_{j}}=\frac{1}{J-1}\sum_{j=1}^{J-1}2\left[\mathbb{I}\left(R_{t+h}\leq\hat{q}^{\alpha_{j}}_{t+h|t}\right)-\alpha_{j}\right]\times\left(\hat{q}^{\alpha_{j}}_{t+h|t}-R_{t+h}\right),$
(1)
where $\mathbb{I}(\cdot)$ denotes the indicator function and
$\hat{q}^{\alpha_{j}}_{t+h|t}$ is the $h$-step ahead quantile forecast for
$R_{t+h}$ at level $\alpha_{j}=j/J$ with $J=20$, which corresponds to
$\alpha_{j}=0.05,0.10,\ldots,0.95$. We construct weighted versions of the
qCRPS, where the weights are selected to emphasize specific regions, such as
the center or one of the tails of the distribution:
$\displaystyle\widehat{wQS}_{t}$
$\displaystyle=\frac{1}{J-1}\sum_{j=1}^{J-1}\nu(\alpha_{j})\widehat{QS}_{t}^{\alpha_{j}},\quad\text{with
}\nu(\alpha_{j})=\begin{cases}\alpha_{j}(1-\alpha_{j})&\text{(center)},\\\
\alpha_{j}^{2}&\text{(right tail)},\\\ (1-\alpha_{j})^{2}&\text{(left
tail)}.\end{cases}$ (2)
In a pairwise comparison, the model with the lowest score is ranked as the
most accurate.
## 4 Results
### 4.1 Univariate time series models
Most previous papers rely on daily or weekly nominal data, therefore in the
case of monthly real carbon prices, it is not clear which benchmark to use. In
the literature concerning the forecasting of commodity prices, particularly
for crude oil, the simple Random Walk (RW) model, or no-change forecast, is
commonly used as a benchmark. In this initial phase of the analysis, we focus
on point and sign forecasts, while we will consider density forecasts at a
later stage. Results are summarized in Table 1, which shows the relative RMFSE
and the SR of different models for selected forecast horizons of
$h=1,2,3,6,9,12$ months ahead.555In the Supplement, we display results for all
forecast horizons from 1 to 12 months ahead and for additional univariate time
series models.
In the short run, including a drift in the RW model (RWD) provides modest
gains in point forecast accuracy, while for forecasts ranging from one-quarter
up to one year ahead, accuracy is comparable to that of the RW model. Coupled
with an SR below 0.5 at the 12-month horizon, this implies that the RWD model
does not qualify as a more accurate benchmark for point and sign forecasting.
If Autoregressive (AR) and/or Moving Average (MA) components are included,
models perform poorly at horizons 1 and 2, but they lead to RMSFE reductions
at horizons from one quarter up to one year that can be as large as 16.98%.
See columns 3 and 4 of Table 1. Looking at the success ratio (panel b), ARIMA
models have some directional accuracy, especially at longer horizons.
Table 1: Relative RMSFE and Success Ratio of univariate time series models (a)
Relative RMSFE
---
| | First-difference, $\Delta r_{t}$
h | RWD | ARIMA(1,1,1) | ARIMA(0,1,1) | BAR(1) | BAR(3) | BAR(12) | BAR(aic)
1 | 0.993 | 1.069 | 1.071 | 1.043 | 1.050 | 1.046 | 1.047
2 | 0.999 | 1.010 | 1.002 | 1.003 | 1.003 | 1.013 | 1.004
3 | 1.0000 | 0.998 | 0.993 | 0.990 | 0.988 | 0.998 | 0.992
6 | 1.0000 | 0.905 | 0.901 | 0.903 | 0.903 | 0.923 | 0.911
9 | 1.0000 | 0.856 | 0.854 | 0.855 | 0.858 | 0.864 | 0.858
12 | 1.0000 | 0.830 | 0.831 | 0.830 | 0.834 | 0.848 | 0.834
(b) Success Ratio
| | First-difference, $\Delta r_{t}$
h | RWD | ARIMA(1,1,1) | ARIMA(0,1,1) | BAR(1) | BAR(3) | BAR(12) | BAR(aic)
1 | 0.603 | 0.535 | 0.535 | 0.517 | 0.535 | 0.517 | 0.517
2 | 0.621 | 0.517 | 0.517 | 0.517 | 0.517 | 0.621 | 0.517
3 | 0.638 | 0.535 | 0.569 | 0.603 | 0.621 | 0.638 | 0.603
6 | 0.776 | 0.759 | 0.759 | 0.776 | 0.776 | 0.776 | 0.776
9 | 0.569 | 0.828 | 0.828 | 0.828 | 0.828 | 0.828 | 0.828
12 | 0.431 | 0.897 | 0.897 | 0.897 | 0.897 | 0.897 | 0.897
Notes: Panel (a) shows the ratio of RMSFE of model $m$ over the RW model.
Values below one suggest superior forecast performance of model $m$ to the RW
(in bold). ∗ denotes that the null hypothesis of the Diebold-Mariano test is
rejected at the 90% (95%) confidence level. Panel (b) reports the success
ratios; entries in bold suggest that the model can accurately predict the
direction of change over 50% of the time. ∗ indicates that the p-value of the
Pesaran and Timmermann, (2009) test of the null of no directional accuracy is
below 0.1, hence providing evidence of statistical accuracy at the 10%
significance level. Models yielding the lowest RMSFE or highest SR are
underlined.
We conclude with Autoregressive, AR($p$), models specified for the log first
difference estimated with Bayesian natural conjugate prior and denoted as BAR
(columns 5-8). We consider fixed lag orders $p$ = 1, 3, and 12, as well as lag
order selection based on the Akaike Information Criterion (AIC). In this case,
we set the maximum lag order to 12 and select $p$ each time a new forecast is
issued. Indeed, BAR models for $\Delta r_{t}$ display directional accuracy and
lead to RMSFE reductions that reach 17% at horizon 12 in the case of a simple
BAR(1). Moreover, it is worth pointing out that fixed and small lag orders are
preferable to either setting $p=12$ or selecting $p$ recursively with the AIC.
To sum up, given that at horizons 1 and 2, not even BAR models for $\Delta
r_{t}$ outperform the RW, we cannot definitively discard it as a benchmark.
In Table 1, we have used the Diebold and Mariano, (1995) test, as modified by
Coroneo and Iacone, (2020), to verify the statistical significance of RMSFE
reductions, and the Pesaran and Timmermann, (2009) test to assess the
statistical significance of SR. Possibly due to the small size of the
evaluation sample or because of instabilities in forecasting performance, we
are never able to provide evidence of statistically significant improvements
over the RW. We address the presence of forecast instabilities in Sections 4.4
and 4.5.
### 4.2 VAR models of the EU ETS carbon market
To link the real price of carbon to its determinants, a natural starting point
is a small-scale VAR(p) model of the EU ETS market:
$\mathbf{y}_{t}=\mathbf{a}+\sum_{j=1}^{p}\mathbf{A}_{j}\mathbf{y}_{t-j}+\mathbf{u}_{t},$
(3)
where $\mathbf{a}$ is a $n\times 1$ vector of intercepts, $\mathbf{A}_{j}$ are
$n\times n$ matrices of coefficients for $j=1,\ldots,p$, and $\mathbf{u}_{t}$
is an $n\times 1$ vector of zero-mean innovations Normally distributed with
covariance matrix $\boldsymbol{\Sigma}$. Following Bjørnland et al., (2023),
we consider a baseline VAR specification in which $\mathbf{y}_{t}$ is a
$3\times 1$ vector including $\Delta r_{t}$ (or alternatively, $r_{t}$), the
first difference of the logarithm of interpolated verified emissions,
$emis_{t}$, and first difference of the logarithm of aggregate industrial
production for EU-19 countries, $\Delta ip_{t}$.
The baseline model does not include many predictors tracked by practitioners,
such as energy prices, technical indicators related to the auctioning of EU
ETS allowances, and weather anomalies that might affect the demand for
electricity and gas. Differently from Bjørnland et al., (2023), we consider
also the Euro Stoxx stock price index as an additional proxy of real economic
activity, as well as a wider set of IP indices. In particular, we add IP
indices for the main sectors covered by the EU ETS to better capture demand-
side pressures affecting the real price of carbon.
Given the relatively small size of our sample of data, including the set of 21
predictors described in Section 3 in the model would not be feasible.
Similarly, we want to avoid sparse model representations implied by the
selection of only a handful of the potentially relevant predictors. Thus, we
consider a Factor-augmented VAR model (FAVAR), where we estimate a modified
version of the baseline VAR specification, in which we replace the aggregate
IP index for EU-19 with up to three factors extracted from the set of 21
predictors.
In detail, we pool the information from the 21 predictors based on principal
component analysis.666Before the analysis, all variables have been transformed
to induce stationarity and then standardized. IP indices, the Euro Stoxx price
index, energy prices, and the auction price are transformed into monthly
growth rates (first differences of logarithms). Clean dark and clean spark
spreads are first-differenced, while the monthly auction price range is log-
transformed. We do not apply any transformation to the auction cover ratio,
temperature, and precipitation anomalies. A preliminary screening for outliers
was also carried out. However, given that only a few extreme observations
(i.e. observations exceeding 20 times the interquartile range from the median)
were detected during the COVID pandemic, we decided to keep them in the
sample. We decided to focus on the first three factors since they account for
48% of the variance of the 21-time series. The percentages of total variance
explained by the first, second, and third factors are 22%, 17.8%, and 8.6%,
respectively.
Figure 3: $R^{2}$ between factors and individual predictors obtained from
regressing a factor on an individual predictor using data from June 2012 to
December 2017.
Figure 3 presents the $R^{2}$ from regressing a factor on an individual
predictor using data from June 2012 to December 2017. Broadly speaking, the
first factor loads mostly on energy prices. The second-factor loads on a
combination of a few IP indices and the cover ratio, while the third loads
heavily on the two spreads, precipitation anomalies, and the remaining IP
indices.
As shown in the coming sections, the forecasting exercise will show that a
BVAR(1) model augmented with one factor predicts both the real price of carbon
and verified emissions adequately and better than alternative models.
Therefore, it is useful to characterize the variables underlying the dynamics
of the first factor. Figure 4 indicates that the first factor tracks the
European business cycle as well as the evolution of energy prices and is
highly correlated with both IP growth and returns of TTF natural gas. Indeed,
during the COVID-19 recession, the first factor is negative and well below its
mean, while it becomes positive in the recovery phase and for most of the
first semester of 2022 when natural gas prices rise due to the Russian
invasion of Ukraine.
Figure 4: Correlation between Factor 1 and industrial production (top) and
growth rate of the price of TTF natural gas (bottom) from December 2017 to
September 2023. Notes: we depict centered 3-month moving average of Factor 1
and predictors (scaled to have same variance as Factor 1).
Figure 5 displays the evolution of the first factor over time, along with bars
representing the contribution of different predictors grouped by class (i.e.
economic activity, energy prices, technical indicators, and weather
anomalies). Interestingly, the drivers underlying the movements of the first
factor change over time, but are always mostly related to economic activity
proxies and energy prices. Measures of real economic activity account for the
largest share of the downward movement of the factor in 2020 when European
economies were frozen during lockdowns. The upward movement of the factor in
2022 and its successive decline in 2023 are attributed to the pressures
related to the war in Ukraine and the subsequent easing of conditions in
energy markets.
Figure 5: Contribution of predictors (grouped by class) to Factor 1,
represented as bars, along with the evolution of Factor 1 (green line) from
December 2017 to September 2023. Notes: we depict the centered 3-month moving
average of both Factor 1 and the predictors’ contributions to it.
### 4.3 Can VAR models forecast the real price of carbon?
In this section, we focus on Bayesian techniques (BVAR or BFAVAR) with
Minnesota prior to forecast the real price of carbon. Overall, both for
baseline BVAR models and factor-augmented specifications, setting the lag
order to 1 or 3 yields the most accurate point and direction-of-change
forecasts. Indeed, inspecting the leftmost column of Table 2, the baseline
BVAR model with $p=12$ – as set by Bjørnland et al., (2023) – is regularly
outperformed by specifications with $p=1$ and $p=3$.777In the Supplement, we
have estimated the same models with the real price of carbon log-levels, and
point forecasts were always less accurate than growth rates estimation. As for
the lag length of BVAR models, we consider pre-specified values of $p=1,3,12$
along with a data-driven approach that sets $p$ by minimizing the AIC at each
forecast origin, selecting among BVAR($p$) models up to order 12.
Table 2 shows that at horizons up to 2 months, BVAR and BFAVAR models are as
accurate as no-change point forecasts. However, as the forecast horizon grows,
BVAR and BFAVAR models lead to sizable accuracy gains both for point and sign
forecasts. For instance, at a one-year horizon, the BVAR(1) and factor-
augmented BVAR(1) model result in a reduction in RMSFE in the range 17.8-18.2%
and can accurately predict the direction of price movements 89.7% of the time.
Table 2: Relative RMSFE and Success Ratio for Bayesian Vector Autoregressive
and Factor models. (a) Relative RMSFE
---
| | 1 Factor | 2 Factors | 3 Factors
h | BVAR(1) | BVAR(3) | BVAR(12) | BVAR(1) | BVAR(3) | BVAR(1) | BVAR(3) | BVAR(1) | BVAR(3)
1 | 1.057 | 1.075 | 1.074 | 1.062 | 1.073 | 1.039 | 1.077 | 1.062 | 1.114
2 | 1.006 | 1.019 | 1.028 | 1.010 | 1.028 | 1.024 | 1.102 | 1.066 | 1.154
3 | 0.995 | 0.996 | 1.002 | 1.002 | 1.019 | 0.999 | 1.052 | 1.015 | 1.062
4 | 0.953 | 0.952 | 0.970 | 0.947 | 0.953 | 0.963 | 0.960 | 0.976 | 0.975
5 | 0.965 | 0.964 | 0.984 | 0.957 | 0.960 | 0.970 | 0.971 | 0.985 | 0.983
6 | 0.915 | 0.916 | 0.937 | 0.906 | 0.916 | 0.921 | 0.919 | 0.935 | 0.927
7 | 0.897 | 0.899 | 0.917 | 0.886 | 0.899 | 0.902 | 0.894 | 0.916 | 0.899
8 | 0.844 | 0.852 | 0.858 | 0.831 | 0.847 | 0.850 | 0.856 | 0.867 | 0.863
9 | 0.857 | 0.865 | 0.873 | 0.851 | 0.872 | 0.856 | 0.901 | 0.863 | 0.894
10 | 0.825 | 0.836 | 0.838 | 0.824 | 0.850 | 0.830 | 0.887 | 0.833 | 0.881
11 | 0.822 | 0.837 | 0.843 | 0.824 | 0.849 | 0.828 | 0.886 | 0.831 | 0.883
12 | 0.818 | 0.834 | 0.848 | 0.822 | 0.849 | 0.828 | 0.892 | 0.834 | 0.891
(b) Success Ratio
| | 1 Factor | 2 Factors | 3 Factors
h | BVAR(1) | BVAR(3) | BVAR(12) | BVAR(1) | BVAR(3) | BVAR(1) | BVAR(3) | BVAR(1) | BVAR(3)
1 | 0.552 | 0.569 | 0.517 | 0.552 | 0.500 | 0.569 | 0.569 | 0.552 | 0.569
2 | 0.535 | 0.552 | 0.569 | 0.535 | 0.569 | 0.552 | 0.569 | 0.552 | 0.535
3 | 0.603 | 0.621 | 0.621 | 0.621 | 0.603 | 0.621 | 0.655∗ | 0.603 | 0.603
4 | 0.707 | 0.672 | 0.672 | 0.690 | 0.655 | 0.707 | 0.690 | 0.690 | 0.672
5 | 0.672 | 0.672 | 0.655 | 0.672 | 0.638 | 0.672 | 0.655 | 0.672 | 0.672
6 | 0.776 | 0.759 | 0.759 | 0.776 | 0.741 | 0.776 | 0.724 | 0.776 | 0.759
7 | 0.793 | 0.776 | 0.793 | 0.793 | 0.759 | 0.793 | 0.759 | 0.793 | 0.776
8 | 0.845 | 0.828 | 0.845 | 0.845 | 0.810 | 0.845 | 0.810 | 0.845 | 0.828
9 | 0.828 | 0.810 | 0.828 | 0.828 | 0.793 | 0.828 | 0.793 | 0.828 | 0.810
10 | 0.862 | 0.828 | 0.862 | 0.862 | 0.828 | 0.862 | 0.810 | 0.862 | 0.828
11 | 0.914 | 0.879 | 0.914 | 0.914 | 0.879 | 0.914 | 0.862 | 0.914 | 0.874
12 | 0.897 | 0.879 | 0.897 | 0.897 | 0.862 | 0.897 | 0.845 | 0.897 | 0.862
Notes: see notes to Table 1
At intermediate forecast horizons (i.e., 4 up to 10 months ahead), the BVAR(1)
model augmented with a single factor regularly yields the most accurate point
forecasts, and its performance tends to improve as the horizon grows.
Increasing the number of factors above one generally leads to less accurate
point forecasts. In particular, adding the third factor does not seem to
provide any advantage to models with one or two factors. The model with two
factors, on the other hand, seems to be slightly better than the single-factor
specification at shorter forecast horizons.
### 4.4 Are there instabilities in forecasting performance?
There is widespread evidence that the relative forecasting performance of
models changes over time due to parameter instability, shocks with time-
varying volatilities, and changes in the variance of the predictors (Giacomini
and Rossi,, 2010; Rossi,, 2021). In such cases, averaging the difference in
forecasting performance over the full evaluation sample, as we did in the
previous sections, results in a loss of information that might lead to
standard tests of predictive ability to conclude that two competing models are
equally accurate.
In Figure 6, we investigate the existence of instabilities in forecasting
performance. We do so by implementing the fluctuation test statistic,
$\mathcal{F}_{t}^{OOS}$, of Giacomini and Rossi, (2010) considering a centered
moving average over a $19$-month window.888The test has a nonstandard
distribution, and the critical values provided by Giacomini and Rossi, (2010)
depend on the ratio between the size of the window used to compute the moving
average and the number of observations in the evaluation sample. In the
Supplement, we show that the results are robust to changes in the size of the
window. The test is based on the (standardized) difference between the MSFE of
a benchmark model and of the factor-augmented BVAR(1) model with one or two
factors. Positive values of the test statistic indicate that the BFAVAR model
has a lower MSFE than the benchmark. All tests are one-sided: the null
hypothesis is the factor VAR model has the same MSFE as the benchmark, while
the alternative is that the former is more accurate than the latter. The
dashed line indicates the 5% critical value, $CV_{0.05}$, and the null
hypothesis is rejected when $\max\mathcal{F}_{t}^{OOS}>CV_{0.05}$.
In our comparison, we focus on one month, one quarter, and one year ahead
forecast horizons. To raise the bar of forecast evaluation, we do not solely
focus on the RW model. Instead, we compare the performance of factor models
against the BAR(1) and the BVAR(1) models, which, as shown in previous
sections, appear to be more competitive benchmarks than the RW model.
Figure 6: Fluctuation test statistic for a BFAVAR with 1 factor (left) and 2
factors (right) against different benchmarks for forecast horizons $h=1,3,12$
months from September 2019 to December 2022. Notes: the fluctuation test
statistic, $\mathcal{F}_{t}^{OOS}$, of Giacomini and Rossi, (2010), is
calculated with a 19-month centered rolling window. Positive values indicate
that the BFAVAR model is better than the benchmark. All tests are one-sided,
with the null hypothesis being that the BFAVAR(1) model has the same MSFE as
the benchmark; the alternative is that the BFAVAR(1) model forecasts better
than the benchmark. The dashed line indicates the one-sided 5% critical value,
$CV_{0.05}$. The null hypothesis is rejected when
$\max\mathcal{F}_{t}^{OOS}>CV_{0.05}$.
Several interesting results emerge from Figure 6. First, regardless of the
benchmark model, the relative forecasting performance of BVAR specifications –
augmented with either one or two factors – changes over time and tends to
deteriorate at the end of 2022. Indeed, especially at shorter forecast
horizons, the test statistic often becomes negative in this period. Second,
the model with one factor (left panel) is usually preferable to the model with
two factors (right panel), in that the latter rarely beats the benchmarks.
Third, at a forecast horizon of one month, the single-factor model performs
better than the RW model until the beginning of 2021; while it never beats the
BAR(1). However, at longer forecast horizons and especially for $h=12$, the
single-factor model performs better than the benchmarks during most of the
evaluation sample. Moreover, for $h=12$, the fluctuation test leads to a
rejection of the null hypothesis of indistinguishable forecasting performance;
in all cases, the test statistic lies above the 5% critical value in late 2020
and in part of 2021.
Thus the single-factor BVAR(1) model, condensing information of a broad set of
predictors related to the EU ETS, is a promising specification for forecasting
the magnitude as well as the direction-of-change of the real price of carbon
at longer horizons. At horizons shorter than a quarter, VAR models do not
offer a real advantage over simple univariate specifications, especially in
point forecasting.
### 4.5 Density forecasts and the role of stochastic volatility
To gauge the uncertainty associated with point forecasts and following
evidence that modeling SV improves density forecasts of macroeconomic
aggregates (Clark and Ravazzolo,, 2015; Chan,, 2023), we evaluate density
forecasting for future values of the real price of carbon.
We also decided to include in BVAR models a Choleski multivariate SV process
(Chan,, 2023, for details):
$\displaystyle\mathbf{u}_{t}$ $\displaystyle\sim
N(\mathbf{0},\boldsymbol{\Sigma}_{t}),\quad\boldsymbol{\Sigma}_{t}^{-1}=\mathbf{B}_{0}^{\prime}\mathbf{D}_{t}^{-1}\mathbf{B}_{0},$
(4) $\displaystyle h_{i,t}$
$\displaystyle=\mu_{i}+\phi_{i}\left(h_{i,t-1}-\mu_{i}\right)+\varepsilon_{i,t},\quad\varepsilon_{i,t}\sim
N(0,\sigma^{2}_{\varepsilon}),$ (5)
where $\mathbf{B}_{0}$ is a lower triangular matrix of size $n\times n$ with
ones along the main diagonal and
$\mathbf{D}_{t}=\text{diag}\left(e^{h_{1,t}},\cdots,e^{h_{n,t}}\right)$, and
Equation (5) specifies an independent autoregressive volatility process for
each variable in the model for $t=2,\ldots,T$ with initial condition
$h_{i,1}\sim N(\mu_{i},\sigma^{2}_{i}/(1-\phi_{i}^{2}))$.
To evaluate the density performance, we rely on qCRPS (Gneiting and Ranjan,,
2011), where the model with the lowest qCRPS is ranked as the most
accurate.999Note that, contrary to the previous tables, these numbers are not
ratios to a benchmark but are expressed in the same scale as real prices. In
Panel (a) of Table 3, we focus on the center of the distribution, while the
accuracy in forecasting the right and left tails of the distribution, which
are of interest to assess the probability of extreme price movements, is
evaluated in Panel (b) and (c), respectively.
Table 3: Quantile-weighted Continuous Ranked Probability Score (qCRPS). (a)
Quantile-weighted CRPS: center
---
| | 1 Factor | 2 Factors | | 1 Factor | 2 Factors
h | BVAR(1) | BVAR(1) | BVAR(1) | BVAR-SV(1) | BVAR-SV(1) | BVAR-SV(1)
1 | 0.729 | 0.721 | 0.692 | 0.712 | 0.714 | 0.704
2 | 0.978 | 0.966 | 0.968 | 0.977 | 0.974 | 0.983
3 | 1.231 | 1.226 | 1.200 | 1.257 | 1.252 | 1.253
4 | 1.322 | 1.310 | 1.318 | 1.349 | 1.341 | 1.355
5 | 1.625 | 1.611 | 1.623 | 1.648 | 1.639 | 1.659
6 | 1.670 | 1.649 | 1.663 | 1.701 | 1.689 | 1.712
7 | 1.790 | 1.761 | 1.782 | 1.804 | 1.793 | 1.819
8 | 1.860 | 1.831 | 1.861 | 1.886 | 1.877 | 1.905
9 | 2.121 | 2.102 | 2.109 | 2.144 | 2.139 | 2.161
10 | 2.180 | 2.169 | 2.185 | 2.245 | 2.248 | 2.277
11 | 2.325 | 2.325 | 2.336 | 2.407 | 2.404 | 2.432
12 | 2.454 | 2.472 | 2.480 | 2.587 | 2.585 | 2.612
(b) Quantile-weighted CRPS: right tail
| | 1 Factor | 2 Factors | | 1 Factor | 2 Factors
h | BVAR(1) | BVAR(1) | BVAR(1) | BVAR-SV(1) | BVAR-SV(1) | BVAR-SV(1)
1 | 1.286 | 1.217 | 1.124 | 1.237 | 1.220 | 1.198
2 | 1.733 | 1.676 | 1.663 | 1.681 | 1.664 | 1.678
3 | 2.207 | 2.131 | 2.088 | 2.166 | 2.139 | 2.170
4 | 2.392 | 2.315 | 2.327 | 2.358 | 2.332 | 2.389
5 | 2.915 | 2.835 | 2.847 | 2.845 | 2.815 | 2.880
6 | 3.111 | 3.042 | 3.064 | 3.036 | 3.015 | 3.084
7 | 3.324 | 3.219 | 3.268 | 3.235 | 3.208 | 3.299
8 | 3.515 | 3.394 | 3.452 | 3.462 | 3.437 | 3.544
9 | 3.984 | 3.890 | 3.917 | 3.914 | 3.900 | 4.008
10 | 4.304 | 4.230 | 4.264 | 4.302 | 4.293 | 4.417
11 | 4.630 | 4.576 | 4.596 | 4.638 | 4.634 | 4.757
12 | 5.015 | 4.9996 | 5.008 | 5.090 | 5.098 | 5.222
(b) Quantile-weighted CRPS: left tail
| | 1 Factor | 2 Factors | | 1 Factor | 2 Factors
h | BVAR(1) | BVAR(1) | BVAR(1) | BVAR-SV(1) | BVAR-SV(1) | BVAR-SV(1)
1 | 1.270 | 1.282 | 1.235 | 1.278 | 1.288 | 1.268
2 | 1.599 | 1.605 | 1.588 | 1.677 | 1.671 | 1.680
3 | 1.897 | 1.914 | 1.834 | 2.046 | 2.026 | 2.001
4 | 1.943 | 1.958 | 1.953 | 2.116 | 2.091 | 2.097
5 | 2.404 | 2.412 | 2.410 | 2.606 | 2.572 | 2.598
6 | 2.303 | 2.292 | 2.306 | 2.506 | 2.464 | 2.491
7 | 2.471 | 2.467 | 2.471 | 2.642 | 2.609 | 2.629
8 | 2.534 | 2.506 | 2.562 | 2.689 | 2.656 | 2.686
9 | 2.901 | 2.888 | 2.863 | 3.082 | 3.052 | 3.048
10 | 2.806 | 2.803 | 2.811 | 3.021 | 2.998 | 3.001
11 | 2.946 | 2.960 | 2.963 | 3.179 | 3.155 | 3.165
12 | 3.007 | 3.029 | 3.053 | 3.289 | 3.266 | 3.282
Notes: The best forecasts, associated with the lowest scores, are underlined.
Figure 7: Fluctuation test statistic for the right tail for BFAVAR with 1
factor (left) and 2 factors (right). September 2019 to December 2022.
With very few exceptions, homoskedastic factor-augmented BVAR models – with
either one or two factors – are more accurate at forecasting the center and
both tails than the alternative specifications. In the case of the real price
of carbon, incorporating SV does not seem to offer any sizable advantage on
any horizon. Interestingly, the homoskedastic BVAR(1) model augmented with two
factors is more accurate than any other specifications at horizons up to a
quarter-ahead. This result stands in contrast to what is observed for point
and sign forecasts, for which factor-augmented models do not yield accuracy
gains at shorter horizons. Moreover, it emphasizes that, for short-term
density forecasts, the second factor, which captures predictors beyond energy
prices and economic activity that are the main drivers of the first factor
mostly, might be relevant.101010See the Supplement for results concerning the
inclusion of SV in BVAR models.
We complement these results with evidence of instabilities in the (right) tail
forecasting. When evaluating the single-factor model, against the BAR(1) and
BVAR(1) models, results in the first column of Figure 7 allow rejecting the
null of equal accuracy in right tail forecasting at all forecast horizons.
Therefore, the relative forecasting performance is not stable, but in several
months, the single-factor model offers accuracy gains. As for the
specification with two factors, we can see that the null is rejected at all
horizons when the benchmark is the BAR(1) model and at horizons 1 and 12 when
the benchmark is the BVAR(1) model. Results concerning the center of the
distribution largely mimic what is observed for point forecasts: the single-
factor model does better than the benchmarks at horizons of one quarter and
one year.111111In the Supplement, we consider the center and the left tail as
well as robustness checks involving changes in the implementation of the
fluctuation test. We have considered if dropping verified emissions from BVAR
and BFAVAR models could alter our main conclusions, but the relative ranking
of models is not affected.
## 5 Expert forecasts, verified emissions and market monitoring
### 5.1 Expert forecasts of the nominal price of carbon
As a further step, we provide a qualitative comparison of the single-factor
BVAR(1) point forecasts against those issued by the Carbon Team at the London
Stock Exchange Group (LSEG; formerly Refinitiv), and its survey forecasts. In
both cases, LSEG provides forecasts expressed in current euros; thus, we need
to transform our real price forecasts into nominal terms by producing
inflation forecasts from an Unobserved Component SV model and using them to
get nominal price forecasts.121212Given the short time span of our evaluation
sample, using different inflation forecasts, such as RW forecasts, does not
alter the results.
LSEG Carbon Team’s forecasts. The Carbon Team at LSEG produces forecasts of
nominal EU ETS price with an irregular cadence for the period 2014-2023
ranging from 3 to 6 times per year, where the forecast horizon can be either
the current year or several years in the future. One challenge of working with
these data is that they are “fixed event forecasts”, while our models produce
“fixed horizon forecasts”. The characteristic of “fixed event forecasts” is
that the forecast horizon changes as the forecast origin moves forward. At
each forecast origin, the LSEG team produces forecasts for the current year,
$f^{FE}_{t+k|t}$, and for the next year, $f^{FE}_{t+k+12|t}$ where
$k=1,...,12$ represents the number of months until the end of the year (e.g.
$k=12$ in January and $k=1$ in December). To approximate one year ahead fixed
horizon forecasts, $f_{t+12|t}^{FH}$, using LSEG’s fixed event forecasts, we
follow Dovern et al., (2012):
Figure 8: Nominal EU ETS price and forecasts. Notes: Nominal EU ETS price (red
line), LSEG (blue dots), RW (dashed black line), BFAVAR(1) one-year-ahead
forecasts (dotted blue line). The bars represent the difference between the
RMSFE of the LSEG forecast and RW (black) or BFAVAR(1) (blue). Positive values
indicate that LSEG is less accurate than the alternative forecast.
$f_{t+12|t}^{FH}=\frac{k}{12}f^{FE}_{t+k|t}+\frac{12-k}{12}f^{FE}_{t+k+12|t},$
(6)
where weights are proportional to the degree of overlap of the two fixed event
forecasts.131313A fixed horizon forecast issued in January, $k=12$, would
therefore be equal to $f_{t+12|t}^{FH}=f^{FE}_{t+12|t}$. We obtain a set of 39
one-year-ahead forecasts irregularly spaced over the period January 2015 - May
2023; only 19 of these forecasts overlap with those in our evaluation period
spanning from December 2018 to September 2023.
Figure 8 shows that LSEG and RW forecasts are remarkably similar; indeed, the
correlation of the respective forecast errors is 0.95 over the period January
2015 - May 2023. For the period when the single-factor BVAR(1) model forecasts
overlap with those from LSEG, the model-based RMSFE is smaller than LSEG’s
RMSFE in 15 cases out of 19. All in all, while the short sample of data only
allows for a qualitative comparison, these results show that the model-based
forecasts are strikingly different from those issued by LSEG, which, on the
other hand, are similar to those from a RW model.
Figure 9: Survey and model-based one-year-ahead density forecasts of the
nominal EU ETS price from 2021 to 2023. Notes: each year the survey provides a
different set of price ranges across which respondents can choose. The title
indicates the date of the forecast, while the data come from the survey of the
previous year. The 2020 survey – leftmost plot – provided only three
categories “$<25$”, “About 25” and “$>25$” Euro per tCO2e.
Survey forecasts. Carbon Market Surveys run by LSEG each year (from 2020 to
2022) between February and April (in 2020 it was closed in March) capture the
market sentiment of respondents from a multitude of countries who are mostly
stakeholders with tangible and financial interests in carbon markets (e.g.,
traders, firms covered by an ETS). Participants who have answered the section
of the survey concerning EU ETS price expectations are 60 in 2020, 119 in
2021, and 88 in 2022. Since also survey forecasts are of the fixed event type,
we approximate fixed horizon one-year-ahead prediction densities derived from
surveys and compare them with prediction densities from the BFAVAR model and
the realized price at the target date (see Figure 9).
Only in 2023 survey and model-based forecasts are aligned and centered around
the realized price. In 2021 and 2022, both the survey and the single-factor
BVAR model tend to under-forecast and hence largely miss the run-up in the
European carbon price over this period. Nevertheless, we notice that the
realized prices lie in the right tail of the one-year-ahead prediction density
of the BFAVAR model. The fact that forecasting the price for March 2021 a year
in advance was a hard task appears evident, noticing that the price lies
outside the 95% BFAVAR prediction interval only in this case. The 2021
forecast is based on survey data collected in March 2020, at the beginning of
the COVID-19 pandemic, when overall macroeconomic uncertainty was at a record
level, especially in European countries. Similarly, also BFAVAR’s one-year-
ahead forecasts for 2021 and 2022 are based on macroeconomic data for the
COVID-19 recession period, without making any adjustments to them. These
factors should be kept in mind when evaluating the prediction accuracy for
2021 and 2022 in absolute terms.
### 5.2 Verified emissions forecasts
Forecasts of verified emissions are as crucial as price forecasts, and they
have received essentially no attention in the academic literature. Entities
participating in the EU ETS need reliable emission forecasts to plan emission
reduction strategies and ensure compliance with regulatory requirements, while
environmental agencies rely on these forecasts to assess the impact of
emissions reduction initiatives. BVAR models considered in this paper might
represent a valuable tool to produce verified emission forecasts.
Table 4 shows that, as far as point forecasts are concerned, the baseline
BVAR(1)-SV model represents the most accurate model at all horizons above 1
month. Interestingly, since data on verified emissions are published once a
year in April, the one-year-ahead forecast horizon appears particularly
relevant. Indeed, the baseline BVAR(1)-SV model and the factor-augmented
BFAVAR(1)-SV models yield RMSFE reductions over 7%. Moving to density
forecasting, Table 5 highlights that the BVAR(1)-SV and the single-factor
BFAVAR(1)-SV models yield the most accurate forecasts of the center of the
distribution of verified emissions.
Table 4: Relative RMSFE for verified emissions. | | | 1 Factor | 2 Factors | | | 1 Factor | 2 Factors
---|---|---|---|---|---|---|---|---
h | BAR(1) | BVAR(1) | BVAR(1) | BVAR(1) | BAR(1)-SV | BVAR(1)-SV | BVAR(1)-SV | BVAR(1)-SV
1 | 1.102 | 1.241 | 1.182 | 1.237 | 1.005 | 1.019 | 1.029 | 1.018
2 | 1.055 | 1.230 | 1.166 | 1.204 | 0.999 | 0.997 | 1.002 | 0.998
3 | 1.046 | 1.252 | 1.174 | 1.218 | 0.998 | 0.986 | 0.989 | 0.988
4 | 1.042 | 1.268 | 1.175 | 1.226 | 0.997 | 0.975 | 0.979 | 0.978
5 | 1.033 | 1.247 | 1.154 | 1.204 | 0.996 | 0.963 | 0.965 | 0.965
6 | 1.029 | 1.238 | 1.141 | 1.193 | 0.996 | 0.956 | 0.958 | 0.957
7 | 1.023 | 1.209 | 1.117 | 1.166 | 0.995 | 0.948 | 0.950 | 0.950
8 | 1.021 | 1.195 | 1.104 | 1.153 | 0.996 | 0.944 | 0.946 | 0.946
9 | 1.018 | 1.173 | 1.086 | 1.133 | 0.996 | 0.939 | 0.942 | 0.941
10 | 1.014 | 1.149 | 1.071 | 1.112 | 0.996 | 0.935 | 0.938 | 0.938
11 | 1.014 | 1.143 | 1.065 | 1.106 | 0.996 | 0.930 | 0.934 | 0.933
12 | 1.012 | 1.129 | 1.053 | 1.093 | 0.996 | 0.926 | 0.929 | 0.929
Notes: see notes to Table 1 Table 5: Quantile-weighted Continuous Ranked
Probability Score (qCRPS) for verified emissions. (a) Quantile-weighted CRPS:
center
---
| | | 1 Factor | 2 Factors | | | 1 Factor | 2 Factors
h | BAR(1) | BVAR(1) | BVAR(1) | BVAR(1) | BAR(1)-SV | BVAR-SV(1) | BVAR-SV(1) | BVAR-SV(1)
1 | 154.876 | 154.834 | 154.813 | 154.799 | 154.837 | 154.819 | 154.812 | 154.824
2 | 154.875 | 154.813 | 154.787 | 154.774 | 154.791 | 154.775 | 154.759 | 154.781
3 | 154.890 | 154.810 | 154.786 | 154.771 | 154.757 | 154.743 | 154.725 | 154.750
4 | 154.929 | 154.836 | 154.814 | 154.798 | 154.746 | 154.734 | 154.716 | 154.742
5 | 154.975 | 154.868 | 154.853 | 154.839 | 154.743 | 154.730 | 154.712 | 154.741
6 | 155.076 | 154.958 | 154.946 | 154.931 | 154.795 | 154.782 | 154.765 | 154.795
7 | 155.180 | 155.055 | 155.046 | 155.035 | 154.851 | 154.839 | 154.825 | 154.856
8 | 155.332 | 155.200 | 155.194 | 155.191 | 154.953 | 154.9380 | 154.928 | 154.966
9 | 155.424 | 155.284 | 155.282 | 155.301 | 154.997 | 154.983 | 154.974 | 155.020
10 | 155.583 | 155.442 | 155.441 | 155.451 | 155.109 | 155.093 | 155.091 | 155.134
11 | 155.698 | 155.551 | 155.550 | 155.568 | 155.175 | 155.156 | 155.160 | 155.207
12 | 155.792 | 155.632 | 155.633 | 155.658 | 155.226 | 155.201 | 155.208 | 155.257
Notes: see notes to Table 3
### 5.3 Market monitoring
Following Baumeister et al., (2022), we construct indices of demand pressure,
upward, and downward price pressure for the EU ETS market. For approximating
demand pressure, we take the difference between one-year and one-month-ahead
verified emission forecasts from the BFAVAR(1)-SV model. A negative value of
the proxy signals expectations of loosening market conditions over the next
year.
Using the predictive densities delivered by Bayesian estimation of the single-
factor BVAR(1) model, upward and downward price pressure indices are defined
as follows:
$\displaystyle PP^{+}_{t}$
$\displaystyle=\frac{1}{12}\sum_{h=1}^{12}\mathbb{I}\left[\hat{R}_{t+h|t}>\max\left(R_{t},R_{t-1},\ldots,R_{t-11}\right)\right],$
(7) $\displaystyle PP^{-}_{t}$
$\displaystyle=\frac{1}{12}\sum_{h=1}^{12}\mathbb{I}\left[\hat{R}_{t+h|t}<\min\left(R_{t},R_{t-1},\ldots,R_{t-11}\right)\right].$
(8)
These proxies estimate the probability that over the next 12 months, the real
price of carbon is above (below) the maximum (minimum) value observed in the
previous year.
The upper panel of Figure 10 displays the demand pressure proxy as a set of
bars along with a line representing verified emissions. Note that, although
verified emissions are available at an annual sampling frequency, BVAR models
are estimated with monthly data; therefore, the demand pressure index allows
monitoring expectations about the EU ETS market in real time each month. The
index is always negative to track the long-term decline of verified emissions,
interspersed with an increase in 2021 in the aftermath of the COVID-19
recession.
Figure 10: Demand and price pressure indices for the EU ETS market from March
2018 to August 2023. Notes: we plot the backward 3-month moving average of
price and demand pressure indices. Indices are aligned with the forecast
origin.
As shown by Bjørnland et al., (2023), the dynamics of verified emissions are
elicited by three main forces: a supply shock and two distinct demand shocks.
On the supply side, we have EU ETS regulation: the cap and trade scheme has
progressively tightened the limit of total greenhouse gas emissions of covered
entities. This is the first element factored in the negative demand pressure
index. The two demand shocks are related to economic activity and transition
demand. The first essentially depends on the business cycle and, therefore,
can be associated with positive or negative changes in the demand pressure
index. For instance, during the COVID pandemic, expectations of a future quick
recovery associated with an increase in industrial activity are captured by a
rebound of the demand pressure index. Much like the supply-side shock, the
transition-demand shock – capturing, among other things, the increased usage
of renewables – contributes to the steady decline in verified emissions and
therefore keeps the pressure index in negative territory.
The two price pressure indices, along with the evolution of the real EU ETS
prices, are shown in the middle and bottom panels of Figure 10. Given that
between 2018 and 2023, the real price has steadily increased – peaking at
almost 120 Euros in early 2023 – the upward pressure index in the middle panel
always signals expectations of soaring prices. Two exceptions are recorded in
2020 and in early 2023 when the downward price pressure index rises
temporarily, capturing expectations of price decreases.
## 6 Conclusions
The fact that the EU ETS regulation is strengthening over time calls for even
more analyses focusing on the macroeconomic effects of carbon price shocks.
Indeed, the ECB already embeds technical assumptions on carbon pricing in its
projections (European Central Bank, 2021b, ). Technical assumptions boil down
to setting the trajectory for key variables entering the ECB’s macroeconomic
models over the projection horizon and are derived in a variety of ways,
including univariate and multivariate econometric models and using the price
of futures contracts (European Central Bank,, 2016).
In this paper, we have identified carbon price drivers and methodological
choices that can directly inform projections and scenario analyses used to
gauge the macroeconomic effect of carbon price shocks. Our results show that
EU ETS prices and verified emissions can be forecasted with relatively simple
BVAR models estimated with monthly data.
There are at least two aspects of our analysis that deserve further
investigation. First, the use of time series sampled at different frequencies.
Mixed Frequency and Mixed-data sampling (MIDAS) models could be used to
further improve monthly forecasts relying on data sampled at a higher
frequency, such as weather and financial variables. Moreover, given that
several key predictors of real carbon prices – such as the level of verified
emissions and some macroeconomic aggregates – are available only at the lower
sampling frequency, it is also interesting to rely on reverse-MIDAS approaches
to exploit these data (Foroni et al.,, 2023). Lastly, our results could be
extended in the direction of real-time data to assess the impact of data
revisions on monthly forecasts of the price of carbon.
## Supplementary Materials
An online Supplement provides further forecasting results and different
fluctuation tests for various models and rolling windows.
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# On the Kohayakawa–Kreuter conjecture
Eden Kuperwasser School of Mathematical Sciences, Tel Aviv University, Tel
Aviv 6997801, Israel<EMAIL_ADDRESS>,
Wojciech Samotij and Yuval Wigderson
###### Abstract.
Let us say that a graph $G$ is Ramsey for a tuple $(H_{1},\dots,H_{r})$ of
graphs if every $r$-coloring of the edges of $G$ contains a monochromatic copy
of $H_{i}$ in color $i$, for some $i\in\llbracket{r}\rrbracket$. A famous
conjecture of Kohayakawa and Kreuter, extending seminal work of Rödl and
Ruciński, predicts the threshold at which the binomial random graph $G_{n,p}$
becomes Ramsey for $(H_{1},\dots,H_{r})$ asymptotically almost surely. In this
paper, we resolve the Kohayakawa–Kreuter conjecture for almost all tuples of
graphs. Moreover, we reduce its validity to the truth of a certain
deterministic statement, which is a clear necessary condition for the
conjecture to hold. All of our results actually hold in greater generality,
when one replaces the graphs $H_{1},\dots,H_{r}$ by finite families
$\mathcal{H}_{1},\dots,\mathcal{H}_{r}$. Additionally, we pose a natural
(deterministic) graph-partitioning conjecture, which we believe to be of
independent interest, and whose resolution would imply the Kohayakawa–Kreuter
conjecture.
EK, WS, and YW are supported by ERC Consolidator Grant 101044123
(RandomHypGra), by Israel Science Foundation Grant 2110/22, and by NSF–BSF
Grant 2019679. YW is additionally supported by ERC Consolidator Grant 863438
(LocalGlobal).
## 1\. Introduction
### 1.1. Symmetric Ramsey properties of random graphs
Given graphs $G$ and $H_{1},\dotsc,H_{r}$, one says that $G$ is _Ramsey for
the tuple $(H_{1},\dotsc,H_{r})$_ if, for every $r$-coloring of the edges of
$G$, there is a monochromatic copy of $H_{i}$ in some color
$i\in\llbracket{r}\rrbracket$. In the symmetric case $H_{1}=\dotsb=H_{r}=H$,
we simply say that $G$ is _Ramsey for $H$ in $r$ colors_. Ramsey’s theorem
[24] implies that the complete graph $K_{n}$ is Ramsey for
$(H_{1},\dotsc,H_{r})$ whenever $n$ is sufficiently large. The fundamental
question of graph Ramsey theory is to determine, for a given tuple
$(H_{1},\dotsc,H_{r})$, which graphs $G$ are Ramsey for it. For more on this
question, as well as the many fascinating sub-questions it contains, we refer
the reader to the survey [3].
In this paper, we are interested in Ramsey properties of random graphs, a
topic that was initiated in the late 1980s by Frankl–Rödl [6] and
Łuczak–Ruciński–Voigt [31]. The main question in this area is, for a given
tuple $(H_{1},\dotsc,H_{r})$, which functions $p=p(n)$ satisfy that $G_{n,p}$
is Ramsey for $(H_{1},\dots,H_{r})$ a.a.s.111As usual, $G_{n,p}$ denotes the
binomial random graph with edge probability $p$ and we say that an event
happens _asymptotically almost surely (a.a.s.)_ if its probability tends to
$1$ as $n\to\infty$. In the case $H_{1}=\dotsb=H_{r}$, this question was
resolved in the remarkable work of Rödl and Ruciński [25, 26, 27]. In order to
state their result, we need the following terminology and notation. For a
graph $J$, we denote by $v_{J}$ and $e_{J}$ the number of vertices and edges,
respectively, of $J$. The _maximal $2$-density_ of a non-empty graph $H$ with
$v_{H}\geqslant 3$ is then defined222We also define $m_{2}(K_{2})\coloneqq
1/2$ and $m_{2}(H)\coloneqq 0$ if $H$ has no edges. to be
$m_{2}(H)\coloneqq\max\left\\{\frac{e_{J}-1}{v_{J}-2}:J\subseteq
H,v_{J}\geqslant 3\right\\}.$
With this notation, we can state the random Ramsey theorem of Rödl and
Ruciński [27].
###### Theorem 1.1 (Rödl–Ruciński [27]).
For every graph $H$ which is not a forest333Rödl and Ruciński also determined
the Ramsey threshold when $H$ is a forest, but for simplicity we do not state
this more general result. and every integer $r\geqslant 2$, there exist
constants $c,C>0$ such that
$\lim_{n\to\infty}\operatorname{Pr}(G_{n,p}\text{ is Ramsey for $H$ in $r$
colors})=\begin{cases}1&\text{if }p\geqslant Cn^{-1/m_{2}(H)},\\\ 0&\text{if
}p\leqslant cn^{-1/m_{2}(H)}.\end{cases}$
As with many such threshold results for random graph properties, Theorem 1.1
really consists of two statements: the _$1$ -statement_, which says that
$G_{n,p}$ satisfies the desired property a.a.s. once $p$ is above some
threshold, and the _$0$ -statement_, which says that $G_{n,p}$ a.a.s. fails to
satisfy the desired property if $p$ is below some threshold.
In recent years, there has been a great deal of work on transferring
combinatorial theorems, such as Ramsey’s theorem or Turán’s theorem [30], to
sparse random settings. As a consequence, several new proofs of the
$1$-statement of Theorem 1.1 have been found. Two such proofs were first given
by Conlon–Gowers [4] and, independently, by Friedgut–Rödl–Schacht [8] (see
also Schacht [29]) with the use of their transference principles. More
recently, Nenadov and Steger [22] found a very short proof of the 1-statement
of Theorem 1.1 that uses the hypergraph container method of Saxton–Thomason
[28] and Balogh–Morris–Samotij [1].
However, these techniques are not suitable for proving the respective
0-statements such as that in Theorem 1.1. Furthermore, whereas the 0-statement
of the aforementioned sparse random analogue of Turán’s theorem is very easy
to establish, proving the 0-statement of Theorem 1.1 requires a significant
amount of work. To understand this, suppose that $G$ is some graph that is
Ramsey for $H$ in $r$ colors. As is well-known (see e.g. [14, Theorem 3.4]),
the probability that $G_{n,p}$ contains $G$ as a subgraph is bounded away from
zero if (and only if) $p=\Omega(n^{-1/m(G)})$, where $m(G)$ is the _maximal
density_ of $G$, defined by
$m(G)\coloneqq\max\left\\{\frac{e_{J}}{v_{J}}:J\subseteq G,v_{J}\geqslant
1\right\\}.$
In particular, if $m(G)\leqslant m_{2}(H)$, then the $0$-statement of Theorem
1.1 cannot hold. Therefore, a prerequisite for any proof of the $0$-statement
is the following result, which Rödl–Ruciński [25] termed the _deterministic
lemma_ : If $G$ is Ramsey for $H$ in $r$ colors, then $m(G)>m_{2}(H)$. We
stress that this result is by no means trivial; in particular, it turns out to
be false if we remove the assumption that $H$ is not a forest [27, 7], or if
we move from graphs to hypergraphs [9].
To complement the deterministic lemma, Rödl–Ruciński also proved what they
termed a _probabilistic lemma_. Loosely speaking, this is a result that says
that the $0$-statement of Theorem 1.1 is actually _equivalent_ to the
deterministic lemma. In other words, an obvious necessary condition for the
validity of the $0$-statement—the non-existence of a graph $G$ that is Ramsey
for $H$ and satisfies $m(G)\leqslant m_{2}(H)$—is also a sufficient condition.
### 1.2. Asymmetric Ramsey properties of random graphs
Given our good understanding of Ramsey properties of random graphs in the
symmetric case, provided by Theorem 1.1, it is natural to ask what happens if
we remove the assumption that $H_{1}=\dotsb=H_{r}$. This question was first
raised by Kohayakawa and Kreuter [15], who proposed a natural conjecture for
the threshold controlling when $G_{n,p}$ is Ramsey for an arbitrary tuple
$(H_{1},\dotsc,H_{r})$. To state their conjecture, we need the notion of the
_mixed $2$-density_: For graphs $H_{1},H_{2}$ with $m_{2}(H_{1})\geqslant
m_{2}(H_{2})$, their mixed $2$-density is defined as
$m_{2}(H_{1},H_{2})\coloneqq\max\left\\{\frac{e_{J}}{v_{J}-2+1/m_{2}(H_{2})}:J\subseteq
H_{1},v_{J}\geqslant 2\right\\}.$
With this terminology, we may state the conjecture of Kohayakawa and Kreuter
[15].
###### Conjecture 1.2 (Kohayakawa–Kreuter [15]).
Let $H_{1},\dots,H_{r}$ be graphs satisfying
$m_{2}(H_{1})\geqslant\dotsb\geqslant m_{2}(H_{r})$ and $m_{2}(H_{2})>1$.
There exist constants $c,C>0$ such that
$\lim_{n\to\infty}\operatorname{Pr}(G_{n,p}\text{ is Ramsey for
}(H_{1},\dotsc,H_{r}))=\begin{cases}1&\text{if }p\geqslant
Cn^{-1/m_{2}(H_{1},H_{2})},\\\ 0&\text{if }p\leqslant
cn^{-1/m_{2}(H_{1},H_{2})}.\end{cases}$
The assumption $m_{2}(H_{2})>1$ is equivalent to requiring that $H_{1}$ and
$H_{2}$ are not forests; it was added by Kohayakawa, Schacht, and Spöhel [16]
to rule out sporadic counterexamples, in analogy with the assumption that $H$
is not a forest in Theorem 1.1.
The role of the mixed $2$-density $m_{2}(H_{1},H_{2})$ in the context of
Conjecture 1.2 can seem a little mysterious at first, but there is a natural
(heuristic) explanation. Since one can color all edges that do not lie in a
copy of $H_{1}$ with color $1$, the only important edges are those that do lie
in copies of $H_{1}$. The mixed $2$-density is defined in such a way that
$p=\Theta(n^{-1/m_{2}(H_{1},H_{2})})$ is the threshold at which the number of
copies of (the densest subgraph of) each of $H_{2},\dotsc,H_{r}$ is at least
of the same order of magnitude as the number of edges in the union of all
copies of (the densest subgraph of) $H_{1}$ in $G_{n,p}$. Since at least one
edge in each copy of $H_{1}$ must receive a color from $\\{2,\dotsc,r\\}$,
this is the point where avoiding monochromatic copies of $H_{2},\dotsc,H_{r}$
becomes difficult.
Conjecture 1.2 has received a great deal of attention over the years, and has
been proved in a number of special cases. Following a sequence of partial
results [15, 19, 16, 9, 11], the $1$-statement of Conjecture 1.2 was proved by
Mousset, Nenadov, and Samotij [20] with the use of the container method as
well as a randomized “typing” procedure. We henceforth focus on the
$0$-statement, where progress has been more limited.
Note that, in order to prove the $0$-statement, one can make several
simplifying assumptions. First, one can assume that $r$, the number of colors,
is equal to $2$. Indeed, if one can a.a.s. $2$-color the edges of $G_{n,p}$
and avoid monochromatic copies of $H_{1},H_{2}$ in colors $1,2$, respectively,
then certainly $G_{n,p}$ is not Ramsey for $(H_{1},\dots,H_{r})$. Furthermore,
if $H_{2}^{\prime}\subseteq H_{2}$ is a subgraph satisfying
$m_{2}(H_{2}^{\prime})=m_{2}(H_{2})$, then the $0$-statement for the pair
$(H_{1},H_{2}^{\prime})$ implies the $0$-statement for $(H_{1},H_{2})$, as any
coloring with no monochromatic copy of $H_{2}^{\prime}$ in particular has no
monochromatic copy of $H_{2}$. Thus, we may assume that $H_{2}$ is _strictly
$2$-balanced_, meaning that $m_{2}(H_{2}^{\prime})<m_{2}(H_{2})$ for any
$H_{2}^{\prime}\subsetneq H_{2}$. For exactly the same reason, we may assume
that $H_{1}$ is _strictly $m_{2}(\cdot,H_{2})$-balanced_, meaning that
$m_{2}(H_{1}^{\prime},H_{2})<m_{2}(H_{1},H_{2})$ for any
$H_{1}^{\prime}\subsetneq H_{1}$. Let us say that the pair $(H_{1},H_{2})$ is
_strictly balanced_ if $H_{2}$ is strictly $2$-balanced and $H_{1}$ is
strictly $m_{2}(\cdot,H_{2})$-balanced. Additionally, let us say that
$(H_{1}^{\prime},H_{2}^{\prime})$ is a _strictly balanced pair of subgraphs_
of $(H_{1},H_{2})$ if $(H_{1}^{\prime},H_{2}^{\prime})$ is strictly balanced
and satisfies $m_{2}(H_{2}^{\prime})=m_{2}(H_{2})$ and
$m_{2}(H_{1}^{\prime},H_{2}^{\prime})=m_{2}(H_{1},H_{2})$. All previous works
on the $0$-statement of Conjecture 1.2 have made these simplifying
assumptions, working in the case $r=2$ and with a strictly balanced pair
$(H_{1},H_{2})$.
The original paper of Kohayakawa and Kreuter [15] proved the $0$-statement of
Conjecture 1.2 when $H_{1}$ and $H_{2}$ are cycles. This was extended to the
case when both $H_{1}$ and $H_{2}$ are cliques in [19], and to the case when
$H_{1}$ is a clique and $H_{2}$ is a cycle in [18]. To date, the most general
result is due to Hyde [13], who proved the $0$-statement of Conjecture 1.2 for
almost all pairs of regular graphs $(H_{1},H_{2})$; in fact, this follows from
Hyde’s main result [13, Theorem 1.9], which establishes a certain
deterministic condition whose validity implies the $0$-statement of Conjecture
1.2. Finally, the first two authors [17] recently proved the $0$-statement of
Conjecture 1.2 in the case where $m_{2}(H_{1})=m_{2}(H_{2})$. Because of this,
we henceforth focus on the case that $m_{2}(H_{1})>m_{2}(H_{2})$.
### 1.3. New results
As in the symmetric setting, a necessary prerequisite for proving the
$0$-statement of Conjecture 1.2 is proving the following _deterministic lemma_
: If $G$ is Ramsey for $(H_{1},H_{2})$, then $m(G)>m_{2}(H_{1},H_{2})$. The
main result in this paper is a corresponding probabilistic lemma, which states
that this obvious necessary condition is also sufficient.
###### Theorem 1.3.
The $0$-statement of Conjecture 1.2 holds if and only if, for every strictly
balanced pair $(H_{1},H_{2})$, every graph $G$ that is Ramsey for
$(H_{1},H_{2})$ satisfies $m(G)>m_{2}(H_{1},H_{2})$.
More precisely, we prove that if $(H_{1},H_{2})$ is any pair of graphs and
$(H_{1}^{\prime},H_{2}^{\prime})$ is a strictly balanced pair of subgraphs of
$(H_{1},H_{2})$, then the $0$-statement of Conjecture 1.2 holds for
$(H_{1},H_{2})$ if every graph $G$ which is Ramsey for
$(H_{1}^{\prime},H_{2}^{\prime})$ satisfies
$m(G)>m_{2}(H_{1}^{\prime},H_{2}^{\prime})=m_{2}(H_{1},H_{2})$.
While we believe that the probabilistic lemma, Theorem 1.3, is our main
contribution, we are able to prove the deterministic lemma in a wide range of
cases. This implies that the $0$-statement of Conjecture 1.2 is true for
almost all pairs of graphs. The most general statement we can prove is
slightly tricky to state because of the necessity of passing to a strictly
balanced pair of subgraphs; however, here is a representative example of our
results, which avoids this technicality and still implies Conjecture 1.2 for
almost all pairs of graphs. We state the more general result in Theorem 1.7
below.
###### Theorem 1.4.
Conjecture 1.2 holds for all sequences $H_{1},\dots,H_{r}$ of graphs
satisfying $m_{2}(H_{1})\geqslant\dotsb\geqslant m_{2}(H_{r})$ and
$m_{2}(H_{2})>\frac{11}{5}$.
As discussed above, Theorem 1.4 follows easily from Theorem 1.3 and a
deterministic lemma for strictly balanced pairs $(H_{1},H_{2})$ satisfying
$m_{2}(H_{1})\geqslant m_{2}(H_{2})>\frac{11}{5}$. The deterministic lemma in
this setting is actually very straightforward and follows from standard
coloring techniques.
Using a number of other coloring techniques, we can prove the deterministic
lemma (and thus Conjecture 1.2) in several additional cases, which we discuss
below. However, let us first propose a conjecture, which we believe to be of
independent interest, and whose resolution would immediately imply Conjecture
1.2 in all cases.
###### Conjecture 1.5.
For any graph $G$, there exists a forest $F\subseteq G$ such that
$m_{2}(G\setminus F)\leqslant m(G).$
Here, $G\setminus F$ denotes the graph obtained from $G$ by deleting the edges
of $F$ (but not deleting any vertices). To give some intuition for Conjecture
1.5, we note that $m(G)\leqslant m_{2}(G)\leqslant m(G)+1$ for any graph $G$,
and that $m_{2}(F)=1$ for any forest $F$ which is not a matching. Thus, it is
natural to expect that by deleting the edges of a forest, we could decrease
$m_{2}(G)$ by roughly $1$. Conjecture 1.5 says that this is roughly the case,
in that the deletion of an appropriately-chosen forest can decrease $m_{2}(G)$
to lie below $m(G)$.
Moreover, we note that Conjecture 1.5 easily implies the deterministic lemma
in all cases444Recall that the case of $m_{2}(H_{1})=m_{2}(H_{2})$ was settled
in [17], so we may freely make this assumption. with
$m_{2}(H_{1})>m_{2}(H_{2})$, and thus implies Conjecture 1.2. Indeed, it is
straightforward to verify in this case that $m_{2}(H_{1})>m_{2}(H_{1},H_{2})$
(see Lemma 3.4 below). Now, suppose that $G$ is some graph with $m(G)\leqslant
m_{2}(H_{1},H_{2})<m_{2}(H_{1})$. If Conjecture 1.5 is true, we may partition
the edges of $G$ into a forest $F$ and a graph $K$ with $m_{2}(K)\leqslant
m(G)<m_{2}(H_{1})$. This latter condition implies, in particular, that $K$
contains no copy of $H_{1}$. Additionally, by the assumption $m_{2}(H_{2})>1$
in Conjecture 1.2, we know that $H_{2}$ contains a cycle and thus $F$ contains
no copy of $H_{2}$. In other words, coloring the edges of $K$ with color $1$
and the edges of $F$ with color $2$ witnesses that $G$ is not Ramsey for
$(H_{1},\dots,H_{r})$.
Because of this, it would be of great interest to prove Conjecture 1.5.
Somewhat surprisingly, we know how to prove Conjecture 1.5 under the extra
assumption that $m(G)$ is an integer. This extra condition seems fairly
artificial, but we do not know how to remove it—our technique uses tools from
matroid theory that seem to break down once $m(G)$ is no longer an integer. We
present this proof in Appendix B, in the hope that it may serve as a first
step to the full resolution of Conjecture 1.5, and thus Conjecture 1.2.
Although we are not able to resolve Conjecture 1.5, we do have a number of
other techniques for proving the deterministic lemma, and thus Conjecture 1.2,
under certain assumptions. First, we are able to resolve the case when the
number of colors is at least three and $m_{2}(H_{2})=m_{2}(H_{3})$.
###### Theorem 1.6.
Let $H_{1},\dots,H_{r}$ be a sequence of graphs with $r\geqslant 3$ and
suppose that $m_{2}(H_{1})\geqslant
m_{2}(H_{2})=m_{2}(H_{3})\geqslant\dotsb\geqslant m_{2}(H_{r})$ and
$m_{2}(H_{2})>1$. Then Conjecture 1.2 holds for $H_{1},\dots,H_{r}$.
We can also prove Conjecture 1.2 in a number of additional cases, expressed in
terms of the properties of (a strictly balanced pair of subgraphs of) the pair
$(H_{1},H_{2})$ of two densest graphs.
Recall that the _degeneracy_ of $H$ is the maximum over all $J\subseteq H$ of
the minimum degree of $J$.
###### Theorem 1.7.
Suppose that $(H_{1},H_{2})$ is strictly balanced. Suppose additionally that
one of the following conditions holds:
1. (a)
$\chi(H_{2})\geqslant 3$, or
2. (b)
$H_{2}$ is not the union of two forests, or
3. (c)
$\chi(H_{1})>m_{2}(H_{1},H_{2})+1$, or
4. (d)
$H_{1}$ has degeneracy at least $\lfloor 2m_{2}(H_{1},H_{2})\rfloor$, or
5. (e)
$H_{1}=K_{s,t}$ for some $s,t\geqslant 2$, or
6. (f)
$m_{2}(H_{1})>\lceil m_{2}(H_{1},H_{2})\rceil$.
In any of these cases, Conjecture 1.2 holds for $(H_{1},H_{2})$.
###### Remark.
The only graphs $H_{2}$ which do not satisfy (a) or (b) are sparse bipartite
graphs, such as even cycles. On the other hand, (c) applies whenever $H_{1}$
is a clique555Note that $m_{2}(H_{1},H_{2})\leqslant m_{2}(H_{1})$, hence (c)
holds if $\chi(H_{1})>m_{2}(H_{1})+1$, and cliques satisfy
$m_{2}(K_{k})=\frac{k+1}{2}$. or, more generally, a graph obtained from a
clique by deleting few edges. Moreover, (d) applies to reasonably dense
graphs, as well as all $d$-regular bipartite graphs with $d\geqslant 2$, and
(e) handles all cases when $H_{1}$ is a biclique666In fact, our proof of (e)
applies to a larger class of graphs, which we call _$(s,t)$ -graphs_; see
Section 5 for details.. Thus, very roughly speaking, the strictly balanced
cases that remain open in Conjecture 1.2 are those in which $H_{2}$ is
bipartite and very sparse and $H_{1}$ is not “too dense”.
Case (f) is somewhat stranger and it is not obvious that there exist graphs to
which it applies. However, one can check that, for example, it applies if
$H_{1}=K_{3,3,3,3}$ and $H_{2}=C_{8}$, and that none of the other cases of
Theorem 1.7 (or any of the earlier results on Conjecture 1.2) apply in this
case. However, the main reason we include (f) is that it is implied by our
partial progress on Conjecture 1.5; since we believe that this conjecture is
the correct approach to settling Conjecture 1.2 in its entirety, we wanted to
highlight (f).
We remark that, unfortunately, the conditions in Theorem 1.7 do not exhaust
all cases. While it is quite likely that simple additional arguments could
resolve further cases, Conjecture 1.5 remains the only (conjectural) approach
we have found to resolve Conjecture 1.2 in all cases. Moreover, our proof of
the probabilistic lemma implies that, in order to prove Conjecture 1.2 for a
pair $(H_{1},H_{2})$, it is enough to prove the deterministic lemma for graphs
$G$ of order not exceeding an explicit constant $K=K(H_{1},H_{2})$. In
particular, the validity of Conjecture 1.2 for any specific pair of graphs
reduces to a finite computation.
### 1.4. Ramsey properties of graph families
All of the results discussed in the previous subsection hold in greater
generality, when we replace $H_{1},\dots,H_{r}$ with $r$ finite families of
graphs. In addition to being interesting in its own right, such a
generalization also has important consequences in the original setting of
Conjecture 1.2; indeed, our proof of the three-color result, Theorem 1.6,
relies on our ability to work with graph families. Before we state our more
general results, we need the following definitions.
###### Definition 1.8.
Let $\mathcal{H}_{1},\dots,\mathcal{H}_{r}$ be finite families of graphs. We
say that a graph $G$ is _Ramsey_ for $(\mathcal{H}_{1},\dots,\mathcal{H}_{r})$
if every $r$-coloring of $E(G)$ contains a monochromatic copy of some
$H_{i}\in\mathcal{H}_{i}$ in some color $i\in\llbracket{r}\rrbracket$.
We now define the appropriate generalizations of the notions of maximum
$2$-density and mixed $2$-density to families of graphs. First, given a finite
family of graphs $\mathcal{H}$, we let
$m_{2}(\mathcal{H})\coloneqq\min_{H\in\mathcal{H}}m_{2}(H).$
Second, given a graph $H$ and a (finite) family $\mathcal{L}$ of graphs, we
let
$m_{2}(H,\mathcal{L})\coloneqq\max\left\\{\frac{e_{J}}{v_{J}-2+1/m_{2}(\mathcal{L})}:J\subseteq
H,v_{J}\geqslant 2\right\\}.$
Third, given two finite families of graphs $\mathcal{H}$ and $\mathcal{L}$
with $m_{2}(\mathcal{H})\geqslant m_{2}(\mathcal{L})$, we define
$m_{2}(\mathcal{H},\mathcal{L})\coloneqq\min_{H\in\mathcal{H}}m_{2}(H,\mathcal{L}).$
Finally, continuing the terminology above, let us say that the pair
$(\mathcal{H},\mathcal{L})$ is _strictly balanced_ if every graph in
$\mathcal{L}$ is strictly $2$-balanced and every graph in $\mathcal{H}$ is
strictly $m_{2}(\cdot,\mathcal{L})$-balanced.
The following conjecture is a natural generalization of Conjecture 1.2 to
families of graphs.
###### Conjecture 1.9 (Kohayakawa–Kreuter conjecture for families).
Let $\mathcal{H}_{1},\dots,\mathcal{H}_{r}$ be finite families of graphs with
$m_{2}(\mathcal{H}_{1})\geqslant\dotsb\geqslant m_{2}(\mathcal{H}_{r})$ and
suppose that $m_{2}(\mathcal{H}_{2})>1$. There exist constants $c,C>0$ such
that
$\lim_{n\to\infty}\operatorname{Pr}(G_{n,p}\text{ is Ramsey for
}(\mathcal{H}_{1},\dots,\mathcal{H}_{r}))=\begin{cases}1&\text{if }p\geqslant
Cn^{-1/m_{2}(\mathcal{H}_{1},\mathcal{H}_{2})},\\\ 0&\text{if }p\leqslant
cn^{-1/m_{2}(\mathcal{H}_{1},\mathcal{H}_{2})}.\end{cases}$
Note that, for any $H_{1}\in\mathcal{H}_{1},\dots,H_{r}\in\mathcal{H}_{r}$,
the property of being Ramsey for $(H_{1},\dots,H_{r})$ implies the property of
being Ramsey for $(\mathcal{H}_{1},\dots,\mathcal{H}_{r})$. Therefore, the
$1$-statement of Conjecture 1.9 follows from the $1$-statement of Conjecture
1.2, which we know to be true by the result of Mousset, Nenadov, and Samotij
[20].
The $0$-statement of Conjecture 1.9 remains open; the only progress to date is
due to the first two authors [17], who proved Conjecture 1.9 whenever
$m_{2}(\mathcal{H}_{1})=m_{2}(\mathcal{H}_{2})$. We make further progress on
this conjecture: as in the case of single graphs, we prove a probabilistic
lemma that reduces the $0$-statement to a deterministic lemma, which is
clearly a necessary condition.
###### Theorem 1.10 (Probabilistic lemma for families).
The $0$-statement of Conjecture 1.9 holds if and only if, for every strictly
balanced pair $(\mathcal{H}_{1},\mathcal{H}_{2})$ of finite families of
graphs, every graph $G$ that is Ramsey for $(\mathcal{H}_{1},\mathcal{H}_{2})$
satisfies $m(G)>m_{2}(\mathcal{H}_{1},\mathcal{H}_{2})$.
As in Theorems 1.4 and 1.7, we can prove the deterministic lemma for families
in a wide variety of cases, namely when every graph $H_{1}\in\mathcal{H}_{1}$
or every graph $H_{2}\in\mathcal{H}_{2}$ satisfies one of the conditions in
Theorem 1.7. In particular, we resolve Conjecture 1.9 in many cases. However,
we believe that the right way to resolve Conjecture 1.9 in its entirety is the
same as the right way to resolve the original Kohayakawa–Kreuter conjecture,
Conjecture 1.2. Namely, if Conjecture 1.5 is true, then Conjecture 1.9 is true
for all families of graphs.
### 1.5. Organization
Most of the rest of this paper is dedicated to proving Theorem 1.10, and thus
also Theorem 1.3. Our technique is inspired by recent work of the first two
authors [17], who proved Conjecture 1.9 in the case
$m_{2}(\mathcal{H}_{1})=m_{2}(\mathcal{H}_{2})$. Therefore, we assume
henceforth that $m_{2}(\mathcal{H}_{1})>m_{2}(\mathcal{H}_{2})$. We will now
change notation and denote $\mathcal{H}_{1}=\mathcal{H}$ and
$\mathcal{H}_{2}=\mathcal{L}$. The names stand for _heavy_ and _light_ ,
respectively, and are meant to remind the reader that
$m_{2}(\mathcal{L})<m_{2}(\mathcal{H})$. We also assume henceforth that
$(\mathcal{H},\mathcal{L})$ is a strictly balanced pair of families.
The rest of this paper is organized as follows. In Section 2, we present a
high-level overview of our proof of Theorem 1.10. Section 3 contains a number
of preliminaries for the proof, including the definitions and basic properties
of _cores_ —a fundamental notion in our approach—as well as several simple
numerical lemmas. The proof of Theorem 1.10 is carried out in detail in
Section 4. In Section 5, we prove the deterministic lemma under various
assumptions, which yields Theorems 1.7 and 1.4 as well as their
generalizations to families. We conclude with two appendices: Appendix A
proves Theorem 1.6 by explaining what in our proof needs to be adapted to deal
with the three-color setting; and Appendix B presents our partial progress on
Conjecture 1.5.
#### Additional note
As this paper was being written, we learned that very similar results were
obtained independently by Bowtell, Hancock, and Hyde [2], who also resolve
Conjecture 1.2 in the vast majority of cases. As with this paper, they first
prove a probabilistic lemma, showing that resolving the Kohayakawa–Kreuter
conjecture is equivalent to proving a deterministic coloring result. By using
a wider array of coloring techniques, they are able to prove more cases of
Conjecture 1.2 than we can. Additionally, they consider a natural
generalization of the Kohayakawa–Kreuter to uniform hypergraphs (a topic that
we chose not to pursue here) and establish its $0$-statement for almost all
pairs of hypergraphs; see also [9] for more on such hypergraph questions. In
contrast, their work does not cover families of graphs, a generalization that
falls out naturally from our approach.
#### Acknowledgments
We would like to thank Anita Liebenau and Letícia Mattos for fruitful
discussions on Ramsey properties of random graphs. We are also indebted to
Candida Bowtell, Robert Hancock, and Joseph Hyde for sharing an early draft of
their paper [2] with us, and for their many invaluable comments.
## 2\. Proof outline
We now sketch, at a very high level, the proof of the probabilistic lemma. Let
us fix a strictly balanced pair of families $(\mathcal{H},\mathcal{L})$. We
wish to upper-bound the probability that $G_{n,p}$ is Ramsey for
$(\mathcal{H},\mathcal{L})$, where $p\leqslant
cn^{-1/m_{2}(\mathcal{H},\mathcal{L})}$ for an appropriately chosen constant
$c=c(\mathcal{H},\mathcal{L})>0$. Our approach is modeled on the recent proof
of the $0$-statement of Theorem 1.1 due to the first two authors [17];
however, there are substantial additional difficulties that arise in the
asymmetric setting.
One can immediately make several simplifying assumptions. First, if $G_{n,p}$
is Ramsey for $(\mathcal{H},\mathcal{L})$, then there exists some $G\subseteq
G_{n,p}$ that is _minimally_ Ramsey for $(\mathcal{H},\mathcal{L})$, in the
sense that any proper subgraph $G^{\prime}\subsetneq G$ is not Ramsey for
$(\mathcal{H},\mathcal{L})$. It is not hard to show (see Lemma 3.2 below) that
every minimally Ramsey graph has a number of interesting properties. In
particular, if $G$ is minimally Ramsey, then every edge of $G$ lies in at
least one copy of some $H\in\mathcal{H}$, and at least one copy of some
$L\in\mathcal{L}$. Our arguments will exploit a well-known strengthening of
this property, which we call _supporting a core_ ; see Definition 3.1 for the
precise definition.
We would ideally like to union-bound over all possible minimally Ramsey graphs
$G$ in order to show that a.a.s. none of them appears in $G_{n,p}$.
Unfortunately, there are potentially too many minimally Ramsey graphs for this
to be possible. To overcome this, we construct a smaller family $\mathcal{S}$
of subgraphs of $K_{n}$ such that every Ramsey graph $G$ contains some element
of $\mathcal{S}$ as a subgraph. Since $\mathcal{S}$ is much smaller than the
family of minimally Ramsey graphs, we can effectively union-bound over
$\mathcal{S}$. This basic idea also underlies the container method [28, 1] and
the recent work of Harel, Mousset, and Samotij on the upper tail problem for
subgraph counts [12]. The details here, however, are slightly subtle; there
are actually three different types of graphs in $\mathcal{S}$ and a different
union-bound argument is needed to handle each type.
We construct our family $\mathcal{S}$ with the use of an exploration process
on minimally Ramsey graphs, each of which supports a core. This exploration
process starts with a fixed edge of $K_{n}$ and gradually adds to it copies of
graphs in $\mathcal{H}\cup\mathcal{L}$. As long as the subgraph
$G^{\prime}\subseteq G$ of explored edges is not yet all of $G$, we add to
$G^{\prime}$ a copy of some graph in $\mathcal{H}\cup\mathcal{L}$ that
intersects $G^{\prime}$ but is not fully contained in it. By choosing this
copy in a principled manner (more on this momentarily), we can ensure that
$\mathcal{S}$ satisfies certain conditions which enable this union-bound
argument.
Since our goal is to show that the final graph $G^{\prime}$ is rather dense
(and thus unlikely to appear in $G_{n,p}$), we always prefer to add copies of
graphs in $\mathcal{H}$, as these boost the density of $G^{\prime}$. If there
are no available copies of $H\in\mathcal{H}$, we explore along some
$L\in\mathcal{L}$. As $L$ may be very sparse, this can hurt us; however, the
“core” property guarantees that each copy of $L$ comes with at least one copy
of some $H\in\mathcal{H}$ per new edge. An elementary (but fairly involved)
computation shows that the losses and the gains pencil out, which is the key
fact showing that $\mathcal{S}$ has the desired properties.
## 3\. Preliminaries
### 3.1. Ramsey graphs and cores
Given a graph $G$, denote by
$\mathcal{F}_{\mathcal{H}}[G],\mathcal{F}_{\mathcal{L}}[G]$ the set of all
copies of members of $\mathcal{H},\mathcal{L}$, respectively, in $G$. We think
of $\mathcal{F}_{\mathcal{H}}[G],\mathcal{F}_{\mathcal{L}}[G]$ as hypergraphs
on the ground set $E(G)$; in particular, we think of an element of
$\mathcal{F}_{\mathcal{H}}[G],\mathcal{F}_{\mathcal{L}}[G]$ as a collection of
edges of $G$ that form a copy of some $H\in\mathcal{H},L\in\mathcal{L}$,
respectively. To highlight the (important) difference between the members of
$\mathcal{H}\cup\mathcal{L}$ and their copies (i.e. the elements of
$\mathcal{F}_{\mathcal{H}}[G]\cup\mathcal{F}_{\mathcal{L}}[G]$), we will
denote the former by $H$ and $L$ and the latter by $\widehat{H}$ and
$\widehat{L}$.
Given a graph $G$ and
$\mathcal{F}_{\mathcal{H}}\subseteq\mathcal{F}_{\mathcal{H}}[G],\mathcal{F}_{\mathcal{L}}\subseteq\mathcal{F}_{\mathcal{L}}[G]$,
we say that the tuple
$(G,\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$ is _Ramsey_ if, for
every two-coloring of $E(G)$, there is an element of
$\mathcal{F}_{\mathcal{H}}$ that is monochromatic red or an element of
$\mathcal{F}_{\mathcal{L}}$ that is monochromatic blue. In particular, we see
that $G$ is Ramsey for $(\mathcal{H},\mathcal{L})$ if and only if
$(G,\mathcal{F}_{\mathcal{H}}[G],\mathcal{F}_{\mathcal{L}}[G])$ is Ramsey.
Having said that, allowing tuples
$(G,\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$ where
$\mathcal{F}_{\mathcal{H}}$ and $\mathcal{F}_{\mathcal{L}}$ are proper subsets
of $\mathcal{F}_{\mathcal{H}}[G]$ and $\mathcal{F}_{\mathcal{L}}[G]$,
respectively, enables us to deduce further useful properties. These are
encapsulated in the following definition.
###### Definition 3.1.
An _$(\mathcal{H},\mathcal{L})$ -core_ (or _core_ for short) is a tuple
$(G,\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$, where $G$ is a
graph and
$\mathcal{F}_{\mathcal{H}}\subseteq\mathcal{F}_{\mathcal{H}}[G],\mathcal{F}_{\mathcal{L}}\subseteq\mathcal{F}_{\mathcal{L}}[G]$,
with the following properties:
* •
The hypergraph $\mathcal{F}_{\mathcal{H}}\cup\mathcal{F}_{\mathcal{L}}$ is
connected and spans $E(G)$.
* •
For every $\widehat{H}\in\mathcal{F}_{\mathcal{H}}$ and every edge
$e\in\widehat{H}$, there exists an $\widehat{L}\in\mathcal{F}_{\mathcal{L}}$
such that $\widehat{H}\cap\widehat{L}=\\{e\\}$.
* •
For every $\widehat{L}\in\mathcal{F}_{\mathcal{L}}$ and every edge
$e\in\widehat{L}$, there exists an $\widehat{H}\in\mathcal{F}_{\mathcal{H}}$
such that $\widehat{H}\cap\widehat{L}=\\{e\\}$.
We say that $G$ _supports a core_ if there exist
$\mathcal{F}_{\mathcal{H}}\subseteq\mathcal{F}_{\mathcal{H}}[G],\mathcal{F}_{\mathcal{L}}\subseteq\mathcal{F}_{\mathcal{L}}[G]$
such that $(G,\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$ is a core.
The reason we care about cores is that minimal Ramsey graphs support cores, as
shown in the following lemma. Essentially the same lemma appears in the work
of Rödl and Ruciński [25], where it is given as an exercise. The same idea was
already used in several earlier works, including [15, Claim 6] and [18, Lemma
4.1].
###### Lemma 3.2.
Suppose that a graph $G$ is Ramsey for $(\mathcal{H},\mathcal{L})$, but none
of its proper subgraphs are Ramsey for $(\mathcal{H},\mathcal{L})$. Then $G$
supports an $(\mathcal{H},\mathcal{L})$-core.
###### Proof.
As $G$ is Ramsey for $(\mathcal{H},\mathcal{L})$, we know that
$(G,\mathcal{F}_{\mathcal{H}}[G],\mathcal{F}_{\mathcal{L}}[G])$ is a Ramsey
tuple. Let
$\mathcal{F}_{\mathcal{H}}\subseteq\mathcal{F}_{\mathcal{H}}[G],\mathcal{F}_{\mathcal{L}}\subseteq\mathcal{F}_{\mathcal{L}}[G]$
be inclusion-minimal subfamilies such that
$(G,\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$ is still a Ramsey
tuple. In other words, this tuple is Ramsey, but for any
$\mathcal{F}_{\mathcal{H}}^{\prime}\subseteq\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}}^{\prime}\subseteq\mathcal{F}_{\mathcal{L}}$
such that at least one inclusion is strict, the tuple
$(G,\mathcal{F}_{\mathcal{H}}^{\prime},\mathcal{F}_{\mathcal{L}}^{\prime})$ is
not Ramsey. We will show that
$(G,\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$ is a core.
If some $e\in E(G)$ is not contained in any edge of
$\mathcal{F}_{\mathcal{H}}\cup\mathcal{F}_{\mathcal{L}}$, then $(G\setminus
e,\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$ is still Ramsey, and
thus $G\setminus e$ is Ramsey for $(\mathcal{H},\mathcal{L})$, contradicting
the minimality of $G$. Furthermore, if
$\mathcal{F}_{\mathcal{H}}\cup\mathcal{F}_{\mathcal{L}}$ is not connected,
then at least one of its connected components induces a Ramsey tuple, which
contradicts the minimality of
$(\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$. Thus, the first
condition in the definition of a core is satisfied. We now turn to the next
two parts of the definition.
To see that the second condition in the definition of a core is satisfied, fix
some $\widehat{H}\in\mathcal{F}_{\mathcal{H}}$ and some $e\in\widehat{H}$. By
minimality, we can find a two-coloring of $E(G)$ such that no element of
$\mathcal{F}_{\mathcal{L}}$ is blue and no element of
$\mathcal{F}_{\mathcal{H}}\setminus\\{\widehat{H}\\}$ is red. Note that all
edges of $\widehat{H}$ are colored red, as otherwise our coloring would
witness $(G,\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$ being not
Ramsey. Flip the color of $e$ from red to blue. Since $\widehat{H}$ is now no
longer monochromatic red, we must have created a monochromatic blue element
$\widehat{L}$ of $\mathcal{F}_{\mathcal{L}}$. As all edges of
$\widehat{H}\setminus e$ are still red, we see that
$\widehat{H}\cap\widehat{L}=\\{e\\}$, as required. Interchanging the roles of
$\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}}$, and the colors yields
the third condition in the definition of a core. ∎
### 3.2. Numerical lemmas
In this section, we collect a few useful numerical lemmas, all of which are
simple combinatorial facts about vertex- and edge-counts in graphs. We begin
with the following well-known result, which we will use throughout.
###### Lemma 3.3 (The mediant inequality).
Let $a,c\geqslant 0$ and $b,d>0$ be real numbers with $a/b\leqslant c/d$. Then
$\frac{a}{b}\leqslant\frac{a+c}{b+d}\leqslant\frac{c}{d}.$
Moreover, if one inequality is strict, then so is the other (which happens if
and only if $a/b<c/d$).
###### Proof.
Both inequalities are easily seen to be equivalent to the inequality
$ad\leqslant bc$, which is itself the same as $a/b\leqslant c/d$. ∎
###### Lemma 3.4.
Let $(\mathcal{H},\mathcal{L})$ be a strictly balanced pair. If
$m_{2}(\mathcal{L})<m_{2}(\mathcal{H})$, then
$m_{2}(\mathcal{L})<m_{2}(\mathcal{H},\mathcal{L})<m_{2}(\mathcal{H})$.
###### Proof.
To see the second inequality, let $H\in\mathcal{H}$ be a graph with
$m_{2}(H)=m_{2}(\mathcal{H})$ and observe that the strict
$m_{2}(\cdot,\mathcal{L})$-balancedness of $H$ implies that
$m_{2}(H,\mathcal{L})=\frac{e_{H}}{v_{H}-2+1/m_{2}(\mathcal{L})}=\frac{(e_{H}-1)+1}{(v_{H}-2)+1/m_{2}(\mathcal{L})}\leqslant\frac{m_{2}(H)\cdot(v_{H}-2)+1}{(v_{H}-2)+1/m_{2}(\mathcal{L})}.$
Since $m_{2}(H)=m_{2}(\mathcal{H})>m_{2}(\mathcal{L})$, Lemma 3.3 implies that
$m_{2}(\mathcal{H},\mathcal{L})\leqslant
m_{2}(H,\mathcal{L})<m_{2}(\mathcal{H})$.
For the first inequality, let $H\in\mathcal{H}$ be a graph for which
$m_{2}(H,\mathcal{L})=m_{2}(\mathcal{H},\mathcal{L})$ and let $J\subseteq H$
be its subgraph with $\frac{e_{J}-1}{v_{J}-2}=m_{2}(H)$. By the strict
$m_{2}(\cdot,\mathcal{L})$-balancedness of $H$, we have
$m_{2}(H,\mathcal{L})\geqslant
m_{2}(J,\mathcal{L})=\frac{(e_{J}-1)+1}{(v_{J}-2)+1/m_{2}(\mathcal{L})}=\frac{m_{2}(H)\cdot(v_{J}-2)+1}{(v_{J}-2)+1/m_{2}(\mathcal{L})}.$
Since $m_{2}(H)>m_{2}(\mathcal{L})$, Lemma 3.3 implies that
$m_{2}(\mathcal{H},\mathcal{L})=m_{2}(H,\mathcal{L})\geqslant
m_{2}(J,\mathcal{L})>m_{2}(\mathcal{L})$. ∎
###### Lemma 3.5.
Let $H\in\mathcal{H}$ be strictly $m_{2}(\cdot,\mathcal{L})$-balanced. Then
for any $F\subsetneq H$ with $v_{F}\geqslant 2$, we have
$e_{H}-e_{F}>m_{2}(H,\mathcal{L})\cdot(v_{H}-v_{F})\geqslant
m_{2}(\mathcal{H},\mathcal{L})\cdot(v_{H}-v_{F}).$
###### Proof.
The second inequality follows from the definition of
$m_{2}(\mathcal{H},\mathcal{L})$. Since $e_{F}<e_{H}$, we may assume that
$v_{F}<v_{H}$, as otherwise the claimed inequality holds vacuously. Since $H$
is strictly $m_{2}(\cdot,\mathcal{L})$-balanced, we have
$m_{2}(H,\mathcal{L})=\frac{e_{H}}{v_{H}-2+1/m_{2}(\mathcal{L})}=\frac{(e_{H}-e_{F})+e_{F}}{(v_{H}-v_{F})+(v_{F}-2+1/m_{2}(\mathcal{L}))}$
whereas
$\frac{e_{F}}{v_{F}-2+1/m_{2}(\mathcal{L})}<m_{2}(H,\mathcal{L}).$
Since $v_{H}>v_{F}$, we may use Lemma 3.3 to conclude that
$(e_{H}-e_{F})/(v_{H}-v_{F})>m_{2}(H,\mathcal{L})$. ∎
###### Lemma 3.6.
Let $L\in\mathcal{L}$ be strictly 2-balanced. Then for any $J\subsetneq L$
with $e_{L}\geqslant 1$, we have
$e_{L}-e_{J}\geqslant m_{2}(L)\cdot(v_{L}-v_{J})\geqslant
m_{2}(\mathcal{L})\cdot(v_{L}-v_{J}).$
Moreover, the first inequality is strict unless $J=K_{2}$.
###### Proof.
The second inequality is immediate since $m_{2}(\mathcal{L})\leqslant
m_{2}(L)$. Since $e_{J}<e_{L}$, we may assume that $v_{J}<v_{L}$, as otherwise
the claimed (strict) inequality holds vacuously. We clearly have equality if
$J=K_{2}$ and strict inequality if $v_{J}=2$ and $e_{J}=0$, so we may assume
henceforth that $v_{J}>2$. Since $L$ is strictly $2$-balanced,
$m_{2}(L)=\frac{e_{L}-1}{v_{L}-2}=\frac{(e_{L}-e_{J})+(e_{J}-1)}{(v_{L}-v_{J})+(v_{J}-2)}$
whereas $(e_{J}-1)/(v_{J}-2)<m_{2}(L)$. Since $v_{J}>2$, we may apply Lemma
3.3 to conclude the desired result, with a strict inequality. ∎
###### Lemma 3.7.
Suppose that $(\mathcal{H},\mathcal{L})$ is a strictly balanced pair. Defining
$\alpha\coloneqq m_{2}(\mathcal{H},\mathcal{L})$ and
$X\coloneqq\min_{H\in\mathcal{H}}\\{(e_{H}-1)-\alpha\cdot(v_{H}-2)\\}$, we
have that
$X+(v_{K}-2)(\alpha-1)\geqslant
e_{K}\cdot\left(\frac{\alpha}{m_{2}(L)}-1\right)$
for every $L\in\mathcal{L}$ and every non-empty $K\subseteq L$. Moreover, the
inequality is strict unless $K=K_{2}$.
###### Proof.
Without loss of generality, we may assume that $m_{2}(L)<\alpha$ and that
$v_{K}>2$, as otherwise the statement holds vacuously (recall from Lemma 3.4
that $\alpha=m_{2}(\mathcal{H},\mathcal{L})>m_{2}(\mathcal{L})>1$). Fix some
$L\in\mathcal{L}$ and a nonempty $K\subseteq L$. Recall that each
$H\in\mathcal{H}$ is strictly $m_{2}(\cdot,\mathcal{L})$-balanced and
satisfies $m_{2}(H,\mathcal{L})\geqslant
m_{2}(\mathcal{H},\mathcal{L})=\alpha$. This implies that
$\frac{e_{H}}{v_{H}-2+1/m_{2}(\mathcal{L})}\geqslant\alpha$
or, equivalently,
$e_{H}\geqslant\alpha\cdot(v_{H}-2)+\frac{\alpha}{m_{2}(\mathcal{L})}.$
Consequently,
$X=\min_{H\in\mathcal{H}}\\{(e_{H}-1)-\alpha\cdot(v_{H}-2)\\}\geqslant\frac{\alpha}{m_{2}(\mathcal{L})}-1\geqslant\frac{\alpha}{m_{2}(L)}-1,$
where the final inequality uses that $m_{2}(L)\geqslant m_{2}(\mathcal{L})$.
Since $L$ is strictly $2$-balanced and we assumed that $m_{2}(L)<\alpha$, we
have
$(e_{K}-1)\cdot\left(\frac{\alpha}{m_{2}(L)}-1\right)\leqslant
m_{2}(L)\cdot(v_{K}-2)\cdot\left(\frac{\alpha}{m_{2}(L)}-1\right)=(v_{K}-2)(\alpha-
m_{2}(L)).$
Rearranging the above inequality, we obtain
$\displaystyle
e_{K}\cdot\left(\frac{\alpha}{m_{2}(L)}-1\right)-(v_{K}-2)(\alpha-1)$
$\displaystyle\leqslant(1-m_{2}(L))(v_{K}-2)+\left(\frac{\alpha}{m_{2}(L)}-1\right)$
$\displaystyle<\frac{\alpha}{m_{2}(L)}-1\leqslant X,$
where the penultimate inequality uses the assumption that $v_{K}>2$. ∎
## 4\. Proof of the probabilistic lemma
In this section, we prove Theorem 1.10. We in fact prove the following more
precise statement.
###### Lemma 4.1 (Theorem 1.10, rephrased).
Let $(\mathcal{H},\mathcal{L})$ be a strictly balanced pair of finite families
of graphs satisfying $m_{2}(\mathcal{H})>m_{2}(\mathcal{L})$. There exists a
constant $c>0$ such that the following holds. If $p\leqslant
cn^{-1/m_{2}(\mathcal{H},\mathcal{L})}$, then a.a.s. every $G\subseteq
G_{n,p}$ which supports a core satisfies $m(G)\leqslant
m_{2}(\mathcal{H},\mathcal{L})$.
Note that this immediately implies the difficult direction in Theorem 1.10.
Indeed, suppose that the $0$-statement of 1.9 fails for some tuple
$(\mathcal{H}_{1},\dotsc,\mathcal{H}_{r})$, i.e., the random graph $G_{n,p}$
is Ramsey for $(\mathcal{H}_{1},\dotsc,\mathcal{H}_{r})$ with probability
bounded away from zero when
$p=cn^{-1/m_{2}(\mathcal{H}_{1},\mathcal{H}_{2})}$, for an arbitrarily small
constant $c>0$. In particular, with probability bounded away from zero,
$G_{n,p}$ contains a graph that is also Ramsey for any pair
$(\mathcal{H},\mathcal{L})$ of families of subgraphs of
$(\mathcal{H}_{1},\mathcal{H}_{2})$. For an appropriately chosen pair
$(\mathcal{H},\mathcal{L})$, Lemma 3.2 implies that some subgraph $G\subseteq
G_{n,p}$ supports an $(\mathcal{H},\mathcal{L})$-core. By the assumed
assertion of Lemma 4.1, a.a.s. any such $G\subseteq G_{n,p}$ satisfies
$m(G)\leqslant m_{2}(\mathcal{H},\mathcal{L})$. However, by the deterministic
lemma (i.e. the assumption of Theorem 1.10), we know that no such $G$ can be
Ramsey for $(\mathcal{H},\mathcal{L})$, a contradiction.
Our proof of Lemma 4.1 follows closely the proof of the probabilistic lemma in
recent work of the first two authors [17]. Fix a strictly balanced pair
$(\mathcal{H},\mathcal{L})$ of families satisfying
$m_{2}(\mathcal{H})>m_{2}(\mathcal{L})$, and let $\alpha\coloneqq
m_{2}(\mathcal{H},\mathcal{L})$. Let $\mathcal{G}_{\mathrm{bad}}$ denote the
set of graphs $G\subseteq K_{n}$ which support a core and satisfy
$m(G)>m_{2}(\mathcal{H},\mathcal{L})$. The key lemma, which implies Lemma 4.1,
is as follows.
###### Lemma 4.2.
There exist constants $\Lambda,K>0$ and a collection $\mathcal{S}$ of
subgraphs of $K_{n}$ satisfying the following properties:
1. (a)
Every element of $\mathcal{G}_{\mathrm{bad}}$ contains some $S\in\mathcal{S}$
as a subgraph.
2. (b)
Every $S\in\mathcal{S}$ satisfies at least one of the following three
conditions:
1. (i)
$v_{S}\geqslant\log n$ and $e_{S}\geqslant\alpha\cdot(v_{S}-2)$;
2. (ii)
$v_{S}<\log n$ and $e_{S}\geqslant\alpha\cdot v_{S}+1$;
3. (iii)
$v_{S}\leqslant K$ and $m(S)>\alpha$.
3. (c)
For every $k\in\llbracket{n}\rrbracket$, there are at most $(\Lambda n)^{k}$
graphs $S\in\mathcal{S}$ with $v_{S}=k$.
Before we prove Lemma 4.2, let us see why it implies Lemma 4.1.
###### Proof of Lemma 4.1.
Recall that $p\leqslant cn^{-1/\alpha}$, for a small constant
$c=c(\mathcal{H},\mathcal{L})$ to be chosen later. We wish to prove that
a.a.s. $G_{n,p}$ contains no element of $\mathcal{G}_{\mathrm{bad}}$. By Lemma
4.2(a), it suffices to prove that a.a.s. $G_{n,p}$ contains no element of
$\mathcal{S}$. By (b), the elements of $\mathcal{S}$ are of three types, each
of which we deal with separately. First, recall that for any fixed graph $S$
with $m(S)>\alpha$, we have that $\operatorname{Pr}(S\subseteq G_{n,p})=o(1)$
(see e.g. [14, Theorem 3.4]). As there are only a constant number of graphs on
at most $K$ vertices, we may apply the union bound and conclude that a.a.s. no
graph $S$ satisfying $v_{S}\leqslant K$ and $m(S)>\alpha$ appears in
$G_{n,p}$. This deals with the elements of $\mathcal{S}$ corresponding to case
(b)(iii).
Let $\mathcal{S}^{\prime}\subseteq\mathcal{S}$ be the set of $S\in\mathcal{S}$
which lie in cases (b)(i) or (b)(ii). We have that
$\displaystyle\operatorname{Pr}(S\subseteq G_{n,p}\text{ for some
}S\in\mathcal{S}^{\prime})$
$\displaystyle\leqslant\sum_{S\in\mathcal{S}^{\prime}}p^{e_{S}}$
$\displaystyle\leqslant\sum_{k=1}^{\lceil\log n\rceil-1}(\Lambda
n)^{k}p^{\alpha k+1}+\sum_{k=\lceil\log n\rceil}^{\infty}(\Lambda
n)^{k}p^{\alpha(k-2)}$ $\displaystyle\leqslant p\sum_{k=1}^{\infty}(\Lambda
c^{\alpha})^{k}+c^{-2\alpha}n^{2}\sum_{k=\lceil\log n\rceil}^{\infty}(\Lambda
c^{\alpha})^{k}$
We now choose $c$ so that $\Lambda c^{\alpha}=e^{-3}$. Then the first sum
above can be bounded by $p$, which tends to $0$ as $n\to\infty$. The second
term can be bounded by $2c^{-2\alpha}n^{-1}$, which also tends to $0$ as
$n\to\infty$. All in all, we find that a.a.s. $G_{n,p}$ does not contain any
graph in $\mathcal{S}$, as claimed. ∎
### 4.1. The exploration process and the proof of Lemma 4.2
In this section, we prove Lemma 4.2. We will construct the family
$\mathcal{S}$ by considering an exploration process on the set $\mathcal{G}$
of graphs $G\subseteq K_{n}$ which support a core. For each such
$G\in\mathcal{G}$, let us arbitrarily choose collections
$\mathcal{F}_{\mathcal{H}}\subseteq\mathcal{F}_{\mathcal{H}}[G]$ and
$\mathcal{F}_{\mathcal{L}}\subseteq\mathcal{F}_{\mathcal{L}}[G]$ such that
$(G,\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$ is a core. From now
on, by copies of graphs from $\mathcal{H},\mathcal{L}$ in $G$, we mean only
those copies that belong to the families
$\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}}$, respectively. This
subtlety will be extremely important in parts of the analysis.
We first fix arbitrary orderings on the graphs in $\mathcal{H}$ and
$\mathcal{L}$. Additionally, we fix a labeling of the vertices of $K_{n}$,
which induces an ordering of all subgraphs according to the lexicographic
order. Together with the ordering on $\mathcal{H},\mathcal{L}$, we obtain a
lexicographic ordering on all copies in $K_{n}$ of graphs in
$\mathcal{H},\mathcal{L}$. Now, given a $G\in\mathcal{G}$, we build a sequence
$G_{0}\subsetneq G_{1}\subsetneq\dotsb\subseteq G$ as follows. We start with
$G_{0}$ being the graph comprising only the smallest edge of $G$. As long as
$G_{i}\neq G$, do the following: Since $G\neq G_{i}$ and
$(G,\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$ is a core, there
must be some copy of a graph from $\mathcal{H}\cup\mathcal{L}$ which belongs
to $\mathcal{F}_{\mathcal{H}}\cup\mathcal{F}_{\mathcal{L}}$ that intersects
$G_{i}$ but is not fully contained in $G_{i}$. Call such an _overlapping_ copy
_regular_ if it intersects $G_{i}$ in exactly one edge, called its _root_ ;
otherwise, call the copy _degenerate_. We form $G_{i+1}$ from $G_{i}$ as
follows:
1. (1)
Suppose first that there is an overlapping copy of some graph in
$\mathcal{H}$. We form $G_{i+1}$ by adding to $G_{i}$ the smallest (according
to the lexicographic order) such copy. We call $G_{i}\to G_{i+1}$ a
_degenerate $\mathcal{H}$-step_.
2. (2)
Otherwise, there must be an overlapping copy $\widehat{L}$ of some
$L\in\mathcal{L}$. Note that, for every edge $e\in\widehat{L}\setminus G_{i}$,
there must be a copy of some $H\in\mathcal{H}$ that meets $\widehat{L}$ only
at $e$, as $(G,\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$ is a
core. Note further that this copy of $H$ does not intersect $G_{i}$, as
otherwise we would perform a degenerate $\mathcal{H}$-step. We pick the
smallest such copy for every $e\in\widehat{L}\setminus G_{i}$, and call it
$\widehat{H_{e}}$ (note that the graphs $H_{e}\in\mathcal{H}$ such that
$H_{e}\cong\widehat{H_{e}}$ may be different for different choices of $e$). We
say that $\widehat{L}$ is _pristine_ if it is regular and the graphs
$\\{\widehat{H_{e}}\\}_{e\in\widehat{L}\setminus G_{i}}$ are all vertex-
disjoint (apart from the intersections that they are forced to have in
$V(\widehat{L})$).
1. (2.1)
If there is a pristine copy of some graph in $\mathcal{L}$, we pick the
smallest one in the following sense: First, among all edges of $G_{i}$ that
are roots of a pristine copy of some graph in $\mathcal{L}$, we choose the one
that arrived to $G_{i}$ earliest. Second, among all pristine copies that are
rooted at this edge, we pick the smallest (according to the lexicographic
order). We then form $G_{i+1}$ by adding to $G_{i}$ this smallest copy
$\widehat{L}$ as well as all $\widehat{H_{e}}$ where $e\in\widehat{L}\setminus
G_{i}$. We call $G_{i}\to G_{i+1}$ a _pristine step_.
2. (2.2)
If there are no pristine copies of any graph in $\mathcal{L}$, we pick the
smallest (according to the lexicographic order) overlapping copy $\widehat{L}$
of a graph in $\mathcal{L}$ and we still form $G_{i+1}$ by adding to $G_{i}$
the union of $\widehat{L}$ and all its $\widehat{H_{e}}$ with
$e\in\widehat{L}\setminus G_{i}$. We call $G_{i}\to G_{i+1}$ a _degenerate
$\mathcal{L}$-step_.
We define the _balance_ of $G_{i}$ to be
$b(G_{i})\coloneqq e_{G_{i}}-\alpha\cdot v_{G_{i}},$
where we recall that $\alpha=m_{2}(\mathcal{H},\mathcal{L})$. The key result
we will need in order to prove (b) is the following lemma. We remark that a
similar result was proved by Hyde [13, Claims 6.2 and 6.3]; it plays an
integral role in his approach to the Kohayakawa–Kreuter conjecture.
###### Lemma 4.3.
For every $i$, we have that $b(G_{i+1})\geqslant b(G_{i})$. Moreover, there
exists some $\delta=\delta(\mathcal{H},\mathcal{L})>0$ such that
$b(G_{i+1})\geqslant b(G_{i})+\delta$ if $G_{i+1}$ was obtained from $G_{i}$
by a degenerate step.
As the proof of Lemma 4.3 is somewhat technical, we defer it to Section 4.2.
For the moment, we assume the result and continue the discussion of how we
construct the family $\mathcal{S}$. We now let $\Gamma\coloneqq\lceil
2\alpha/\delta\rceil$, where $\delta$ is the constant from Lemma 4.3. For
$G\in\mathcal{G}$, let
$\tau(G)\coloneqq\min\\{i:v_{G_{i}}\geqslant\log n\text{ or }G_{i}=G\text{ or
}G_{i-1}\to G_{i}\text{ is the $\Gamma$th degenerate step}\\}$
and let
$\mathcal{S}\coloneqq\\{G_{\tau(G)}:G\in\mathcal{G}_{\mathrm{bad}}\\}.$ (1)
Having defined the family $\mathcal{S}$, we are ready to prove Lemma 4.2.
Since the definition of $\mathcal{S}$ clearly guarantees property (a), it
remains to establish properties (b) and (c). We begin by showing that, if $K$
is sufficiently large (depending only on $\mathcal{H}$ and $\mathcal{L}$),
then (b) holds.
###### Proof of Lemma 4.2(b).
Let $\delta$ be the constant from Lemma 4.3, let $M\coloneqq\max\\{e_{L}\cdot
v_{H}:H\in\mathcal{H},L\in\mathcal{L}\\}$, and let $K\coloneqq 2M^{2}\Gamma$;
note that each of these parameters depends only on $\mathcal{H}$ and
$\mathcal{L}$.
Every $S\in\mathcal{S}$ is of the form $G_{\tau(G)}$ for some
$G\in\mathcal{G}_{\mathrm{bad}}$. We split into cases depending on which of
the three conditions defining $\tau(G)$ caused us to stop the exploration.
Suppose first that we stopped the exploration because $v_{S}\geqslant\log n$.
By Lemma 4.3, we have that
$e_{S}-\alpha\cdot v_{S}=b(S)=b(G_{\tau(G)})\geqslant b(G_{0})=1-2\alpha,$
and therefore $e_{S}\geqslant\alpha\cdot(v_{S}-2)$. This yields case (b)(i).
Next, suppose we stopped the exploration because step $G_{\tau(G)-1}\to
G_{\tau(G)}$ was the $\Gamma$th degenerate step. As we are not in the previous
case, we may assume that $v_{S}<\log n$. By Lemma 4.3 and our choice of
$\Gamma$, we have that
$e_{S}-\alpha\cdot v_{S}=b(S)=b(G_{\tau(G)})\geqslant
b(G_{0})+\Gamma\delta\geqslant 1-2\alpha+2\alpha=1.$
Rearranging, we see that $e_{S}\geqslant\alpha\cdot v_{S}+1$, yielding case
(b)(ii).
The remaining case is when we stop because $S=G\in\mathcal{G}_{\mathrm{bad}}$.
Since the definition of $\mathcal{G}_{\mathrm{bad}}$ implies that
$m(G)>\alpha$, in order to establish (b)(iii), we only need to show that
$v_{G}\leqslant K$. For this proof, we need to keep track of another parameter
during the exploration process, which we term the _pristine boundary_. Recall
that at every pristine step, we add to $G_{i}$ a copy $\widehat{L}$ of some
$L\in\mathcal{L}$ that intersects $G_{i}$ in a single edge (the root), and
then add copies $\widehat{H_{e}}$ of graphs $H_{e}\in\mathcal{H}$, one for
every edge of $\widehat{L}$ apart from the root. Let us say that the
_boundary_ of this step is the set of all newly added vertices that are not in
$\widehat{L}$, that is, the set $V(G_{i+1})\setminus(V(G_{i})\cup
V(\widehat{L}))=(\bigcup_{e\in\widehat{L}\setminus
G_{i}}V(\widehat{H_{e}}))\setminus V(\widehat{L})$. Note that the size of the
boundary is equal to
$Y_{i}\coloneqq\sum_{e\in\widehat{L}\setminus G_{i}}(v_{H_{e}}-2);$
indeed, by the definition of pristine steps, the copies $\widehat{H_{e}}$ are
vertex-disjoint outside of $V(\widehat{L})$.
We claim that $Y_{i}\geqslant 3$. To see this, note first that $L$ has at
least three edges, as it is not a forest. Similarly, each $H_{e}$ has at least
three vertices. Putting these together, we see that there are at least two
terms in the sum, and every term in the sum is at least one. Thus,
$Y_{i}\geqslant 3$ unless $e_{L}=3$ and $v_{H_{e}}=3$ for all $e$. But in this
case, $L=K_{3}=H_{e}\in\mathcal{H}$ for all $e$, which means that
$\widehat{L}$ should have been added to $G_{i}$ as a degenerate
$\mathcal{H}$-step.
We now inductively define the pristine boundary $\partial G_{i}$ of $G_{i}$ as
follows. We set $\partial G_{0}\coloneqq\varnothing$. If $G_{i}\to G_{i+1}$ is
a pristine step, then we delete from $\partial G_{i}$ the two endpoints of the
root and add to $\partial G_{i}$ the boundary of this pristine step. Note that
$\lvert\partial G_{i+1}\rvert\geqslant\lvert\partial
G_{i}\rvert+Y_{i}-2\geqslant\lvert\partial G_{i}\rvert+1$. On the other hand,
if $G_{i}\to G_{i+1}$ is a degenerate step, then we only remove vertices from
$\partial G_{i}$, without adding any new vertices. Namely, we remove from
$\partial G_{i}$ all the vertices which are included in the newly added
graphs. In other words, if we performed a degenerate $\mathcal{H}$-step by
adding a copy $\widehat{H}$ of some graph in $\mathcal{H}$, we set $\partial
G_{i+1}\coloneqq\partial G_{i}\setminus V(\widehat{H})$. Similarly, if we
performed a degenerate $\mathcal{L}$-step by adding a copy $\widehat{L}$ of
some graph in $\mathcal{L}$ along with the graphs $\widehat{H_{e}}$ for all
$e\in\widehat{L}\setminus G_{i}$, we set $\partial G_{i+1}\coloneqq\partial
G_{i}\setminus(V(\widehat{L})\cup\bigcup_{e}V(\widehat{H_{e}}))$. Note that in
either case $\lvert\partial G_{i+1}\rvert\geqslant\lvert\partial
G_{i}\rvert-M$, as the union of all graphs added in each degenerate step can
have at most $M$ vertices.
We now argue that $\partial G_{\tau(G)}=\varnothing$. Indeed, suppose we had
some vertex $v\in\partial G_{\tau(G)}$. By definition, $v$ was added during a
pristine step, as a vertex of a copy $\widehat{H_{e}}$ of some graph
$H_{e}\in\mathcal{H}$, and was never touched again. Observe that $v$ is
incident to some edge $uv$ of $\widehat{H_{e}}$ that was not touched by any
later step of the exploration. However, as
$(G,\mathcal{F}_{\mathcal{H}},\mathcal{F}_{\mathcal{L}})$ is a core and
$\widehat{H_{e}}\in\mathcal{F}_{\mathcal{H}}$, there must be some
$\widehat{L_{uv}}\in\mathcal{F}_{\mathcal{L}}$ that intersects
$\widehat{H_{e}}$ only at $uv$. Moreover, as $\widehat{L_{uv}}$ has minimum
degree at least two (by the strict $2$-balancedness assumption), there is some
edge $vw\in\widehat{L_{uv}}\setminus uv$ that is incident to $v$. Since we
assumed that $G_{\tau(G)}=G$, the edge $vw$ must have been added at some
point, a contradiction to the assumption that $v$ was never touched again.
Finally, since $\lvert\partial G_{i}\rvert$ increases by at least one during
every pristine step and decreases by at most $M$ during each of the at most
$\Gamma$ degenerate steps, in order to achieve $\partial
G_{\tau(G)}=\varnothing$, there can be at most $M\Gamma$ pristine steps. In
particular, the total number of exploration steps is at most $M\Gamma+\Gamma$.
As each exploration step adds at most $M$ vertices to $G_{i}$, we conclude
that $v_{G}\leqslant M(M\Gamma+\Gamma)+2\leqslant K$. This completes the proof
of (b)(iii). ∎
###### Proof of Lemma 4.2(c).
Suppose $S$ has $k$ vertices and let $G\in\mathcal{G}_{\mathrm{bad}}$ be such
that $S=G_{\tau(G)}$. We consider the exploration process on $G$. Note that in
every step we add an overlapping copy of a graph from a finite family
$\mathcal{F}$ that comprises all graphs in $\mathcal{H}$ (for the cases where
we made a degenerate $\mathcal{H}$-step) and graphs in $\mathcal{L}$ that have
graphs from $\mathcal{H}$ glued on subsets of their edges, with all
intersection patterns (for the pristine and degenerate $\mathcal{L}$-steps).
Let $\mathcal{F}^{\times}$ denote the graphs in $\mathcal{F}$ that correspond
to a pristine step.
Now, every degenerate step can be described by specifying the graph
$F\in\mathcal{F}$ whose copy $\widehat{F}$ we are adding, the subgraph
$F^{\prime}\subseteq F$ and the embedding $\varphi\colon V(F^{\prime})\to
V(G_{i})$ that describe the intersection $\widehat{F}\cap G_{i}$, and the
sequence of $v_{F}-v_{F^{\prime}}$ vertices of $K_{n}$ that complete $\varphi$
to an embedding of $F$ into $K_{n}$. Every pristine step is uniquely described
by the root edge in $G_{i}$, the graph $F\in\mathcal{F}^{\times}$, the edge of
$F$ corresponding to the root, and the (ordered sequence of) $v_{F}-2$
vertices of $K_{n}$ that complete the root edge to a copy of $F$ in $K_{n}$.
There are at most $n^{k}$ ways to choose the sequence of vertices that were
added through this exploration process, in the order that they are introduced
to $G$. Each pristine step adds at least one new vertex, so there are at most
$k$ pristine steps. Furthermore, there are always at most $\Gamma$ degenerate
steps, meaning that $\tau(G)\leqslant k+\Gamma$. In particular, there are at
most $(k+\Gamma)\cdot 2^{k+\Gamma}$ ways to choose $\tau(G)$ and to specify
which steps were pristine.
For every degenerate step, there are at most
$\sum_{F\in\mathcal{F}}\sum_{\ell=2}^{v_{F}}\binom{v_{F}}{\ell}k^{\ell}\leqslant\lvert\mathcal{F}\rvert\cdot(k+1)^{M_{v}}$
ways of choosing $F\in\mathcal{F}$ and describing the intersection of its copy
$\widehat{F}$ with $G_{i}$ (the set $V(F^{\prime})\subseteq V(F)$ and the
embedding $\varphi$ above), where $M_{v}\coloneqq\max\\{v_{F}\colon
F\in\mathcal{F}\\}$. As for the pristine steps, note that, in the course of
our exploration, the sequence of the arrival times of the roots to
$G_{\tau(G)}$ must be non-decreasing. This is because as soon as an edge
appears in some $G_{i}$, every pristine step that includes it as a root at any
later step is already available, and we always choose the one rooted at the
edge that arrived to $G$ the earliest. Therefore, there are at most
$\binom{e_{S}+k}{k}$ possible sequences of root edges, since this is the
number of non-decreasing sequences of length $k$ in
$\llbracket{e_{S}}\rrbracket$. To supplement this bound, remember that every
step increases the number of edges in $G_{i}$ by at most
$M_{e}\coloneqq\max\\{e_{F}:F\in\mathcal{F}\\}$, which means that
$e_{S}\leqslant 1+\tau(G)\cdot M_{e}\leqslant 1+(k+\Gamma)\cdot M_{e}.$
To summarize, the number of $S\in\mathcal{S}$ with $k$ vertices is at most
$n^{k}\cdot(k+\Gamma)\cdot
2^{k+\Gamma}\cdot\left(\lvert\mathcal{F}\rvert\cdot(k+1)^{M_{v}}\right)^{\Gamma}\cdot\binom{(k+\Gamma)\cdot
M_{e}+k+1}{k}\cdot\left(\lvert\mathcal{F}\rvert\cdot M_{e}\right)^{k}.$
Every term in this product, apart from the first, is bounded by an exponential
function of $k$, since $\Gamma,\lvert\mathcal{F}\rvert,M_{v}$, and $M_{e}$ are
all constants. Therefore, if we choose
$\Lambda=\Lambda(\mathcal{H},\mathcal{L})$ sufficiently large, we find that
the number of $S\in\mathcal{S}$ with $v_{S}=k$ is at most $(\Lambda n)^{k}$,
as claimed. ∎
### 4.2. Proof of Lemma 4.3
In this section, we prove Lemma 4.3. The proof is divided into a number of
claims. Recall Lemma 3.5, which asserts that
$e_{H}-e_{F}>m_{2}(\mathcal{H},\mathcal{L})\cdot(v_{H}-v_{F})=\alpha\cdot(v_{H}-v_{F})$
for all $H\in\mathcal{H}$ and all $F\subsetneq H$. This implies that we can
choose some $\delta_{1}=\delta_{1}(\mathcal{H},\mathcal{L})>0$ so that
$e_{H}-e_{F}\geqslant\alpha\cdot(v_{H}-v_{F})+\delta_{1}$ (2)
for all $H\in\mathcal{H}$ and all $F\subsetneq H$; we henceforth fix such a
$\delta_{1}>0$.
Our first claim deals with the (easy) case that $G_{i}\to G_{i+1}$ is a
degenerate $\mathcal{H}$-step.
###### Claim 4.4.
If $G_{i}\to G_{i+1}$ is a degenerate $\mathcal{H}$-step, then
$b(G_{i+1})\geqslant b(G_{i})+\delta_{1}$.
###### Proof.
Suppose we add to $G_{i}$ a copy of some $H\in\mathcal{H}$ that intersects
$G_{i}$ on a subgraph $F\subseteq H$. This means that
$e_{G_{i+1}}=e_{G_{i}}+(e_{H}-e_{F})\qquad\text{ and }\qquad
v_{G_{i+1}}=v_{G_{i}}+(v_{H}-v_{F})$
and thus
$b(G_{i+1})-b(G_{i})=(e_{H}-e_{F})-\alpha\cdot(v_{H}-v_{F})\geqslant\delta_{1},$
where the inequality follows from (2), as $F$ must be a proper subgraph of
$H$. ∎
Now, suppose that $G_{i}\to G_{i+1}$ is an $\mathcal{L}$-step, either
degenerate or pristine, which means that we add a copy $\widehat{L}$ of some
$L\in\mathcal{L}$ and then add, for every edge $e\in\widehat{L}\setminus
G_{i}$, a copy $\widehat{H_{e}}$ of some $H_{e}\in\mathcal{H}$. Let
$G_{i}^{\prime}\coloneqq G_{i}\cup\widehat{L}$ and let $\widehat{J}\coloneqq
G_{i}\cap\widehat{L}$, so that $\widehat{J}\cong J$ for some $J\subsetneq L$
with at least one edge. Note that
$b(G_{i}^{\prime})-b(G_{i})=(e_{L}-e_{J})-\alpha\cdot(v_{L}-v_{J}),$ (3)
as we add $e_{L}-e_{J}$ edges and $v_{L}-v_{J}$ vertices to $G_{i}$ when
forming $G_{i}^{\prime}$.
In order to analyze $b(G_{i+1})-b(G_{i}^{\prime})$, we now define an auxiliary
graph $\mathcal{I}$ as follows. Its vertices are the edges of
$\widehat{L}\setminus\widehat{J}$. Recall that, for every such edge $e$, the
graph $\widehat{H_{e}}\cong H_{e}$ intersects $G_{i}^{\prime}$ only in the
edge $e$. A pair $e,f$ of edges of $\widehat{L}\setminus\widehat{J}$ will be
adjacent in $\mathcal{I}$ if and only if their corresponding graphs
$\widehat{H_{e}}$ and $\widehat{H_{f}}$ share at least one edge (equivalently,
the graphs $\widehat{H_{e}}\setminus e$ and $\widehat{H_{f}}\setminus f$ share
an edge).
Denote the connected components of $\mathcal{I}$ by $K_{1},\dotsc,K_{m}$ and
note that each of them corresponds to a subgraph of
$\widehat{L}\setminus\widehat{J}$. For each $j\in\llbracket{m}\rrbracket$, let
$U_{j}\coloneqq\bigcup_{e\in K_{j}}(\widehat{H_{e}}\setminus e).$
Note that the graphs $G_{i}^{\prime}$ and $U_{1},\dotsc,U_{m}$ are pairwise
edge-disjoint and that each $U_{j}$ shares at least $v_{K_{j}}$ vertices (the
endpoints of all the edges of $K_{j}$) with $G_{i}^{\prime}$. It follows that
$b(G_{i+1})-b(G_{i}^{\prime})\geqslant\sum_{j=1}^{m}(e_{U_{j}}-\alpha\cdot(v_{U_{j}}-v_{K_{j}}))=\sum_{j=1}^{m}(b(U_{j})+\alpha\cdot
v_{K_{j}}).$ (4)
Finally, as in the statement of Lemma 3.7, define
$X\coloneqq\min\\{(e_{H}-1)-\alpha\cdot(v_{H}-2):H\in\mathcal{H}\\}.$
The following claim lies at the heart of the matter.
###### Claim 4.5.
For every $j\in\llbracket{m}\rrbracket$, we have
$b(U_{j})\geqslant
X-2\alpha-(v_{K_{j}}-2)+\min\\{\delta_{1},1\\}\cdot\mathbf{1}_{v_{K_{j}}>2}.$
###### Proof.
Since $K_{j}$ is connected in $\mathcal{I}$, we may order its edges as
$e_{1},\dotsc,e_{\ell}$ so that, for each $r\in\llbracket{\ell-1}\rrbracket$,
the edge $e_{r+1}$ is $\mathcal{I}$-adjacent to $\\{e_{1},\dotsc,e_{r}\\}$.
Letting $F\subseteq H_{e_{r+1}}$ be the subgraph corresponding to this
intersection, we define, for each $r\in\\{0,\dotsc,\ell\\}$,
$U_{j}^{r}\coloneqq\bigcup_{s=1}^{r}(\widehat{H_{e_{s}}}\setminus e_{s}),$
so that $\varnothing=U_{j}^{0}\subseteq\dotsb\subseteq U_{j}^{\ell}=U_{j}$.
Observe that
$b(U_{j}^{1})=e_{U_{j}^{1}}-\alpha\cdot
v_{U_{j}^{1}}=(e_{H_{e_{1}}}-1)-\alpha\cdot v_{H_{e_{1}}}\geqslant X-2\alpha,$
where the inequality follows from the definition of $X$.
Suppose now that $r\geqslant 1$ and let $\widehat{F}$ be the intersection of
$\widehat{H_{e_{r+1}}}\setminus e_{r+1}$ with $U_{j}^{r}$; note that this
intersection is non-empty as $e_{r+1}$ is $\mathcal{I}$-adjacent to
$\\{e_{1},\dots,e_{r}\\}$. We have
$b(U_{j}^{r+1})-b(U_{j}^{r})=(e_{H_{e_{r+1}}}-1-e_{F})-\alpha\cdot(v_{H_{e_{r+1}}}-v_{F}).$
Let $t_{r+1}$ be the number of endpoints of $e_{r+1}$ that are not in
$U_{j}^{r}$. Suppose first that $t_{r+1}=0$, that is, both endpoints of
$e_{r+1}$ are already in $U_{j}^{r}$. In this case, both endpoints of
$e_{r+1}$ also belong to $\widehat{F}$ and thus $\widehat{F}\cup e_{r+1}$ is
isomorphic to a subgraph $F^{+}\subseteq H_{e_{r+1}}$ with $e_{F}+1$ edges and
$v_{F}$ vertices, which means that
$b(U_{j}^{r+1})-b(U_{j}^{r})=(e_{H_{e_{r+1}}}-e_{F^{+}})-\alpha\cdot(v_{H_{e_{r+1}}}-v_{F^{+}})\geqslant
0,$
by Lemma 3.5. In case $t_{r+1}>0$, $F$ is a proper subgraph of $H_{e_{r+1}}$
and thus we have
$b(U_{j}^{r+1})-b(U_{j}^{r})\geqslant\delta_{1}-1\geqslant\delta_{1}-t_{r+1},$
see (2). We may thus conclude that
$b(U_{j})=b(U_{j}^{1})+\sum_{r=1}^{\ell-1}(b(U_{j}^{r+1})-b(U_{j}^{r}))\geqslant
X-2\alpha-\sum_{r=1}^{\ell-1}t_{r+1}+\delta_{1}\cdot\mathbf{1}_{t_{2}+\dotsb+t_{\ell}>0}.$
The desired inequality follows as $t_{2}+\dotsb+t_{\ell}=\lvert
V(K_{j})\setminus V(U_{j}^{1})\rvert\leqslant v_{K_{j}}-2$ and, further,
$v_{K_{j}}>2$ implies that the sum $t_{2}+\dotsb+t_{r}$ is either positive or
at most $v_{K_{j}}-3$. ∎
We are now ready to show that the balance only increases when we perform an
$\mathcal{L}$-step.
###### Claim 4.6.
If $G_{i}\to G_{i+1}$ is an $\mathcal{L}$-step, then $b(G_{i+1})\geqslant
b(G_{i})$. Moreover, if this $\mathcal{L}$-step is degenerate, then
$b(G_{i+1})\geqslant b(G_{i})+\delta_{2}$ for some $\delta_{2}>0$ that depends
only on $\mathcal{H}$ and $\mathcal{L}$.
###### Proof.
By (3), (4), and Claim 4.6, we have
$\displaystyle b$
$\displaystyle(G_{i+1})-b(G_{i})=b(G_{i}^{\prime})-b(G_{i})+b(G_{i+1})-b(G_{i}^{\prime})$
$\displaystyle\geqslant(e_{L}-e_{J})-\alpha\cdot(v_{L}-v_{J})+\sum_{j=1}^{m}(b(U_{j})+\alpha\cdot
v_{K_{j}})$
$\displaystyle\geqslant(e_{L}-e_{J})-\alpha\cdot(v_{L}-v_{J})+\sum_{j=1}^{m}\left(X+(v_{K_{j}}-2)(\alpha-1)\right)+\min\\{\delta_{1},1\\}\cdot\mathbf{1}_{\mathcal{I}\neq\varnothing},$
since $\mathcal{I}$ is nonempty only if one of its components has more than
two vertices. We now apply Lemma 3.7 to each component $K_{j}$ to conclude
that
$\sum_{j=1}^{m}\left(X+(v_{K_{j}}-2)(\alpha-1)\right)\geqslant\sum_{j=1}^{m}e_{K_{j}}\cdot\left(\frac{\alpha}{m_{2}(L)}-1\right)=(e_{L}-e_{J})\left(\frac{\alpha}{m_{2}(L)}-1\right).$
Therefore,
$b(G_{i+1})-b(G_{i})\geqslant(e_{L}-e_{J})\cdot\frac{\alpha}{m_{2}(L)}-\alpha\cdot(v_{L}-v_{J})+\min\\{\delta_{1},1\\}\cdot\mathbf{1}_{\mathcal{I}\neq\varnothing}\geqslant\min\\{\delta_{1},1\\}\cdot\mathbf{1}_{\mathcal{I}\neq\varnothing},$
where the last inequality follows from Lemma 3.6. This implies the desired
result if the $\mathcal{L}$-step is pristine. If the $\mathcal{L}$-step is not
pristine but $\mathcal{I}$ has no edges, it means that some vertex was
repeated between different $\widehat{H_{e}}$. In that case, the first
inequality in (4) is strict (we assumed there that the graphs $U_{j}$ share no
vertices outside of $V(K_{j})$). All in all, we obtain the desired boost in
the degenerate case. ∎
Combining Claims 4.4 and 4.6, we obtain Lemma 4.3. This completes the proof of
the probabilistic lemma.
## 5\. Proof of the deterministic lemma
Given the probabilistic lemma and the work of the first two authors on the
symmetric case [17], in order to prove Conjecture 1.9, which generalizes the
Kohayakawa–Kreuter conjecture, we only need to show the following. For every
strictly balanced pair $(\mathcal{H},\mathcal{L})$ of finite families of
graphs with $m_{2}(\mathcal{H})>m_{2}(\mathcal{L})>1$, we can two-color the
edges of every graph $G$ satisfying $m(G)\leqslant
m_{2}(\mathcal{H},\mathcal{L})$ so that there are neither red monochromatic
copies of any $H\in\mathcal{H}$ nor blue monochromatic copies of any
$L\in\mathcal{L}$. As discussed in the introduction, we do not know how to do
this in all cases. However, the following proposition lists a number of extra
assumptions under which we are able to find such a coloring. We recall the
notion of the _$1$ -density_ (or _fractional arboricity_) of a graph $L$,
defined by
$m_{1}(L)\coloneqq\max\left\\{\frac{e_{J}}{v_{J}-1}:J\subseteq
L,v_{J}\geqslant 2\right\\}.$
We also make the following definition.
###### Definition 5.1.
Given positive integers $s\leqslant t$, we say that a graph is an _$(s,t)$
-graph_ if its minimum degree is at least $s$, and every edge contains a
vertex of degree at least $t$. We say that a graph is _$(s,t)$ -avoiding_ if
none of its subgraphs is an $(s,t)$-graph.
###### Proposition 5.2.
Let $(\mathcal{H},\mathcal{L})$ be a strictly balanced pair of finite families
of graphs satisfying $m_{2}(\mathcal{H})>m_{2}(\mathcal{L})$ and suppose that
at least one of the following conditions holds:
1. (a)
$\chi(L)\geqslant 3$ for all $L\in\mathcal{L}$;
2. (b)
$\chi(H)>m_{2}(\mathcal{H},\mathcal{L})+1$ for every $H\in\mathcal{H}$;
3. (c)
$m_{1}(L)>2$ for all $L\in\mathcal{L}$;
4. (d)
every $H\in\mathcal{H}$ contains an $(s,t)$-graph as a subgraph, for some
integers $s\leqslant t$ satisfying
$\frac{1}{s+1}+\frac{1}{t+1}<\frac{1}{m_{2}(\mathcal{H},\mathcal{L})};$
5. (e)
$\lceil m_{2}(\mathcal{H},\mathcal{L})\rceil<m_{2}(\mathcal{H})$;
Then any graph $G$ with $m(G)\leqslant m_{2}(\mathcal{H},\mathcal{L})$ is not
Ramsey for $(\mathcal{H},\mathcal{L})$.
Cases (a)–(c) all follow fairly easily from known coloring techniques; we
supply the details in the remainder of this section. Case (d) is proved by a
short inductive argument, see below. Case (e) follows from our partial
progress on Conjecture 1.5, namely, that we are able to prove it when $m(G)$
is an integer; we present the proof of this result in Appendix B. We end this
section with short derivations of Theorems 1.7 and 1.4 from the proposition.
###### Proof of Theorem 1.4.
Assume that $m_{2}(L)>\frac{11}{5}$. By passing to a subgraph with the same
$2$-density, we may assume that $L$ is strictly $2$-balanced. Thanks to cases
(a) and (c) of Proposition 5.2, we are done unless $m_{1}(L)\leqslant 2$ and
$L$ is bipartite. The bounds on $m_{1}(L)$ and $m_{2}(L)$ imply that
$2v_{L}-2\geqslant e_{L}>\frac{11}{5}(v_{L}-2)+1$, which yields $v_{L}<7$.
However, as $L$ is bipartite on at most six vertices, we have
$m_{2}(L)\leqslant m_{2}(K_{3,3})=2$, a contradiction. ∎
###### Proof of Theorem 1.7.
Cases (a), (b), (c), and (f) follow immediately777Proposition 5.2(c) implies
Theorem 1.7(b) thanks to Nash-Williams’s theorem (Theorem 5.7 below). from
Proposition 5.2. For Theorem 1.7(d), note that a graph with minimum degree $d$
is a $(d,d)$-graph. Thus, if $H_{1}$ has degeneracy at least $d$, then it
contains some $(d,d)$-graph as a subgraph. Similarly, Theorem 1.7(e) follows,
since if $s\leqslant t$, then $K_{s,t}$ is an $(s,t)$-graph satisfying
$1/m_{2}(K_{s,t})=(s+t-2)/(st-1)\geqslant 1/(s+1)+1/(t+1)$. ∎
### 5.1. Auxiliary results
We start with a helpful observation relating $m(G)$ and the degeneracy of $G$.
We say that a graph is _$d$ -degenerate_ if its degeneracy is at most $d$.
###### Lemma 5.3.
Every graph $G$ is $\lfloor 2m(G)\rfloor$-degenerate.
###### Proof.
For every $G^{\prime}\subseteq G$, we have
$\delta(G^{\prime})\leqslant\left\lfloor\frac{2e_{G^{\prime}}}{v_{G^{\prime}}}\right\rfloor\leqslant\lfloor
2m(G)\rfloor,$
where $\delta(G^{\prime})$ is the minimum degree of $G^{\prime}$. ∎
Our second lemma allows us to compare between the various densities.
###### Lemma 5.4.
For every graph $H$, we have $m_{2}(H)\leqslant m_{1}(H)+\frac{1}{2}\leqslant
m(H)+1$.
###### Proof.
Notice that both $\frac{e-1}{v-2}\leqslant\frac{e}{v-1}+\frac{1}{2}$ and
$\frac{e}{v-1}\leqslant\frac{e}{v}+\frac{1}{2}$ are equivalent to
$e\leqslant\binom{v}{2}$, so both inequalities hold whenever $v,e$ are the
numbers of vertices and edges, respectively, of any graph. In particular, if
$v,e$ correspond to the subgraph of $H$ that achieves $m_{2}(H)$, we find that
$m_{2}(H)=\frac{e-1}{v-2}\leqslant\frac{e}{v-1}+\frac{1}{2}\leqslant
m_{1}(H)+\frac{1}{2}$. The second inequality follows in the same way, now
passing to the subgraph that achieves $m_{1}(H)$. ∎
Our next lemma gives a lower bound on the average degree of an $(s,t)$-graph.
We remark that this inequality is tight for $K_{s,t}$ and that it can be
restated as $e_{H}/v_{H}\geqslant m(K_{s,t})$.
###### Lemma 5.5.
If $H$ is an $(s,t)$-graph, then
$\frac{1}{s}+\frac{1}{t}\geqslant\frac{v_{H}}{e_{H}}.$
###### Proof.
The assumption that $H$ is an $(s,t)$-graph implies that, for every $uv\in
E(H)$, we have $1/\deg(u)+1/\deg(v)\leqslant 1/s+1/t$. This means that
$e_{H}\cdot\left(\frac{1}{s}+\frac{1}{t}\right)\geqslant\sum_{uv\in
H}\left(\frac{1}{\deg(u)}+\frac{1}{\deg(v)}\right)=v_{H}.\qed$
The next lemma supplies a decomposition of a graph of bounded degeneracy.
###### Lemma 5.6.
If a graph $G$ is $(dk-1)$-degenerate, for some positive integers $d,k$, then
there is a partition $V(G)=V_{1}\cup\dotsb\cup V_{k}$ such that the graphs
$G[V_{1}],\dotsc,G[V_{k}]$ are all $(d-1)$-degenerate.
###### Proof.
We may construct the desired partition in the following way. Initialize
$V_{1}=\dotsb=V_{k}=\varnothing$ and let $v_{1},\dotsc,v_{n}$ be an ordering
of the vertices of $G$ such that every $v_{i}$ has at most $dk-1$ neighbors
preceding it. We distribute the vertices one-by-one, each time putting $v_{i}$
in a set $V_{j}$ where, at the time, $v_{i}$ has the smallest number of
neighbors. By the pigeonhole principle, this number is at most
$\lfloor\frac{dk-1}{k}\rfloor=d-1$. ∎
Finally, we quote Nash-Williams’s theorem on partitions of graphs into
forests.
###### Theorem 5.7 (Nash-Williams [21]).
A graph $G$ can be partitioned into $t$ forests if and only if $\lceil
m_{1}(G)\rceil\leqslant t$.
### 5.2. Proof of Proposition 5.2
We are now ready to prove Proposition 5.2. Denote $\alpha\coloneqq
m_{2}(\mathcal{H},\mathcal{L})$ and let $G$ be an arbitrary graph satisfying
$m(G)\leqslant\alpha$. We will argue that (the edge set of) $G$ can be
partitioned into an $\mathcal{H}$-free graph and an $\mathcal{L}$-free graph.
We split into cases, depending on which condition is satisfied by the pair
$(\mathcal{H},\mathcal{L})$.
#### Cases (a) and (b).
Let $k\coloneqq\lfloor\alpha\rfloor+1$, so that $m(G)\leqslant\alpha<k$, and
note that Lemma 5.3 implies that $G$ is $(2k-1)$-degenerate. Consequently,
Lemma 5.6 yields two partitions of the edges of $G$: a partition into a
$1$-degenerate graph and a $k$-colorable graph; and a partition into a
$(k-1)$-degenerate graph and a bipartite graph. The existence of the first
partition proves (b), as every $1$-degenerate graph is $\mathcal{L}$-free
whereas the assumption on $\mathcal{H}$ implies that $\chi(H)>k$ for every
$H\in\mathcal{H}$. We now argue that the existence of the second partition
proves (a). To this end, note that the assumption there implies that every
bipartite graph is $\mathcal{L}$-free, so it is enough to show that
$\delta(H)\geqslant k$ for every $H\in\mathcal{H}$ and thus every
$(k-1)$-degenerate graph is $\mathcal{H}$-free. To see that this is the case,
consider an arbitrary $H\in\mathcal{H}$ and let $v\in V(H)$ be its vertex with
smallest degree. As $H$ is strictly $m_{2}(\cdot,\mathcal{L})$-balanced, Lemma
3.5 gives $\delta(H)=e_{H}-e_{H\setminus v}>\alpha$, unless $v_{H}=3$, in
which case $H=K_{3}$ and we still have $\delta(H)\geqslant 2=m_{2}(H)\geqslant
m_{2}(\mathcal{H})>\alpha$. Since $\delta(H)$ is an integer, we actually have
$\delta(H)\geqslant\lfloor\alpha\rfloor+1=k$, as needed.
#### Case (c).
It is enough to show that $G$ can be partitioned into an $\mathcal{H}$-free
graph and a union of two forests; indeed, if $m_{1}(L)>2$ for all
$L\in\mathcal{L}$, then no union of two forests can contain a member of
$\mathcal{L}$ as a subgraph, by (the easy direction of) Theorem 5.7. Let
$m_{1}(\mathcal{H})\coloneqq\min\\{m_{1}(H):H\in\mathcal{H}\\}$. By Lemma 5.4
and the assumption $m(G)\leqslant
m_{2}(\mathcal{H},\mathcal{L})<m_{2}(\mathcal{H})$, we find that
$m_{1}(G)\leqslant m(G)+\frac{1}{2}\leqslant
m_{2}(\mathcal{H})+\frac{1}{2}\leqslant m_{1}(\mathcal{H})+1.$
As a result, if we let $t\coloneqq\lceil m_{1}(\mathcal{H})\rceil$, we find
that $\lceil m_{1}(G)\rceil\leqslant t+1$ and therefore Theorem 5.7 supplies a
partition $G$ into $t+1$ forests $G_{1},\dots,G_{t+1}$. Taking
$G^{\prime}\coloneqq G_{1}\cup\dots\cup G_{t-1}$, we arrive at a partition
$G=G^{\prime}\cup(G_{t}\cup G_{t+1})$. By (the easy direction of) Theorem 5.7,
we know that $m_{1}(G^{\prime})\leqslant t-1<m_{1}(\mathcal{H})$, so
$G^{\prime}$ is $\mathcal{H}$-free. As $G_{t}$ and $G_{t+1}$ are forests, we
get the desired decomposition.
#### Case (d).
It is enough to show that $G$ can be decomposed into a forest and an
$(s,t)$-avoiding graph. Assume that this is not the case and let $G$ be a
smallest counterexample with $m(G)\leqslant\alpha$. It is enough to show that
$G$ is an $(s+1,t+1)$-graph, as then Lemma 5.5 gives
$\frac{1}{s+1}+\frac{1}{t+1}\geqslant\frac{v_{G}}{e_{G}}\geqslant\frac{1}{m(G)}\geqslant\frac{1}{\alpha},$
a contradiction.
Suppose first that $G$ has a vertex $v$ of degree at most $s$. By minimality
of $G$, we can decompose the edges of $G\setminus v$ into an $(s,t)$-avoiding
graph $K$ and a forest $F$. Adding an arbitrary edge incident with $v$ to $F$
and the remaining edges to $K$ maintains $F$ being a forest and $K$ being
$(s,t)$-avoiding, as any $(s,t)$-subgraph of $K$ would have to use $v$, which
has degree at most $s-1$ in $K$. This contradicts our assumption on
indecomposability of $G$.
Second, suppose that $G$ contains an edge $uv$ with $\deg(u),\deg(v)\leqslant
t$. By minimality of $G$, we can decompose $G^{\prime}\coloneqq G\setminus uv$
into a forest $F$ and an $(s,t)$-avoiding graph $K$. Adding $uv$ to $F$ must
close a cycle, meaning that both $u$ and $v$ are incident to at least one
$F$-edge of $G^{\prime}$ and thus the $K$-degrees of $u$ and $v$ in
$G^{\prime}$ are at most $t-2$. This means, however, that we can add $uv$ to
$K$ while still keeping the degrees of both its endpoints strictly below $t$.
Again, we find that $K$ contains no $(s,t)$-subgraph, a contradiction.
#### Case (e).
Let $k\coloneqq\lceil m_{2}(\mathcal{H},\mathcal{L})\rceil$. Since we assume
that $m_{2}(\mathcal{H})>k$, it is enough to decompose $G$ into a forest and a
graph $K$ with $m_{2}(K)\leqslant k$. The following theorem, which implies
Conjecture 1.5 in the case that $m(G)$ is an integer, supplies such a
decomposition.
###### Theorem 5.8.
Let $k$ be an integer, and let $G$ be a graph with $m(G)\leqslant k$. Then
there exists a forest $F\subseteq G$ such that $m_{2}(G\setminus F)\leqslant
k$.
The proof of Theorem 5.8 is substantially more involved, as it relies on
techniques from matroid theory. We are hopeful that similar techniques may be
used to prove Conjecture 1.5 in its entirety. We defer the proof of Theorem
5.8 to Appendix B.
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## Appendix A The three-color setting
In this section, we explain what about the proof needs to change to handle the
case $r\geqslant 3$, and prove Theorem 1.6. As many of these results are
essentially identical to the results discussed previously, we omit or shorten
several of the proofs. We begin by defining a natural three-color analogue of
cores.
###### Definition A.1.
Let $\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3}$ be three finite families
of graphs. A tuple $(G,\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{F}_{3})$ is an
_$(\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3})$ -core_ if $G$ is a graph
and $\mathcal{F}_{i}\subseteq\mathcal{F}_{\mathcal{H}_{i}}[G]$ for all
$i\in\llbracket{3}\rrbracket$ are families satisfying the following
properties:
* •
The hypergraph $\mathcal{F}_{1}\cup\mathcal{F}_{2}\cup\mathcal{F}_{3}$ is
connected and spans $E(G)$.
* •
For every $i\in\llbracket{3}\rrbracket$, every
$\widehat{H_{i}}\in\mathcal{F}_{i}$, every edge $e\in\widehat{H_{i}}$, and
every $j\in\llbracket{3}\rrbracket\setminus\\{i\\}$, there is some
$\widehat{H_{j}}\in\mathcal{F}_{j}$ with
$\widehat{H_{i}}\cap\widehat{H_{j}}=\\{e\\}$.
We say that $G$ _supports a core_ if there exists a core
$(G,\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{F}_{3})$.
The following simple lemma is a straightforward generalization of Lemma 3.2,
so we omit the proof.
###### Lemma A.2.
Let $G$ be a graph that is minimally Ramsey for
$(\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3})$, in the sense that any
proper subgraph $G^{\prime}\subsetneq G$ is not Ramsey for
$(\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3})$. Then $G$ supports a core.
It would be very convenient if every
$(\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3})$-core were also an
$(\mathcal{H}_{1},\mathcal{H}_{2})$-core. At first glance this seems true,
since the intersection property in Definition A.1 easily implies the
intersection property in Definition 3.1. Unfortunately, it may be the case
that the hypergraph $\mathcal{F}_{1}\cup\mathcal{F}_{2}\cup\mathcal{F}_{3}$ is
connected, but that the hypergraph $\mathcal{F}_{1}\cup\mathcal{F}_{2}$ is
disconnected. Nonetheless, this is the only obstruction, and the following
result is true.
###### Lemma A.3.
Let $(G,\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{F}_{3})$ be
$(\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3})$-core for some families of
graphs $\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3}$. Then
$(G,\mathcal{F}_{1},\mathcal{F}_{2}\cup\mathcal{F}_{3})$ is an
$(\mathcal{H}_{1},\mathcal{H}_{2}\cup\mathcal{H}_{3})$-core.
###### Proof.
First note that the hypergraph
$\mathcal{F}_{1}\cup(\mathcal{F}_{2}\cup\mathcal{F}_{3})$ is simply the same
as the hypergraph $\mathcal{F}_{1}\cup\mathcal{F}_{2}\cup\mathcal{F}_{3}$, so
it is connected and spans $E(G)$ by assumption. For every
$\widehat{H_{1}}\in\mathcal{F}_{1}$ and every edge $e\in\widehat{H_{1}}$, we
may apply Definition A.1 with $j=2$ to see that there exists some
$\widehat{H_{2}}\in\mathcal{F}_{2}\subseteq\mathcal{F}_{2}\cup\mathcal{F}_{3}$
such that $\widehat{H_{1}}\cap\widehat{H_{2}}=\\{e\\}$. Similarly, applying
Definition A.1 with $j=1$, we see that for every
$\widehat{H_{23}}\in\mathcal{F}_{2}\cup\mathcal{F}_{3}$ and every edge
$e\in\widehat{H_{23}}$, there is some $\widehat{H_{1}}\in\mathcal{F}_{1}$ such
that $\widehat{H_{1}}\cap\widehat{H_{23}}=\\{e\\}$. Thus,
$(G,\mathcal{F}_{1},\mathcal{F}_{2}\cup\mathcal{F}_{3})$ is an
$(\mathcal{H}_{1},\mathcal{H}_{2}\cup\mathcal{H}_{3})$-core. ∎
The key (trivial) observation is that if
$m_{2}(\mathcal{H}_{2})=m_{2}(\mathcal{H}_{3})$, then
$m_{2}(\mathcal{H}_{2}\cup\mathcal{H}_{3})$ is also equal to both these
numbers, as
$m_{2}(\mathcal{H}_{2}\cup\mathcal{H}_{3})=\min\\{m_{2}(\mathcal{H}_{2}),m_{2}(\mathcal{H}_{3})\\}$.
Now, suppose we are given families
$\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3}$ with
$m_{2}(\mathcal{H}_{1})>m_{2}(\mathcal{H}_{2})=m_{2}(\mathcal{H}_{3})$. By
passing to families of subgraphs, we may assume that
$\mathcal{H}_{2},\mathcal{H}_{3}$ are strictly 2-balanced and that
$\mathcal{H}_{1}$ is strictly $m_{2}(\cdot,\mathcal{H}_{2})$-balanced. We now
define $\mathcal{H}=\mathcal{H}_{1}$ and
$\mathcal{L}=\mathcal{H}_{2}\cup\mathcal{H}_{3}$. By Lemma 4.1, we know that
there exists some $c>0$ such that if $p\leqslant
cn^{-1/m_{2}(\mathcal{H},\mathcal{L})}$, then a.a.s. $G_{n,p}$ contains no
subgraph $G$ which supports an $(\mathcal{H},\mathcal{L})$-core and satisfies
$m(G)>m_{2}(\mathcal{H},\mathcal{L})$.
On the other hand, if $G_{n,p}$ is Ramsey for
$(\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3})$, then it must contain some
minimally Ramsey subgraph $G$. By Lemmas A.3 and A.2, $G$ supports an
$(\mathcal{H},\mathcal{L})$-core. Moreover, by the above, we must have
$m(G)\leqslant
m_{2}(\mathcal{H},\mathcal{L})=m_{2}(\mathcal{H}_{1},\mathcal{H}_{2})$, for
otherwise $G\nsubseteq G_{n,p}$ a.a.s. Given this, the following deterministic
lemma concludes the proof.
###### Lemma A.4.
Let $\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3}$ satisfy
$m(\mathcal{H}_{1})\geqslant m(\mathcal{H}_{2})\geqslant
m(\mathcal{H}_{3})>1$. If $G$ is Ramsey for
$(\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3})$, then
$m(G)>m_{2}(\mathcal{H}_{1},\mathcal{H}_{2})$.
###### Proof.
We will actually prove that $m(G)>m_{2}(\mathcal{H}_{1})$, which implies the
desired result since $m_{2}(\mathcal{H}_{1})\geqslant
m_{2}(\mathcal{H}_{1},\mathcal{H}_{2})$. Suppose for contradiction that
$m(G)\leqslant m_{2}(\mathcal{H}_{1})$. By Theorem 5.7 (cf. the proof of
Proposition 5.2(c)), we know that $G$ is the union of an
$\mathcal{H}_{1}$-free graph and two forests. As
$m_{2}(\mathcal{H}_{2})\geqslant m_{2}(\mathcal{H}_{3})>1$, every graph in
$\mathcal{H}_{2}\cup\mathcal{H}_{3}$ contains a cycle, and hence each of these
forests is $\mathcal{H}_{2}\cup\mathcal{H}_{3}$-free. Using one color for the
$\mathcal{H}_{1}$-free graph and one color for each of the two forests, we see
that $G$ is not Ramsey for
$(\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3})$. ∎
## Appendix B Proof of Conjecture 1.5 in the integer case
In this section, we present the proof of Theorem 5.8, which implies Conjecture
1.5 in the case that $m(G)$ is an integer. We will use some well-known results
from matroid theory; all definitions and proofs can be found in any standard
reference on matroid theory, such as Oxley’s book [23].
The main result we will need is the following matroid partitioning theorem,
originally due to Edmonds [5]. We remark that this theorem easily implies
Nash-Williams’s theorem (Theorem 5.7), which was used in the proof of
Proposition 5.2(c).
###### Theorem B.1.
Let $M_{1},M_{2}$ be matroids on the same ground set $E$, with rank functions
$r_{1},r_{2}$, respectively. Then $E$ can be partitioned as $E=I_{1}\cup
I_{2}$, with $I_{i}$ independent in $M_{i}$ for $i=1,2$, if and only if
$r_{1}(X)+r_{2}(X)\geqslant\lvert X\rvert$
for every $X\subseteq E$.
A slightly weaker statement appears as [23, Theorem 11.3.12], where the result
is only stated when $M_{1}=M_{2}$. However, it is clear and well-known that
the same proof proves Theorem B.1, using the formula for the rank of a matroid
union, as given in [23, Theorem 11.3.1].
In our application, we will set $E=E(G)$ and let $M_{1}$ be the graphic
matroid of $G$, whose independent sets are precisely the acyclic subgraphs of
$G$. We may view any subset of $E(G)$ as a subgraph $J$ of $G$; we then use
$e_{J}$ rather than $\lvert J\rvert$ to denote the size of this subset of
$E(G)$. Additionally, we use $v_{J}$ to denote the number of vertices incident
to any edge of $J$, and $\omega_{J}$ to denote the number of connected
components of $J$. It is well-known (e.g. [23, equation 1.3.8]) that the rank
function of $M_{1}$ is given by $r_{1}(J)=v_{J}-\omega_{J}$ for all
$J\subseteq E(G)$.
The second matroid we use will be one whose independent sets are precisely
those subgraphs $K\subseteq G$ with $m_{2}(K)\leqslant k$. The fact that this
is a matroid is the content of the next lemma.
###### Lemma B.2.
Let $G$ be a graph and let $k$ be a positive integer. Then the family of
subgraphs $K\subseteq G$ with $m_{2}(K)\leqslant k$ is the collection of
independent sets of a matroid.
###### Proof.
Define a function $f\colon 2^{E(G)}\to\mathbb{Z}$ by $f(J)=k(v_{J}-2)+1$, for
every $J\subseteq E(G)$. Note that this function is integer-valued since
$k\in\mathbb{Z}$. Additionally, it is clear that $f$ is increasing, in the
sense that $f(J)\leqslant f(J^{\prime})$ whenever $J\subseteq J^{\prime}$.
Finally, we claim that $f$ is submodular. This is easiest to see by recalling
that the function $g(J)=v_{J}$ is submodular (see e.g. [23, Proposition
11.1.6]); as $f$ is obtained from $g$ by multiplying by a positive constant
and adding a constant, we find that $f$ is submodular as well.
Now, by [23, Corollary 11.1.2], we find that there exists a matroid $M(f)$ on
$E(G)$ whose independent sets are precisely those $K\subseteq E(G)$ with the
property that $e_{J}\leqslant f({J})$ for all non-empty $J\subseteq K$. Note
that, for a graph $J$ with at least three vertices, the inequality
$e_{J}\leqslant f(J)$ is equivalent to $d_{2}(J)\leqslant k$, where
$d_{2}(J)=(e_{J}-1)/(v_{J}-2)$. If $J$ is non-empty and has only two vertices,
then it must have one edge and $e_{J}\leqslant f(J)$ always holds. Thus, we
see that $K$ is independent in $M(f)$ if and only if
$\max\\{(e_{J}-1)/(v_{J}-2):J\subseteq K,v_{J}\geqslant 3\\}\leqslant k$. This
condition is precisely the condition that $m_{2}(K)\leqslant k$. ∎
In order to apply Theorem B.1 to the matroids $M_{1},M_{2}$, we need a way of
lower-bounding the rank function of $M_{2}$. This is achieved by the following
lemma.
###### Lemma B.3.
Let $k$ be a positive integer. If $J$ is a graph with $m(J)\leqslant k$, then
there is a subgraph $J^{\prime}\subseteq J$ with $m_{2}(J^{\prime})\leqslant
k$ and $e_{J}\leqslant e_{J^{\prime}}+v_{J}-1.$
###### Proof.
A well-known theorem of Hakimi [10], which is itself a simple consequence of
Theorem B.1, implies that since $m(J)\leqslant k$, we can partition $J$ into
graphs $J_{1},\dots,J_{k}$, with $m(J_{i})\leqslant 1$ for all $i$ (i.e. every
component of every $J_{i}$ has at most one cycle). We may assume without loss
of generality that $J_{k}$ is non-empty. Let $e$ be an edge of $J_{k}$ and
define $J^{\prime}=J_{1}\cup\dotsb\cup J_{k-1}\cup\\{e\\}$. We claim that
$m_{2}(J^{\prime})\leqslant k$ and $e_{J}\leqslant e_{J^{\prime}}+v_{J}-1$.
The second claim is fairly easy to see, as
$e_{J^{\prime}}=1+\sum_{i=1}^{k-1}e_{J_{i}}=1+(e_{J}-e_{J_{k}})\geqslant
1+e_{J}-v_{J_{k}}\geqslant e_{J}-v_{J}+1,$
where the second equality uses the fact that $J_{1},\dots,J_{k}$ partition
$J$, and the two inequalities follow from $e_{J_{k}}\leqslant
v_{J_{k}}\leqslant v_{J}$, since $m(J_{k})\leqslant 1$ and $J_{k}\subseteq J$.
So it remains to prove that $m_{2}(J^{\prime})\leqslant k$, i.e. that
$d_{2}(L)\leqslant k$ for all $L\subseteq J^{\prime}$. If $v_{L}\leqslant
2k-1$, then
$d_{2}(L)\leqslant\frac{\binom{v_{L}}{2}-1}{v_{L}-2}=\frac{1}{2}\cdot\frac{v_{L}^{2}-v_{L}-2}{v_{L}-2}=\frac{1}{2}(v_{L}+1)\leqslant
k,$
as claimed. So we may assume that $v_{L}\geqslant 2k$. As $m(J_{i})\leqslant
1$ for all $i$, we see that $e_{L}\leqslant(k-1)v_{L}+1$. Therefore,
$d_{2}(L)=\frac{e_{L}-1}{v_{L}-2}\leqslant\frac{(k-1)v_{L}}{v_{L}-2}\leqslant\frac{kv_{L}-2k}{v_{L}-2}=k.\qed$
With all of these preliminaries, we are ready to prove Theorem 5.8.
###### Proof of Theorem 5.8.
Let $G$ be a graph with $m(G)\leqslant k$ and let $E=E(G)$. Let $M_{1}$ be the
graphic matroid on the ground set $E$ and let $M_{2}$ be the matroid given by
Lemma B.2, whose independent sets are those $K\subseteq G$ with
$m_{2}(K)\leqslant k$. We wish to prove that $E$ can be partitioned into an
independent set from $M_{1}$ and an independent set from $M_{2}$; by Theorem
B.1, it suffices to prove that $r_{1}(J)+r_{2}(J)\geqslant e_{J}$ for all
$J\subseteq G$.
So fix some $J\subseteq G$, and let its connected components be
$J_{1},\dots,J_{t}$. We then have that $r_{1}(J)=v_{J}-\omega_{J}=v_{J}-t$. As
$m(G)\leqslant k$, we certainly have that $m(J_{i})\leqslant k$ for all $i$,
and hence Lemma B.3 implies that there exist $J^{\prime}_{i}\subseteq J_{i}$
with $m_{2}(J^{\prime}_{i})\leqslant k$ and $e_{J_{i}}\leqslant
e_{J^{\prime}_{i}}+v_{J_{i}}-1$. Let $J^{\prime}=J_{1}^{\prime}\cup\dotsb\cup
J_{t}^{\prime}$. If $J^{\prime}$ is a matching, then
$m_{2}(J^{\prime})\leqslant 1\leqslant k$. If not, then its maximal
$2$-density is attained on some connected component, hence
$m_{2}(J^{\prime})=\max_{i}m_{2}(J_{i}^{\prime})\leqslant k$. Therefore,
$J^{\prime}$ is independent in $M_{2}$, which implies that
$r_{2}(J)\geqslant
r_{2}(J^{\prime})=e_{J^{\prime}}=\sum_{i=1}^{t}e_{J_{i}^{\prime}}\geqslant\sum_{i=1}^{t}(e_{J_{i}}-(v_{J_{i}}-1))=e_{J}-(v_{J}-t).$
Recalling that $r_{1}(J)=v_{J}-t$, we conclude that
$r_{1}(J)+r_{2}(J)\geqslant e_{J}$, as claimed. ∎
|
1
# Dependence-Aware, Unbounded Sound Predictive Race Detection
This extended arXiv version of an OOPSLA 2019 paper adds Appendices A–C
Kaan Genç Ohio State UniversityUSA<EMAIL_ADDRESS>, Jake Roemer Ohio State
UniversityUSA<EMAIL_ADDRESS>, Yufan Xu Ohio State UniversityUSA
<EMAIL_ADDRESS>and Michael D. Bond Ohio State UniversityUSA
<EMAIL_ADDRESS>
(2019)
# Dependence-Aware, Unbounded Sound Predictive Race Detection
This extended arXiv version of an OOPSLA 2019 paper adds Appendices A–C
Kaan Genç Ohio State UniversityUSA<EMAIL_ADDRESS>, Jake Roemer Ohio State
UniversityUSA<EMAIL_ADDRESS>, Yufan Xu Ohio State UniversityUSA
<EMAIL_ADDRESS>and Michael D. Bond Ohio State UniversityUSA
<EMAIL_ADDRESS>
(2019)
###### Abstract.
Data races are a real problem for parallel software, yet hard to detect. Sound
predictive analysis observes a program execution and detects data races that
exist in some _other, unobserved_ execution. However, existing predictive
analyses miss races because they do not scale to full program executions or do
not precisely incorporate data and control dependence.
This paper introduces two novel, sound predictive approaches that incorporate
data and control dependence and handle full program executions. An evaluation
using real, large Java programs shows that these approaches detect more data
races than the closest related approaches, thus advancing the state of the art
in sound predictive race detection.
data race detection, dynamic predictive analysis
††copyright: rightsretained††doi: 10.1145/3360605††journalyear: 2019††journal:
PACMPL††journalvolume: 3††journalnumber: OOPSLA††article:
179††publicationmonth: 10††ccs: Software and its engineering Dynamic
analysis††ccs: Software and its engineering Software testing and debugging
## 1\. Introduction
With the rise in parallel software, _data races_ represent a growing hazard.
Programs with data races written in shared-memory languages including Java and
C++ have weak or undefined semantics, as a result of assuming data race
freedom for performance reasons (Manson et al., 2005; Boehm and Adve, 2008;
Adve and Boehm, 2010). Data races are culprits in real software failures,
resulting in substantial financial losses and even harm to humans (Boehm,
2011; Kasikci et al., 2015; Lu et al., 2008; Kasikci et al., 2012;
Narayanasamy et al., 2007; Cao et al., 2016; Flanagan and Freund, 2010a; Sen,
2008; Burnim et al., 2011; Zhivich and Cunningham, 2009; U.S.–Canada Power
System Outage Task Force, 2004; Leveson and Turner, 1993; PCWorld, 2012).
Writing scalable, data-race-free code is challenging, as is detecting data
races, which occur nondeterministically depending on shared-memory
interleavings and program inputs and environments. The most common approach
for dealing with data races is to detect them during in-house testing using
dynamic _happens-before (HB)_ analysis (Flanagan and Freund, 2009; Pozniansky
and Schuster, 2007; Elmas et al., 2007; Serebryany and Iskhodzhanov, 2009;
Serebryany et al., 2012; Intel Corporation, 2016), which detects conflicting
accesses (two memory accesses, at least one of which is a write, to the same
variable by different threads) unordered by the HB partial order (Lamport,
1978). However, HB analysis misses data races when accesses _could_ race in
some _other_ execution but are ordered by critical sections on the same lock
in the observed execution.
A promising alternative to HB analysis is _sound predictive analysis_ , which
detects additional predictable data races from an observed execution (Huang et
al., 2014; Said et al., 2011; Huang and Rajagopalan, 2016; Chen et al., 2008;
Şerbănuţă et al., 2013; Liu et al., 2016; Smaragdakis et al., 2012; Kini et
al., 2017; Roemer et al., 2018; Pavlogiannis, 2019); an analysis is _sound_ if
it detects no false races (Section 2). Some predictive analyses rely on
generating and solving SMT constraints, so in practice they cannot scale to
full program executions and instead analyze _bounded windows_ of execution,
missing races between accesses that do not execute close together (Huang et
al., 2014; Said et al., 2011; Huang and Rajagopalan, 2016; Chen et al., 2008;
Şerbănuţă et al., 2013; Liu et al., 2016) (Section 8). In contrast,
_unbounded_ predictive analyses avoid this limitation by detecting races based
on computing a partial order weaker than HB, using analyses with linear
running time in the length of the trace (Kini et al., 2017; Roemer et al.,
2018). However, these partial-order-based analyses miss predictable races
because they do not incorporate precise notions of _data and control
dependence_. More precisely, existing predictive partial orders do _not_
encode the precise conditions for reordering memory accesses to expose a race:
The reordering can change the last writer of a memory read (data dependence)
if the read in turn cannot affect whether the racing accesses execute (control
dependence). Encoding data and control dependence precisely in a partial order
is fundamentally challenging (Section 2).
##### Contributions.
This paper designs and evaluates new predictive analyses, making the following
contributions:
* •
A partial order called _strong-dependently-precedes (SDP)_ that improves over
the highest-coverage sound partial order from prior work (Kini et al., 2017)
by incorporating data dependence more precisely (Section 4).
* •
A proof that SDP is sound, i.e., detects no false races (Section 4.2).
* •
A partial order called _weak-dependently-precedes (WDP)_ that improves over
the previous highest-coverage partial order from prior work (Roemer et al.,
2018) by incorporating data and control dependence precisely (Section 4).
* •
A proof that WDP is complete (sometimes called _maximal_ (Huang et al., 2014;
Şerbănuţă et al., 2013)), detecting all races knowable from an observed
execution (Section 4.2).
* •
Dynamic analyses that compute SDP and WDP and detect SDP- and WDP-races
(Section 5).
* •
An algorithm for filtering out WDP-races that are false races, by extending
prior work’s _vindication_ algorithm (Roemer et al., 2018), yielding an
overall sound approach (Section 6).
* •
An implementation and evaluation of SDP and WDP analyses and WDP-race
vindication on benchmarked versions of real, large Java programs (Section 7).
The evaluation shows that the analyses find predictable races missed by the
closest related approaches (Kini et al., 2017; Roemer et al., 2018; Huang et
al., 2014).
## 2\. Background and Motivation
Recent partial-order-based predictive analyses can scale to full program
executions, enabling detection of predictable races that are millions of
executed operations apart (Kini et al., 2017; Roemer et al., 2018). However,
these partial orders are fundamentally limited and miss predictable races, as
this section explains. First, we introduce formalisms used throughout the
paper.
### 2.1. Execution Model
An _execution trace_ $\mathit{tr}$ is a sequence of events, ordered by the
total order $<_{\textsc{$\mathit{tr}$}}$, that represents a multithreaded
execution without loss of generality, corresponding to a linearization of a
sequentially consistent (SC) execution.111Although programs with data races
may violate SC (Manson et al., 2005; Boehm and Adve, 2008; Adve and Boehm,
2010; Dolan et al., 2018), dynamic race detection analyses (including ours)
add synchronization instrumentation before accesses, generally ensuring SC. We
assume every event in $\mathit{tr}$ is a unique object (e.g., has a unique
identifier), making it possible to identify the same event $e$ across other,
predicted traces. Each event has two attributes: (1) an identifier for the
thread that executed the operation; and (2) an operation, which is one of
wr(x), rd(x), acq(m), rel(m), or br, where x is a program memory location and
m is a program lock. (Later we consider how to extend analyses to handle lock-
free accesses and Java volatile / C++ atomic accesses.) An execution trace
must be _well formed_ : a thread may only acquire an unheld lock and may only
release a lock it has acquired.
Each br (branch) event $b$ represents an executed conditional operation—such
as a conditional jump, polymorphic call, or array element access—that may be
dependent on some prior read event(s) by the same thread. We assume a helper
function $\mathit{brDepsOn}(b,r)$ exists that returns true if the value read
by read event $r$ may affect $b$’s outcome. An implementation could use static
dependence analysis to identify reads on which a branch is data dependent. For
simplicity, the paper’s examples assume $\mathit{brDepsOn}(b,r)$ always
returns true, i.e., every branch is assumed dependent on preceding reads by
the same thread. Our implementation and evaluation make the same assumption,
as explained later. This assumption limits the capability of predictive
analysis to predict different executions; in other words, it limits the number
of knowable data races from a single execution.
Three example traces are shown in Figures 1(b), 1(c), and 1(e), in which top-
to-bottom order represents trace order, and column placement denotes an
event’s executing thread. We discuss these examples in detail later.
Two read or write events to the same variable are _conflicting_ , notated
$e\asymp e^{\prime}$, if the events are executed by different threads and at
least one is a write.
_Program-order (PO)_ is a partial order that orders events in the same thread:
$e\prec_{\textsc{\tiny{PO}}}e^{\prime}$ if
$e<_{\textsc{$\mathit{tr}$}}e^{\prime}$ and the events are executed by the
same thread.
The function $\mathit{CS(}e)$ returns the set of events in the critical
section started or ended by acquire or release event $e$, including the
bounding acquire and release events. $\mathit{R}(a)$ returns the release event
ending the critical section started by acquire event $a$, and $\mathit{A}(r)$
returns the acquire event starting the critical section ended by release event
$r$. The function $\mathit{lockset}(e)$ returns the set of locks held at a
read or write event $e$ by its executing thread.
### 2.2. Predictable Traces and Predictable Races
⬇ int z = 0, y = 0; Object m = new Object(); new Thread(() -> { synchronized
(m) { int t = z; y = 1; } }).start(); new Thread(() -> { synchronized (m) { z
= 1; x = 1; } }).start(); new Thread(() -> { synchronized (m) { int t = x; if
(t == 0) return; } int t = y; } }).start();
((a)) Java code that could lead to the executions in (b) and (c).
Thread 1 Thread 2 Thread 3 acq(m) rd(z) wr(y) rel(m) acq(m) wr(z) wr(x) rel(m)
acq(m) rd(x) br rel(m) rd(y)
((b)) Execution with a predictable race
Thread 1 Thread 2 Thread 3 acq(m) wr(z) wr(x) rel(m) acq(m) rd(x) br rel(m)
acq(m) rd(z) wr(y) rd(y)
((c)) Predictable trace of (b)
⬇ int z = 0, y = 0; Object m = new Object(); new Thread(() -> { synchronized
(m) { int t = z; if (t == 0) y = 1; } }).start(); new Thread(() -> {
synchronized (m) { z = 1; x = 1; } }).start(); new Thread(() -> { synchronized
(m) { int t = x; if (t == 0) return; } int t = y; } }).start();
((d)) Java code that could lead to the execution in (e).
Thread 1 Thread 2 Thread 3 acq(m) rd(z) br wr(y) rel(m) acq(m) wr(z) wr(x)
rel(m) acq(m) rd(x) br rel(m) rd(y)
((e)) Execution with no predictable race
Figure 1. Two code examples, with potential executions they could lead to. The
execution in (b) has a predictable race, as demonstrated by the predictable
trace in (c). The execution in (e) has no predictable race.
By observing one execution of a program, it is possible to predict data races
in both the observed execution and some _other_ executions of the program. The
information present in the observed execution implies the existence of other,
different executions, called _predictable traces_. To define what traces can
be predicted from an observed trace, we first define several relevant
concepts.
###### Definition 2.1 (Last writer).
Given a trace $\mathit{tr}$, let $\mathit{lastwr}_{\mathit{tr}}(r)$ for a read
event $r$ be the last write event before $r$ in $\mathit{tr}$ that accesses
the same variable as $r$, or $\varnothing$ if no such event exists.
To ensure that a predictable trace is feasible, each read in a predictable
trace must have the same last writer as in the observed trace—with one
exception: a read can have a different last writer if the read cannot take the
execution down a different control-flow path than the observed execution. An
example of such a read is Thread 1’s rd(z) event in Figure 1(c). Next, we
introduce a concept that helps in identifying reads whose last writer must be
preserved in a predictable trace.
###### Definition 2.2 (Causal events).
Given a trace $\mathit{tr}$, set of events $\mathit{S}$, and event $e$, let
$\mathit{causal(\mathit{tr},\mathit{S},e)}$ be a function that returns true if
at least one of the following properties holds, and false otherwise.
* •
$e$ is a read, and there exists a branch event $b$ such that
$b\in\mathit{S}\land e\prec_{\textsc{\tiny{PO}}}b\land\mathit{brDepsOn}(b,e)$.
* •
$e$ is a write, and there exists a read event $e^{\prime}$ such that
$e^{\prime}\in\mathit{S}\land e=\mathit{lastwr}_{\mathit{tr}}(e^{\prime})$.
* •
$e$ is a read, and there exists a write event $e^{\prime}$ such that
$e^{\prime}\in\mathit{S}\land e\prec_{\textsc{\tiny{PO}}}e^{\prime}$ ($e$ and
$e^{\prime}$ may access different variables).
Intuitively, $\mathit{causal(\mathit{tr},\mathit{S},e)}$ tells us whether an
event $e$ could have affected some event $e^{\prime}$ in $\mathit{S}$
directly. For example, $\mathit{causal(\mathit{tr},\mathit{S},e)}$ if read $e$
may affect a branch event in $\mathit{S}$; if $e$ writes a variable later read
by an event in $\mathit{S}$; or if $e$ reads a value that may affect a later
write by the same thread in $\mathit{S}$ (even if the read and write are to
different variables, to account for intra-thread data flow).
We can now define a predictable trace of an observed trace, which is a trace
that is definitely a feasible execution of the program, given the existence of
the observed execution.
###### Definition 2.3 (Predictable trace).
An execution trace $\mathit{tr^{\prime}}$ is a _predictable trace_ of trace
$\mathit{tr}$ if $\mathit{tr^{\prime}}$ contains only events in $\mathit{tr}$
(i.e., $\forall e:e\in\mathit{tr^{\prime}}\implies e\in\mathit{tr}$) and all
of the following rules hold:
_Program order (PO) rule:_ For any events $e_{1}$ and $e_{2}$, if
$e_{1}\prec_{\textsc{\tiny{PO}}}e_{2}$, then
$e_{1}<_{\textsc{$\mathit{tr^{\prime}}$}}e_{2}\lor
e_{2}\notin\mathit{tr^{\prime}}$.
_Last writer (LW) rule:_ For every read event $e$ such that
$\mathit{causal(\mathit{tr},\mathit{tr^{\prime}},e)}$,
$\mathit{lastwr}_{\mathit{tr^{\prime}}}(e)=\mathit{lastwr}_{\mathit{tr}}(e)$.
(In this context, $\mathit{tr^{\prime}}$ means the set of events in the trace
$\mathit{tr^{\prime}}$.)
_Lock semantics (LS) rule:_ For acquire events $e_{1}$ and $e_{2}$ on the same
lock, if $e_{1}<_{\textsc{$\mathit{tr^{\prime}}$}}e_{2}$ then
$e_{1}<_{\textsc{$\mathit{tr^{\prime}}$}}\mathit{R}(e_{1})<_{\textsc{$\mathit{tr^{\prime}}$}}e_{2}$.
The PO and LW rules ensure key properties from $\mathit{tr}$ also hold in
$\mathit{tr^{\prime}}$, while the LS rule ensures that $\mathit{tr^{\prime}}$
is well formed. The intuition behind the LW rule is that any read that may
(directly or indirectly) affect the control flow of the program must have the
same last writer in predictable trace $\mathit{tr^{\prime}}$ as in observed
trace $\mathit{tr}$.
Note that throughout the paper, partial ordering notation such as $e\prec
e^{\prime}$ refers to the order of $e$ and $e^{\prime}$ in the _observed_
trace $\mathit{tr}$ (not a predictable trace $\mathit{tr^{\prime}}$).
Predictable traces do not in general contain every event in the observed trace
they are based on. For the purposes of race detection, a predictable trace
will _conclude with_ a pair of conflicting events, which are preceded by
events necessary according to the definition of predictable trace. For
example, consider Figure 1(c), which is a predictable trace of Figure 1(b)
that excludes Thread 1’s event after wr(y). The PO rule is satisfied, and the
LS rule is satisfied after reordering Thread 2 and 3’s critical sections
before Thread 1’s. The LW rule is satisfied because rd(z) is not a causal
event in $\mathit{tr^{\prime}}$.
###### Definition 2.4 (Predictable race).
An execution $\mathit{tr}$ has a predictable race if a predictable trace
$\mathit{tr^{\prime}}$ of $\mathit{tr}$ has two _conflicting_ , _consecutive_
events: $e_{1}\asymp e_{2}\land
e_{1}<_{\textsc{$\mathit{tr^{\prime}}$}}e_{2}\land(\nexists
e:e_{1}<_{\textsc{$\mathit{tr^{\prime}}$}}e<_{\textsc{$\mathit{tr^{\prime}}$}}e_{2})$.
Figure 1(b) has a predictable race, as demonstrated by the predictable trace
in Figure 1(c). In contrast, Figure 1(e) has no predictable race. The
difference between Figures 1(b) and 1(e) is the br event in Thread 1. Since no
br exists in Thread 1 in Figure 1(b), rd(z) is not a causal event, which in
turn allows the critical sections in Threads 2 and 3 to be reordered above the
critical section in Thread 1, allowing wr(y) and rd(y) to be consecutive in
the predictable trace. In contrast, a br event executes before wr(y) in Figure
1(e). No predictable trace of this example can exclude br without excluding
wr(y); otherwise the PO rule would be violated. rd(z) is a causal event in any
predictable trace where wr(y) is included, which makes it impossible to
reorder the critical sections. As a result, no predictable trace exists in
which wr(y) and rd(y) are consecutive. Figures 1(a) and 1(d) show source code
that could lead to the executions in Figures 1(b) and 1(e), respectively. The
code in Figure 1(d) has no race; in fact, any deviation of critical section
ordering from Figure 1(e)’s causes wr(y) or rd(y) _not_ to execute.
### 2.3. Existing Predictive Partial Orders
Here we overview three relations introduced in prior work, called _happens-
before (HB)_ , _weak-causally-precedes (WCP)_ , and _doesn’t-commute (DC)_ ,
that can be computed in time linearly proportional to the length of the
execution trace (Kini et al., 2017; Roemer et al., 2018). Intuitively, each
relation orders events that may not be legal to reorder in a predictable
trace, so that two unordered conflicting events represent a true or potential
data race (depending on whether the relation is sound). An execution trace has
an _HB-race_ , _WCP-race_ , or _DC-race_ if it contains two conflicting events
that are unordered by HB, WCP, or DC, respectively.
##### Definitions of relations.
Table 1 gives definitions of HB, WCP, and DC by presenting their properties
comparatively. The first two rows of the table say how the relations order
critical sections on the same lock. HB orders all critical sections on the
same lock, and it orders the first critical section’s rel(m) to the second
critical section’s acq(m). WCP and DC order only _conflicting_ critical
sections (critical sections on the same lock containing conflicting events),
and they order from the first critical section’s rel(m) to the second critical
section’s conflicting access event. That is, if $r_{1}$ and $r_{2}$ are
release events on the same lock such that
$r_{1}<_{\textsc{$\mathit{tr}$}}r_{2}$, and $e_{1}$ and $e_{2}$ are
conflicting events ($e_{1}\asymp e_{2}$) such that
$e_{1}\in\mathit{CS(}r_{1})\land e_{2}\in\mathit{CS(}r_{2})$, then
$r_{1}\prec_{\textsc{\tiny{WCP}}}e_{2}$ and
$r_{1}\prec_{\textsc{\tiny{DC}}}e_{2}$. The intuition behind these properties
of WCP and DC is that non-conflicting critical sections can generally be
reordered in a predictable trace; and even in the case of conflicting critical
sections, the second critical section can be “reordered” so that it executes
only up to its conflicting access and the first critical section does not
execute at all in the predictable trace.
Property | $\prec_{\textsc{\tiny{HB}}}$ | $\prec_{\textsc{\tiny{WCP}}}$ | $\prec_{\textsc{\tiny{DC}}}$
---|---|---|---
Same-lock critical section ordering | All | Confl. | Confl.
Orders rel to… | acq | wr/rd | wr/rd
Includes $\prec_{\textsc{\tiny{PO}}}$? | Yes | No | Yes
Left-and-right composes with $\prec_{\textsc{\tiny{HB}}}$? | Yes | Yes | No
$\textsf{acq(m)}\prec\textsf{rel(m)}$ implies $\textsf{rel(m)}\prec\textsf{rel(m)}$? | Yes | Yes | Yes
Transitive? | Yes | Yes | Yes
Table 1. Definitions of three strict partial orders over events in an
execution trace. Each order is the minimum relation satisfying the listed
properties.
The next two table rows show whether the relations include PO or compose with
HB. HB and DC include (i.e., are supersets of) PO: if
$e_{1}\prec_{\textsc{\tiny{PO}}}e_{2}$, then
$e_{1}\prec_{\textsc{\tiny{HB}}}e_{2}$ and
$e_{1}\prec_{\textsc{\tiny{DC}}}e_{2}$. In contrast, WCP does not include PO
but instead _composes with_ the stronger HB: if
$e_{1}\prec_{\textsc{\tiny{WCP}}}e_{2}\prec_{\textsc{\tiny{HB}}}e_{3}$ or
$e_{1}\prec_{\textsc{\tiny{HB}}}e_{2}\prec_{\textsc{\tiny{WCP}}}e_{3}$, then
$e_{1}\prec_{\textsc{\tiny{WCP}}}e_{2}$. (By virtue of being transitive, HB
composes with itself.) The intuition behind including or composing with PO (a
subset of HB) is that PO-ordered events cannot be reordered in a predictable
trace. The intuition behind WCP composing with HB, in essence, is to avoid
predicting traces that violate the LS rule of predictable traces. As a result,
WCP is sound while DC is unsound, as we will see.
The last two rows show properties shared by all relations. First, if two
critical sections on the same lock
$a_{1}\prec_{\textsc{\tiny{PO}}}r_{1}<_{\textsc{$\mathit{tr}$}}a_{2}\prec_{\textsc{\tiny{PO}}}r_{2}$
are ordered at all (meaning simply $a_{1}\prec_{\ast}r_{2}$ because all
relations minimally compose with PO), then their release events are ordered
($r_{1}\prec_{\ast}r_{2}$). Second, all of the relations are transitive. As a
result of being transitive, antisymmetric, and irreflexive, all of the
relations are strict partial orders.
##### Example
As an example of WCP and DC ordering, consider the execution in Figure 2(b).
Both relations order Thread 1’s rel(m) to Thread 2’s wr(x) because the
critical sections on m contain conflicting accesses to x. By WCP’s composition
with HB (and thus PO) and DC’s inclusion of PO, both WCP and DC transitively
order rd(x) to wr(x) and wr(y) to rd(y)
($\textsf{wr(y)}\prec_{\textsc{\tiny{WCP}}}\textsf{rd(y)}$ and
$\textsf{wr(y)}\prec_{\textsc{\tiny{DC}}}\textsf{rd(y)}$), so the execution
has no WCP- or DC-races.
##### Soundness and completeness.
A relation or analysis is _sound_ if it detects a race only for an execution
trace with a predictable race or deadlock.222A trace has a _predictable
deadlock_ if there exists a valid reordering with a deadlock. We define
soundness to include predictable deadlocks because prior work’s WCP relation
(Kini et al., 2017) and our SDP relation are sound in this way. A relation or
analysis is _complete_ if it detects a race for every execution trace with a
predictable race.
WCP (and HB) are sound: a WCP-race (HB-race) indicates a predictable race or
deadlock. DC is unsound: an execution with a DC-race may have no predictable
race or deadlock. However, prior work shows that DC-races are generally true
predictable races in practice, and an efficient _vindication_ algorithm can
verify DC-races as predictable races by computing additional constraints and
building a predictable trace exposing the race (Roemer et al., 2018). Later in
the paper, we provide more details about vindication, when introducing a new
relation that (like DC) is unsound and makes use of a vindication algorithm.
### 2.4. Limitations of Existing Predictive Partial Orders
WCP and DC analyses are the state of the art in detecting as many predictable
races as possible using online, unbounded analysis (Kini et al., 2017; Roemer
et al., 2018). However, WCP and DC are _incomplete_ , failing to detect some
predictable races. WCP and DC are overly strict because they order all
conflicting accesses, conservatively ruling out some predictable traces that
still preserve the last writer of each causal read. This strictness arises
from imprecise handling of data and control dependence:
##### Data dependence:
WCP and DC order all conflicting accesses, which is imprecise because the
order of a write–write or read–write conflict does not necessarily need to be
preserved to satisfy the last-writer (LW) rule of predictable traces.
Thread 1 Thread 2 acq(m) rd(x) br wr(y) rel(m) acq(m) wr(x) rel(m) rd(y)
((a)) WDP-race but no WCP-, DC-, or SDP-race. No predictable race exists.
Thread 1 Thread 2 acq(m) wr(y) rd(x) br rel(m) acq(m) wr(x) rel(m) rd(y)
((b)) WDP-race but no WCP-, DC-, or SDP-race. A predictable race exists, as
(c) shows.
Thread 1 Thread 2 acq(m) wr(x) rel(m) acq(m) wr(y) rd(y)
((c)) Predictable trace of (b) showing that (b) has a predictable race
DCSDPDCSDP
Figure 2. Executions showing WCP and DC’s overly strict handling of read–write
dependencies. Edges represent ordering, labeled using the weakest applicable
relation(s) (and omitting ordering established by HB alone), implying ordering
by strictly stronger relations (see Figure 3(a) for comparison of relations).
Thread 1 Thread 2 acq(m) wr(x) wr(y) rel(m) acq(m) wr(x) rel(m) rd(y) acq(m)
rd(x) rel(m) br
((a)) Execution with SDP- and WDP-race but no WCP- or DC-race
DC SDP DC WDP SDP
Thread 1 Thread 2 acq(m) wr(x) rel(m) acq(m) wr(x) wr(y) rd(y)
((b)) Valid reordering of (a) showing that (a) has a predictable race
Thread 1 Thread 2 acq(m) wr(x) sync(o) wr(y) rel(m) acq(m) wr(x) rel(m)
sync(o) rd(x) br rd(y)
((c)) Execution with WDP-race but no SDP-race or predictable race
Figure 3. Executions showing WCP and DC’s overly strict handling of
write–write dependencies. sync(o) is an abbreviation for the sequence acq(o);
rd(oVar); br; wr(oVar); rel(o).
Consider the executions in Figures 2(a) and 2(b), in which WCP and DC order
rd(x) to wr(x). WCP and DC’s read–write ordering assumes that no predictable
trace exists where a pair of conflicting accesses are reordered. This
rationale works for Figure 2(a), in which no predictable race exists. However,
conflicting accesses may be reordered as long as the LW rule of predictable
traces is satisfied. Figure 2(c) is a predictable trace of Figure 2(b) that
reorders the critical sections and exposes a race on y.
Similarly for write–write conflicts, consider Figure 2(a), in which WCP and DC
order the two wr(x) events, leading to no WCP- or DC-race on accesses to y.
However, the wr(x) events can be reordered, as the predictable trace in Figure
2(b) shows, exposing a race. (The reader can ignore Figure 2(c) until Section
4.)
It is difficult to model read–write and write–write dependencies more
precisely using a partial order. In the case of a read–write dependency, the
accesses can be reordered _as long as the read cannot impact a branch’s
outcome in the predictable trace_ (i.e., the read is not a causal event in the
predictable trace, or is not part of the predictable trace). For a write–write
dependency, the accesses can be reordered _as long as they do not change a
causal read’s last writer in the predictable trace._ Incorporating either kind
of constraint into a partial order is challenging but also desirable because
partial orders can be computed efficiently.
##### Control dependence:
Thread 1 Thread 2 wr(y) acq(m) wr(x) rel(m) acq(m) rd(x) rel(m) rd(y)
((a)) WDP-race but no WCP-, DC-, or SDP-race
Thread 1 Thread 2 acq(m) rd(x) rel(m) wr(y) rd(y)
((b)) Valid reordering of (a) showing that (a) has a predictable race
Thread 1 Thread 2 wr(y) acq(m) wr(x) rel(m) acq(m) rd(x) rd(y) br rel(m)
((c)) WDP-race but no WCP-, DC-, or SDP-race
Thread 1 Thread 2 acq(m) rd(x) wr(y) rd(y)
((d)) Valid reordering of (c) showing that (c) has a predictable race
DC
SDP
DC
SDP
WDP
Figure 4. Executions showing WCP and DC’s overly strict handling of control
dependencies.
WCP and DC order true (write–read) dependencies even when the read may not
affect a branch outcome that affects whether a race happens.
Figure 4(a) shows an execution with a predictable race, as the predictable
trace in Figure 4(b) demonstrates. Note that in Figure 4(b), rd(x) has a
different last writer than in Figure 4(a), but the lack of a following br
event means that rd(y) is still guaranteed to happen (i.e., rd(x) is not a
causal event). A variant of this example is in Figure 4(c), which has a branch
event dependent on a read outcome, but the branch can be absent from a
predictable trace demonstrating a predictable race (Figure 4(d)).
Thread 1 | Thread 2 | Thread 3
---|---|---
this.method = …; | |
acq(this); | |
this.time += time | |
rel(this); | |
| acq(this); |
| this.time += time; |
| rel(this); |
| br |
| | acq(this);
| | this.time += time;
| | rel(this);
| | br
| | …= this.method;
DCDCWDPWDP
Figure 5. A predictable race in the Java program pmd that was detected by WDP,
but not WCP, DC, or SDP. Note the transitive edges formed by DC, which WDP
avoids as the branches are outside the critical sections. The code has been
simplified and abbreviated.
WCP and DC miss the predictable races in Figures 4(a) and 4(c) by
conservatively assuming that any event after a rd(x) may be control dependent
on the read value. Similarly, WCP and DC miss the predictable race in Figure
5, which our implementation found in the Java program pmd (Section 7).
Essentially, WCP and DC conservatively assume that a dependent branch
immediately follows each read. This limitation is unsurprising considering the
challenge of modeling control dependencies using a partial order. In
particular, it is difficult for a partial order to model the fact that _a read
must have the same last writer only if the read may affect a branch in the
predictable trace_.
This work develops partial orders that are weaker than WCP and DC and thus
predict more races. At the same time, these new partial orders retain key
properties of the existing relations: WCP’s soundness and DC’s amenability to
a vindication algorithm that ensures soundness, respectively.
## 3\. Overview
The previous section introduced prior work’s weak-causally-precedes (WCP)
(Kini et al., 2017) and doesn’t-commute (DC) (Roemer et al., 2018), and
explained their limitations that lead to missing predictable races. The next
two sections introduce new relations and analyses that overcome these
limitations. Section 4 introduces the _strong-dependently-precedes (SDP)_ and
_weak-dependently-precedes (WDP)_ relations, which are weaker than WCP and DC,
respectively. Section 5 presents online dynamic analyses for computing SDP and
WDP and detecting SDP- and WDP-races.
## 4\. New Dependence-Aware Predictive Relations
This section introduces new partial orders called _strong-dependently-precedes
(SDP)_ and _weak-dependently-precedes (WDP)_ that overcome the limitations of
prior work’s predictive relations (Roemer et al., 2018; Kini et al., 2017)
(Section 2.4) by incorporating more precise notions of data and control
dependence.
### 4.1. The SDP and WDP Partial Orders
SDP is weaker than WCP 333It may seem confusing that SDP is _weaker_ than WCP.
SDP is so named because it is stronger than WDP, while WCP is so named because
it is weaker than prior work’s _causally-precedes (CP)_ (Smaragdakis et al.,
2012). by not ordering write–write conflicts, based on the insight that writes
can be unordered unless they can affect the outcome of a read, but write–read
and read–write ordering already handles that ordering soundly. WDP only orders
the last writer of a read to a branch that depends on that read, which is the
only reordering constraint that does not lead to missing predictable races.
| Prior work | This paper
---|---|---
Property | $\prec_{\textsc{\tiny{HB}}}$ | $\prec_{\textsc{\tiny{WCP}}}$ | $\prec_{\textsc{\tiny{DC}}}$ | $\prec_{\textsc{\tiny{SDP}}}$ | $\prec_{\textsc{\tiny{WDP}}}$
Same-lock critical section ordering | All | Confl. | Confl. | Confl. | Last wr–rd only
Orders rel to… | acq | wr/rd | wr/rd | wr/rd or to next rd* | Next br
Includes $\prec_{\textsc{\tiny{PO}}}$? | Yes | No | Yes | No | Yes
Left-and-right composes with $\prec_{\textsc{\tiny{HB}}}$? | Yes | Yes | No | Yes | No
$\textsf{acq(m)}\prec\textsf{rel(m)}$ implies $\textsf{rel(m)}\prec\textsf{rel(m)}$? | Yes | Yes | Yes | Yes | Yes
Transitive? | Yes | Yes | Yes | Yes | Yes
Table 2. Definitions of five strict partial orders over events in an execution
trace. Each order is the minimum relation satisfying the listed properties.
This table adds columns $\prec_{\textsc{\tiny{SDP}}}$ and
$\prec_{\textsc{\tiny{WDP}}}$ to Table 1 (page 1).
* As the text explains, SDP adds release–access ordering for write–read and read–write conflicts, and adds ordering from the release to the next read for write–write conflicts.
Table 2 defines SDP and WDP. The table shows that SDP is like WCP and WDP is
like DC except in how they order critical sections (first two table rows).
##### The SDP relation.
SDP only orders conflicting critical sections when one critical section
contains a read. Like WCP, SDP orders the first critical section’s rel(m) to
the second critical section’s access. The intuition behind this property is
that write–write conflicts generally do not impose any limitations on what
traces can be predicted. Figure 2(a) shows an example in which two conflicting
writes can be safely reordered in a predictable trace.
However, ignoring write–write conflicts altogether would be unsound, as Figure
2(c) shows: the execution has no predictable race. To ensure soundness, SDP
handles write–write conflicts by ordering the first critical section to the
second thread’s next _read_ to the same variable.
More formally, SDP handles conflicting critical sections as follows. If
$r_{1}$ and $r_{2}$ are release events on the same lock,
$r_{1}<_{\textsc{$\mathit{tr}$}}r_{2}$, $e_{1}$ and $e_{2}$ are write events
and $e_{3}$ is a read event, $e_{1}\asymp e_{2}$, $e_{1}\asymp e_{3}$,
$e_{2}\prec_{\textsc{\tiny{PO}}}e_{3}$, $e_{1}\in\mathit{CS(}r_{1})$, and
$e_{2}\in\mathit{CS(}r_{2})$, then $r_{1}\prec_{\textsc{\tiny{SDP}}}e_{3}$.
SDP addresses a limitation of WCP via more precise handling of data
dependencies. SDP certainly does not address all imprecise data dependencies
(e.g., read–write dependencies), and it does not address control dependence.
SDP is the weakest known sound partial order.
##### The WDP relation
A separate but worthwhile goal is to develop a partial order that is weaker
than DC but produces few false positives so that it is practical to vindicate
potential races. WDP achieves this goal and is in fact complete, detecting all
predictable races. WDP orders the last writer of each read to the earliest
branch that depends on that read (and orders no other conflicting critical
sections). The intuition behind this behavior is that the only constraint that
is universally true for all predictable traces is that the last writer of a
read must not occur after the read if there is a branch that depends on the
read.
More formally, if $r_{1}$ and $r_{2}$ are releases on the same lock,
$e_{1}\in\mathit{CS(}r_{1})$, $e_{2}\in\mathit{CS(}r_{2})$,
$e_{1}=\mathit{lastwr}_{tr}(e_{2})$, $e_{2}\prec_{\textsc{\tiny{PO}}}b$, and
$\mathit{brDepsOn}(b,e_{2})$, then $r_{1}\prec_{\textsc{\tiny{WDP}}}b$.
Unlike DC, WDP integrates control dependence by ordering the write’s critical
section to the first branch dependent on the read. WDP does not model _local_
data dependencies, where a read affects the value written by a write in the
same thread. As a result, WDP may find false races, but Section 6 describes a
method for ruling out such false races. These properties make WDP complete (as
we show). WDP is the strongest known complete partial order.
##### SDP- and WDP-races.
Unlike WCP and DC, SDP and WDP do not inherently order all conflicting
accesses that hold a common lock. Thus the following definition of SDP- and
WDP-races explicitly excludes conflicting accesses holding a common lock.
A trace has a _SDP-race_ (or _WDP-race_) if it has two conflicting events
unordered by SDP (WDP) that hold no lock in common. That is, $\mathit{tr}$ has
an SDP-race (WDP-race) on events $e$ and $e^{\prime}$ if
$e<_{\textsc{$\mathit{tr}$}}e^{\prime}$, $e\asymp e^{\prime}$,
$e\not\prec_{\textsc{\tiny{SDP}}}e^{\prime}$
($e\not\prec_{\textsc{\tiny{WDP}}}e^{\prime}$), and
$\mathit{lockset}(e)\cap\mathit{lockset}(e^{\prime})=\emptyset$.
##### Examples.
To illustrate SDP and WDP, we refer back to examples from pages 3–5.
The executions in Figures 2(a) and 2(b) have no SDP-races: SDP orders the
read–write conflicts. In contrast, these executions have WDP-races: there is
no cross-thread WDP ordering because the executions have no lock-protected
write–read conflicts.
Figure 2(a) has SDP- and WDP-races. SDP and WDP do not order the write–write
conflict on x. Nor does WDP order events on the write–read conflict on x,
since Thread 1’s wr(x) is not the last writer of rd(x).
Figure 2(c), which has a WDP-race but no SDP-race or predictable race, shows
the need for SDP’s release–read ordering for write–write conflicts.
The executions in Figures 4(a) and 4(c) have no SDP-race since SDP does not
take branches into account. On the other hand, both executions have WDP-races:
Figure 4(a) has no branch dependent on the read, and Figure 4(c) has a branch,
but it occurs after rd(y).
WDP analysis discovers the predictable race in Figure 5. In this case, the
fact that there is no branch within the critical section allows WDP to avoid
creating an unnecessary transitive edge that otherwise would hide the race.
$\prec_{\textsc{\tiny{HB}}}$$(\prec_{\textsc{\tiny{WCP}}}\cup\prec_{\textsc{\tiny{PO}}})$$\prec_{\textsc{\tiny{DC}}}$$(\prec_{\textsc{\tiny{SDP}}}\cup\prec_{\textsc{\tiny{PO}}})$$\prec_{\textsc{\tiny{WDP}}}$
((a)) Weaker above stronger relations
HB-racesWCP-racesDC-racesSDP-racesWDP-races ((b)) Supersets above subsets
Table 3. Lattices showing the relationships among the relations and
corresponding kinds of races. Only WCP and SDP do not include PO and thus do
not in general order events within the same thread, but this property is
irrelevant for comparing relation strength because same-thread accesses cannot
race in any case, so the relation lattice uses
$(\prec_{\textsc{\tiny{WCP}}}\cup\prec_{\textsc{\tiny{PO}}})$ and
$(\prec_{\textsc{\tiny{SDP}}}\cup\prec_{\textsc{\tiny{PO}}})$ to make the
relations more directly comparable. (WCP and SDP analyses in fact detect races
by comparing access events using
$(\prec_{\textsc{\tiny{WCP}}}\cup\prec_{\textsc{\tiny{PO}}})$ and
$(\prec_{\textsc{\tiny{SDP}}}\cup\prec_{\textsc{\tiny{PO}}})$, respectively.)
| Sound? | Complete?
---|---|---
HB | Yes (Lamport, 1978) | No (Fig. 2(b), 2(a), 4(a), 4(c))
WCP | Yes (Kini et al., 2017) | No (Fig. 2(b), 2(a), 4(a), 4(c))
DC | No (Roemer et al., 2018) | No (Fig. 2(b), 2(a), 4(a), 4(c))
SDP | Yes (Section 4.2) | No (Fig. 2(b), 4(a), 4(c))
WDP | No (Fig. 2(a) and 2(c)) | Yes (Section 4.2)
Table 4. Soundness and completeness of each relation.
### 4.2. Soundness and Completeness
Figure 4 and Table 4 illustrate the relationships among the different
relations and corresponding race types. SDP never misses a race that WCP
finds, and WDP never misses a race that DC or SDP finds. SDP is sound but
incomplete (never reports a false race but may miss predictable races), while
WDP is unsound but complete (may report false races but never misses a
predictable race).
Here we prove that SDP is sound and WDP is complete. The proofs are manual and
have not been verified by a theorem prover.
###### Theorem 0 (SDP soundness).
If an execution trace has a SDP-race, then it has a predictable race or a
predictable deadlock.
###### Proof.
We define $\prec_{\textsc{\tiny{SDP}}(i)}$ to be a variant of
$\prec_{\textsc{\tiny{WCP}}}$ and $\prec_{\textsc{\tiny{SDP}}}$ that orders
critical sections like SDP for the first $i$ conflicting writes in
$\mathit{tr}$, and orders critical sections like WCP otherwise (Table 2).
Formally, for conflicting events $e$ and $e^{\prime}$ in critical sections on
the same lock, $e\prec_{\textsc{\tiny{SDP}}(i)}e^{\prime}$ if either:
* •
$e$ or $e^{\prime}$ is a read; or
* •
there are $i$ many conflicting pairs of write events (i.e., two conflicting
write events without an intervening conflicting write event) before the write
pair $(e,e^{\prime})$ (a write pair $(w,w^{\prime})$ is before a write pair
$(e,e^{\prime})$ if $w^{\prime}<_{tr}e^{\prime}$).
Note that $\prec_{\textsc{\tiny{SDP}}(0)}\equiv\prec_{\textsc{\tiny{WCP}}}$
and $\prec_{\textsc{\tiny{SDP}}(\infty)}\equiv\prec_{\textsc{\tiny{SDP}}}$.
The rest of the proof proceeds by induction to show that SDP($i$) is sound for
all $i$, i.e., if an execution trace has an SDP($i$)-race, then it has a
predictable race or deadlock.
##### Base case:
Since $\prec_{\textsc{\tiny{SDP}}(0)}\equiv\prec_{\textsc{\tiny{WCP}}}$ and
WCP is sound (Kini et al., 2017), SDP($0$) is sound.
##### Inductive step:
Let $\sigma$ be an execution trace whose first SDP($i$)-race is between events
$e_{1}$ and $e_{2}$, where _first_ means that $e_{2}$ is as early as possible
in $\sigma$, and among SDP($i$)-races whose second event is $e_{2}$, $e_{1}$
is as late as possible in $\sigma$. Proceeding with proof by contradiction,
suppose $\sigma$ has no predictable race or deadlock.
Now let $\mathit{tr}$ be a trace equivalent to $\sigma$ that moves all events
between $e_{1}$ and $e_{2}$ that are not HB ordered with both events, to
outside of $e_{1}$ and $e_{2}$, and additionally removes all events after
$e_{2}$. Specifically:
* •
if $e_{1}<_{\sigma}e\land
e_{1}\not\prec_{\textsc{\tiny{HB}}}e\prec_{\textsc{\tiny{HB}}}e_{2}$, move $e$
before $e_{1}$ in $\mathit{tr}$;
* •
if
$e_{1}<_{\sigma}e<_{\sigma}e_{2}\land{e\not\prec_{\textsc{\tiny{HB}}}e_{2}}$,
omit $e$ from $\mathit{tr}$;
* •
if $e_{2}<_{\sigma}e$, omit $e$ from $\mathit{tr}$.
Thus the last event in $\mathit{tr}$ is $e_{2}$. Like $\sigma$,
$\mathit{tr}$’s first SDP($i$)-race is between events $e_{1}$ and $e_{2}$, and
$\mathit{tr}$ has no predictable race or deadlock. That is,
$e_{1}<_{\textsc{$\mathit{tr}$}}e_{2}$, $e_{1}\asymp e_{2}$, and
$e_{1}\not\prec_{\textsc{\tiny{SDP}}}e_{2}$.
Because $\mathit{tr}$ has no predictable race or deadlock,
$e_{1}\prec_{\textsc{\tiny{SDP}}(i-1)}e_{2}$ by the induction hypothesis.
Because of the disparity between $e_{1}\prec_{\textsc{\tiny{SDP}}(i-1)}e_{2}$
and $e_{1}\not\prec_{\textsc{\tiny{SDP}}(i)}e_{2}$, it must be that
$l\prec_{\textsc{\tiny{SDP}}(i-1)}w^{\prime}$ and
$l\not\prec_{\textsc{\tiny{SDP}}(i)}w^{\prime}$, where $w^{\prime}$ is the
$i$th conflicting write in $\mathit{tr}$, $w$ is the latest write before
$w^{\prime}$ such that $w\asymp w^{\prime}$, and $l$ is the outermost release
event of a critical section containing $w$ that releases the same lock as any
critical section containing $w^{\prime}$.
If there is a read event $r$ that reads the same variable as $w$ and
$w^{\prime}$ such that
$w<_{\textsc{$\mathit{tr}$}}r<_{\textsc{$\mathit{tr}$}}e_{2}$, then either
* •
$w^{\prime}\asymp r\land r<_{\textsc{$\mathit{tr}$}}w^{\prime}$, in which case
$e_{1}\prec_{\textsc{\tiny{HB}}}r\prec_{\textsc{\tiny{SDP}}(i)}w^{\prime}\prec_{\textsc{\tiny{HB}}}e_{2}$;
* •
$w^{\prime}\asymp r\land w^{\prime}<_{\textsc{$\mathit{tr}$}}r$, in which case
$e_{1}\prec_{\textsc{\tiny{HB}}}w^{\prime}\prec_{\textsc{\tiny{SDP}}(i)}r\prec_{\textsc{\tiny{HB}}}e_{2}$;
* •
$w\asymp r\land w^{\prime}\prec_{\textsc{\tiny{PO}}}r$, in which case
$e_{1}\prec_{\textsc{\tiny{HB}}}l\prec_{\textsc{\tiny{SDP}}(i)}r\prec_{\textsc{\tiny{HB}}}e_{2}$;
or
* •
$w\asymp r\land r\prec_{\textsc{\tiny{PO}}}w^{\prime}$, in which case
$e_{1}\prec_{\textsc{\tiny{HB}}}w\prec_{\textsc{\tiny{SDP}}(i)}r\prec_{\textsc{\tiny{HB}}}e_{2}$.
In each of these cases, by SDP($i$)’s composition with HB,
$e_{1}\prec_{\textsc{\tiny{SDP}}(i)}e_{2}$, a contradiction.
Therefore there is no read event $r$ that reads the same variable as $w$ and
$w^{\prime}$ such that $w<_{\textsc{$\mathit{tr}$}}r$. Any read event that
reads the same variable as $w$ and $w^{\prime}$ must occur _before_ $w$.
Now consider the trace $\mathit{tr^{\prime}}$ that is equivalent to
$\mathit{tr}$ except:
* •
$w^{\prime}$ is replaced by a wr(x) event, where x is a brand-new variable not
used in $\mathit{tr}$.
* •
For every read $r$ in $\mathit{tr}$ that reads the same variable as
$w^{\prime}$, an event $r^{\prime}$ is appended immediately after $r$ such
that $r^{\prime}$ is a rd(x) event and $r\prec_{\textsc{\tiny{PO}}}r^{\prime}$
in $\mathit{tr^{\prime}}$.
Note that the SDP($i-1$) ordering for $\mathit{tr^{\prime}}$ is the same as
the SDP($i$) ordering for $\mathit{tr}$: the rd(x)–wr(x) conflicts introduce
the same ordering in $\mathit{tr^{\prime}}$ as the original read–write
conflicts between $w^{\prime}$ and its prior reads in $\mathit{tr}$, and
$\mathit{tr^{\prime}}$ does not contain the write–write conflict on $w$ and
$w^{\prime}$ found in $\mathit{tr}$. Thus in $\mathit{tr^{\prime}}$,
$e_{1}\not\prec_{\textsc{\tiny{SDP}}(i-1)}e_{2}$. By the induction hypothesis,
$\mathit{tr^{\prime}}$ has a predictable race or deadlock. Let
$\mathit{tr^{\prime\prime}}$ be a predictable trace of $\mathit{tr^{\prime}}$
that exposes a race or deadlock. However, if we modify
$\mathit{tr^{\prime\prime}}$ by removing the rd(x) events and replacing the
wr(x) event with $w^{\prime}$, the resulting trace is a predictable trace of
$\mathit{tr}$ that exposes a race or deadlock. Thus $\mathit{tr}$ has a
predictable race or deadlock, which is a contradiction.
Thus for all $i$, SDP($i$) is sound. Since
$\prec_{\textsc{\tiny{SDP}}(\infty)}\equiv\prec_{\textsc{\tiny{SDP}}}$,
therefore SDP is sound. ∎
###### Theorem 4.1 (WDP completeness).
If an execution trace has a predictable race, then it has a WDP-race.
To prove the theorem, we use the following helper lemma:
###### Lemma 4.2 (WDP-ordered events cannot be reordered).
Given an execution trace $\mathit{tr}$, for any events $e_{1}$ and $e_{2}$ in
$\mathit{tr}$ such that $e_{1}\prec_{\textsc{\tiny{WDP}}}e_{2}$, let
$\mathit{tr^{\prime}}$ be a reordering of $\mathit{tr}$ where $e_{1}$ and
$e_{2}$ have been reordered: either
$e_{2}<_{\textsc{$\mathit{tr^{\prime}}$}}e_{1}$ or
$e_{2}\in\mathit{tr^{\prime}}\land e_{1}\notin\mathit{tr^{\prime}}$. Then,
$\mathit{tr^{\prime}}$ must not be a valid predictable trace of $\mathit{tr}$.
The overall proof strategy is analogous to a corresponding proof for DC
(Roemer et al., 2018), so we have relegated the proof of Lemma 4.2 to Appendix
A.
###### Proof of Theorem 4.1.
Let us prove this theorem by contradiction. Let $\mathit{tr}$ be a trace with
a predictable race on conflicting events $e_{1}$ and $e_{2}$ such that
$e_{1}<_{\textsc{$\mathit{tr}$}}e_{2}$, but no WDP-race. Let
$\mathit{tr^{\prime}}$ be a predictable trace of $\mathit{tr}$ in which
$e_{1}$ and $e_{2}$ are consecutive:
$e_{1}<_{\textsc{$\mathit{tr^{\prime}}$}}e_{2}$ and $\nexists
e:e_{1}<_{\textsc{$\mathit{tr^{\prime}}$}}e<_{\textsc{$\mathit{tr^{\prime}}$}}e_{2}$.
Applying the definition of a WDP-race (Section 4), either
$e_{1}\prec_{\textsc{\tiny{WDP}}}e_{2}$ or
$\mathit{lockset}(e_{1})\cap\mathit{lockset}(e_{2})\neq\emptyset$. If
$\mathit{lockset}(e_{1})\cap\mathit{lockset}(e_{2})\neq\emptyset$, then
$\mathit{tr^{\prime}}$ violates the LS rule of predictable traces.
Thus $e_{1}\prec_{\textsc{\tiny{WDP}}}e_{2}$. By the definition of a
predictable race, $e_{1}$ and $e_{2}$ must be read or write events, and must
be on different threads. As a result, the WDP ordering between $e_{1}$ and
$e_{2}$ cannot be established by WDP conflicting critical section ordering or
“$\textsf{acq(m)}\prec_{\textsc{\tiny{WDP}}}\textsf{rel(m)}\implies\textsf{rel(m)}\prec_{\textsc{\tiny{WDP}}}\textsf{rel(m)}$,”
which require $e_{1}$ to be a release, and not by PO since the events are on
different threads. Therefore, $e_{1}\prec_{\textsc{\tiny{WDP}}}e_{2}$ by WDP
transitivity, so there must exist an event $e$ such that
$e_{1}\prec_{\textsc{\tiny{WDP}}}e\prec_{\textsc{\tiny{WDP}}}e_{2}$.
Since $e_{1}$ and $e_{2}$ are consecutive in $\mathit{tr^{\prime}}$, either
$e<_{\textsc{$\mathit{tr^{\prime}}$}}e_{1}$,
$e_{2}<_{\textsc{$\mathit{tr^{\prime}}$}}e$, or
$e_{2}\in\mathit{tr^{\prime}}\land e\notin\mathit{tr^{\prime}}$. By Lemma 4.2,
any of these possibilities implies $\mathit{tr^{\prime}}$ is an invalid
predictable trace of $\mathit{tr}$, a contradiction. ∎
### 4.3. Using Precise Dependence Information
Up to this point, we have assumed that a branch event depends on every
preceding read event in the same thread, meaning that the condition
$\mathit{brDepsOn}(b,e_{2})$ in WDP’s handling of write–read critical sections
holds for every read $e_{2}$ and branch $b$. This assumption is needed unless
static control dependence information is available from conservative static
program analysis (e.g., (Huang and Huang, 2017; Ferrante et al., 1987)). We
tried out one kind of static analysis to compute static control dependencies
but found it provided no benefit, so our experiments do not use it (Section
7). Here we show some examples of how WDP uses static control dependence
information if it is available.
Figure 6 shows two executions that differ only in whether precise static
control dependency information is available. Figure 6(a) has no control
dependency information available, so each branch is conservatively dependent
on all prior reads, or the information is available but the branch outcome
_may depend_ on the prior read. Figure 6(b) has control dependency information
that says that the branch outcome does _not_ depend on the prior read. As a
result, Figure 6(b) has weaker WDP ordering than Figure 6(a), leading to a
detected WDP-race in Figure 6(b) only.
Thread 1 Thread 2 wr(y) acq(m) wr(x) rel(m) acq(m) rd(x) br rel(m) rd(y)
((a)) No WDP-race exists if the branch event is dependent on the prior read.
Thread 1 Thread 2 wr(y) acq(m) wr(x) rel(m) acq(m) rd(x) br rel(m) rd(y)
((b)) A WDP-race exists if the branch event does not depend on the prior read.
Thread 1 Thread 2 acq(m) rd(x) rel(m) br wr(y) rd(y)
((c)) A valid reordering of (b) demonstrating that (b) has a race.
WDP
Figure 6. Example executions that differ only in the static control
dependencies between branches and reads. Dotted edges indicate reads that a
branch depends on, i.e., $\mathit{brDepsOn}(b,e)$. If precise control
dependence information rules out read–branch dependencies, WDP can find
additional races, such as the race on y in (b).
## 5\. SDP and WDP Analyses
Algorithm 1 SDP analysis at each event type, with differences from WCP
analysis
1:procedure acquire($t,l$)
2:
$\use@mathgroup\M@U\symAMSb{H}_{t}\leftarrow\use@mathgroup\M@U\symAMSb{H}_{t}\sqcup\use@mathgroup\M@U\symAMSb{H}_{l}$
3:
$\use@mathgroup\M@U\symAMSb{C}_{t}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}\sqcup\use@mathgroup\M@U\symAMSb{C}_{l}$
4: foreach $t^{\prime}\neq t$ do
$Acq_{l}(t^{\prime}).Enque(\use@mathgroup\M@U\symAMSb{C}_{t}[t:=\use@mathgroup\M@U\symAMSb{H}_{t}(t)])$
5:procedure release($t,l,R,W$)
6: while
$Acq_{l}(t).Front()\sqsubseteq\use@mathgroup\M@U\symAMSb{C}_{t}[t:=\use@mathgroup\M@U\symAMSb{H}_{t}(t)]$
do
7: $Acq_{l}(t).Deque()$
8:
$\use@mathgroup\M@U\symAMSb{C}_{t}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}\sqcup
Rel_{l}(t).Deque()$
9: foreach $x\in R$ do
$\use@mathgroup\M@U\symAMSb{L}^{r}_{l,x}\leftarrow\use@mathgroup\M@U\symAMSb{L}^{r}_{l,x}\sqcup\use@mathgroup\M@U\symAMSb{H}_{t}$
10: foreach $x\in W$ do
$\use@mathgroup\M@U\symAMSb{L}^{w}_{l,x}\leftarrow\use@mathgroup\M@U\symAMSb{L}^{w}_{l,x}\sqcup\use@mathgroup\M@U\symAMSb{H}_{t}$
11:
$\use@mathgroup\M@U\symAMSb{H}_{l}\leftarrow\use@mathgroup\M@U\symAMSb{H}_{t}$
12:
$\use@mathgroup\M@U\symAMSb{C}_{l}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}$
13: foreach $t^{\prime}\neq t$ do
$Rel_{l}(t^{\prime}).Enque(\use@mathgroup\M@U\symAMSb{H}_{t})$
14:
$\use@mathgroup\M@U\symAMSb{H}_{t}(t)\leftarrow\use@mathgroup\M@U\symAMSb{H}_{t}(t)+1$
15:procedure read($t,x,L$)
16: $+$
$\use@mathgroup\M@U\symAMSb{C}_{t}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}\sqcup\use@mathgroup\M@U\symAMSb{B}_{t,x}$$\triangleright$
Apply prior write–write conflict
17:
$\use@mathgroup\M@U\symAMSb{C}_{t}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}\sqcup_{l\in
L}\use@mathgroup\M@U\symAMSb{L}^{w}_{l,x}$
18: check
$\use@mathgroup\M@U\symAMSb{W}_{x}\sqsubseteq\use@mathgroup\M@U\symAMSb{C}_{t}[t:=\use@mathgroup\M@U\symAMSb{H}_{t}(t)]$$\triangleright$
Write–read race?
19:
$\use@mathgroup\M@U\symAMSb{R}_{x}(t)\leftarrow\use@mathgroup\M@U\symAMSb{H}_{t}(t)$
20:procedure write($t,x,L$)
21:
$\use@mathgroup\M@U\symAMSb{C}_{t}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}\sqcup_{l\in
L}\use@mathgroup\M@U\symAMSb{L}^{r}_{l,x}$
22:$-$
$\use@mathgroup\M@U\symAMSb{C}_{t}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}\sqcup_{l\in
L}\use@mathgroup\M@U\symAMSb{L}^{w}_{l,x}$
23: check
$\use@mathgroup\M@U\symAMSb{W}_{x}\sqsubseteq\use@mathgroup\M@U\symAMSb{C}_{t}[t:=\use@mathgroup\M@U\symAMSb{H}_{t}(t)]\sqcup_{l\in
L}\use@mathgroup\M@U\symAMSb{L}^{w}_{l,x}$$\triangleright$ Write–write race?
24: $+$ $\use@mathgroup\M@U\symAMSb{B}_{t,x}\leftarrow\sqcup_{l\in
L}\use@mathgroup\M@U\symAMSb{L}^{w}_{l,x}$$\triangleright$ Records write–write
conflict for future read
25: check
$\use@mathgroup\M@U\symAMSb{R}_{x}\sqsubseteq\use@mathgroup\M@U\symAMSb{C}_{t}[t:=\use@mathgroup\M@U\symAMSb{H}_{t}(t)]$$\triangleright$
Read–write race?
26:
$\use@mathgroup\M@U\symAMSb{W}_{x}(t)\leftarrow\use@mathgroup\M@U\symAMSb{H}_{t}(t)$
27:procedure branch($t,L$)
28: skip $\triangleright$ No analysis at branch events
Algorithm 2 WDP analysis at each event type, with differences from DC analysis
1:procedure acquire($t,l$)
2: foreach $t^{\prime}\neq t$ do
$Acq_{l,t^{\prime}}(t).Enque(\use@mathgroup\M@U\symAMSb{C}_{t})$
3:procedure release($t,l,R,W$)
4: foreach $t^{\prime}\neq t$ do
5: while
$Acq_{l,t}(t^{\prime}).Front()\sqsubseteq\use@mathgroup\M@U\symAMSb{C}_{t}$ do
6: $Acq_{l,t}(t^{\prime}).Deque()$
7:
$\use@mathgroup\M@U\symAMSb{C}_{t}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}\sqcup
Rel_{l,t}(t^{\prime}).Deque()$
8: foreach $x\in W$ do
$\use@mathgroup\M@U\symAMSb{L}_{l,x}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}$
$\triangleright$ Record release time for writes in critical section
9:$-$ foreach $x\in R$ do
$\use@mathgroup\M@U\symAMSb{L}^{r}_{l,x}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}$
10: foreach $t^{\prime}\neq t$ do
$Rel_{l,t^{\prime}}(t).Enque(\use@mathgroup\M@U\symAMSb{C}_{t})$
11:
$\use@mathgroup\M@U\symAMSb{C}_{t}(t)\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}(t)+1$
12:procedure read($t,x,e,L$)
13:$-$
$\use@mathgroup\M@U\symAMSb{C}_{t}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}\bigsqcup_{l\in(L\cap
L^{w}_{x,t^{\prime}})}\use@mathgroup\M@U\symAMSb{L}_{l,x}$
14:$-$ check
$\use@mathgroup\M@U\symAMSb{W}_{x}\sqsubseteq\use@mathgroup\M@U\symAMSb{C}_{t}$
15: $+$ foreach thread $t^{\prime}\neq t$ check
$\use@mathgroup\M@U\symAMSb{W}_{x}(t^{\prime})\leq\use@mathgroup\M@U\symAMSb{C}_{t}(t^{\prime})\lor
L^{w}_{x,t^{\prime}}\cap L\neq\emptyset$ $\triangleright$ Check write–read
race
16: $+$ let $t^{\prime}\leftarrow T_{x}$ $\triangleright$ Get last writer
thread of $x$
17: if $+$ $t^{\prime}\notin\\{\varnothing,t\\}\land L\cap
L^{w}_{x,t^{\prime}}\neq\emptyset$ then $\triangleright$ Write–read conflict
18: $+$ $\use@mathgroup\M@U\symAMSb{B}_{t,x}\leftarrow\bigsqcup_{l\in(L\cap
L^{w}_{x,t^{\prime}})}\use@mathgroup\M@U\symAMSb{L}_{l,x}$ $\triangleright$
Record time of writer thread’s release for later use
19: if $+$
$\use@mathgroup\M@U\symAMSb{B}_{t,x}\nsqsubseteq\use@mathgroup\M@U\symAMSb{C}_{t}$
then $D_{t}\leftarrow D_{t}\cup\\{\langle x,e\rangle\\}$ $\triangleright$
Record read
20:
$\use@mathgroup\M@U\symAMSb{R}_{x}(t)\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}(t)$
21: $+$ $L^{r}_{x,t}\leftarrow L$
22:procedure write($t,x,L$)
23:$-$
$\use@mathgroup\M@U\symAMSb{C}_{t}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}\sqcup\bigsqcup_{l\in(L\cap
L^{w}_{x,t^{\prime}})}\use@mathgroup\M@U\symAMSb{L}_{l,x}\sqcup\bigsqcup_{l\in(L\cap
L^{r}_{x,t^{\prime}})}\use@mathgroup\M@U\symAMSb{L}^{r}_{l,x}$
24:$-$ check
$\use@mathgroup\M@U\symAMSb{W}_{x}\sqsubseteq\use@mathgroup\M@U\symAMSb{C}_{t}$
25: $+$ foreach thread $t^{\prime}\neq t$ check
$\use@mathgroup\M@U\symAMSb{W}_{x}(t^{\prime})\leq\use@mathgroup\M@U\symAMSb{C}_{t}(t^{\prime})\lor
L^{w}_{x,t^{\prime}}\cap L\neq\emptyset$ $\triangleright$ Check write–write
race
26:$-$ check
$\use@mathgroup\M@U\symAMSb{R}_{x}\sqsubseteq\use@mathgroup\M@U\symAMSb{C}_{t}$
27: $+$ foreach thread $t^{\prime}\neq t$ check
$\use@mathgroup\M@U\symAMSb{R}_{x}(t^{\prime})\leq\use@mathgroup\M@U\symAMSb{C}_{t}(t^{\prime})\lor
L^{r}_{x,t^{\prime}}\cap L\neq\emptyset$ $\triangleright$ Check read–write
race
28:
$\use@mathgroup\M@U\symAMSb{W}_{x}(t)\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}(t)$
29: $+$ $L^{w}_{x,t}\leftarrow L$
30: $+$ $T_{x}\leftarrow t$ $\triangleright$ Set last writer thread of $x$
31:procedure branch($t,e,L$)
32:$-$ skip
33: $+$ $\textbf{foreach}\langle x,r\rangle\in
D_{t}:\mathit{brDepsOn}(e,r)\textbf{ do
}\use@mathgroup\M@U\symAMSb{C}_{t}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}\sqcup\use@mathgroup\M@U\symAMSb{B}_{t,x}$
$\triangleright$ Add release–branch ordering
34: $+$ $D_{t}\leftarrow D_{t}\setminus\\{\langle x,r\rangle\in
D_{t}:\mathit{brDepsOn}(e,r)\\}$ $\triangleright$ Remove dependencies that
were applied
_SDP analysis_ and _WDP analysis_ are new online dynamic program analyses that
compute SDP and WDP and detect SDP- and WDP-races, respectively. Algorithms 1
and 2 show SDP and WDP analyses, respectively, for each kind of event. This
section’s notation and terminology follow the WCP and DC papers’ to some
extent (Roemer et al., 2018; Kini et al., 2017).
Both algorithms show the differences relative to prior analyses (SDP versus
WCP and WDP versus DC) by labeling lines with “$+$” to show logic added by our
analyses and “$-$” with grayed-out text to show lines removed by our analyses.
Algorithm 1 shows that SDP analysis requires few changes to WCP analysis.
These changes are for tracking write–write conflicts to add ordering when a
future read is detected on the second write’s thread. In addition, SDP
analysis avoids reporting write–write races for writes in critical sections on
the same lock by using $\use@mathgroup\M@U\symAMSb{L}^{w}_{l,x}$ at line 23.
Algorithm 2 shows that WDP analysis makes several significant changes to DC
analysis. These changes are primarily to deal with branches, by recording
information about write–read dependencies at read events (lines 16–19) and
using the recorded information at branch events (lines 33–34). Unlike DC
analysis, WDP analysis does not establish ordering at any conflicting
accesses, and it never needs to track ordering from a read to another access
(since it detects only write–read conflicts). In addition, WDP analysis
ensures it does not report races on accesses in critical sections on the same
lock, by maintaining and using the locksets $L^{w}_{x,t^{\prime}}$ and
$L^{r}_{x,t^{\prime}}$ when checking for races.
##### Analysis details
In both SDP and WDP analyses, the procedural parameters $t$ and $l$ are the
current thread and lock; $L$ is the set of locks held by the thread performing
the current event; $R$ and $W$ are the sets of variables that were read and
written in the ending critical section on $l$; and $e$ represents the current
read or branch event (for detecting branch dependencies).
The analysis uses _vector clocks_ (Mattern, 1988) to represent logical SDP or
WDP time. A vector clock $C:\mathit{Tid}\mapsto\mathcal{N}$ maps each thread
to a nonnegative integer. Operations on vector clocks are pointwise comparison
($C_{1}\sqsubseteq C_{2}\iff\forall t.C_{1}(t)\leq C_{2}(t)$) and pointwise
join ($C_{1}\sqcup C_{2}\equiv\lambda t.\mathit{\max(C_{1}(t),C_{2}(t))}$):
Both analyses maintain the following state:
* •
$\use@mathgroup\M@U\symAMSb{C}_{t}$ is a vector clock that represents the
current SDP or WDP time for thread $t$.
* •
$\use@mathgroup\M@U\symAMSb{R}_{x}$ and $\use@mathgroup\M@U\symAMSb{W}_{x}$
are vector clocks that represent the SDP or WDP time of the last reads and
writes to $x$.
* •
$\mathit{Acq}_{l}(t)$ and $\mathit{Rel}_{l}(t)$ (SDP) and
$\mathit{Acq}_{l,t^{\prime}}(t)$ and $\mathit{Rel}_{l,t^{\prime}}(t)$ (WDP)
are queues of vector clocks that help compute the
“$\textsf{acq(m)}\prec\textsf{rel(m)}$ implies
$\textsf{rel(m)}\prec\textsf{rel(m)}$” property (Table 2).
In addition, SDP analysis maintains the following state:
* •
$\use@mathgroup\M@U\symAMSb{H}_{t}$ is a vector clock that represents the
current HB time for thread $t$.
$\use@mathgroup\M@U\symAMSb{C}_{t}[t:=\use@mathgroup\M@U\symAMSb{H}_{t}(t)]$,
which evaluates to a vector clock with every element equal to
$\use@mathgroup\M@U\symAMSb{C}_{t}$ except that element $t$ is equal to
$\use@mathgroup\M@U\symAMSb{H}_{t}(t)$, represents
$\prec_{\textsc{\tiny{SDP}}}\cup\prec_{\textsc{\tiny{PO}}}$.
* •
$\use@mathgroup\M@U\symAMSb{B}_{t,x}$ is a vector clock that represents the
SDP time of a release event $e$ of a critical section on lock $m$ containing a
write event $w$ to $x$ such that a later write event $w^{\prime}$ to $x$ by
$t$ conflicts with the write and
$m\in\mathit{lockset}(w)\cap\mathit{lockset}(w^{\prime})$.
* •
$\use@mathgroup\M@U\symAMSb{L}^{r}_{l,x}$ and
$\use@mathgroup\M@U\symAMSb{L}^{w}_{l,x}$ are vector clocks that represent the
HB times of critical sections on $l$ that read and wrote $x$, respectively.
WDP analysis maintains the following additional state:
* •
$\use@mathgroup\M@U\symAMSb{L}_{l,x}$ is a vector clock that represents the
WDP time of critical sections on $l$ that wrote $x$.
* •
$\use@mathgroup\M@U\symAMSb{B}_{t,x}$ is a vector clock that represents the
WDP time of the last write event $e$ to $x$ such that a later read event
$e^{\prime}$ to $x$ by $t$ conflicts with $e$ and
$\mathit{lockset}(e)\cap\mathit{lockset}(e^{\prime})\neq\emptyset$.
* •
$T_{x}$ is the last thread to write to $x$, or $\varnothing$ if no thread has
yet written $x$.
* •
$D_{t}$ is a set of pairs $\langle x,e\rangle$ such that event $e$ is a read
to $x$ by a thread that has not (yet) executed a branch $b$ such that
$\mathit{brDepsOn}(b,e)$.
* •
$L^{r}_{x,t}$ and $L^{w}_{x,t}$ are sets of locks that were held by thread $t$
when it last read and wrote variable $x$, respectively.
Initially, every vector clock maps every thread to 0, except $\forall
t.\use@mathgroup\M@U\symAMSb{H}_{t}(t)=1$ for SDP analysis, and $\forall
t.\use@mathgroup\M@U\symAMSb{C}_{t}(t)=1$ for WDP analysis. Every queue and
set is initially empty.
The analyses provide SDP and WDP’s handling of conflicting critical sections
by detecting some kinds of conflicts on accesses holding a common lock. SDP
analysis orders the earlier release of a common lock to the current event for
write–read and read–write conflicts using
$\use@mathgroup\M@U\symAMSb{L}^{r}_{l,x}$ and
$\use@mathgroup\M@U\symAMSb{L}^{w}_{l,x}$ (lines 17 and 21 in Algorithm 1).
For write–write conflicts, SDP analysis stores the time of the earlier release
of a common lock in $\use@mathgroup\M@U\symAMSb{B}_{t,x}$ (line 24) to order
the earlier write to a later read of $x$ by the current thread (line 16).
WDP analysis orders the release on the writer’s executing thread to a later
branch dependent on the read. The analysis does so by recording the time of
the last writer’s release in $\use@mathgroup\M@U\symAMSb{L}_{l,x}$ (line 8 in
Algorithm 2). Later, when a conflicting read occurs on thread $t$ holding $l$,
the analysis uses $\use@mathgroup\M@U\symAMSb{L}_{l,x}$ to get the time for
the last conflicting writer $T_{x}$’s release, and stores this time in
$\use@mathgroup\M@U\symAMSb{B}_{t,x}$ (line 18). When $t$ executes a branch
dependent on the prior conflicting read, WDP adds ordering from the release to
the current branch (line 33). The analysis detects the dependent branch using
$D_{t}$, which contains a set of $\langle x,e\rangle$ pairs for which a branch
dependent on read event $e$ has not yet executed (line 30). The exact
representation of $e$ and behavior of $\mathit{brDepsOn}(b,e)$ are
implementation dependent.
The analyses compute the “$\textsf{acq(m)}\prec\textsf{rel(m)}$ implies
$\textsf{rel(m)}\prec\textsf{rel(m)}$” property (Table 2) in the same way as
WCP and DC analyses, respectively (Kini et al., 2017; Roemer et al., 2018).
Briefly, $\mathit{Acq}_{l}(t)$ and $\mathit{Rel}_{l}(t)$ contain times of
acq(l) and rel(l) operations (by any thread other than $t$) such that the
acq(l) operation is not yet SDP ordered to a following rel(l) by thread $t$.
$\mathit{Acq}_{l,t^{\prime}}(t)$ and $\mathit{Rel}_{l,t^{\prime}}(t)$ contain
times of acq(l) and rel(l) operations by thread $t$ such that the acq(l)
operation is not yet WDP ordered to a following rel(l) by thread $t^{\prime}$.
SDP analysis provides composition with HB using
$\use@mathgroup\M@U\symAMSb{C}_{t}$, $\use@mathgroup\M@U\symAMSb{H}_{t}$, and
$\use@mathgroup\M@U\symAMSb{C}_{t}[t:=\use@mathgroup\M@U\symAMSb{H}_{t}(t)]$.444In
Algorithm 1, $\use@mathgroup\M@U\symAMSb{C}_{t}$ and
$\use@mathgroup\M@U\symAMSb{C}_{t}[t:=\use@mathgroup\M@U\symAMSb{H}_{t}(t)]$
are analogous to the WCP paper’s $\use@mathgroup\M@U\symAMSb{P}_{t}$ and
$\use@mathgroup\M@U\symAMSb{C}_{t}$, respectively (Kini et al., 2017). WDP
analysis includes PO with the increment of
$\use@mathgroup\M@U\symAMSb{C}_{t}(t)$ at line 11.
The analyses check the conditions for a SDP- or WDP-race by using
$\use@mathgroup\M@U\symAMSb{R}_{x}$ and $\use@mathgroup\M@U\symAMSb{W}_{x}$.
Since the analyses do not order all pairs of conflicting accesses, unordered
conflicting accesses are not sufficient to report a race. SDP analysis uses
the vector clock $\use@mathgroup\M@U\symAMSb{L}^{w}_{l,x}$ and WDP analysis
uses the locksets $L^{r}_{x,t}$ and $L^{w}_{x,t}$ to check if the current and
prior conflicting accesses’ held locks overlap (lines 18, 23, and 25 in
Algorithm 1; lines 15, 25, and 27 in Algorithm 2).
##### Atomic accesses and operations.
We extend SDP and WDP analyses to handle accesses that have ordering or
atomicity semantics: _atomic accesses_ that introduce ordering such as Java
volatile and C++ atomic accesses, and _atomic read-modify-write operations_
such as atomic test-and-set.
The following pseudocode shows how we extend WDP analysis (Algorithm 2) to
handle atomic reads and writes and atomic operations. (The extensions to SDP
analysis are similar but also add conflicting read–write and write–write–read
ordering.)
1:procedure atomicRead($t,x,e$)
2: let $t^{\prime}\leftarrow T_{x}$ $\triangleright$ Get last writer thread of
$x$
3: if
$t^{\prime}\notin\\{\varnothing,t\\}\land\use@mathgroup\M@U\symAMSb{W}_{x}\nsqsubseteq\use@mathgroup\M@U\symAMSb{C}_{t}$
then $\triangleright$ Write–read conflict
4:
$\use@mathgroup\M@U\symAMSb{B}_{t,x}\leftarrow\use@mathgroup\M@U\symAMSb{W}_{x}$
$\triangleright$ Record the write
5: if
$\use@mathgroup\M@U\symAMSb{B}_{t,x}\nsqsubseteq\use@mathgroup\M@U\symAMSb{C}_{t}$
then $D_{t}\leftarrow D_{t}\cup\\{\langle x,e\rangle\\}$
6:procedure atomicWrite($t,x$)
7:
$\use@mathgroup\M@U\symAMSb{W}_{x}\leftarrow\use@mathgroup\M@U\symAMSb{C}_{t}$
8: $T_{x}\leftarrow t$ $\triangleright$ Set last writer thread of $x$
9:procedure atomicReadModifyWrite($t,x,e$)
10: atomicRead($t$, $x$, $e$)
11: atomicWrite($t$, $x$)
In essence, the analysis handles atomic accesses like regular accesses
contained in single-access critical sections on a unique lock to the accessed
variable. The analysis treats an atomic operation as an atomic read followed
by an atomic write.
##### Handling races.
The behavior of programs with data races is unreliable (Adve and Boehm, 2010;
Dolan et al., 2018), but our analyses’ instrumentation performs
synchronization operations before accesses, which generally ensures sequential
consistency (SC) for all program executions. A different problem is that if an
analysis continues detecting races after the first race, then additional
detected races are not necessarily real races because they may depend on an
earlier race (i.e., if the earlier race were ordered, the later race would not
exist). Our implementation (Section 7.1) addresses this issue by treating
racing accesses as if they were contained in single-access critical sections
on the same lock. Specifically, SDP analysis orders one racing event to the
other for write–read and read–write races, and WDP analysis orders write–read
races to a branch that depends on the read if the write is the last writer of
the read. For example, after detecting a race in line 15, WDP analysis
performs the following:
$\textbf{if}\>t^{\prime}=T_{x}\>\textbf{then}\;\use@mathgroup\M@U\symAMSb{B}_{t,x}\leftarrow\use@mathgroup\M@U\symAMSb{W}_{x}(t^{\prime});\>D_{t}\leftarrow
D_{t}\cup\\{\langle x,e\rangle\\}$.
##### Time and space complexity.
SDP analysis and WDP analysis’s run times are each linear in the number of
events. Like WCP and DC analyses (Kini et al., 2017; Roemer et al., 2018), for
$N$ events, $L$ locks, and $T$ threads, time complexity for an entire
execution trace is $O(N\times(L\times T+T^{2}))$. Briefly, SDP or WDP analysis
at a read or write takes $O(L\times T+T^{2})$ time even considering the
additional computation it performs compared with WCP or DC analysis; and WDP
analysis’s run time at branch events can be amortized over read events.
## 6\. Verifying WDP-Races
WDP analysis is unsound, so a WDP-race may not indicate a predictable race
(i.e., there may be no race exposed in any predictable trace). To avoid
reporting false races, our approach post-processes each detected WDP-race with
an algorithm called VindicateWDPRace. Here we overview VindicateWDPRace;
Appendix B presents VindicateWDPRace in detail with an algorithm and examples.
To support performing VindicateWDPRace on WDP-races, WDP analysis builds a
_constraint graph_ in which execution events are nodes, and initially edges
represent WDP ordering. VindicateWDPRace discovers and adds additional
constraints to the graph that enforce lock semantics (LS) and last-writer (LW)
rules. VindicateWDPRace uses the constraint graph to attempt to construct a
predictable trace that exposes the WDP-race as a predictable race.
VindicateWDPRace extends prior work’s VindicateDCRace algorithm for checking
DC-races (Roemer et al., 2018). VindicateWDPRace differs from VindicateDCRace
primarily in the following way. VindicateWDPRace computes and adds LW
constraints to the constraint graph for all reads that must be causal events
in the predictable trace. Importantly, VindicateWDPRace computes causal reads
and adds LW constraints at each iteration of adding constraints and at each
attempt at building a predictable trace.
Algorithm 3 shows VindicateWDPRace at a high level; Appendix B presents
VindicateWDPRace in detail. VindicateWDPRace takes the initial constraint
graph ($G$) and a WDP-race ($e_{1},e_{2}$) as input (line 1). It first calls
AddConstraints (line 2), which adds necessary constraints to $G$ and returns
an updated constraint graph. AddConstraints first adds _consecutive-event_
constraints (i.e., edges) to $G$ to enforce that any predictable trace must
execute $e_{1}$ and $e_{2}$ consecutively (line 9). AddConstraints then
computes the set of causal events for any predictable trace constrained by
$G$, which it uses to add LW constraints to $G$, ensuring that every causal
read in a predictable trace can have the same last writer as in the original
trace (line 11). Next, AddConstraints adds LS constraints to $G$, by
identifying critical sections on the same lock that are partially ordered and
thus must be fully ordered to obey LS rules (line 12). Since added LW and LS
constraints may lead to new LS and LW constraints being detected,
respectively, AddConstraints iterates until it finds no new constraints to add
(lines 10–13).
Algorithm 3 Check if WDP-race is a true predictable race (high-level version
of algorithm)
1:procedure VindicateWDPRace($G,e_{1},e_{2}$) $\triangleright$ Inputs:
constraint graph and WDP-race events
2: $G\leftarrow\textsc{AddConstraints}(G,e_{1},e_{2})$
3: if $G$ has a cycle reaching $e_{1}$ or $e_{2}$ then return No predictable
race
4: else
5:
$\mathit{tr^{\prime}}\leftarrow\textsc{ConstructReorderedTrace}(G,e_{1},e_{2})$
$\triangleright$ Non-empty iff predictable trace constructed
6: if $\mathit{tr^{\prime}}\neq\langle\,\rangle\,$ then return Predictable
race witnessed by $\mathit{tr^{\prime}}$ $\triangleright$ Check for non-empty
trace
7: else return Don’t know
8:procedure AddConstraints($G,e_{1},e_{2}$)
9: Add consecutive-event constraints to $G$
10: do
11: Compute causal reads and add last-writer (LW) constraints to $G$
12: Add lock-semantics (LS) constraints to $G$
13: while $G$ has changed
14: return $G$
The constraints added by AddConstraints are necessary but insufficient
constraints on any trace exposing a predictable race on $e_{1}$ and $e_{2}$.
Thus if $G$ has a cycle that must be part of any predictable trace, then the
original trace has no predictable race on $e_{1}$ and $e_{2}$ (line 3).
Otherwise, AddConstraints calls ConstructReorderedTrace (line 5), which
attempts to construct a legal predictable trace $\mathit{tr^{\prime}}$.
ConstructReorderedTrace is a greedy algorithm that starts from $e_{1}$ and
$e_{2}$ and works backward, adding events in reverse order that satisfy $G$’s
constraints and also conform to LS and LW rules ($G$’s constraints are
necessary but insufficient). If ConstructReorderedTrace returns a (non-empty)
trace $\mathit{tr^{\prime}}$, it is a legal predictable trace exposing a race
on $e_{1}$ and $e_{2}$ (line 6). Otherwise, ConstructReorderedTrace returns an
empty trace, which means that it could not find a predictable race, although
one may exist (line 7).
## 7\. Evaluation
This section evaluates the predictive race detection effectiveness and run-
time performance of this paper’s approaches.
### 7.1. Implementation
We implemented SDP and WDP analyses and VindicateWDPRace by extending the
publicly available _Vindicator_ implementation, which includes HB, WCP, and DC
analyses and VindicateDCRace (Roemer et al.,
2018).555https://github.com/PLaSSticity/Vindicator Vindicator is built on top
of _RoadRunner_ , a dynamic analysis framework for concurrent Java programs
(Flanagan and Freund,
2010b).666https://github.com/stephenfreund/RoadRunner/releases/tag/v0.5 We
extended RoadRunner to instrument branches to enable WDP analysis at program
branches. RoadRunner operates on the Java bytecode of analyzed programs, so
analysis properties such as SDP soundness and WDP completeness hold with
respect to the execution of the bytecode, even if the JVM compiler optimizes
away control or data dependencies.
Our implementation of SDP and WDP analyses and VindicateWDPRace is publicly
available.777https://github.com/PLaSSticity/SDP-WDP-implementation
We evaluated _Joana_ to perform static analysis for detecting whether a branch
depends on prior reads or not (Giffhorn and Hammer, 2008), following the
system dependency graphs used in MCR-S (Huang and Huang, 2017). We found no
practical advantages to using Joana. In most programs, for the vast majority
of the write–read branch dependencies executed, the next branch after the read
is dependent on the read according to Joana. In pmd and sunflow, static
analysis reported many write–read dependencies where the following branch did
not depend on the read, but this did not lead to any additional WDP-races
being detected. It is unclear whether these results are mainly due to
properties of the evaluated programs (i.e., if almost all branches do depend
on prior reads) or imprecision of Joana’s static analysis. Our implementation
and evaluation do not use static analysis, and instead assume that branches
always depend on prior reads.
##### SDP and WDP analyses.
We implemented a single analysis tool within RoadRunner that can perform HB,
WCP, DC, SDP, and WDP analyses on a single observed execution. The
implementation of HB, WCP, and DC analyses are taken from the Vindicator
implementation, and implementation of SDP and WDP analyses follows Algorithms
1 and 2. For thread fork and join (including implicitly forked threads (Roemer
et al., 2018)) and static class initializer edges (Lindholm and Yellin, 1999),
each analysis adds appropriate ordering between the two synchronizing events.
The analyses treat calls to m.wait() as a release of m followed by an acquire
of m. The analyses instrument volatile variable accesses as _atomic accesses_
as described in Section 5. The analyses can in theory handle lock-free data
structures, such as data structures in java.util.concurrent, by handling
atomic operations as in Section 5. However, RoadRunner instruments only
application code, not Java library code, and it does not intercept underlying
atomic operations (e.g., by instrumenting calls to atomic sun.misc.Unsafe
methods). The analyses may thus miss some synchronization in the evaluated
programs.
The analyses can determine that some observed events are “redundant” and
cannot affect the analysis results. For a read or write event, if the same
thread has performed a read or write, respectively, to the same variable
without an intervening synchronization operation, then the access is
redundant. For a branch event, if the same thread has not performed a read
event since the last branch event, then the branch is redundant (since our
implementation assumes that a branch is dependent on all prior reads). The
implementation “fast path” detects and filters redundant events, and does not
perform analysis for them.
The implementation is naturally parallel because application threads running
in parallel perform analysis. The implementation uses fine-grained
synchronization on metadata to ensure atomicity of the analysis for an event.
For WDP analysis, to obtain an approximation of $<_{\textsc{$\mathit{tr}$}}$
(needed by vindication; see line 44 of Algorithm 4 in Appendix B), the
implementation assigns each event node in the constraint graph a Lamport
timestamp (Lamport, 1978) that respects HB order:
$e\prec_{\textsc{\tiny{HB}}}e^{\prime}\implies\mathit{ts}(e)<\mathit{ts}(e^{\prime})$.
##### Handling races.
To keep finding real races after the first detected race, whenever an analysis
detects a race, it updates vector clocks (and WDP’s constraint graph) so that
the execution so far is race free. SDP and WDP analyses treat racing accesses
as though minimal critical sections on the same lock protected them, as
described in Section 5. HB, WCP, and DC analyses handle detected races by
adding ordering between all accesses.
If an analysis detects multiple races involving the current access, it reports
only one of the races but adds ordering to eliminate all of the races.
##### Vindication.
WDP analysis constructs a constraint graph representing the observed
execution’s WDP ordering. When the execution completes, the implementation
calls VindicateWDPRace on a configurable subset of the WDP-races, e.g., each
WDP-race that is not also a SDP-race.
### 7.2. Methodology
The experiments execute large, real Java programs harnessed as the DaCapo
benchmarks (Blackburn et al., 2006), version 9.12-bach. We use a version of
the DaCapo programs that the RoadRunner authors have modified to work with
RoadRunner;888https://github.com/stephenfreund/RoadRunner/releases/tag/v0.5
the resulting workloads are approximately equal to DaCapo’s default workload.
The experiments exclude DaCapo programs eclipse, tradebeans, and tradesoap,
which the RoadRunner authors have not modified to run with RoadRunner; jython,
which failed to run with RoadRunner in our environment; and the single-
threaded program fop.
The experiments execute on a quiet Intel Xeon E5-4620 with four 8-core
processors with hyperthreading disabled and 256 GB of main memory, running
Linux 3.10.0. We execute RoadRunner with the HotSpot 1.8.0 JVM and set the
maximum heap size to 245 GB.
We run various combinations of the analyses to collect race results and
statistics and measure performance. To account for run-to-run variation, each
reported result is the mean of five trials.
Each WDP-race in an execution is a _dynamic_ WDP-race (similarly for SDP-,
DC-, WCP-, and HB-races). Among dynamic WDP-races, some may be detected at the
same static accesses. If two dynamic WDP-races have the same two static source
location regardless of order, then they are the same _static_ WDP-race
(similarly for SDP-, DC-, WCP-, and HB-races).
### 7.3. Dynamic Characteristics
Table 5 shows properties of the analyzed programs. The _#Thr_ column reports
total threads created by an execution and, in parentheses, threads active at
termination. The rest of the columns count events from WDP analysis; other
analyses are similar but exclude branch events. _Total events_ are all
executed events instrumented by the analysis.
| | | Total | Analyzed events
---|---|---|---|---
| #Thr | events | All ( | acq/rel | wr | rd | br )
avrora | 7 | (7) | 2,400 M | 260 M ( | 1.2% | 17.7% | 42.2% | 38.4% )
batik | 7 | (7) | 490 M | 17 M ( | 0.6% | 26.3% | 38.4% | 34.0% )
h2 | 34 | (33) | 9,368 M | 768 M ( | 0.5% | 17.1% | 43.1% | 39.1% )
luindex | 3 | (3) | 910 M | 72 M ( | 0.6% | 20.1% | 42.4% | 36.9% )
lusearch | 34 | (34) | 2,746 M | 301 M ( | 0.9% | 19.5% | 43.6% | 35.6% )
pmd | 33 | (33) | 403 M | 41 M ( | $<\,$0.1% | 28.5% | 37.3% | 34.2% )
sunflow | 65 | (33) | 14,452 M | 887 M ( | $<\,$0.1% | 44.7% | 41.4% | 13.8% )
tomcat | 106 | (67) | 113 M | 29 M ( | 2.8% | 18.7% | 42.1% | 36.1% )
xalan | 33 | (33) | 1,306 M | 436 M ( | 2.1% | 12.0% | 48.8% | 37.1% )
Table 5. Dynamic characteristics of the analyzed programs. Event counts (shown
in millions) and percentages are collected from WDP analysis; other analyses
do not analyze branch events.
_Analyzed events_ are the events _not_ filtered by the fast path that detects
redundant events. The rest of the columns show the breakdown of analyzed
events by event type. The percentages do not add up to 100% because they do
not include other events (e.g., fork, join, wait, volatile access, and static
class initializer events), which are always less than 1% of analyzed events.
Unsurprisingly, most analyzed events are memory accesses or branches.
### 7.4. Race Detection Effectiveness
Table 6 reports detected races for two different experiments that each run a
combination of analyses on the same executions. Table 6(a)’s results are from
an experiment that runs HB, WCP, and SDP analyses together on the same
executions, to compare these analyses’ race detection capabilities directly.
Likewise, a separate experiment runs DC and WDP analyses together on the same
executions to make them directly comparable, resulting in Table 6(b)’s
results.
For each race count, the first value is static races, followed by dynamic
races in parentheses. For example, on average over the five trials, the
analysis detects about 406,000 WDP-races for avrora, which each correspond to
one of 5 different unordered pairs of static program locations.
Program | HB-races | WCP-races | SDP-races | |
---|---|---|---|---|---
avrora | 5 | (205 K) | 5 | (206 K) | 5 | (206 K) | |
batik | 0 | (0) | 0 | (0) | 0 | (0) | |
h2 | 9 | (52 K) | 9 | (52 K) | 9 | (53 K) | |
luindex | 1 | (1) | 1 | (1) | 1 | (1) | |
lusearch | 0 | (0) | 0 | (0) | 0 | (0) | |
pmd | 6 | (351) | 6 | (354) | 8 | (562) | |
sunflow | 2 | (19) | 2 | (25) | 2 | (25) | |
tomcat | 85 | (34 K) | 86 | (34 K) | 91 | (38 K) | |
xalan | 6 | (203) | 21 | (520 K) | 52 | (2.2 M) | |
((a)) HB, WCP, and SDP analyses on the same executions.
Program | | | | DC-races | WDP-races
---|---|---|---|---|---
avrora | | | | 5 | (203 K) | 5 | (406 K)
batik | | | | 0 | (0) | 0 | (0)
h2 | | | | 11 | (54 K) | 12 | (63 K)
luindex | | | | 1 | (1) | 1 | (1)
lusearch | | | | 0 | (0) | 1 | (30)
pmd | | | | 9 | (2 K) | 10 | (3 K)
sunflow | | | | 2 | (49) | 2 | (100)
tomcat | | | | 94 | (36 K) | 284 | (125 K)
xalan | | | | 17 | (649 K) | 170 | (15 M)
((b)) DC and WDP analyses on the same executions.
Table 6. Static and dynamic (in parentheses) race counts from two different
experiments.
Table 6(a) shows that SDP analysis finds significantly more races than not
only HB analysis but also WCP analysis—the state of the art in unbounded sound
predictive race detection (Flanagan and Freund, 2017; Kini et al., 2017).
These additional races are due to SDP incorporating data dependence more
precisely than WCP by not ordering write–write conflicting critical sections,
essentially permitting predictable traces that swap writes without changing a
causal read’s last writer.
Likewise, Table 6(b) shows that WDP analysis finds more races than DC
analysis, the state of the art in high-coverage unbounded predictive race
detection (Roemer et al., 2018). These additional races result from WDP being
more precise with respect to both data and control dependence than DC, and in
fact being complete.
The counts of HB-, WCP-, and DC-races we report here are significantly
different from those reported by the Vindicator paper (Roemer et al., 2018).
(While the counts are not directly comparable, both papers show similar trends
between relations.) The most significant cause of this effect is that
RoadRunner stops tracking a field after the field has 100 races, a behavior
that Vindicator used but that we disabled for these results to avoid
artificially underreporting race counts. Furthermore, our analyses do not use
a Vindicator optimization that merges events, reducing the number of races
reported when there are multiple races between synchronization-free regions.
We disabled this optimization because WDP analysis must track variable access
information for each event, negating the advantages of this optimization.
Another difference is that the Vindicator experiments spawned fewer threads
for some benchmarks by setting the number of available cores to 8.
Table 8 reports results from an experiment that runs SDP and WDP analyses
together and then performs VindicateWDPRace on _WDP-only races_ , which are
WDP-races that are not also SDP-races. The _SDP-races_ and _WDP-races_ columns
report static and dynamic races, as in Table 6. The _WDP-only_ column is
_static_ WDP-only races, which are static WDP-races that have no dynamic
instances that are SDP-races. The last column, _WDP-only $\rightarrow$
Verified_, reports how many static WDP-only races are detected and how many
are successfully vindicated as true races by VindicateWDPRace. In this
experiment, the implementation tries to vindicate up to 10 dynamic instances
of each static WDP-only race. The implementation first attempts to vindicate
the five earliest dynamic instances of a static WDP-only race, then five
random dynamic instances, stopping as soon as it verifies any dynamic instance
of the static race.
Program | SDP-races | WDP-races | WDP-only$\;\;\rightarrow\;\;$ | Verified
---|---|---|---|---
avrora | 5 | (202 K) | 5 | (407 K) | 0$\;\;{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\rightarrow}\;\;$ |
batik | 0 | (0) | 0 | (0) | 0$\;\;{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\rightarrow}\;\;$ |
h2 | 12 | (53 K) | 13 | (63 K) | 1$\;\;\rightarrow\;\;$ | 0
luindex | 1 | (1) | 1 | (1) | 0$\;\;{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\rightarrow}\;\;$ |
lusearch | 0 | (0) | 1 | (30) | 1$\;\;\rightarrow\;\;$ | 0
pmd | 9 | (456) | 10 | (3 K) | 1$\;\;\rightarrow\;\;$ | 1
sunflow | 2 | (32) | 2 | (100) | 0$\;\;{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\rightarrow}\;\;$ |
tomcat | 98 | (37 K) | 334 | (128 K) | 236$\;\;\rightarrow\;\;$ | 60
xalan | 31 | (1.7 M) | 170 | (15 M) | 139$\;\;\rightarrow\;\;$ | 137
Table 7. Static and dynamic (in parentheses) race counts from an experiment
running SDP and WDP analyses together and vindicating dynamic instances of
static WDP-only races. The _WDP-only $\rightarrow$ Verified_ column reports
static WDP-only races, followed by how many static WDP-only races were
verified as predictable races by VindicateWDPRace.
| Mean$\;\;\pm\;$ | Stdev | Max
---|---|---|---
pmd | 24,200$\;\;\pm\;$ | 14,100 | 40,294
tomcat | 4,830,000$\;\;\pm\;$ | 5,540,000 | 28,734,020
xalan | 52,100$\;\;\pm\;$ | 90,500 | 751,701
Table 8. Characteristics of the distribution of event distances of WDP-only
races that are verified predictable races. The table rounds the mean and
standard deviation to three significant digits.
Around half of the static WDP-only races are verified predictable races: out
of 378 static WDP-only races on average, 198 are verified predictable races.
As Section 7.5 shows, it can take a few minutes for VindicateWDPRace to check
a WDP-race. Given the difficulty and importance of detecting unknown, hard-to-
expose data races in mature software—and the amount of time developers
currently spend on testing and debugging—the time for VindicateWDPRace is
reasonable.
We confirmed that in all of the experiments, SDP analysis detected every race
detected by WCP analysis, and WDP analysis detected every race detected by DC
or SDP analysis.
##### Race characteristics.
SMT-solver-based predictive race detectors can be as precise as SDP and WDP
analyses, but cannot scale to unbounded program executions (Huang et al.,
2014; Said et al., 2011; Huang and Rajagopalan, 2016; Chen et al., 2008;
Şerbănuţă et al., 2013; Liu et al., 2016) (Section 8). These approaches
typically analyze bounded windows of an execution trace, missing races
involving “far apart” events. We can estimate whether SMT-based approaches
would miss a predictable race by computing the race’s _event distance_ , which
is the number of events in the execution trace between the race’s two events.
Since our implementation does not compute a total order of events, it
approximates event distance using Lamport timestamps: event distance is the
number of events $e$ such that
$\mathit{ts}(e_{1})<\mathit{ts}(e)<\mathit{ts}(e_{2})$.
Table 8 reports the distribution of event distances between accesses in each
successfully vindicated WDP-only race (i.e., the last column of Table 8). The
average distance and standard deviation are across all trials. The last column
reports the greatest distance found among all trials.
### 7.5. Performance
Table 9 reports the run-time performance of various combinations of analyses.
_Base_ is native execution time without any instrumentation. Other columns
(excluding _Failed_ and _Verified_) are slowdowns relative to _Base_.
The _Instr. only_ columns are RoadRunner configurations that instrument events
(excluding or including branches) but perform no analysis in the
instrumentation.
The _Analyses w/o constraint graph_ show configurations that do not construct
a constraint graph. Only configurations including WDP analysis instrument
branch events. The _WCP_ and _SDP_ columns show the slowdowns from running the
WCP and SDP analyses independently; the performance difference between them is
modest, suggesting that there is no significant performance penalty from using
SDP analysis over WCP analysis. (SDP’s performance improvement over WCP for h2
is not statistically significant, according to confidence intervals in Table
14 in Appendix C.)
| | Instr. only | Analyses w/o constraint graph | SDP+WDP+graph
---|---|---|---|---
| Base | w/o br | w/ br | WCP | SDP | SDP+DC | SDP+WDP | Slowdown | Failed | Verified
avrora | 6.0 s | 2.7$\;\\!\times$ | 3.4$\;\\!\times$ | 21$\;\\!\times$ | 23$\;\\!\times$ | 32$\;\\!\times$ | 38$\;\\!\times$ | 50$\;\\!\times$ | - | -
batik | 4.2 s | 3.2$\;\\!\times$ | 4.4$\;\\!\times$ | 14$\;\\!\times$ | 14$\;\\!\times$ | 16$\;\\!\times$ | 22$\;\\!\times$ | 25$\;\\!\times$ | - | -
h2 | 9.0 s | 6.7$\;\\!\times$ | 9.4$\;\\!\times$ | 125$\;\\!\times$ | 135$\;\\!\times$ | 213$\;\\!\times$ | 208$\;\\!\times$ | 241$\;\\!\times$ | 386 s | -
luindex | 1.6 s | 5.0$\;\\!\times$ | 9.5$\;\\!\times$ | 66$\;\\!\times$ | 69$\;\\!\times$ | 82$\;\\!\times$ | 100$\;\\!\times$ | 120$\;\\!\times$ | - | -
lusearch | 4.1 s | 3.8$\;\\!\times$ | 4.5$\;\\!\times$ | 17$\;\\!\times$ | 17$\;\\!\times$ | 21$\;\\!\times$ | 23$\;\\!\times$ | 32$\;\\!\times$ | ¡ 0.1 s | -
pmd | 3.0 s | 6.4$\;\\!\times$ | 8.7$\;\\!\times$ | 21$\;\\!\times$ | 22$\;\\!\times$ | 23$\;\\!\times$ | 27$\;\\!\times$ | 29$\;\\!\times$ | - | 0.9 s
sunflow | 2.8 s | 8.6$\;\\!\times$ | 12$\;\\!\times$ | 104$\;\\!\times$ | 106$\;\\!\times$ | 116$\;\\!\times$ | 124$\;\\!\times$ | 189$\;\\!\times$ | - | -
tomcat | 1.9 s | 5.1$\;\\!\times$ | 5.4$\;\\!\times$ | 23$\;\\!\times$ | 22$\;\\!\times$ | 37$\;\\!\times$ | 40$\;\\!\times$ | 43$\;\\!\times$ | 1.8 s | 49 s
xalan | 5.5 s | 2.4$\;\\!\times$ | 3.0$\;\\!\times$ | 34$\;\\!\times$ | 35$\;\\!\times$ | 51$\;\\!\times$ | 66$\;\\!\times$ | 113$\;\\!\times$ | 29 s | 0.2 s
Table 9. Slowdowns of program instrumentation and various analyses over
uninstrumented execution, and the average time taken to vindicate WDP-only
races.
_SDP+DC_ represents performing SDP and DC analyses together. We run DC
analysis with SDP analysis to minimize DC-races that need vindication.
(Vindicator combined DC analysis with WCP analysis for this purpose (Roemer et
al., 2018), but SDP analysis is more powerful.)
_SDP+WDP+graph_ represents the canonical use case for WDP analysis. This
configuration performs SDP and WDP analyses and constructs the constraint
graph to enable vindication. It uses SDP to reduce how many WDP-races need
vindication. For comparison purposes, _SDP+WDP_ forgoes constructing the
constraint graph, showing the cost of constructing the graph, which we have
not optimized. _SDP+WDP_ is slower than _SDP+DC_ because WDP analysis is
generally more complex than DC analysis.
Finally, _Failed_ and _Verified_ are the average times taken for each dynamic
race that VindicateWDPRace fails to verify or successfully verifies,
respectively. Vindication times vary significantly across programs;
vindication is particularly slow for tomcat because most of its racing
accesses are separated by millions of events (Table 8). Vindication is slow
for h2, even though its races are not far apart, because VindicateWDPRace
discovers new critical section constraints that require analyzing over 500
million events.
### 7.6. Summary and Discussion
Our SDP- and WDP-based approaches are slower than other predictive approaches,
but they find more races, some of which are millions of events apart. SMT-
based approaches would not be able to find these far-apart races because they
cannot scale past analyzing bounded windows of executions (Huang et al., 2014;
Said et al., 2011; Huang and Rajagopalan, 2016; Chen et al., 2008; Şerbănuţă
et al., 2013; Liu et al., 2016) (Section 8). Notably, _RVPredict_ , which
(like WDP) incorporates precise control and data dependence, uses an analysis
window of 10,000 events (Huang et al., 2014), meaning it would miss many of
the predictable races detected and verified by our approach.
Our evaluation demonstrates the power of SDP and WDP analyses to find more
races than prior approaches _in a single execution_. A potential limitation of
the evaluation is that it does not compare our analyses with approaches that
perform HB analysis on multiple executions (e.g., using one of the many
schedule exploration approaches; Section 8). Our work’s aim is to push the
limit on what can be found in a single execution, which is essentially
complementary to approaches that explore multiple schedules. No other known
sound technique could have predicted all of these races from the observed
executions.
## 8\. Related Work
This section describes and compares with prior work, starting with the most
closely related work.
##### Unbounded predictive analysis.
Prior work introduces unbounded predictive analyses, weak-causally-precedes
(WCP) and doesn’t-commute (DC) analyses (Kini et al., 2017; Roemer et al.,
2018), which Sections 2 and 7 covered and evaluated in detail. SDP and WDP
analyses predict more races in real programs than WCP and DC analyses,
respectively (Section 7).
The WCP relation is weaker (i.e., detects more races) than Smaragdakis et
al.’s earlier _causally-precedes (CP)_ relation (Smaragdakis et al., 2012).
Smaragdakis et al.’s implementation detects races within bounded windows of
500 events because of the difficulty of developing an efficient unbounded
analysis for CP (Smaragdakis et al., 2012; Roemer and Bond, 2019).
Recent work introduces the _afterward-confirm (AC)_ relation and an approach
called _DigHR_ for computing AC (Luo et al., 2018). AC is the same as _CP_
except that it removes write–write conflicting critical section ordering.
Despite this similarity with our work, the contributions differ significantly,
and the DigHR work has major correctness issues. Foremost, the DigHR paper
claims incorrectly that AC is sound. AC is, to the best of our understanding,
unsound: its removal of write–write ordering leads to detecting false races,
including for the execution in Figure 2(c) (with the br event omitted; DigHR’s
event model does not include branches). The DigHR paper provides a soundness
proof, which we believe is incorrect as a result of informality leading to not
covering cases such as Figure 2(c). In contrast with DigHR, our work
introduces a sound relaxation of WCP (SDP). Additionally, our work introduces
a complete relation (WDP), handles control dependencies (br events), and
presents linear-time analyses for SDP and WDP (DigHR is superlinear, like
existing CP analyses (Smaragdakis et al., 2012; Roemer and Bond, 2019)).
Concurrently with our work, Pavlogiannis introduces a predictive race
detection approach called _M2_ that is related to vindication (Pavlogiannis,
2019). Pavlogiannis uses lockset analysis as an imprecise filter for potential
races checked by M2, while our work introduces linear-time WDP analysis as a
less-imprecise filter for potential races checked by VindicateWDPRace.
Although Pavlogiannis reports performance that is sometimes competitive with
the performance of HB, WCP, and DC analyses, Pavlogiannis’s implementations of
these analyses perform extra passes over execution traces in addition to the
efficient single-pass vector-clock-based analyses from prior work (Kini et
al., 2017; Roemer et al., 2018). It is unclear to us how M2 and
VindicateWDPRace would compare in terms of detection capability (aside from
the fact that only VindicateWDPRace takes branches and control dependence into
account). In addition to these differences, our work incorporates branches and
control dependence sensitivity, while Pavlogiannis’s work does not and thus
would miss races such as Figures 1(b) and 5; and our work introduces a sound
partial order and linear-time analysis (SDP and SDP analysis).
Our concurrent work introduces the _SmartTrack_ algorithm, which optimizes the
performance of WCP and DC analyses (Roemer et al., 2019). SmartTrack’s
optimizations apply to analyses that compute predictive relations that order
all pairs of conflicting accesses—a property that SDP and WDP do not conform
to. In any case, optimizing WDP analysis would have limited benefit because
the analysis still must construct a constraint graph in order to perform
vindication (a necessity considering that so many WDP-races fail vindication
in practice). The SmartTrack paper also introduces a new relation _weak-
doesn’t-commute (WDC)_ that is a weak variant of DC (Roemer et al., 2019).
Unlike SDP and WDP, WDC does _not_ help find more races than DC, but rather
serves to improve analysis performance.
##### Bounded predictive approaches.
Other approaches predict data races by generating and solving satisfiability
modulo theories (SMT) constraints (Huang et al., 2014; Said et al., 2011;
Huang and Rajagopalan, 2016; Chen et al., 2008; Şerbănuţă et al., 2013; Liu et
al., 2016). These SMT-based approaches cannot analyze long execution traces in
reasonable time, because constraints are quadratic or cubic in trace length,
and constraint-solving time often grows superlinearly with constraints. These
approaches thus break traces into bounded “windows” of traces (e.g.,
500–10,000 events each), missing predictable races involving accesses that
occur further apart in the trace.
One advantage of SMT-based approaches is that they can be both sound and
complete (within a bounded window) by encoding precise constraints. Notably,
_RVPredict_ includes branches in its execution model (Huang et al., 2014).
(RVPredict also incorporates _values_ into its event model, so a read in a
predictable trace can have a different last writer as long as it writes the
same value (Huang et al., 2014).) RVPredict is thus complete by Section 2’s
definition, except that it is practically limited to bounded windows of
execution. WDP analysis, on the other hand, is complete without the windowing
limitation, but VindicateWDPRace is not guaranteed to vindicate a WDP-race
even when a predictable race exists.
##### Schedule exploration.
In contrast to predictive analysis, _schedule exploration_ approaches execute
the program multiple times to explore more program behaviors (Huang, 2015;
Huang and Huang, 2017; Musuvathi and Qadeer, 2007; Burckhardt et al., 2010;
Eslamimehr and Palsberg, 2014; Sen, 2008; Cai and Cao, 2015; Henzinger et al.,
2004). These approaches may be systematic (often called _model checking_) or
be based on randomness or heuristics. Schedule exploration is complementary to
predictive analysis, which aims to glean as much as possible from a given
execution. _Maximal causality reduction (MCR)_ combines schedule exploration
with predictive analysis (Huang, 2015; Huang and Huang, 2017). _MCR-S_
incorporates static control flow information to reduce the complexity of MCR’s
generated SMT constraints (Huang and Huang, 2017).
##### Other analyses.
Widely used data race detectors typically use dynamic _happens-before (HB)_
analysis (Lamport, 1978; Flanagan and Freund, 2009; Pozniansky and Schuster,
2007; Elmas et al., 2007; Serebryany and Iskhodzhanov, 2009; Serebryany et
al., 2012; Intel Corporation, 2016). HB analysis cannot predict races
involving reordered critical sections on the same lock. The detected races
thus depend heavily on the scheduling of the analyzed program. Other analyses
find a subset of HB-races by detecting simultaneously executing conflicting
accesses or regions (Veeraraghavan et al., 2011; Biswas et al., 2015;
Effinger-Dean et al., 2012; Erickson et al., 2010; Sen, 2008; Biswas et al.,
2017).
_Lockset_ analysis checks a locking discipline, ensuring that all pairs of
conflicting accesses hold some common lock (Savage et al., 1997; Nishiyama,
2004; Choi et al., 2002; von Praun and Gross, 2001). Lockset analysis is
predictive but unsound, reporting false races for synchronization patterns
other than the locking discipline. _Hybrid_ lockset–HB analyses generally
incur disadvantages of one or both kinds of analysis (Dinning and Schonberg,
1991; O’Callahan and Choi, 2003; Yu et al., 2005; Pozniansky and Schuster,
2007).
_Sampling-based_ analysis trades coverage for performance (opposite of
predictive analysis) in order to detect data races in production (Bond et al.,
2010; Erickson et al., 2010; Marino et al., 2009; Kasikci et al., 2013; Biswas
et al., 2017; Sheng et al., 2011; Zhang et al., 2017). Custom hardware support
can detect data races with low performance cost but has not been implemented
(Devietti et al., 2012; Zhou et al., 2007; Wood et al., 2014; Segulja and
Abdelrahman, 2015; Peng et al., 2017; Lucia et al., 2010; Marino et al., 2010;
Singh et al., 2011).
Dynamic analysis can estimate the likely harm of a data race (Boehm, 2011;
Kasikci et al., 2015; Narayanasamy et al., 2007; Cao et al., 2016; Flanagan
and Freund, 2010a; Burnim et al., 2011), which is orthogonal to detection. All
data races are erroneous under language memory models that ascribe them
undefined semantics (Adve and Boehm, 2010; Boehm and Adve, 2008; Manson et
al., 2005; Boehm and Demsky, 2014; Boehm, 2012; Boehm and Adve, 2012; Ševčík
and Aspinall, 2008). Java’s memory model defines weak semantics for data races
(Manson et al., 2005), but inadvertently prohibits common JVM optimizations
(Ševčík and Aspinall, 2008; Boehm and Demsky, 2014).
_Static_ program analysis can detect all races across all feasible executions
of a program (von Praun and Gross, 2003; Naik et al., 2006; Naik and Aiken,
2007; Pratikakis et al., 2006; Engler and Ashcraft, 2003; Voung et al., 2007),
but it reports thousands of false races for real programs (Biswas et al.,
2017; Lee et al., 2012).
##### Avoiding or tolerating data races.
New languages and type systems can ensure data race freedom, but require
significant programmer effort (Bocchino et al., 2009; Boyapati et al., 2002;
Rinard and Lam, 1998; Matsakis and Klock, 2014; Abadi et al., 2006; Flanagan
and Freund, 2007). Compilers and hardware can provide well-defined semantics
for data races, but incur high run-time costs or hardware complexity (Segulja
and Abdelrahman, 2015; Lucia et al., 2010; Marino et al., 2010; Singh et al.,
2011; Marino et al., 2011; Singh et al., 2012; Sura et al., 2005; Sengupta et
al., 2015; Ahn et al., 2009; Ouyang et al., 2013).
## 9\. Conclusion
SDP and WDP analyses improve over existing predictive analyses by
incorporating precise notions of data and control dependence, finding more
races both in theory and in practice while retaining linear (in trace length)
run time and thus unbounded analysis. SDP analysis maintains WCP analysis’s
soundness while increasing race coverage. WDP analysis finds all data races
that can be predicted from an observed execution; not all WDP-races are
predictable races, but VindicateWDPRace can efficiently filter false races.
Experiments show that our new approaches find many predictable races in real
programs that prior approaches are unable to find. These properties and
results suggest that our contributions advance the state of the art in
predictive race detection analysis.
## Acknowledgments
We thank Rob LaTour for early help with this project. Thanks to Steve Freund
for making RoadRunner publicly available and providing help with using and
modifying it. Thanks to the anonymous reviewers for their thorough and
insightful feedback.
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## Appendix A Proof of WDP Completeness Helper Lemma
###### Proof of Lemma 4.2.
We proceed by induction on the _WDP-distance_ of two events,
$\mathit{d(e,e^{\prime})}$, defined as follows (the WDP properties refer to
Table 2):
$\displaystyle\mathit{d(e,e^{\prime})}=\min\begin{cases}0&\textnormal{ if
}e\prec_{\textsc{\tiny{WDP}}}e^{\prime}\textnormal{ by WDP conflicting
critical section ordering}\\\
1+\mathit{d(\mathit{A}(e),e^{\prime})}&\textnormal{ if
}e\prec_{\textsc{\tiny{WDP}}}e^{\prime}\textnormal{ by
``$\textsf{acq(m)}\prec_{\textsc{\tiny{WDP}}}\textsf{rel(m)}\implies\textsf{rel(m)}\prec_{\textsc{\tiny{WDP}}}\textsf{rel(m)}$''}\\\
0&\textnormal{ if }e\prec_{\textsc{\tiny{WDP}}}e^{\prime}\textnormal{ by
PO}\\\ \lx@intercol
1+\min_{e^{\prime\prime}}(\mathit{d(e,e^{\prime\prime})}+\mathit{d(e^{\prime\prime},e^{\prime})})\hfil\lx@intercol\\\
&\textnormal{ if }\exists
e^{\prime\prime}:e\prec_{\textsc{\tiny{WDP}}}e^{\prime\prime}\land
e^{\prime\prime}\prec_{\textsc{\tiny{WDP}}}e^{\prime}\textnormal{ by WDP
transitivity}\\\ \infty&\textnormal{ otherwise}\end{cases}$
##### Base case
Let $e_{1}$ and $e_{2}$ be events in a trace $\mathit{tr}$ such that
$\mathit{d(e_{1},e_{2})}=0$ and $e_{1}\prec_{\textsc{\tiny{WDP}}}e_{2}$. Since
$\mathit{d(e_{1},e_{2})}=0$, $e_{1}$ and $e_{2}$ are ordered directly, using
PO or WDP’s conflicting critical section ordering:
If $e_{1}\prec_{\textsc{\tiny{WDP}}}e_{2}$ by PO, then
$e_{1}\prec_{\textsc{\tiny{PO}}}e_{2}$. Let $\mathit{tr^{\prime}}$ be a
predictable trace of $\mathit{tr}$ in which
$e_{2}<_{\textsc{$\mathit{tr^{\prime}}$}}e_{1}$ or
$e_{2}\in\mathit{tr^{\prime}}\land e_{1}\notin\mathit{tr^{\prime}}$, either of
which violates the PO rule of a predictable traces.
If $e_{1}\prec_{\textsc{\tiny{WDP}}}e_{2}$ by WDP’s conflicting critical
section ordering, then $e_{1}$ is a release event, $e_{2}$ is a branch event,
there exists a release event $r_{2}$ over the same lock as $e_{1}$, and there
exists write event $e$ and read event $e^{\prime}$ such that
$e\in\mathit{CS(}e_{1})$, $e^{\prime}\in\mathit{CS(}r_{2})$, $e\asymp
e^{\prime}$, $\mathit{lastwr}_{\mathit{tr}}(e^{\prime})=e$, and
$\mathit{brDepsOn}(e_{2},e^{\prime})$. Let $\mathit{tr^{\prime}}$ be a
predictable trace of $\mathit{tr}$ where
$e_{2}<_{\textsc{$\mathit{tr^{\prime}}$}}e_{1}$ or
$e_{2}\in\mathit{tr^{\prime}}\land e_{1}\notin\mathit{tr^{\prime}}$. Then in
either case, $r_{2}<_{\textsc{$\mathit{tr^{\prime}}$}}a$ or
$r_{2}\in\mathit{tr^{\prime}}\land a\notin\mathit{tr^{\prime}}$, where $a$ is
the matching acquire of $e_{1}$; otherwise $\mathit{tr^{\prime}}$ would be an
invalid predictable trace due to the LS rule of predictable traces. As a
result, $e^{\prime}<_{\textsc{$\mathit{tr^{\prime}}$}}e$ or
$e^{\prime}\in\mathit{tr^{\prime}}\land e\notin\mathit{tr^{\prime}}$, which
means
$\mathit{lastwr}_{\mathit{tr^{\prime}}}(e^{\prime})\neq\mathit{lastwr}_{\mathit{tr}}(e^{\prime})$.
|
# Imperfect Narrow Escape problem
T. Guérin Laboratoire Ondes et Matière d’Aquitaine, CNRS, UMR 5798,
Université de Bordeaux, F-33400 Talence, France M. Dolgushev O. Bénichou
Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière
Condensée, LPTMC, F-75005 Paris, France R. Voituriez Sorbonne Université,
CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMC,
F-75005 Paris, France Sorbonne Université, CNRS, Laboratoire Jean Perrin,
LJP, F-75005 Paris, France
###### Abstract
We consider the kinetics of the imperfect narrow escape problem, i.e. the time
it takes for a particle diffusing in a confined medium of generic shape to
reach and to be adsorbed by a small, imperfectly reactive patch embedded in
the boundary of the domain, in two or three dimensions. Imperfect reactivity
is modeled by an intrinsic surface reactivity $\kappa$ of the patch, giving
rise to Robin boundary conditions. We present a formalism to calculate the
exact asymptotics of the mean reaction time in the limit of large volume of
the confining domain. We obtain exact explicit results in the two limits of
large and small reactivities of the reactive patch, and a semi-analytical
expression in the general case. Our approach reveals an anomalous scaling of
the mean reaction time as the inverse square root of the reactivity in the
large reactivity limit, valid for an initial position near the extremity of
the reactive patch. We compare our exact results with those obtained within
the “constant flux approximation”; we show that this approximation turns out
to give exactly the next-to-leading order term of the small reactivity limit,
and provides a good approximation of the reaction time far from the reactive
patch for all reactivities, but not in the vicinity of the boundary of the
reactive patch due to the above mentioned anomalous scaling. These results
thus provide a general framework to quantify the mean reaction times for the
imperfect narrow escape problem.
## I Introduction
How much time does it take for a random walker to reach a target point? The
answer to this question has received a lot of attention in the last decade in
the physics literature [1, 2, 3, 4, 5, 6, 7, 8, 9]. First passage problems
appear in various areas of biological and soft matter physics and are in
particular relevant to the problem of reaction kinetics, since two reactants
have to meet before being able to react [10, 11]. When the reaction is
“perfect”, i.e. when it occurs instantaneously upon each encounter, its
kinetics is controlled by the first passage statistics of one reactant
molecule, seen as a random walker, to the second reactant, seen as a “target”.
However, many reactions do not occur at first contact between the random
walker and the targets, leading to imperfect reactivity. Imperfect reactivity
can have diverse origins at the microscopic scale, such as orientational
constraints on the reactive particles [11], the fact that the surface of the
reactive particles is not entirely covered by reactive patches (such as in the
chemoreception problem [12]), the need to overcome an energetic [13] (or
entropic [14]) activation barrier before reaction, the presence of a gate that
can be randomly closed or opened when the reactant meets the target [15, 16],
etc (see Ref. [17] and references therein for a recent review on imperfect
reactivity).
Imperfect reactivity was early investigated for molecules diffusing in
infinite space [18, 19, 11, 20] (with an imposed concentration at infinity).
The search problem for a single random walker moving in a confining volume for
an imperfect target, initially considered in Ref. [21] for centered spheres,
has also attracted recent attention and several asymptotic results for
imperfect search kinetics have been derived [22, 23, 24, 25, 26, 27, 28, 29,
30]. Recently, explicit asymptotics of the reaction time statistics have been
obtained for general Markovian random walks [31]. Besides the case of reactive
targets located in the bulk of a confining domain, the narrow escape problem
(NEP) consists in calculating the escape time of a random walker out of a
confining domain, through a small window at the boundary of the domain (see
Figure 1(a)). While the NEP is now well characterized for perfect reactions,
for spherical domains [8, 32, 33, 34, 35] and large domains of arbitrary
shapes [36], fewer results are available for imperfect reactions (i.e. for a
partially adsorbing patch). The imperfect narrow escape problem has been
investigated for particular geometries in cylindrical [37, 38] or spherical
domains [39][40] in which case the analysis depends on the eigenfunctions of
the particular confining volume that is considered and relies on the so-called
uniform flux approximation introduced in Ref. [20].
Figure 1: (a) Illustration of the imperfect narrow escape problem. A
partially reactive patch (thick red line) is embedded in the boundary of a
confining domain. A random walker, starting from the initial position (red
sphere) diffuses in the domain and is eventually adsorbed on the patch. (b)
Zoom on the portion of space delimited by the dashed blue lines around the
reactive patch.
The aim of the present paper is to apply the formalism introduced in Ref. [31]
to cover the case of the imperfect narrow escape problem in a domain of
generic shape. Our formalism is asymptotically exact in the limit of large
confining volume – it does not involve the constant flux approximation – and
provides explicit results in both regimes of small and large reactivity. Of
particular interest for imperfect reaction problems is the mean reaction time
when the initial position is located on the reactive patch; this time is
exactly zero for perfect reactions and scales as $1/\kappa$ for targets in the
interior of the volume. We identify a region for which the reaction time
behaves anomalously with the reactivity $\kappa$. This region, which does not
exist for targets in the bulk of the confining domain, is located at the
boundary of the imperfectly reactive patch. While one would naively expect
this time to be inversely proportional to $\kappa$, we find instead that when
the initial position is at the boundary of the reactive domain, the mean
reaction time $\langle T\rangle_{e}$ is actually anomalously high
($\propto\kappa^{-1/2}$) and follows the exact asymptotics
$\displaystyle\langle
T\rangle_{e}\underset{\kappa\to\infty}{\sim}\frac{V}{(d-1)(2\pi\ \kappa
Da)^{1/2}a^{d-2}}.$ (1)
Here, $d=2$ or $d=3$ is the spatial dimension, $D$ is the diffusion
coefficient, $a$ is the radius of the reactive patch, and $V$ the volume of
the confining domain. Here, we assume that the confining volume is taken large
enough, and the patch small enough, so that the confining boundaries at the
vicinity of the target can be considered as a flat wall in which the reactive
patch is embedded, the latter being considered as a line segment of length
$2a$ in $d=2$ or as a flat disk of radius $a$ in $d=3$. We show below how to
obtain this anomalous scaling relation by solving a Wiener-Hopf integral
equation. We will also show how this “anomalous” behavior (1) of $\langle
T\rangle$ with $\kappa$ can be related to the divergence of fields in
Laplacian problems near surfaces presenting asperities, as occurs in
electrostatics near conducting edges [41] or in the coffee ring effect [42].
More generally, we show that the mean escape time for an arbitrary initial
position far from the target, and for any finite reactivity $\kappa$ satisfies
the following exact asymptotics:
$\displaystyle\langle T\rangle/V\underset{r\gg
a}{\sim}\begin{cases}\frac{1}{\pi D}\ln(r/a)+C_{\infty}&(d=2)\\\
-\frac{1}{2\pi Dr}+C_{\infty}&(d=3)\end{cases},$ (2)
where $C_{\infty}$ is independent of the initial distance $r$ from the target.
For finite values of the reactivity $\kappa$, we show that $C_{\infty}$ can be
obtained through a semi analytical procedure. In the limit of large
reactivity, we show that $C_{\infty}$ can be determined explicitly and is
given by :
$\displaystyle C_{\infty}\underset{\kappa a\gg D}{\sim}\begin{cases}\frac{\ln
2}{\pi D}+\frac{1}{\pi^{2}\kappa a}\left(\ln\frac{8\kappa
a}{D}+\gamma_{e}+1\right)&(d=2)\\\ \frac{1}{4Da}+\frac{1}{4\pi\kappa
a^{2}}\left(\ln\frac{2\kappa a}{D}+\gamma_{e}+1\right)&(d=3)\end{cases}$ (3)
where $\gamma_{e}$ is Euler’s constant. This expression is understood as the
first two terms in the expansion of $C_{\infty}$ in powers of $1/\kappa$.
Interestingly, this result shows that the term $C_{\infty}$ is not analytic in
powers of $\kappa$, which originates from the anomalous scaling (1). Finally,
we also give exact results in the small reactivity limit, which will be found
to be exactly the same (at first order) as the results obtained within the
self-consistent, “constant flux approximation” that has been invoked to study
the imperfect narrow escape in the literature [20, 39, 40, 37]. It is found
that, far from the reactive patch this approximation is very accurate (for any
reactivity), while it fails for initial positions close to the reactive patch,
and in particular does not predict the behavior (1) near the boundary of the
patch.
The outline of the paper is as follows. First, we recall the formalism of Ref.
[31] in the particular case of the imperfect narrow escape problem for
diffusing particles to obtain equations for the mean escape time in the large
volume limit (Section II.1). We show how the formalism can be presented under
the form of an integral equation that is suitable for studying the large and
small reactivity limits in Section II.2. The large reactivity limit is
investigated in Section III where Eqs. (1) and (3) are derived. In this
Section, we also give a simple scaling argument that relates the anomalous
behavior of $\langle T\rangle$ near the extremity of the patch to the
divergence of electric fields near the edges of conducting objects. The small
reactivity limit is examined in Section IV. Last, we study briefly how the
constant flux approximation can be implemented within our formalism in Section
V. An exact, but formal solution for any reactivity parameter (that requires
numerical tools, however) is presented in Appendix A.
## II Formalism for the imperfect narrow escape problem in the large volume
limit
### II.1 General formalism
We consider the stochastic motion of an overdamped particle moving with
diffusion coefficient $D$ in a confining volume $\Omega$. The boundary of the
volume is $\partial\Omega$ and contains a small window $S_{r}$ which is
partially reactive, the rest of the confining boundary is assumed to be smooth
and perfectly reflecting, see Fig. 1(a). We assume that $S_{r}$ is formed by
the region of the surface at geodesic distance less than $a$ from the center,
and $a$ is called the radius of the patch $S_{r}$. The Fokker-Planck equation
for the probability density $p(\mathbf{r},t)$ to observe the particle at
position $\mathbf{r}$ and time $t$ is
$\displaystyle\partial_{t}p=D\ \nabla^{2}p$
$\displaystyle(\mathbf{r}\in\Omega),$ (4) $\displaystyle\mathbf{n}\cdot\nabla
p=0$ $\displaystyle(\mathbf{r}\in\partial\Omega\backslash S_{r}),$ (5)
$\displaystyle D\ \mathbf{n}\cdot\nabla p+\kappa\ p=0$
$\displaystyle(\mathbf{r}\in S_{r}).$ (6)
where $\mathbf{n}$ is the unit vector normal to the surface, pointing to the
exterior of the volume. For a partially reactive surface, the reactivity
parameter $\kappa$ is defined in such a way that the probability that the
particle is absorbed by an infinitesimal surface element $dS$ located around
$\mathbf{r}_{s}$ during $dt$ is $\kappa\ p(\mathbf{r}_{s},t)dSdt$. We assume
that the space dimension is $d=2$ (2D) or $d=3$ (3D). It is very well known
that an equation for the mean first passage time can be obtained by
identifying the adjoint transport operator [1], which in our case leads to the
following equation for the mean reaction time $\langle T\rangle(\mathbf{r})$
to the target, where $\mathbf{r}$ now represents the initial position of the
particle:
$\displaystyle D\nabla^{2}\langle T\rangle=-1$
$\displaystyle(\mathbf{r}\in\Omega),$ (7)
$\displaystyle\mathbf{n}\cdot\nabla\langle T\rangle=0$
$\displaystyle(\mathbf{r}\in\partial\Omega\backslash S_{r}),$ (8)
$\displaystyle D\ \mathbf{n}\cdot\nabla\langle T\rangle+\kappa\langle
T\rangle=0$ $\displaystyle(\mathbf{r}\in S_{r}).$ (9)
Integrating Eq. (7) over the whole volume, and using the divergence formula
and the boundary conditions leads to the exact integral relation:
$\displaystyle\kappa\int_{S_{r}}dS(\mathbf{r})\langle T\rangle=V,$ (10)
where $V=|\Omega|$ is the volume of the domain. Now, we consider the large
volume limit, which is obtained when the confining volume extends without
changing its shape, keeping constant the size of the target and the initial
distance to the target. We define the rescaled mean escape time $\Phi$ by
$\displaystyle\Phi(\mathbf{r})=\lim_{V\to\infty}\langle
T(\mathbf{r})\rangle/V.$ (11)
In the large volume limit, the boundary at the vicinity of the reactive target
becomes increasingly similar to a flat surface in which the reactive patch is
a flat disk of radius $a$ in 3D (or a flat segment of length $2a$ in 2D).
Here, we denote the distance to the reflecting surface containing reactive
patch as $z$, see Fig. 1(b). With this in mind, inserting the ansatz (11) into
the above equations yields a closed system of equations in the large volume
limit:
$\displaystyle\nabla^{2}\Phi=0\ \ (\text{if }z>0),$ (12)
$\displaystyle\kappa\int_{S_{r}}dS(\mathbf{r})\Phi=1,$ (13) $\displaystyle
D\partial_{z}\Phi=\begin{cases}0&(\text{if }\ z=0,|\mathbf{r}|>a),\\\ \kappa\
\Phi&(\text{if }\ z=0,|\mathbf{r}|<a).\end{cases}$ (14)
Importantly, we see that in the large volume limit, Eq. (11), the obtained
equations are independent of the shape of the confining volume, which is
present only though the scale factor $V$ in the definition of $\Phi$. We have
directly controlled this aspect by performing numerical stochastic simulations
of trajectories in the confined domain. The results of such simulations are
shown on Fig. 2 and confirm that our formalism correctly predicts the mean
reaction time in the large volume limit, independently on the shape of the
confining domain.
Equations (12), (13), (14) generalize the formalism of Ref. [36] to the case
of imperfect reactions. The fact that the mean first reaction time scales with
the volume is actually more general than the specific diffusive walk that we
have considered here [31]. To solve the above equations, we may be tempted to
use spheroidal coordinates, which can be used to solve the problem for either
infinite or vanishing reactivity. For finite reactivity however, the resulting
equations in such coordinates involve Robin conditions with non-uniform
coefficients, so that the mean reaction time can be obtained only in terms of
the solution of an infinite linear system. This procedure is described in
Appendix A, and it indeed leads to a generic numerical solution that will be
useful to test our analytical insights in all the paper. However, it is not
suitable for analytical calculations. For this reason, we adopt a different
approach, consisting in deriving an integral equation satisfied by $\Phi$ on
the reactive patch.
Figure 2: (a) Geometry of the confining domains (called $A$ and $B$) that are
considered for stochastic simulations. In 2D, these domains are defined in
polar coordinates by $r(\theta)=Rf(\theta)$ with $f=1.6(1+0.5\cos^{2}\theta)$
for domain $A$ and $f=1.6(1+0.1\sin\theta+0.3\sin 3\theta)$ for domain $B$.
Domains in 3D are obtained by considering revolution of 2D curves around the
vertical dashed line. The reactive patch is indicated by a thick red line, and
the initial position is taken at a distance $r$ from the center of the patch
along the black dashed line. In the figure, we have used $R=6a$. (b),(c)
Results of Brownian dynamics simulations for the mean reaction time in 2D/3D
(parameters are indicated in the legend) compared to general theoretical
expressions as obtained in Appendix A. In all simulations, we used a time step
$\Delta t=10^{-4}a^{2}/D$. Boundary conditions are implemented as follows: if,
at the end of a time step, the random walker falls outside the domain, then if
it is “behind” a reflecting wall it is reflected with respect to this wall,
and if it falls “behind” the reactive patch, it is absorbed with probability
$P_{a}=\kappa\sqrt{\pi dt/D}$ (in which case the trajectory ends) and it is
reflected with probability $1-P_{a}$, see Ref. [43].
### II.2 Obtaining an integral equation for the mean reaction time
Let us first characterize the large distance behavior of the rescaled mean
first reaction time $\Phi$. The condition (13), combined with the boundary
condition at $z=0$, implies that
$\displaystyle\int_{S_{0}}dS\ \mathbf{n}\cdot\nabla\Phi=1/D,$ (15)
for any surface $S_{0}$ whose intercept with the plane $z=0$ encloses the
reactive patch. Taking such surface $S_{0}$ to be a half-disk of radius $R$
(in 2D) or a half-sphere (in 3D), we see that
$\displaystyle\partial_{r}\Phi\underset{r\to\infty}{\sim}\begin{cases}1/[\pi
Dr]&(d=2),\\\ 1/[2\pi Dr^{2}]&(d=3),\end{cases}$ (16)
where $r$ is the distance to the center of the reactive patch. Hence, the
behavior of $\Phi$ for large $r$ takes the form
$\displaystyle\Phi(\mathbf{r})\underset{|\mathbf{r}|\to\infty}{\sim}\begin{cases}\frac{1}{\pi
D}\ln|\mathbf{r}|+C_{\infty}+o(1)&(d=2),\\\ -\frac{1}{2\pi
Dr}+C_{\infty}+o(1)&(d=3),\end{cases}$ (17)
where $C_{\infty}$ does not depend on $\mathbf{r}$. The quantity $C_{\infty}$
thus characterizes the behavior of the mean reaction time far from the target,
it could be used in matched asymptotics expansions if one aims to identify the
first passage times distributions, as in ref. [22] (which deals with interior
targets).
Let us now introduce the Green’s function $G_{N}$ for the Laplace problem with
Neumann boundary conditions at $z=0$ (including on the reactive region). Such
Green’s function satisfy
$\displaystyle\nabla^{2}G_{N}(\mathbf{r}|\mathbf{r}_{0})=-\delta(\mathbf{r}-\mathbf{r}_{0}),$
(18) $\displaystyle\mathbf{n}\cdot\nabla
G_{N}(\mathbf{r}|\mathbf{r}_{0})=0\hskip 56.9055pt(z=0).$ (19)
The expression of $G_{N}$ is easily found by using the image method [44]:
$\displaystyle
G_{N}(\mathbf{r}|\mathbf{r}_{0})=\begin{cases}-\frac{1}{2\pi}[\ln|\mathbf{r}-\mathbf{r}_{0}|+\ln|\mathbf{r}-\mathbf{r}_{0}^{*}|]&(d=2),\\\
\frac{1}{4\pi}\left[\frac{1}{|\mathbf{r}-\mathbf{r}_{0}|}+\frac{1}{|\mathbf{r}-\mathbf{r}_{0}^{*}|}\right]&(d=3),\end{cases}$
(20)
where $\mathbf{r}_{0}^{*}$ represents the symmetric image of $\mathbf{r}_{0}$
with respect to the plane $z=0$. We note that the large $r$ behavior of
$G_{N}$ is
$\displaystyle
G_{N}(\mathbf{r}|\mathbf{r}_{0})\underset{|\mathbf{r}|\to\infty}{=}\begin{cases}-\frac{1}{\pi}\ln|\mathbf{r}|+o(1/r)&(d=2),\\\
\frac{1}{2\pi r}+o(1/r^{2})&(d=3),\end{cases}$ (21)
We now use manipulations that are standard in Green’s function problems [44]
to put the problem for $\Phi$ on the form of an integral equation. Using the
Eq. (18) and $\nabla^{2}\Phi=0$ we see that the following equality holds:
$\displaystyle\Phi(\mathbf{r}_{0})=\int_{z\geq
0}d\mathbf{r}[G_{N}(\mathbf{r}|\mathbf{r}_{0})\nabla_{\mathbf{r}}^{2}\Phi-\Phi(\mathbf{r})\nabla_{\mathbf{r}}^{2}G_{N}(\mathbf{r}|\mathbf{r}_{0})].$
(22)
Using the divergence formula, we obtain
$\displaystyle\Phi(\mathbf{r}_{0})=$
$\displaystyle\int_{S}dS(\mathbf{r})\mathbf{n}\cdot[G_{N}(\mathbf{r}|\mathbf{r}_{0})\nabla_{\mathbf{r}}\Phi(\mathbf{r})-\Phi(\mathbf{r})\nabla_{\mathbf{r}}G_{N}(\mathbf{r}|\mathbf{r}_{0})],$
(23)
where $S$ is any closed surface in the half-space $z\geq 0$. Taking this
surface to be a half-circle (or half-sphere in 3D) of radius $R$ joined with a
segment of size $2R$ on the axis $z=0$, we see that in the limit $R\to\infty$
$\displaystyle\Phi(\mathbf{r}_{0})=C_{\infty}-\frac{\kappa}{D}\int_{S_{r}}dS(\mathbf{r}_{s})\Phi(\mathbf{r}_{s})G_{N}(\mathbf{r}_{s}|\mathbf{r}_{0}),$
(24)
where we have used Eqs. (17),(21),(19) and (14) to simplify the integrals over
the surfaces in Eq. (23). The above equation means that $\Phi(\mathbf{r})$ can
be constructed for any position as soon as one knows its value on the reactive
patch. Taking $\mathbf{r}_{0}$ to be on the reactive patch yields an integral
equation for $\Phi_{s}$, defined to be the value of $\Phi$ on the reactive
patch. Since the above equation involves an unknown constant $C_{\infty}$ it
must be accompanied by a supplementary condition, which is provided by the
relation (13). Let us finally write explicitly the integral equations for
$\Phi$ for the 2D and the 3D cases. For $d=2$, we obtain:
$\displaystyle\Phi_{s}(x_{0})=C_{\infty}+\frac{\kappa}{D\pi}\int_{-a}^{a}dx\
\Phi_{s}(x)\ \ln|x-x_{0}|,$ (25) $\displaystyle\int_{-a}^{a}dx\
\Phi_{s}(x)=1/\kappa,$ (26)
In the case $d=3$, we note that $\Phi_{s}(\mathbf{r})$ depends only on the
radial distance to the disk center, $\Phi_{s}(\mathbf{r})=\Phi_{s}(r)$. The
kernel of the integral equation can be simplified after a few algebraic
manipulations detailed in Appendix B, leading to
$\displaystyle\Phi_{s}(r_{0})=C_{\infty}-\frac{\kappa}{D}\int_{0}^{a}dr\frac{2\
r\ \Phi_{s}(r)}{\pi(r+r_{0})}K\left(\frac{2\sqrt{rr_{0}}}{r+r_{0}}\right),$
(27) $\displaystyle 2\pi\int_{0}^{a}dr\ r\ \Phi_{s}(r)=1/\kappa,$ (28)
where $K(k)$ is the complete elliptic integral of the first kind, defined as
$K(k)=\int_{0}^{1}dt[(1-t^{2})(1-k^{2}t^{2})]^{-1/2}$ with $k$ the Elliptic
modulus (to be distinguished from the parameter $m=k^{2}$). These integral
equations admit no known analytical solution in general. In the next sections,
we focus on their asymptotic study. From now on, without loss of generality,
we set the units of length and time so that $a=1$ and $D/a^{2}=1$. The
remaining parameter $\kappa$ then represents $\kappa a/D$ in full units.
## III The limit of large reactivity
### III.1 A scaling argument for the anomalous behavior of the mean reaction
time for large reactivity.
Here we present a brief scaling argument that leads to the anomalous scaling
(1). In the case of perfect reactions $\kappa=\infty$, it is clear from Eqs.
(12),(14),(15) that $\Phi$ can be seen as the electrostatic potential
generated by a charged conducting disk that is embedded in an insulating
surface, with the prescription that $\Phi=0$ on this disk. In fact, using the
image methods, it is easy to show that $\Phi$ is also the electrostatic
potential in infinite space generated by an infinitely thin disk. It is well
known [41] that, for this electrostatic problem, the “electric field”
$-\nabla\Phi$ diverges near the edge as $1/\rho^{1/2}$, where $\rho$ is the
distance from the disk extremity. Therefore, at a distance $\rho\ll a$ from
the edges, $\Phi\propto\rho^{1/2}$. In the case that $\kappa$ is large but
finite, since the natural length scale associated to finite reactivity is
$\ell^{*}=D/\kappa$ [17], we may therefore assume that the mean reaction time
is comparable to the mean first passage time when the starting position is
located at a distance $\ell^{*}$ from the reactive patch. In this condition,
with $\rho=\ell^{*}$ we obtain the anomalous scaling $\Phi\propto
1/\sqrt{\kappa}$, as announced in Eq. (1). In what follows, we show how to
rigorously derive this scaling law, with the prefactor.
### III.2 2D case
#### III.2.1 Leading order
Consider now the limit $\kappa\to\infty$ in the case $d=2$. Since this
situation corresponds to a first passage problem, we know that the asymptotic
value of $C_{\infty}$ does not depend on $\kappa$. The fact that $\Phi_{s}$
vanishes in the large $\kappa$ limit leads us to postulate in line with Eqs.
(25)-(26) that
$\displaystyle\Phi_{s}(x_{0})\underset{\kappa\to\infty}{\sim}\frac{1}{\kappa}\Phi_{1}(x_{0}),\hskip
28.45274ptC_{\infty}\underset{\kappa\to\infty}{\sim}C_{1},$ (29)
where $C_{1}$ and $\Phi_{1}$ do not depend on $\kappa$. Inserting these ansatz
into the integral equation (25) and the normalisation condition (26), we
obtain
$\displaystyle\int_{-1}^{1}dx\ \Phi_{1}(x)\ln|x-x_{0}|=-\pi C_{1},$ (30)
$\displaystyle\int_{-1}^{1}dx\ \Phi_{1}(x)=1.$ (31)
The integral equation (30) for $\Phi_{1}(x)$ is known as Carleman’s equation
and its analytical solution is known explicitly [45]. Using also the
normalisation condition, we obtain the final expression for $C_{1}$ and
$\Phi_{1}$:
$\displaystyle C_{1}=\frac{\ln 2}{\pi},\
\Phi_{1}(x)=\frac{1}{\pi\sqrt{1-x^{2}}}.$ (32)
#### III.2.2 Boundary layer near the extremities of the reactive patch
We now note that $\Phi_{1}(x)$ is formally infinite at $x=\pm 1$, i.e. near
the boundary of the reactive patch. This means that our expansion (29) is not
valid near these points, suggesting a behavior similar to those obtained for
boundary layer problems. Since the reaction length [17] is $1/\kappa$ in our
units, we expect processes happening at such scales. Therefore, we assume the
behavior
$\displaystyle\Phi_{s}(x)=\kappa^{\alpha}\psi((1-|x|)\kappa),\hskip
28.45274pt(1-|x|)\ll 1,$ (33)
with $\psi$ a scaling function. The exponent $\alpha$ will be set such that
the behavior of $\Phi$ in the boundary layer matches with that far from the
boundary layer. Namely, the compatibility of the above ansatz with Eq. (32)
imposes the choice
$\displaystyle\alpha=-1/2,\hskip
28.45274pt\psi(X)\underset{X\to\infty}{\sim}\frac{1}{\pi\sqrt{2X}}.$ (34)
Hence, the structure of the solution in the limit $\kappa\to\infty$ is
$\displaystyle\Phi_{s}(x)=\begin{cases}\kappa^{-1}\Phi_{1}(x)+...&(1-|x|)\gg
1/\kappa\\\ \kappa^{-1/2}\psi((1-|x|)\kappa)+...&(1-|x|)\ll 1\end{cases}$ (35)
and the condition (34) ensures that these two expressions give the same result
in their common validity regime $\kappa^{-1}\ll 1-|x|\ll 1$. A key point here
is that the mean return time, starting from the boundary of the reactive
region scales as $1/\kappa^{1/2}$ and is thus infinitely larger than the mean
return time starting from the center of the target, which scales as
$1/\kappa$. This suggests to set $C_{\infty}=C_{1}+C_{1/2}/\sqrt{\kappa}+...$
in the limit of large reactivity (even though the constant $C_{1/2}$ will turn
out to vanish).
Let us now find the set of equations satisfied by $\psi$. First, we consider
the normalisation condition (26), which we write under the form
$\displaystyle\int_{-1}^{1}dx\left[\Phi_{s}(x)-\frac{\Phi_{1}(x)}{\kappa}\right]=0.$
(36)
Here we remark that the integrand in the above integral is maximal near $x=\pm
1$. Let us define an intermediate length scale $\ell$ such that
$\displaystyle 1/\kappa\ll\ell\ll 1.$ (37)
We will keep this notation in the whole paper. We write Eq. (36) by separating
the integral into two regions: when $(1-|x|)>\ell$ then we approximate
$\Phi_{s}(x)$ by the first line of Eq. (35) and if $(1-|x|)<\ell$ we use the
expressions on the second line of Eq. (35) for $\Phi_{s}$ and we approximate
by its behavior near $x=\pm 1$, which reads $\Phi_{1}\simeq
1/[\pi\sqrt{2(1-|x|)}]$. This leads to
$\displaystyle 0$
$\displaystyle=\int_{-1}^{-1+\ell}dx\left[\frac{\psi((1-|x|)\kappa)}{\sqrt{\kappa}}-\frac{1}{\kappa\pi\sqrt{2(1-|x|)}}\right]$
$\displaystyle+\int_{1-\ell}^{1}dx\left[\frac{\psi((1-|x|)\kappa)}{\sqrt{\kappa}}-\
\frac{1}{\kappa\pi\sqrt{2(1-|x|)}}\right].$
Setting $X=(1-|x|)\kappa$, and using $\ell\kappa\gg 1$, we obtain
$\displaystyle\int_{0}^{\infty}dX\left[\psi(X)-\frac{1}{\pi\sqrt{2X}}\right]=0.$
(38)
In order to find the equation satisfied by $\psi$ it is useful to write the
difference between the general equation (25) and the equation satisfied by
$\Phi_{1}$, to find
$\displaystyle\Phi_{s}(x_{0})=$ $\displaystyle\ C_{\infty}-C_{1}$
$\displaystyle+\frac{\kappa}{\pi}\int_{-1}^{1}dx\left[\Phi_{s}(x)-\frac{\Phi_{1}(x)}{\kappa}\right]\ln|x-x_{0}|.$
(39)
This leads to
$\displaystyle\Phi_{s}(x_{0})=\ C_{\infty}-C_{1}$
$\displaystyle+\frac{\kappa}{\pi}\int_{-1}^{-1+\ell}dx\left[\frac{\psi((1-|x|)\kappa)}{\sqrt{\kappa}}-\frac{\Phi_{1}(x)}{\kappa}\right]\ln|x-x_{0}|$
$\displaystyle+\frac{\kappa}{\pi}\int_{1-\ell}^{1}dx\left[\frac{\psi((1-|x|)\kappa)}{\sqrt{\kappa}}-\frac{\Phi_{1}(x)}{\kappa}\right]\ln|x-x_{0}|.$
(40)
For $x_{0}=1-X_{0}/\kappa$, if we set $X=(1-|x|)\kappa$, this yields
$\displaystyle\psi(X_{0})=C_{1/2}+\int_{0}^{\ell\kappa}\frac{dX}{\pi}\left[\psi(X)-\frac{1}{\pi\sqrt{2X}}\right]\ln\frac{|X-X_{0}|}{\kappa}$
$\displaystyle+\frac{1}{\pi}\int_{0}^{\ell\kappa}dX\left[\psi(X)-\frac{1}{\pi\sqrt{2X}}\right]\ln\left|2-\frac{X+X_{0}}{\kappa}\right|.$
(41)
Now, in the limit $\kappa\to\infty$, noting that $\ell\kappa\to\infty$ [by
definition of $\ell$, see Eq. (37)] and using the previously found condition
(38) we obtain
$\displaystyle\psi(X_{0})$ $\displaystyle=C_{1/2}$
$\displaystyle+\int_{0}^{\infty}dX\left[\psi(X)-\frac{1}{\pi\sqrt{2X}}\right]\frac{\ln|X-X_{0}|}{\pi}.$
(42)
This is a Wiener-Hopf integral equation for the unknown function $\psi(X)$. We
solve it in the next section.
#### III.2.3 Solution of the Wiener-Hopf equation (42)
We solve the Wiener-Hopf integral equation with Carleman’s method, as
described in Ref. [45]. We note that a similar equation has appeared in
viscous flow theory [46, 47, 48] but the differences between our equations and
the equation studied in these references justify the fact to solve it here in
detail. First, let us introduce the following notations. We denote $f_{+}(X)$
all functions (depending on the real variable $X$) that vanish for all $X<0$,
and $f_{-}(X)$ all functions that vanish for $X>0$. For any function $f(X)$
one can write $f(X)=f_{+}(X)+f_{-}(X)$, with $f_{+}(X)=f(X)\theta(X)$ and
$f_{-}(X)=f(X)\theta(-X)$, where $\theta$ is the Heaviside step function. We
introduce the complex Fourier transform and its inverse:
$\displaystyle\hat{f}(z)=\int_{-\infty}^{\infty}dXf(X)e^{-izX},$ (43)
$\displaystyle f(X)=\frac{1}{2\pi}\int_{-\infty}^{\infty}du\hat{f}(u)e^{+iuX}$
(44)
where $z$ represents a complex number and $u$ a real number. We denote
$\hat{f}_{+}(z)$ the Fourier transform of the function $f_{+}(X)$, and
$\hat{f}_{-}(z)$ the Fourier transform of $f_{-}(z)$. Typically, Fourier
transforms of the form $\hat{f}_{+}(z)$ are defined in the lower complex half-
plane $\text{Im}(z)\leq 0$, Fourier transforms of the form $\hat{f}_{-}(z)$
are defined in the upper complex half-plane $\text{Im}(z)\geq 0$ (as long as
$f_{\pm}(x)$ does not diverge exponentially at $x\to\pm\infty$). Now, we can
define $\psi_{+}(X)\equiv\psi(X)\theta(X)$, and we introduce
$\displaystyle K(X)=\frac{1}{\pi}\ln|X|,\hskip
28.45274pt\chi_{+}(X)=\frac{1}{\pi\sqrt{2X}}\theta(X).$ (45)
The integral equation (42) can be generalized for negative $X_{0}$ by writing
$\displaystyle\psi_{+}(X_{0})=$
$\displaystyle\int_{0}^{\infty}dX\left[\psi_{+}(X)-\chi_{+}(X)\right]K(X-X_{0})+y_{-}(X_{0}),$
(46)
where the only remarkable property of $y_{-}(X_{0})$ is that it vanishes for
positive $X_{0}$. Note that we have assumed that $C_{1/2}=0$, this will be
justified at the end of the calculation by the fact that the obtained solution
satisfies the normalization condition (38) for this value of $C_{1/2}$. Taking
the Fourier transform of the above equation, we obtain
$\displaystyle\hat{\psi}_{+}(u)=[\hat{\psi}_{+}(u)-\hat{\chi}_{+}(u)]\hat{K}(u)+\hat{y}_{-}(u).$
(47)
Calculating the Fourier transforms leads to
$\displaystyle\hat{\psi}_{+}(u)\left(1+\frac{1}{|u|}\right)=-\frac{1-i\
\text{sign}(u)}{2\sqrt{\pi}|u|^{3/2}}+\hat{y}_{-}(u),$ (48)
where $\text{sign}(u)=\theta(u)-\theta(-u)$ is the sign function. This
equation can be considerably simplified by introducing an auxilliary function
$S_{-}(X)$ defined by
$\displaystyle S_{-}(X)=\theta(-X)\frac{\sqrt{2\left|X\right|}}{\pi},\
\hat{S}_{-}(u)=\frac{1-i\ \text{sign}(u)}{2\sqrt{\pi}\left|u\right|^{3/2}},$
(49)
so that the Wiener-Hopf equation can be written as
$\displaystyle\hat{\psi}_{+}(u)\left(1+\frac{1}{|u|}\right)=-\hat{S}_{-}(u)+\hat{y}_{-}(u)=\hat{f}_{-}(u),$
(50)
where $\hat{f}_{-}(u)$ is the Fourier transform of another unknown function
$f_{-}(X)$, whose only remarkable property is to vanish for positive $X$. This
kind of equations is known as a homogeneous Wiener-Hopf equation, the method
to solve it consists in obtaining a factorization of the form
$\hat{\psi}_{+}(u)\hat{g}_{+}(u)=\hat{g}_{-}(u)$. To this end, we write the
Wiener-Hopf equation under the form
$\displaystyle\hat{\psi}_{+}(u)e^{\hat{W}(u)}=\hat{f}_{-}(u),$ (51)
with
$\displaystyle\hat{W}(u)=\ln[1+1/|u|],$ (52)
and we seek a factorization $\hat{W}(u)=\hat{W}_{+}(u)+\hat{W}_{-}(u)$. This
can be done by calculating its inverse Fourier transform:
$\displaystyle W(X)=\frac{\cos(X)[2\
\text{Si}(\left|X\right|)-\pi]+\pi}{2\pi\left|X\right|}-\frac{\text{Ci}(\left|X\right|)\sin(X)}{\pi
X}$ (53)
where Ci and Si are the integral cosine and integral sine functions
$\displaystyle\text{Ci}(X)=-\int_{X}^{\infty}dt\frac{\cos(t)}{t},\hskip
8.5359pt\text{Si}(X)=\int_{0}^{X}dt\frac{\sin(t)}{t}.$ (54)
A factorization may thus be obtained by setting $W(x)=W_{+}(x)+W_{-}(x)$, i.e
$W_{+}(x)=W(x)\theta(x)$ and $W_{-}(x)=W(x)\theta(-x)$. Now, we write the
equation (51) as
$\displaystyle\hat{\psi}_{+}(u)e^{\hat{W}_{+}(u)}=\hat{f}_{-}(u)e^{-\hat{W}_{-}(u)}.$
(55)
We are now in the favorable case: the terms on left-hand-side are analytic
functions in the upper complex plane, those on the right are analytic in the
lower complex plane except for one pole at $z=0$, and these terms are equal on
the real axis. According to the theorem of analytic continuation, combined
with the Cauchy theorem, we conclude that both terms are equal to a constant
plus a $1/z$ term on the whole complex plane [45]. We thus have
$\displaystyle\hat{\psi}_{+}(u)e^{\hat{W}_{+}(u)}=a_{0}+\frac{a_{1}}{u},$ (56)
where the constants $a_{0},a_{1}$ will be found by requiring that $\psi(X)$ is
a solution to our problem. Since $\hat{W}_{+}(z)$ is defined on the lower
complex plane, we may consider the above equations on the lower imaginary axis
$u=-is$, in which case the above equality can be written in terms of Laplace
transforms, with the usual notation
$\tilde{f}_{+}(s)=\int_{0}^{\infty}dte^{-st}f_{+}(t)=\hat{f}_{+}(-is)$:
$\displaystyle\tilde{\psi}_{+}(s)=\left(a_{0}+\frac{i\
a_{1}}{s}\right)e^{-\tilde{W}^{+}(s)}.$ (57)
The Laplace transform $\tilde{W}^{+}(s)$ can be identified by calculating its
derivative, i.e. the Laplace transform of $-xW(x)$, and then by integrating
over $s$; this leads to
$\displaystyle\tilde{W}^{+}(s)=\frac{1}{4}\ln\frac{1+s^{2}}{s^{2}}+m(s),$ (58)
with
$\displaystyle m(s)=-\int_{0}^{s}\frac{dw\ \ln w}{\pi(1+w^{2})}.$ (59)
We know that the behavior of $\psi$ for large arguments is given by the
matching condition Eq. (34), which translates to the small-$s$ behavior:
$\displaystyle\tilde{\psi}_{+}(s)\underset{s\to 0}{\sim}\frac{1}{\sqrt{2\pi
s}}.$ (60)
Inserting Eq. (58) into (57) and taking the small-$s$ limit, we see that the
above behavior is obtained for $ia_{1}=1/\sqrt{2\pi}$. Next, the value of
$a_{0}$ is found by requiring that $\psi(0)$ is finite, so that
$\tilde{\psi}(s)$ vanishes in the limit $s\to\infty$; this leads to $a_{0}=0$.
Hence, the final expression for the function $\psi$ is given in the Laplace
domain by
$\displaystyle\tilde{\psi}_{+}(s)=\frac{1}{\sqrt{2\pi
s}(1+s^{2})^{1/4}}e^{-m(s)}.$ (61)
Finally, we must check that the normalization condition (38) holds, this is
readily done by noting that $\tilde{\psi}(s)-1/\sqrt{2\pi
s}=\mathcal{O}(\sqrt{s}\ln s)$ vanishes for small $s$. This justifies our
hypothesis $C_{1/2}=0$. Unfortunately, the Laplace inversion cannot be
performed so that we know only $\psi(X)$ in closed form, however we can easily
derive the asymptotic behavior of $\psi(X)$ for small and large arguments. The
study of the asymptotic behavior of $\tilde{\psi}(s)$ for large $s$ leads
readily to the initial value of $\psi(X)$:
$\psi(0)=\lim_{s\to\infty}s\tilde{\psi}(s)=\frac{1}{\sqrt{2\pi}},$ (62)
this justifies the previously announced result (1). The large $X$ behavior can
be computed by noting that the Laplace transform of $X\psi(X)$ is
$d\tilde{\psi}/ds$ and by expanding this one for small $s$, with the result:
$\displaystyle\psi(X)\underset{X\to\infty}{\simeq}\frac{1}{\sqrt{2X}\pi}+\frac{\ln(4X)+\gamma_{e}-1}{\pi^{2}(2X)^{3/2}}+...$
(63)
with $\gamma_{e}$ the Euler-Mascheroni constant. A formula for $\psi(X)$ can
be obtained by considering the inverse Laplace transform of $\tilde{\psi}(s)$
with the Mellin’s inverse formula, by using a contour that follows the
negative real axis (above and below), such Laplace inversion is obtained in
Appendix 4 of Ref. [48]:
$\displaystyle\psi(X)=\frac{1}{\pi}\int_{0}^{\infty}dp\
\frac{e^{-pX+m(p)}}{\sqrt{2\pi p}(1+p^{2})^{3/4}}.$ (64)
In summary, here we have obtained an analytic expression in Laplace space for
the scaling function $\psi(X)$ which characterizes the behavior of the mean
first passage time near the extremities of the reactive patch in two
dimensions. The validity of our approach is checked on Figure 3 by comparing
with exact numerical results obtained from the general form of the solution.
Figure 3: Behavior of the mean first reaction time near the extremities of the
reactive patch. Symbols: exact general solution obtained numerically in
Appendix A in 2D (upper symbols) and 3D (lower symbols). We also represent the
values of $\psi$ and $\psi^{3d}$ obtained from Eqs. (64) and (78).
#### III.2.4 Next-to-leading order expansion
Up to now, the constant $C_{\infty}$, which characterizes the behavior of the
mean reaction time when the initial position is far from the target, has been
obtained at leading order only in the large reactivity limit, with the same
result as in the case of a perfectly reactive patch. Here we show how to
obtain the first non-trivial correction for $C_{\infty}$ for large reactivity,
with the result that $C_{\infty}(\kappa)$ is not analytic in $\kappa$. First,
we note that when $1/\kappa\ll 1-|x|\ll 1$, Eqs. (63) and (35) indicate that
$\displaystyle\Phi_{s}(x)\simeq\frac{1}{\kappa\pi\sqrt{2(1-|x|)}}+\frac{{\color[rgb]{0,0,0}\ln[4(1-|x|)\kappa]}+\gamma_{e}-1}{\kappa^{2}\pi^{2}[2(1-|x|)]^{3/2}}+...$
(65)
This suggests that, outside the boundary layer, the next-to-leading order
behavior of $\Phi_{s}$ reads:
$\displaystyle\Phi_{s}(x)=\frac{\Phi_{1}}{\kappa}+\frac{\Phi_{2}^{*}\ln\kappa+\Phi_{2}}{\kappa^{2}}+...,$
(66)
because this expression can be matched with (65) by imposing that
$\displaystyle\Phi_{2}^{*}(x)\simeq\frac{1}{\pi^{2}[2(1-|x|)]^{3/2}},$
$\displaystyle(x\to\pm 1),$ (67)
$\displaystyle\Phi_{2}(x)\simeq\frac{\ln[4(1-|x|)]+\gamma_{e}-1}{\pi^{2}[2(1-|x|)]^{3/2}},$
$\displaystyle(x\to\pm 1).$ (68)
These expansions also lead us to assume that, for large reactivity the
constant $C_{\infty}$ behaves as
$\displaystyle C_{\infty}=C_{1}+\frac{C_{2}+C_{2}^{*}\ln\kappa}{\kappa}+...$
$\displaystyle(\kappa\to\infty)$ (69)
The equation for $\Phi_{2}$ and $\Phi_{2}^{*}$ can be identified as follows.
We consider again the intermediate length scale $\ell$ satisfying (37), and we
start from the integral equation (39), which we write as
$\displaystyle\int_{-1+\ell}^{1-\ell}dx\left[\frac{\Phi_{2}^{*}(x)\ln\kappa+\Phi_{2}(x)}{\kappa}\right]\ln|x-x_{0}|=$
$\displaystyle-\pi\frac{C_{2}+C_{2}^{*}\ln\kappa}{\kappa}+\frac{\pi\phi_{1}(x_{0})}{\kappa}+B(x_{0})+B(-x_{0}),$
(70)
where $B$ contains all the terms which appear due to the fact that the
integral over $\Phi_{2}$ in the above equation is evaluated over a truncated
interval $]-1+\ell;1+\ell[$ instead of $]-1;1[$, so that
$\displaystyle B=$
$\displaystyle\int_{0}^{\ell\kappa}\frac{dX}{\sqrt{\kappa}}\left(\frac{1}{\pi\sqrt{2X}}-\psi(X)\right)\ln\left(1-x_{0}-\frac{X}{\kappa}\right).$
(71)
To proceed further, we consider (70) as an integral equation for
$\Phi_{2}+\ln\kappa\Phi_{2}^{*}$ over the truncated interval
$]-1+\ell;1+\ell[$. Its solution is analytically known and we identify the
constants $C_{2}$ and $C_{2}^{*}$ by requiring that the normalisation
condition is satisfied at this order of $\kappa$. This procedure requires to
evaluate $B$ in the limit $\ell\to 0$ without assuming that $\ell\ll
1-|x_{0}|$, and it turns out to be relatively tedious. The calculation is
described in Appendix C.1, and leads to the explicit results
$\displaystyle C_{2}^{*}=\frac{1}{\pi^{2}},$ $\displaystyle
C_{2}=\frac{1+\gamma_{e}+\ln 8}{\pi^{2}}.$ (72)
These values of $C_{2}$ and $C_{2}^{*}$ are in excellent agreement with the
exact solution for $\Phi_{s}$ obtained numerically, as shown in Fig. 4(a).
Figure 4: Comparison of the values of $C_{\infty}$ obtained numerically
(symbols, see Appendix A for details), with the analytical predictions in Eqs.
(72) and (81) (black lines), for the two-dimensional (a) and three-dimensional
(b) domains.
### III.3 3D case
We now adapt the approach to the 3D case. It turns out that the solution
admits the same scaling behaviors than in 2D:
$\displaystyle\Phi_{s}(r)=\begin{cases}\frac{\Phi_{1}(r)}{\kappa}+\frac{\Phi_{2}^{*}(r)\ln\kappa+\Phi_{2}(r)}{\kappa^{2}}&(1-r)\gg
1/\kappa,\\\ \frac{1}{\sqrt{\kappa}}\psi^{3d}((1-r)\kappa)&(1-r)\ll
1,\end{cases}$ (73)
where the first line is the expansion of $\Phi_{s}(r)$ in powers of $\kappa$
at fixed $r$, and the second line the expansion of $\Phi$ in powers of
$\kappa$ at fixed $X=(1-r)\sqrt{\kappa}$. At leading order, the integral
equation for $\Phi_{1}$ reads
$\displaystyle 0=C_{1}-\int_{0}^{1}dr\
\frac{2r}{\pi(r+r_{0})}K\left(\frac{2\sqrt{rr_{0}}}{r+r_{0}}\right)\Phi_{1}(r),$
(74) $\displaystyle 2\pi\int_{0}^{1}dr\ r\ \Phi_{1}(r)=1.$ (75)
The solution of the above integral equation (where $r\ \Phi_{1}(r)$ is
considered to be the unknown function) is known [45] and this leads to the
solution
$\displaystyle\Phi_{1}(r)=\frac{1}{2\pi\sqrt{1-r^{2}}},\hskip
28.45274ptC_{1}=\frac{1}{4}.$ (76)
We thus note that
$\displaystyle\Phi_{1}(r\to 1)\sim\frac{1}{2\pi\sqrt{2(1-r)}}.$ (77)
In the boundary layer near $r=1$, we set $r=1-X/\kappa$,
$r_{0}=1-X_{0}/\kappa$, and $\Phi_{s}(r)=1/\sqrt{\kappa}\psi(X)$. With these
scalings we can expand the integral equation (27) with the result that
$\psi^{3d}$ satisfies exactly the same equation than in 2D, the only
difference is that it has to match with $\psi^{3d}(X)\sim 1/(2\pi\sqrt{2X})$
for large $X$ [due to Eq. (77)] and there is thus a factor of 2 that arises
when we compare to the 2D case:
$\displaystyle\psi^{3d}(X)=\frac{1}{2}\psi(X).$ (78)
This relation is checked on Fig. 3. Let us now identify the next order terms
in 3D. Inserting the ansatz (73) into the integral equation (27) and expanding
at second order, we obtain
$\displaystyle\int_{0}^{1-\ell}\ \frac{dr\
r}{r+r_{0}}K\left(\frac{2\sqrt{rr_{0}}}{r+r_{0}}\right)[\Phi_{2}(r)+\Phi_{2}^{*}(r)\ln\kappa]=$
$\displaystyle-\frac{\pi\Phi_{1}}{2}+\frac{\pi}{2}[C_{2}^{*}\ln\kappa+C_{2}]+B$
(79)
where the term $B$ compensates the fact that the above integrals are evaluated
over the truncated interval $[0;1-\ell[$, so that:
$\displaystyle B(r_{0},\ell)=-\sqrt{\kappa}\int_{1-\ell}^{1}\ \frac{dr\
r}{r+r_{0}}K\left(\frac{2\sqrt{rr_{0}}}{r+r_{0}}\right)$
$\displaystyle\times\left[\psi^{3d}((1-r)\kappa)-\frac{1}{2\pi\sqrt{2(1-r)\kappa}}\right].$
(80)
As in the 2D situation, we consider (79) as an integral equation for which the
solution is analytically known; and we then chose $C_{2}$ and $C_{2}^{*}$ so
that the normalisation condition for $\Phi_{s}$ holds at all orders of
$\kappa$. The final result is
$\displaystyle C_{2}^{*}=\frac{1}{4\pi},\
C_{2}=\frac{\gamma_{e}+1+\ln(2)}{4\pi},$ (81)
and it agrees perfectly with numerical solutions, as shown in Fig. 4(b).
## IV The limit of small reactivity
Let us now consider the limit $\kappa\to 0$. At leading order, the mean
reaction time is homogeneous. We seek a solution under the form
$\displaystyle\Phi_{s}(x)=\frac{1}{\kappa}\sum_{n\geq 0}f_{n}(x)\kappa^{n},\
C_{\infty}=\frac{1}{\kappa}\sum_{n\geq 0}c_{n}\kappa^{n}.$ (82)
At leading order, we obtain
$\displaystyle f_{0}=c_{0}=1/|S_{r}|.$ (83)
where $S_{r}$ is the length (in 2D) or the area (in 3D) of the reactive patch.
Furthermore, next-orders can be found iteratively by using
$\displaystyle
f_{n}(\mathbf{r})=c_{n}-\frac{1}{D}\int_{S_{r}}dS(\mathbf{r}^{\prime})f_{n-1}(\mathbf{r^{\prime}})G_{N}(\mathbf{r}|\mathbf{r}^{\prime})$
(84)
with the condition for $n\geq 1$:
$\displaystyle\int_{S_{r}}dSf_{n}=0.$ (85)
For $d=2$, the explicit computations can be done for the first orders, and we
find
$\displaystyle c_{0}=1/2;\ c_{1}=\frac{3-\ln 4}{2\pi},\
c_{2}=\frac{2}{9}-\frac{7}{3\pi^{2}}.$ (86)
In the 3D situation, the leading order is simply
$\displaystyle f_{0}=c_{0}=1/\pi;$ (87)
and the recurrence relation is
$\displaystyle
f_{n}(x)=c_{n}-\frac{2}{\pi}\int_{0}^{1}dyf_{n-1}(y)\frac{y}{x+y}K\left(\frac{2\sqrt{xy}}{x+y}\right).$
(88)
Unfortunately, it seems very difficult to calculate these integrals, and even
at first order the coefficient $c_{1}$ can be calculated only numerically:
$\displaystyle
c_{1}=\frac{4}{\pi^{2}}\int_{0}^{1}dx\int_{0}^{1}dy^{\prime}\frac{xy}{x+y}K\left(\frac{2\sqrt{xy}}{x+y}\right)\simeq
0.27.$ (89)
## V Comparison with the constant flux approximation
The constant flux approximation (CFA) [20] has been used in many recent
studies [39, 40, 37] on imperfect reactivity in confinement, and here we
consider how this approximation compares to the exact results in our
formalism. First, we need to adapt this approximation to our situation of
large volume limit. In the CFA, one replaces the Robin condition (14) by
inhomogeneous Neumann conditions:
$\displaystyle D\partial_{z}\Phi=\begin{cases}0&(z=0,r>a)\\\
-Q&(z=0,r<a)\end{cases},$ (90)
where the flux $Q$ is assumed to be constant on the reactive patch and will be
determined self-consistently with a closure relation. A natural choice of
closure relation is to impose that Robin condition is satisfied on average,
hence
$\displaystyle Q=\kappa\int_{S_{r}}dS\ \Phi_{s},$ (91)
but we also have the normalization condition (13), so that
$\displaystyle Q=1.$ (92)
Now, inserting (90) into (23) leads directly to a solution for $\Phi$ within
CFA:
$\displaystyle\Phi(\mathbf{r}_{0})=C_{\infty}+\frac{Q}{D}\int_{S_{r}}dS(\mathbf{r}_{s})G_{N}(\mathbf{r}_{s}|\mathbf{r}_{0}).$
(93)
Integrating over $S$ and using (92) and (91), we obtain the CFA value of
$C_{\infty}$:
$\displaystyle C_{\infty}^{\text{cfa}}=\frac{1}{\kappa
S_{r}}\left(1-\frac{\kappa}{D}\int_{S_{r}}dS(\mathbf{r})\int_{S_{r}}dS(\mathbf{r}_{0})G_{N}(\mathbf{r}|\mathbf{r}_{0})\right)$
(94)
Comparing with the results of Sec. IV, we see that in the CFA, $C_{\infty}$ is
exactly the same as the next-to-leading order expansion of $C_{\infty}$ in the
limit of low reactivity, i.e.,
$\displaystyle C_{\infty}^{\text{cfa}}=\frac{c_{0}}{\kappa}+c_{1}$ (95)
It may be therefore surprising that CFA works for $C_{\infty}$ even for rather
large values of the reactivity (Fig. 5), but this comes from the fact that the
value of $c_{1}$ turns out to be extremely close to the exact value of
$C_{\infty}(\kappa=\infty)$ (the difference is of the order of a few
percents). This might be the reason why the CFA approach can be implemented to
yield accurate results in other contexts. However, the value of the mean first
passage time near the extremity of the reactive patch is not well captured by
this approximation, since it is obvious in Eq. (93) that it does not scale as
$1/\sqrt{\kappa}$ for large $\kappa$, contrary to what we have found.
Figure 5: Values of $C_{\infty}$ in 2D (a) and 3D (b), as found from the exact
numerical solution compared to the exact large and small reactivity
asymptotics. Note that the constant flux approximation (CFA) is exactly
equivalent to the first order expansion in the limit of low reactivity. Here
we use the units so that $D=1$, $a=1$.
## VI Conclusion
In this paper we have considered the imperfect narrow escape problem for
diffusive particles in confinement. We have established a general formalism
which provides the mean reaction time in the large volume limit for any value
of the reactivity parameter. We have obtained explicit results in $d=2$ and
$d=3$ in the respective limits of low and large reactivity parameter. Our most
surprising result is the scaling of the mean reaction time when the initial
position is at the extremity of the imperfect patch; this mean return time
scales as $\kappa^{-1/2}$ and is thus much larger than the naively expected
scaling $1/\kappa$. Interestingly, we have shown that this anomalous scaling
is closely related to the divergence of the electric field near corners and
edges of conducting objects [41], which is also responsible for the existence
of coffee rings [42] or the crispiness of the extremities of cooked potatoes
[49]. We have explicitly identified the prefactor of this scaling law by
solving a Wiener-Hopf equation. We have also identified a non-analytic
behavior for the capacitances of the imperfect patches as a function of the
reactivity. We note that we have restricted ourselves to the case of circular
patches, but we believe that for the more general patches with a smooth
boundary the asymptotic scaling laws should remain unchanged. Finally, we have
made a link between the results obtained within the Constant Flux
Approximation (CFA) and the low reactivity limit. It turns out that the CFA
gives very accurate predictions of the mean reaction time when the initial
position is far from the target, but fails to predict the correct behavior of
the mean return times; when the initial position is on the reactive patch. In
the future, one could adapt our formalism to multiple targets, for example to
generalize the classical calculation [12] of the absorption time by a sphere
covered by reactive patches to imperfect patches. Our results provide a
general framework to quantify the mean reaction times for the imperfect narrow
escape problem.
###### Acknowledgements.
Computer time for this study was provided by the computing facilities MCIA
(Mesocentre de Calcul Intensif Aquitain) of the Université de Bordeaux and of
the Université de Pau et des Pays de l’Adour. T. G. acknowledges the support
of the grant ComplexEncounters, ANR-21-CE30-0020.
## Appendix A Exact general form of the solution $\Phi(\mathbf{r})$
### A.1 Imperfect narrow escape problem in 2D
Here we describe a way to obtain the general solution of the problem formed by
Eqs. (12)-(14). It consists in writing the equations in a set of orthogonal
coordinates and using the standard method of separation of variables. We first
describe this approach in the 2D case, for which we use the elliptic
coordinates $\mu,\nu$ defined as
$\displaystyle x=a\ \mathrm{ch}(\mu)\cos(\nu),\ y=a\
\mathrm{sh}(\mu)\sin(\nu).$ (96)
We calculate the scale factors $h_{i}=|\partial_{i}\mathbf{r}|$ with
$i\in\\{\mu,\nu\\}$:
$\displaystyle h_{\mu}=h_{\nu}=a\sqrt{\mathrm{ch}^{2}(\mu)-\cos^{2}\nu}.$ (97)
The Laplace equation satisfied by $\Phi$ and the reflecting boundary
conditions outside the reactive patch are written in this coordinate system as
$\displaystyle\partial_{\nu}^{2}\Phi+\partial_{\mu}^{2}\Phi=0,\ $
$\displaystyle\partial_{\nu}\Phi|_{\nu=0}=\partial_{\nu}\Phi|_{\nu=\pi/2}=0.$
(98)
The general solution for these equations can be written by using the method of
separation of variables, which leads to
$\displaystyle\Phi=B\mu+\sum_{n=0}^{\infty}\phi_{n}\ e^{-2n\mu}\cos(2n\nu).$
(99)
Furthermore, the normalization condition (13) can also be written $D\int
dS\partial_{n}\Phi=1$ for any surface surrounding the target. Far from the
target, this means that $\partial_{r}\Phi=1/(\pi rD)$. Noting that
$\mu\simeq\ln(2r/a)$ for large $r$, we thus find
$\displaystyle B=1/(\pi D).$ (100)
We also note that the quantity $C_{\infty}$ is given, in this mode
decomposition, by
$\displaystyle C_{\infty}=\frac{\ln 2}{\pi D}+\Phi_{0}.$ (101)
Finally, the Robin condition at the target surface reads
$\displaystyle
D\partial_{n}\Phi+\kappa\Phi=\left(-\frac{D}{h_{\mu}}\partial_{\mu}\Phi+\kappa\Phi\right)_{\mu=0}=0,$
(102)
so that
$\displaystyle D\partial_{\mu}\Phi|_{\mu=0}=\kappa\ a\ \sin\nu\
\Phi|_{\mu=0}.$ (103)
Using this condition and the form of the general solution (99), we find that
the coefficients $\phi_{n}$ are solution of the infinite linear system
$\displaystyle\pi m\phi_{m}+\frac{\kappa
a}{D}\sum_{n=0}^{\infty}A_{mn}\phi_{n}=\delta_{m,0},$ (104)
which is satisfied for all positive integers $m$, with
$\displaystyle A_{nm}=\int_{0}^{\pi}d\nu\sin\nu\cos(2m\nu)\cos(2n\nu)$
$\displaystyle=\frac{2[1-4(m^{2}+n^{2})]}{16(m^{4}+n^{4})+1-8(m^{2}+n^{2})-32m^{2}n^{2}}.$
(105)
In practice, this linear system (104) can be solved numerically by taking into
account only a finite number of modes $N$, and checking that the obtained
quantities do not depend on $N$ for large $N$. Note also that $C_{\infty}$ can
be directly calculated by using Eq. (101).
### A.2 3D case
This approach can be adapted to the 3D case, for which we use orthogonal
coordinates defined as
$\displaystyle x=a\sqrt{(1+\alpha^{2})(1-\beta^{2})}\cos\varphi,$ (106)
$\displaystyle y=a\sqrt{(1+\alpha^{2})(1-\beta^{2})}\sin\varphi,$ (107)
$\displaystyle z=a\alpha\beta,$ (108)
where $\varphi$ is the azimuthal angle. Note that $\alpha>0$ and
$\beta\in[0;1]$ are related to the standard oblate spheroidal coordinates
($\mu,\nu,\varphi)$ by $\alpha=\sinh(\mu)$ and $\beta=\sin(\nu)$. Inversion
formulas read
$\displaystyle\alpha=\sqrt{\frac{\left(\frac{r}{a}\right)^{2}-1+\sqrt{1+\left(\frac{r}{a}\right)^{4}+2\left(\frac{r}{a}\right)^{2}\cos(2\theta)}}{2}},$
(109)
$\displaystyle\beta=\sqrt{\frac{1-\left(\frac{r}{a}\right)^{2}+\sqrt{1+\left(\frac{r}{a}\right)^{4}+2\left(\frac{r}{a}\right)^{2}\cos(2\theta)}}{2}},$
(110)
with $(r,\theta,\varphi)$ the usual spherical coordinates. It is useful to
calculate the scale factors $h_{i}=|\partial{\mathbf{r}}/\partial i|$, with
$i=\\{\alpha,\beta,\varphi\\}$,
$\displaystyle
h_{\alpha}=a\left(\frac{\alpha^{2}+\beta^{2}}{1+\alpha^{2}}\right)^{1/2},\
h_{\beta}=a\left(\frac{\alpha^{2}+\beta^{2}}{1-\beta^{2}}\right)^{1/2},$
$\displaystyle h_{\varphi}=a\left[(1+\alpha^{2})(1-\beta^{2})\right]^{1/2}.$
(111)
For axisymmetric functions, the Laplacian reads in this orthogonal coordinates
$\displaystyle\nabla^{2}\Phi=\frac{1}{h_{\alpha}h_{\beta}h_{\varphi}}\left(\frac{\partial}{\partial\alpha}\frac{h_{\beta}h_{\varphi}}{h_{\alpha}}\frac{\partial\Phi}{\partial\alpha}+\frac{\partial}{\partial\beta}\frac{h_{\alpha}h_{\varphi}}{h_{\beta}}\frac{\partial\Phi}{\partial\beta},\right),$
(112)
so that $\Phi$ satisfies the equation
$\displaystyle\frac{\partial}{\partial\alpha}(1+\alpha^{2})\frac{\partial\Phi}{\partial\alpha}+\frac{\partial}{\partial\beta}(1-\beta^{2})\frac{\partial\Phi}{\partial\beta}=0.$
(113)
We impose Neumann conditions for $\beta=0$ and $\beta=1$, at which
$\partial_{\beta}\Phi=0$. With these conditions, the general solution can be
found by the method of separation of variables, which leads to
$\displaystyle\Phi(\alpha,\beta)=\Phi_{\infty}+\sum_{q=0}^{\infty}a_{q}\
g_{q}(\alpha)\ P_{2q}(\beta),$ (114)
where $P_{2q}$ are even Legendre polynomials (satisfying both Neumann
conditions at $\beta=0$ and $\beta=1$), and
$\displaystyle
g_{q}(\alpha)=\frac{1}{i}Q_{2q}(i\alpha)-\frac{\pi}{2}P_{2q}(i\alpha),$ (115)
where $i^{2}=-1$ and $Q_{2q}$ are Legendre functions of the second kind. Let
us give here additional details on the function $g_{q}$. To see that $g_{q}$
is real it is useful to write $Q_{2q}$ as [50]
$\displaystyle Q_{2q}(x)=\frac{P_{2q}(x)}{2}\ln\frac{1+x}{1-x}-W_{2q-1}(x),$
(116)
where $W$ is the polynomial
$\displaystyle
W_{2q-1}(x)=\sum_{m=1}^{q}\frac{4q-(1+4(m-1))}{(2m-1)(2q-m+1)}P_{2q-(2m-1)}(x).$
(117)
For purely imaginary arguments $x=i\alpha$, we have
$\displaystyle Q_{2q}(i\alpha)=iP_{2q}(i\alpha)\arctan(x)-W_{2q-1}(i\alpha),$
(118)
and we thus see that
$\displaystyle
g_{q}(\alpha)=P_{2q}(i\alpha)\arctan(\alpha)-\frac{W_{2q-1}(i\alpha)}{i}-\frac{\pi}{2}P_{2q}(i\alpha).$
(119)
Using the parity of $P$ and $W$, it becomes clear that $g_{q}$ is real.
Furthermore it can be checked that it decreases to zero at infinity (and
$g_{0}\sim 1/\alpha$ for large $\alpha$).
Now, the equation satisfied by the coefficients $a_{q}$ is identified by using
the Robin condition. In these coordinates, the partially absorbing disk
corresponds to $\alpha=0$, and the Robin conditions can be deduced from
$\partial_{n}\Phi=-\left(h_{\alpha}^{-1}\partial_{\alpha}\Phi\right)_{\alpha=0}$
so that the boundary conditions read
$\displaystyle(D\partial_{\alpha}\Phi-a\beta\kappa\Phi)_{\alpha=0}=0.$ (120)
Inserting the general solution (114) into the above boundary condition,
multiplying by $P_{2k}(\beta)$ and integrating, we obtain the linear system:
$\displaystyle\sum_{q=0}^{\infty}a_{q}g_{q}^{\prime}(0)\int_{0}^{1}d\beta
P_{2q}(\beta)P_{2k}(\beta)=\kappa\Phi_{\infty}\int_{0}^{1}d\beta\beta
P_{2k}(\beta)$
$\displaystyle+\kappa\sum_{q=0}^{\infty}a_{q}g_{q}(0)\int_{0}^{1}d\beta\beta
P_{2q}(\beta)P_{2k}(\beta),$ (121)
for all positive integers $k$. Finally, we calculate the surface element at
$\alpha=0$, $dS_{\alpha}=h_{\varphi}h_{\beta}d\varphi d\beta$ so that the
normalization condition reads
$\displaystyle 2\pi\kappa a^{2}\int_{0}^{1}\Phi(0,\beta)\beta d\beta=1,$ (122)
which leads to the equation
$\displaystyle\kappa a^{2}\pi\Phi_{\infty}+2\pi\kappa
a^{2}\sum_{q=0}^{\infty}a_{q}g_{q}(0)\int_{0}^{1}d\beta\beta P_{2q}(\beta)=1.$
(123)
A numerical solution for $\Phi_{\infty}$ can thus be found by solving the
linear system (121) for the coefficients $a_{q}$ (completed by the above
normalisation condition). Note also that $C_{\infty}=\Phi_{\infty}$.
## Appendix B Identification of the integral equation (27) in 3D
Here we briefly show how to obtain the integral equation (27). Using Eq. (20)
for $d=3$, we see that Eq. (24) writes
$\displaystyle\Phi(r_{0})=C_{\infty}-\frac{\kappa}{D}\int_{0}^{a}drK(r,r_{0})\Phi(r),$
(124)
with
$\displaystyle K(r,r_{0})$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{r\
d\theta}{\sqrt{r^{2}+r_{0}^{2}-2rr_{0}\cos\theta}}.$ (125)
The quantity $K(r,r_{0})$ can be simplified as follows:
$\displaystyle K(r,r_{0})$ $\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}\frac{r\
d\theta}{\sqrt{r^{2}+r_{0}^{2}-2rr_{0}\cos\theta}}$
$\displaystyle=\frac{2r}{\pi}\int_{0}^{\pi/2}\frac{du}{\sqrt{r^{2}+r_{0}^{2}-2rr_{0}\cos(2u)}}$
$\displaystyle=\frac{2r}{\pi}\int_{0}^{\pi/2}\frac{du}{\sqrt{r^{2}+r_{0}^{2}-2rr_{0}[2\cos^{2}u-1]}}$
$\displaystyle=\frac{2r}{\pi(r+r_{0})}\int_{0}^{\pi/2}\frac{du}{\sqrt{1-\frac{4rr_{0}}{(r+r_{0})^{2}}\cos^{2}u}}$
$\displaystyle=\frac{2r}{\pi(r+r_{0})}K\left(\frac{2\sqrt{rr_{0}}}{r+r_{0}}\right),$
(126)
where $u=\theta/2$ and in the last line we have recognized the definition of
the elliptic function $K$. Inserting the above result into Eq. (124) finally
leads to Eq. (27).
## Appendix C Calculation details for the identification of $C_{2}$ and
$C_{2}^{*}$
### C.1 2D situation
Here we describe the details of calculations leading to the identification of
the constants $C_{2}$ and $C_{2}^{*}$ in the 2D situation. Let us first
evaluate the term $B$ in Eq. (71):
$\displaystyle
B\simeq-\frac{1}{\sqrt{\kappa}}\int_{0}^{\ell\kappa}dX\left(\psi(X)-\frac{1}{\pi\sqrt{2X}}\right)\ln(1-x_{0})$
$\displaystyle-\int_{0}^{\ell\kappa}\frac{dX}{\sqrt{\kappa}}\left(\psi(X)-\frac{1}{\pi\sqrt{2X}}\right)\ln\frac{1-x_{0}-X/\kappa}{1-x_{0}}.$
(127)
Here we have only assumed that one can use the leading order of the scaling
form for the solution $\Phi$ for arguments lower than $\ell$. Now, to evaluate
the first line we use the normalisation condition (38), and to evaluate the
terms on the second line, we change variable $X=u\kappa\ell$:
$\displaystyle
B\simeq\frac{1}{\sqrt{\kappa}}\int_{\ell\kappa}^{\infty}dX\left(\psi(X)-\frac{1}{\pi\sqrt{2X}}\right)\ln(1-x_{0})$
$\displaystyle-\sqrt{\kappa}\ell\int_{0}^{1}du\left(\psi(\kappa
u\ell)-\frac{1}{\pi\sqrt{2\kappa
u\ell}}\right)\ln\frac{1-x_{0}-u\ell}{1-x_{0}}.$ (128)
Using Eq. (63), we can write $B$ under the form
$\displaystyle B$
$\displaystyle\simeq\frac{f_{2}^{*}(x_{0})\ln\kappa+f_{2}(x_{0})}{\kappa},$
(129)
with
$\displaystyle f_{2}^{*}(x_{0})$
$\displaystyle=\frac{\ln(1-x_{0})}{\sqrt{\ell}\sqrt{2}\pi^{2}}-\int_{0}^{1}\frac{du\
\ell}{\pi^{2}(2u\ell)^{\frac{3}{2}}}\ln\frac{1-x_{0}-u\ell}{1-x_{0}},$ (130)
$\displaystyle f_{2}(x_{0})$
$\displaystyle=\frac{\ln(4\ell)+\gamma_{e}+1}{\sqrt{2\ell}\pi^{2}}\ln(1-x_{0})$
$\displaystyle-\ell\int_{0}^{1}du\frac{\ln(4u\ell)+\gamma_{e}-1}{\pi^{2}(2u\ell)^{3/2}}\ln\frac{1-x_{0}-u\ell}{1-x_{0}}.$
(131)
Let us precise a few properties of $f_{2}^{*}$ (similar properties hold for
$f_{2}$). In the limit $\ell\to 0$ at fixed $x_{0}$, we see that
$\displaystyle f_{2}^{*}(x_{0})$ $\displaystyle\underset{\ell\to
0}{\sim}\frac{\ln(1-x_{0})}{\sqrt{\ell}\sqrt{2}\pi^{2}}$ (132)
Near the extremity of the patch, we set $x_{0}=1-v\ell$ to determine the
behavior of $f_{2}^{*}$. In the limit $\ell\to 0$ at fixed $v=(1-x_{0})/\ell$,
we obtain
$\displaystyle f_{2}^{*}(x_{0}=1-v\ell)$ $\displaystyle\underset{\ell\to
0}{\sim}\frac{\ln(v\ell)}{\sqrt{\ell}\sqrt{2}\pi^{2}}-\int_{0}^{1}\frac{du\
\ell}{\pi^{2}(2u\ell)^{\frac{3}{2}}}\ln\frac{v-u}{v}$ (133)
Collecting the terms $O(\ln\kappa/\kappa)$ in the integral equation (70) leads
to
$\displaystyle\int_{-1+\ell}^{1-\ell}dx\
\Phi_{2}^{*}(x)\ln|x-x_{0}|=F_{2}^{*}(x_{0}),$ (134) $\displaystyle
F_{2}^{*}(x_{0})=-\pi C_{2}^{*}+f_{2}^{*}(x_{0})+f_{2}^{*}(-x_{0}).$ (135)
We consider this equation as an integral equation over the interval
$[-1+\ell;1-\ell]$, for which the solution is analytically known [45]:
$\displaystyle\Phi_{2}^{*}(x)=\frac{1}{\pi^{2}\sqrt{b^{2}-x^{2}}}\Bigg{[}\int_{-b}^{b}dt\frac{\sqrt{b^{2}-t^{2}}\partial_{t}F_{2}^{*}(t)}{t-x}$
$\displaystyle+\frac{1}{\ln[b/2]}\int_{-b}^{b}dt\frac{F_{2}^{*}(t)}{\sqrt{b^{2}-t^{2}}}\Bigg{]},$
(136)
where we have set $b=1-\ell$. As a consequence, the integral of $\Phi_{2}^{*}$
over the truncated interval $]-b;b[$ reads
$\displaystyle I_{2}^{*}(\ell)$ $\displaystyle=\int_{-b}^{b}dx\
\Phi_{2}^{*}(x)=\frac{1}{\pi\ln[b/2]}\int_{-b}^{b}\frac{dx\
F_{2}^{*}(x)}{\sqrt{b^{2}-x^{2}}}.$ (137)
When $\ell\to 0$, the behavior of $I_{2}^{*}(\ell)$ is obtained by inserting
the small $\ell$ limit of $f_{2}^{*}(x_{0})$ at fixed $x_{0}$ given by Eq.
(132) into Eq. (135) and inserting the result into the above equation, leading
to
$\displaystyle I_{2}^{*}(\ell)\underset{\ell\to 0}{\simeq}\frac{(-1)}{\pi\ln
2}\int_{-1}^{1}\frac{dt\
\ln\left(1-t^{2}\right)}{\sqrt{2}\pi^{2}\sqrt{\ell}\sqrt{1-t^{2}}}=\frac{\sqrt{2}}{\pi^{2}\sqrt{\ell}},$
(138)
this result is consistent with the fact that the behavior of $\Phi_{2}^{*}$
near $x=\pm 1$ is given by (67). Now, the fact that the normalisation
condition (26) holds at all powers of $\kappa$ implies that
$\displaystyle\lim_{\ell\to
0}\left[I_{2}^{*}(\ell)-\frac{\sqrt{2}}{\pi^{2}\sqrt{\ell}}\right]=0.$ (139)
We thus evaluate
$\displaystyle\Delta
I_{2}^{*}(\ell)=I_{2}^{*}(\ell)-\frac{\sqrt{2}}{\pi^{2}\sqrt{\ell}}$
$\displaystyle\simeq\frac{(-1)}{\pi\ln
2}\int_{-1}^{1}dt\left[\frac{F_{2}^{*}(t)\theta(b-|t|)}{\sqrt{b^{2}-t^{2}}}-\frac{\
\ln\left(1-t^{2}\right)}{\pi^{2}\sqrt{2\ell(1-t^{2})}}\right].$ (140)
The contributions of $f_{2}^{*}(t)$ in this integral are negligible except for
$x_{0}$ at the vicinity of $1$. Thus, we set $t=1-v\ell$ to evaluate the above
integral, and the integral can be evaluated by integrating $v$ over
$[0,\infty]$ (except for the term $C_{2}^{*}$ which is evaluated without this
change of variable). Using (133) yields
$\displaystyle\Delta I_{2}^{*}(\ell)\simeq\frac{(-1)}{\pi\ln
2}\Bigg{[}\int_{-1}^{1}dt\frac{-\pi C_{2}^{*}}{\sqrt{1-t^{2}}}$
$\displaystyle+\frac{2}{\pi^{2}\sqrt{2}}\int_{0}^{\infty}dv\ln(2v\ell)\left(\frac{\theta(v-1)}{\sqrt{2(v-1)}}-\frac{1}{\sqrt{2v}}\right)$
$\displaystyle-\int_{1}^{\infty}dv\int_{0}^{1}d{u}\frac{2}{\pi^{2}(2{u})^{3/2}\sqrt{2(v-1)}}\ln\frac{v-{u}}{v}.\Bigg{]}$
(141)
All the integrals appearing in the above equations can be evaluated. Imposing
$\Delta I_{2}^{*}(\ell)=0$ then leads to
$\displaystyle C_{2}^{*}=1/\pi^{2}.$ (142)
To identify $C_{2}$ we proceed in the same way. The integral equation is
$\displaystyle\int_{-1+\ell}^{1-\ell}dx\ \Phi_{2}(x)\ln|x-x_{0}|=F_{2}(x),$
(143) $\displaystyle F_{2}(x)=-\pi
C_{2}+\pi\phi_{1}+f_{2}(x_{0})+f_{2}(-x_{0}),$ (144)
so that
$\displaystyle I_{2}(\ell)=\int_{-b}^{b}dx\
\Phi_{2}(x)=\frac{1}{\pi\ln(b/2)}\int_{-b}^{b}\frac{dt\
F_{2}(t)}{\sqrt{b^{2}-t^{2}}}.$ (145)
As before we need to evaluate the behavior of $F_{2}(x,\ell)$ for small $\ell$
$\displaystyle F_{2}(x)\underset{\ell\to
0}{\sim}\frac{\ln(4\ell)+\gamma+1}{\sqrt{2\ell}\pi^{2}}\ln(1-x_{0}^{2})=\frac{F_{2}^{0}(x)}{\sqrt{\ell}}.$
(146)
At leading order for small $\ell$ we obtain
$\displaystyle I_{2}(\ell)\underset{\ell\to
0}{\sim}\frac{1}{\pi\ln(1/2)}\int_{-1}^{1}dt\frac{F_{2}^{0}(t)}{\sqrt{\ell}\sqrt{1-t^{2}}}=\frac{I_{2}^{0}}{\sqrt{\ell}}.$
(147)
The next-to-leading order is
$\displaystyle I_{2}-\frac{I_{2}^{0}}{\sqrt{\ell}}\simeq\frac{(-1)}{\pi\ln
2}\int_{-1}^{1}dt\left[\frac{F_{2}(t)\theta(b-|t|)}{\sqrt{b^{2}-t^{2}}}-\frac{F_{2}^{0}(t)}{\sqrt{\ell(1-t^{2})}}\right]$
(148)
Again, we evaluate it by setting $t=1-v\ell$ and taking the small $\ell$ limit
of the obtained integrand at fixed $v$, leading to
$\displaystyle I_{2}(\ell)-I_{2}^{0}(\ell)\simeq\frac{(-1)}{\pi\ln
2}\Bigg{\\{}\int_{-b}^{b}dx\frac{-\pi C_{2}+\Phi_{1}(x)}{\sqrt{b^{2}-x^{2}}}$
$\displaystyle+2\int_{0}^{\infty}dv\frac{\ln(4\ell)+\gamma+1}{\sqrt{2}\pi^{2}}\ln(2v\ell)\left[\frac{\theta(v-1)}{\sqrt{2(v-1)}}-\frac{1}{\sqrt{2v}}\right]$
$\displaystyle-2\int_{1}^{\infty}dv\int_{0}^{\infty}du\frac{\ln(4{u}\ell)+\gamma_{e}-1}{\pi^{2}(2{u})^{3/2}\sqrt{2(v-1)}}\ln\frac{v-{u}}{v}\Bigg{\\}}$
(149)
To evaluate the term involving $\Phi_{1}$ we introduce a variable
$\varepsilon$ so that $\ell\ll\varepsilon\ll 1$ and we calculate
$\displaystyle\int_{-b}^{b}dx$
$\displaystyle\frac{\Phi_{1}(x)}{\sqrt{b^{2}-x^{2}}}=\int_{-b}^{b}\frac{dx}{\pi\sqrt{1-x^{2}}\sqrt{b^{2}-x^{2}}}$
$\displaystyle\simeq\int_{-1+\varepsilon}^{1-\varepsilon}\frac{dx}{\pi(1-x^{2})}+\int_{1}^{\varepsilon/\ell}\frac{2\ell\
dv}{2\pi\sqrt{v(v-1}}$ $\displaystyle\simeq\frac{\ln(8/\ell)}{\pi},$ (150)
where the last equality follows from the evaluation of the integrals with
$\ell\ll\varepsilon\ll 1$. Finally, evaluating all terms in Eq. (149) and
requiring that the $I_{2}(\ell)-I_{2}^{0}(\ell)\to 0$ for small $\ell$, we
obtain
$\displaystyle C_{2}=\frac{1+\gamma_{e}+\ln 8}{\pi^{2}},$ (151)
as announced in the main text.
### C.2 Second-order calculation in the limit of large reactivity in 3D
We evaluate the term $B$ in Eq. (80) by writing
$\displaystyle
B(r_{0},\ell)\simeq\sqrt{\kappa}\frac{K\left(\frac{2\sqrt{r_{0}}}{1+r_{0}}\right)}{(1+r_{0})}\int_{\ell\kappa}^{\infty}dX\left[\psi(X)-\frac{1}{2\pi\sqrt{2X}}\right]$
$\displaystyle-\ell\int_{0}^{1}\frac{du}{\sqrt{\kappa}(1+r_{0})}\left[\psi(u\ell\kappa)-\frac{1}{2\pi\sqrt{2u\ell\kappa}}\right]$
$\displaystyle\times\left[K\left(\frac{2\sqrt{(1-u\ell)r_{0}}}{1-u\ell+r_{0}}\right)-K\left(\frac{2\sqrt{r_{0}}}{1+r_{0}}\right)\right].$
(152)
Hence
$\displaystyle
B(r_{0},\ell)=\frac{K\left(\frac{2\sqrt{r_{0}}}{1+r_{0}}\right)}{(1+r_{0})}\frac{1+\gamma_{e}+\ln(4\ell\kappa)}{2\sqrt{2\ell}\pi^{2}}$
$\displaystyle-\ell\int_{0}^{1}du\frac{[-1+\gamma_{e}+\ln(4u\ell\kappa)]}{2\pi^{2}(2u\ell)^{3/2}(1+r_{0})}$
$\displaystyle\times\left[K\left(\frac{2\sqrt{(1-u\ell)r_{0}}}{1-u\ell+r_{0}}\right)-K\left(\frac{2\sqrt{r_{0}}}{1+r_{0}}\right)\right].$
(153)
Note that here, for conciseness we will treat $\ln\kappa$ as being of order
$1$ in powers of $\kappa$, the result will be exactly the same as in the case
where one separates the $\ln\kappa$ terms and the $O(1)$ terms.
In the small $\ell$ limit at fixed $r_{0}$ we obtain
$\displaystyle B(r_{0},\ell)\underset{\ell\to
0}{\sim}\frac{K\left(\frac{2\sqrt{r_{0}}}{1+r_{0}}\right)}{1+r_{0}}\frac{1+\gamma_{e}+\ln(4\ell\kappa)}{2\sqrt{2\ell}\pi^{2}}=\frac{B^{0}(r_{0})}{\sqrt{\ell}},$
(154)
whereas if we set $r_{0}=1-\ell v$, in the limit $\ell\to 0$ at fixed $v$ we
obtain
$\displaystyle B(1-v\ell,\ell)\underset{\ell\to
0}{\sim}\frac{1}{8}\frac{1+\gamma_{e}+\ln(4\ell\kappa)}{\sqrt{2\ell}\pi^{2}}\ln\frac{8^{2}}{(v\ell)^{2}}$
$\displaystyle-\int_{0}^{1}d{u}\
\frac{[-1+\gamma_{e}+\ln(4\ell\kappa{u})]}{4\pi^{2}(2{u})^{3/2}}\ln\frac{v}{v-{u}},$
(155)
where we have used $K(1-y)\simeq\frac{1}{2}\ln(8/y)$ for small $y$. Let us
write the integral equation (79) under the form
$\displaystyle\int_{0}^{1-\ell}\ \frac{dr\ r\
\tilde{\Phi}_{2}(r)}{r+r_{0}}K\left(\frac{2\sqrt{rr_{0}}}{r+r_{0}}\right)=\frac{\pi[\tilde{C}_{2}-\Phi_{1}]}{2}+B$
(156)
with $\tilde{\Phi}_{2}=\Phi_{2}(r)+\Phi_{2}^{*}(r)\ln\kappa$,
$\tilde{C}_{2}=C_{2}^{*}\ln\kappa+C_{2}$. Let us define
$\displaystyle I_{2}(r,\ell)=\int_{0}^{1-\ell}dr\ r\ \tilde{\Phi}_{2}(r).$
(157)
Using the analytically known solution [45] of the integral equation (156), we
obtain
$\displaystyle
I_{2}(\ell)=\frac{4}{\pi^{2}}\int_{0}^{1-\ell}ds\frac{s\left(\frac{\pi}{2}(C_{2}-\Phi_{1})+B(s,\ell)\right)}{\sqrt{(1-\ell)^{2}-s^{2}}}.$
(158)
When $\ell\to 0$ we obtain at leading order
$\displaystyle I_{2}(r,\ell)\underset{\ell\to
0}{\sim}\frac{4}{\pi^{2}\sqrt{\ell}}\int_{0}^{1}ds\frac{s}{\sqrt{1-s^{2}}}B^{0}(s,\ell)$
(159)
and this integral diverges for $\ell\to 0$, as it should due to the known
behavior for $\tilde{\Phi}_{2}(r)$ when $r$ approaches $1$. At next-to-leading
order, we evaluate the terms involving $B$ by setting $s=1-v\ell$ and take the
small $\ell$ limit at fixed $v$, so that we can use Eq. (155):
$\displaystyle I_{2}(\ell)-\frac{4}{\pi^{2}\sqrt{\ell}}\int_{0}^{1}\frac{ds\
sB^{0}}{\sqrt{1-s^{2}}}\simeq\frac{4}{\pi^{2}}\Bigg{\\{}\int_{0}^{1}\frac{ds\
s}{\sqrt{1-s^{2}}}\frac{\pi}{2}C_{2}$
$\displaystyle-\int_{1}^{\infty}dv\int_{0}^{1}d{u}\
\frac{[-1+\gamma_{e}+\ln(4\ell\kappa{u})]}{4\pi^{2}(2u)^{3/2}\sqrt{2(v-1)}}\ln\frac{v}{v-{u}}$
$\displaystyle+\int_{0}^{\infty}dv\frac{1+\gamma_{e}+\ln(4\ell\kappa)}{4\sqrt{2}\pi^{2}}\ln\frac{8}{v\ell}\left(\frac{\theta(v-1)}{\sqrt{2(v-1)}}-\frac{1}{\sqrt{2v}}\right)$
$\displaystyle+\int_{0}^{1-\ell}ds\frac{s\pi\Phi_{1}(s)}{2\sqrt{(1-\ell)^{2}-s^{2}}}\Bigg{\\}}.$
(160)
To evaluate the term containing $\Phi_{1}$, defined in Eq. (76) we can use
again a trick where we use a variable $\varepsilon$ with
$\ell\ll\varepsilon\ll 1$:
$\displaystyle\int_{0}^{1-\ell}\frac{dr\ r\
\Phi_{1}(r)}{\sqrt{(1-\ell)^{2}-r^{2}}}=\int_{0}^{1-\varepsilon}dr\frac{r\Phi_{1}(r)}{\sqrt{1-r^{2}}}$
$\displaystyle+\int_{1}^{\varepsilon/\ell}\frac{dv\
\ell}{4\pi\sqrt{v(v-1}}=\frac{\ln(2/\ell)}{4\pi}.$ (161)
Finally, all the integrals in (160) can be evaluated, requiring that it
vanishes for small $\ell$ leads to
$\displaystyle\tilde{C}_{2}=\frac{\gamma_{e}+1+\ln(2\kappa)}{4\pi},$ (162)
which is exactly Eq. (81).
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|
# A kinematic calibration of the O-rich Mira variable period–age relation from
Gaia
Hanyuan Zhang and Jason L. Sanders
Department of Physics and Astronomy, University College London, London WC1E
6BT, UK
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
Empirical and theoretical studies have demonstrated that the periods of Mira
variable stars are related to their ages. This, together with their brightness
in the infrared, makes them powerful probes of the formation and evolution of
highly-extincted or distant parts of the Local Group. Here we utilise the Gaia
DR3 catalogue of long-period variable candidates to calibrate the period–age
relation of the Mira variables. Dynamical models are fitted to the O-rich Mira
variable population across the extended solar neighbourhood and then the
resulting solar neighbourhood period–kinematic relations are compared to
external calibrations of the age–kinematic relations to derive a Mira variable
period–age relation of $\tau\approx(6.9\pm
0.3)\,\mathrm{Gyr}(1+\tanh((330\,\mathrm{d}-P)/(400\pm 90)\mathrm{d})$. Our
results compare well with previous calibrations using smaller datasets as well
as the period–age properties of Local Group cluster members. This calibration
opens the possibility of accurately characterising the star formation and the
impact of different evolutionary processes throughout the Local Group.
###### keywords:
stars: variables: general – stars: AGB – Galaxy: disc – Galaxy: kinematics and
dynamics – Galaxy: evolution
††pubyear: 2023††pagerange: A kinematic calibration of the O-rich Mira
variable period–age relation from Gaia–References
## 1 Introduction
In the study of the formation and evolution of the Milky Way, one crucial
ingredient is accurate stellar ages (Freeman & Bland-Hawthorn, 2002; Bland-
Hawthorn & Gerhard, 2016). With this information, we can begin disentangling
the series of events that have led to the observed Milky Way today, as well as
directly measure the dynamical restructuring of the Galaxy. However, despite
their clear advantages in analysing the Galaxy, stellar ages are awkward
quantities due to their indirect measurement only via stellar models. Many
stellar age indicators exist (Soderblom, 2010) which often provide different
levels of accuracy for different stellar types and different stellar
populations. With the availability of Gaia astrometry (Gaia Collaboration et
al., 2016, 2018) and complementary large-scale spectroscopic surveys (e.g. De
Silva et al., 2015; Majewski et al., 2017), two methods applicable to large
collections of stars are comparisons to isochrone models (e.g. Xiang et al.,
2017; Sanders & Das, 2018; Xiang & Rix, 2022, which operates most successfully
for subgiant stars that have recently turned off the main sequence), and
indirect mass measurements of giant stars through spectroscopic measurements
of the products of dredge-up episodes calibrated via asteroseismology (e.g.
Masseron & Gilmore, 2015; Martig et al., 2016).
Mira variables are high-amplitude thermally-pulsing asymptotic giant branch
(AGB) stars. Their study in the Large Magellanic Cloud (e.g. Glass & Evans,
1981; Wood et al., 1999; Groenewegen, 2004) demonstrated that they follow a
tight period–luminosity relation (believed to be associated with fundamental
mode pulsation) making them interesting tracers both for local Galactic and
cosmological studies (Catchpole et al., 2016; Grady et al., 2019, 2020; Huang
et al., 2020). The chemistry of Mira variables is either oxygen or carbon-
dominated depending on the C/O ratio (Höfner & Olofsson, 2018), but O-rich
Mira variables are significantly more common in the Milky Way and are found to
follow tighter period–luminosity relations due potentially to less
circumstellar dust (Ita & Matsunaga, 2011). It has long been empirically known
that groups of Mira variables binned by period show correlations between
period and scaleheight/velocity dispersion (Merrill, 1923; Feast, 1963), which
is typically interpreted as a correlation between the period and age of a Mira
variable where the older stars have longer periods. This opens the possibility
of using Mira variables as age indicators within the Galaxy and beyond (e.g.
Grady et al., 2020). A limited number of Mira variables in clusters also
validate the period–age connection although confident assignment of membership
has only been possible recently with Gaia data (Grady et al., 2019; Marigo et
al., 2022). Although the period–age relation has been approximately calibrated
empirically (Feast & Whitelock, 2000b), relatively few theoretical models
reproducing the behaviour exist (Wyatt & Cahn, 1983; Eggen, 1998; Trabucchi &
Mowlavi, 2022) and the lack of detailed reproduction of the period–luminosity
relations of fundamental mode pulsation from theoretical models suggests the
period–age relations still have some associated uncertainty and there is a
need for accurate data-driven calibrations.
Encounters in the stellar discs of galaxies cause stellar populations to
slowly kinematically heat giving rise to age–velocity dispersion relations
(Wielen, 1977) such as those suggested for Mira variable stars (Feast, 1963).
There are multiple suggested perturbers that give rise to disc heating
including molecular clouds, spiral arms or merger events (Spitzer &
Schwarzschild, 1951, 1953; Barbanis & Woltjer, 1967; Velazquez & White, 1999;
Hänninen & Flynn, 2002; Aumer et al., 2016) that likely have differing
relative contributions across the Galactic disc (Mackereth et al., 2019). In
the solar neighbourhood, the stellar velocity dispersion is approximately a
power law in age with exponent $\sim 0.3$ for the radial dispersion and $\sim
0.5$ for the vertical dispersion (Holmberg et al., 2009; Aumer & Binney, 2009;
Sharma et al., 2021). A common picture (Binney & Tremaine, 2008) for this
behaviour is that the spiral arms are efficient in-plane heating sources
giving rise to the increase in radial velocity dispersion and molecular clouds
are efficient in converting this radial energy into vertical energy (Aumer et
al., 2016). There is the further complication that the stellar populations
could have been born hotter in the past, which could play a part in the
observed correlations (Bird et al., 2021). Now with Gaia data, the
age–velocity dispersion relations can be inspected across the Galactic disc
(Sanders & Das, 2018; Mackereth et al., 2019; Sharma et al., 2021; Gaia
Collaboration et al., 2021). For our purposes, the fact that correlations
between age and kinematics exist is sufficient and we need not necessarily
understand the underlying cause. In this way, kinematics can be used as an age
proxy for groups of stars. Note that for this procedure to operate well, we
are perhaps implicitly assuming that the kinematic–age relations are monotonic
as evidenced in the solar neighbourhood (e.g. Holmberg et al., 2009).
With the publication of large catalogues of variable stars from Gaia with
associated proper motions (Eyer et al., 2022), there is now the possibility of
thorough characterizations of the dynamical properties of different families
of Mira variable stars (Alvarez et al., 1997). Kinematic characterization then
opens up the possibility of mutual age calibration of different age tracers.
By assuming kinematics are solely a function of age, we can anchor different
age indicators to each other by requiring they all reproduce the same
age–kinematic relations (e.g Angus et al., 2015; Angus et al., 2020). In this
way, we can characterise the Mira variable period–age relation. This
simplifying assumption can be complicated by metallicity dependence,
particularly if different tracers are biased toward different metallicity
populations. The Mira variable stage occurs in stars of all metallicities
although C-rich Mira variables are only formed through dredge-up in young,
metal-poor stars (Boyer et al., 2013). This strategy of mutual age calibration
via age–kinematic relations has been utilised successfully in the study of
gyrochronology (Angus et al., 2015) and chromospheric activity in late-type
stars (Wilson & Woolley, 1970; West et al., 2015), and promises a route to the
mutual calibration of all stellar age indicators.
In this work, we utilise the astrometry of the latest Gaia DR3 long-period
variable candidate catalogue to characterise the kinematic behaviour of O-rich
Mira variables separated by period and combine this information with
literature age–velocity dispersion relations in the solar neighbourhood to
characterise the period–age relation for O-rich Mira variable stars. In
Section 2 we describe the dataset we use focusing on the cuts required to
isolate both O-rich AGB stars and those high-amplitude long-period variables
that are likely Mira variables. In Section 3 we describe our modelling
procedure and tests on mock data, before showing the results applied to data
in Section 4 and the resulting period–age relation in Section 5. We critically
discuss our approach and compare to other Mira variable period–age relations
in Section 6 before summarising our conclusions in Section 7.
## 2 The Gaia DR3 O-rich Mira variable sample
Figure 1: Colour–magnitude diagrams computed using a $3\sigma$-adjusted
parallax, $\varpi-3\sigma_{\varpi}$. We define the region occupied by AGB
stars as
$G-5\log_{10}(100/(\varpi-3\sigma_{\varpi}))<2.5(G_{\mathrm{BP}}-G_{\mathrm{RP}})-5$:
any star outside this is likely a YSO. The right panel shows those only those
stars with $\texttt{best\\_class\\_score}>0.8$ which effectively removes any
likely YSO contaminants. Figure 2: Properties of our O-rich Mira sample: the
top left panel shows the distribution of the Wesenheit index difference from
Lebzelter et al. (2018) used to separate O-rich and C-rich Mira. The lower
left panel shows the distribution of this quantity vs. period. The right two
panels show the period and distance error for the O-rich Mira sample.
Figure 3: The contour plot of the C-rich (black) and O-rich (red) Mira
variable population selected by their spectrum on period–amplitude plane and
period–colour plane respectively. Candidates below the blue line were removed
from the sample.
We begin by describing how we form our O-rich Mira variable sample. It is
important to note that our analysis relies on characterising the velocity
distributions at each Galactic location. In this way, considerations on the
completeness of our sample are unimportant provided we do not perform any
specific selections on the velocities of the stars. Our primary objective with
the selection is to form a low-contamination subset.
We use the long period variable (LPV) candidate catalogue from Gaia DR3
(Lebzelter et al., 2022). This catalogue has been constructed in a two-stage
process – likely variable stars are identified by comparison to literature
variable sources and reference non-variable Gaia sources, and then classified
based on literature classifications and features including the Lomb-Scargle
period, time summary statistics, colours and parallax (Holl et al., 2018;
Rimoldini et al., 2019, 2022). Stars classified as LPVs with $G$ $5$th$-95$th
percentile greater than $0.1\,\mathrm{mag}$ and
$G_{\mathrm{BP}}-G_{\mathrm{RP}}>0.5$ (along with other less important cuts
for our purposes) were further considered by the specific object study (SOS).
Candidate LPVs from the SOS have published generalised Lomb–Scargle periods
(and Fourier amplitudes) in Gaia DR3 if the period is greater than
$35\,\mathrm{day}$ and shorter than the $34$ month time series duration, the
$G$-band signal-to-noise $>15$ and there is no correlation between the image
determination parameters and the time series. Infrared photometric
measurements were acquired from the 2MASS catalogue (Skrutskie et al., 2006)
using the precomputed cross-match provided on the Gaia archive. There are
$1\,657\,987$ variable star observations in the Gaia DR3 LPV candidate SOS
catalogue after the cross-match with 2MASS. We first remove stars without
measured periods or without $J$ and $K_{s}$ photometric measurements which are
needed for later selection pipelines. These requirements reduce the size of
the sample to $387\,419$ objects.
To isolate a sample of likely Mira variables, we employ cuts in period and
magnitude. We retain stars with $80<\mathrm{Period}/\,\mathrm{day}<1000$
(Matsunaga et al., 2009) and in amplitude we employ a similar cut to Grady et
al. (2019), which removes stars with $\texttt{amplitude}<0.5\,\mathrm{mag}$
(compared to Grady et al. 2019 cut at $0.43\,\mathrm{mag}$). Here amplitude is
the $G$ semi-amplitude computed from a Fourier fit. Note that around the
problematic period of $190$ day, the Fourier fit can significantly
overestimate the amplitude of the LPVs leading to lower-amplitude semi-regular
variable contaminants in a Mira variable selection. We remove stars with
$170<\mathrm{Period(days)}<200$ and $\texttt{amplitude}>1.3$, and
$350<\mathrm{Period(days)}<400$ and $\texttt{amplitude}>1.6$ to mitigate
against this.
As highlighted by Mowlavi et al. (2018), young stellar objects (YSOs) can be a
contaminant in the LPV processing as they have similar colours, amplitudes and
periods to LPVs. In the classification pipeline from Holl et al. (2018) and
Rimoldini et al. (2019), the probability of the object being of the reported
class, best_class_score, seems an effective indicator of YSOs. In Fig. 1, we
show the colour–absolute magnitude diagram for our sample computed using a
parallax adjusted by $3$ times the parallax uncertainty. This gives the
brightest possible magnitude for each star within the parallax uncertainties
so any star consistent with being near the main sequence using this measure is
likely a YSO. Many of these objects also have
$\texttt{best\\_class\\_score}<0.8$ so we choose to only consider stars with
$\texttt{best\\_class\\_score}>0.8$. From this series of cuts, we end up with
$75\,874$ Mira variable star candidates.
### 2.1 O-rich/C-rich classification
LPVs can be either oxygen-rich or carbon-rich depending on the metallicity and
the strength of the dredge-ups which is controlled by the initial mass (Höfner
& Olofsson, 2018). The O-rich stars follow a tighter period-luminosity
relation (due to increased circumstellar dust in the C-rich stars, Ita &
Matsunaga, 2011) and are significantly more common in the Milky Way (with
C-rich stars contributing more in the outer disc, Blanco et al., 1984;
Ishihara et al., 2011). As shown by Lebzelter et al. (2022), the Gaia DR3
BP/RP (XP) spectra can be used to effectively separate O-rich and C-rich AGB
stars due to the differing set of band heads and features in their spectra
arising primarily from the TiO and CN absorption features. Sanders & Matsunaga
(submitted) have provided an unsupervised classification approach for these
spectra that effectively separates O-rich and C-rich LPV stars and performs
better than the Gaia DR3 classifications for highly-extincted sources. We
adopt their classifications where Gaia DR3 XP spectra are available. Lebzelter
et al. (2018) showed that, within the LMC, O-rich and C-rich Mira variables
can be separated in the plane of $W_{\mathrm{BPRP}}-W_{JK_{s}}$ vs. $K_{s}$.
Here the two Wesenheit indices are
$W_{\mathrm{BPRP}}=G_{\mathrm{RP}}-1.3(G_{\mathrm{BP}}-G_{\mathrm{RP}})$ and
$W_{JK_{s}}=K_{s}-0.686(J-K_{s})$. Although the boundary employed by Lebzelter
et al. (2018) is slightly curved, we can employ a very similar cut to select
O-rich Mira as $W_{\mathrm{BPRP}}-W_{JK_{s}}<1$. The left two panels of Fig. 2
show that this Wesenheit index difference against period for the selected Mira
sample, whilst the right panels are the period and distance percentage error
of the O-rich Mira after further selections. The performance and purpose of
these two cuts are very alike, but we employed both cuts here to maximally
remove C-rich Mira contamination.
Aided by the XP spectrum classifications, we have found that O-rich and C-rich
sources are separated in the period–amplitude plane and period–colour plane as
shown in Fig. 3. Hence, we make a further two cuts to remove those C-rich Mira
variables when an XP classification is not available:
$\texttt{amplitude}>1.2\log_{10}(\mathrm{Period}/\mathrm{days})-2.22$;
$G_{\mathrm{BP}}-G_{\mathrm{RP}}>7\log_{10}(\mathrm{Period}/\mathrm{days})-13.20$.
The resulting number of O-rich Mira variable candidates was $46\,107$.
### 2.2 Assigning distances
The distance modulus, $m$, of O-rich Mira stars are estimated from the
period–luminosity relation
$M_{KJK}=\left\\{\begin{array}[]{rcl}-7.53-4.05(\log_{10}P-2.3),&&\log_{10}P<2.6,\\\
-8.75-6.99(\log_{10}P-2.6),&&\log_{10}P\geq 2.6,\end{array}\right.$ (1)
where $P$ is the period in days and $M_{KJK}$ the absolute Wesenheit
magnitude, and the corresponding apparent Wesenheit magnitude $W_{KJK}$ is
$W_{KJK}=K_{s}-0.473(J-K_{s}).$ (2)
The extinction coefficient is taken from Wang & Chen (2019). This extinction
coefficient does not include the reddening caused by the circumstellar dust if
its properties are different from the interstellar dust. Instead, because the
period-luminosity relation is calibrated with respect to the O-rich Mira
variables in the LMC, the reddening from circumstellar dust has already been
considered in equation (1). The only caveat left is the potential difference
in properties of the circumstellar dust between O-rich Mira variables in the
LMC and the Milky Way possibly arising due to the difference in metallicity.
We consider this a minor effect in our analysis, particularly at shorter
periods where significant circumstellar dust is uncommon (Ita & Matsunaga,
2011).
The intrinsic scatter $\sigma$ of the period-luminosity relation is
$\sigma=\left\\{\begin{array}[]{rcl}\sigma_{23}+m_{\sigma_{1}}(\log_{10}P-2.3),&&\log_{10}P<2.6,\\\
\sigma_{23}+0.3m_{\sigma_{1}}+m_{\sigma_{2}}(\log_{10}P-2.6),&&\log_{10}P\geq
2.6,\end{array}\right.$ (3)
where $\ln\sigma_{23}=-1.47$, $m_{\sigma_{1}}=0.20$ and $m_{\sigma_{2}}=0.89$.
These relationships are taken from fits of the single-epoch 2MASS data for
Mira variables in the LMC (Sanders, in prep.). The scatter is a combination of
the single-epoch scatter and the intrinsic scatter due to variance in the
population. Whitelock et al. (2008) has argued from a comparison of LMC Mira
variables with local Mira variables with Hipparcos and VLBI parallaxes that
the Mira variable period-luminosity relation is metallicity-independent,
validating our use of the LMC relations for the Milky Way disc Mira variables.
Sanders (in prep.) has shown that the $W_{KJK}$ relations for the Milky Way
are quite similar to the LMC relations. To compute the uncertainties in
distance modulus, $\sigma_{m}$, we combine in quadrature the intrinsic scatter
of the period–luminosity relation from equation (3) with the uncertainty
propagated from the photometric and period measurement uncertainties. The
typical period uncertainties give rise to a median scatter of
$0.06\,\mathrm{mag}$ but the scatter arising from the single-epoch
measurements is $\gtrsim 0.22\,\mathrm{mag}$. Note that the period
uncertainties are only meaningful if the correct periodogram peak has been
identified. In the case of aliases, the reported period can be formally
inconsistent with the true period. Lebzelter et al. (2022) show the impact of
aliasing is low. Additionally, in our modelling, we allow for the possibility
of a star to be an ‘outlier’ which will capture any incorrectly assigned
periods.
### 2.3 Gaia astrometric data quality
LPV stars are one of the most challenging regimes for the Gaia astrometric
pipeline for a number of reasons. First, these sources are very red and Gaia’s
image parameter determination is not well characterised for sources redder
than $\nu_{\mathrm{eff}}=1.24\,\mu\mathrm{m}^{-1}$ (Rowell et al., 2021).
Secondly, LPVs are variable whilst the current Gaia astrometric pipelines
utilise a fixed colour in the modelling that could lead to systematics
(Pourbaix et al., 2003). Finally and possibly most importantly, LPVs can have
radii of $1\,\mathrm{AU}$ or larger, and in the optical the photocentres
wobble on the order of $\lesssim 10\,\mathrm{per\,cent}$ the radius of the
star Chiavassa et al. (2011); Chiavassa et al. (2018). This additional
photocentre wobble can lead to biases in the recovered astrometry (e.g.
Andriantsaralaza et al., 2022) but as the motion is somewhat random and
importantly not aligned in any special directions with respect to the
parallactic and proper motion directions, particularly when averaging over
many stars, the predominant effect is that the reported astrometric
uncertainties are underestimates of the true uncertainties.
Sanders (in prep.) has looked at the expected performance of Gaia on a set of
modelled Mira variable stars and found that the parallax uncertainties must be
inflated for higher parallax objects. This analysis agreed approximately with
a full characterisation of the period–luminosity relation and Gaia parallaxes
for the Mira variable stars for which Sanders (in prep.) measured an inflation
factor of $1+\exp[-(m-8.5)/0.8]$ for the parallax uncertainties. Here, we
assume that the proper motion uncertainties must be inflated by the same
factor (as validated by Sanders, in prep.). We do not consider the parallaxes
in this work.
In addition to the inflation of the astrometric uncertainties on purely
physical grounds, any mischaracterisation of Gaia’s performance gives rise to
misestimated astrometric uncertainties. Steps are taken to mitigate against
this in the Gaia pipeline (Lindegren et al., 2012) but several studies have
shown that problems likely still exist (e.g. El-Badry et al., 2021; Maíz
Apellániz, 2022). Again, this is particularly a concern for the redder sources
due to the image parameter determination. Sanders (in prep.) has modelled the
period–luminosity relation using the Gaia parallaxes including a flexible
model for the factor by which Gaia’s parallax errors must be inflated. The
model is two quadratics in $G$ and $\nu_{\mathrm{eff}}$ for the $5-$ or
$6-$parameter astrometric solutions respectively. We adopt their models for
the $W_{KJK}$ period–luminosity fits which typically require the parallax
uncertainties to be inflated by a factor $\sim 1.5$. Although the inflation
factor is appropriate for parallax errors, the astrometric modelling is a
linear regression so underestimates in the output parameters reflect
misestimates of the individual epoch astrometric (along-scan) measurements. It
is therefore appropriate to assume all the astrometric uncertainties must be
scaled in a similar way to the parallax uncertainties.
### 2.4 Final spatial cuts
We adopt a final series of spatial cuts to focus on Galactic disc members. We
remove stars with $270^{\circ}<\ell<290^{\circ}$, $-42^{\circ}<b<-22^{\circ}$,
and $40<\mathrm{heliocentric\ distance}\,(\mathrm{kpc})<60$ to remove
potential LMC candidates. As we only consider Mira variables from the Galactic
disc, we removed possible bar-bulge contribution by cutting stars with
$R<5\,\mathrm{kpc}$, where $R$ is the galactocentric radius. For the interest
of kinematic modelling, we only looked at stars with $\mathrm{heliocentric\
distance}<8\,\mathrm{kpc}$ and $R<10\,\mathrm{kpc}$. Stars with
$\sigma_{m}>0.6$ are removed to avoid stars with extremely large spatial
uncertainties. With all of the cuts described in this section, there remain
$8\,290$ O-rich Mira variable star candidates in the sample.
## 3 Kinematic modelling using dynamical models
Due primarily to the specifics of the scanning law, Gaia’s detection of
variable stars is a strong function of on-sky location and magnitude. This
makes fitting density, or full dynamical, models to any Gaia variable dataset
difficult without a careful characterisation of the selection function. Here
we employ a simpler approach by only considering the velocity,
$\boldsymbol{v}$, distribution of our sample at each observed Galactic
location, $\boldsymbol{x}$ i.e. $p(\boldsymbol{v}|\boldsymbol{x})$. Except in
the most extreme cases, a Mira variable star will not fail to be in the
catalogue as a result of its proper motion such that we can safely model the
conditional distribution of the proper motions given position. We opt to work
with full dynamical models $f(\boldsymbol{J})$ expressed as functions of the
actions $\boldsymbol{J}$ due to their ability to capture the detailed shapes
of the velocity distributions and their necessary linking of the radial and
azimuthal velocity profiles.
In Fig. 4, we plot the latitudinal velocity dispersion profile for several
period bins of the selected O-rich Mira as shown. A clear trend in
period–dispersion relation is seen implying that the O-rich Mira variables
follow a period–age relationship. In our modelling procedure, we will model
populations of stars in period bins. Note that the periods are uncertain (as
described in the previous section), but typically the uncertainty in the
period is small ($\sim 10\,\mathrm{day}$, except in the case of aliases) and
mixing between bins is a small effect. Working with binned data significantly
simplifies our procedure and allows us to fully explore the kinematics with
period rather than imposing some functional form. We discuss this latter
possibility later.
Figure 4: The transverse latitudinal velocity, $v_{b}$, dispersion profiles of
O-rich Mira separated into different period bins. Stars in this figure are
only from $|b|<5^{\circ}$, so $v_{b}$ is approximately equal to the Galactic
vertical velocity, or $v_{z}$, dispersion.
For a given population of stars with similar periods, we wish to fit the
probability distribution function $p(\boldsymbol{\mu}|\ell,b,m)$ where
$\boldsymbol{\mu}$ is the proper motion vector, $(\ell,b)$ the Galactic
coordinates and $m$ the distance modulus (as described in the previous
section). We begin by writing
$p(\boldsymbol{\mu}|\ell,b,m)=\frac{p(\ell,b,m,\boldsymbol{\mu})}{p(\ell,b,m)}=\frac{\int\mathrm{d}v_{||}p(\ell,b,m,\boldsymbol{\mu},v_{||})}{\int\mathrm{d}^{2}\boldsymbol{\mu}\,\mathrm{d}v_{||}\,p(\ell,b,m,\boldsymbol{\mu},v_{||})}.$
(4)
The proper motions and distance moduli are measured quantities with some
associated uncertainties characterised by the proper motion covariance matrix
$\boldsymbol{\Sigma}_{\mu}$ and the uncertainty in distance modulus
$\sigma_{m}$. We, therefore, marginalize over the uncertainties by writing
$\begin{split}p(\ell,&b,m,\boldsymbol{\mu},v_{||})\\\
&=\int\mathrm{d}^{2}\boldsymbol{\mu}^{\prime}\mathrm{d}m^{\prime}\mathcal{N}(\boldsymbol{\mu}|\boldsymbol{\mu}^{\prime},\boldsymbol{\Sigma}_{\mu})\mathcal{N}(m|m^{\prime},\sigma^{2}_{m})p(\ell,b,m^{\prime},\boldsymbol{\mu}^{\prime},v_{||}),\end{split}$
(5)
where $\mathcal{N}(x|\mu,\sigma^{2})$ are Gaussians with mean $\mu$ and
variance $\sigma^{2}$. We then relate the distribution in observable
coordinates to the dynamical distribution function in actions as
$p(\ell,b,m^{\prime},\boldsymbol{\mu}^{\prime},v_{||})=\left|\frac{\partial(\boldsymbol{J},\boldsymbol{\theta})}{\partial(\ell,b,m,\boldsymbol{\mu},v_{||})}\right|f(\boldsymbol{J})\propto
s^{5}\cos b\,f(\boldsymbol{J}),$ (6)
where $\boldsymbol{J}=(J_{r},J_{\phi},J_{z})$ is the set of actions
corresponding to the observed 6d coordinate (with corresponding angle
coordinates $\boldsymbol{\theta}$) and $s$ is the distance corresponding to
distance modulus $m$. Note the Jacobian between
$(\boldsymbol{x},\boldsymbol{v})$ and $(\boldsymbol{J},\boldsymbol{\theta})$
is unity due to the canonical nature of the action-angle coordinates.
We choose $f(\boldsymbol{J})$ as a quasi-isothermal distribution function,
which is suitable for warm discs (Binney, 2010). We follow the implementation
in Agama (Vasiliev, 2019) which has a functional form given by
$\displaystyle
f(\boldsymbol{J})=\frac{\tilde{\Sigma}\,\Omega}{2\pi^{2}\,\kappa^{2}}\times\frac{\kappa}{\tilde{\sigma}_{r}^{2}}\exp\left(-\frac{\kappa\,J_{r}}{\tilde{\sigma}_{r}^{2}}\right)\times\frac{\nu}{\tilde{\sigma}_{z}^{2}}\exp\left(-\frac{\nu\,J_{z}}{\tilde{\sigma}_{z}^{2}}\right)\times
B(J_{\phi}),$ $\displaystyle B(J_{\phi})=\left\\{\begin{array}[]{ll}1&\mbox{if
}J_{\phi}\geq 0,\\\
\exp\left(\frac{2\Omega\,J_{\phi}}{\tilde{\sigma}_{r}^{2}}\right)&\mbox{if
}J_{\phi}<0,\end{array}\right.,$ (9)
$\displaystyle\tilde{\Sigma}(R_{\mathrm{c}})\equiv\Sigma_{0}\exp(-R_{\mathrm{c}}/R_{\mathrm{disc}}),$
$\displaystyle\tilde{\sigma}_{r}^{2}(R_{\mathrm{c}})\equiv\sigma_{r,0}^{2}\exp(-2(R_{\mathrm{c}}-R_{0})/R_{\sigma,r}),$
$\displaystyle\tilde{\sigma}_{z}^{2}(R_{c})\equiv\sigma_{z,0}^{2}\exp(-2(R_{c}-R_{0})/R_{\sigma,z}),$
(10)
where $R_{\mathrm{c}}$ is the radius corresponding to a circular orbit of
angular momentum $J_{\phi}\equiv L_{z}$ and $(\kappa,\Omega,\nu)$ are the
epicyclic frequencies at this angular momentum. This distribution function
describes an approximately exponential disc in radius which is
broadened/warmed vertically and radially by two exponential terms. There are
five key free parameters for the model: (i) the scalelength of the disc,
$R_{\mathrm{disc}}$, (ii) the radial ($\sigma_{r,0}$) and vertical
($\sigma_{z,0}$) normalizations of the velocity dispersions at the Sun
($R=R_{0}$), and (iii) their corresponding scalelengths ($R_{\sigma,r}$ and
$R_{\sigma,z}$). The actions are evaluated using the ‘Stäckel fudge’ algorithm
described by Binney (2012), summarized and critically assessed against
alternatives in Sanders & Binney (2016) and implemented in Agama (Vasiliev,
2019). We adopt a fixed axisymmetric gravitational potential for the Galaxy
from McMillan (2017). Fixing the potential could lead to sub-optimal model
fits (as we will discuss later) but it significantly simplifies the
computation and incorporates external constraints from the analysis of other
datasets.
### 3.1 Computational considerations
The computational difficulty in evaluating equation (4) is computing the
integrals efficiently. Here we use Monte Carlo integration. For the numerator,
we generate a set of $N$ samples for each star from the proper motion and
distance modulus error ellipses. The unknown $v_{||}$ is sampled from a
probability distribution $G(v_{||}|\ell,b,m,\boldsymbol{\mu})$ which is
proportional to a quasi-isothermal distribution function with fixed parameters
$f^{\prime}(\boldsymbol{J})$ at a given $(\ell,b,m,\boldsymbol{\mu})$,
$G(v_{||}|\ell,b,m,\boldsymbol{\mu})=\frac{p(\ell,b,m,\boldsymbol{\mu},v_{||})}{\int\mathrm{d}v_{||}\,p(\ell,b,m,\boldsymbol{\mu},v_{||})}=A_{v_{||}}f^{\prime}(\boldsymbol{J}).$
(11)
Samples are generated from this distribution using the inverse cumulative
distribution. The value of $f^{\prime}(\boldsymbol{J}_{i})$ for each sample is
stored to reweight the Monte Carlo sum. For the denominator, we sample
$\boldsymbol{v}=(v_{x},v_{y},v_{z})$ directly at a given observed position
$(\ell,b,m)$ in a similar way to the numerator as
$G(\boldsymbol{v}|\ell,b,m)=\frac{p(\ell,b,m,\boldsymbol{v})}{\int\mathrm{d^{3}}\boldsymbol{v}\,p(\ell,b,m,\boldsymbol{v})}=A_{\boldsymbol{v}}f^{\prime}(\boldsymbol{J}),$
(12)
from which samples are generated using Markov Chain Monte Carlo (MCMC,
Foreman-Mackey et al., 2013), and once again $f^{\prime}(\boldsymbol{J}_{i})$
are stored. $A_{v_{||}}$ and $A_{\boldsymbol{v}}$ defined in equation (11) and
(12) are constant factors which can be computed for each individual star. Only
the ratio of these two normalisation factors is important:
$A\equiv\frac{A_{\boldsymbol{v}}}{A_{v_{||}}}=\frac{\int\mathrm{d}v_{||}\,p(\ell,b,m,\boldsymbol{\mu},v_{||})}{\int\mathrm{d^{3}}\boldsymbol{v}\,p(\ell,b,m,\boldsymbol{v})}=\frac{\int\mathrm{d}v_{||}\,f^{\prime}(\boldsymbol{J})}{\int\mathrm{d^{3}}\boldsymbol{v}\,f^{\prime}(\boldsymbol{J})}.$
(13)
$A$ is evaluated using Monte Carlo integration: $v_{||}$ and $\boldsymbol{v}$
are sampled from a Gaussian distribution centred on zero in the radial and
vertical velocities, and on the rotation curve in the azimuthal velocity. As
$f^{\prime}(\boldsymbol{J})$ is fixed, $A$ can be precomputed once for each
individual star to a desired accuracy. The $f^{\prime}(\boldsymbol{J})$ we use
throughout this paper has fixed parameters:
$R_{\mathrm{disc}}=2.5\,\mathrm{kpc}$, $\sigma_{r,0}=50\,\mathrm{km/s}$,
$\sigma_{z,0}=50\,\mathrm{km/s}$, $R_{\sigma,r}=5.0\,\mathrm{kpc}$ and
$R_{\sigma,z}=5.0\,\mathrm{kpc}$. These parameters are chosen such that the
distributions of the integration samples are typically broader than the
modelled distributions to minimise bias in the Monte Carlo integration.
Sampling from the distribution $G$, instead of a Gaussian distribution
increases the computational efficiency by reducing the noise in the Monte
Carlo integration for a fixed number of sampling. Now for each star, the
integrals (up to a normalization constant) are given by
$p(\ell,b,m,\boldsymbol{\mu})\approx\frac{1}{NA_{v_{||}}}\color[rgb]{0,0,0}\sum^{\begin{subarray}{c}\mathrm{errors\,in\,}m,\boldsymbol{\mu}\\\
v_{||}\mathrm{\,from\,}G(v_{||}|\dots)\end{subarray}}_{i}s_{i}^{5}\cos
b\frac{f(\boldsymbol{J}_{i})}{f^{\prime}(\boldsymbol{J}_{i})},$ (14)
and
$p(\ell,b,m)\approx\frac{1}{NA_{\boldsymbol{v}}}\color[rgb]{0,0,0}\sum^{\begin{subarray}{c}\mathrm{errors\,in\,}m\\\
\boldsymbol{v}\mathrm{\,from\,}G(\boldsymbol{v}|\dots)\end{subarray}}_{i}s_{i}^{3}\cos
b\frac{f(\boldsymbol{J}_{i})}{f^{\prime}(\boldsymbol{J}_{i})}.$ (15)
Note in the second expression we only have $3$ powers of $s$ as the integral
has been rewritten in terms of the 3d space velocity $\boldsymbol{v}$ (as
opposed to the observable space of proper motion and radial velocity). As we
are using a fixed potential, we precompute $\boldsymbol{J}_{i}$,
$R_{\mathrm{c},i}$ and the epicyclic frequencies for all samples using the
routines from Vasiliev (2019) and Bovy (2015).
### 3.2 Outlier component
Another complexity is to introduce an outlier distribution to overcome the
contamination of samples by stars which are members of the halo, are possibly
not Mira variable stars or have poorly-measured periods. To do this, we assume
that the velocity distribution of the contamination stars is described by a 3D
spherically symmetric Gaussian distribution that is centred on Galactocentric
$\boldsymbol{v}=\boldsymbol{0}$ with standard deviation in each dimension
$\sigma_{v}$. Similar to the previous approach, we calculate the
$p_{\mathrm{outlier}}(\boldsymbol{\mu}|\ell,b,m)$ using equation (4) and (5),
but replacing $p(\ell,b,m^{\prime},\boldsymbol{\mu}^{\prime},v_{||})$ with
$p_{\mathrm{outlier}}(\ell,b,m^{\prime},\boldsymbol{\mu}^{\prime},v_{||})$
which is chosen to be
$p_{\mathrm{outlier}}(\ell,b,m^{\prime},\boldsymbol{\mu}^{\prime},v_{||})=s^{5}\cos
b\,\mathcal{N}(\boldsymbol{v}|\boldsymbol{0},\sigma_{{v}}^{2}\boldsymbol{I})\mathcal{U}(x,y,z),$
(16)
where $\mathcal{U}(x,y,z)$ is the uniform distribution in Galactocentric
Cartesian spatial coordinates $(x,y,z)$. For each star,
$p_{\mathrm{outlier}}(\boldsymbol{\mu}|\ell,b,m)$ is evaluated numerically by
$p_{\mathrm{outlier}}(\boldsymbol{\mu}|\ell,b,m)=\frac{\sum^{\mathrm{errors\,in\,}m,\boldsymbol{\mu}}_{i}s_{i}^{5}\mathcal{N}(s_{i}\boldsymbol{\mu}_{i}+\boldsymbol{v}_{t,\odot,i}|\boldsymbol{0},\sigma^{2}_{v}\boldsymbol{I})}{\sum^{\mathrm{errors\,in\,}m}_{i}s_{i}^{3}},$
(17)
where $v_{t,\odot}$ is the solar velocity in the Galactocentric frame
projected in the plane perpendicular to the line-of-sight between the star and
the Sun. To include this distribution in the log-likelihood, we rewrite the
probability for each individual star as
$p_{\mathrm{tot},j}=(1-\epsilon)p_{j}(\boldsymbol{\mu}|\ell,b,m)+\epsilon
p_{\mathrm{outlier},j}(\boldsymbol{\mu}|\ell,b,m).$ (18)
Note with this definition, the outlier fraction at each spatial location,
$\epsilon$, is approximately constant. We choose the Gaussian because the
contamination could come from a variety of sources, and the Gaussian
distribution is a general, easily-computed way to characterise those sources.
### 3.3 Likelihood
We have now fully specified our model. The full log-likelihood for each
population of stars is
$\ln L=\sum^{\mathrm{stars}}_{j}\ln
p_{\mathrm{tot}}(\boldsymbol{\mu}_{j}|\ell_{j},b_{j},m_{j}),$ (19)
For each population of stars, we optimize the likelihood with respect to the
five parameters of the quasi-isothermal ($R_{\mathrm{disc}}$, $\sigma_{r,0}$,
$\sigma_{z,0}$, $R_{\sigma,r}$, $R_{\sigma,z}$) and the two parameters of the
outlier distribution $(\epsilon,\sigma_{v})$. The log-likelihood is explored
using MCMC performed with emcee (Foreman-Mackey et al., 2013). We adopt priors
on the radial scale lengths as
$R_{\mathrm{disc}}\sim\mathcal{N}(3.8\,\mathrm{kpc},(2\,\mathrm{kpc})^{2})$
and $R_{\sigma,r/z}\sim\mathcal{N}(4.5\,\mathrm{kpc},(3\,\mathrm{kpc})^{2})$
and a prior for velocity dispersion of the outlier component is a normal
distribution
$\sigma_{\boldsymbol{v}}\sim\mathcal{N}(200\,\mathrm{km\,s}^{-1},(150\,\mathrm{km\,s}^{-1})^{2})$.
The priors for the other three parameters are uniform:
$\sigma_{r/z,0}\sim\mathcal{U}(0,120\,\mathrm{km\,s}^{-1})$ and
$\epsilon\sim\mathcal{U}(0,1)$.
A final step in our procedure is converting the modelled distribution function
parameters to the physical measures of the velocity dispersion in the solar
neighbourhood. It is these quantities we compare with previous
characterisations of the age–velocity dispersion relation. For each set of
($R_{\mathrm{disc}}$, $\sigma_{r,0}$, $\sigma_{z,0}$, $R_{\sigma,r}$,
$R_{\sigma,z}$), we generate mock stars using the Agama DF sampling routines
and fit an exponential profile
$\sigma_{i}=\widetilde{\sigma}_{i,0}\exp[{(R_{0}-R)/\widetilde{R}_{\sigma,i}}]$
to the radial and vertical velocity dispersions binned in radius. The
normalization $\widetilde{\sigma}_{i,0}$ and scalelength
$\widetilde{R}_{\sigma,i}$ give the physical velocity dispersion and its
radial gradient in the solar neighbourhood.
### 3.4 Mock samples and validation
Figure 5: Results of fits on mock data: the lower left corner plot is the
posterior of the fitting parameters from the test on mock data including an
outlier distribution. The red lines are the parameters that generated the mock
sample, and the black dashed lines are the $16$th, $50$th and $86$th
percentiles of the posterior, respectively. The upper right corner plot gives
the posteriors of the physical velocity dispersion parameters corresponding to
the sets of fitted parameters. The physical velocity dispersion parameters are
propagated from the fitted parameters using the routine described in Section
4.
Given a fitted $f(\boldsymbol{J})$ model, we wish to draw mock samples to
compare with the data and validate our fitting procedure. We use the Agama DF
sampling routine to generate a large number of mock stars. For each generated
mock star, we find the nearest observed star in our dataset in $(R,z)$, place
the mock star at the azimuth $\phi$ of the real star and transform the mock
polar velocities to $\boldsymbol{\mu}$. This procedure exploits the
axisymmetry of the models. We further scatter the proper motions and distance
moduli of the mock stars by the corresponding uncertainties of the real stars.
The previous Mira selection criteria in the heliocentric distance and $R$ are
also applied to the mock sample. Note this procedure produces a mock dataset
with each real star corresponding to multiple mock stars in proportion to the
local stellar density at the location of the real star. This reduces the shot
noise in our mock samples but means the mock sample has a different spatial
density to the data. To reproduce the spatial distribution of the dataset, we
record the index of the closest matched real star for each mock star and then
count the number of times that this real star is the closest match to any mock
star. A weight is calculated for each mock star as the reciprocal of this
number count. The weight will be used when we compare our fitted model to the
dataset. When directly comparing to a fitted dataset, we further remove mock
stars which do not reside within $100\,\mathrm{pc}$ of any real star (this
requirement is not imposed on the mock test set described below but makes
little practical difference). Our procedure does not fully generate the data
as we have not accounted for uncertainty in the data $(R,z)$. However, it is
sufficient for validation purposes.
We can use the generated mock observations to test the validity of our method.
We generate a mock sample of $614$ stars from $f(\boldsymbol{J})$ with known
parameters chosen arbitrarily as $R_{\mathrm{disc}}=3.8\,\mathrm{kpc}$,
$\sigma_{r,0}=45.0\,\mathrm{km\,s}^{-1}$,
$\sigma_{z,0}=35.0\,\mathrm{km\,s}^{-1}$, $R_{\sigma,r}=4.5\,\mathrm{kpc}$ and
$R_{\sigma,z}=4.4\,\mathrm{kpc}$. We then replace velocities of $10\%$ of the
generated data with $\boldsymbol{v}$ sampled from a spherically symmetric
Gaussian
$\mathcal{N}(\boldsymbol{v}|\boldsymbol{0},(100\,\mathrm{km/s})^{2}\boldsymbol{I})$
which is the assumed velocity distribution of outlier stars. Stars sampled
from the outlier distribution have a chance to be unbound from the potential,
so after removing those unbound stars, the actual proportion of outlier stars
can be smaller than $10\%$, i.e. $\epsilon<10\%$. Without those high-velocity
stars in the mock sample, the velocity dispersion of the generated outlier
stars is reduced so a fitted $\sigma_{{v}}<100\,\mathrm{km/s}$ is expected but
the recovered parameters of the $f(\boldsymbol{J})$ model should be unbiased.
The posteriors from the MCMC are shown in the low left of Fig. 5. The
parameters $\sigma_{z,0}$ and $R_{\sigma_{z,0}}$ both deviate slightly from
the default parameters but only around the 1$\sigma$ level. In the upper right
corner of Fig. 5, we convert each set of fitted parameters into the physical
velocity dispersion profile parameters, $\widetilde{\sigma}_{i,0}$ and
$\widetilde{R}_{\sigma,i}$. Although there are small differences in the
distribution function parameters, the resulting physical velocity dispersions
and scalelengths at the solar position are well recovered. We also produced
the posterior of the same sample using the log-likelihood without the outlier
distribution. The medians of the parameters are
$R_{\mathrm{disc}}=3.56\,\mathrm{kpc}$,
$\sigma_{r,0}=65.74\,\mathrm{km\,s}^{-1}$,
$\sigma_{z,0}=43.16\,\mathrm{km\,s}^{-1}$, $R_{\sigma,r}=3.14\,\mathrm{kpc}$
and $R_{\sigma,z}=2.48\,\mathrm{kpc}$. As expected, $\sigma_{r,0}$ and
$\sigma_{z,0}$ are overestimated. This demonstrates that adding the outlier
distribution is necessary when the contamination of the sample is significant.
## 4 Velocity dispersion of O-rich Mira variable stars in different age bins
Figure 6: Velocity histograms for O-rich Mira variables separated by period (as given in days above each column). The top panels show $v_{\ell}$ and bottom $v_{b}$. The points are data and black lines the models. Table 1: Distribution function parameter estimates for the Mira variable model fits. The left column gives the considered period bin and the other columns show the median and uncertainties estimated from the $16$th and $84$th percentiles. Period range (day) | Mean period (day) | Number of stars | $R_{\mathrm{disc}}\,\mathrm{(kpc)}$ | $\sigma_{r,0}\,\mathrm{(km/s)}$ | $\sigma_{z,0}\,\mathrm{(km/s)}$ | $R_{\sigma,r}\,\mathrm{(kpc)}$ | $R_{\sigma,z}\,\mathrm{(kpc)}$ | $\epsilon$ | $\sigma_{\boldsymbol{v}}\,\mathrm{(km/s)}$
---|---|---|---|---|---|---|---|---|---
$80-150$ | 126.4 | 230 | $3.55^{+2.40}_{-1.19}$ | $48.05^{+6.17}_{-4.83}$ | $30.45^{+2.14}_{-2.01}$ | $8.32^{+3.29}_{-1.64}$ | $9.65^{+2.28}_{-2.41}$ | $0.06^{+0.02}_{-0.02}$ | $151.49^{+30.72}_{-22.91}$
$150-200$ | 179.3 | 430 | $3.56^{+1.27}_{-0.74}$ | $40.59^{+3.06}_{-3.26}$ | $41.07^{+3.62}_{-3.46}$ | $3.54^{+0.25}_{-0.19}$ | $6.47^{+1.14}_{-1.10}$ | $0.02^{+0.02}_{-0.02}$ | $105.58^{+61.84}_{-40.22}$
$200-225$ | 212.8 | 442 | $5.11^{+1.00}_{-1.35}$ | $53.14^{+5.51}_{-5.39}$ | $51.72^{+3.39}_{-2.89}$ | $5.05^{+1.19}_{-0.60}$ | $9.52^{+2.06}_{-1.94}$ | $0.01^{+0.04}_{-0.01}$ | $46.46^{+111.19}_{-21.72}$
$225-250$ | 237.7 | 494 | $3.79^{+1.43}_{-0.82}$ | $37.66^{+2.66}_{-2.18}$ | $55.99^{+3.74}_{-4.19}$ | $3.97^{+0.32}_{-0.31}$ | $10.36^{+2.89}_{-2.19}$ | $0.01^{+0.01}_{-0.01}$ | $78.35^{+71.38}_{-36.07}$
$250-275$ | 263.3 | 708 | $3.53^{+1.95}_{-0.91}$ | $52.38^{+3.59}_{-3.72}$ | $42.22^{+1.83}_{-2.10}$ | $12.16^{+3.37}_{-2.72}$ | $7.41^{+1.30}_{-0.92}$ | $0.03^{+0.02}_{-0.01}$ | $79.23^{+28.35}_{-12.22}$
$275-300$ | 287.2 | 909 | $4.47^{+1.55}_{-0.98}$ | $51.79^{+2.49}_{-2.11}$ | $39.57^{+2.00}_{-1.75}$ | $13.57^{+3.08}_{-2.17}$ | $7.24^{+1.27}_{-0.87}$ | $0.02^{+0.01}_{-0.01}$ | $104.36^{+22.33}_{-15.63}$
$300-325$ | 313.0 | 907 | $2.72^{+0.71}_{-0.58}$ | $46.46^{+2.04}_{-2.18}$ | $34.53^{+1.90}_{-1.72}$ | $11.68^{+2.13}_{-1.49}$ | $8.12^{+1.50}_{-1.15}$ | $0.00^{+0.01}_{-0.00}$ | $109.87^{+45.84}_{-36.79}$
$325-350$ | 337.7 | 970 | $2.49^{+0.67}_{-0.47}$ | $43.15^{+1.68}_{-1.75}$ | $32.94^{+1.34}_{-1.18}$ | $12.10^{+2.50}_{-1.54}$ | $9.08^{+1.55}_{-1.17}$ | $0.01^{+0.01}_{-0.00}$ | $114.63^{+37.64}_{-21.27}$
$350-375$ | 362.3 | 861 | $5.29^{+1.78}_{-1.38}$ | $42.44^{+1.82}_{-1.78}$ | $28.84^{+1.26}_{-1.38}$ | $13.52^{+4.08}_{-2.42}$ | $11.35^{+2.55}_{-2.39}$ | $0.01^{+0.01}_{-0.01}$ | $79.10^{+79.99}_{-45.18}$
$375-400$ | 387.3 | 784 | $4.69^{+2.07}_{-1.41}$ | $42.33^{+2.26}_{-1.62}$ | $23.89^{+1.54}_{-1.49}$ | $12.01^{+3.08}_{-1.96}$ | $7.89^{+2.12}_{-1.27}$ | $0.01^{+0.01}_{-0.01}$ | $79.19^{+48.55}_{-32.91}$
$400-450$ | 422.5 | 1015 | $2.87^{+1.40}_{-0.69}$ | $41.45^{+1.81}_{-1.61}$ | $25.77^{+0.86}_{-1.04}$ | $14.43^{+3.11}_{-2.01}$ | $13.69^{+2.48}_{-1.58}$ | $0.00^{+0.01}_{-0.00}$ | $88.41^{+66.37}_{-43.88}$
$450-500$ | 470.9 | 396 | $3.18^{+2.57}_{-1.28}$ | $37.42^{+1.90}_{-2.27}$ | $19.56^{+1.59}_{-1.71}$ | $13.22^{+2.70}_{-2.64}$ | $15.03^{+4.98}_{-4.03}$ | $0.04^{+0.02}_{-0.02}$ | $82.44^{+21.10}_{-18.52}$
$500-600$ | 527.5 | 144 | $4.68^{+2.68}_{-2.10}$ | $34.27^{+3.22}_{-2.85}$ | $16.85^{+1.92}_{-2.41}$ | $11.45^{+3.82}_{-3.06}$ | $11.34^{+5.52}_{-3.99}$ | $0.02^{+0.03}_{-0.01}$ | $121.84^{+69.88}_{-44.77}$
Table 2: Solar neighbourhood velocity dispersions and local spatial gradients of the velocity dispersions for the Mira variable fits. The age estimations are also provided, where $\tau_{r}$ is the age estimation from the radial velocity dispersion while $\tau_{z}$ is that from the vertical velocity dispersion. Period (days) | $\widetilde{\sigma}_{r,0}\,\mathrm{(km/s)}$ | $\widetilde{\sigma}_{z,0}\,\mathrm{(km/s)}$ | $\widetilde{R}_{\sigma,r}\,\mathrm{(kpc)}$ | $\widetilde{R}_{\sigma,z}\,\mathrm{(kpc)}$ | $\tau_{r}\,\mathrm{(Gyr)}$ | $\tau_{z}\,\mathrm{(Gyr)}$
---|---|---|---|---|---|---
$80-150$ | $49.83^{4.39}_{3.78}$ | $24.59^{1.47}_{1.38}$ | $10.54^{3.94}_{2.13}$ | $9.04^{1.64}_{1.64}$ | $8.57^{+1.13}_{-0.98}$ | $6.34^{+0.36}_{-0.34}$
$150-200$ | $67.20^{4.70}_{4.20}$ | $34.44^{1.86}_{1.43}$ | $6.88^{0.62}_{0.61}$ | $7.46^{0.85}_{0.92}$ | $10.82^{+0.56}_{-0.52}$ | $8.07^{+0.82}_{-0.75}$
$200-225$ | $62.96^{2.99}_{3.11}$ | $38.24^{1.55}_{1.56}$ | $7.92^{1.47}_{0.91}$ | $9.37^{1.48}_{1.27}$ | $10.41^{+0.45}_{-0.45}$ | $9.34^{+0.64}_{-0.66}$
$225-250$ | $52.83^{3.00}_{3.20}$ | $40.24^{1.67}_{1.78}$ | $6.16^{0.53}_{0.53}$ | $10.10^{1.66}_{1.42}$ | $9.25^{+0.73}_{-0.89}$ | $9.72^{+0.61}_{-0.67}$
$250-275$ | $51.90^{2.48}_{2.29}$ | $32.92^{1.14}_{1.16}$ | $15.31^{4.51}_{3.61}$ | $7.78^{0.94}_{0.77}$ | $9.09^{+0.72}_{-0.78}$ | $7.58^{+0.60}_{-0.55}$
$275-300$ | $50.54^{2.07}_{1.91}$ | $31.08^{1.21}_{1.08}$ | $16.61^{4.20}_{2.97}$ | $7.54^{1.05}_{0.76}$ | $8.75^{+0.78}_{-0.70}$ | $7.25^{+0.52}_{-0.44}$
$300-325$ | $46.67^{1.73}_{1.72}$ | $27.57^{1.11}_{1.01}$ | $14.35^{2.78}_{2.08}$ | $8.05^{1.15}_{0.87}$ | $7.80^{+0.55}_{-0.57}$ | $6.77^{+0.35}_{-0.35}$
$325-350$ | $43.27^{1.50}_{1.54}$ | $26.27^{0.86}_{0.88}$ | $14.41^{3.15}_{2.00}$ | $8.66^{1.10}_{0.91}$ | $7.01^{+0.59}_{-0.54}$ | $6.61^{+0.34}_{-0.34}$
$350-375$ | $41.74^{1.67}_{1.69}$ | $23.11^{0.89}_{0.93}$ | $15.35^{4.79}_{2.96}$ | $10.24^{1.74}_{1.75}$ | $6.66^{+0.52}_{-0.49}$ | $6.20^{+0.33}_{-0.30}$
$375-400$ | $41.98^{1.84}_{1.47}$ | $20.02^{1.00}_{0.94}$ | $13.73^{3.78}_{2.23}$ | $7.82^{1.65}_{1.12}$ | $6.67^{+0.51}_{-0.49}$ | $5.66^{+0.34}_{-0.32}$
$400-450$ | $41.08^{1.58}_{1.50}$ | $20.92^{0.71}_{0.76}$ | $16.45^{3.56}_{2.46}$ | $11.61^{1.65}_{1.17}$ | $6.43^{+0.47}_{-0.43}$ | $5.86^{+0.31}_{-0.31}$
$450-500$ | $37.21^{1.87}_{2.03}$ | $16.46^{1.21}_{1.16}$ | $14.60^{3.38}_{2.91}$ | $12.58^{3.00}_{2.60}$ | $5.52^{+0.57}_{-0.57}$ | $4.60^{+0.61}_{-0.96}$
$500-600$ | $34.24^{2.99}_{2.78}$ | $14.65^{1.34}_{1.85}$ | $12.44^{4.03}_{3.23}$ | $10.46^{3.60}_{3.11}$ | $4.50^{+0.86}_{-1.12}$ | $3.62^{+0.76}_{-1.05}$
Figure 7: Velocity dispersion profiles as a function of Galactocentric radius
for O-rich Mira variables separated by period (as given in days above each
column). The top panels show longitudinal, $\ell$, and bottom latitudinal,
$b$. The points are data and black lines the models. Figure 8: Velocity
histograms for O-rich Mira variables with periods in the range $275-300$ day
separated into bins of Galactocentric radius (as given above each column). The
top panels show the longitudinal velocity $v_{\ell}$ and the bottom the
latitudinal velocity, $v_{b}$. The red points are data and the black lines are
the models. Figure 9: Vertical density distribution profile for O-rich Mira
variables separated by period bins (as given in days above each panel). Each
panel shows the dataset (points) compared to the unweighted distribution of
mock samples (black lines). All histograms are normalised, and subplots do not
share the same y-axis. The discrepancy between the distributions is a
reflection of the completeness of the dataset.
To investigate the kinematic properties of the sample defined in Section 2, we
put the O-rich Mira variables into period bins and treat stars in each bin as
a sub-population drawn from the same DF. We choose the period bins to be wider
than the typical uncertainties in the period measurements, and hence we
neglect the period uncertainties that scatter stars from bin to bin (the
impact of the period uncertainties on the distance uncertainties _have_ been
considered). The median of the period uncertainties is $11.6$ days and $7.1$
days for those stars with periods less than $300$ days. We have also tried to
bin stars with a wider period bin ($50$ days instead of $25$), which gives
very similar results to the presented binning strategy. The adopted priors on
the radial scale lengths are
$R_{\mathrm{disc}}\sim\mathcal{N}(4\,\mathrm{kpc},(3\,\mathrm{kpc})^{2})$ and
$R_{\sigma,r/z}\sim\mathcal{N}(10\,\mathrm{kpc},(6\,\mathrm{kpc})^{2})$, the
prior for velocity dispersion of the outlier component is a normal
distribution
$\sigma_{{v}}\sim\mathcal{N}(100\,\mathrm{km\,s}^{-1},(80\,\mathrm{km\,s}^{-1})^{2})$
and the other priors are uniform as defined in the previous section. The
posterior distributions for the fits of each period bin are given in the
supplementary material and are summarised by the medians and percentiles in
Table 1. The contamination fraction $\epsilon$ is generally small and
$\sigma_{{v}}$ generally large for all period bins. Table 2 reports the
physical radial and vertical velocity dispersion normalization and scalelength
in the solar neighbourhood, $\widetilde{\sigma}_{i,0}$ and
$\widetilde{R}_{\sigma,i}$ respectively.
To verify the results of the MCMC fitting, we generate mock samples for the
best-fit parameters according to the procedure from Section 3.4, and we make
use of the weights for the mock sample to compare the kinematics of the fitted
model with the dataset under the same spatial distribution. In Fig. 6, we have
plotted the $v_{\ell}$ and $v_{b}$ distributions of these mock samples
compared to that of the observations, where $v_{\ell/b}=s\cdot\mu_{\ell/b}$.
We have chosen to omit the lowest period bin ($80-150$ days) from this plot
and in later plots and analysis because the contamination rate, $\epsilon$ is
the highest among other period bins (see Table 1) and it is likely it does not
follow the broad trend of increasing dispersion with decreasing period due to
contamination from short-period-red stars as we will discuss in Section 6.3.
For the displayed period bins, the mock samples generally agree with the
observations. For some period bins, the shape of the observed $v_{b}$ is
sharper than the mock sample implying that our modelling has some caveats.
Three reasons could lead to this: first, the assumed outlier distribution did
not characterise the contamination accurately and underestimated the outlier
star contribution consequently. Secondly, the period binning strategy needs to
be improved. Bins at long periods cover Mira variables of a broader range of
ages than the bins at short periods. Hence, if the younger stars in the period
bin have much smaller velocity dispersion than the average of the bin, the
sharper peak in observation would appear while the general shape of the
overall distribution is still correct. Thirdly, the assumed functional form
for the velocity dispersion parameters
$\sigma_{i}=\sigma_{i,0}\exp{(R_{0}-R)/R_{\sigma,i}}$ may be inappropriate. We
illustrate this final possibility by plotting the radial profile of the
longitudinal and latitudinal velocity dispersions $\sigma_{\ell}$ and
$\sigma_{b}$ in Fig. 7. For one or two period bins, the large $R$ radial
behaviour of $\sigma_{b}$ is not completely in agreement with the
observations. The $\sigma_{\ell}$ distribution is relatively more poorly
fitted than the $v_{b}$ distribution. Again, this could be due to the adopted
form of the distribution function. However, apart from these very minor
discrepancies, our modelling is in agreement with the observations. This is
reinforced by the comparison of the $v_{\ell}$ and $v_{b}$ distribution for
$275<\mathrm{Period/day}<300$ in Fig. 8. The model is in good agreement with
the observations. We will discuss further limitations of our approach in
Section 6.
As noted previously, the spatial distribution of stars has not been considered
in the modelling as it is subject to completeness effects arising from Gaia’s
scanning law and the effects of extinction. As a result, the spatial
distribution of the (unweighted) mock samples and the observations are in
disagreement when the completeness of the dataset is not considered. Our
weighting of the mock samples reproduces the spatial distribution of the data
enabling comparison of the kinematic fits as shown in Fig. 6, for example.
When the weights are not considered, the mock sample distribution can be
considered as the approximate underlying completeness-corrected distribution
of the data (only up to a point as according to our procedure, where there is
no data there will also be no mock stars). The weights are thus giving the
proportion of stars at each $\boldsymbol{x}$ that have been observed. This is
demonstrated in Fig. 9 by comparing the unweighted Galactic height
distribution of the mock sample to the dataset. Note that our procedure only
gives access to the relative completeness so the histograms have been chosen
to be normalized. The distributions of the data points are generally broader
than the unweighted mock distributions, which we interpret as incompleteness
in the dataset towards the Galactic midplane, possibly arising from
extinction. This interpretation of the unweighted mock samples assumes the
distribution functions well describe the Milky Way sub-populations. We discuss
the shortcomings of the approach later, but the good agreement in Fig. 9 also
demonstrates that even without considering incompleteness, the distribution
functions do a good job of describing the data.
## 5 Period–age relationship
Figure 10: The calibrated age–period relationship of the O-rich Mira
variables. The orange and violet points are the velocity dispersion from the
kinematic modelling. The orange, purple, and black lines are the fitted
period–age relations using radial, vertical velocity dispersions, and two
together respectively, with fitted parameters given in Table 3.
With the dynamical distribution functions in each Mira variable period bin
well characterised, we now turn to what this implies for the corresponding age
of each period bin. To do this we must adopt an age–velocity dispersion
relation (AVR). We choose the AVR measured by Yu & Liu (2018) from LAMOST data
of $\sim 3500$ sub-giant/red giant stars. Yu & Liu (2018) characterised the
velocity dispersions of their sample split into age bins using the entirety of
their dataset and also for two sets split by Galactic height:
$|z|<0.27\,\mathrm{kpc}$ and $|z|>0.27\,\mathrm{kpc}$. The ages of stars in Yu
& Liu (2018) were estimated by comparing the stellar parameters
($[\mathrm{Fe/H}]$, $\mathrm{T_{eff}}$, $\log g$) measured by LAMOST to a grid
of isochrone models. Age estimates were found by marginalizing the likelihood
over initial mass and absolute magnitude. The AVRs were produced by further
binning stars in their sample by age. This procedure accounts for
uncertainties arising from the velocities but _not_ the ages. We discuss the
impact of this later.
We estimate the corresponding AVR of our sample by averaging the two
$|z|$-separated AVRs in Yu & Liu (2018) weighted by the number of stars in our
sample that are above and below $|z|=0.27\,\mathrm{kpc}$ in each bin.
Consequently, the final AVR was slightly different for each bin. At low ages,
the corresponding AVR is not monotonic due in part to uncertainties and the
low numbers of stars in some low-age bins. Thus, we remove points in the AVR
if the age is less than that of the previous age bin so that we could
interpolate a monotonic AVR to find an age at each radial and vertical
dispersion, $\widetilde{\sigma}_{r,0}$ and $\widetilde{\sigma}_{z,0}$. The
uncertainty is again propagated using Monte Carlo samples. The final
calibrated age–period relationship is shown in Fig. 10.
Yu & Liu (2018) discussed that the uncertainties in the estimated ages of
stars would broaden the measured AVR. Liu et al. (2015) argued that the age
estimation method used in Yu & Liu (2018) could have uncertainties at the
$30\,\mathrm{per\,cent}$ level which propagate from the uncertainties of the
LAMOST stellar parameters. Here, we will discuss how much this effect would
affect the period–age relationship. We generate $500\,000$ stars with
uniformly distributed ages and assign each star a radial and vertical velocity
from a Gaussian distribution centred at $0$ with standard deviations of
$\sigma_{r}$ and $\sigma_{z}$ calculated from the AVR. Then, the ages of the
stars are scattered by $(10,20,30)\%$ uncertainties. We then bin the stars
with the scattered age and calculate the measured radial and vertical velocity
dispersion. The ratio of the measured to actual velocity dispersion for the
AVR is given in Fig. 11, where the left and right panels are made for the AVR
of $|z|<0.27\,\mathrm{kpc}$ and $|z|>0.27\,\mathrm{kpc}$ respectively. We
divide this ratio by the corresponding velocity dispersions in the AVR as a
correction. In Fig. 12 we show the period–age relations calibrated using AVRs
with different levels of age uncertainty. We see that with $30\%$ uncertainty
in AVR the maximum correction could be up to $20\%$ in age as calibrated from
$\sigma_{R,0}$ and $34\%$ from $\sigma_{z,0}$.
Figure 11: The ratio of the age–velocity dispersion relation broadened by
different age uncertainties ($10,20$ and $30\,\mathrm{per\,cent}$ denoted by
dotted, dashed and solid) relative to the ‘true’ age–velocity dispersion
relation without age uncertainties. The left panel shows results for the
$|z|<0.27\,\mathrm{kpc}$ AVR from Yu & Liu (2018) and the right panel their
age–velocity dispersion relation for $|z|>0.27\,\mathrm{kpc}$. Yellow lines
correspond to $\sigma_{r}$ and blue $\sigma_{z}$. Figure 12: The calibrated
period–age relationship using age–velocity dispersion relations broadened by
different age uncertainties (as labelled in the legend). The relation
calibrated by $\widetilde{\sigma}_{r,0}$ is shown in the left panel while
$\widetilde{\sigma}_{z,0}$ is on the right. The error bars are not shown in
this figure. The black dashed lines in both panels are the fitted period–age
relations shown by the orange and pink lines in Fig. 10 respectively.
We have also considered other recent AVR calibrations available in the
literature. For example, Sharma et al. (2021) have provided a fit of the
radial and vertical dispersions in a separable form in terms of the age,
angular momentum, metallicity and Galactic height. Their relations produce
significantly smaller dispersions at fixed age such that the derived
period–age relation will assign significantly larger ages at fixed period
which in the extreme can be $\gg 14\,\mathrm{Gyr}$. We are therefore inclined
to use the Yu & Liu (2018) relations and the applicability of the Sharma et
al. (2021) relations merits further investigation.
## 6 Discussion
We now turn to the interpretation and understanding of our results, in
particular concentrating on the comparison with previous period–age estimates
for Mira variable stars and possible future model improvements.
### 6.1 Comparison with Mira variable cluster members and previous results
Figure 13: Comparison of the derived period–age relations with other
literature results. The orange squares and pink triangles show our Mira
variable period–age measurements from Table 2. The small grey points are from
the models of Wyatt & Cahn (1983), the green short-dashed line from the model
of Eggen (1998) and the orange long-dashed line from the model of Trabucchi &
Mowlavi (2022, along with the associated scatter shown by the shaded region).
The light blue squares are Mira variable globular cluster members from Clement
et al. (2001), the brown diamonds C-rich Mira variable open cluster members
from Marigo et al. (2022) and the light-blue triangles LMC cluster members.
The solid blue line is a fit from Grady et al. (2019) to a broader sample of
LMC cluster members. The grey points are period–age estimates for disc
populations from Feast et al. (2006), Feast (2009) and Feast & Whitelock
(2014). The black line is the joint fit of our results and the globular
cluster members from Table 3 and the thinner orange, pink and grey lines show
the other three fits from that same table.
In Fig. 10 we display a series of period–age indicators of Mira variable
stars. The age–kinematic method for period–age calibration has been utilised
by Feast et al. (2006), Feast (2009) and Feast & Whitelock (2014). Feast &
Whitelock (2000b) demonstrated that Mira variables in the solar neighbourhood
exhibited clear correlations between period and kinematics. These have been
translated approximately into period–age measurements using results from the
solar neighbourhood in the cited works. However, it should be said that all of
the quoted results are only approximate due to the absence of robust
age–kinematics calibrations.
Mira variables in clusters give a more direct measurement of the period–age
relation than the indirect method using the age–kinematic calibrations.
Unfortunately, there are comparatively few cluster Mira variables. Those in
globular clusters have been studied by Sloan et al. (2010) whilst those with
good evidence of Milky Way open cluster membership from Gaia have been studied
by Marigo et al. (2022). There are also many candidates for LMC cluster
membership as studied by Grady et al. (2019). However, membership of an LMC
cluster is difficult to discern purely from projected coordinates (as used by
Grady et al., 2019) and proper motion data. We compile Mira variable globular
cluster members using the globular cluster variable star compilation from
Clement et al. (2001). We consider all stars flagged as ‘M’ or ‘M?’, and not
flagged as a likely field star (‘f’ or ‘f?’). Furthermore, if available, we
ensure the Gaia DR3 proper motion is within $3\sigma$ of the measured cluster
mean proper motion from Vasiliev & Baumgardt (2021). Here $\sigma$ is a
quadrature sum of the measurement uncertainty and the central velocity
dispersion. We complement with ages primarily from VandenBerg et al. (2013)
and Dotter et al. (2010), and from Beaulieu et al. (2001) for NGC 6553,
Geisler et al. (2007) for Terzan 7, Ortolani et al. (1999) for Terzan 1,
Marín-Franch et al. (2009) and Forbes & Bridges (2010) for NGC 6441 and Santos
& Piatti (2004) for NGC 6356, NGC 6388, NGC 6642 and NGC 6760. Terzan 5 has
evidence of multiple star formation events (Ferraro et al., 2016) so we assign
stars with periods $<400$ day an age of $12\,\mathrm{Gyr}$ and longer-period
stars an age of $4.5\,\mathrm{Gyr}$. There is a carbon-rich Mira variable in
the old globular cluster Lyngå 7 that has been suggested as a product of
binary evolution (Feast et al., 2013). However, its Gaia DR3 proper motion is
not consistent with being a cluster member. Its radial velocity is perfectly
consistent so one possibility is that the Gaia measurement is spurious. This
seems quite likely as there are two nearby Gaia DR3 sources with only two-
parameter astrometric solutions suggesting contamination in the Lyngå 7 C-rich
Mira variable measurement.
For Mira variable open cluster members, we use the compilation from Marigo et
al. (2022) adopting their measured periods and the cluster ages from Cantat-
Gaudin et al. (2020). Marigo et al. (2022) identify some cluster members on
the fundamental period–luminosity relation followed by Mira variable stars but
with too low an amplitude for traditional Mira variable classification. We
consider all stars that Marigo et al. (2022) identify as fundamental pulsators
and with $G$ band amplitudes greater than $0.865\,\mathrm{mag}$ (Grady et al.,
2019) estimated from the photometric uncertainties. There are two such stars
with are both C-rich.
Finally, we consider possible LMC and SMC cluster members from the Gaia DR3
LPV candidate catalogue. We combine the list of cluster ages from Baumgardt et
al. (2013) and Bonatto & Bica (2010). To limit contaminants, we conservatively
find all Gaia DR3 LPV candidates within one cluster radius as determined by
Bica et al. (2008) (adopting the median cluster radius of
$0.45\,\mathrm{arcmin}$ when a radius is not available). We further limit to
those with proper motions within $3\sigma$ of
$(\mu_{\alpha}*,\mu_{\delta})=(1.910,0.229)\mathrm{mas\,yr}^{-1}$
(Kallivayalil et al., 2013) where $\sigma$ is the quadrature sum of the
uncertainties and $100\,\mathrm{km\,s}^{-1}$ at the distance of the LMC, and
those with distances between $30$ and $70\,\mathrm{kpc}$ as determined from
equation (1). We isolate Mira variables by restricting to stars with $G$
amplitudes $>0.865\,\mathrm{mag}$ as determined by the $G$ photometric
uncertainties and the Fourier light curve fits. This results in $4$ high-
confidence LMC cluster members.
The described combination of cluster measurements is shown in Fig. 13. We see
in general the good agreement between the results derived from the
age–kinematic relation and the cluster members. There are some globular
cluster members with longer periods but higher ages (most notably the $312$
day period Mira in NGC 5927 which has an age of $12.25\,\mathrm{Gyr}$ from
Dotter et al. 2010 and $10.75\,\mathrm{Gyr}$ from VandenBerg et al. 2013).
This may reflect metallicity dependence in the period–age relation or these
could be the results of binary evolution in these clusters producing slightly
more massive AGB stars than expected at fixed age.
There are several theoretical period–age relations from the literature. The
earliest of these are the results from Wyatt & Cahn (1983) who found ages for
local Mira variable stars via main-sequence mass estimates derived from models
of Mira variables as fundamental pulsators which were fitted to optical and
infrared photometry and periods. Eggen (1998) similarly provided a
theoretically-motivated period–age relation by supposing fundamental Mira-like
pulsations occur once a star of a given mass (age) reaches some critical
radius. Most recently, Trabucchi & Mowlavi (2022) have used theoretical models
to produce period–age calibrations for O-rich and C-rich Mira variable stars.
They highlighted one expectation of the models is a large spread of age at
fixed period. Furthermore, their period–age relations agreed very well with
the cluster member measurements mostly compiled by Grady et al. (2019).
However, as we have hinted at above, there is perhaps good reason to believe
that the LMC cluster members are quite a contaminated set and that LMC field
stars coincident on the sky with the clusters are likely to be incorrectly
identified as cluster members. The field stars will typically be older than
the cluster members, having already left their parent clusters, and so these
contaminants will act to decrease the typical age at fixed period. It could be
that there is an additional variable controlling the period–age relation that
produces the discrepancy between the LMC clusters and the local age–kinematic
relations. The spread in models from Trabucchi & Mowlavi (2022) is almost
consistent with the measurements made here. However, the globular clusters
suggest any metallicity dependence would go the other way. Furthermore, binary
evolution produces higher periods at fixed age so would not explain the
discrepancy.
A further supporting piece of evidence for the age–period relation we have
derived here is the properties of the LMC population as a whole and the
Galactic bulge sample. In both sets, there are stars with $\sim
500-600\,\mathrm{day}$ periods. From our calibrations, these stars are $\sim
3-4\,\mathrm{Gyr}$ old. The LMC has a tail towards longer-periods consistent
with even more recent star formation. The Galactic bulge is primarily
considered as an old population (Zoccali et al., 2003) although there has been
significant evidence that there are intermediate-age populations as young as
$3\,\mathrm{Gyr}$ (Bensby et al., 2013, 2017; Nataf, 2016). Our calibration is
entirely consistent with these results. A lower age–period relation would mean
a significant population of stars in the Galactic bulge with $\lesssim
1\,\mathrm{Gyr}$ old populations although again we should stress the expected
spread in ages at each period could still produce some consistency in the
results.
### 6.2 A parametric period–age relation
Our fitting has provided the approximate ages of O-rich Mira variable
populations in a series of period bins. It is more convenient to work with an
analytic relation that approximately fits the results. The flexible form
$\tau=\tau_{0}\frac{1}{2}\left(1+\tanh\Big{[}\frac{330-P(\mathrm{days})}{P_{s}}\Big{]}\right),$
(20)
provides an approximate fit to the data. We take the data reported in Table 2
and fit equation (20) allowing for an additional fractional scatter in the
ages of $f_{\tau}$ such that the age errors are
$\sqrt{\sigma_{\tau}^{2}+f_{\tau}^{2}\tau^{2}}$. $(\tau_{0},P_{s},f_{\tau})$
are given logarithmic flat priors and we sample using emcee (Foreman-Mackey et
al., 2013). We fit for $\sigma_{r}$ and $\sigma_{z}$ both separately and
jointly and report the results in Table 3. Although the dispersion parameters
are derived from the same model fit, the corner plots in the supplementary
material demonstrate the parameter constraints are uncorrelated for nearly all
period bins validating treating the results in this way. We also perform a
joint fit of the dispersion results together with the globular cluster member
compilation described in the previous section, again reporting the results in
Table 3. All four sets of results are quite consistent with $\sigma_{z}$-only
fits producing the lowest age at fixed period and the combination with the
globular clusters producing the highest. As expected, the scatter is largest
for the combined fit with the globular clusters but nevertheless, the scatter
is only around $10\,\mathrm{per\,cent}$ in age.
Table 3: Functional form for the period–age relation fitted to our results. We adopt the form $\tau=(\tau_{0}/2)\left(1+\tanh((330-P(\mathrm{day}))/P_{s}\right)$ with a fractional age uncertainty of $f_{\tau}$. Subset | $\tau_{0}$ | $P_{s}$ | $\ln f_{\tau}$
---|---|---|---
$\sigma_{r}$ | $14.9\pm 0.7$ | $389\pm 77$ | $-6.70\pm 0.01$
$\sigma_{z}$ | $13.0\pm 0.5$ | $404\pm 111$ | $-5.71\pm 0.03$
Both | $13.7\pm 0.6$ | $401\pm 88$ | $-2.63\pm 0.04$
With GC | $14.7\pm 0.7$ | $308\pm 54$ | $-2.17\pm 0.04$
### 6.3 Model limitations and future improvements
Before concluding, we will discuss some of the limitations of our modelling
and possible improvements that could be adopted in future analyses.
_Binning in period_ : We have opted to bin our data in period and analyse each
period bin independently. This is a valid approach as the period uncertainties
are typically quite small: the median period uncertainty is $11.6$ day and
$7.1$ day for $\mathrm{period}<300\,\mathrm{day}$. Hence, our strategy is
valid for most of the period bins considered. A further generalization is to
express the models in terms of the period as a continuous subpopulation label.
We then have to introduce hyper-parametrizations for the parameters in
$f(\boldsymbol{J})$ to express $f(\boldsymbol{J}|P)$. The integrals would
involve an additional integral over the label $P$ and we would have a
weighting of the populations $f(P)$ (which if we are considering periods as
proxies for age is akin to a star formation rate and could be an exponential
in age, for example). The advantage of this approach is a more principled
accounting of the period uncertainties as well as providing a route to
consider the spread in age (kinematics) at each period that might arise from
helium flashes, hot-bottom burning or the presence of short-period red stars.
The downside of such an approach is that we would have to fit a parametrized
form for the parameters as a function of $P$ making the models significantly
more complicated and potentially producing biased by our choice of functional
form.
_Velocity dispersion profile_ : We have here adopted a simple pure exponential
decay for the velocity dispersion of each period bin. This form gives a good
fit of the models to the data, particularly as we have chosen a rather limited
Galactocentric radial range. It has been suggested that the velocity
dispersion in the outer disc flattens or even increases with radius (Sanders &
Das, 2018; Mackereth et al., 2019). For example, Sharma et al. (2021) argues
that the pure exponential decay of the velocity dispersions is not well
motivated by the data, which shows signs of a rising dispersion beyond the
solar radius. To incorporate this possibility, one possible change is to
modify $\tilde{\sigma}_{i}(R_{c})$ as
$\tilde{\sigma}_{i}(R_{c})\equiv\sigma_{i,0}(\exp[-(R_{c}-R_{0})/R_{\sigma,i}]+\alpha_{i}(R_{i}/R_{0})^{2})/(1+\alpha_{i}),$
(21)
with the additional fitting parameters $\alpha_{i}$ to match the
flattening/upturning dispersion profiles in the outer disc as suggested by
Sharma et al. (2021). This may be a necessary enhancement when modelling the
data beyond the extended solar neighbourhood. For example, if one were to
consider investigating possible metallicity dependence of the period–age
relation. However, such an enhancement does not seem necessary for our data.
_Limitations of equilibrium axisymmetric distribution function approach_ : It
is reassuring to note that the age estimates from the radial and vertical
dispersions separately give very similar results for the period–age relation
of the O-rich Mira variables. However, the relation derived from the radial
dispersion is consistently higher than that derived from the vertical
dispersion. We have seen how our dynamical models capture well both the
longitudinal and latitudinal velocity distributions of the sample but
typically the latitudinal distributions are better modelled suggesting our
results are more reliable for the period–age relation derived from
$\sigma_{z}$. This occasional mismatch of the longitudinal dispersion in Fig.
7 could be a shortcoming of the use of a quasi-isothermal distribution
function. There are other action-based disc models available in the literature
(e.g. Binney & Vasiliev, 2023) which could be explored. As mentioned
previously, using a dynamical distribution function simply incorporates the
required asymmetry in the azimuthal component as well as necessarily linking
together the radial and azimuthal dispersions due to the requirement of
dynamical equilibrium. There could also be inconsistencies arising from this
assumption of equilibrium as it is known that the Galactic disc shows non-
equilibrium structure at the $5-10\,\mathrm{per\,cent}$ level. Any inflation
of the velocity dispersion as a result of this is not a concern as we have
anchored to tracers that will also display this inflation. The assumption of
axisymmetry could also be giving rise to similar variations. We are using the
velocity dispersion at the solar radius from a range of different azimuths but
if the velocity dispersion is varying significantly with azimuth (e.g. Gaia
Collaboration et al., 2022), the comparison between our sample and the
age–velocity dispersion results from Yu & Liu (2018) may be inappropriate.
Furthermore, our model has assumed a fixed Milky Way potential from McMillan
(2017). Whilst this potential captures many of the global features of the
Milky Way, it may not in detail be appropriate across the entirety of the
Galactic disc region considered here. In the wrong potential, it may be very
difficult to fully match the full velocity distribution of the data at every
spatial location. Reasonable variations of the potential will likely inflate
the uncertainties in our derived parameters. We should also note that although
we have inflated the Gaia astrometric uncertainties in our analysis to reflect
shortcomings of the current Gaia data processing, it is likely that future
Gaia data releases will improve the uncertainty estimates providing a better
handle on the underlying dispersions of the disc populations. This may
decrease the dispersion for the youngest populations (e.g. the $500-600$ day
period bin) but the dispersions of the oldest populations are very insensitive
to the uncertainties so we believe our measurements are reliable.
_SP-red stars_ : We found the stars in our lowest considered period bin
($80-150$ day) have significantly lower dispersions and hence lower ages than
the neighbouring $150-200$ day bin (see Table 1). This bucks the broad trend
seen in e.g. Fig. 13 and for this reason, as well as the fact that this bin
requires the largest outlier fraction of all modelled bins, we decided to
neglect these results in our period–age relation fits. Feast & Whitelock
(2000b) found a similar effect from Hipparcos data that they attribute to
short-period(SP)-red stars which contaminate the short-period end and are
kinematically more similar to the longer-period Mira variables. It is not
clear exactly what the origin of these stars is and they could represent a
different evolutionary stage to the bulk Mira variable population. Feast &
Whitelock (2000b) hypothesise they could be stars on their way to becoming
longer-period Mira variables or temporarily dimmed during their helium-shell
flash cycle (Trabucchi et al., 2017). From Gaia-2MASS colour-colour diagrams,
we did not clearly identify a distinct population of SP-red-like stars in the
short-period bin but it is likely they are present and potentially also more
weakly contaminating the $150-200$ day bin which also shows a slightly lower
$\sigma_{z}$ than perhaps expected. It is known that Mira variables in
globular clusters follow a period–metallicity relation with shorter-period
stars more prevalent in metal-poor clusters (Feast & Whitelock, 2000a). This
then suggests that the shortest period bin we considered has significant
contamination from more metal-poor objects and is not representative of the
broader solar neighbourhood samples used to calibrate the period–age
relations. However, it is then surprising that a more metal-poor population
would have a lower than expected dispersion as in both in-situ and accreted
scenarios the opposite is likely the case. More generally, our methodology
could be impacted by metallicity effects. We have already limited to O-rich
Mira variables which should preferentially remove metal-poor stars. Further
investigation is required to separate out the degeneracies between period, age
and metallicity, and a possible avenue is to consider the variation of
kinematics with unextincted colour as a metallicity proxy (e.g. Alvarez et
al., 1997).
_Hot-bottom burning_ : From equation (1), the slope of the period–luminosity
relation changes after $\mathrm{Period(days)}>400$. This hints that our O-rich
Mira variable star sample with periods above $400$ days is a mixture of hot-
bottom burning (HBB) stars and low-mass fundamental pulsators right at the end
of their lifetime (Whitelock et al., 2003; Trabucchi et al., 2019). The
balance of these two kinematically distinct populations depends on the star-
formation history (e.g. the HBB population would be reduced if there is no
recent star formation). Hence, as we are measuring the average age at a fixed
period, our result is somewhat related to the star-formation history of the
Milky Way. This mixing of HBB stars likely also broadens the period–age
relation for $\mathrm{Period(days)}>400$ (as it perhaps does the
period–luminosity relation e.g. Ita & Matsunaga, 2011), and it might address
the small discrepancy between our relation and the literature results shown in
Fig. 13. We hypothesise that the period–age relation is more universal and
reliable for periods under $400$ days.
_C-rich stars_ : Finally, a further direction is to consider the C-rich Mira
variables from Gaia. C-rich Mira variables also follow period–luminosity
relations that are typically broader than that for the O-rich Mira variables
due to circumstellar dust (Ita & Matsunaga, 2011). They also appear to trace
period–age relations (e.g. Feast et al., 2006, and evidenced in Fig. 13).
Typically they are less abundant in the Galaxy than the O-rich counterparts
(Ishihara et al., 2011) but importantly are biased towards younger ages (and
lower metallicities, e.g. Boyer et al., 2013) so present a route to better
constraining the longer-period end of the Mira variable period–age relation.
## 7 Conclusions
We have used the Gaia DR3 long-period variable candidate catalogue to produce
a calibration of the Mira variable period–age relation. Using a carefully
selected population of likely O-rich Mira variable stars, we have fitted a
series of action-based dynamical models to the stars separated by period. We
have found very good model fits for the velocity distributions of our sample
from which we have derived period–kinematic relations for the solar
neighbourhood. Comparison with an age–velocity dispersion relation for sub-
giant/red giant stars in the solar neighbourhood has allowed us to provide a
calibration of the Mira variable period–age relation.
Our derived relation agrees well with previous literature approaches using a
similar methodology and with the members of clusters with known ages. Some
theoretical models agree well with the derived relation but more recent
calibrations appear to be consistently younger at fixed period than our
relations suggest. Consideration of the age distribution of Mira variable
stars in the Galactic bar-bulge produces a consistent picture with other bar-
bulge age tracers using our relation.
This new period–age relation opens the possibility of inspecting the star
formation history and evolutionary properties of distant and/or highly-
extincted regions of our Galaxy and the Local Group. Mira variables are some
of the brightest stars in an intermediate-age population, their infrared
brightness makes them ideal tracers of dusty environments, and their high
amplitude and long periods mean they suffer low contamination. For these
reasons, in the era of JWST, Mira variables will provide us with a new window
of the evolution of the Universe.
## Data availability
All data utilised in this work are in the public domain. In the supplementary
material, we provide corner plots showing the posterior distributions of the
dynamical model parameters for each period bin.
## Acknowledgements
We thank the anonymous referee for a careful reading of the paper and
thoughtful comments that improved the presentation. JLS thanks the support of
the Royal Society (URF\R1\191555). This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular, the institutions
participating in the Gaia Multilateral Agreement. This publication makes use
of data products from the Two Micron All Sky Survey, which is a joint project
of the University of Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the National Aeronautics
and Space Administration and the National Science Foundation. This paper made
use of numpy (van der Walt et al., 2011), scipy (Virtanen et al., 2020),
matplotlib (Hunter, 2007), seaborn (Waskom et al., 2017), pandas (McKinney,
2010), corner (Foreman-Mackey, 2016) astropy (Astropy Collaboration et al.,
2013; Price-Whelan et al., 2018), galpy (Bovy, 2015), and Agama (Vasiliev,
2019).
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|
# Dynamic Price of Parking Service based on Deep Learning
Alejandro Luque-Cerpa Department of Computer Science and Artificial
Intelligence
University of Seville, Seville, Spain<EMAIL_ADDRESS>Miguel A.
Gutiérrez-Naranjo Department of Computer Science and Artificial Intelligence
University of Seville, Seville, Spain<EMAIL_ADDRESS>Miguel Cárdenas-Montes
Department of Fundamental Research, Centro de Investigaciones Energéticas
Medioambientales y Tecnológicas, Madrid, Spain<EMAIL_ADDRESS>
###### Abstract
The improvement of air-quality in urban areas is one of the main concerns of
public government bodies. This concern emerges from the evidence between the
air quality and the public health. Major efforts from government bodies in
this area include monitoring and forecasting systems, banning more pollutant
motor vehicles, and traffic limitations during the periods of low-quality air.
In this work, a proposal for dynamic prices in regulated parking services is
presented. The dynamic prices in parking service must discourage motor
vehicles parking when low-quality episodes are predicted. For this purpose,
diverse deep learning strategies are evaluated. They have in common the use of
collective air-quality measurements for forecasting labels about air quality
in the city. The proposal is evaluated by using economic parameters and deep
learning quality criteria at Madrid (Spain).
###### keywords:
Air Quality, Deep Learning , Convolutional Neural Networks , LSTM , U-Net
## 1 Introduction
Air pollution is one of the most critical health issues in urban areas with a
tough concern for governments and citizens. The scientific literature shows
its relation with the population health [1, 2, 3, 4, 5, 6, 7, 8, 9, 10].
Furthermore, the population growth in urban areas in the forthcoming years
will aggravate this issue.
At this point, a major contributor to the air pollution is the motor vehicles.
German Environment Agency and other studies estimate that road transportation
is responsible for about 60% of emissions of $NO_{2}$ in cities [11, 12]. For
monitoring and mitigating the adverse effect of air pollution, governments
have proposed a wide variety of actions, including: the creation of networks
of monitoring stations in the cities and outskirts, the establishing of
grading protocol scenarios, the speed limit reduction in the accesses to the
city centers, promotion of public transport, the creation of Low Emission
Zones, and prohibition of use of public parking when the air quality falls
below critical levels. In the past, some of these actions have demonstrated
their efficiency [11, 13]. However, to the best of our knowledge, a dynamic
price for public parking service as a function of the pollution level has not
been proposed yet.
For cities where it is illegal to forbid the parking during the low-quality
air episodes, this application aims at discouraging the use of public parking,
and therefore the private vehicles, during these episodes. By dynamically
adjusting the price of public parking during the high air pollution days,
restrictive actions —over the whole city or by district— can be implemented.
In the current work, diverse deep learning algorithms aimed at providing the
necessary support for dynamic pricing of public parking are proposed and
evaluated, namely, different models based on Convolutional Neural Networks,
LSTM layers and U-Time architecture are compared. Our proposals endorse
important features such as a collective behavior —the prediction is based on
measurements of multiple monitoring stations—, and measurability —both from
accuracy of prediction and economic side—. In order to fix a realistic
prediction adapted to the use of motor vehicles, days are divided in four
blocks of 6 hours each one, while the prediction focuses on the block II (from
06:00 to 12:00) and III (from 12:00 to 18:00). The blocks I (from 00:00 to
06:00) and IV (from 18:00 to 24:00) are discarded for this study due to the
low outdoor activities during those periods.
This proposal aligns with the Sustainable Development Goals (SDGs) of the
United Nations and concretely it may act as enabler for the goals Goal 3
”Ensure healthy lives and promote well-being for all at all ages” and
particularly for the Target 3.9. By 2030, substantially reduce the number of
deaths and illnesses from hazardous chemicals and air, water and soil
pollution and contamination” [14].
Concerning the data, it has been extracted from the Air Quality Monitoring
Network of Madrid [15]. This public repository offers hourly and daily data
from more than 24 monitoring stations, including three categories: suburban
(stations in parks in urban areas), traffic (term for stations affected by
traffic and close to a principal street or road), and background (urban
background station affected by both traffic and background pollution). For
this work, 12 monitoring stations have been selected.
The paper is organized as follows: In Section 2 the main background of the
work, including a description of the dataset, the deep architectures used are
presented. The Results and the Analysis are shown in Section 3. Finally, the
Conclusions are presented in Section 4.
## 2 Methods and Materials
### 2.1 Data Acquisition and Preprocessing
The datasets used in this work are obtained from Madrid’s City Council Open
Data website111https://datos.madrid.es/portal/site/egob/. They include daily
and hourly measurements of the concentration of diverse pollutants, such as
$CO$, $NO$ or $CH_{4}$. We will study the hourly data related to $NO_{2}$,
measured in $\mu$g/m3, between 2010/01 and 2019/12. The dataset contains some
missing values due to instrument failure and other reasons. In order to fill
the missing data, they are replaced with the corresponding monthly mean. A
min-max normalization method for scaling the data to the [0,1] range is also
used.
The data are obtained from 24 air quality stations distributed in 5 different
zones. Our data is collected from 12 of these stations: Escuelas Aguirre (EA)
Barrio del Pilar (BP), Plaza del Carmen (PlC), Retiro (Re), Ensanche de
Vallecas (EV), Arturo Soria (AS), Barajas (Ba), Juan Carlos I (PJCI), El Pardo
(ElP), Casa de Campo (CC), Fernández Ladreda (FL) y Farolillo (Faro). The list
of stations includes suburban, traffic, and background (Fig. 1).
Figure 1: Partial map of Madrid. The monitoring stations used in this study
are marked.
Since the goal of this work is to compute a dynamic price for the regulated
parking service during the periods where the main outdoors activities are
undertaken, as pointed above, the day is divided in four blocks: one block for
each quarter of the day. The proposal focuses on the air quality prediction
for the second (6:00-12:00) and third (12:00-18:00) blocks.
Regarding the air-quality levels, Madrid City Council through the Action
protocol for nitrogen dioxide pollution
episodes222https://www.madrid.es/UnidadesDescentralizadas/AreasUrbanas_EducacionAmbiental/
Catalogo/AirQualityPlan2011-15.pdf defines three alert levels:
* 1.
Pre-warning: when in any two stations of the same area, the 180 $\frac{\mu
g}{m^{3}}$ of $NO_{2}$ level is exceeded for two consecutive hours
simultaneously or in any three stations of the surveillance network the 180
$\frac{\mu g}{m^{3}}$ level is exceeded for three consecutive hours
simultaneously.
* 2.
Warning: when in any two stations of the same zone the 200 $\frac{\mu
g}{m^{3}}$ of $NO_{2}$ level is exceeded during two consecutive hours
simultaneously or in any three stations of the surveillance network the 200
$\frac{\mu g}{m^{3}}$ are exceeded during three consecutive hours
simultaneously.
* 3.
Alert: when any three stations in the same zone (or two if it is zone 4)
exceed 400 $\frac{\mu g}{m^{3}}$ for three consecutive hours simultaneously.
Let us remark that these pollution levels are not coherent with the air
quality of a city like Madrid: the 200 $\frac{\mu g}{m^{3}}$ of $NO_{2}$ level
is hardly reached, and the 400 $\frac{\mu g}{m^{3}}$ of $NO_{2}$ level is
never reached. Because of this, two different rules with four alert levels
each of them are proposed in this work. Instead of levels based on absolute
values, our proposal is based on annual percentiles. The first rule is the
following:
Rule I
* 1.
Pre-warning: when the 75th annual percentile is exceeded for three consecutive
hours simultaneously in any three stations of the surveillance network.
* 2.
Warning: when the 90th annual percentile is exceeded for three consecutive
hours simultaneously in any three stations of the surveillance network.
* 3.
Alert: when the 95th annual percentile is exceeded for three consecutive hours
simultaneously in any three stations of the surveillance network.
* 4.
No alert: if none of the previous conditions is satisfied.
Every 6-hours group is classified in one of these four alert levels. Since the
warning and alert level are so similar, it is expected that deep learning
algorithms will hardly differentiate them. Because of this, a second rule
—less stringent— with another four pollution levels are proposed:
Rule II
* 1.
Pre-warning: when the 50th annual percentile is exceeded for three consecutive
hours simultaneously in any three stations of the surveillance network.
* 2.
Warning: when the 75th annual percentile is exceeded for three consecutive
hours simultaneously in any three stations of the surveillance network.
* 3.
Alert: when the 95th annual percentile is exceeded for three consecutive hours
simultaneously in any three stations of the surveillance network.
* 4.
No alert: if none of the previous conditions is satisfied.
### 2.2 Convolutional Neural Networks
Convolutional Neural Networks (CNN) is one of the most popular Deep Learning
architectures [16]. They are designed to deal with pieces of information
placed on a $n$-dimensional grid and, therefore, they are profusely used to
deal with digital images, although nowadays they have applications in a wide
range of research areas including time series analysis [17] or navigation in
indoor environments [18]. Many different architectures have been defined in
the class of CNN, but all of them consist of a sequence of convolutional
layers, which deal with local environments of the grid by using the so-called
kernels together with other kind ways of processing information, which can
involve pooling-based layers, dropout and other regularization techniques.
Although CNN are usually associated with image or audio classification —2D
grid examples—- or video sequence —3D grid examples—, it can also be applied
to univariate time series analysis —1D grid examples— or multivariate time
series —2D grid examples—. The key idea in all these applications is the use
of several convolutional layers in order to abstract local semantic features
by the use of kernels which can be optimized in the training process.
In this paper, our target is to predict the alert level of pollution in a city
from a dataset which includes spatio-temporal information. The data has a
time-line dimension but also a spatial dimension due the geographical
distribution of the weather stations. Such distribution produces some kind of
correlation in the data, since close stations report close pollution levels.
In order to handle such complexity, several Deep Learning architectures have
been considered. Data have been presented as a two-dimensional matrix with one
column for each weather station and a row for each time unit. Since CNN is
adapted to data placed on a grid, one of the considered options has been to
apply this architecture to our 2D matrix.
### 2.3 LSTM
If CNN has become the standard tool for studying images with Deep Learning
techniques, Long short-term memory (LSTM) [19] architecture has become a
standard for the study of sequential information. This architecture belongs to
the so-called gated recurrent neural networks and the main motivation for
using them to create paths along time to avoid the vanishing gradient problem.
For this purpose, instead of considering simple neurons which apply an
activation function to the affine transformation of inputs, LSTM architectures
consider the so-called LSTM cells which have internal recurrence in addition
to the outer recurrence of the recurrent neural network.
The key idea on these LSTM cells is to keep information in a cell state for
later, preventing older signals from vanishing during processing. LSTM systems
can add or remove information of such cell states by structures called gates.
In LSTM, there are three types of gates: input gate, forget gate and output
gate. In 2014, Cho et al. [20] presented a new type of gated recurrent neural
network with fewer parameters than LSTM, as it lacks an output gate.
As pointed above, collected pollution data are sequentially grouped by time
units. Such a temporal dimension has been exploited in our analysis of the
data by using a LSTM architecture.
### 2.4 U-Time
In 2015, Ronneberger al [21] presented a new Deep Learning architecture for
biomedical image segmentation called U-net. The networks with this
architecture are basically CNN endowed with new features. Technically, a
contracting path and an expansive path are considered, which give to the model
the U-shaped architecture. The contracting path is a standard CNN which
includes convolutions layers, ReLU as activation functions and max pooling
operations. As usual, in the convolutional layers, the spatial information is
reduced while feature information is increased. The original part of the U-net
architecture is the expansive pathway which combines the feature and spatial
information through a sequence of up-convolutions and concatenations with
high-resolution features from the contracting path.
In 2019, Perslev et al. [22] presented a fully convolutional encoder-decoder
network called U-Time inspired by the U-Net architecture. U-Time adopts basic
concepts from U-Net for 1D time-series segmentation by mapping a whole
sequence to a dense segmentation in a single forward pass. Originally, U-Time
was used for studying time series segmentation applied to sleep staging. In
this paper, U-time is used as inspiration for our study of the air-quality in
Madrid. In a similar way to the study based on simple CNN architectures, the
U-net study also exploits the 2D representation of the data on space and time.
## 3 Results and Analysis
Since the goal of this paper is to predict the pollution level of six-hours
blocks knowing the previous pollution levels, different models and initial
conditions are considered. As pointed above, two different rules of pollution
alert levels are considered: Rule I for the 95th, 90th and 75th percentiles,
and Rule II for the 95th, 75th and 50th percentiles. Three distinct sizes of
input data are considered: one day of data per station (24x12 values), three
days of data per station (72x12 values) and a week of data per station (168x12
values).
Some considerations must be taken into account. Firstly, the classes are
highly unbalanced: most of the days, the pollution alert level is null.
Secondly, the 95th, 90th and 75th percentiles are closer together than the
95th, 75th and 50th percentiles, so we expect the models to obtain worse
results for the first pollution alert levels than for the second ones.
Finally, it is better to obtain false positives than false negatives, given
that the ultimate objective is reducing pollution.
### 3.1 Confusion Matrix based Analysis
In this work, four different pollution alert levels are defined: Pre-Warning,
Warning, Alert or No-alert. To correctly evaluate the performance of a model
predicting the pollution alert level, it is necessary to obtain the accuracy
of such model for each of these levels. The results are presented using
confusion matrices, where the performance of the different models can be
correctly visualized, and more detailed analysis is allowed. Each row of a
matrix represents the instances in an actual class, while each column
represents the instances in a predicted class.
#### 3.1.1 LSTM
In this first approach, a model based on LSTM cells is considered, namely a
model with two LSTM layers, the first one with 50 neurons and the second one
with 10 neurons. After these layers a Dropout layer [23] with index 0.05 was
included. Finally, an output layer with four neurons and softmax as activation
function is added. Cross-entropy loss has been used for training. After
training the LSTM with different data blocks, input sizes and pollution
levels, the confusion matrices are obtained. Let us remark that the classes
have been balanced before training in order to avoid overfitting.
As it is shown in Figs. 2a, 2b, and 2c; and in Figs. 4a, 4b, and 4c for the
normalized confusion matrices, for the Rule I and for the second block
(6:00-12:00), positive results are achieved. There is a significant difference
between using 24 hours, 72 hours or 168 hours of input data, obtaining better
results when using 72 hours of input data, the accuracy ranges from 63% to
80%; and when using 168 hours of input data, the accuracy ranges from 53% to
83%. In both cases, the adjacent errors —labels erroneously predicted as a
contiguous label— and the non-adjacent errors —labels predicted as a non-
contiguous label— are high. They rise up to 26%.
In a similar way, as detailed in Figs. 2d, 2e, and 2f; and in Figs. 4d, 4e,
and 4ffor the normalized confusion matrices for the Rule I and for the third
block (12:00-18:00), the results are confusing, without a obvious best
configuration, with low accuracy in the main diagonal, and large adjacent and
non-adjacent errors. It is important to remark that for both blocks the
results are not concentrated around the main diagonal, but scattered. This is
probably due to the closeness of the percentiles values, and as a consequence
of the imbalanced labels.
In the case of the pollution alert levels for the Rule II, as illustrated in
Figs. 3a, 3b, and 3c for the second block, and Figs. 3d, 3e, and 3f for the
third block; and for the normalized confusion matrices in Figs. 5a, 5b, and 5c
for the second block, and Figs. 5d, 5e, and 5f for the third block, the
results for both blocks are better than those obtained for Rule I. This stems
from the less imbalance labels under the Rule II percentiles. For the second
block and the third block, it is difficult to ascertain which configuration
produces the best results. For the second block, the accurate results —main
diagonal— range from 74% to 89% for the 72 hours configuration to the range
from 64% to 92% for the 128 hours configuration. For the third block, the
accurate results overlap: from 69% to 83% when using the configuration of 24
hours, from 49% to 91% when using the configuration of 72 hours, and from 69%
to 87% when using the configuration of 168 hours. At the same time large
adjacent errors are also produced, for instance the 28% of Pre-Warning are
incorrectly predicted as No alert when using the configuration of 72 hours.
(a) Second block. 24 hours.
(b) Second block. 72 hours.
(c) Second block. 168 hours.
(d) Third block. 24 hours.
(e) Third block. 72 hours.
(f) Third block. 168 hours.
Figure 2: Confusion matrices obtained from the LSTM model for the Rule I.
Three input sizes are considered: 24 hours, 72 hours and 168 hours of data per
station. Two blocks are considered: the second block (6:00-12:00) and the
third block (12:00-18:00). Each row represents the actual class and each
column represents the predicted class.
(a) Second block. 24 hours.
(b) Second block. 72 hours.
(c) Second block. 168 hours.
(d) Third block. 24 hours.
(e) Third block. 72 hours.
(f) Third block. 168 hours.
Figure 3: Confusion matrices obtained from the LSTM model for the Rule II.
Three input sizes are considered: 24 hours, 72 hours and 168 hours of data per
station. Two blocks are considered: the second block (6:00-12:00) and the
third block (12:00-18:00). Each row represents the actual class and each
column represents the predicted class.
(a) Second block. 24 hours.
(b) Second block. 72 hours.
(c) Second block. 168 hours.
(d) Third block. 24 hours.
(e) Third block. 72 hours.
(f) Third block. 168 hours.
Figure 4: Normalized confusion matrices obtained from the LSTM model for the
Rule I. Three input sizes are considered: 24 hours, 72 hours and 168 hours of
data per station. Two blocks are considered: the second block (6:00-12:00) and
the third block (12:00-18:00). Each row represents the actual class and each
column represents the predicted class.
(a) Second block. 24 hours.
(b) Second block. 72 hours.
(c) Second block. 168 hours.
(d) Third block. 24 hours.
(e) Third block. 72 hours.
(f) Third block. 168 hours.
Figure 5: Normalized confusion matrices obtained from the LSTM model for the
Rule II. Three input sizes are considered: 24 hours, 72 hours and 168 hours of
data per station. Two blocks are considered: the second block (6:00-12:00) and
the third block (12:00-18:00). Each row represents the actual class and each
column represents the predicted class.
#### 3.1.2 CNN
In this approach, a model of CNN networks is considered. The architecture of
the model can be summarized as follows: (1) a first convolutional layer with
16 kernels, kernel size 5 and ReLU as activation function; (2) a max-pooling
layer; (3) a second convolutional layer with 64 kernels, kernel size 3 and
ReLU as activation function; (4) another max-pooling layer; (5) A dense layer
with 20 neurons and ReLU as activation function; (6) a dropout layer with
index 0.05; (7) a final dense output layer with 20 neurons and softmax as
activation function. Cross-entropy loss has been used for training. The
confusion matrices are shown in Figs. 6, and 7; while the next comparisons are
undertaken by the normalized ones.
Following the same line of experiments, the CNN model is trained for the same
conditions that are established in the case of the LSTM model. Comparing Figs.
4a, 4b, 4c and Figs. 8a, 8b, 8c, it is shown that better results are obtained
in the case of the CNN than in the LSTM one for the second block (6:00-12:00)
and the Rule I. When using CNN model, there is a significant difference
between using 24, 72 or 168 hours of input data, obtaining better results for
greater amounts of hours. For 168 hours of input data, the accuracy ranges
from 73% to 94%. For 71 hours, the accuracy has a higher inferior accuracy
74%, but also exhibits a much lower higher accuracy 86%. For the three
configurations of number of hours, CNN outperforms the equivalent LSTM model
for the second block and the Rule I.
Comparing now Figs. 4d, 4e, 4f and Figs. 8d, 8e, 8f, we obtain considerably
better results for the Rule I and for the third block (12:00-18:00) from the
CNN model. As for the second block, there is a significant difference between
using 24, 72 or 168 hours of input data, obtaining better results for those
configurations with larger amounts of hours. The accuracy for 168 hours
configuration ranges from 80% to 98%. Besides, the CNN model shows the lowest
high-adjacent errors and non-adjacent errors than their equivalents in the
LSTM model.
Considering now the Rule II and second block, the results of LSTM (Figs. 5a,
5b, 5c) and CNN (Figs. 9a, 9b, 9c) models are difficult to ascertain a best
configuration and model. Ranges of accuracy overlapping, rawly being better
the high accuracy of CNN model —up to 95% for the configuration of 168 hours—,
but being in many cases the low accuracy worse than their counterpart
configuration at LSTM model. In this case, the exploration of diverse models
takes a great value, and the evaluation of an ensemble model could be proposed
as future work. The lowest high-errors, both adjacent and non-adjacent, are
achieved for the configuration of 168 hours, being respectively 11% for the
Alert labels predicted as Warning, and 2% for two cases.
Finally, for the Rule II and for the third block (12:00-18:00), we can observe
that the CNN model (Figs. 9d, 9e, 9f) obtains again better results than the
LSTM model (Figs. 5d, 5e, 5f). The accuracy for the CNN model for 72 hours
configuration ranges from 82% to 93%, at the same time that it keeps low the
highest errors: adjacent and non-adjacent.
(a) Second block. 24 hours.
(b) Second block. 72 hours.
(c) Second block. 168 hours.
(d) Third block. 24 hours.
(e) Third block. 72 hours.
(f) Third block. 168 hours.
Figure 6: Confusion matrices obtained from the CNN model for the Rule I. Three
input sizes are considered: 24 hours, 72 hours and 168 hours of data per
station. Two blocks are considered: the second block (6:00-12:00) and the
third block (12:00-18:00). Each row represents the actual class and each
column represents the predicted class.
(a) Second block. 24 hours.
(b) Second block. 72 hours.
(c) Second block. 168 hours.
(d) Third block. 24 hours.
(e) Third block. 72 hours.
(f) Third block. 168 hours.
Figure 7: Confusion matrices obtained from the CNN model for the Rule II.
Three input sizes are considered: 24 hours, 72 hours and 168 hours of data per
station. Two blocks are considered: the second block (6:00-12:00) and the
third block (12:00-18:00). Each row represents the actual class and each
column represents the predicted class.
(a) Second block. 24 hours.
(b) Second block. 72 hours.
(c) Second block. 168 hours.
(d) Third block. 24 hours.
(e) Third block. 72 hours.
(f) Third block. 168 hours.
Figure 8: Normalized confusion matrices obtained from the CNN model for the
Rule I and the Rule II. Three input sizes are considered: 24 hours, 72 hours
and 168 hours of data per station. Two blocks are considered: the second block
(6:00-12:00) and the third block (12:00-18:00). Each row represents the actual
class and each column represents the predicted class.
(a) Second block. 24 hours.
(b) Second block. 72 hours.
(c) Second block. 168 hours.
(d) Third block. 24 hours.
(e) Third block. 72 hours.
(f) Third block. 168 hours.
Figure 9: Normalized confusion matrices obtained from the CNN model for the
Rule II. Three input sizes are considered: 24 hours, 72 hours and 168 hours of
data per station. Two blocks are considered: the second block (6:00-12:00) and
the third block (12:00-18:00). Each row represents the actual class and each
column represents the predicted class.
#### 3.1.3 U-Time
Due to the structural needs of U-Time, we decide to only use the configuration
of 168 hours of input data for all the experiments performed with this model
(Figs. 10 and 11). U-Time is a deep autoencoder that reduces 1.920 values of
the input data into one single value of the latent vector. If we want to
benefit from the properties of U-Time, the input size must be as large as
possible. Shorter configurations are unable to pass through the U-Time model.
By analysing the results obtained from the U-Time model for the second and the
third blocks, and for the Rule I and the Rule II, it can be observed that they
are worse than those obtained from the other models. The accuracy of the
U-Time model ranges from 43% to 74% for the second block and the Rule I (Fig.
10c), from 34% to 85% for the third block and the Rule I (Fig. 10d), from 46%
to 89% for the second block and the Rule II (Fig. 11c), and from 24% to 91%
for the third block and the Rule II (Fig. 11d). The predicted values are also
scattered and not concentrated around the main diagonal. Adjacent errors grow
up to 41% for the Pre-Warning labels predicted as No-alert (Fig. 11d). And for
the non-adjacent errors, they rise up to 36% for the pre-warning labels
predicted as Alert (Fig. 10d). The CNN model clearly outperforms the U-Time
model for both blocks and rules.
(a) Second block. 168 hours.
(b) Third block. 168 hours.
(c) Second block. 168 hours. Normalized.
(d) Third block. 168 hours. Normalized.
Figure 10: Confusion matrices obtained from the U-Time model for the Rule I.
Only one input size is considered: 168 hours of data per station. Two blocks
are considered: the second block (6:00-12:00) and the third block
(12:00-18:00). Each row represents the actual class and each column represents
the predicted class.
(a) Second block. 168 hours.
(b) Third block. 168 hours.
(c) Second block. 168 hours. Normalized.
(d) Third block. 168 hours. Normalized.
Figure 11: Confusion matrices obtained from the U-Time model for the Rule II.
Only one input size is considered: 168 hours of data per station. Two blocks
are considered: the second block (6:00-12:00) and the third block
(12:00-18:00). Each row represents the actual class and each column represents
the predicted class.
### 3.2 Economic Analyses
Depending on the error direction, the parking regulatory system could
overcharge over the customers. This scenario corresponds when the predicted
labels have a higher criticality than the final true labels. Oppositely, when
the predicted labels have a lower criticality that the true labels two
pernicious effects appear. On the one hand, the charge for the use of the
parking is undercharged; and on the other hand, the deterrent effect over the
use of private motor vehicles vanish.
An additional consideration is the fairness of the prediction. A well-balanced
deep learning architecture with null economic impact in a temporal period, but
with large errors in both directions is not acceptable. This kind of error
leads to distrust of the citizens on IA applications. For this reason, a rate
for the hours of each criticality level is proposed, the application of this
rate is evaluated. The evaluation is undertaken by adding the incorrect charge
in each direction of the errors and then their absolute values added.
The prices per hour established are for the criticality levels area: 0.40,
0.60, 1.20 and 2.40 euros. Currently the price per one hour of public parking
at Madrid is 0.40 euros. In our proposal, this price is increased by 50% when
the criticality level goes up from No-alert to Pre-Warning, and then doubled
for the two next levels: Warning and Alert. The aim of doubling the price for
the two last criticality levels is to simulate the deterrent effect in the use
of private transport.
Then the economic impact —per 6 hours block— in each direction is evaluated
and their absolute values added (Tables 1 and 2). The lower the values, the
fairness of the architecture. Except for the case of the prediction for 1 day
of the criticality level in Rule II and Block II, the CNN-based architecture
produces the lowest errors and the highest fairness among the architectures
tested.
Table 1: Evaluation of the deep architectures for block II based on economic criterion. | | CNN | LSTM | UTime
---|---|---|---|---
Rule I | 1d | 872.2 | 1044.2 | NaN
3d | 354.8 | 808.6 | NaN
7d | 337.6 | 1006.6 | 816.0
Rule II | 1d | 837.8 | 708.2 | NaN
3d | 566.4 | 662.6 | NaN
7d | 493.2 | 833.4 | 1583.2
Table 2: Evaluation of the deep architectures for block III based on economic criterion. | | CNN | LSTM | UTime
---|---|---|---|---
Rule I | 1d | 246.8 | 522.8 | NaN
3d | 174.0 | 643.8 | NaN
7d | 114.0 | 593.2 | 1213.2
Rule II | 1d | 306.4 | 393.0 | NaN
3d | 187.0 | 501.8 | NaN
7d | 202.6 | 362.4 | 660.2
## 4 Conclusions
In summary, air pollution is a critical problem in densely populated areas.
Some measures have been implemented, like traffic limitations during periods
of low-quality air or forecasting systems. In our research, we conclude that
deep learning models can be useful to implement another kind of measure: some
dynamic regulated parking services that would discourage motor vehicles
parking when low-quality episodes are predicted. To achieve this objective,
three different proposals are considered: one based on LSTM, one based on CNN
and one based on the U-Time model.
While U-Time was a promising architecture, the results obtained were less than
ideal. Due to its size, the computation time needed is between three and four
times the computation time that the LSTM or the CNN models need, and the CNN
model outperforms both of them. Analysing the results obtained by these
models, we can draw some conclusions. First, it is better to use the
percentile levels 95th, 75th and 50th than the 95th, 90th and 75th. This must
be because of the fact that the last percentiles are closer together than the
first ones. Second, there is a significant difference between the results
obtained with the selected input sizes: usually it is better to use greater
input sizes. Third, it is a good sign that the amount of false negatives is
lesser than the amount of false positives. This is important for us due to the
nature of our proposal: it is more desirable to have less motor vehicles in a
regular episode, than more motor vehicles in a low-quality episode.
## 5 Future Work
In this work, the input data is taken from 12 of the 24 different air quality
stations. There is a simple reason for this: only these 12 stations have been
extracting data since 2010/01. Once there is enough data, it would be
interesting to train again the models with data taken from the 24 air quality
stations. In the case of the U-Time model, it is possible that, once there is
more data, the model could be deeper and consequently get better results.
We have decided to use three different sizes of input data: 24 hours, 72 hours
and 168 hours. It would be interesting to use greater sizes of input data,
such as 336 hours (two weeks) or even 672 hours (a month). With greater input
sizes it would be coherent to also train models with more parameters, and
because of this it would be reasonable to train CNN or LSTM models with more
layers; more filters or bigger kernels in the case of the CNN models; or more
neurons per layer in the case of the LSTM model.
Considering our goal, we have proposed a dynamic parking regulatory system
considering both the fairness and the deterrent effect in the use of private
transport, and we have defined four different alert levels based on
percentiles: No-alert, Pre-Warning, Warning and Alert. It would be interesting
to work with expert environmental scientists that could establish more useful
and realistic thresholds and pollution alert levels.
## Acknowledgements
MCM is co-funded by the Spanish Ministry of Science and Innovation for funding
support through the grant PID2020-113807RA-I00 ”SERVICIOS INNOVADORES DE
ANALISIS DE DATOS PARA EL EXPERIMENTO CMS”. MAGN is partially supported by the
Spanish Ministry of Science and Innovation through the project
PID2019-107339GB-I00 ”Advances in Computational Topology and Applications”.
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# On Gravitational Stefan-Boltzmann Law and Casimir Effect in FRW Universe
A. F. Santos<EMAIL_ADDRESS>Instituto de Física,
Universidade Federal de Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil S. C. Ulhoa<EMAIL_ADDRESS>International Center of Physics, Instituto de Física, Universidade de
Brasília, 70910-900, Brasília, DF, Brazil Canadian Quantum Research Center,
204-3002 32 Ave Vernon, BC V1T 2L7 Canada E. P. Spaniol<EMAIL_ADDRESS>UDF Centro Universitário and Centro Universitário de Brasília UniCEUB,
Brasília, DF, Brazil. Faqir C. Khanna111Professor Emeritus - Physics
Department, Theoretical Physics Institute, University of Alberta
Edmonton, Alberta, Canada<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Victoria,
3800 Finnerty Road Victoria, BC, Canada
###### Abstract
Both Stefan-Boltzmann law and the Casimir effect, in a universe described by
the FRW metric with zero curvature, are calculated. These effects are
described by Thermo Field Dynamics (TFD). The gravitational energy-momentum
tensor is defined in the context of Teleparallel Equivalent to General
Relativity (TEGR). Each of the two effects gives a consistent prediction with
what is observed on a cosmological scale. One of the effect establishes a
minimum range for the deceleration parameter. While another leads to the
conclusion that a possible cosmological constant has a very small order of
magnitude.
## I Introduction
The introduction of temperature in the gravitational field has been
successfully implemented recently gravTFD . Thermo Field Dynamics (TFD) was
used for this purpose which is an approach that allows both a temporal
evolution of the field at finite temperature. It is an advantage over the
historical approach that associates time with temperature matsubara . A
gravitational field at finite temperature is a theory of quantum gravity since
TFD uses creation and annihilation operators. The field propagator is the
fundamental entity of the thermalization process. It is interesting to note
that such an approach associates the temperature with space such that a
universe with zero temperature will not be expected flatTFD . Absolute zero
temperature will not be natural even in flat space. Within the scope of TFD,
there is a topological structure that allows treating effects such as diverse
as the Stefan-Boltzmann law and the Casimir effect on an equal footing. An
area of intense investigation into the implications of various aspects of
quantum gravity is black hole thermodynamics. Whether in the investigation of
entropy of black holes, or in the understanding of the information paradox. It
was recently investigated how the evaporation process of a black hole
generates an entanglement between quantum fields and geometry, this yields a
modified Page curve that can have implications for several theories of quantum
gravity 4 . It has also been shown that the structure of TFD plays a key role
in this approach 5 . The TFD appears to be a promising theory of quantum
gravity.
It is necessary to thermalize the energy-momentum tensor of the field in
addition to a propagator. The standard model of gravitation is problematic. In
the construction of gravity at finite temperature, an alternative theory of
gravitation is used, Teleparallelism Equivalent to General Relativity (TEGR)
maluf . In TEGR the problem of gravitational energy is well established, as
well as other conserved quantities. As a result, gravitational entropy is
introduced as a direct consequence of Maxwell’s relationships involving
gravitational pressure. Normally this gravitational entropy may be seen as a
fundamental quantity when made equal to Hawking’s expression induces a
temperature of the black hole event horizon different from that commonly
accepted. The whole space-time has a finite temperature, not just the event
horizon of a black hole. Then there is a smooth transition from singularity to
infinity entropy . We must note that TEGR is a formulation of gravitation that
takes into account local Lorentz’s symmetry, such a dependence appears in the
field equations that are entirely equivalent to Einstein’s equations. On the
other hand, recently, proposals have emerged that attribute the local Lorentz
symmetry to the spin connection 7 ; 8 ; 9 , this line of investigation has
received some criticism and in our opinion still requires further
investigation 10 . In TEGR the conserved quantities are sensitive to the
global Lorentz transformations and that is the limit of our approach.
One of the major problems in cosmology is why there is an accelerated
expansion of the universe. Usually the explanation given is an exotic energy
known as dark energy. On the other hand, instead of looking for candidates for
such energy, alternative explanations can be tried. This last chain of thought
will be used. The more interesting features of the universe are analyzed.
There is a non-zero temperature other than zero even at the most distant point
in interstellar space. In addition, it has an observable dynamic horizon
increasing with time. Such a horizon works as a causal barrier to events
within it. Mainly this system behaves like a spherical Casimir effect. There
appears to be two associated phenomena observed in the universe: i) a thermal
radiation like Stefan-Boltzmann’s law and ii) a force a la Casimir effect
responsible for an accelerated expansion of the system. This leads us to
consider that gravitation at finite temperature explains such a phenomena.
This hypothesis is explored here.
This article is divided as follows. In section II TFD is introduced briefly.
In section III the TEGR is presented and the thermal expressions are
calculated. In section IV the energy-momentum tensor at finite temperature is
applied to the FRW universe. With this both the Stefan-Boltzmann law and the
Casimir effect for a zero curvature in the metric are calculated. Finally
conclusions are presented in the last section.
## II Thermo Field Dynamics (TFD)
A quantum field theory at finite temperature is developed by two distinct, but
equivalent, approaches: (i) the imaginary time formalism matsubara and (ii)
the real time formalism Schwinger ; Umezawa1 ; Umezawa2 ; Umezawa22 ; Khanna1
; Khanna2 . TFD is a real-time finite temperature formalism. The temperature
dependent vacuum is defined such that the vacuum expectation value of an
arbitrary operator $\displaystyle A$ agrees with the statistical average,
i.e.,
$\displaystyle\displaystyle\langle A\rangle=\langle
0(\beta)|A|0(\beta)\rangle,$ (1)
where $\displaystyle|0(\beta)\rangle$ is the thermal vacuum and
$\displaystyle\beta=\frac{1}{k_{B}T}$, with $\displaystyle T$ being the
temperature and $\displaystyle k_{B}$ the Boltzmann constant. To construct
this thermal state two elements are necessary: the doubling of the original
Hilbert space and the Bogoliubov transformation. This doubling is defined by
$\displaystyle{\cal S}_{T}={\cal S}\otimes\tilde{\cal S}$, where
$\displaystyle{\cal S}$ is the Hilbert space and $\displaystyle\tilde{\cal S}$
is the dual (tilde) space. The map between the non-tilde
$\displaystyle{A_{i}}$ and tilde $\displaystyle\tilde{A_{i}}$ operators is
given by tilde (or dual) conjugation rules. These rules are
$\displaystyle\displaystyle(A_{i}A_{j})^{\thicksim}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\tilde{A_{i}}\tilde{A_{j}},$
$\displaystyle\displaystyle(cA_{i}+A_{j})^{\thicksim}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
c^{*}\tilde{A_{i}}+\tilde{A_{j}},$
$\displaystyle\displaystyle(A_{i}^{\dagger})^{\thicksim}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\tilde{A_{i}}^{\dagger},$
$\displaystyle\displaystyle(\tilde{A_{i}})^{\thicksim}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle-\xi A_{i},$ (2)
with $\displaystyle\xi=-1(+1)$ for bosons (fermions). In addition, the tilde
conjugation rules associate each operator in $\displaystyle{\cal S}$ to two
operators in $\displaystyle{\cal S}_{T}$. Considering $\displaystyle a$ as an
operator leads to
$\displaystyle\displaystyle A=a\otimes
1,\quad\quad\quad\quad\tilde{A}=1\otimes a.$ (3)
TFD and Bogoliubov transformations introduce thermal effects through a
rotation between tilde ($\displaystyle\tilde{\cal S}$) and non-tilde
($\displaystyle{\cal S}$) operators. With an arbitrary operator
$\displaystyle{\cal O}$, the Bogoliubov transformation is defined as
$\displaystyle\displaystyle\left(\begin{array}[]{cc}{\cal O}(k,\alpha)\\\
\xi\tilde{\cal O}^{\dagger}(k,\alpha)\end{array}\right)={\cal
B}(\alpha)\left(\begin{array}[]{cc}{\cal O}(k)\\\ \xi\tilde{\cal
O}^{\dagger}(k)\end{array}\right),$ (8)
where the $\displaystyle\alpha$ is called the compactification parameter
defined by $\displaystyle\alpha=(\alpha_{0},\alpha_{1},\cdots\alpha_{D-1})$
and $\displaystyle{\cal B}(\alpha)$ is
$\displaystyle\displaystyle{\cal
B}(\alpha)=\left(\begin{array}[]{cc}u(\alpha)&-w(\alpha)\\\ \xi
w(\alpha)&u(\alpha)\end{array}\right),$ (11)
with $\displaystyle u^{2}(\alpha)+\xi w^{2}(\alpha)=1$. These quantities
$\displaystyle u(\alpha)$ and $\displaystyle w(\alpha)$ are related to the
Bose distribution. For the case $\displaystyle\alpha_{0}\equiv\beta$ and
$\displaystyle\alpha_{1},\cdots\alpha_{D-1}=0$, the temperature effect is
introduced. Using such formalism, a topological quantum field theory is
considered. A topology
$\displaystyle\Gamma_{D}^{d}=(\mathbb{S}^{1})^{d}\times\mathbb{R}^{D-d}$ with
$\displaystyle 1\leq d\leq D$ is used. Here $\displaystyle D$ is the space-
time dimensions and $\displaystyle d$ is the number of compactified
dimensions. Any set of dimensions of the manifold
$\displaystyle\mathbb{R}^{D}$ can be compactified.
In the TFD formalism, all propagators are written in terms of the
compactification parameter $\displaystyle\alpha$. Here the scalar field
propagator is defined as
$\displaystyle\displaystyle G_{0}^{(AB)}(x-x^{\prime};\alpha)=i\langle
0,\tilde{0}|\tau[\phi^{A}(x;\alpha)\phi^{B}(x^{\prime};\alpha)]|0,\tilde{0}\rangle,$
(12)
where $\displaystyle\tau$ is the time ordering operator and $\displaystyle
A\,\mathrm{and}\,B=1,2$. The Bogoliubov transformation is used to write as
$\displaystyle\displaystyle\phi(x;\alpha)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle{\cal B}(\alpha)\phi(x){\cal B}^{-1}(\alpha).$ (13)
In the thermal vacuum, which is defined as
$\displaystyle|0(\alpha)\rangle={\cal U}(\alpha)|0,\tilde{0}\rangle$, the
propagator becomes
$\displaystyle\displaystyle G_{0}^{(AB)}(x-x^{\prime};\alpha)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle i\langle
0(\alpha)|\tau[\phi^{A}(x)\phi^{B}(x^{\prime})]|0(\alpha)\rangle,$ (14)
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ik(x-x^{\prime})}G_{0}^{(AB)}(k;\alpha),$
where
$\displaystyle\displaystyle G_{0}^{(AB)}(k;\alpha)={\cal
B}^{-1}(\alpha)G_{0}^{(AB)}(k){\cal B}(\alpha),$ (15)
with
$\displaystyle\displaystyle
G_{0}^{(AB)}(k)=\left(\begin{array}[]{cc}G_{0}(k)&0\\\ 0&\xi
G^{*}_{0}(k)\end{array}\right),$ (18)
and
$\displaystyle\displaystyle G_{0}(k)=\frac{1}{k^{2}-m^{2}+i\epsilon},$ (19)
where $\displaystyle m$ is the mass. The Green function becomes
$\displaystyle\displaystyle G_{0}^{(11)}(k;\alpha)=G_{0}(k)+\xi
w^{2}(k;\alpha)[G^{*}_{0}(k)-G_{0}(k)].$ (20)
Here the physical quantities are given by the non-tilde variables, i.e.
$\displaystyle A=B=1$. In addition, $\displaystyle w^{2}(k;\alpha)$ is the
generalized Bogoliubov transformation GBT given as
$\displaystyle\displaystyle
w^{2}(k;\alpha)=\sum_{s=1}^{d}\sum_{\\{\sigma_{s}\\}}2^{s-1}\sum_{l_{\sigma_{1}},...,l_{\sigma_{s}}=1}^{\infty}(-\xi)^{s+\sum_{r=1}^{s}l_{\sigma_{r}}}\,\exp\left[{-\sum_{j=1}^{s}\alpha_{\sigma_{j}}l_{\sigma_{j}}k^{\sigma_{j}}}\right],$
(21)
where $\displaystyle\\{\sigma_{s}\\}$ denotes the set of all combinations with
$\displaystyle s$ elements and $\displaystyle k$ is the 4-momentum.
## III Teleparallel Gravity
Teleparallelism Equivalent to General Relativity (TEGR) is dynamically
equivalent to the standard theory of gravitation formulated in a Riemann
space. However, TEGR is described in terms of torsion in the Weitzenböck
space. The connection in such a space is
$\displaystyle\Gamma_{\mu\lambda\nu}=e^{a}\,_{\mu}\partial_{\lambda}e_{a\nu}\,,$
where $\displaystyle e^{a}\,_{\mu}$ is the tetrad field. It is the dynamical
variable of the theory. The relationship between the metric tensor and the
tetrad field is $\displaystyle g_{\mu\nu}=e^{a}\,_{\mu}e_{a\nu}\,.$ The tetrad
contains two important symmetries, that is the bridge between them. Lorentz
symmetry (Latin indices) and the transformation of coordinates (Greek
indices). It is interesting to note that the Weintzenböck connection is
curvature free, while the anti-symmetric part establishes the following
torsion tensor
$T^{a}\,_{\lambda\nu}=\partial_{\lambda}e^{a}\,_{\nu}-\partial_{\nu}e^{a}\,_{\lambda}\,.$
(22)
This connection is related to the Christoffel symbols by
$\Gamma_{\mu\lambda\nu}={}^{0}\Gamma_{\mu\lambda\nu}+K_{\mu\lambda\nu}\,,$
(23)
where the contortion tensor, $\displaystyle K_{\mu\lambda\nu}$, is given by
$\displaystyle\displaystyle K_{\mu\lambda\nu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2}(T_{\lambda\mu\nu}+T_{\nu\lambda\mu}+T_{\mu\lambda\nu})\,,$
(24)
with $\displaystyle T_{\mu\lambda\nu}=e_{a\mu}T^{a}\,_{\lambda\nu}$. The above
identity leads to the relation
$eR(e)\equiv-e(\frac{1}{4}T^{abc}T_{abc}+\frac{1}{2}T^{abc}T_{bac}-T^{a}T_{a})+2\partial_{\mu}(eT^{\mu})\,.$
(25)
The Lagrangian density for TEGR is
$\displaystyle\displaystyle\mathfrak{L}(e_{a\mu})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\kappa\,e\,(\frac{1}{4}T^{abc}T_{abc}+\frac{1}{2}T^{abc}T_{bac}-T^{a}T_{a})-\mathfrak{L}_{M}$
(26) $\displaystyle\displaystyle\equiv$
$\displaystyle\displaystyle-\kappa\,e\Sigma^{abc}T_{abc}-\mathfrak{L}_{M}\;,$
where $\displaystyle\kappa=1/(16\pi)$, $\displaystyle\mathfrak{L}_{M}$ is the
Lagrangian density of matter fields and $\displaystyle\Sigma^{abc}$ is given
by
$\Sigma^{abc}=\frac{1}{4}(T^{abc}+T^{bac}-T^{cab})+\frac{1}{2}(\eta^{ac}T^{b}-\eta^{ab}T^{c})\;,$
(27)
with $\displaystyle T^{a}=e^{a}\,_{\mu}T^{\mu}$. If a derivative of eq. (26)
with respect to the tetrad field is performed, the field equation reads
$\partial_{\nu}\left(e\Sigma^{a\lambda\nu}\right)=\frac{1}{4\kappa}e\,e^{a}\,_{\mu}(t^{\lambda\mu}+T^{\lambda\mu})\;,$
(28)
where
$t^{\lambda\mu}=\kappa\left[4\,\Sigma^{bc\lambda}T_{bc}\,^{\mu}-g^{\lambda\mu}\,\Sigma^{abc}T_{abc}\right]\,,$
(29)
is the gravitational energy-momentum tensor. Such an expression is frame
dependent and to calculate its average a class of observers must be chosen,
that is, certain conditions must be imposed on the tetrad field . It is to be
noted that the skew-symmetry in $\displaystyle\Sigma^{a\lambda\nu}$ leads to
$\partial_{\lambda}\partial_{\nu}\left(e\Sigma^{a\lambda\nu}\right)\equiv
0\,.$ (30)
This is the conservation law. It is then possible to establish the energy-
momentum vector as
$P^{a}=\int_{V}d^{3}x\,e\,e^{a}\,_{\mu}(t^{0\mu}+T^{0\mu})\,,$ (31)
or with the help of eq. (28), it reads
$P^{a}=4k\,\int_{V}d^{3}x\,\partial_{\nu}\left(e\,\Sigma^{a0\nu}\right)\,.$
(32)
This is the total energy vector. It is interesting to note that it is a vector
under global Lorentz transformation which implies that energy, as the zero
component of this 4-vector, is not an invariant. In fact, it depends on the
choice of tetrad, which determines the very choice of the observer. On the
other hand the quantity is not dependent on the coordinate choice. These are
indeed desirable features for any definition of energy-momentum.
With the well-established definition of an energy-moment tensor, the first
element necessary for the application of TFD is defined. It is still necessary
to obtain a propagator for the field. Using the weak field approximation
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},$ (33)
which in eq. (26) leads to
$\langle
e_{b\lambda},e_{d\gamma}\rangle=\Delta_{bd\lambda\gamma}=\frac{\eta_{bd}}{\kappa
q^{\lambda}q^{\gamma}}.$ (34)
This is the graviton propagator usk . Then the Green function is
$\displaystyle\displaystyle
G_{0}(x,x^{\prime})=-i\Delta_{bd\lambda\gamma}\,g^{\lambda\gamma}\eta^{bd}.$
(35)
Explicitly it is
$G_{0}(x,x^{\prime})=-\frac{i64\pi}{q^{2}}\,,$ (36)
with $\displaystyle q=x-x^{\prime}$, where $\displaystyle x$ and
$\displaystyle x^{\prime}$ are four vectors. With the weak field approximation
the gravitational energy-momentum tensor $\displaystyle t^{\lambda\mu}$
becomes
$\displaystyle\displaystyle t^{\lambda\mu}(x)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\kappa\Bigl{[}g^{\mu\alpha}\partial^{\gamma}e^{b\lambda}\partial_{\gamma}e_{b\alpha}-g^{\mu\gamma}\partial^{\alpha}e^{b\lambda}\partial_{\gamma}e_{b\alpha}-g^{\mu\alpha}(\partial^{\lambda}e^{b\gamma}\partial_{\gamma}e_{b\alpha}-\partial^{\lambda}e^{b\gamma}\partial_{\alpha}e_{b\gamma})$
(37)
$\displaystyle\displaystyle-2g^{\lambda\mu}\partial^{\gamma}e^{b\alpha}(\partial_{\gamma}e_{b\alpha}-\partial_{\alpha}e_{b\gamma})\Bigl{]}\,.$
For dealing with the mean of the energy-moment tensor the standard procedure
is to consider it at different points of the space and then take the limit.
This avoids divergences. Hence
$\displaystyle\displaystyle\langle t^{\lambda\mu}(x)\rangle$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\langle
0|t^{\lambda\mu}(x)|0\rangle,$ (38) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\lim_{x^{\mu}\rightarrow
x^{\prime\mu}}4i\kappa\left(-5g^{\lambda\mu}\partial^{\prime\gamma}\partial_{\gamma}+2g^{\mu\alpha}\partial^{\prime\lambda}\partial_{\alpha}\right)G_{0}(x-x^{\prime})\,,$
where $\displaystyle\langle
e_{c}^{\,\,\,\lambda}(x),e_{b\alpha}(x^{\prime})\rangle=i\eta_{cb}\,\delta^{\lambda}_{\alpha}\,G_{0}(x-x^{\prime})$.
This average applies to any metric that is related to the linearized
Einstein’s equations. On the other hand, the validity of this expression is
restricted to stationary observers.
## IV Stefan-Boltzmann law and Casimir effect in FRW universe
The TEGR expression in the weak field approximation leads to the TFD
framework. The mean value of the energy-moment tensor becomes
$\displaystyle\displaystyle\langle
t^{\lambda\mu(AB)}(x;\alpha)\rangle=\lim_{x\rightarrow
x^{\prime}}4i\kappa\left(-5g^{\lambda\mu}\partial^{\prime\gamma}\partial_{\gamma}+2g^{\mu\alpha}\partial^{\prime\lambda}\partial_{\alpha}\right)G_{0}^{(AB)}(x-x^{\prime};\alpha).$
(39)
If we use the Casimir prescription,
$\displaystyle\displaystyle{\cal T}^{\lambda\mu(AB)}(x;\alpha)=\langle
t^{\lambda\mu(AB)}(x;\alpha)\rangle-\langle t^{\lambda\mu(AB)}(x)\rangle\,,$
(40)
then
$\displaystyle\displaystyle{\cal
T}^{\lambda\mu(AB)}(x;\alpha)=\lim_{x\rightarrow
x^{\prime}}\Gamma^{\lambda\nu}(x,x^{\prime})\overline{G}_{0}^{(AB)}(x-x^{\prime};\alpha),$
(41)
where
$\displaystyle\displaystyle\Gamma^{\lambda\nu}=4i\kappa\left(-5g^{\lambda\mu}\partial^{\prime\gamma}\partial_{\gamma}+2g^{\mu\alpha}\partial^{\prime\lambda}\partial_{\alpha}\right)\,,$
(42)
and
$\displaystyle\displaystyle\overline{G}_{0}^{(AB)}(x-x^{\prime};\alpha)=G_{0}^{(AB)}(x-x^{\prime};\alpha)-G_{0}^{(AB)}(x-x^{\prime})\,.$
(43)
It is necessary to establish the appropriate space-time geometry i.e.,
analysing the result of such expressions on cosmological scales. A homogeneous
and isotropic universe is chosen. The suitable line element is
$\displaystyle\displaystyle
ds^{2}=-dt^{2}+a\left(t\right)\left(dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}{\theta}d\phi^{2}\right)\,,$
(44)
which is the FRW line element of zero curvature. This metric respects the
approach used, as well as the constraints arising from the experiments. If eq.
(42) is used together with eq. (44), then
$\displaystyle\displaystyle\Gamma^{00}=\frac{i}{4\pi}\left[-3\partial^{\prime}_{0}\partial_{0}+\frac{5}{a^{2}}\left(\partial^{\prime}_{1}\partial_{1}+\frac{1}{r^{2}}\partial^{\prime}_{2}\partial_{2}+\frac{1}{r^{2}\sin^{2}{\theta}}\partial^{\prime}_{3}\partial_{3}\right)\right]$
(45)
and
$\displaystyle\displaystyle\Gamma^{11}=\frac{i}{4\pi
a^{2}}\left[5\partial^{\prime}_{0}\partial_{0}-\frac{5}{a^{2}}\left(\frac{3}{5}\partial^{\prime}_{1}\partial_{1}+\frac{1}{r^{2}}\partial^{\prime}_{2}\partial_{2}+\frac{1}{r^{2}\sin^{2}{\theta}}\partial^{\prime}_{3}\partial_{3}\right)\right]\,.$
(46)
Using these relations to calculate the energy and pressure for Stefan-
Boltzmann law and the Casimir effect according to the Bogoliubov
transformation is desired.
### IV.1 Gravitational Stefan-Boltzmann Law
To calculate the Stefan-Boltzmann law, $\displaystyle\alpha=(\beta,0,0,0)$ is
chosen, which leads to the Bogoliubov transformation
$\displaystyle\displaystyle v^{2}(\beta)=\sum_{j_{0}=1}^{\infty}e^{-\beta
k^{0}j_{0}}\,,$ (47)
where $\displaystyle\beta=\frac{1}{T}$. Then the Green function is
$\displaystyle\displaystyle\overline{G}_{0}^{(11)}(x-x^{\prime};\beta)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\sum_{j_{0}=1}^{\infty}G_{0}^{(11)}\left(x-x^{\prime}-i\beta
j_{0}n_{0}\right),$ (48)
where $\displaystyle n_{0}=(1,0,0,0)$ and the physical component
$\displaystyle(AB)=(11)$ is chosen, then
$\displaystyle\displaystyle E=\frac{32\pi^{4}}{15}T^{4}\,,$ (49)
and
$\displaystyle\displaystyle P=\frac{32\pi^{4}}{45a^{2}}T^{4}\,,$ (50)
with $\displaystyle E=\langle t^{00(11)}(x;\beta)\rangle$ and $\displaystyle
P=\langle t^{11(11)}(x;\beta)\rangle$. It is interesting to note that the
pressure is dependent on the scale factor which in turn is expanded as
$\displaystyle\displaystyle
a=1+H_{0}\left(t-t_{0}\right)-\frac{q_{0}H_{0}^{2}}{2}\left(t-t_{0}\right)^{2}\,,$
(51)
where $\displaystyle H_{0}$ and $\displaystyle q_{0}$ refer to the Hubble
constant and the deceleration parameter respectively. So when $\displaystyle
a=1$, the state equation becomes $\displaystyle P=\frac{E}{3}$, which is to be
the expected state equation for the graviton. Taking into account the relation
$\displaystyle\left(\frac{\partial P}{\partial
T}\right)_{V}=\left(\frac{\partial S}{\partial V}\right)_{T}$, the entropy
density is
$\displaystyle\displaystyle s=\frac{S}{V}=\frac{128\pi^{4}}{45a^{2}}T^{3}\,.$
(52)
Using the expansion for the scale factor above, the second time derivative of
the entropy density is
$\displaystyle\displaystyle\ddot{s}=-\frac{256\pi^{4}T^{3}}{45a^{2}}\left[\frac{\ddot{a}}{a}-3\left(\frac{\dot{a}}{a}\right)^{2}\right]\,,$
(53)
where dot means a time derivative. Here the Landau theory of second order
phase transition is involved. A divergence in the second derivative of the
entropy determines a critical quantity that characterizes the phase
transition. Here it is assumed that time is the dynamic variable. Hence
$\displaystyle s\rightarrow\infty$ implies $\displaystyle a=0$. If
$\displaystyle\tau=H_{0}\left(t-t_{0}\right)$ is defined as an auxiliary
variable, then it follows that
$\displaystyle\displaystyle\frac{q_{0}}{2}\tau^{2}-\tau-1=0\,.$ (54)
This imposes a constraint on the current deceleration parameter, such that
$\displaystyle q_{0}\geq-\frac{1}{2}\,.$ This is an interesting result
considering that the deceleration parameter is written in terms of the main
cosmological parameters. Results obtained in Feeney showed that using the
broad (truncated) Gaussian $\displaystyle q_{0}=-0.5\pm 1$, it is indeed
possible to obtain a competitive constraint on the Hubble constant. These
results are consistent with phenomenological models of the interaction rates
Pan using the latest microwave background observations from Planck 2018 and
baryon acoustic oscillations measurements.
### IV.2 Casimir effect
The Casimir effect is described in TFD with the choice
$\displaystyle\alpha=(0,i2d,0,0)$, where $\displaystyle d$ is the radius of
the outer spherical surface. This leads to the Bogoliubov transformation
$\displaystyle\displaystyle
v^{2}(d)=\sum_{l_{1}=1}^{\infty}e^{-i2dk^{1}l_{1}}.$ (55)
If the Green function is given by
$\displaystyle\displaystyle\overline{G}_{0}^{(11)}(x-x^{\prime};d)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\sum_{l_{1}=1}^{\infty}G_{0}^{(11)}\left(x-x^{\prime}-2dl_{1}n_{1}\right),$
(56)
with $\displaystyle n_{1}=(0,1,0,0)$, then
$\displaystyle\displaystyle E_{c}=-\frac{2\pi^{4}}{45d^{4}a^{4}}\,,$ (57)
and
$\displaystyle\displaystyle P_{c}=-\frac{2\pi^{4}}{15d^{4}a^{6}}\,.$ (58)
This result is obtained by choosing the physical component of Green’s function
$\displaystyle(AB)=(11)$. The same identification for the average energy-
momentum tensor, $\displaystyle E_{c}=\langle t^{00(11)}(x;d)\rangle$ and
$\displaystyle P_{c}=\langle t^{11(11)}(x;d)\rangle$. Two important features
need to be highlighted. The first is the Casimir energy and pressure are
obtained in a vacuum. A time-dependent negative pressure is consistent with an
accelerating expanding universe. The second is that Casimir pressure is
associated with the cosmological constant $\displaystyle\Lambda$. In natural
units, the pressure and the cosmological constant have the same dimension.
Thus the cosmological constant is understood as a fluid with the following
pressure
$\displaystyle p=-\frac{c^{4}\Lambda}{G}\,,$
thus this given by, in unities of the international system,
$\displaystyle\displaystyle\Lambda=\frac{2\pi^{4}G\hbar}{15d^{4}c^{3}}\,,$
(59)
in the present time. In this estimate the outer surface is used as the
observable radius of the universe, this determines $\displaystyle d$ as
$\displaystyle 10^{10}$ light years. The cosmological constant is of the order
of $\displaystyle 10^{-180}\,m^{-2}$. It is interesting to note that in an
incipient universe $\displaystyle\Lambda$ was much larger than it is today.
## V Conclusion
The Stefan-Boltzmann law and the Casimir effect are analyzed in a homogeneous
and isotropic universe. The FRW metric for zero curvature is used. From the
Stefan-Boltzmann law it is possible to understand that there is an energy and
pressure from strictly gravitational thermal radiation. The entropy density
provides for a phase transition that limits the range of the deceleration
parameter. The Casimir effect establishes a negative pressure consistent with
an accelerated expanding universe. Such a quantity when interpreted as the
observable radius of the universe leads to the conclusion that the
cosmological constant is small. It is important to note that due to its
temporal evolution, the cosmological constant played a more relevant role in a
primordial universe. When Casimir effect is established at finite temperature,
imaginary quantities are obtained. This leads to interpret that the
temperature effect in the universe is independent of the pressure exerted by
the vacuum. Perhaps both effects are linked on a smaller scale when quantum
effects are more relevant.
## Acknowledgments
This work by A. F. S. is supported by CNPq projects 308611/2017-9 and
430194/2018-8.
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|
# Binary evolution, gravitational-wave mergers and explosive transients
in multiple-populations gas-enriched globular-clusters
Mor Rozner Technion - Israel Institute of Technology, Haifa, 3200002, Israel
Hagai B. Perets Technion - Israel Institute of Technology, Haifa, 3200002,
Israel
###### Abstract
Most globular clusters (GCs) show evidence for multiple stellar populations,
suggesting the occurrence of several distinct star-formation episodes. The
large fraction of second population (2P) stars observed requires a very large
2P gaseous mass to have accumulated in the cluster core to form these stars.
Hence the first population of stars (1P) in the cluster core has had to become
embedded in 2P gas, just prior to the formation of later populations. Here we
explore the evolution of binaries in ambient 2P gaseous media of multiple-
population GCs. We mostly focus on black hole binaries and follow their
evolution as they evolve from wide binaries towards short periods through
interaction with ambient gas, followed by gravitational-wave (GW) dominated
inspiral and merger. We show this novel GW-merger channel could provide a
major contribution to the production of GW-sources. We consider various
assumptions and initial conditions and calculate the resulting gas-mediated
change in the population of binaries and the expected merger rates due to gas-
catalyzed GW-inspirals. For plausible conditions and assumptions, we find an
expected GW merger rate observable by aLIGO of the order of up to a few tens
of $\rm{Gpc^{-3}yr^{-1}}$, and an overall range for our various models of
$0.08-25.51\ \rm{Gpc^{-3}yr^{-1}}$. Finally, our results suggest that the
conditions and binary properties in the early stage of GCs could be critically
affected by gas-interactions and may require a major revision in the current
modeling of the evolution of GCs.
## 1 Introduction
Stars are thought to form following the collapse of giant molecular clouds
(GMCs), and further grow and evolve through accretion from, and interaction
with the GMC ambient gaseous environment during their early evolution, of up
to a few Myrs. Following the gas dispersal and depletion, the later long-term
evolution of stars and multiple systems is thought to be dominated by their
gas-free stellar evolution and their dynamical interactions with other stellar
companions and/or stars in the cluster. However, some environments can be
replenished with gas leading to late epochs of stellar and binary evolution of
stars embedded in gas. Already decades ago, Bahcall & Ostriker (1976) have
suggested that stellar compact objects can interact with gaseous disks around
massive black holes (active galactic nuclei; AGNs), accrete and give rise to
X-ray flarings. Ostriker (1983) suggested that stars and compact objects
embedded in AGNs disks can accrete gas from the ambient gaseous medium, grow
to Chandrasekhar mass and explode as type Ia supernovae (SNe), and later
Artymowicz et al. (1993) discussed accretion onto stars in AGN disks giving
rise to massive stars exploding as core-collapse (CC) SNe and polluting the
AGN disks.
The dynamical evolution of _b_ inary gravitating objects embedded in a large-
scale gaseous environment could be altered through gas-dynamical friction and
accretion that change their orbit and masses and potentially catalyze their
merger. We have first discussed binary evolution in gaseous media in the
context of catalyzed mergers of binary planetesimals in a protoplanetary disks
(Perets & Murray-Clay, 2011; Grishin & Perets, 2016), and later in the context
of compact object binaries in AGN disks (McKernan et al., 2012), where the
latter have been extensively studied since then (e.g. Stone et al., 2017;
McKernan et al., 2018; Roupas & Kazanas, 2019; Tagawa et al., 2020, and
references therein). Baruteau et al. (2011) explored the evolution of binary
_m_ ain-sequence stars (MS) in gas disks around massive black holes (MBHs),
suggesting they harden and merger through the interaction with the gas.
Various studies followed the evolution of pre-MS/MS binaries embedded in gas
just following their formation during the star-formation epoch of stars in
molecular clouds/young clusters, also suggesting that binaries can shrink and
merge through the process (Gorti & Bhatt, 1996; Er et al., 2009; Korntreff et
al., 2012). It was also suggested that the evolution of embedded binaries
could be driven by the formation of a circumbinary disk, which torques the
binary. The evolution of binaries in circumbinary disks have been more
extensively studied over a wide range of scales from planets, to stars and
MBHs (though typically not in the context of a large-scale gaseous
environment), but the exact evolution and even the direction of the binary
migration in such circumbinary disks are still debated (e.g. Artymowicz et
al., 1991; Artymowicz & Lubow, 1994; Bate, 2000; Tang et al., 2017; Moody et
al., 2019; Muñoz et al., 2019; Duffell et al., 2020; Muñoz et al., 2020, and
references therein).
Although the evolution of stars, binaries and compact objects embedded in
gaseous (typically AGN) disks near MBHs have been extensively studied over the
last few years, other gas-embedded stellar environments received far less
attention. Here and in a companion paper (Perets, 2022) we study the evolution
of single and binary compact-object binaries in the early gas-rich
environments that likely existed in multiple-population globular clusters
(GCs) and other young massive clusters (YMCs). We also briefly discuss other
(non compact-object - main-sequence and evolved) stars and binaries in such
environments, but postpone detailed study of the latter to future exploration.
As we discuss below, such gas-rich environments are likely to be far more
ubiquitous than AGN disks and potentially play a key role in the the
production of compact binaries, binary mergers, gravitational-waves sources
and explosive transients.
For decades, GCs were thought to host simple stellar populations formed
through a single star-formation episode. However, detailed observations over
the last decade (see e.g. Carretta et al., 2009; Bastian & Lardo, 2018, and
references therein) have shown that the vast majority of galactic GCs host
multiple stellar populations showing different light elements content. The
origins of multiple populations have been extensively studied, but no clear
solution has yet been found (see Renzini et al., 2015; Bastian & Lardo, 2018;
Gratton et al., 2019, for summaries of the scenarios and their caveats). The
current thought is that GCs experienced two or more star formation episodes,
in which second generation/population (2P) stars formed from processed (2P)
gas lost from earlier generation/population (1P) stars, and/or accreted
external gas. Kinematics show that 2P stars are more centrally concentrated
and were likely formed in the inner region of the GC where the 2P gas is
expected to have accumulated.
While the source of the 2P gas is debated, the late formation of 2P stars
require that tens up to hundreds of Myrs after their formation, 1P stars had
become embedded in a highly gas-rich environment that later produced the 2P
stars. The evolution of stars, binaries and compact objects embedded in gas
could therefore be significantly altered in such gaseous environments,
following similar processes as discussed for AGN disks and pre-MS stars
embedded in the progenitor GMCs. Such processes were little studied in the
context gas-embedded multiple-population GCs (Vesperini et al., 2010;
Maccarone & Zurek, 2012; Leigh et al., 2013, 2014; Roupas & Kazanas, 2019;
Perets, 2022, but see works by us and others on some aspects of such
evolution) which is the focus of the the study below. In particular, in this
paper, we introduce the effect of gas-catalyzed hardening (shrinkage of the
orbit) of binaries in GCs, and discuss its implications for GCs (and YMCs)
binary population and binary mergers, the production of GW sources, and the
formation of other merger products, compact binaries and explosive transient
events catalyzed by binary interactions with gas.
In section 2 we briefly discuss the gas replenishment in multiple-population
GCs. In section 3 we describe the hardening processes of binaries in globular
clusters due to gas-dynamical friction, and its relation to dynamical
hardening by stars and GW inspirals. In section 4 we introduce our results: in
subsection 4.1 we focus on the evolution of an individual binary under the
effect of gas hardening and in subsection 4.4 we estimate the expected merger
rate from the channel we proposed. In section 5 we discuss our results and
additional implications. In section 6 we summarize and conclude.
## 2 Multiple stellar populations and early gas-replenishment in GCs
As discussed above and in Perets (2022), gas could be replenished in GCs (and
YMCs) through mass lost from evolved stars and binaries and/or through
accretion of external gas onto the clusters (see a detailed review in Bastian
& Lardo, 2018).
The formation channel sets the amount of gas and hence the dynamics and
evolution of embedded stars/binaries. Given the correlation between the
fractions of 2P stars and GCs properties, it is likely that a large fraction
of 2P stars correspond to higher masses of the clusters, larger escape
velocities (Mastrobuono-Battisti & Perets, 2020), and hence larger mass of
replenished gas.
Given the observed kinematics and concentrations of 2P stars, and theoretical
models for the formation and evolution of 2P stars, it is thought that the
replenished gas is concentrated in the central part of GCs, where 2P stars are
concentrated. It is likely that the remnant angular momentum of replenished
gas gives rise to the formation of 2P in gaseous disks, rather than spherical
distribution (Bekki, 2010, 2011; Mastrobuono-Battisti & Perets, 2013, 2016).
The total mass of 2P gas in GCs is highly uncertain, but given reasonable
assumptions on the relation between the gas and the observed populations of 2P
stars in GCs one can provide an estimate the amount of replenished gas and its
density. The typical gas density in star forming regions is usually
constrained in the range $10^{2}-10^{6}\ M_{\odot}\rm pc^{-3}$ (Leigh et al.,
2014). Estimates for the 2P gas densities could be obtained from a simple
order of magnitude calculations, assuming 2P stars were formed from
replenished gas. The gas density is then $\rho_{g}\sim M_{g}/V_{\rm 2P}$ where
$M_{g}$ is the mass of the gas and $V_{\rm 2P}$ is the typical volume in which
the 2P stars reside. Following Bekki (2017), $M_{\rm 2P}\sim 10^{5}M_{\odot}$
and $\epsilon_{g}=0.3$, then $M_{g}\sim 3\times 10^{5}M_{\odot}$, where
$\epsilon$ is the star-formation efficiency. The infalling replenished gas is
likely concentrated in a compact region in the central parts of GCs, such that
the typical effective radius that encloses the 2P population is of the order
of $1\ \rm{pc}$ (Bekki, 2017). Taken together, the typical density of the
replenished gas is $\sim 3\times 10^{5}\ M_{\odot}\ \rm{pc}^{-3}$, which lies
within the expected range for gas densities in star-forming regions. From this
density, we will consider scaling to different gas masses, considering $R_{\rm
core}=1\ \rm{pc}$ and take $\rho_{g}\sim M_{\rm g}/R_{\rm core}^{3}$
accordingly. In particular, as we discuss below, the 2P gas is likely enclosed
in a disk-like configuration, in which case the expected gas densities are
higher. A priori, the binary hardening releases energy that could heat the gas
significantly, but from a crude calculation, the cooling rate is high enough
to compensate for it (see also Tagawa et al., 2020 for a similar calculation
in AGN disks). We also note that the possible production of jets could
potentially unbind gas from the disk (Soker, 2016; Tagawa et al., 2022), but
the study of this possibility is beyond the scope of the current paper.
The total amount of gas is depleted in time, due to formation of stars and/or
accretion onto stars, and later gas ejection through possible radiation
pressure processes and SNe. For simplicity we assume an exponential decay,
i.e. $\rho_{g}(t)=\rho_{g,0}\exp(-t/\tau_{\rm gas})$ and consider several
possible options for the gas lifetime, to account for uncertainties in the
possible gas-depletion processes involved.
### 2.1 Disk configuration
Gas replenishment leading to the formation of 2P stars in GCs might form a
disk-like structure in the cluster nuclei (e.g. Bekki, 2010; Mastrobuono-
Battisti & Perets, 2013).
Following Bekki (2010), we consider a flat disk, i.e. with a constant aspect
ratio. We estimate the aspect ratio by $h/r\sim c_{s}/v_{K}$ where
$v_{K}=\sqrt{G(M_{\rm gas}+M_{\star})/R_{\rm core}}$ is the typical velocity
in the central parsec. The speed of sound $c_{s}=\sqrt{k_{B}T_{\rm gas}/\mu
m_{p}}$ ranges between $0.1-10\ \rm{km/sec}$ (e.g. Bekki, 2010; Leigh et al.,
2013), in correspondence to the gas temperature $T_{\rm gas}$, such that
$c_{s}\approx 0.6\ \rm{km/sec}$ corresponds to temperature of $100\ K$, which
is the typical temperature in star formation areas, where $\mu=2.3$, and
$m_{p}$ is the proton mass. Exponential disk models were also considered
(Hénault-Brunet et al., 2015), but here we focus on simple models.
In our fiducial model, we consider $c_{s}=10\ \rm{km/sec}$, unless stated
otherwise. Then, the aspect ratio $h/r\approx 0.23$. Following Bekki (2010),
we consider a velocity dispersion of $\sigma_{\rm disk}=10\ \rm{km/sec}$ for
stars embedded in the disk. As a conservative assumption, we consider the
stellar/massive objects density in the disk to be the same as in the core i.e.
$n_{\star,disk}\approx n_{\star}=10^{5}\ \rm{pc^{-3}}$. However, it should be
noted that due to gas dynamical friction, stars will migrate and experience
inclination damping, and the effective density in the disk is expected to be
higher (e.g. Artymowicz et al., 1993; Leigh et al., 2014; Grishin & Perets,
2016).
We can estimate the volume ratio between the disk and the core volume by $\pi
R_{\rm core}^{2}h/(4\pi R_{\rm core}^{3}/3)\sim 0.75h/r$. Then, under the
assumption that all the second generation gas is concentrated in the disk, we
get a typical gas density of $\rho_{\rm g,disk}\sim 1.74\times 10^{6}\
M_{\odot}\ \rm{pc^{-3}}$. The fraction of stars in the disk will change for
thinner/thicker disks correspondingly.
The evolution of binaries in disks differs in several aspects from the
evolution in a spherical configuration. For our discussion, the major ones
are: the velocity dispersion decreases, the gas density increases and the
total number of stars contained in the disk is only the volumetric fraction of
the disk compared with the volume of the spherical core. The fraction might
change with time due to the interaction with gas.
## 3 Dynamics of binaries and their interaction with gas: binary hardening
and mergers
Binaries embedded in gas interact with it, exchange angular momentum and
energy and possibly accrete gas. These processes are quite complex; here we
focus on the interaction through gas-dynamical friction (GDF), while other
suggested processes for interaction with gas are discussed in subsec. 4.2.
Besides interaction with gas, binaries in GCs can interact with other stars
through dissipative effects such as GW inspirals or tidal evolution and
through dynamical encounters with other stars through three (or more)-body
encounters (Heggie, 1975).
The semimajor axis (SMA) of a given massive binary in a gas-enriched
environment evolves through the combined effect of the above-mentioned
processes:
$\displaystyle\frac{da_{\rm bin}}{dt}=\frac{da_{\rm bin}}{dt}\bigg{|}_{\rm
3-body}+\frac{da_{\rm bin}}{dt}\bigg{|}_{\rm GDF}+\frac{da_{\rm
bin}}{dt}\bigg{|}_{\rm GW}$ (1)
where $a_{\rm bin}$ is the binary SMA.
A priori, all the three mechanisms contribute to the evolution of the SMA.
However, in practice, each of these process dominates in a specific regime,
and can be typically neglected in other regimes. Binaries could shrink to
shorter periods (harden) due to the effect of gas-interaction or GWs inspiral,
and get harder or softer (wider) due to three-body interactions with other GC
stars. As we discuss in the following, the evolution of hard binaries is
dominated by gas-interactions at large separations and by GW-emission at small
separation, while dynamical hardening and softening through three-body
encounters (Heggie, 1975) can be neglected in these regimes. Nevertheless,
binary softening and evaporation before the gas-replenishment episode can
destroy the widest binaries in the clusters, and hence determine the largest
possible initial SMAs for binaries in the cluster at the beginning of the gas-
interaction epoch. Moreover, it could play a role in hardening binaries that
did not merge within the gas epoch.
The interaction with gas can also give rise to the formation of new wide
binaries through two-body and three-body encounters in gas (Goldreich et al.,
2002; Tagawa et al., 2020), allowing for replenishment of binaries in
clusters.
In the following we discuss these various processes, while we neglect the
effect of direct accretion onto compact objects and their growth, which is
beyond the scope of the current paper (though generally such accretion, if
effective likely further accelerates binary hardening (e.g. Roupas & Kazanas,
2019).
### 3.1 Hardening and softening through dynamical encounters with stars
Due to interactions with other stars, hard binaries tend to get harder, while
soft binaries tend to get softer (Heggie, 1975); see updated discussion and
overview of these issues in Ginat & Perets (2021a, b). Hence, in the absence
of a gaseous environment stellar dynamical hardening plays an important role
in binary evolution and in catalyzing binary mergers.
#### 3.1.1 Hard binaries
For hard binaries, the dynamical hardening rate (up to order unity corrections
calibrated usually from numerical simulations) is given by (Spitzer, 1987)
$\displaystyle\frac{da_{\rm bin}}{dt}\bigg{|}_{3-body}=-\frac{2\pi
Gn_{\star}m_{\rm pert}(2m+m_{\rm pert})a_{\rm bin}^{2}}{mv_{\infty}}$ (2)
where we consider a binary with equal mass components, $m=m_{1}=m_{2}$ and an
external perturber with mass $m_{\rm pert}$. For interactions with other
massive objects only, $n_{\star}$ and $m_{\rm pert}$ should be taken as
$n_{\bullet}$ and $\bar{m}_{\bullet}$ correspondingly.
#### 3.1.2 Soft binaries
A binary is called a soft binary if its energy is lower than
$\bar{m}\sigma^{2}$. This condition sets a critical SMA
$\displaystyle a_{\rm SH}=\frac{2Gm^{2}}{\bar{m}\sigma^{2}}\approx$ (3)
$\displaystyle\approx 200.53\rm{AU}\left(\frac{m}{10\
M_{\odot}}\right)^{2}\left(\frac{43.2\
\rm{km/sec}}{\sigma}\right)^{2}\left(\frac{0.5\ M_{\odot}}{\bar{m}}\right)$
As can be seen, massive stars tend to be hard relative to the background stars
in the cluster, due to the scaling $a_{\rm SH}\propto m^{2}/\bar{m}$. Hence,
one should define the hardness of massive binaries relative to both low mass
and high mass stars, in particular, the latter will give rise to softer
binaries. We then get the following modified expression (Quinlan, 1996; Kritos
& Cholis, 2020),
$\displaystyle a_{\rm SH,\bullet}\approx\frac{Gm}{4\sigma^{2}}\approx
1.25\rm{AU}\left(\frac{m}{10\ M_{\odot}}\right)\left(\frac{43.2\
\rm{km/sec}}{\sigma}\right)^{2}$ (4)
Soft wide binaries are prone to destruction due to encounters with other
stars. The dynamical evolution of massive binaries is dominated by
interactions with other massive stars and their number density in the core is
elevated due to mass segregation (Sigurdsson & Phinney, 1995).
As to bracket the effect of softening, we consider two possibilities. (1)
Softening is dominated by encounters with stellar BHs, where we assume the
number density of such objects to be $n_{\rm\rm b}=n_{\bullet}=10^{3}\
\rm{pc^{-3}}$, due to mass segregation to the core, where
$\bar{m}_{\bullet}=10\ M_{\odot}$ (see a discussion in Miller & Hamilton,
2002). (2) Softening is dominated by low-mass $0.5\ M_{\odot}$ mass stars, if
the cluster is not well segregated, and the $n_{\rm b}=n_{\star}=10^{5}\
\rm{pc^{-3}}$.
Hence, the typical lifetime of a soft massive binary is given by (e.g. Binney
& Tremaine, 2008),
$\displaystyle\tau_{\rm evap,massive}$
$\displaystyle\approx\frac{(m_{1}+m_{2})\sigma}{16\sqrt{\pi}n_{\rm
b}\bar{m}^{2}_{b}Ga\ln\Lambda}$ (5)
where $\ln\Lambda$ is the Coulomb logarithm and $n_{b}$ and $\bar{m}_{b}$ are
the number density and the mass of the background stars and change according
to our choice between (1) and (2). The widest binaries that survive
evaporation until the formation of second generation stars, signed as
$\tau_{\rm SG}$, taken here to be $100\ \rm{Myr}$
$\displaystyle a_{\rm widest}$ $\displaystyle=\max\left\\{a_{\rm
SH,\bullet},\frac{(m_{1}+m_{2})\sigma}{16\sqrt{\pi}n_{b}\bar{m}^{2}_{b}G\tau_{\rm
SG}\ln\Lambda}\right\\}$ (6)
For our fiducial parameters, $a_{\rm widest}=24.9\ \rm{AU}$ for the segregated
case and $200.53\ \rm{AU}$ for the non-segregated case. In principle, binaries
could soften and be disrupted via encounters during the gas-replenishment
episode, however, the GDF hardening described in the following is more
efficient at this stage. Therefore binary evaporation due to encounters sets
the stage, and determines the SMA of the widest binaries at the beginning of
the gas-enrichment stage, but can be neglected during the the time binaries
are embedded in gas.
### 3.2 Gas dynamical friction
In gas-rich environments, such as the 2P gas environment of multiple-
population GCs/YMCs (and AGN disks), GDF can play a major role in hardening.
The evolution of binaries in gaseous media has been studied over a wide range
of astrophysical scales from asteroids to MBHs (as discussed in the
introduction).
The effect of gas was suggested to be modeled mainly via several approaches.
One suggestion is the accretion of gas onto a binary forms a circumbinary
minidisk, due to accretion to the Hill sphere. In such disks, torques similar
to the ones described type I/II migration of planets in protoplanetary disks
could lead to the shrinkage of the binary SMA (e.g. Artymowicz et al., 1991;
McKernan et al., 2012; Stone et al., 2017; Tagawa et al., 2020). Such
migration leads to very efficient mergers, far more efficient than the case of
interaction dominated by GDF, as we discuss below. However, these issues are
still debated, and some hydrodynamical simulations show that such torques
might lead to outward migration (e.g. Moody et al., 2019; Duffell et al.,
2020; Muñoz et al., 2020), while other hydrodynamical studies indicate that in
thin disks one should have inward migration (Duffell et al., 2020; Tiede et
al., 2020). We do note that most studies consider initially circular orbits,
and generally follow circular orbits, while eccentric orbits could evolve
differently, with their orbital eccentricity possibly excited into very high
eccentricities, as we discuss below in the context of modeling the evolution
through GDF.
Therefore, the approach on which we focus here, considers the effects of GDF
(Ostriker, 1999). When an object has a non-zero velocity relative to the
background gas, the interaction with the gas reduces the relative velocity and
therefore hardens binaries (e.g. Escala et al., 2004; Baruteau et al., 2011).
The binary hardening induced by GDF for the circular case, with binary
components with the same mass $m_{1}=m_{2}=m$ is given by (Grishin & Perets,
2016),
$\displaystyle\frac{da_{\rm bin}}{dt}\bigg{|}_{\rm GDF}$
$\displaystyle=-\frac{8\pi G^{3/2}a_{\rm
bin}^{3/2}}{\sqrt{m_{1}+m_{2}}}\rho_{g}(t)\frac{m}{v_{\rm
rel}^{2}}f\left(\frac{v_{\rm rel}}{c_{s}}\right);$ (7) $\displaystyle f(x)$
$\displaystyle=\begin{cases}\frac{1}{2}\log\frac{1+x}{1-x}-x,\ 0<x<1,\\\
\frac{1}{2}\log\left(x^{2}-1\right)+\log\Lambda_{g},\ x>1\end{cases}$ (8)
where $f$ is a dimensionless function derived in Ostriker (1999), $v_{\rm
rel}$ is the velocity of the binary relative to the gas, taken as the
Keplerian velocity of the binary, i.e. $v_{K}=\sqrt{G(m_{1}+m_{2})/a_{\rm
bin}}$, which dominates the relative velocity throughout most of the
evolution.
Under this assumption, eq. 7 could be written as
$\displaystyle\frac{da_{\rm bin}}{dt}\bigg{|}_{\rm GDF}$
$\displaystyle=-8\pi\sqrt{\frac{Ga_{\rm
bin}^{5}}{2m}}\rho_{g}(t)f\left(\frac{v_{K}}{c_{s}}\right)$ (9)
For massive binaries, the effect of stellar hardening will be weaker than the
effect on less massive stars, as can be seen directly from eq. 2. In contrast,
the effect of gas hardening increases with mass (eq. 7). Comparison of the two
shows that hardening is dominated by gas hardening rather than stellar
hardening. Moreover, although the effect of GDF decreases as the binary
hardens, it decays more slowly than the three-body hardening, as could be seen
from the scaling $\dot{a}_{\rm hard,\star}\propto a^{2}$ and $\dot{a}_{\rm
GDF}\propto a^{3/2}$, and therefore GDF dominates the evolution over stellar-
hardening throughout the evolution. After gas depletion, three-body hardening
becomes the dominant dynamical process for wide binaries, while for
sufficiently small separations, the evolution is GWs-dominated.
### 3.3 Gravitational-wave inspiral
For stellar mass objects GW inspiral becomes important only at very small
separations, and can be neglected in regard to main-sequence (or evolved)
stellar binaries that merge before GW emission becomes important. However, GW
inspiral plays a key-role in the evolution of binaries composed of compact
objects.
For a circular binary in the quadruple approximation, the GWs inspiral rate is
given by (Peters, 1964),
$\displaystyle\frac{da}{dt}\bigg{|}_{\rm
GW}=-\frac{64G^{3}m_{1}m_{2}(m_{1}+m_{2})}{5c^{5}a^{3}}$ (10)
where $G$ is the gravitational constant and $c$ is the speed of light.
Without gas dissipation, the maximal SMA for GW merger within a Hubble time is
given by
$\displaystyle a_{max,GW}=\left(\frac{64\tau_{\rm
Hubble}G^{3}m_{1}m_{2}(m_{1}+m_{2})}{5c^{5}}\right)^{1/4}\approx$
$\displaystyle\approx 0.07\ \rm{AU}\left(\frac{m}{10\ M_{\odot}}\right)^{3/4}$
(11)
A compact binary that is driven by GDF to separations below $a_{max,GW}$ would
eventually inspiral and merge, even if it survived the gas-replenishment
stage, and would produce a GW-source.
## 4 Results
Accounting for the effects of the various processes discussed above, we can
follow the evolution of binaries in clusters during the gas epoch and assess
its outcomes. Overall we find that under plausible conditions all black-hole
binaries initially existing in the cluster inner regions that become embedded
in gas during the gas-replenishment phase could be driven to short separations
and merge within a Hubble time.
These results suggest that gas-catalyzed GW-mergers in GCs and YMCs, not
considered at all in current modeling of GCs, could serve as an important
channel for the production of GW-sources, and plays a key role in the
evolution of binaries in such clusters.
Both the GDF and GW-inspiral timescales for lower mass compact objects such as
neutron stars (NSs) and white dwarfs (WDs), are longer (as can be seen in eq.
3.3), but they are also expected to modify their semimajor axis distribution.
Here we focus on mergers of BHs, and postpone a detailed discussion of NS and
WD mergers for a follow-up paper, but we should already note that potential WD
mergers could give rise to the production of explosive events such as type Ia
supernovae from mergers of massive white-dwarfs (see also Perets, 2022), and
could produce GW sources observable by planned GW-detection space missions. NS
mergers could produce short gamma-ray bursts and aLIGO GW sources. Combined
BH-NS or BH-WD binaries with their high mass but lower mass-ratio could be
driven to mergers at intermediate timescales between highest and lowest
timescales considered here giving rise to WD/NS disruptions by the BH possibly
producing rapid faint SNe (e.g. Zenati et al., 2019, 2020; Bobrick et al.,
2022, and references therein) or short-GRBs accompanied by a potential GW
aLIGO-source. The dynamics of binaries with non-equal masses could be however
more complicated and is not explored here.
In the following we discuss our results in detail.
### 4.1 Gas-assisted GW-mergers
Figure 1: The effects of gas hardening, GWs and three-body hardening. The blue
dashed line represents the maximal SMA in which GW emission catalyzes a binary
merger within a Hubble time. We consider the evolution of a binary with masses
$m_{1}=m_{2}=10\ M_{\odot}$ and initial separation of $a_{0}=1\ \rm AU$. We
consider an exponential decaying background gas density
$\rho_{g}=\rho_{g,0}\exp(-t/\tau_{\rm gas})$ with $\rho_{g,0}=1.74\times
10^{6}\ M_{\odot}\rm{pc}^{-3}$ and $\tau_{\rm gas}=50\ \rm Myr$. Figure 2:
The combined effect of gas hardening, three-body hardening and GWs on a
binary, for different background gas masses (and corresponding gas densities).
The blue dashed line represents the maximal SMA in which GW emission catalyzes
a binary merger within a Hubble time. The solid lines corresponds to the
evolution of the SMA, starting from an initial separation of $a_{0}=1\
\rm{AU}$, and given different background densities, with an exponential
decaying gas density $\rho_{g}=\rho_{g,0}\exp(-t/\tau_{\rm gas})$ with
$\tau_{\rm gas}=50\ \rm Myr$ (that corresponds to $M_{\rm gas,0}=3\times
10^{5}\ M_{\odot}$). The velocity dispersions are calculated given the total
mass of the gas and stars. Figure 3: The combined effect of gas hardening,
three-body hardening and GWs on a binary, for different initial separations.
The blue dashed line represents the maximal SMA in which GW emission catalyzes
a binary merger within a Hubble time. The purple dashed line corresponds to
the widest binary allowed by evaporation considerations. The solid lines
corresponds to the evolution of the SMA, starting from an initial separations
of $a_{0}=1,10,100,200\ \rm{AU}$, and given an exponential decaying gas
density $\rho_{g}=\rho_{g,0}\exp(-t/\tau_{\rm gas})$ with $\tau_{\rm gas}=50\
\rm Myr$.
In Fig. 1 we compare the different hardening processes of binaries in gas-
embedded regions. As can be seen, for large separations, the evolution is
dominated by the gas hardening, while for smaller separations (at late times
after the gas depletion), three-body hardening and finally GWs dominate the
evolution. The transition between the different regimes is determined by the
gas density in the cluster, as well as stellar density. Unless stated
otherwise, we consider for our fiducial model a background of stars with
typical masses of $\bar{m}=0.5\ M_{\odot}$.
In Fig. 2 we present the evolution of binaries with an initial separation of
$a_{0}=1\rm AU$, due to GDF, for different ambient gas-densities. The gas
hardening mechanism is generally very effective and leads to binary migration
to small separations within short timescales, given a sufficiently dense
gaseous environment. As we discuss below, such gas-assisted evolution would
then give rise to high rates of GW-mergers of BH binaries, comparable with the
BH merger rates inferred from aLIGO-VIRGO-KAGRA (LVK) collaboration (Abbott et
al., 2016, 2021).
It should be noted that the gas could still dominate the evolution even after
reaching $a_{\rm GW}$, as long as the gas was not depleted and the timescale
for GWs mergers is larger than the GDF induced merger timescale. In principle,
GDF-dominated evolution might even be identified in the GW inspiral (in future
space missions) before the merger, under appropriate conditions, if GDF still
dominates the evolution in LISA frequencies.
We find that circular binaries shrink and reach final small separations,
dictated by the initial conditions, which are not sufficiently small as to
allow for GW emission alone to drive the binaries to merger even after a
Hubble time. Nevertheless, at such short period, these very-hard binaries are
more likely to merge due to dynamical encounters on the long-term compared
with the primordial population of binary-BHs, and should be appropriately
accounted for in simulations of GC stellar populations.
In Fig. 3, we introduce the evolution of binaries with different initial
separations under the combined effect of GDF, three-body hardening and GWs. It
could be seen that although the merger timescales of wider binaries are
slightly larger, all the binaries are expected to merge within a Hubble time.
Hence, the effect of the presence of gas in the initial stages is robust
across all separations and will modify the binary population. For wide enough
binaries, we enter the subsonic range. In order to avoid the discontinuity in
eq. 8, we take it as a constant in a small environment around Mach $1$ – for
$\mathcal{M}<1.01$, we consider $f(\mathcal{M})\equiv f(1.01)$, where the
widest binary we consider corresponds to $\mathcal{M}\approx 0.97$.
Figure 4: The evolution of the binary separation for different sound speeds.
We consider equal mass binaries with initial separation of $a=1\ \rm{AU}$,
masses $m=m_{1}=m_{2}=10\ M_{\odot}$ and an exponential decaying background
density with $\rho_{\rm g,0}=1.74\times 10^{6}\ M_{\odot}\rm{pc^{-3}}$. The
blue dashed line corresponds to the maximal separation from which a GWs merger
is expected. Figure 5: The effect of gas hardening on a binary, as dictated
by GDF, for different masses of binaries. The different curves correspond to
the evolution of the SMA for different binary masses, starting from an initial
separation of $a_{0}=1\ \rm{AU}$, given a background density with an
exponential decaying gas density $\rho_{g}=\rho_{g,0}\exp(-t/\tau_{\rm gas})$
with $\tau_{\rm gas}=50\ \rm Myr$ and $\rho_{g,0}=1.74\times
10^{6}M_{\odot}\rm{pc}^{-3}$.
In Fig. 5, we demonstrate the dependence of gas hardening on different binary
masses. As can be seen from eq. 7, lower-mass binaries harden over longer
timescales, due to the dependence on the mass that scales as
$\propto\sqrt{m}$, for an equal mass binary with companions $m_{1}=m_{2}=m$.
The final SMA of the binary also depends on the mass of the binary, such that
more massive binaries will attain smaller final SMAs.
In Fig. 4 we consider different sound speeds, all of them in the supersonic
regime. Higher sound speed lead to larger merger timescales, although the
results are robust and do not change steeply between the different choices of
sound speed in this regime.
### 4.2 Comparison with other gas hardening models
Heretofore, we considered gas hardening induced by GDF. However, there are
other approaches to model gas hardening.
In AGN disks, gas hardening is also modeled using processes similar to
migration models in protoplanetary disks (as was suggested in the context of
AGN disks McKernan et al., 2012; Stone et al., 2017; Tagawa et al., 2020). Gas
is captured in the Hill sphere of a binary and leads to the formation of a
circumbinary minidisk. The disk applies a torque on the binary that leads to
separation decay similar to migration type I/II in protoplanetary disks,
although there were studies that pointed out that this torque could lead to a
softening rather than hardening (Moody et al., 2019). Notwithstanding, we will
assume that the formation of a minidisk can take place in GCs and compare the
resulted hardening with our GDF model. The typical timescale for hardening due
to migration torques is given by (e.g. McKernan et al., 2012), modify to the
other type II migration equation
$\displaystyle\tau_{\rm typeII}\sim 46\
\rm{yr}\left(\frac{0.01}{\alpha}\right)\left(\frac{0.23}{h/r}\right)^{2}\left(\frac{40\rm{yr^{-1}}}{\Omega_{\rm
bin}(a_{\rm bin})}\right)$ (12)
where $\alpha$ is the Shakura Sunayev parameter, $h/r$ is the aspect ratio and
$\Omega_{\rm bin}=\sqrt{G(m_{1}+m_{2})/a_{\rm bin}^{3}}$ is the angular
frequency of the binary. We adopt values of $h/r=0.23$ and $\alpha=0.01$ as a
conservative value for the viscosity parameter of the disk. We substitute the
$\Omega_{\rm bin}$ that corresponds to a binary with a separation of $1\
\rm{AU}$. Under these assumptions, the migration timescales, which could be
used to approximate the hardening timescales, are shorter than the typical
migration timescales we derived using the GDF model. These timescales are also
shorter then the ones obtained in AGN disks (e.g. Stone et al., 2017; Tagawa
et al., 2020), as expected. We therefore expect the merger rates we derived to
be similar in this case, and even higher for the lowest gas-densities models,
where the rates were limited by slower hardening. There were more recent
studies that suggested modified migration timescales, here taken for an equal
mass binary
$\displaystyle\tau_{\rm{typeII,K}}=\frac{\Sigma_{\rm disk}}{\Sigma_{\rm
disk,min}}\tau_{\rm typeII},$ (13) $\displaystyle\Sigma_{\rm
disk,min}=\frac{\Sigma_{\rm disk}}{1+0.04K},$ $\displaystyle
K=\left(\frac{m_{1}}{m_{1}+m_{2}}\right)^{2}\left(\frac{h}{r}\right)^{-5}\alpha^{-1}$
These factors lengthen significantly the typical migration timescales, such
that for our fiducial model we expect $\tau_{\rm typeII,K}\approx 71515\
\rm{yr}$. This timescale is still much shorter than the expected timescale
calculated via the gas dynamical friction model.
Another approach to modeling gas-induced inspirals is discussed in Antoni et
al. (2019). They simulate Bondi-Hoyle-Lyttelton (BHL) supersonic flows and
derive the corresponding energy dissipation, fitted to an analytical theory.
While the overall gas hardening timescales could be comparable or shorter for
the parameters that are in our major interest, there are significant
differences in the scaling. The typical inspiral timescale is given by (eq.
$52$ in Antoni et al. (2019)),
$\displaystyle\tau_{\rm BHL}=61\ \rm{Myr}$
$\displaystyle\left(\frac{a_{0}}{AU}\right)^{0.19}\left(\frac{v_{\rm
rel}}{\rm{100\ km\times sec^{-1}}}\right)^{3.38}\times$ (14)
$\displaystyle\times\left(\frac{20\
M_{\odot}}{m_{1}+m_{2}}\right)^{1.19}\left(\frac{7.72\times 10^{7}\
\rm{cm}^{-3}}{n_{\rm gas}}\right)$
where $a_{0}$ is the initial separation of the binary and $n_{\rm{gas}}$ is
the number density of the gas, such that $\rho_{\rm gas}=n_{\rm gas}m_{p}$
where $m_{p}$ is the proton mass.
Each model for gas hardening sets a different critical initial separation from
which the binary will merge within a Hubble time. The timescales dictated both
from the type II migration and BHL mechanism are even shorter than the ones
expected by our fiducial model.
Hence, we will conclude that in all the approaches that we considered to model
gas hardening, the process is very efficient and leads to a robust rate of
mergers, that modifies significantly the binaries’ population, while the major
difference between them is the time of the merger, dictated by the different
gas hardening timescales.
### 4.3 Eccentric evolution
The evolution of binaries in a gaseous medium is significantly different for
non-circular binaries. Here we derive and solve the equations for an orbit-
averaged eccentric evolution of an initially eccentric binary embedded in gas,
but leave a more detailed discussion on the implications for the dynamical
3-body hardening of eccentric binaries to future study.
For simplicity, we will assume that the Keplerian velocity of the binary
components dominate the relative velocity to the gas, and that the gas
velocity is zero relative to the center of mass of the binary. Hence, the
relative velocity between the binary and the gas in the center of mass frame
is given by
$\displaystyle\textbf{v}_{\rm rel}=\frac{\Omega a}{2\sqrt{1-e^{2}}}\left[e\sin
f\hat{r}+(1+e\cos f)\hat{\varphi}\right]$ (15)
The orbit equations for the GDF for a binary with two equal masses are then
given by
$\displaystyle\frac{{da}}{dt}\bigg{|}_{\rm GDF}=\frac{2a^{3/2}}{m_{\rm
bin}\sqrt{Gm_{\rm bin}(1-e^{2})}}\left[F_{r}e\sin f+F_{\varphi}(1+e\cos
f)\right],$ (16) $\displaystyle\frac{{de}}{dt}\bigg{|}_{\rm
GDF}=\frac{2}{m}\sqrt{\frac{a(1-e^{2})}{Gm_{\rm bin}}}\left[F_{r}\sin
f+F_{\varphi}(\cos f+\cos E)\right]$ (17)
where $\textbf{F}_{\rm drag}=F_{r}\hat{r}+F_{\varphi}\hat{\varphi}$, $f$ is
the true anomaly and $E$ is the eccentric anomaly. The orbit-averaged
equations are given by
$\displaystyle\frac{\overline{da}}{dt}\bigg{|}_{\rm
GDF}=\frac{4F_{0}(1-e^{2})^{2}}{\pi m_{\rm
bin}\Omega^{3}a^{2}}\int_{0}^{2\pi}\frac{Idf}{(1+e\cos f)^{2}\sqrt{1+2e\cos
f+e^{2}}},$ (18) $\displaystyle\frac{\overline{de}}{dt}\bigg{|}_{\rm
GDF}=\frac{4F_{0}(1-e^{2})^{3}}{\pi m_{\rm
bin}\Omega^{3}a^{3}}\int_{0}^{2\pi}\frac{I(e+\cos f)df}{(1+e\cos
f)^{2}(1+2e\cos f+e^{2})^{3/2}}$ (19)
where $F_{0}$ is given by $\textbf{F}_{\rm drag}=F_{0}I\textbf{v}_{\rm
rel}/v_{\rm rel}^{3}$. The orbit-averaged equations for GWs are given by
$\displaystyle\frac{\overline{da}}{dt}\bigg{|}_{\rm
GW}=-\frac{64G^{3}m_{1}m_{2}(m_{1}+m_{2})}{5c^{5}a^{3}(1-e^{2})^{7/2}}\left(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}\right),$
(20) $\displaystyle\frac{\overline{de}}{dt}\bigg{|}_{\rm
GW}=-\frac{304G^{3}em_{1}m_{2}(m_{1}+m_{2})}{15c^{5}a^{4}(1-e^{2})^{5/2}}\left(1+\frac{121}{304}e^{2}\right)$
(21)
Figure 6: The effects of gas hardening on eccentric orbit. We consider the
evolution of a binary with masses $m_{1}=m_{2}=10\ M_{\odot}$ and initial
separation of $a_{0}=1\ \rm AU$. We consider an exponential decaying
background gas density $\rho_{g}=\rho_{g,0}\exp(-t/\tau_{\rm gas})$ with
$\rho_{g,0}=1.74\times 10^{6}\ M_{\odot}\rm{pc}^{-3}$ and $\tau_{\rm gas}=50\
\rm Myr$. The solid lines correspond to semimajor axis evolution and the
dashed lines to pericenter evolution. Figure 7: The effects of gas hardening
and GWs on eccentric orbit. We consider the evolution of a binary with masses
$m_{1}=m_{2}=10\ M_{\odot}$ and initial separation of $a_{0}=1\ \rm AU$. We
consider an exponential decaying background gas density
$\rho_{g}=\rho_{g,0}\exp(-t/\tau_{\rm gas})$ with $\rho_{g,0}=1.74\times
10^{6}\ M_{\odot}\rm{pc}^{-3}$ and $\tau_{\rm gas}=50\ \rm Myr$. The solid
lines correspond to semimajor axis evolution and the dashed lines to
pericenter evolution.
In Fig. 6 and Fig. 7 we introduce the evolution of eccentric binaries. In Fig.
6, we present the evolution due only to GDF, and in Fig. 7, we also introduce
the effect of GW-emission. As can be seen, the eccentricities become extremely
high within short timescales, indicating that the pericenter shrinks
significantly. Once the pericenters are sufficiently small, the effect of GWs
becomes more significant, and the orbit shrinkage is accompanied by
eccentricity damping, and the binaries are driven into approximately circular
orbit when entering the VLK GW-bands.
Such eccentric evolution could play a key role in the evolution of the binary
populations, as eccentric binaries merge within potentially far shorter
timescales than circular binaries. We note, however, that some studies of a
circumbinary gas-disk evolution of binaries, suggest they are only excited to
moderate eccentricities Tiede et al., 2020, $\sim 0.45$. Nevertheless, if
binaries migration occurs through such processes the overall shrinkage is
rapid irrespective of the eccentricity, leading to fast migration timescale
(see previous subsec.).
We further discuss these issues, and in particular the implications for the
delay time distribution of GW sources from this channel in subsec. 4.4. We
note that the consideration of eccentric binaries gas-hardening, little
studied before should play a similarly important role in binary evolution in
AGN disks, possibly in a different manner than in cases where circumbinary-
disk evolution is assumed Samsing et al., 2020; Tagawa et al., 2021.
### 4.4 Gravitational-waves merger rate
In the following we estimate the GW mergers rate of binary black holes from
the gas-catalyzed channel studied here. We will consider old-formed GCs and
YMCs separately, given their different formation history.
In all the models we considered for gas hardening, all the binaries are
expected to merge within a Hubble time. However, different gas hardening
models suggest different merger timescales. As discussed above, our GDF models
suggest that eccentric binaries merge rapidly, and some of the hydrodynamical
studies discussed above suggest that even circular binaries merge during the
early gas phase. Since most GCs formed very early, such mergers would not be
detected by VLK, given the effectively limited lookback time. However, the
younger equivalents of GCs, so called YMCs, continue to form and generally
follow the star-formation history in the universe. Hence, mergers in such YMC
could occur sufficiently late (and hence closer by) and be detected by VLK and
the contribution of YMCs to the total VLK rate will be the dominant one for
the eccentric cases (or for all binaries, according to e.g. the circumbinary
disk migration models. It should be noted that there is an observational
evidence for gas replenishment also in YMCs (e.g. Li et al., 2016). If,
however, gas densities are lower or the binaries are initially circular/in low
eccentricity, the final SMA of the binaries could be larger, leading to longer
GW-merger time catalyzed by three-body hardening (driving the delay time
distribution to longer timescales), in which case the contribution from old
GCs would be the dominant one.
The rates as a function of the redshift change according to the geometric
structure of the 2P stars. Formation of 2P stars in disks is characterized by
lower velocity dispersions, that lead to earlier mergers, where for the case
of spherical constellation, the higher velocity dispersion leads to later
mergers.
We will start by estimating the number of mergers per cluster,
$\displaystyle N_{\rm merge}\sim f_{\rm disk}f_{\rm bin,surv}f_{\geq
20M_{\odot}}f_{\rm ret}f_{\rm merge}N_{\star}$ (22)
where $f_{\rm disk}$ is the fraction of stars that reside in the disk, $f_{\rm
bin,surv}$ is the fraction of binaries among massive stars that will survive
stellar evolution (i.e. SNe), $f_{\geq 20M_{\odot}}$ is the fraction of stars
with masses that exceed $20\ M_{\odot}$, $f_{\rm ret}$ is the retention
fraction of BHs in the cluster, $f_{\rm merge}$ is the fraction of binaries
that merge among the surviving binaries embedded in the disk and $N_{\star}$
is the number of stars in the cluster.
Following our geometrical considerations in subsec. 2.1, we set $f_{\rm disk}$
in the range $[2\%,20\%]$. However, even large fractions could be taken into
account if there is a significant capture of objects to the disk.
The binarity fraction of massive BHs is $\sim 0.7$, although even higher
values are quite plausible for the massive star progenitors of black holes
(e.g. Sana et al., 2012), stellar evolution may reduce this fraction to a
typical value of $f_{\rm bin,surv}=0.1$ (e.g. Antonini & Perets, 2012). We use
a Kroupa mass function for the cluster, such that the fraction of stars with
masses larger than $20\ M_{\odot}$ is $2\times 10^{-3}$ for non-segregated
environment, for segregated ones we take a fraction of $0.01$. The retention
fraction from the cluster is taken to be $10\%$ (e.g. Kritos & Cholis, 2020
and references therein). Taking into account the initial survival fraction of
wide binaries, we consider $f_{\rm merge}\approx 0.49-0.61$ for our fiducial
model. The lower value corresponds to massive background stars and the upper
limit to low mass background stars ($\bar{m}=0.5\ M_{\odot}$), see a
discussion below eq. 6.
Following Rodriguez et al. (2016), we consider logarithmically flat
distribution of initial SMA in the range $[10^{-2},10^{5}]\ \rm{AU}$ where the
lower limit is close to the point of stellar contact and the upper one to the
Hill radius. It should be noted that although the choice of logarithmically
flat is common, there were other choices of distribution considered, based on
observational data (see Antonini & Perets, 2012 for further discussion).
For our fiducial model, $N_{\star}=10^{5}$ and $M_{\rm cluster}=10^{5}\
M_{\odot}$.
In order to calculate the GWs merger from old GCs, we follow the calculation
of Rodriguez et al. (2016); Kritos & Cholis (2020),
$\displaystyle\mathcal{R}_{\rm old}(z)=\frac{1}{V_{c}(z)}\int_{z_{\rm
min}}^{z}\Gamma_{\rm old}(z^{\prime})n_{\rm
old}(z^{\prime})\frac{dV_{c}}{dz^{\prime}}(1+z^{\prime})^{-1}dz^{\prime}$ (23)
where $\Gamma_{\rm old}$ is the rate of mergers in old GCs, $n_{\rm old}$ is
the GCs number density, which is taken to be in the range
$[0.33,2.57]E^{3}(z)\ \rm{Mpc}^{-3}$ (Portegies Zwart & McMillan, 2000;
Rodriguez et al., 2016; Kritos & Cholis, 2020), $dV_{c}/dz$ is the comoving
volume and $(1+z)^{-1}$ accounts for the time dilation. The comoving volume is
given by (Hogg, 1999),
$\displaystyle\frac{dV_{c}}{dz}$ $\displaystyle=\frac{4\pi
c^{3}}{H_{0}^{3}E(z)}\left(\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}\right)^{2},$
(24) $\displaystyle E(z)$
$\displaystyle=\sqrt{\Omega_{M}(1+z^{\prime})^{3}+\Omega_{\Lambda}}$ (25)
where $\Omega_{K}=0,\ \Omega_{M}=0.3$ and $\Omega_{\Lambda}=0.7$ (Planck
Collaboration et al., 2016).
As a conservative estimate, we take the mergers rate $\Gamma_{\rm old}$ to be
$\Gamma_{\rm old}\sim N_{\rm merge}/\tau_{\rm GC}$ where $\tau_{\rm GC}$ is
taken to be $10\ \rm{Gyr}$. In Fig. 8 we present the cumulative rate of
expected mergers in old GCs (in blue). There are two types of contributions to
the rate: eccentric binaries, such as these with initial eccentricity of $2/3$
that corresponds to the mean value of a thermal eccentricity distribution,
that will merge within short timescales, i.e. with negligible delay time.
These practically follow the star formation rate. In this case observed
contributions are likely to rise from YMCs. The second case corresponds to low
eccentricity/circular binaries, in which there will be a delay time that
corresponds to a typical time of $\sim 10^{4}\rm{Myr}$. These contributions
will be observed in old GCs.
Figure 8: The cumulative contribution to GWs rate from YMCs (in red) and old
GCs (in blue), from the gas hardening channel, as derived from the GDF. The
shaded area relates to the range of parameters. The black line relates to the
range of rates inferred by LVK. In the case of circular binaries, the rate
will be dominated by old GCs, while for eccentric it will be dominated by
YMCs.
In this case, the major contribution from our channel to currently observable
GW-sources would not originate from old GCs, but from YMCs. We define a YMC as
a cluster formed later than redshift 2 and mass $>10^{4}\ M_{\odot}$ such that
we assume for that case that the 2P formation already occurred. The formation
rate of YMCs follows the star formation rate (SFR), which enables us to write
the merger rate from YMC as (Banerjee, 2021)
$\displaystyle\mathcal{R}_{\rm young}(z)=\frac{N_{\rm mrg}}{N_{\rm
samp}}\frac{1}{2\Delta t_{\rm obs}}\frac{\int_{M_{\rm cl,low}}^{M_{\rm
cl,high}}\Phi_{\rm CLMF}(M)dM}{\int_{M_{\rm GC,low}}^{M_{\rm
GC,high}}\Phi_{\rm CLMF}(M)dM}\times$ (26)
$\displaystyle\times\frac{\int_{0}^{z}\Psi_{\rm
SFR}(z_{f})dz_{f}}{\int_{3}^{6}\Psi_{\rm SFR}(z_{f})dz_{f}}\rho_{\rm GC}$
$N_{\rm mrg}$ is the number of mergers expected in $N_{\rm samp}$ clusters,
$\Delta t_{\rm obs}=0.15\ \rm{Gyr}$ (Banerjee, 2021) is the uncertainty in the
cluster formation epoch, $\Phi_{\rm CLMF}\propto M^{-2}$ (e.g. Portegies Zwart
et al., 2010) is the cluster mass function and we consider $[M_{\rm
cl,low},M_{\rm cl,max}]=[10^{4},10^{5}]\ M_{\odot}$ as the available mass
range for YMCs and $[M_{\rm GC,low},M_{\rm GC,high}]=[10^{5},10^{6}]\
M_{\odot}$ as the typical present-day masses for GCs. $\rho_{\rm GC}$ is the
observed number density of GCs per unit comoving volume. $\Psi_{\rm SFR}(z)$
is the cosmic star formation rate, which is given by (Madau & Dickinson,
2014),
$\displaystyle\Psi_{\rm
SFR}(z)=0.01\frac{(1+z)^{2.6}}{1+[(1+z)/3.2]^{6.2}}M_{\odot}\rm{Mpc}^{-3}\rm{yr}^{-1}$
(27)
We consider $N_{\rm mrg}/N_{\rm samp}=N_{\rm merge}$ and spatial densities in
the range $[0.33,2.57]\ \rm{Mpc}^{-3}$, following (Banerjee, 2021) and
references therein. In Fig. 8, we present the cumulative rate of expected
mergers in YMCs and GCs. For YMCs, the rate follows the star formation rate
(in general, with a small correction due to the delay time – which is short),
and hence peaks in relatively low redshifts. For the eccentric case, the
dominant contribution will rise from YMC, while for circular ones the dominant
contribution is from GCs.
It should be noted that in general, there could be a non-negligible delay time
for the binaries merger. However, for all the parameters we checked for the
disk configuration, the merger timescales are extremely short and are
negligible in terms of redshifts.
The total contribution to the GWs merger rate from YMCs is in the range
$\mathcal{R}_{\rm young}\approx[0.08,25.51]\ \rm{Gpc}^{-3}\rm{yr}^{-1}$, which
intersects the expected range of LVK, i.e. $23.9_{-8.6}^{+14.3}\ \rm{Gpc^{-3}\
yr^{-1}}$ (Abbott et al., 2021), where the range is bracket by the models with
lowest and and highest rates (see Table 1).
model | $\mathcal{R}_{\rm YMC}(z\leq 1)\ [\rm{Gpc^{-3}\ yr^{-1}}]$ | model | $\mathcal{R}_{\rm YMC}(z\leq 1)\ [\rm{Gpc^{-3}\ yr^{-1}}]$
---|---|---|---
$\rho_{-}c_{s-}n_{+}$ | 0.32 | $\rho_{+}c_{s-}n_{+}$ | 2.55
$\rho_{-}c_{s-}n_{-}$ | 0.08 | $\rho_{+}c_{s-}n_{-}$ | 0.64
$\rho_{-}c_{s+}n_{+}$ | 3.28 | $\rho_{+}c_{s+}n_{+}$ | 25.51
$\rho_{-}c_{s+}n_{-}$ | 0.82 | $\rho_{+}c_{s+}n_{-}$ | 6.35
Table 1: Rates from YMCs for redshifts $z\leq 1$, for different choices of
parameters. $\rho_{\pm}$ correspond to $\rho_{\rm GC}=0.33E^{3}(z)\
\rm{Mpc}^{-3}$ and $\rho_{\rm GC}=2.57E^{3}(z)\ \rm{Mpc}^{-3}$, $c_{s\pm}$
correspond to $c_{s-}=1\ \rm{km/sec}$ and $c_{s+}=10\ \rm{km/sec}$ and
$n_{\pm}$ corresponds to high density of progenitors and low fraction of hard
binaries ($n_{+}$, segregated environment) and low density of progenitors and
high fraction of hard binaries ($n_{-}$, non-segregated environment). These
correspond also to different fractions of soft/hard binaries, see subsec.
3.1.2. Here we present the rates expected for initially eccentric binaries
(e.g. $e_{0}=0.66$)
In table 1 we present our calculated rates for different choices of
parameters. As expected, higher gas densities lead to larger merger rates and
higher sound speeds correspond to thicker disks that host more stars and hence
yield more mergers.
### 4.5 GW merger properties
Given the early epoch of gas replenishment, gas-catalyzes mergers operate on
primordial binaries in the clusters. The merging components are therefore
likely distributed similar to the primordial distribution of binary
components. However, even very wide binaries can merge in this channel
compared with only relatively close binaries merging in e.g. isolated binary
evolution channels for GW mergers. This could give rise to significant
differences in the expected masses and mass-ratios of the merger objects.
Interaction with gaseous media could excite binaries to high eccentricities,
due to the dependence of the drag force on the relative velocity between the
gas and the binary, which changes along the orbit such that the effect is the
strongest at the apocenter. Evolution of eccentric binaries hence shorten
significantly the expected merger timescales, as larger separations correspond
to small pericenters, in which GWs could dominate the evolution. In this case
GW-emission would dump the eccentricites and GW-mergers would generally be
circular in the VLK bands. However, if the combined gas-catalyzed and GW-
emission binary shrinkage is slower (e.g. for circular-orbits or lower gas-
densities), where 3-body encounters dominate the final evolution, and higher
eccentricities can be achieved for at least a small fraction of the mergers,
similar to the dynamical channels for GW-sources explored in the past.
We should remark in passing on the possibility of triples. In triples, the
outer component migrates faster than the inner binary, potentially leading to
an unstable configuration and effective chaotic three body interaction (see
e.g. a the reversed case of triples expanding due to mass-loss, leading to
similar instability in Perets & Kratter, 2012), such chaotic encounters could
give rise to eccentric mergers. This possibility and its potential
contribution will be discussed elsewhere.
## 5 Discussion
In the following we discuss our results and implications for the evolution of
binaries and singles in gas-enriched GCs.
### 5.1 Other aspects of binary evolution
As we showed, the presence of gas modifies the binary population in GCs. It
leads to an efficient merger of binaries, together with the formation of
binaries via the L2 and L3 mechanism (which was initially used to study the
formation of Kuiper-belt binaries (Goldreich et al., 2002) and recently was
applied to calculate the formation rate in AGN disks Tagawa et al., 2020).
After the gas dissipation, the initial properties of the binary, as well as
the gas, dictate the final separation, to which all the binaries with initial
separations larger than the final separations will converge.
Therefore, gas hardening leaves a significant signature on the binary
population and its properties, which sets the ground for further dynamical
processes in general and specifically for later dynamical mergers.
In addition to the contribution of the channel to the total rate of GWs, the
modification of the properties of binaries (mass, separation etc.) caused by
the gas hardening, sets unique initial conditions for the other GWs channels.
This will induce an indirect signature of the gas hardening on the expected
observed mergers. We introduced analytical results that could in principle be
plugged in as initial conditions for the later evolution of GCs, and the
dynamical channels for GW production in such environments. The binary
abundance changes due to the gas hardening, since a significant fraction of
binaries could merge, while others form. Furthermore, additional L2/L3-formed
binaries could participate and produce GW sources, beyond the primordial
binaries considered here. Nevertheless, since stars might be far more abundant
than BHs, L2/L3 processes might mostly produce mixed BH-star binaries and may
not contribute to the GW merger rate, but may form other exotic binaries such
as X-ray sources etc., and/or produce micro tidal disruption events (Perets et
al., 2016, disruption of stars by stellar black holes).
### 5.2 Implications for other gas-rich environments
The gas-catalyzed dynamics discussed here could take place in any other gas-
rich environments, with the proper scaling. While enhanced GW merger rates
were discussed in the context of AGN disks (McKernan et al., 2012; Stone et
al., 2017; Tagawa et al., 2020, and references therein), they are usually
discussed in the context of the evolution of a particular binary or the
overall BH-merger rate. However, in those cases too, the whole binary
populations, of both compact objects and stars will change their properties.
A very similar process could take place for young binaries embedded in star
formation regions Korntreff et al. (2012). In this case, the effect is limited
to a shorter timescale and compact objects might not yet have formed, and are
therefore not directly affected (but their progenitor massive stars are).
### 5.3 YMCs and very massive clusters
YMCs are still relatively little studied in the context of the production of
GW sources, although their contribution to the total estimated rate of GWs is
potentially not negligible (Portegies Zwart & McMillan, 2000; Banerjee, 2021).
In these clusters, gas can be present up to smaller redshifts, such that the
effect from the channel we suggested for GWs could potentially be observed.
Hence, their overall contribution to the currently observed merger rates in
LVG will be more significant (as can be seen also in Fig. 8). Our rate
estimates discussed below, account for both GCs and their younger
counterparts, YMCs.
### 5.4 Dynamics in gas enriched clusters
All the dynamical processes that take place in the early stages of GCs
evolution might be affected by the presence of gas, e.g. few-body dynamics.
One aspect is that wide binaries that formed during the gas epoch are
protected from evaporation by the gas hardening, as they harden within
timescales shorter than the typical evaporation/ionization timescales.
GDF could also enhance mass segregation (Indulekha, 2013; Leigh et al., 2014).
The energy dissipation leads to a change in the velocity dispersion in short
timescales, such that massive objects will fall towards the center of the
cluster. Moreover, since the more massive objects are prone to merge (as can
be seen from eq. 7, or visually from Fig. 5), the relaxation will be affected
by the modified mass function induced by the gas hardening.
#### 5.4.1 GW recoils, spins and mass-gap objects
It is possible that gas-accretion onto binaries and not only GDF (e.g Roupas &
Kazanas, 2019) could affect their evolution. In particular, sufficient
accretion might align the BH spins and orbits, especially if some circumbinary
disk forms around the binaries, in which case the GW-recoil velocity following
mergers is likely to be small, and allow a larger fraction of merged, now more
massive BHs to be retained in the cluster. This in turn would affect the later
dynamics in the clusters, and the resulting mergers in the dynamical formation
channels operating in the clusters. This could then potentially give rise to
higher fraction of BHs reaching high (even mass-gap) masses following repeated
mergers. The spin evolution and accretion, however, require more detailed
study, which is beyond the current scope.
The spin evolution of binaries will be affected by the role played by
dynamical encounters, as well as the direction of the gas relative to the
binary. In some cases, initially misaligned binaries could be aligned later
due to gas accretion, but when dynamical encounters are dominant, the spins
won’t be aligned.
### 5.5 Implications for neutron stars and white dwarfs: accretion &
explosive transients
The focus of the current paper is the merger of BHs and the production of GW
sources due to gas interactions in multiple population clusters. However, the
evolution of stars and other compact objects such as WDs and NSs could be
significantly affected in similar ways. Though some of these aspects are
discussed in a companion paper (Perets, 2022)), we postpone a detailed
exploration of these objects to a later stage, and only briefly mention
qualitatively some potentially interesting implications.
A fraction of the gas could be accreted on objects in the cluster. Gas
accretion changes the velocities of the accretors and the overall mass
function of objects in the cluster, such that there is a shift towards higher
masses (e.g. Leigh et al., 2014), that might affect the dynamical GWs channels
in clusters that operate after the gas-replenishment epoch, since we enrich
the abundance of massive objects which are likely to be the progenitors of
GWs. Stars that accrete gas could evolve into compact objects that in turn
might produce novae. Enhanced accretion in the early stages of the cluster
evolution could potentially modify the novae rates and properties (Maccarone &
Zurek, 2012) and the production of accretion-induced collapse of WDs into NSs
(Perets, 2022).
We should point out that our scenario suggests a robust merger not only of
BHs, but also of neutron stars (NSs) and white dwarfs (WDs). These mergers
might leave unique signatures. Besides their contribution to the production of
short GRBs and GW sources, binary NSs mergers are a promising channel to the
production of heavy elements via r-process (e.g. Freiburghaus et al., 1999),
and would affect the chemical evolution of the clusters.
Thermonuclear explosions of WDs could produce type Ia SNe, whether via single
degenerate channel (WD and a non-degenerate companion, Whelan & Iben, 1973) or
double degenerate (two WDs, Iben & Tutukov, 1984). Both of these channels will
be affected by the gas accretion. First, as we mentioned (Leigh et al., 2014
and references therein), the mass function will change. This is turn might
change the characteristics of the SNe and their rate. Furthermore, regardless
of the mass variation, a large fraction of the compact object binaries are
expected to merge within short timescales, which will also affect the SNe
rate.
Mergers of WDs could yield a remnant merged object with small or absent natal
kick and hence constitute another channel for NS formation. Accretion could
potentially change the retention fraction, and potentially explain the
retention problem in the formation of pulsars (Perets, 2022).
### 5.6 Constraining the parameters of the cluster
The amount and origin of gas in GCs during the formation of 2P stars are still
uncertain (Bekki, 2017). In this channel, we suggest that the amount of gas
dictates a final SMA, such that the separation distribution/GW rate could be
used to constrain the gas abundance in the cluster and its lifetime.
For sufficiently low gas densities (or lower densities following gas
depletion), gas hardening is not efficient enough to lead to a merger. In this
case, the terminal SMA of the binary will exceed $a_{\rm GW}$, such that GWs
will not be emitted without a further dissipation process. However, if the gas
remains for longer timescales, further hardening will occur. For the whole
parameter space we considered, the early stages of the hardening process are
very efficient, i.e. wide binaries harden and become hard binaries on short
timescales.
This channel of production of GWs-sources could serve as a tracer to later
star formation, as it is coupled to the gas that accompanies this formation.
The amount of gas, its decay with time are determined by the star formation
history. Since these parameters play a role in gas hardening and hence on the
final separation distribution at the end of the gas epoch, they could
potentially serve to constrain the 2P gas and star-formation phase, and may
help explain some of the differences between 1P and 2P stellar populations.
For example, we might speculate that the inferred difference between the 1P
and 2P binary fractions (e.g. Lucatello et al. (2015)) could be explained by
gas-catalyzed hardening and mergers of main-sequence stars residing in the
gaseous region. Such 1P binaries which also accrete significant mass of 2P gas
would appear and be part of the 2P populations, while outside the gas regions
binaries are not affected. In this case some of the 2P binaries preferentially
merge compared with 1P stars outside the 2P gas region, leading to an overall
smaller binary fractions.
That being said, the many uncertainties and degeneracies involved might be
challenging in directly connecting current populations with the early
conditions directly.
### 5.7 Caveats/future directions
In the following we discuss potential caveats of our model/scenario.
$\bullet$ The specific scenario for formation of 2P stars is still
unknown/debated, and hence there are large uncertainties in the amount of gas
in the cluster, its source during the different stages of evolution. Moreover,
some explanations for the different chemical composition of the so-called 2P
stars might require lower gas masses than the total mass of 2P stars. In these
cases, the phenomena we described might be somewhat suppressed, though, as we
have shown even lower gas densities could be highly effective, and will not
qualitatively change the results.
$\bullet$ The expected production rates of GW sources depend on the initial
parameters of the clusters we consider, including the gas densities, stellar
and binary populations, star formation histories etc. All of these contain
many uncertainties, which we did not directly address in this initial study,
limited to a small number of models as to provide an overall estimate to
bracket the expected GW rates from this channel. Nevertheless, all of our
models show that gas-catalyzed mergers in multiple population clusters could
produce a significant and even major contribution to the GW-merger rate, and
could play a key role in the general evolution of stars and binaries in such
clusters.
$\bullet$ The interaction of gas with binaries is complex and includes many
physical aspects. Here we assumed that the gas density in the cluster, or at
least in the region in which the binaries evolve, is spatially constant. Most
of the gas should be concentrated in the star-forming region, preferentially
towards the inner parts of the cluster. Outer parts of the cluster might be
more dilute. Future study could relax the simplified assumption of a constant
spatial density and account for a more detailed distribution of gas, stars and
binaries.
$\bullet$ We assumed that the relative velocity between the objects and the
gas is dominated by the Keplerian velocity of the binary component become
dominant. A more realistic approach, but requiring a detailed Monte-Carlo or
N-body simulation could account for the detailed velocity distribution of
binaries in the cluster.
$\bullet$ As we mentioned in subsection 5.5, objects embedded in gas could
accrete from it and change their mass over time. As a result, their dynamics
will change both in the cluster and as binaries (Roupas & Kazanas, 2019). Here
we considered constant masses throughout the evolution, and neglected the
effects of gas accretion. This is a somewhat conservative assumption, in
regard to catalysis of mergers, as more massive objects are prown to merge
even faster in gas (see eq. 7 and Fig. 5).
$\bullet$ We considered several choices for the gas depletion, assuming an
exponential decay, with a fiducial model of $50\ \rm{Myr}$ and a lifetime of
$100\ \rm{Myr}$. However, the formation epochs of stars could set different
scenarios, e.g. in which gas is abundant in the cluster for longer timescales
of $\sim 100\ \rm{Myr}$, but only intermittently (Bekki, 2017), which will
change the picture, or when several wide scale gas replenishmet episode occur
over timescale of even many hundreds of Mys or even Gyrs, as might be the case
for nuclear clusters.
$\bullet$ In our analysis we considered for simplicity only equal mass
binaries. Though we don’t expect a major change in the results, the
generalization to binaries with different masses is more complex and requires
more detailed population studies, beyond the scope of the current study.
$\bullet$ It should be noted that there were studies that suggested limited
efficiency of gas dynamical friction (e.g. Li et al., 2020; Toyouchi et al.,
2020) than considered here. A more detailed comparison is left out for further
studies.
$\bullet$ Although the initial parameters of our disk suggest a thick disk, in
later stages the disk will be thinner and finally fragment if it to enable
star formation. Hence, for these stages/initial thin disks, the gas hardening
epoch should be limited to the regime in which the disk is stable.
$\bullet$ We restrict ourselves to binaries which are not likely to be
disrupted by interactions with other stars. Further disruptions could take
place and are encapsulated in $f_{\rm bin,surv}$ (see eq. 22.
## 6 Summary
In this paper we discussed the evolution of binaries in gas-enriched
environments which likely existed at the early stage multiple-population
clusters. We showed the binary interaction with the ambient gas-environment
significantly affects their evolution and give rise to major changes in binary
population in the cluster and its properties.
Binaries interaction with gas has been extensively studied over the last few
years in the context of AGN disks. Here we show that the environments of
multiple population GCs and YMCs similarly give rise to important effects. In
particular, focusing on the production of GW sources from binary BH mergers,
we find that gas-enriched multiple population clusters could provide a
significant and possibly major contribution to the production of GW sources of
up to a few tens of Gpc-1yr-1, comparable with the GW-sources production rate
inferred by VLK for the local universe. These might even be higher once
formation of new binaries due to gas-assisted capture is considered (to be
discussed in a follow-up paper).
Moreover, we expect catalyzed mergers of other compact objects such as NSs and
WDs, and of binary main-sequence and evolved stars to give rise to the
enhanced rate of a wide range of merger outcomes, producing a range of
transient events such as supernovae, GRBs and the formation of X-ray binaries
and stellar mergers, which will be discussed elsewhere.
Furthermore, our findings on the overall evolution of binary populations are
relevant for other gas-enriched environments such as AGN disks.
Finaly, our focus here was on binary BH mergers in multiple-population cluster
environments, but we point out that the early gas enriched phase of such
clusters (which in practice is relevant to the vast majority of GCs, given
that most GCs show multiple populations) significantly affect all the stellar
and binary populations, and the overall dynamics inside GCs. Hence the current
modeling of the typical initial conditions in GCs and their evolution might
need to be fundamentally revised.
## ACKNOWLEDGMENTS
We like to thank to the anonymous referee for important comments and points
that significantly improved the manuscript. We would also like to thank
Aleksey Generozov, Johan Samsing ,Jim Fuller, Kyle Kremer, Noam Soker and
Evgeni Grishin for fruitful discussions. MR acknowledges the generous support
of Azrieli fellowship. HBP and MR acknowledge support from the from the
European Union’s Horizon 2020 research and innovation program under grant
agreement No 865932-ERC-SNeX.
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## Appendix A Fiducial Parameters
Symbol | Definition | Fiducial Value
---|---|---
$\tau_{\rm gas}$ | gas lifetime | $50\ \rm{Myr}$
$\tau_{\rm SG}$ | formation time of SG | $100\ \rm{Myr}$
$M_{\star}$ | total mass of stars in cluster | $10^{5}\ M_{\odot}$
$M_{\rm gas}$ | gas mass in the cluster | $3\times 10^{5}\ M_{\odot}$
$\rho_{\rm g,disk}$ | initial gas density in disk | $1.74\times 10^{6}\ M_{\odot}\rm{pc^{-3}}$
$h/r$ | scale-height | $0.23$
$\sigma_{\rm disk}$ | disk velocity dispersion | $10\ \rm{km/sec}$
$\bar{m}$ | average stellar mass | $0.5\ M_{\odot}$
$n_{\star}$ | stellar density | $10^{5}\ \rm{pc^{-3}}$
$n_{\star,disk}$ | stellar density in disk | $10^{5}\ \rm{pc^{-3}}$
$c_{s}$ | sound speed | $10\ \rm km/sec$
$\log\Lambda_{g}$ | gas Coulomb logarithm | 3.1
|
# The Attractor of the Replicator Dynamic in Zero-Sum Games
Oliver Biggar
CIICADA Lab
Australian National University
Acton, ACT, Australia
<EMAIL_ADDRESS>
&Iman Shames
CIICADA Lab
Australian National University
Acton, ACT, Australia
<EMAIL_ADDRESS>
###### Abstract
In this paper we characterise the long-run behaviour of the replicator dynamic
in zero-sum games (symmetric or non-symmetric). Specifically, we prove that
every zero-sum game possesses a unique global replicator attractor, which we
then characterise. Most surprisingly, this attractor depends _only_ on each
player’s preference order over their own strategies and not on the cardinal
payoff values, defined by a finite directed graph we call the game’s
_preference graph_. When the game is symmetric, this graph is a tournament
whose nodes are strategies; when the game is not symmetric, this graph is the
game’s response graph. We discuss the consequences of our results on chain
recurrence and Nash equilibria.
## 1 Introduction
Learning in the presence of other learning agents is an increasingly
fundamental topic in modern machine learning, motivated by its role at the
core of cutting-edge techniques like _learning from self-play_ (38; 39) and
_Generative Adversarial Networks_ (18). The challenge of these systems is
analysing their collective behavior, which is where learning theory intersects
with game theory. To quote Hofbauer and Sigmund (21), “a major task of game
theory [is] to describe the dynamical outcome of model games described by
strategies, payoffs and adaptive mechanisms." That is, when agents learn
collectively, what do they learn?
[scale=1]figs/OD2
Figure 1: A zero-sum game (left) and its associated preference graph (right).
The sink component of the graph consists of all profiles other than $(a,a)$.
Under the replicator dynamic, this game has a unique attractor which is the
_content_ of the sink component (Theorem 3.3). The attractor is the union of
the strategy spaces of the subgames $\\{a,b,c\\}\times\\{b,c\\}$ and
$\\{b,c\\}\times\\{a,b,c\\}$, represented by the shaded region on the graph.
Note that the strategy space of the game is 4-dimensional, with the attractor
a 3-dimensional region on the boundary.
In online learning, the best-known approaches use variants of the
Multiplicative Weights Update algorithm (MWU) (5). To achieve the ‘no-regret’
property, these algorithms typically decrease the step size as more samples
are observed. In the long-run, as the step size becomes small, the behavior of
a collection of MWU-playing agents converges to the flow of a famous
differential equation: the _replicator dynamic_ (41; 37; 22; 44; 36). This
model was originally inspired by biological models of evolution (40), and is a
central object of study in _evolutionary game theory_ , the subfield of game
theory which focuses on dynamic processes. Since its discovery, the replicator
has been extensively analysed by biologists, mathematicians, economists and
computer scientists (22; 36; 25). Indeed, just as MWU is the flagship
algorithm in online learning, the replicator is the flagship dynamic in
evolutionary game theory (21; 36).
figs/RPS
(a) Rock-Paper-Scissors.
figs/MP
(b) Matching Pennies
Figure 2: The preference graphs (Definition 2.2) of two zero-sum games: (2(a))
Rock-Paper-Scissors (symmetric), and (2(b)) Matching Pennies (non-symmetric).
Describing the ‘dynamical outcome’ of games under the replicator dynamic (and
hence over MWU) involves answering a basic question: _to which strategy
profiles do we converge over time?_ In dynamical systems, a system’s long-run
behavior is defined by its _attractors_ (35; 27). Attractors are sets of
points which are _invariant_ (points inside the set remain there for all
time), _asymptotically stable_ (points in some neighbourhood converge to the
set) and _minimal_ (they do not contain a smaller set with the first two
properties). Attractors are fundamental objects in dynamical systems theory,
and so since the discovery of the replicator, identifying its attractors has
been a major topic of research (45; 2; 44; 22; 25; 32; 29; 26; 42). A number
of now-classical partial results are known, establishing that _pure Nash
equilibria_ (28) are always replicator attractors, and in _symmetric_ games
the related concept of _Evolutionary Stable Strategies_ (40) are likewise
attractors111In zero-sum games, a related result proves that the _time-
average_ of the replicator (and MWU) always converges to the Nash equilibrium
(20; 23; 16). However, the time-average behavior is distinct from the day-to-
day or _last-iterate_ behavior (32).. More strongly, a strategy profile is an
attractor _if and only if_ it is a _strict (pure) Nash equilibrium_ (44; 36).
However, this result only describes the simplest attractors—those which
contain only a single point. Most games—especially zero-sum games—don’t have
pure Nash equilibria, and most attractors contain more than one point!
Instead, the trajectories of the replicator in zero-sum games are typically
periodic (26), and under MWU they are often _chaotic_ (11; 12; 6).
We conclude that, despite decades of research, the attractors of the
replicator remain largely unknown, even in zero-sum games, arguably the best-
studied special case (2; 20; 21; 22; 36; 33; 30; 6; 11) (see Section 1.1).
This is what we achieve in this paper: we characterise the attractors of the
replicator dynamic in every zero-sum game. Beyond this result, our concepts
and techniques shed new light on equilibria, the graph structure of games (8;
9) and the modelling of payoffs/losses in economics and machine learning.
Characterising the replicator attractors of zero-sum games is a valuable
development, but we believe the most remarkable aspect of this result is the
form that this attractor takes. Specifically, the attractor depends _only_ on
players’ discrete preferences over their strategies, and not on the cardinal
payoff values. These ‘preferences’ are captured in a directed graph we call
the _preference graph_. In non-symmetric games this graph coincides with the
game’s _response graph_ (8; 32)222We use the name _preference graph_ to unify
the symmetric and non-symmetric cases, and because we find the name “response
graph” can be confusing. The word ‘response’ sounds like a turn-based game,
when in fact the name is actually a contraction of the “better-response
relation”, an ordering which defines player’s _preferences_., an object which
has been of increasing interest in algorithmic game theory (10; 8),
particularly in relation to the replicator (32; 29; 9). The nodes of the
preference graph are the profiles of the game, and the arcs represent which
strategies players prefer, given the strategies of the other player. As an
example, consider Figure 2(b), which shows the preference graph of the
Matching Pennies game. In this game, player 1 prefers to match the choice of
player 2, and player 2 prefers to mismatch player 1. The arc
${(T,H)}$${(H,H)}$, for example, captures the fact that, given player 2 plays
Heads, player 1 ‘prefers’ Heads over Tails. In symmetric zero-sum games, like
Rock-Paper-Scissors (Figure 2(a)), the preference graph has an even simpler
form where each arc represents the preferred option between some pair of
strategies. For example, given Rock ‘beats’ Scissors, in a match-up of Rock
and Scissors, one prefers to play Rock, hence the arc ScissorsRock.
Conceptually, the preference graph stores the underlying combinatorial
structure of the game. Most game-theoretic concepts—including the replicator
dynamic and the Nash equilibrium—are defined by cardinal payoffs, which serve
as a numerical instantiation of the underlying preference structure. Our
result shows that the choice of representation of preferences by numbers has a
transient effect: two games with different payoffs but the same preferences
have the same long-run behavior, in that their attractors are identical. One
practical consequence is that computing the attractor is easy (we can do it by
traversing the preference graph). More fundamentally, this lends our
prediction stability in the face of uncertainty in our model, an important and
rare property in game theory (43).
### 1.1 Contributions and Related Work
The main result of the paper is Theorem 3.3, which characterises the
attractors of zero-sum games. For each zero-sum game, we prove that an
attractor exists, is unique and attracts all points on the interior of the
game. This attractor is precisely the _content_ (9) (Definition 3.1) of the
preference graph’s unique (8)333This uniqueness is a zero-sum property; non-
zero-sum games, such as the $2\times 2$ Coordination game (30), may have
preference graphs with multiple sink components, and thus can have multiple
attractors under the replicator dynamic (9). _sink connected component_ ,
which is a strongly connected component with no arcs from a node inside the
component to a node outside. The sink component is a set of pure profiles; its
content is the set of _mixed_ profiles whose support contains only profiles in
this component, which is always an invariant set under the replicator. A
recent result (9) demonstrated that a replicator attractor always contains the
content of a sink component. The challenging part of our proof is showing that
the content of the sink component is also asymptotically stable in any zero-
sum game, and so is an attractor. We demonstrate stability using a potential
function argument. Our choice of function derives from the preference graph:
specifically, we use the total probability mass over all sink component
profiles. The proof then separates into two cases, reflecting two standard
types of zero-sum game: symmetric and non-symmetric. In evolutionary game
theory these are often called _single-_ and _multiple-population_ games.
Single-population games are also called _population games_ (36) or _matrix
games_ (4). The replicator has a slightly different properties in each case.
We show that in _symmetric zero-sum games_ the preference graph possesses a
special simplified form. The remainder of the symmetric case is
straightforward. The non-symmetric case is more complex, and involves
embedding the flow of the replicator on the non-symmetric zero-sum game in the
flow of the replicator on a larger symmetric zero-sum game we call the
_symmetrised game_ (Definition 3.6), restricted to the subspace of product
distributions (Theorem 3.8). This allows for a much simpler definition of the
replicator, and makes clear the dependence on the preferences.
Long-run stability of strategy profiles under the replicator is a core topic
in evolutionary game theory (40), especially with regard to classical solution
concepts, such as Nash equilibria and _evolutionarily stable strategies_. See
the textbooks (21; 44; 36) for a summary. Some particularly relevant works are
that of Akin and Losert (2), who studied zero-sum games under the replicator,
proving a crucial _volume conservation_ property, analysed in depth by
Hofbauer (20). A sequence of further results (33; 30; 26; 42; 9) used this
property to bound asymptotically stable sets of the replicator. Zeeman (45)
performed an early study of replicator attractors, suggesting that the
qualitative behaviour of the replicator can be split into a finite number of
classes; our results show that in zero-sum games the qualitative behaviour is
defined by the preference graph, of which there are a finite number.
Additionally, Ritzberger and Weibull (34) showed that when a _subgame_ is
closed under “weakly better responses", then it is an attractor under the
replicator. The preference graph is defined by the weakly better responses,
and so our results can be considered a generalisation which handles the more
complex case where the sink component is not a subgame.
Despite these results, a general negative conclusion of this line of work was
that the replicator doesn’t converge to mixed equilibria (32), and moreover no
dynamic can converge to equilibria in all games (19; 7). Indeed equilibria are
generally computationally intractable (15; 14). Consequently, the algorithmic
game theory community has increasingly shifted towards new notions of dynamic
outcome which can predict the day-to-day behavior of computational agents in
games (25). To this end, _sink chain components_ were recently proposed (32)
as the outcome of dynamic games, with the replicator used as the motivating
example. Sink chain components are a slight generalisation of attractors; when
a replicator attractor exists, it is a sink chain component (Lemma 4.2). Sink
chain components are built on a concept called _chain recurrence_ (Definition
4.1), a generalisation of periodicity which forms the foundation of the
Fundamental Theorem of Dynamical Systems (13). Crucially, chain components are
grounded in computational considerations. Informally, two points are in the
same chain component if a bounded-precision computational device cannot
determine whether they lie on the same periodic orbit (30). Thus, finding
attractors of the replicator dynamic is motivated not only by dynamical
systems but also computer science: the attractor gives us the strongest
prediction of long-run behavior which is consistent with computation.
This chain recurrence approach (32) has inspired a number of new results on
games and the replicator dynamic (30; 31; 29; 9; 8). In zero-sum games, when a
_fully-mixed_ Nash equilibria exists, the behavior is essentially
unpredictable: the sink chain component is the whole game (33; 26; 30; 31).
Under MWU, we observe chaotic behavior (11; 12) in these games. Further, when
a fully-mixed NE does not exist, (33; 26) showed that all fully-mixed strategy
profiles converge to the subgame containing the equilibrium, called the
_essential subgame_. However, this convergence is not ‘uniform’, and so the
essential subgame is typically _not_ asymptotically stable (see Section 4),
one of the defining properties of attractors and sink chain components (3).
Instead, interior profiles which are _arbitrarily close_ to the essential
subgame may move far away before returning. Consequently, like the mixed Nash
equilibrium itself, the essential subgame may not be a plausible prediction of
bounded-precision computational agents.
The story so far of predicting the replicator seems generally negative. Our
results tell a different, more positive story: we characterise the
attractor/sink chain component of the replicator, which, while larger than the
essential subgame, is the smallest ‘computationally plausible’ outcome of the
game, in the sense of chain recurrence. What’s more, being defined by discrete
preferences, the solution is somehow natural. Our shift away from Nash-based
methods also reveals new insight. Prior approaches suggested a connection
between equilibria and chain recurrence in zero-sum games (30; 26). We find
instead that chain recurrence in zero-sum games is entirely defined by the
preference graph (Corollary 4.3). The previous findings are now explained by a
non-trivial connection between equilibria and the preference graph: the
existence of a fully-mixed equilibrium implies strong connectedness of the
preference graph (Lemma 4.4).
## 2 Preliminaries
In game theory, a game is defined by a triple consisting of the _players_ ,
_strategy sets_ for each player, and _payoffs_. A combination of strategies
for each player is called a _strategy profile_ or simply a profile, and for
each profile there is a real-valued payoff to each player. In this paper we
focus on two-player games, where we denote the players by the integers 1 and 2
and their strategy sets by $S_{1}$ and $S_{2}$. The strategy names are simply
labels, so we assume $S_{1}=[n]:=\\{i\in\mathbb{N}_{0}|\ i<n\\}$ and
$S_{2}=[m]:=\\{i\in\mathbb{N}_{0}|\ i<m\\}$. The profiles are the pairs
$S_{1}\times S_{2}$. We call this an $n\times m$ game. An $n\times m$ game is
defined by a pair $(A,B)$ of matrices, representing the payoffs to players 1
and 2 respectively. We focus on _zero-sum games_ , which we represent by a
single matrix $M\in^{n\times m}$, implicitly assumed to be the payoffs for the
first player, which defines the game $(M,-M^{T})$. That is, in a profile
$(s_{1},s_{2})$, player 1 receives $M_{s_{1},s_{2}}$ and player 2 receives
$(-M^{T})_{s_{2},s_{1}}=-M_{s_{1},s_{2}}$. A _subgame_ of a game is formed by
choosing subsets $T_{1}\subseteq S_{1}$ and $T_{2}\subseteq S_{2}$ of each
player’s strategy sets and restricting the game to the profiles in
$T_{1}\times T_{2}$. We typically represent a subgame by its product set of
profiles $T_{1}\times T_{2}$.
_Symmetric_ games consist of a single set of strategies, denoted $S$, with
$|S|=n$ and a matrix $M\in^{n\times n}$. In evolutionary game theory, we think
of symmetric games as having a single population where different subtypes
evolve in competition with the other subtypes. In symmetric games, ‘strategy
profiles’ are simply individual strategies. Again, the payoffs are defined by
a matrix $M\in^{n\times n}$. A symmetric game $M$ is zero-sum if $(M,M)$
defines a zero-sum game, that is $M=-M^{T}$. This is equivalent to requiring
that $M$ is _anti-symmetric_. Thus, there is a natural one-to-one
correspondence between anti-symmetric real matrices and symmetric zero-sum
games.
A _mixed strategy_ is a distribution over a player’s strategies, and a _mixed
profile_ is an assignment of a mixed strategy to each player. We sometimes
refer to a profile as a _pure profile_ to distinguish it from a mixed profile.
If $x$ is a mixed strategy for player $i$ and $s$ is a strategy for that
player, we write $x_{s}$ for the $s$-entry of $x$. In non-symmetric games, we
denote mixed strategies by pairs $(x,y)$ where $x$ and $y$ are distributions
over the first and second player’s strategies, respectively. In symmetric
games, mixed profiles are just distributions over $S$, considered as the
‘population distribution’. The _support_ of a mixed strategy $x$ is the set of
strategies $s$ where $x_{s}$ is non-zero. We define the _support_ of a mixed
profile to be the set of profiles whose strategies are in the support of each
player’s mixed strategy. As distributions over a finite set, mixed strategies
can be naturally embedded in the standard probability simplex in Euclidean
space, by choosing some arbitrary ordering of the strategies in $S_{1}$ and
$S_{2}$. We denote these spaces by $\Delta(S_{1})$ and $\Delta(S_{2})$. The
set of mixed profiles is the product $\Delta(S_{1})\times\Delta(S_{2})$, which
we call this the _strategy space_ of the game. In a symmetric game this is
$\Delta(S)$. It also has a natural embedding in Euclidean space, so we can
talk about geometric properties of sets of mixed profiles, which we also refer
to as ‘points’ in strategy space.
The payoffs of a game extends naturally to mixed profiles. In a non-symmetric
zero-sum game $(M,-M^{T})$, the _expected payoff_ of a mixed profile $(x,y)$
is $y^{T}Mx$ to player 1 and $x^{T}(-M^{T})y=-(y^{T}Mx)$ to player 2. In a
symmetric zero-sum game, the expected payoff of a mixed profile $x$ is
$x^{T}Mx$.
### 2.1 Preference graphs
Two profiles are _$i$ -comparable_ if they differ only in the strategy of
player $i$; they are _comparable_ if they are $i$-comparable for some player
$i$. If two profiles are comparable, then there is exactly one $i$ such that
they are $i$-comparable. In symmetric games, profiles and strategies are the
same, and we define all profiles to be comparable. Up to strategic
equivalence, the game is defined by the _payoff differences_ between
comparable profiles (10). We store this in a matrix we call the _weight
matrix_ $W$ of the game.
###### Definition 2.1.
Let $M$ be a zero-sum game, and let $p$ and $q$ be comparable profiles. If $M$
is a symmetric game, then profiles and strategies are the same, and we define
$W_{p,q}$ to be the same as $M_{p,q}$. If $M$ is non-symmetric, then
$p=(p_{1},p_{2})$ and $q=(q_{1},q_{2})$ and
$W_{p,q}=\begin{cases}M_{p_{1},p_{2}}-M_{q_{1},q_{2}}&\text{$p$ and $q$ are
$1$-comparable}\\\ M_{q_{1},q_{2}}-M_{p_{1},p_{2}}&\text{$p$ and $q$ are
$2$-comparable}\end{cases}$
If $p$ and $q$ are not comparable, then $W_{p,q}$ _is undefined_.
We deliberately leave the payoff differences between incomparable profiles
undefined, so that it is clear to the reader that we will only reference $W$
when the associated profiles are comparable. Note that for any comparable
profiles $p$ and $q$, $W_{p,q}=-W_{q,p}$.
###### Definition 2.2.
Let $M$ be a zero-sum game. The _preference graph_ of $M$ is the graph whose
nodes are the profiles of the game and where there is an arc ${p}$${q}$
between profiles $p$ and $q$ if and only if they are comparable and
$W_{p,q}\leq 0$ (equivalently, $W_{q,p}\geq 0$).
While the definition is the same for symmetric and non-symmetric games, the
resultant graphs are not the same, because the weight matrix is defined
differently. In symmetric zero-sum games all profiles are comparable, so the
preference graph is a _tournament_ (a directed graph with an arc between every
pair of nodes). In non-symmetric games the preference graph is the game’s
response graph (8), which is never a tournament because not all profiles are
comparable. We think of the entries in the weight matrix as being weights on
the associated arc, as in ${p}$${q}$$\scriptstyle{|W_{p,q}|}$.
### 2.2 Dynamical Systems and the Replicator
The replicator dynamic is a continuous-time dynamical system (36; 22), defined
by an ordinary differential equation. Let $x\in\Delta(S_{1})$ and
$y\in\Delta(S_{2})$ be mixed strategies, and let $s\in S_{1}$ and $t\in S_{2}$
be pure strategies. Then, for a (non-symmetric) zero-sum game $M$ we have
###### Definition 2.3 (Non-Symmetric Zero-Sum Replicator Equation).
$\displaystyle\dot{x}_{s}$ $\displaystyle=x_{s}((My)_{s}-x^{T}My)$
$\displaystyle\dot{y}_{t}$ $\displaystyle=-y_{t}((M^{T}x)_{t}-x^{T}My)$
In a symmetric game $M$, the replicator is defined a similar way:
$\dot{x}_{s}=x_{s}((Mx)_{s}-x^{T}Mx)$
If $M$ is also zero-sum, then $x^{T}Mx=0$ (by anti-symmetry), and this reduces
to
###### Definition 2.4 (Symmetric Zero-Sum Replicator Equation).
$\dot{x}_{s}=x_{s}(Mx)_{s}$
Note that the symmetric (2.4) and non-symmetric (2.3) replicator, while
similar, are distinct equations with different properties. The solutions to
these equations define a _flow_ (36) on the strategy space of the game, which
is a function $\phi:X\times\to X$ that is a continuous group action of the
reals on $X$. We call this the (symmetric or non-symmetric) _replicator flow_.
The forward orbit of the flow from a given point is called a _trajectory_ of
the system. A set of points $Y$ is called _invariant_ under $\phi$ if
$\phi(Y,t)=Y$ for any $t\in$.
###### Definition 2.5.
Let $A$ be a compact set under a flow $\phi$ on a compact space $X$. If there
is a neighbourhood $U$ of $A$ such that
$\lim_{t\to\infty}\sup_{x\in U}\inf_{y\in A}\mathbf{d}(\phi(x,t),y)=0$
where $\mathbf{d}$ is a metric, then we say $A$ is _asymptotically stable_. If
$A$ is also invariant, we call it an _attracting set_. If $A$ is an attracting
set, and no smaller attracting set is contained within it, we call it an
_attractor_. Attracting sets of the time-reversed flow $\phi^{-1}$ we call
_repelling sets_ , and attractors of the reversed flow we call _repellors_.
There are some differences in terminology in the literature to be wary of.
What we call attracting sets are sometimes called attractors (e.g. (13; 36;
9)), in which case what we call an attractor is a _minimal attractor_.
Otherwise, our definition is the same as (36; 9). The set of points which
approach an attractor $A$ in the limit $t\to\infty$ is called the attractor’s
_basin of attraction_. We call an attractor _global_ if its basin of
attraction includes all points in $\operatorname{int}(X)$.
## 3 The Attractor of the Replicator
In this section we prove Theorem 3.3, which characterises the attractor of the
replicator in zero-sum games. We begin by noting that any mixed profile $x$
naturally defines a distribution over profiles, with $x_{p}$ denoting the mass
on a profile $p$. If the game is symmetric, profiles and strategies are the
same and so this is trivial. In a non-symmetric game, $x=(x_{1},x_{2})$ is a
pair of mixed strategies, and the distribution is defined by the _product_ ,
with the mass on a profile $p=(p_{1},p_{2})$ defined by
$x_{p}:={x_{1}}_{p_{1}}{x_{2}}_{p_{2}}$ (1)
This distribution over profiles is used to define an important concept: the
_content_ of a set of profiles.
###### Definition 3.1 (Biggar and Shames (9)).
Let $H$ be a set of profiles in a game. The _content_ of $H$, denoted
$\operatorname{content}(H)$, is the set of all mixed profiles $x$ where all
profiles in the support of $x$ are in $H$.
Equivalently, $x\in\operatorname{content}(H)$ if and only if
$x_{H}:=\sum_{h\in H}x_{h}=1$, that is, $x$ defines a distribution whose mass
is entirely distributed over profiles in $H$. The content is a union of
subgames, and so is an invariant set under the replicator (36). We will show
that the unique global attractor is the content of the unique sink component
of the preference graph. Uniqueness follows from graph structure, established
originally in (8).
###### Lemma 3.2 (Uniqueness).
The preference graph of a zero-sum game has a unique sink component.
The proof of Lemma 3.2 can be found in the appendix. Now we can prove our main
theorem.
###### Theorem 3.3 (The Attractor of the Replicator).
In a (symmetric or non-symmetric) zero-sum game $M$, the content of the unique
sink component $H$ of its preference graph is the unique global attractor of
the (respectively symmetric or non-symmetric) replicator dynamic.
###### Proof.
Proving Theorem 3.3 requires showing (i) $\operatorname{content}(H)$ is an
invariant set (_invariance_), (ii) every attracting set contains
$\operatorname{content}(H)$ (_minimality_) and (iii)
$\operatorname{content}(H)$ is asymptotically stable and its basin of
attraction is $\operatorname{int}(X)$ (_global asymptotic stability_). We
establish (i) and (ii) in Lemma 3.4.
###### Lemma 3.4 (Invariance and Minimality).
If $H$ is the sink component of the preference graph of a (symmetric or non-
symmetric) zero-sum game $M$, then $\operatorname{content}(H)$ is invariant
under the replicator. Further, for any attracting set $A$,
$\operatorname{content}(H)\subseteq A$.
Lemma 3.4 largely follows similar results in the literature, so we defer its
proof to the appendix. The challenge and main contribution of Theorem 3.3 lies
in (iii): showing asymptotic stability of $\operatorname{content}(H)$. We do
this by demonstrating that $x_{H}$, the total mass on the sink component $H$,
increases over time (Lemmas 3.5 and 3.9). That is,
$\dot{x}_{H}=\frac{\mathrm{d}}{\mathrm{d}t}x_{H}=\sum_{h\in
H}\frac{\mathrm{d}}{\mathrm{d}t}x_{h}=\sum_{h\in H}\dot{x}_{h}>0$, for any
$x\in\operatorname{int}(X)$. The function $x_{H}$ is a natural choice, because
$x_{H}$ is uniformly continuous, bounded in $[0,1]$ and $x_{H}=1$ if and only
if $x\in\operatorname{content}(H)$, so $x_{H}$ can be thought of as a metric
for the distance between $x$ and the content. Showing $\dot{x}_{H}>0$ requires
different arguments for the symmetric and non-symmetric cases. The symmetric
case is straightforward, and we complete it below (Lemma 3.5).
###### Lemma 3.5.
In a symmetric zero-sum game $M$, under the symmetric replicator (2.4), if
$x_{H}\in(0,1)$ then $\dot{x}_{H}>0$.
###### Proof.
First, $\dot{x}_{H}=\sum_{h\in H}x_{h}(Mx)_{h}$. As $M$ is anti-symmetric,
$x_{h}x_{q}M_{h,q}+x_{q}x_{h}M_{q,h}=0$ if $q\in H$. Hence,
$\dot{x}_{H}=\sum_{q\in S_{1}\times S_{2}}\sum_{h\in
H}x_{q}x_{h}M_{h,q}=\sum_{q\not\in H}\sum_{h\in H}x_{q}x_{h}M_{h,q}.$
For any $q\notin H$ and $h\in H$, as $H$ is a sink component, there is an arc
${q}$${h}$ in the preference graph. That is, $M_{h,q}>0$. Hence, all summands
are nonnegative. Moreover, due to the fact that $x_{H}\in(0,1)$, there must
exist a $q\not\in H$ and an $h\in H$ such that $x_{q}x_{h}>0$. Thus,
$\dot{x}_{H}>0$. ∎
The more challenging case of the proof is showing that $\dot{x}_{H}>0$ in non-
symmetric games (Lemma 3.9), which we will solve by embedding the flow of the
non-symmetric replicator on our non-symmetric game $M$ into a that flow of the
symmetric replicator on a larger symmetric game (Theorem 3.8) which we call
the _symmetrised game_.
###### Definition 3.6.
Let $M$ be a zero-sum game, with $M\in^{n\times m}$. The _symmetrised game_ ,
written $\mathcal{S}_{M}$, is defined as the following (nm)×(nm) matrix, which
we index by profiles $p=(p_{1},p_{2})$ and $q=(q_{1},q_{2})$:
$(\mathcal{S}_{M})_{p,q}=M_{p_{1},q_{2}}-M_{q_{1},p_{2}}.$
The strategy space of $\mathcal{S}_{M}$ is $S_{1}\times S_{2}$. That is, the
_strategy profiles_ of the original game $M$ become _strategies_ of the
symmetrised game. Mixed profiles likewise become _mixed strategies_ , using
the production distribution as in equation (1):
$x_{p}:={x_{1}}_{p_{1}}{x_{2}}_{p_{2}}$.
This construction is a natural choice, and related constructions have appeared
before in the literature on zero-sum games (17). The symmetrised game has two
important properties: (1) it is anti-symmetric, and so is a symmetric zero-sum
game, and (2) it can be viewed as an extension of the weight matrix $W$,
previously only defined on comparable profiles, to a relation over all
profiles. In particular, if $p$ and $q$ are comparable then
$(\mathcal{S}_{M})_{p,q}=W_{p,q}$. More generally,
###### Lemma 3.7.
Let $p=(p_{1},p_{2})$ and $q=(q_{1},q_{2})$ be profiles. Then:
$(\mathcal{S}_{M})_{p,q}=W_{(p_{1},q_{2}),p}+W_{(q_{1},p_{2}),p}=W_{(p_{1},q_{2}),q}+W_{(q_{1},p_{2}),q}.$
Using Theorem 3.8, we now show that the symmetrised game allows us to analyse
the flow of the replicator on $M$ by considering the flow of the replicator on
$\mathcal{S}_{M}$.
###### Theorem 3.8 (Symmetrising the Replicator Dynamic).
Let $M$ be a non-symmetric zero-sum game. Let $x=(x_{1},x_{2})$ be a mixed
profile and $p=(p_{1},p_{2})$ a pure profile. Write
$x_{p}:={x_{1}}_{p_{1}}{x_{2}}_{p_{2}}$ as in equation (1). Then, under the
non-symmetric replicator (2.3),
$\dot{x}_{p}=x_{p}(\mathcal{S}_{M}x)_{p}.$
The proofs of Lemma 3.7 and Theorem 3.8 can be found in the appendix. Notice
that the resultant expression is exactly the definition of the symmetric
replicator (2.4) on the symmetric zero-sum game $\mathcal{S}_{M}$. Using this
transformation, we can now complete the proof of Theorem 3.3 on non-symmetric
games by showing again that $x_{H}$ is increasing in $x_{H}\in(0,1)$.
###### Lemma 3.9.
In a non-symmetric zero-sum game $M$, under the non-symmetric replicator
(2.3), if $x_{H}\in(0,1)$ then $\dot{x}_{H}$ > 0.
###### Proof.
If $h=(h_{1},h_{2})$ is a pure profile, then by Theorem 3.8,
$\dot{x}_{h}=\frac{\mathrm{d}}{\mathrm{d}t}({x_{1}}_{h_{1}}{x_{2}}_{h_{2}})=x_{h}(\mathcal{S}_{M}x)_{h}$.
Because $\mathcal{S}_{M}$ is symmetric, by the same argument as in Lemma 3.5,
we can show that
$\dot{x}_{H}=\sum_{q\not\in H}\sum_{h\in H}x_{q}x_{h}(\mathcal{S}_{M})_{h,q}.$
Now pick some profiles $q\not\in H$ and $h\in H$ with $x_{q}x_{h}>0$. Since
$x_{H}>0$, at least one such pair exist. We will show that the sum above is
strictly positive. Firstly, observe that if $q$ and $h$ are comparable, the
arc ${q}$${h}$ of the preference graph of $M$ goes from $q$ to $h$ (and not
vice versa), and so $(\mathcal{S}_{M})_{h,q}=W_{h,q}>0$. Otherwise, suppose
that $q=(q_{1},q_{2})$ and $h=(h_{1},h_{1})$ are not comparable, and let
$a=(q_{1},h_{2})$ and $b=(h_{1},q_{2})$. We have the following three cases:
1. 1.
$a,b\in H$. Then the arcs ${q}$${b}$ and ${q}$${a}$ are directed towards $a$
and $b$ because $q$ is outside the sink component $H$. By Lemma 3.7,
$(\mathcal{S}_{M})_{h,q}=W_{q,a}+W_{q,b}>0$.
2. 2.
$a,b\not\in H$. Then the arcs ${a}$${h}$ and ${b}$${h}$ are directed towards
$h$ because $a$ and $b$ are outside the sink component. By Lemma 3.7,
$(\mathcal{S}_{M})_{h,q}=W_{a,h}+W_{b,h}>0$.
3. 3.
$a\in H$, $b\not\in H$ (the case $b\in H$, $a\not\in H$ is identical). The sum
$\sum_{q\not\in H}\sum_{h\in H}x_{q}x_{h}(\mathcal{S}_{M})_{h,q}$ includes the
terms $x_{q}x_{h}(\mathcal{S}_{M})_{h,q}$ and
$x_{b}x_{a}(\mathcal{S}_{M})_{a,b}$. However,
$x_{q}x_{h}=q_{1}q_{2}h_{1}h_{2}=x_{b}x_{a}$, and so
$x_{q}x_{h}(\mathcal{S}_{M})_{h,q}+x_{b}x_{a}(\mathcal{S}_{M})_{a,b}=x_{q}x_{h}((\mathcal{S}_{M})_{h,q}+(\mathcal{S}_{M})_{b,a})=x_{q}x_{h}(W_{q,a}+W_{q,b}+W_{b,q}+W_{b,h})=x_{q}x_{h}(W_{q,a}+W_{b,h})$
by Lemma 3.7. Since $a$ and $h$ are inside the sink component and $q$ and $b$
are outside, and the arcs ${q}$${a}$ and ${b}$${h}$ must be directed into the
component, so $W_{q,a}>0$ and $W_{b,h}>0$ and thus
$x_{q}x_{h}(\mathcal{S}_{M})_{h,q}+x_{b}x_{a}(\mathcal{S}_{M})_{a,b}>0$.
Overall, we conclude that $\dot{x}_{H}>0$. ∎
Finally, asymptotic stability of $\operatorname{content}(H)$ follows easily
from the fact that $x_{H}$ is uniformly continuous. Pick any
$0<\alpha<\beta<1$. Then for any $x$ with $x_{H}\in[\alpha,\beta]$,
$\dot{x}_{H}>\epsilon$ for some $\epsilon$ (uniform continuity), so after some
finite time $x_{H}>\beta$. Repeating this argument for any $\alpha,\beta$
shows that $\operatorname{content}(H)$ is asymptotically stable. ∎
## 4 Chain Recurrence and Nash Equilibria
In this section we discuss some consequences of Theorem 3.3. The first
concerns chain recurrence, which is defined by $(\epsilon,T)$-chains.
###### Definition 4.1 (Chain Recurrence, (3)).
Let $\phi$ be a flow on a compact metric space $X$, with $x$ and $y$ in $X$.
An _$(\epsilon,T)$ -chain_ from $x$ to $y$ is a finite sequence of points
$x_{1},x_{2},\dots,x_{n}$ with $x=x_{1}$ and $y=x_{n}$, and times
$t_{1},\dots,t_{n}\in[T,\infty)$ such that
$\mathbf{d}(\phi(x_{i},t_{i}),x_{i+1})<\epsilon$. If there is an
$(\epsilon,T)$-chain from $x$ to $y$ for _all_ $\epsilon>0$ and $T>0$ we say
there is a _pseudo-orbit_ from $x$ to $y$.
A point is called _chain recurrent_ if it has a pseudo-orbit to itself. Two
points are _chain equivalent_ if there are pseudo-orbits between them in both
directions, and equivalent chain recurrent points are grouped in topologically
connected components called _chain components_ (3). Reachability under pseudo-
orbits provides an ordering on the chain components, and sink chain components
are those which are minimal in this order. Sink chain components have been
increasingly studied in algorithmic game theory (30; 31; 32). A connection
between the preference graph and chain components of the replicator was
demonstrated in (32; 42; 9), with (9) using this to prove that sink chain
components always exist. However, sink chain components have not generally
been characterised. Attractors, when they exist, are sink chain components, so
our results present the first characterisation of sink chain components of
zero-sum games.
###### Lemma 4.2.
In any flow, every attractor is a sink chain component.
###### Corollary 4.3.
The content of the sink component of the preference graph is the unique sink
chain component of a zero-sum game.
The proofs of Lemma 4.2 and Corollary 4.3 can be found in the appendix. In
zero-sum games, previous state-of-the-art results on chain recurrence made use
of the Nash equilibrium (33; 30; 26). This line of inquiry established first
(1) that all points in the essential subgame (the subgame containing the
equilibrium in its interior) are contained within the sink chain component.
Secondly (2), if the essential subgame is not the whole game, all interior
starting points converge to the essential subgame. This suggests a connection
between chain recurrence and equilibria in zero-sum games. Corollary 4.3 comes
to a different conclusion; it proves that sink chain components are
characterised solely by the preference graph. Lemma 4.4 resolves this seeming
discrepancy: chain components are determined by the preference graph, but the
presence of an equilibrium in a subgame forces some structure on the induced
preference graph of that subgame—in particular, it must be strongly connected
and contained within the sink component of the whole game’s preference graph.
###### Lemma 4.4.
In a zero-sum game, any subgame with a Nash equilibrium in its interior must
(i) be contained within the sink component of the preference graph and (ii)
the subgraph which this subgame induces in the preference graph must be
strongly connected.
###### Proof.
(i) Assume the attractor is not the whole game, in which case it is on the
boundary. By Theorem 3.3, all interior points are in the basin of attraction,
and by Theorem 3.4 of (33), trajectories starting from interior points
converge to the essential subgame in the limit, and so the essential subgame
must be within the attractor. (ii) Each subgame is independent, so we can
assume we are working with the whole game and the equilibrium $x$ is fully-
mixed. For contradiction, assume the preference graph is not strongly
connected. Consequently, there is an attractor $A$ on the boundary of the
strategy space (Theorem 3.3). The point $x$ is in the basin of attraction of
$A$, so $x$ converges to $A$, which contradicts the fact that $x$ is a Nash
equilibrium, which are fixed points under the replicator (36). ∎
This Lemma rephrases our understanding of equilibria and chain recurrence,
highlighting the key role of the preference graph. As an example, Corollary
4.3 implies that sink chain component is the whole game _if and only if_ the
preference graph is strongly connected. When a fully-mixed equilibrium exists,
the preference graph must be connected, and so the sink chain component is the
whole game. Most interestingly, even though we prove Lemma 4.4 using the
replicator dynamic, this lemma is a _purely game-theoretic result_ which only
relates equilibria and the preference graph. This connection between the
preference graph and equilibria is useful for analysing games. For instance,
examining Figure 1 we find the Nash equilibrium is $((0,0.5,0.5),(0,0.5,0.5))$
(both players play $b$ and $c$ half the time), which has support
$\\{b,c\\}\times\\{b,c\\}$. As Lemma 4.4 predicts, this is within the
attractor and its induced preference graph (a 4-cycle) is strongly connected.
## 5 Conclusions and Future Work
In this paper we gave the first characterisation of the unique attractor of
the replicator in zero-sum games, thereby describing the long-term behaviour
of the dynamic in these games. As a secondary result, we have demonstrated the
importance of the preference graph as a tool for analysing game dynamics. In
particular, in the long run, only the _preferences_ dictates the outcome of
the game. This surprising conclusion has potentially significant consequences
for modelling strategic interactions. In game theory—especially evolutionary
game theory—precise knowledge of utilities is difficult to achieve; our
results show that modelling only the preferences for each player is sufficient
to characterise the attractor of the replicator dynamic.
An important goal for future research is to characterise the attractors of the
replicator in _all_ games, not just zero-sum ones. Currently, the attractors
of the replicator have been characterised in only a few classes of games, such
as potential games and $2\times n$ games without dominated strategies (9), as
well as some individual games, like the Asymmetric Cyclic Matching Pennies
game from (25). Our paper adds zero-sum games to this list of solved classes.
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## Appendix A Proofs
###### Lemma A.1 (Lemma 3.2).
The preference graph of a zero-sum game has a unique sink component.
###### Proof.
In the non-symmetric case, the preference graph is the response graph, and the
result follows from Theorem 4.10 of (8). In the symmetric case, the preference
graph is a tournament, and all tournaments have one sink component, as they
are orientations of complete graphs. ∎
###### Lemma A.2 (Lemma 3.4).
If $H$ is the sink component of the preference graph of a (symmetric or non-
symmetric) zero-sum game $M$, then $\operatorname{content}(H)$ is invariant
under the replicator. Further, for any attracting set $A$,
$\operatorname{content}(H)\subseteq A$.
###### Proof.
(_Invariance_ :) Observe that if $x\in\operatorname{content}(H)$, then the
support of $x$ is contained in $H$, and because all mixed profiles in the
subgame $\Delta(\operatorname{support}(x))$ have the same support,
$\Delta(\operatorname{support}(x))\subseteq\operatorname{content}(H)$. It
follows that $\operatorname{content}(H)$ is a union of subgames. By Theorem
5.4.7 of (36), all subgames are invariant sets under the replicator, and
unions of invariant sets are invariant.
($\operatorname{content}(H)\subseteq A$:) By Theorems 9.1.2 and 9.1.6 of (36),
no asymptotically stable set can exist in the interior of the strategy space
of a symmetric or non-symmetric zero-sum game. Subgames have the same
properties as the whole game under the replicator so the same is true of all
subgames. Dually, no repelling set can exist in the interior of any subgame.
The remainder of the proof follows that of Theorem 5.2 of (9).
(_Claim_ : every attracting set contains a profile.) This follows by
induction, using the fact that the replicator dynamic on a subgame has the
same properties as on the whole game. In the whole game, an asymptotically
stable set intersects the boundary. This intersection with the boundary must
also be asymptotically stable in any subgame it intersects on the boundary,
and so it intersects the boundary of this smaller subgame, and so on. We
conclude that such a set contains a pure profile, the smallest possible
subgame.
(_Claim_ : every attracting set contains all profiles in $H$.) An arc
${p}$${q}$ of the preference graph is also a subgame, where only the profiles
$p$ and $q$ are in the support. The (symmetric or non-symmetric) replicator
reduces to $\dot{x}_{p}=x_{p}(1-x_{p})W_{q,p}$ on this subgame, where
$W_{q,p}\geq 0$ (for the player for which these profiles are comparable). If
$p$ is contained in an asymptotically stable set, then $q$ must also be
contained in this set, because points near $p$ move to $q$ along this arc. We
know that asymptotically stable sets contain a pure profile—by this argument
we deduce that they contain all pure profiles reachable from that one in the
preference graph. Such a set of profiles always contains the sink component
$H$.
(_Claim_ : every attracting set contains $\operatorname{content}(H)$.) Let $Y$
be a subgame, where the pure profiles in $Y$ are in an attracting set $A$. If
$Y$ is a pure profile, then all mixed profiles in $Y$ are in the set,
trivially. Now suppose for induction that all points on the boundary of $Y$
are in $A$. Suppose for contradiction that there is a point
$x\in\operatorname{int}(Y)$ that is not in $A$. The set $A\cap Y$ is
attracting in $Y$, and the boundary is contained in $A\cap Y$, but this means
that the dual repelling set of $A\cap Y$ is contained in the interior of $Y$,
but no such sets can be contained in the interior. Hence all of $Y$ is within
$A$. By induction on subgames, we find that all of $\operatorname{content}(H)$
is within every attracting set. ∎
###### Lemma A.3.
$\mathcal{S}_{M}$ is anti-symmetric.
###### Proof.
For $p=(p_{1},p_{2})$ and $q=(q_{1},q_{2})$,
$(\mathcal{S}_{M})_{p,q}=M_{p_{1},q_{2}}-M_{q_{1},p_{2}}=-(M_{q_{1},p_{2}}-M_{p_{1},q_{2}})=-(\mathcal{S}_{M})_{q,p}$.
∎
###### Lemma A.4 (Lemma 3.7).
Let $p=(p_{1},p_{2})$ and $q=(q_{1},q_{2})$ be profiles. Then the following
holds:
$(\mathcal{S}_{M})_{p,q}=W_{(p_{1},q_{2}),p}+W_{(q_{1},p_{2}),p}=W_{(p_{1},q_{2}),q}+W_{(q_{1},p_{2}),q}.$
###### Proof.
$\displaystyle W_{p,(p_{1},q_{2})}+W_{p,(q_{1},p_{2})}$
$\displaystyle=(M_{p_{1},q_{2}}-M_{p_{1},p_{2}})+(M_{p_{1},p_{2}}-M_{q_{1},p_{2}})$
$\displaystyle=M_{p_{1},q_{2}}-M_{q_{1},p_{2}}=(\mathcal{S}_{M})_{p,q}\quad\text{and}$
$\displaystyle W_{(p_{1},q_{2}),q}+W_{(q_{1},p_{2}),q}$
$\displaystyle=(M_{p_{1},q_{2}}-M_{q_{1},q_{2}})+(M_{q_{1},q_{2}}-M_{q_{1},p_{2}})$
$\displaystyle=M_{p_{1},q_{2}}-M_{q_{1},p_{2}}=(\mathcal{S}_{M})_{p,q}$
∎
###### Theorem A.5 (Theorem 3.8).
Let $(M,-M^{T})$ be a two-player zero-sum game under the replicator dynamic.
Let $(x,y)$ be a mixed profile and $p=(p_{1},p_{2})$ a pure profile. Write
$z_{p}:=x_{p_{1}}y_{p_{2}}$ as in equation (1). Then
$\dot{z}_{p}=z_{p}(\mathcal{S}_{M}z)_{p}.$
###### Proof.
The two-population replicator dynamic (written for player 1, the player 2 case
is similar) is equivalent to
$\displaystyle\dot{x}_{s}$ $\displaystyle=x_{s}((My)_{s}-x^{T}My)$
$\displaystyle=x_{s}\sum_{t\in S_{1}}x_{t}((My)_{s}-(My)_{t})$
$\displaystyle=x_{s}\sum_{t\in S_{1}}x_{t}\sum_{r\in
S_{2}}y_{r}\left(M_{s,r}-M_{t,r}\right)$ $\displaystyle=x_{s}\sum_{t\in
S_{1}}\sum_{r\in S_{2}}x_{t}y_{r}\left(M_{s,r}-M_{t,r}\right)$
$\displaystyle=x_{s}\sum_{p=(p_{1},p_{2})\in S_{1}\times
S_{2}}z_{p}\left(M_{s,p_{2}}-M_{p_{1},p_{2}}\right).$
Now we observe that for $p=(p_{1},p_{2})$,
$\displaystyle\dot{z}_{p}$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}(x_{p_{1}}y_{p_{2}})$
$\displaystyle=(x_{p_{1}}y_{p_{2}})(\frac{\dot{x}_{p_{1}}}{x_{p_{1}}}+\frac{\dot{y}_{p_{2}}}{y_{p_{2}}})\quad\text{(product
rule)}$ $\displaystyle=z_{p}\left(\sum_{q\in S_{1}\times
S_{2}}z_{q}(M_{p_{1},q_{2}}-M_{q_{1},q_{2}})+\sum_{q\in S_{1}\times
S_{2}}z_{q}(M_{q_{1},q_{2}}-M_{q_{1},p_{2}})\right)\quad\text{(by above)}$
$\displaystyle=z_{p}\sum_{q\in S_{1}\times
S_{2}}z_{q}\left(M_{p_{1},q_{2}}-M_{q_{1},q_{2}}+M_{q_{1},q_{2}}-M_{q_{1},p_{2}}\right)$
$\displaystyle=z_{p}\sum_{q\in S_{1}\times S_{2}}z_{q}(\mathcal{S}_{M})_{p,q}$
$\displaystyle=z_{p}(\mathcal{S}_{M}z)_{p}.$
∎
###### Lemma A.6 (Lemma 4.2).
In any flow, every attractor is a sink chain component.
###### Proof.
We first show that all points in the attractor are chain recurrent. Attracting
sets are closed under intersection (24; 9), so an attractor cannot overlap any
other attracting set—that would contradict the minimality of the attractor.
(13) showed that points are chain recurrent if, for each attracting set $A$,
the point is contained in either $A$ or its dual repelling set $A^{*}$. The
attractor is compact, invariant, and no attracting set overlaps it, so an
attracting or repelling set must contain all points in the attractor. Hence
all points are chain recurrent.
Pseudo-orbits cannot leave attracting sets (2). Consequently, no point outside
the attractor is chain equivalent to a point inside it, and all points in the
attractor are chain equivalent, so it is a chain component. It is a _sink_
chain component because no pseudo-orbits leave the set. ∎
###### Corollary A.7 (Corollary 4.3).
The content of the sink component of the preference graph is the unique sink
chain component of a zero-sum game.
###### Proof.
By Lemma 4.2 and Theorem 3.3, the content of the sink component is a sink
chain component. Uniqueness follows for the same reason as in Theorem 3.3:
distinct sink chain components are disjoint, but every sink chain component
contains the content (9). ∎
|
Given a number of pairwise preferences of items, a common task is to rank all the items. Examples include pairwise movie ratings, New Yorker cartoon caption contests, and many other consumer preferences tasks. What these settings have in common is two-fold: a scarcity of data (it may be costly to get comparisons for all the pairs of items) and additional feature information about the items (e.g., movie genre, director, and cast). In this paper we modify a popular and well studied method, RankCentrality for rank aggregation to account for few comparisons and that incorporates additional feature information. This method returns meaningful rankings even under scarce comparisons. Using diffusion based methods, we incorporate feature information that outperforms state-of-the-art methods in practice. We also provide improved sample complexity for RankCentrality in a variety of sampling schemes.
§ INTRODUCTION
In this paper we are interested in the problem of rank aggregation from pairwise preferences under settings where the amount of data is scarce but we may have additional structural information.
For example, consider a setting where a set of pairwise comparisons on a set of $n$ movies have been collected from a set of critics and the goal is to give an overall ranking. If $n$ is large, for example, all movies released in the last two decades, it may be extremely costly to get a comparison for each of the $\binom{n}{2}$ pairs. A more realistic regime is to hope that each movie has been viewed at least once.
Standard methods of ranking suggest that the number of comparisons needed is roughly $O(n\log(n))$—when $n$ is large, even hoping for $\log(n)$ comparisons may be hopeless!
However, each movie has additional feature information $x_i\in \mathbb{R}^d$. For example, the dimensions could encapsulate the production budget, the number of A-list actors, the writer, studio, animated or live action, etc.
In general, we may suspect that these features inform the comparisons: if movies A and B have the same Oscar-winning director, and movie A beats movie C in a comparison, we may expect movie B to also perform well against movie C. In an extreme setting, even if we don't have any comparisons involving movie B, we may still hope to infer a meaningful ranking.
In this paper we focus on modifying a popular and well studied method arising in the ranking literature for this setting and demonstrate gains in the scarce setting when the number of comparisons is very small.
A common model in the literature of particular interest to us is the Bradley-Terry-Luce (BTL) model. We assume that we have $n$ items and associated to each item $i$ is a positive score $w_i$ so that the probability that $j$ is preferred to $i$ (“$j$ beats $i$”) in a comparison is
\begin{equation}
P_{ij}:=P(i \prec j) = \frac{w_j}{w_i + w_j}\label{eq:Pij},
\end{equation}
and that we see $m$ comparisons. The underlying ranking on the items is then given by the scores $w$, with an item with a larger score being ranked higher than an item with a smaller score. In the structured setting above, we may expect movies with similar features to have similar scores. Traditional methods of learning $w$ using the BTL model, e.g., maximum likelihood estimation (MLE) or spectral methods such as Rank Centrality (both discussed below), do not naturally incorporate this kind of side information.
We have two main contributions.
* Our main contribution is Algorithm <ref>, Regularized RankCentrality, in Section <ref>. We propose a novel method for regularizing the RankCentrality algorithm that returns meaningful rankings even under scarcity. Using diffusion based methods, we propose a way of incorporating feature information that is empirically competitive with other feature based methods such as RankSVM or Siamese Networks on both synthetic and real-world datasets in scarce settings. In a specific context, we provide a sample complexity result for this regularized method.
* Along the way, we discuss traditional RankCentrality and, under a natural sampling scheme extending that in <cit.>, we show an improved sample complexity bound for the RankCentrality algorithm. For example, when pairs are sampled uniformly, we improve the bound from $O(n^5 \log n)$ to $O(n\log n)$.
§ RELATED WORKS
There is an extensive amount of literature on ranking from pairwise comparisons under various models, and we refer the interested reader to the survey in <cit.>. Roughly speaking, most frameworks either fall into the parametric setting, i.e., a model such as BTL is assumed, or non-parametric where general assumptions on the pairwise comparison matrix $P$, where $P_{ij}$ is the probability that $i$ beats $j$ in a comparison, are made.
In the latter setting, several different conditions on $P$, such as stochastic transitivity and low noise described in <cit.>, or low rank as in <cit.>, and generalized low permutation rank models have been proposed (see <cit.>). All of these models include the BTL model as a specific case. Other estimators such as the Borda count and Condorcet winner (for finding the best item rather than a ranking) have been analyzed in <cit.>.
A variant of the ranking problem also falls under the category of active ranking where the comparisons that are queried are chosen by an active ranker rather than passively considered offline, see <cit.>.
A great deal of attention has been paid to the BTL model. A natural approach to this setting is to compute an estimate for $w$ using the MLE. More precisely given a set of comparisons $S = \{(i_k, j_k, y_k)\}_{k=1}^m$ where the $k$-th comparison is between items $i_k$ and $j_k$, and $y_k=0$ denotes that $i_k$ was preferred in this observation, whereas $y_k=1$ denotes that $j_k$ was preferred.
Then the MLE is given by
\begin{align}
& \argmax_{v \in \R^n} \sum_{i=1}^m -\log \left(1 + e^{(2y_k-1)(v_{j_k} - v_{i_k})} \right) \label{eq:btlmle}
\end{align}
and our estimate is $\hat{w}_i = \exp(v_i).$
We can also consider a constrained MLE where we add an additional constraint[Without loss of generality, assume $\sum_i w_i = 1$ because $P_{ij}$ is invariant to scaling $w$.], e.g., on the maximum entry of $w$, $\|w\|_{\infty} < B$, or, alternatively, we can add add an $\ell_2$ regularizer $\lambda \|v\|_2$ to the objective. The BTL-MLE in any of these formulations is a popular objective since it is convex.
We briefly review the known results on the BTL-MLE. <cit.> have shown the constrained BTL-MLE is minimax optimal for the $\ell_2$ error.
Note that low $\ell_2$ loss does not necessarily guarantee a correct recovery of a ranking. <cit.> shows that the (regularized) MLE and spectral ranking methods (discussed below) are minimax optimal for recovery of a ranking.
The critical parameter for recovery is the minimum gap between any two different BTL scores—which does not show up when one is interested in the $\ell_2$ norm only.
In the next section we discuss the class of algorithms that are the main study of this work: spectral methods and the RankCentrality algorithm.
§ SPECTRAL METHODS
We assume that we have access to a collection of $m$ independent and identically distributed pairwise comparisons $S = \{(i_k, j_k, y_k)\}_{k=1}^m$ where each $i_k < j_k\in [n]$. Furthermore we assume that each pair is i.i.d drawn: $(i,j) \sim_{\mu} \{(i,j), 1\leq i< j\leq n\}$, where $\mu$ is an unknown sampling distribution on the set of ordered pairs. Although $\mu_{ij}$ is defined for $i < j$, we assume it is understood that $\mu_{ij} = \mu_{ji}$ when $i > j$. Denote $\mumin := \min_{i<j} \mu_{ij}$ and $\mumax := \max_{i<j} \mu_{ij}$. In addition, we assume that the label is an independent Bernoulli draw, i.e.
\begin{equation*}
y_{k} = \begin{cases}
1 & \text{with probability } P_{i_kj_k}=\tfrac{w_{j_k}}{w_{i_k}+w_{j_k}} \\
0 & \text{otherwise}
\end{cases}
\end{equation*}
according to the BTL model where $(w_1, \cdots, w_n)\in \mathbb{R}_{>0}^n$ is an unknown vector of BTL-scores, i.e., $i_k\prec j_k$ with probability $P_{i_kj_k}$. Note $P_{ij} = 1- P_{ji}$. Additionally define $b := \max_{i,j} w_i/w_j$. Without loss of generality we assume that $w^T\textbf{1} = 1$, indeed scaling the weights has no effect on the comparison probabilities.
Problem. Given $S$, return $\hat{w}$, an estimator for $w$.
Consider the following matrix $Q\in \mathbb{R}^{n\times n}$, defined as
\begin{equation}
Q_{ij} :=
\begin{cases}
\mu_{ij} P_{ij} & \text{ if } i \neq j \\
1 - \sum_{\ell \neq i} \mu_{i\ell} P_{i\ell} & \text{ if } i = j
\end{cases}.
\label{eq:Q}
\end{equation}
Observe $Q_{ij}$ is the transition matrix of a time-reversible Markov chain, where the we transition from $i$ to $j$ with probability proportional to that of $i$ beating $j$ in a comparison (we refer the reader to Chapter 1 of <cit.> for background on Markov Chains), i.e., it satisfies the detailed balance equations: for all $i \neq j$, we have
\begin{equation*}
w_i Q_{ij} = \frac{\mu_{ij} w_i w_j}{w_i + w_j} = w_j Q_{ji}.
\end{equation*}
This implies the vector $w$ is the stationary distribution of $Q$, satisfying $w^TQ = w$, i.e., $w_i$ is the equilibrium probability of being in state $i$.
This motivates using the stationary distribution of an empirical estimator $\hat Q$, with $\E[\hat Q] = Q$ as an estimator $\hat{w}$ for $w$. The impatient reader can skip ahead to the next section for our choice of $\hat{Q}$.
The connection between the BTL model and time-reversible Markov chains was noticed by <cit.> where they proposed the RankCentrality algorithm for estimating $w$ under a slightly different model. In their setting, they assume they have access to a (connected) graph on $n$ vertices $G$, and for each edge in the graph they repeatedly query the associated pairwise comparison $k$ times. In the specific setting of an – graph $\mathcal{G}_{n,p}$ on $n$ vertices,
they construct an estimator $\hat{w}$
and show for $d\geq 10 C^2 \log n$ and $kd \geq 128 C^2 b^5 \log n$, setting $p = \tfrac{d}{n}$ the following bound on the error rate holds with high probability:
\[
\frac{\big\|\hat w-w\big\|_2}{\|w\|_2} \leq 8 C b^{5/2} \sqrt{\frac{\log n}{k\,d}}.
\]
(where we recall $b := \max_{i,j} w_i/w_j$). Noting that the expected number of comparisons is $O(n^2pk) = O(nkd) = O(b^5n\log(n))$ this yields a sample complexity of $O(b^5 n\log n/\epsilon^2)$ for recovering a weight vector with relative error $\epsilon$. Note that in this setting, for $\mathcal{G}_{n,p}$ to even be connected, it is important that $p$ be at least on order $\log(n)/n$, and we must at least observe $O(n\log(n))$ comparisons. In the more general setting, the sample complexity depends on the spectral gap of the graph Laplacian of $G$ ; precise dependencies have been given in <cit.>
Returning to our setting, our sampling scheme, which we refer to as independent sampling was proposed by <cit.>.
Observe that the independent sampling scheme is more natural in many applications, and in particular each observation is made independent of the other observations, which is not true of those in <cit.>.
Rajkumar and Agarwal show that if $O(\tfrac{Cn}{\varepsilon^2P_\mathrm{min}^2\mu_\mathrm{min}^2} b^3 \ln \left( \frac{n^2}{\delta}\right))$ comparisons are made
then with probability at least $1 - \delta$ (over the random draw of $m$ samples from which $\hat P$ is constructed), the score vector $\hat w$ produced by their version of the RankCentrality algorithm satisfies $\|\hat w - w \|_2 \leq \varepsilon$.
The sample complexity here scales as $O(n^5 \log n)$ since $\mu_\mathrm{min}^{-1} \geq \binom{n}{2}$, with equality achieved only when $\mu$ is uniform. In the next section we propose a different estimator from the one given in <cit.> and we are able to give a $O(n\log n)$ sample complexity bound in the case of uniform sampling.
A crucial point to note is that both <cit.> and <cit.> assume that the directed graph of comparisons, where an edge $(i,j)$ represents that $j$ beat $i$ in at least one comparison, is strongly connected. This is because the empirical estimate $\hat Q$ of the Markov transition matrix needs to be ergodic, i.e., irreducible and aperiodic, which ensures that $\hat Q$ has a unique stationary distribution. When the number of comparisons $m$ is small (i.e., $m<n\log(n)$ in the case of <cit.>), this is usually not the case and these algorithms return a default output. In particular, in the setting mentioned in the introduction where the number of comparisons are scarce, these methods will not return a useful ranking. This is a primary motivation for the work in this paper.
§.§ Warm-up: Improved Results for Independent Sampling
In this section we improve the results given in <cit.> by using a different estimator of $Q$ than the one presented there.
Recall the notation of Section <ref>. Given a dataset of comparisons $S$, define
\begin{align*}
C_{ij} = \textstyle \sum_{k=1}^m \Big( \one\{i_k = i, j_k =j, y_k=1\} \\ + \one\{i_k = j, j_k=i, y_k=0\} \Big),
\end{align*}
i.e., $C_{ij}$ is the number of comparisons between $i$ and $j$ that $j$ won.
Additionally define the empirical Markov transition matrix
\begin{equation}
\hat Q_{ij} :=
\begin{cases}
\frac{C_{ij}}{m} & \text{ if } i \neq j \\
1 - \sum_{\ell \neq i} \frac{C_{i\ell}}{m} & \text{ if } i = j
\end{cases}.
\label{eq:Qhat}
\end{equation}
By construction, $Q = \E(\hat Q)$ so $\hat Q$ is an unbiased estimator of $Q$.
Let $\hat{w}$ be the leading left eigenvector of $\hat Q$. When $\hat Q$ is ergodic, $\hat{w}$ is the unique stationary distribution of $\hat Q$.
Fix $\delta \in (0, 1)$ and $\varepsilon \in (0, 1)$. If
\[ m \geq 64b^3 n^{-1} \mumin^{-2} \varepsilon^{-2}(\mumax + n\mumax^2) \log \frac{2n}{\delta} \]
and the empirical Markov chain $\hat{Q}$ constructed as in (<ref>) is ergodic, then with probability at least $1-\delta$, we have
\[\frac{\|\hat w - w\|}{\|w\|} \leq \varepsilon.\]
A complete proof can be found in the supplementary materials. We sketch an outline of the proof here.
We first prove a result on the deviation of left eigenvectors for perturbations of ergodic row stochastic matrices, Proposition <ref> based on ideas from <cit.>. For each observation $k \in [m]$, we define a random i.i.d. matrix $Q_k$ (in terms of $i_k$, $j_k$, and $y_k$) such that $\hat Q = I + \frac{1}{m} \sum_{k=1}^m Q_k$. We can therefore write $\hat Q - Q = \sum_k Z_k$ where each $Z_k$ is an independent random matrix with $\E(Z_k) = 0$ and we can explicitly compute the matrix variance of $Z_k$ (Lemma <ref>). By using matrix Bernstein inequalities given in <cit.> we can derive a central-limit type upper bound on $P(\|\hat w - w\| > \varepsilon)$ (Theorem <ref>). Solving the resulting inequality for $m$, we get the desired result.
Because $\mumin = \mumax = \binom{n}{2}^{-1}$ when $\mu$ is uniform, we have given an $O\left(b^3\varepsilon^{-2}n\log(\tfrac{n}{\delta})\right)$ sample complexity when $\mu$ is uniform. Our argument improves upon that in <cit.> through improved matrix concentration results and a different (unbiased) estimator for $Q$.
§ REGULARIZING RANKCENTRALITY
When the number of pairwise comparison observations we have available is small, the $\hat Q_{ij}$ entries are poor estimators for $Q_{ij}$: there are $n^2 - n$ off-diagonal entries in $\hat Q$ and each observation only affects one off-diagonal entry leaving most entries zero. Furthermore, as described in the previous section, if the graph of pairwise comparisons (given by connecting any two points with an edge) is not strongly connected, may not guarantee that $\hat{Q}$ has a unique stationary distribution.
Motivated by this, we ask a natural question—when the number of pairwise comparisons is small; i.e., data is scarce (for example we have just observed one comparison per item) how can we still obtain a reasonable ranking?
Intuitively, if the items $[n]$ have some inherent structure, we can hope to exploit that structure to infer pairwise comparisons.
Since $Q_{ij} = \mu_{ij}P_{ij}$; i.e., a scaled probability of $i$ beating $j$, even if we have never seen a comparison between $i$ and $j$, it is reasonable to estimate this value by taking a weighted combination of the empirical $\hat Q_{ik}, 1\leq k\leq n$, where the choice of weights perhaps reflect some prior knowledge on the similarity between $j$ and $k$. In an extreme case—if we suspect item $j$ and $k$ would perform the same against item $i$, we may choose the weight on $\hat Q_{ik}$ to be large, and set the weights on all other $\hat Q_{ik'}, k\neq k'$ to zero.
Said more precisely, we choose a row-stochastic matrix $D$ and use the estimator $\hat{Q}D$ whose $ij$-th entry is
\begin{equation} \label{eq:QDdefn}
[\hat Q D]_{ij} = \sum_{k=1}^n D_{kj} \hat Q_{ik}
\end{equation}
How should we choose $D$? We want $\hat QD$ to be ergodic, but it should also reflect some similarity structure between the items. This prior information could take form in many ways—for example we can imagine that associated to item $i$ is a feature vector $x_i\in \mathbb{R}^d$ and intuitively items that are close together perform similarly on a comparison with some other element $j$ (see Section <ref>). An extreme case of this is assuming that the items are in clusters, and items within a cluster rank similarly (or the same). Finally, we can consider forms of $D$ that do not reflect any prior structure but do at least guarantee that $\hat QD$ is ergodic—as we will show these estimators can still perform competitively with other methods (Section <ref>). To recap, our resulting regularized RankCentrality algorithm that we will discuss in the rest of this section is given below in Algorithm <ref>.
Regularized RankCentrality algorithm
[1]
RankCentrality$n,S, D$
compute $\hat Q$ as in (<ref>)
return leading left eigenvector of $\hat Q D$
§.§ Diffusion Based Regularization
Diffusion RankCentrality leverages additional features $x_i \in \R^d$ for each of the items $i \in [n]$ being ranked. We use this to compute pairwise similarities in a manner consistent with the literature (e.g., in $t$-SNE <cit.> and diffusion maps formulated by <cit.>) so that for a fixed $i$, the similarities $D_{ik}$ are proportional to the probability density of a Gaussian centered at $x_i$. Let $D^{(\sigma)}_{ik}$, the similarity between item $i$ and $j$, be defined as
\begin{equation}
D_{ik}^{(\sigma)} := \frac{\exp\left(\frac{-\|x_i -x_k\|^2}{\sigma^2}\right)}{\sum_{l=1}^n \exp\left(\frac{-\|x_i -x_l\|^2}{\sigma^2}\right)},
\label{eq:similarityD}
\end{equation}
where $\sigma$, the kernel width, is an appropriately chosen hyperparameter. The Diffusion RankCentrality algorithm, obtained by using $D^{(\sigma)}$ in Algorithm <ref>, returns the stationary distribution of the Markov chain $\hat Q D^{(\sigma)}$.
As described in equation (<ref>), $[\hat Q D^{(\sigma)}]_{ij} = \sum_{k=1}^n D_{kj}^{(\sigma)} \hat Q_{ik}$,
i.e., the $ij$ entry is a weighted average of $\hat Q_{ik}$'s. $D_{ij}^{(\sigma)}$ is large when $x_i$ is close to $x_j$ and close to 0 when they are far apart. In particular the $\hat{Q}_{jk}$ contribute more when $j$ is close to $i$ and less otherwise.
An alternative interpretation of this procedure is given by considering the Markov chain induced by $\hat{Q}$ and contrasting it with that of $\hat{Q}D^{(\sigma)}$. Consider starting at any item $i$, and repeatedly transitioning according to $\hat{Q}$. If the number of comparisons is small, there may not even be a path from $i$ to any other item $j$. In addition, any additional comparison greatly affects the stationary distribution (i.e. the limiting distribution as we transition according to $\hat{Q}$) of $\hat{Q}$. Contrast this with the stationary distribution of $\hat{Q}D^{(\sigma)}$. By construction, $\hat{Q}D^{(\sigma)}$ will be dense (assuming each element has some neighbor that has a comparison). We can interpret the elements of $\hat{Q}D^{(\sigma)}$ as a Markov chain themselves: first, we make a sub-step (say from $i$ to $k$) according to $\hat Q$, which is based only the pairwise comparison observations, and then we make a sub-step (say from $k$ to $j$) with probability that inversely depends the distance of points to $k$. In, particular, we have imputed a series of transitions from $i$ to other elements $j$, using the underlying geometry of the points along with the pairwise comparisons. This technique is similar to that found in <cit.>, the MAGIC algorithm used in the field of single-cell RNA sequencing, where each entry in $Q$ is an extremely undersampled low integer count.
Consider the following extreme case example. Suppose the 100 points $\{x_i\}_{i=0}^{99}$ lie in 10 tight clusters with cluster $k$ being $\{x_{10k+1}, \cdots, x_{10k+9}\}$ and the clusters are spaced very far apart. Assume the BTL scores of items are constant within clusters; if items $i$ and $j$ are in the same cluster then $x_i = x_j$ and $w_i = w_j$. Set $\|x_i - x_j\| = \infty$ when $i$ and $j$ are in different clusters.
In this case, the matrix $D^{(\sigma)}$ is block diagonal: $D^{(\sigma)}_{ij} = \frac{1}{10}$ when $i$ and $j$ are in the same cluster and $D^{(\sigma)}_{ij} = 0$ otherwise.
Figure <ref> demonstrates the benefit of multiplying $\hat Q$ by $D^{(\sigma)}$. We see that a comparison between $i$ and $j$ does not just affect the $ij$ entry, but those corresponding to neighbors of $i$ and $j$. To visualize the effect of $D^{(\sigma)}$, we also show heatmaps of the 50-th powers of the transition matrices, $\hat Q$ and $\hat Q D^{(\sigma)}$. The checkered patterns in $Q$ and $QD^{(\sigma)}$ are clearly visible in $(\hat QD^{(\sigma)})^{50}$ while $\hat Q^{50}$ is still very sparse. After 50 iterations of $\hat{Q}$ vs. $\hat{Q}D^{(\sigma)}$, we see the impact of regularization, $(\hat QD^{(\sigma)})^{50}$ is far less sparse than $\hat{Q}^{50}$ and reflects a block structure that is imputing comparisons for items that have been compared less often.
Demonstrating the impact of $D^{(\sigma)}$.
The 100 items in this experiment lie in 10 equally sized tight clusters, where BTL scores are constant within clusters and the corresponding $D^{(\sigma)}$ matrix is block diagonal. The $\hat Q$ matrix was computed using 200 pairwise comparisons simulated according to the BTL model.
There are a number of different ways we could have diffused the information across the samples. We could have used $\hat Q D^{(\sigma)}$, $D^{(\sigma)}\hat Q$, or even $D^{(\sigma)} \hat Q D^{(\sigma)}$. In our empirical analysis, however, we found no significant difference in the performance of the algorithm run with these possibilities.
Finally, we note that the running time of the regularized RankCentrality algorithm is dominated by the computation of the leading eigenvector. The matrices $Q$ and $D$ are of size $n \times n$ and we can form the matrix $M = \hat Q D$ in time $O(n^3)$. We then iterate in the power method with $M$, each iteration, requiring a matrix-vector multiply takes time $O(n^2)$.
Our empirical analysis suggests that a few steps of the power method are sufficient. Furthermore, this iterative eigenvector computation on sparse matrices can be faster, than optimization procedures inherent in the MLE.
§.§ Lambda-Regularized RankCentrality
Implicitly, $D$ is chosen so that two properties are satisfied. Firstly, $\hat QD$ will be an ergodic markov chain, and secondly, as in most regularization situations, we choose $D$ to capture some inherent prior structural information we may have about $w$ apriori. In this section we ignore the second motivation and instead focus on a $D$ which just guarantees that former constraint.
In particular, given $\lambda > 0$ we consider $D_\lambda := (1-\lambda) I + \frac{\lambda}{n} \one \one^T$ as a choice of regularizer in Algorithm <ref>. Note that $\hat QD_\lambda = (1-\lambda) \hat Q + \frac{\lambda}{n} \one \one^T$, which ensures that $\hat Q D_\lambda$ is a positive row-stochastic matrix, which must be ergodic.
In particular, we can run Algorithm 1, regardless of the number of samples and we are guaranteed that $\hat Q D_\lambda$ necessarily has a unique stationary distribution. The simple nature of $D_{\lambda}$ allows us to give a precise theoretical characterization of it's performance.
In general, $\E[\hat QD_{\lambda}] = QD_{\lambda}$, but $QD_{\lambda}$ may not have the same left eigenvector as $Q$. This introduces a bias in our estimator.
How can we overcome this bias? Inspecting the form of $D_{\lambda}$, note that if $\lambda \to 0$ as $m\rightarrow \infty$ then $D_{\lambda} \rightarrow I$. The following theorem characterizes the error of this procedure of any $\lambda$ and shows that it is reasonable to take $\lambda = O(1/\sqrt{m})$.
For notational convenience, we let $\gamma := \frac{n\mumin}{2(1+\sqrt{2})b^{3/2}}$.
Note that $\gamma$ is not constant—in fact it is
Let $\lambda \in (0, \frac{\gamma}{2})$. Choose $\delta \in (0, 1)$ and $\varepsilon \in \left(2\lambda\gamma^{-1}, 1\right)$. Let $\hat w_{\lambda}$ be the output of Regularized RankCentrality run with $D=D_{\lambda}$. Then, with probability at least $1-\delta$,
\begin{equation*}
\frac{\|\hat w_{\lambda} - w\|}{\|w\|} < \ 2 \lambda \gamma^{-1} +\sqrt{\tfrac{68(1-\lambda)b^{3}(\mumax + n\mumax^2)}{n\mumin^2 m} \log\frac{2n}{\delta}},
\end{equation*}
In particular, choosing $\lambda = c/\sqrt{m}$, then with probability at least $1-\delta$, we have
\[ \frac{\|\hat{w} - w\|}{\|w\|} = O\left(\frac{b^3\log(2n/\delta)}{n\mu_{\min}m}\right). \]
We give a proof in the supplementary material under Corollary <ref>.
Our empirical experiments run with $\lambda = \eta m^{-1/2}$ for various values of $\eta$ support decaying $\lambda$ in this way. Figure <ref> demonstrates a run of $\lambda$-Regularized RankCentrality on a setting where $w = [i]_{i=1}^{200}$ and the underlying distribution on pairwise comparisons is assumed to be uniform. We compare several choices of $\lambda$ (with $\lambda = 0$ corresponding to normal RankCentrality) and the BTL MLE with an $\ell_2$ regularizer[Without such a regularizer, the BTL-MLE is underdetermined when the number of comparisons is small and cannot be solved.] on the weights (implemented using logistic regression). Note that $\eta = 1/6$ seems to perform the best and even outperforms regularizing the BTL-MLE for small sample sizes where RankCentrality may still be returning a uniform distribution.
For more details and experiments with different choices of $w$ in this setting, see Appendix <ref> in the supplementary materials.
Remark: To connect the diffusion based regularization with $\lambda$-regularization, observe that if we take $\sigma \to 0$ in the definition of $D$ in Equation <ref>, then $D\to D_0 = I_n$ (when the $x_i$'s are all distinct). The kernel width $\sigma$, therefore, determines the bias of Diffusion RankCentrality—small values of $\sigma$ only introduce a small bias in the algorithm while large values of $\sigma$ introduce considerable bias. Motivated by Theorem <ref>, to diminish this bias as $m$ increases, we can use $(1-\tfrac{1}{\sqrt{m}})I + \frac{1}{\sqrt{m}}D^{(\sigma)}$ in Diffusion RankCentrality instead of $D^{(\sigma)}$ directly. We call this Decayed Diffusion RankCentrality. In general, cross-validation could be used to choose the kernel width.
Comparing $\lambda$-Regularized RankCentrality with BTL-MLE and RankCentrality. Here $w = [i]_{i=1}^{200}$.
§ EMPIRICAL RESULTS FOR REGULARIZED RANKCENTRALITY
In this section we do a comparison of the regularized RankCentrality methods in the structured setting to standard methods for ranking on synthetic and real world datasets. The code we used along with additional plots are part of the supplementary material. Although our theoretical analyses do not make assumptions about $\mu$, our experiments focus on the case where $\mu$ is uniform.
§.§ Comparison to Scoring Functions
As discussed in Section <ref>, there is a rich literature of ranking methods, though less so for ranking data that come with features. Recall, we assume for each item $i \in [n]$ there is a vector $x_i \in \R^d$. In past work, the goal is to learn a function $f:\mathbb{R}^d\rightarrow\mathbb{R}$, presumed to be in a specified function class $\mathcal{F}$, such that $\sign(f(x_i) - f(x_j))$ predicts a comparison between item $i$ and item $j$. To learn $f$ given the dataset $S = \{(i_k, j_k, y_k)\}_{k=1}^m$, and a loss function $\ell:\mathbb{R}\times\mathbb{\R}\times \{0,1\}\rightarrow\mathbb{R}$, we can learn the empirical risk minimizer $\argmin_{f\in \mathcal{F}}\sum_{k=1}^n \ell(f(x_i), f(x_j), y_k)$. Two notable examples that focus on learning a scoring function that we compare to are RankSVM by <cit.> and Siamese network based approaches due to <cit.>.
RankSVM assumes that $\mathcal{F} = \{f: x \mapsto w^T x\}$, i.e. linear separators through the origin and choose $\ell(f(x_i), f(x_j), y) = \min(0, 1-(f(x_i)- f(x_j))(2y-1)$.
When testing RankSVM, we used it naively on the original features but also considered a kernelized version using random features, as described in <cit.> and implemented in SkLearn, <cit.>.
Note that when the loss function is the logistic loss, $\ell(f(x_i), f(x_j), y) = \log\left(\frac{\exp(f(x_j))}{\exp(f(x_i))+\exp(f(x_j))}\right)$,
we recover the MLE under the assumption that the BTL scores are given by a transformation of the features. Such an objective has been proposed several times in the literature, e.g. <cit.>.
In the extreme case $f(x_i) = \theta_i$ is the BTL-MLE.
An example of such an approach are Siamese Nets, introduced by in <cit.>.
We implemented a Siamese network using Keras (<cit.>) with two hidden dense layers, each with 20 nodes and a dropout factor of 0.1, and an output dimension of 1. Each layer in the base network used a ReLU activation. The outputs of the right network is subtracted from that of the left and a cross-entropy loss is then used.
We point out that in general both methods described above have a very different goal from what our paper proposes. Our goal is not to learn a scoring function, but instead to use the similarity information to inform the ranking process. In general, learning a scoring function can be expensive in terms of both computation, and samples. In addition, if the features do not actually inform the ranking very well, we want methods that will still learn a reasonable ranking—guaranteed by regularized RankCentrality as $m\rightarrow \infty$. We now demonstrate competitive performance of regularized RankCentrality even when the data is generated by a scoring function.
We constructed two synthetic datasets. We assume that the BTL-score is given by a continuous function of the features; i.e., there is an $f:\mathbb{R}^d\rightarrow \mathbb{R}$ so that the BTL score $w_i = f(x_i)$. This intuitively captures the idea that items which are close in space are close in rank. We consider a few examples of such functions $f$ as given below.
* In Experiment A, we generated 1600 points $\{x_i\}_{i=1}^{1600}$ chosen uniformly at random from $[0, 4]^2$, we chose $\omega_1, \omega_2, \dots, \omega_4 \in \R^2$ at random, each entry chosen independently from a Gaussian. To each $i \in [1600]$ we associate a score $w_i = \sum_{h=1}^2 \exp(\cos(5 \omega_h^T x_i)) + \sum_{h=3}^4 \exp(\omega_h^T x_i / 10)$.
* In Experiment B, we generated 1000 points $\{x_i\}_{i=1}^{1000}\in [0,4]$ chosen uniformly at random and chose $\omega \in \R$ at random from a Gaussian. To each $i \in [1000]$ we associate a score $w_i = \exp(\cos(5\omega x_i))$.
For varying of $m$, we simulated $m$ observations under the BTL-model with uniform $\mu$ and ran various algorithms that have been discussed. We recorded plotted the average Kendal-tau correlation metric (see Section <ref> in the supplementary for details) between the ranking on the synthetic scores we generated and the true ranking on the items. The results of these experiments are summarized in Figures <ref> and <ref>.
Comparison of algorithms in synthetic experiment A. Diffusion RankCentrality was run with kernel width $\sigma = 2^{-4}$.
Comparison of algorithms in synthetic experiment B. Diffusion RankCentrality was run with kernel width $\sigma = 2^{-5}$.
In Experiment A, Diffusion RankCentrality proves to be the best method when the comparisons are scarce. The impact of Diffusion RankCentrality in Experiment B is dramatic when compared to $\lambda$-regularized RankCentrality. While it is true that RankSVM with random features far outperforms other algorithms, it should not come as a surprise given that the BTL scores $w_i$, as a function of $x_i$, come from monotonic transformations of linear combinations of the basis of the RKHS used for the implementation of random Fourier Features in scikit-learn <cit.>.
In both experiments, Diffusion RankCentrality outperforms Siamese Networks.
To choose the kernel width, we ran Decayed Diffusion RankCentrality with several different choices of $\sigma$ on a validation set and chose the best one (see Figure <ref>).
Impact of kernel width on performance of Diffusion RankCentrality.
§.§ New Yorker Caption Competition
It is challenging to find real-life data sets that satisfy all of the following conditions: 1) The data is structured; i.e., has image or text features associated with the items and 2) the number of items compared is moderate to large in size.
The New Yorker Caption Competition dataset consists of a cartoon and a series of associated (supposedly) funny captions submitted by readers (see <cit.> for details on this dataset). Each week, readers vote on whether they think each caption is funny (2 points), somewhat funny(1 point) or unfunny (0 points), and the caption is assigned an average cardinal score based on these points. Included in this dataset are only two contests (#508 and #509), in which there are a large number of pairwise comparisons in addition to cardinal scores generated from user votes on a small number of items ($n = 29$ items for each contest). Each pair of items received roughly 300 comparisons and each item also received roughly 200 cardinal votes. (The associated captions and visuals of the query types are given in Figure <ref>, and Figure <ref> in the supplementary material). Run directly on this dataset, Diffusion Rank Centrality did not show an appreciable advantage since the number of items was so small and hence similarity information provided less leverage over other methods.
New Yorker Caption Competition Interface for pairwise comparisons for #508. Users were asked to click on the caption they thought was funnier.
A sample of the voting user interface presented to readers of the New Yorker Magazine for contest #651
§.§.§ Cardinal Scores model BTL-scores
We generate comparisons on a much larger set of captions for a different contest by transforming the cardinal data to infer pairwise comparisons. To determine this transformation, we used contest #508 for which we had 300 pairwise comparisons and 200 cardinal votes. For each pair of captions $i,j$ in contest #508, we compute $\hat{P}^{\text{emp}}_{ij}$, the empirical probability of item $i$ beating item $j$. In addition, we used the average empirical cardinal scores of items $i$ and $j$ denoted as $\hat{s}_i, \hat{s}_j$ we computed $\hat{P}^{\text{card}}_{ij} = \exp(\hat{s}_i)/(\exp(\hat{s}_i)+\exp(\hat{s}_j))$. In other words, we calculated the empirical probabilities implied by the cardinal scores and compared them to the empirical probabilities from the pairwise comparisons. A resulting scatterplot of the points $(\hat{P}^{\text{emp}}_{ij},\hat{P}^{\text{card}}_{ij})$ is shown in Figure <ref>. Somewhat surprisingly, this plot demonstrates that a monotonic transformation of the cardinal scores seem to model an underlying pairwise probability model fairly well—implying that up to an exponential scaling transformation, the cardinal scores determine underlying BTL scores for the captions. This seems to be an interesting non-trivial result about ranking and humor that has not been previously observed.
Scatter plot demonstrating the relationship between $\hat P^\text{emp}$ and $\hat P^\text{card}$.
§.§.§ Contest #651
Using the observations in the previous section, we chose a contest, #651, that did not have underlying pairwise comparisons but did have a large number of items all with cardinal scores. We then generated pairwise comparisons from these cardinal scores as described in Section <ref>. The cartoon associated to this contest is in Figure <ref>.
More precisely, from the captions available, we took the 400 captions (out of roughly 7000) with largest empirical average cardinal score (each caption had around 250 votes) and generated BTL weights.
We used the Universal Sentence Encoder in <cit.> to generate 512 dimensional embeddings for each of the captions (this yields the additional structural information we need for regularization). The resulting plot contrasting the methods is shown in <ref>, as before the kernel width was chosen on a validation set—in addition we used $(1-\tfrac{1}{\sqrt{m}})I + \frac{1}{\sqrt{m}}D^{(\sigma)}$ as the regularizer in Diffusion RankCentrality to debias the procedure.
In this setting, Diffusion RankCentrality performs extremely well, locking in a significantly better ranking almost immediately with few comparisons.
Test Error for various algorithms for the New Yorker Caption Competition #651 with $\sigma=.25$.
§.§ Place Pulse
Our final example involves comparisons arising from the Place Pulse dataset used in <cit.>. There were 100 images of locations in Chicago in this dataset, and a total of 5750 comparisons where MTurk workers were asked which of the two locations they thought were safer.
We used ResNetV1 <cit.> to generate features for the images of each location and broke the data up into a train, test and validation set (again used to select $\sigma$ and $\lambda$). Since we do not have an underlying ground truth ranking, we instead plot the test error in Figure <ref>.
Performance of various algorithms from the Place Pulse dataset.
Again, Diffusion RankCentrality (a non-classification based method) performed competitively matching the performance of RankSVM.
§ CONCLUSION
In this paper we provided a way to employ structure in the RankCentrality algorithm that provides meaningful results when data is scarce.
Along the way we provided a stronger sample complexity bound for a natural sampling scheme. For future work we hope to provide rigorous sample complexity bounds for diffusion based methods.
§.§.§ Acknowledgements
The first and third authors were supported by the MIDAS Challenge Grant from the University of Michigan. The first author had the initial idea and motivation for this work while at Agero, Inc., and would like to thank Michael Bell.
§.§.§ References
Missing 'biblatex' package
The bibliography requires the 'biblatex' package.
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Symbol Definition
$\|\cdot\|$ unless stated otherwise, vector norms are $\ell_2$ norms, and matrix norms are operator (spectral) norms
$\gamma$ $\frac{n\mumin}{2(1+\sqrt{2})b^{3/2}}$
$w$ stationary distribution of $Q$
$\hat w$ stationary distribution of $\hat Q$
$\lambda$ regularization constant, see $D_\lambda$
$\lmax(R)$ second largest eigenvalue of matrix $R$ (because the largest eigenvalue of an irreducible Markov chain is always 1)
$\mu_{ij}$ probability that pair $(i,j)$ is observed
$\one$ vector of all one entries, usually in $\R^n$
$b$ $\max_{i,j} \frac{w_i}{w_j}$
$k$ number of comparisons per pair in sampling scheme in <cit.>
$n$ number of items being compared
$m$ number of comparisons total
$P$ pairwise preference matrix
$\hat P$ empirical comparison matrix
$Q$ true markov chain (requires knowing $P$)
$\hat Q$ empirical markov chain
$D_\lambda$ $(1-\lambda) I + \frac{\lambda}{n} \one \one^T$
Notation used in this paper.
§ CONVERGENCE OF RANKCENTRALITY
\begin{equation}
Q^{(ij)} := e_{i}e_{j}^T - e_{i} e_{i}^T,
\label{eq:Qij}
\end{equation}
and additionally
\begin{equation}
Q_k =
\begin{cases}
Q^{(j_ki_k)} & \text{ if } y_k=0 \\
Q^{(i_kj_k)} & \text{ if } y_k=1
\end{cases}.\label{eq:Qk}
\end{equation}
We see now that
\begin{equation}
\hat{Q} = I + \frac{1}{m} \sum_{k=1}^m Q_k,\label{eq:Qhatdefnsum}
\end{equation}
and for the remainder of our analysis we shall consider (<ref>) as the definition of $\hat Q$. Recall
\[Q_{ij} =
\begin{cases}
\mu_{ij} P_{ij} & \text{ if } i \neq j \\
1 - \sum_{k\neq i} \mu_{ik} P_{ik} & \text{ if } i = j
\end{cases},
\]
and observe that $\E(\hat Q) = Q$.
We begin our analysis of the RankCentrality algorithm by giving a bound on the spectral gap of the transition matrix $Q$ constructed from pairwise preferences.
The spectral gap $1-\lmax$ of $Q$ is at least $\frac{n\mumin}{2b}$, where $b = \max_{i,j} \frac{w_i}{w_j}$.
We will use the following lemma from <cit.>.
Let $Q,\pi$ and $R,\tau$ be reversible Markov chains on a finite set $[n]$ representing random walks on a graph $G=([n],E)$, i.e.
$R(i,j)=0$ and $Q(i,j)=0$ if $(i,j)\notin E$.
For $\alpha\equiv\min_{(i,j)\in E}\{\pi_iQ_{ij}/\tau_iR_{ij}\}$ and
\[
\frac{1-\lmax(Q)}{1-\lmax(R)} \geq \frac{\alpha}{\beta}
\]
We will invoke the above lemma with $R = \frac{1}{n} \one \one^T = [\frac{1}{n}]_{ij}$, $\tau = \frac{1}{n} \one = [\frac{1}{n}]_i$, $Q$ as we have defined it previously, and $\pi = w$. Observe that these define a reversible Markov chain. Since $R$ has rank 1, we have $\lmax(R) = 0$, which gives us that $1 - \lmax(Q) \geq \frac{\alpha}{\beta}$. Now we bound $\alpha$ and $\beta$.
We have
\begin{align*}
\alpha = & \ \min_{i,j} \frac{w_iQ_{ij}}{\tau_iR_{ij}} = \min_{ij} \frac{w_i \mu_{ij} \frac{w_j}{w_i + w_j}}{\frac{1}{n} \frac{1}{n}} \geq \ \min_{i,j} \frac{n^2\mumin w_iw_j}{(w_i+w_j)} \geq \frac{n^2 \mumin \min_i w_i}{2}
\end{align*}
We also see $\beta = \max_i \frac{w_i}{\tau_i} = n \max_i w_i$. Thus, $\frac{\alpha}{\beta} \geq \frac{n\mumin}{2b}$.
This bound is close to optimal when $\mu$ is uniform. Since the diagonal entries of $Q$ are each at least $1 - \frac{2}{n-1}$, we know $\frac{n-1}{2}(Q - (1-\frac{2}{n-1})I)$ is non-negative and row stochastic. By the Perron-Frobenius Theorem, the eigenvalues of $\frac{n-1}{2}(Q - (1-\frac{2}{n-1})I)$ lie in $[-1,1]$ and the eigenvalues of $Q$ must lie in $[1-\frac{4}{n-1}, 1]$. The difference between 1 and the smallest possible eigenvalue of $Q$ is only a factor of $4b$ larger than our bound on the spectral gap.
Let $Q$ be the true transition matrix as defined in (<ref>). For any ergodic Markov chain on $[n]$ with row-stochastic transition matrix $\tilde Q$ and stationary distribution $\tilde{w}$, if $\|Q - \tilde{Q}\| < \frac{n\mumin}{2 b^{3/2}}$, we have
\[ \frac{\|\tilde{w} - w \|}{\|w\|} \leq \frac{2\|\Delta\|b^{3/2}}{n\mumin - 2\|\Delta\|b^{3/2}},\]
where $\Delta = \tilde{Q} - Q$.
We begin by citing a lemma <cit.>.
For any Markov chain $\tilde Q=Q+\Delta$ with a reversible Markov chain $Q$,
let $p_t$ be the distribution of the Markov chain $\tilde Q$ when
started with initial distribution $p_0$. Then,
\begin{align*}
\frac{\left\|p_t-w\right\|}{\|w\|} \leq \rho^t\frac{\|p_0-w\|}{\|w\|}\sqrt{\frac{w_{\rm max}}{w_{\rm min}}} + \frac{1}{1-\rho}\|\Delta\|_2\sqrt{\frac{w_{\rm max}}{w_{\rm min}}}\;.
\end{align*}
where $w$ is the stationary distribution of $Q$
and $\rho=\lmax(Q)+\|\Delta\|_2\sqrt{w_{\rm max}/w_{\rm min}}$.
As before, let $b = \max_{i,j} \frac{w_i}{w_j}$. Consider the limit as $t\to\infty$:
* when $0 \leq \rho < 1$ we have $\rho^t \to 0$, and
* when the Markov chain $\tilde Q$ is irreducible we have $p_t \to \tilde w$.
In this case,
\begin{align*}
\frac{\left\|\tilde w-w\right\|}{\|w\|} \leq \frac{1}{1-\rho}\|\Delta\|_2\sqrt{b}.
\end{align*}
Recall that $1 - \lmax(Q) > \frac{n\mumin}{2 b}$ by Proposition <ref>. Now we have that $\rho < 1$ when $\|\Delta\| < \frac{n\mumin}{2b^{3/2}}$ because when this is the case, we have $\|\Delta\|\sqrt{b} < \frac{n\mumin}{2b}$ and hence $\rho \leq 1 - \frac{n\mumin}{2b} + \|\Delta\|\sqrt{b} < 1$. Assuming $\|\Delta\| < \frac{n\mumin}{2b^{3/2}}$, we have
\[\frac{\|\tilde w - w\|}{\|w\|} \leq \frac{\|\Delta\|\sqrt{b}}{\frac{n\mumin}{2b} - \|\Delta\|\sqrt{b}} = \frac{2\|\Delta\|b^{3/2}}{n\mumin - 2\|\Delta\|b^{3/2}}.\]
For transition matrices $Q$ and $\hat Q$ we define the centered transition matrices $Q'$ and $\hat Q'$ by subtracting $I$. That is, $Q' = Q - I$ and $\hat Q' = \hat Q - I$. These centered matrices $Q'$ and $\hat Q'$, as well as $Q_k$ and $Q^{(ij)}$ defined previously, have non-negative entries everywhere except on the diagonal (where they are non-positive) and their rows sum to zero. These centered matrices significantly simplify the algebra in the following computations.
The difference $Z_k := \frac{Q_k - Q'}{m}$ is bounded in norm: $\|Z_k\| < \frac{3}{m}$.
To bound $\|Q_k\|$, recall that $Q_k$ is of the form $Q^{(ij)} = (e_ie_j^T - e_i e_i)$.
Observe that $ Q^{(ij)} Q^{(ij) T} = 2e_ie_i^T$. Therefore, $\|Q_k\| \leq \sqrt{2}$. By convexity of norms, $\|Q'\| = \|\E Q_k\| \leq \E \| Q_k\| \leq \sqrt{2}$. Using the triangle inequality we get $\| Q_k - Q'\| \leq 2\sqrt{2} < 3$.
Let $Z_k = \frac{Q_k - Q'}{m}$, as before. We can bound the variance term as:
\[\sigma^2:= \max \left\{ \left \|\sum_{k=1}^m \E Z_kZ_k^*\right\|, \left \|\sum_{k=1}^m \E Z_k^*Z_k\right\| \right\} \leq \frac{3(n-1)\mumin}{m}.\]
To bound $\|\E Z_k Z_k^*\|$, we see
\[\E Z_k Z_k^* = \frac{1}{m^2} \E \left( Q_k Q_k^{T} - Q_k Q^{\prime T} - Q' Q_k^{T} + Q' Q^{\prime T} \right) = \frac{1}{m^2} \E \left( Q_k Q_k^{T} - Q' Q^{\prime T} \right).\]
We can compute these explicitly.
Begin by considering the $Q_k Q_k^{T}$ term. We know $Q^{(ij)}Q^{(ij)T} = 2e_ie_i^T$. By simple algebra, we get $\E Q_k Q_k^{T} = \sum_{i} \sum_{j\neq i} 2\mu_{ij} P_{ji} e_i e_i^T$. Therefore, $\|\E Q_k Q_k^T \| \leq \max_i \sum_{j\neq i} 2 \mu_{ij} P_{ji} \leq 2 (n-1) \mumax$.
Computing $Q'Q^{\prime T}$ is more tedious.
\begin{align*}
Q' Q^{\prime T} = & \ \left( \sum_{i\neq j} \mu_{ij} P_{ij} (e_i e_j^T - e_i e_i^T) \right)\left( \sum_{u\neq v} \mu_{uv} P_{uv} (e_v e_u^T - e_u e_u) \right) \\
= & \sum_{i\neq j, u\neq v} \mu_{ij}\mu_{uv} P_{ij} P_{uv} (e_i e_j^T e_v e_u^T - e_i e_j^T e_u e_u - e_i e_i^T e_v e_u^T + e_i e_i^T e_u e_u^T).
\end{align*}
By ignoring zero terms (notice that the first of four summands is non-zero only when $j=v$, the second when $j=u$, etc.) and re-indexing, we get
\begin{align*}
Q' Q^{\prime T} = & \left( \sum_{i\neq \ell \neq j} \mu_{i\ell} \mu_{j\ell} P_{i\ell}P_{j\ell} e_i e_j^T - \sum_{i \neq j \neq \ell} \mu_{ij} \mu_{j\ell} P_{ij} P_{j\ell} e_i e_j^T - \sum_{j \neq i \neq \ell} \mu_{i\ell} \mu_{ji} P_{i\ell} P_{ji} e_i e_j^T + \sum_{u \neq i \neq v} \mu_{iu} \mu_{iv} P_{iu} P_{iv} e_i e_i^T \right),
\end{align*}
where statements such as $i\neq \ell \neq j$ mean $i \neq \ell$ and $j \neq \ell$ (but $i$ may be equal to $j$).
This is a symmetric matrix, so its singular values are its eigenvalues. We can now invoke the Gershgorin circle theorem, a consequence of which is that $\|M\| < \max_i \sum_j |M_{ij}|$ for symmetric matrices. Therefore, $\|Q' Q^{\prime T}\| \leq 4n^2\mumax^2$. Finally, the triangle inequality gives $\|\E Z_k Z_k^* \| \leq \frac{1}{m^2} \left( 2(n-1)\mumax + 4n^2\mumax^2 \right)$.
We now turn to $ Z_k^* Z_k$. Similar to the calculations above, simple algebra gets us
\[\E Q_k^{T} Q_k = \sum_i \sum_{j\neq i} \mu_{ij}( P_{ij} + P_{ji} ) (e_ie_i^T - e_i e_j^T).\]
As before, this is a symmetric matrix and we can use the Gershgorin circle theorem to give a bound on the largest singular value of $\E Q_k^T Q_k$:
\[ \|\E Q_k^T Q_k\| \leq \max_i \sum_{j\neq i} 2\mu_{ij} \leq 2 (n-1)\mumax.\]
As before computing $Q^{\prime T} Q'$ is more tedious but gives
\begin{align*}
Q^{\prime T} Q' = & \ \sum_{i \neq j} \sum_{u \neq v} \mu_{ij} \mu_{uv} P_{ij} P_{uv} (e_j e_i^T - e_i e_i^T)(e_u e_v^T - e_u e_u^T) \\
= & \ \sum_{i \neq j} \sum_{u \neq v} \mu_{ij} \mu_{uv} P_{ij} P_{uv} (e_j e_i^T e_u e_v^T - e_j e_i^T e_u e_u^T - e_i e_i^T e_u e_v^T + e_i e_i^T e_u e_u^T) \\
= & \ \sum_{i \neq j} \sum_{v \neq i} \mu_{ij} \mu_{uv} P_{ij} P_{iv} (e_j e_v^T - e_j e_i^T - e_i e_v^T + e_i e_i^T) \\
= & \ \sum_{i \neq j} \left( \sum_{\ell \neq i; \ell \neq j} \mu_{\ell i}\mu_{\ell j} P_{\ell i}P_{\ell j} - \mu_{ji} \mu_{j\ell} P_{ji}P_{j\ell} - \mu_{i\ell} \mu_{ij} P_{i\ell} P_{ij} \right)e_i e_j^T \\
& \qquad \qquad + \sum_{i} \left( \sum_{u\neq i,v\neq i} \mu_{iu} \mu_{iv} P_{iu} P_{iv} + \sum_{\ell \neq i} \mu_{\ell i} \mu_{\ell i} P_{\ell i}P_{\ell i} \right)e_i e_i^{T}.
\end{align*}
Again, we can invoke the Gershgorin circle theorem and see that $\|Q' Q^{\prime T}\| \leq 4n^2\mumax^2$. As before, the triangle inequality gives $\|\E Z_k^* Z_k \| \leq \frac{1}{m^2} \left( 2(n-1)\mumax + 4n^2\mumax^2 \right)$.
Finally, note that $Z_k$ are not only independent but also identically distributed and hence
\[\max \left\{ \left \|\E \sum_k Z_k^* Z_k \right \|, \left \|\E \sum_k Z_k Z_k^* \right\| \right\} = m \max\left\{\|\E Z_k^*Z_k\|, \|\E Z_kZ_k^*\| \right\} \leq \frac{4(n-1)\mumax + 4n^2\mumax^2}{m}.\]
We will soon need to use the Matrix Bernstein Inequality from <cit.> and state it here as a lemma.
Consider a finite sequence $\{ \mathbf{Z}_k \}$ of independent, random matrices with dimensions $d_1 \times d_2$. Assume that each random matrix satisfies
\[
\E \; \mathbf{Z}_k = \mathbf{0}
\quad\text{and}\quad
\norm{ \mathbf{Z}_k } \leq R
\quad\text{almost surely}.
\]
\[
\sigma^2 := \max\left\{
\norm{ \sum\nolimits_k \E( \mathbf{Z}_k \mathbf{Z}_k^* ) }, \
\norm{ \sum\nolimits_k \E(\mathbf{Z}_k^* \mathbf{Z}_k) }
\right\}.
\]
Then, for all $t \geq 0$,
\[
\PP{\left( \norm{ \sum\nolimits_k \mathbf{Z}_k } \geq t \right)}
\leq (d_1 + d_2) \cdot \exp\left( \frac{-t^2/2}{\sigma^2 + Rt/3} \right).
\]
Finally, we put this all together.
Let $\hat Q$ be constructed as in (<ref>). If $\hat Q$ is ergodic and $\hat w$ is the stationary distribution of $\hat Q$, then we have (where probability is taken over the $m$ comparisons made under the BTL model and each pair is equally likely to get picked)
\[\PP\left(\frac{\|\hat w - w\|}{\|w\|} \leq \varepsilon\right) > 1 - 2n \exp \left( \frac{-\mumin^2\varepsilon^2 n m}{16b^3(1+\varepsilon)^2(\mumax + n\mumax^2)} \right).\]
Assuming $\|\Delta\| < \frac{1}{nb^{3/2}}$, by Proposition <ref> we have
\[\frac{\|\hat w - w\|}{\|w\|} \leq \frac{2\|\Delta\|b^{3/2}}{n\mumin - 2\|\Delta\|b^{3/2}}.\]
This means we want
\[\frac{2\|\Delta\|b^{3/2}}{n\mumin - 2\|\Delta\|b^{3/2}} < \varepsilon,\]
which happens when $\|\Delta\| \leq \frac{\varepsilon n\mumin}{2b^{3/2}(1 + \varepsilon)}$. Note that this is stronger than $\|\Delta\| < \frac{n\mumin}{2b^{3/2}}$, so our previous assumption will hold.
Finally, we let $t = \frac{\varepsilon n\mumin}{2b^{3/2}(1 + \varepsilon)}$ and use Lemma <ref> to get
\[ \PP \left( \frac{\|\hat w - w\|}{\|w\|} \geq \varepsilon \right) \leq \PP \left( \|\hat Q - Q\| \geq t \right) \leq -2n\exp\left( \frac{-t^2}{\sigma^2 + Rt/3} \right),\]
where we have $\sigma^2 \leq \frac{4(n-1)\mumax + 4n^2\mumax^2}{m}$ by Lemma <ref> and $R < \frac{3}{m}$ by Lemma <ref>. Therefore, we get
\begin{align*}
\PP \left( \frac{\|\hat w - w\|}{\|w\|} \geq \varepsilon \right) & \leq 2n\exp\left(\frac{-\left( \frac{\varepsilon n\mumin}{2b^{3/2}(1+\varepsilon)} \right)^2}{\frac{4(n-1)\mumax + 4n^2\mumax^2}{m} + \frac{\varepsilon n \mumin}{2mb^{3/2}(1+\varepsilon)} } \right) \\
& \leq 2n \exp \left( \frac{-\mumin^2 \varepsilon^2 n^2 m}{4b^3(1 + \varepsilon)^2\left( 2n\mumax + 4n^2\mumax^2 \right) + 2b^{3/2}\varepsilon (1 + \varepsilon)n\mumin} \right) \\
& \leq 2n \exp \left( \frac{-\mumin^2\varepsilon^2 n m}{16b^3(1+\varepsilon)^2(\mumax + n\mumax^2)} \right).
\end{align*}
Fix $\delta \in (0, 1)$ and $\varepsilon \in (0, 1)$. If
\[ m \geq 64b^3 n^{-1} \mumin^{-2} \varepsilon^{-2}(\mumax + n\mumax^2) \log \frac{2n}{\delta} \]
and the empirical Markov chain $\hat{Q}$ constructed as in (<ref>) is ergodic, then with probability at least $1-\delta$, we have
\[\frac{\|\hat w - w\|}{\|w\|} \leq \varepsilon.\]
We need
\[ \PP\left( \frac{\|\hat w - w\|}{\|w\|} \geq \varepsilon \right) \leq 2n \exp \left( \frac{-\mumin^2\varepsilon^2 n m}{16b^3(1+\varepsilon)^2(\mumax + n\mumax^2)} \right) < \delta.\]
By re-writing in terms of $m$, we see that the second inequality is true when
\[ m > 16b^3(1+\varepsilon)^2 n^{-1} \mumin^{-2} \varepsilon^{-2}(\mumax + n\mumax^2) \log \frac{2n}{\delta} . \]
The desired inequality now follows immediately from $\varepsilon < 1$ (we make this assumption for simplicity; the statement of the theorem is not very strong when $\varepsilon > 1$).
When $\mu$ is uniform and $n>4$, the above theorem requires $m > 48 b^3 \varepsilon^{-2} n \log (\frac{2n}{\delta})$. We have given an $O\left( \varepsilon^{-2} n\log \frac{n}{\delta} \right)$ upper bound on the sample complexity. This is a much better bound than in <cit.>. Their $O(\varepsilon^{-2} \mumin^{-2} n \log (\frac{n}{\delta}))$ scales as $O(\varepsilon^{-2} n^5\log(\frac{n}{\delta}))$ when $\mu$ is uniform and worse otherwise.
§ CONVERGENCE OF LAMBDA-REGULARIZED RANKCENTRALITY
This section is devoted to an analysis of the bias-variance trade-off of $\lambda$-Regularized RankCentrality. We will compare
* $\hat{\tilde w}$, the leading left eigenvector of $\hat QD_\lambda$, i.e., the output of $\lambda$-regularized RankCentrality, and
* $\tilde w$, the leading left eigenvector of $QD_\lambda$, i.e., the expected output of $\lambda$-regularized RankCentrality as $m\to\infty$,
* $w$, the leading left eigenvector of $Q$, and the expected output of RankCentrality as $m \to\infty$.
Fix $\lambda \in (0, \gamma)$. The asymptotic ($m\to \infty$) expectation of the output of the $\lambda$-Regularized RankCentrality algorithm is $\tilde w$ and the bias $ \|w - \tilde w\| / \|w\|$ can be bounded as
\[ \frac{\|w - \tilde{w}\|}{\|w\|} \leq \frac{ \lambda}{\gamma - \lambda} \]
Let $\tilde{Q} = QD_\lambda$. We now have $Q - \tilde{Q} = \lambda ( \frac{1}{n}\one \one^T - Q)$ and $\|Q - \tilde{Q}\| \leq \lambda (1 + \sqrt{2})$. Now we apply Proposition <ref> to see that
\[ \frac{\|w - \tilde{w}\|}{\|w\|} \leq \frac{2(1+\sqrt{2})\lambda b^{3/2}}{n\mumin - 2(1+\sqrt{2})\lambda b^{3/2}} = \frac{\lambda}{\gamma - \lambda}.\]
Fix $\lambda \in (0, \frac{\gamma}{2})$ and choose $\varepsilon \in ( 2\lambda\gamma^{-1}, 1)$. We construct $\hat Q$ as before and let $\tilde{\hat{w}}$ be the stationary distribution (leading left eigenvector) of $\hat Q D_\lambda$ (i.e., the output of $\lambda$-regularized RankCentrality). We have
\[
\PP\left(\frac{\|\tilde{\hat{w}} - w\|}{\|w\|} < \varepsilon \right) >
1 - 2n \exp \left( \frac{-(n\mumin \varepsilon - 4(1+\sqrt{2})b^{3/2}\lambda)^2m}{16b^3(1-\lambda)^2 \left( 4(n-1)\mumax + 4n^2 \mumax^2 \right) + 4b^{3/2}(1-\lambda)(n\mumin \varepsilon - 4b^{3/2}(1+\sqrt{2})\lambda)} \right)
\]
As we noted in the proof of Theorem <ref>, to guarantee $\|w - \tilde{\hat{w}}\|/\|w\| \leq \varepsilon$, we need $\|Q - \hat Q D_\lambda\| \leq \frac{\varepsilon n \mumin}{2(1 + \varepsilon)b^{3/2}}$. Using the triangle inequality, we have $\|Q - \hat Q D_\lambda\| \leq \|Q - QD_\lambda\| + \|QD_\lambda + \hat Q D_\lambda\|$. We showed in Proposition <ref> that $\|Q - QD_\lambda\| \leq \lambda(1+ \sqrt{2})$. So we need
\begin{align*}
\|QD_\lambda - \hat Q D_\lambda\| & \ \leq \frac{\varepsilon n \mumin}{2(1 + \varepsilon)b^{3/2}} - \lambda(1+ \sqrt{2}) \leq \frac{\varepsilon n \mumin}{4b^{3/2}} - \lambda(1+ \sqrt{2}) \\
& \ = \frac{(1+\sqrt{2})}{2}\varepsilon\gamma\ - \lambda (1+\sqrt{2}) = \frac{(1+\sqrt{2})}{2} (\varepsilon \gamma - 2\lambda)
\end{align*}
Note that this quantity is positive when $\varepsilon \in (2\lambda\gamma^{-1}, 1)$ (which is precisely the requirement in the hypothesis above). We have required that $\varepsilon < 1$ to simplify algebra; the theorem is not very useful otherwise. We now require that
\[
\|QD_\lambda - \hat Q D_\lambda\| \leq \frac{\varepsilon n \mumin}{4b^{3/2}} - \lambda(1+ \sqrt{2}).
\]
We can now invoke Lemma <ref> with $Z_k = \frac{1}{m} (Q'D_\lambda - Q_kD_\lambda) = \frac{1}{m}(1-\lambda)(Q' - Q_k)$. By our previous calculations in Lemmas <ref> and <ref>, we have the variance term $\sigma^2 \leq (1-\lambda)^2 \frac{4(n-1)\mumax + 4n^2\mumax^2}{m}$ and the norm term $R \leq (1-\lambda)\frac{3}{m}$. The resulting inequality is
\begin{align*}
\PP\left(\|QD_\lambda - \hat QD_\lambda\| \geq \frac{n\mumin \varepsilon}{4b^{3/2}} - (1+\sqrt{2})\lambda \right)
\leq 2n\exp\left( \frac{-\left( \frac{n\mumin \varepsilon}{4b^{3/2}} - (1+\sqrt{2})\lambda \right)^2}{(1-\lambda)^2 \frac{4(n-1)\mumax + 4n^2\mumax^2}{m} + \frac{1-\lambda}{m} \left( \frac{n\mumin\varepsilon}{4b^{3/2}} - (1+\sqrt{2})\lambda \right)} \right),
\end{align*}
which simplifies to the desired inequality.
Recall $\gamma = \frac{n\mumin}{2(1+\sqrt{2})b^{3/2}}$. Let $\lambda \in (0, \frac{\gamma}{2})$. Choose $\delta \in (0, 1)$ and $\varepsilon \in \left(2\lambda\gamma^{-1}, 1\right)$. If
\[ m > \frac{68(1-\lambda)b^{3}(\mumax + n\mumax^2)}{n\mumin^2 \left( \varepsilon - 2\lambda\gamma^{-1} \right)^2} \log\frac{2n}{\delta}\]
then with probability at least $1-\delta$, we have
\[ \frac{\|\tilde{\hat{w}} - w\|}{\|w\|} \leq \varepsilon. \]
As in Corollary <ref>, we need
\[
\PP\left(\frac{\|\tilde{\hat{w}} - w\|}{\|w\|} > \varepsilon \right) < \delta,\]
which we can guarantee when
\[
2n \exp \left( \frac{-(n\mumin \varepsilon - 4(1+\sqrt{2})b^{3/2}\lambda)^2m}{16b^3(1-\lambda)^2 \left( 4(n-1)\mumax + 4n^2 \mumax^2 \right) + 4b^{3/2}(1-\lambda)(n\mumin \varepsilon - 4b^{3/2}(1+\sqrt{2})\lambda)} \right) < \delta.
\]
Rewriting in terms of $m$, we see that the second inequality is true when
\begin{equation*}
m > \frac{16b^3(1-\lambda)^2 \left( 4(n-1)\mumax + 4n^2 \mumax^2 \right) + 4b^{3/2}(1-\lambda)(n\mumin \varepsilon - 4b^{3/2}(1+\sqrt{2})\lambda)}{(n\mumin \varepsilon - 4(1+\sqrt{2})b^{3/2}\lambda)^2} \log \frac{2n}{\delta}\\
\end{equation*}
The desired inequality now follows by replacing various terms in the above inequality with upper bounds for them (e.g., $(1-\lambda)^2 < 1 - \lambda$, $b^{3/2} < b^{3}$, and $\varepsilon < 1$).
Empirical evidence suggests that values of $\lambda$ larger than $\frac{\gamma}{2}$ often yield meaningful results. Future work could include bridging this gap between the theory and application.
§ EMPIRICAL RESULTS: RANKCENTRALITY AND LAMBDA-REGULARIZED RANKCENTRALITY
Our main experiments was to evaluate convergence of these algorithms with synthetic BTL scores and comparisons. We compared (unregularized) RankCentrality, $\lambda$-regularized RankCentrality (with $\lambda$ decaying as $\eta m^{-1/2}$ for different values of $\eta$, as described in Section <ref>), the BTL maximum likelihood estimation (see equation (<ref>)), and regularized BTL-MLE (using the Scikit-Learn <cit.> implementation of logistic regression).
The BTL score $w_i$ for each item $i$ was either
* assigned by choosing $v_i$ uniformly at random from $[0, 5]$ and setting $w_i = \exp(v_i)$, or
* deterministically constructed, e.g., $w_i = i$ for $i \in [200]$.
Then, for various values of $m$, we generated $m$ comparisons (first chose $m$ pairs of items, uniformly at random from all possible pairs, then drew winners with probabilities according to the BTL model) and ran each algorithm on the same set of comparisons. In each of these cases, we record the $\ell_2$ error and the Kendall's Tau correlation metric. We repeat this process of generating comparisons and evaluating algorithms for a total of 40 times and record the mean and standard error of the $\ell_2$ error and the Kendall-Tau correlation metric. The results for some of these experiments are shown in Figure <ref>.
$w\in \R^{200}$ chosen at random.
$w\in \R^{40}$ chosen at random.
Decaying $\lambda$ with a factor of $m^{-1/2}$.
§ KENDALL'S TAU-B
The Kendall-Tau correlation metric we use in our experiments is also know as Kendall's Tau-b, defined as
\begin{equation}
\tau(\alpha, \beta) = \frac{P - Q}{\sqrt{(P + Q + T) * (P + Q + U)}},
\label{eq:kendalltaub}
\end{equation}
where $P$ is the number of concordant pairs (i.e., the number of pairs $i,j$ such that the relative ordering of $\alpha_i$ and $\alpha_j$ is the same as that of $\beta_i$ and $\beta_j$), $Q$ the number of discordant pairs, $T$ the number of ties only in $\alpha$, and $U$ the number of ties only in $\beta$.
Synthetic Experiment B.
New Yorker Caption Competition #651
Place Pulse dataset.
Impact of kernel width on Diffusion RankCentrality for various datasets.
§ NEW YORKER CAPTION CONTEST
New Yorker Caption Competition Interface for pairwise comparisons for 508. Users were asked to vote for each caption.
|
$\displaystyle\leq\frac{\max_{i}\|G^{i}_{n}(x_{t},\xi_{t})\|}{\eta}\left(1+2(m-|\mathcal{I}_{t}|)\right)+\sum_{i\in\mathcal{I}_{t}}\left(\frac{\hat{\sigma}_{i}(n)\sqrt{\ln\frac{1}{\delta}}+\hat{b}_{i}}{\alpha_{t}^{i}}+\|G^{i}_{n}(x_{t},\xi_{t})\|\left|\frac{1}{\bar{\alpha}_{t}^{i}}-\frac{1}{\alpha_{t}^{i}}\right|\right)$
$\displaystyle\leq\frac{\max_{i}\|G^{i}_{n}(x_{t},\xi_{t})\|}{\eta}\left(1+2(m-|\mathcal{I}_{t}|)\right)+\sum_{i\in\mathcal{I}_{t}}\left(\frac{\hat{\sigma}_{i}(n)\sqrt{\ln\frac{1}{\delta}}+\hat{b}_{i}}{\alpha_{t}^{i}}+\|G^{i}_{n}(x_{t},\xi_{t})\|\frac{\sigma_{i}(n)\sqrt{\ln\frac{1}{\delta}}}{\bar{\alpha}_{t}^{i}\alpha_{t}^{i}}\right)$
$\displaystyle\leq\frac{L}{\eta}\left(2m+1\right),$ (42)
for
$\hat{\sigma}_{i}(n)\leq\frac{\alpha_{t}^{i}\max\|G^{i}_{n}(x_{t},\xi_{t})\|}{2\eta\sqrt{\ln\frac{1}{\delta}}}$,
$\hat{b}_{i}\leq\frac{\alpha_{t}^{i}\max\|G^{i}_{n}(x_{t},\xi_{t})\|}{2\eta}$,
and
$\sigma_{i}(n)\leq\frac{(\alpha_{t}^{i})^{2}}{2\eta\sqrt{\ln\frac{1}{\delta}}}$,
implying $\bar{\alpha}_{t}^{i}\geq\alpha_{t}^{i}/2$ and using
$\|G^{i}_{n}(x_{t},\xi_{t})\|\leq L.$ Then, if
$\min\alpha_{t}^{i}\leq\bar{c}\eta$, we have
$\sum_{i\in\mathcal{I}_{t}}\frac{1}{\alpha_{t}^{i}}\geq\frac{1}{\bar{c}\eta}=\frac{L}{l\eta}\left(2m+1\right),$
and therefore with high probability $\|B\|\leq\|A\|$. Then we get (A.5), that
implies
$\displaystyle\prod_{i\in\mathcal{I}_{t}}\alpha^{i}_{t+1}\geq\prod_{i\in\mathcal{I}_{t}}\alpha^{i}_{t}.$
(43)
Moreover, using the same reasoning, we can prove that
$\displaystyle\prod_{i\in\mathcal{I}}\alpha^{i}_{t+1}\geq\prod_{i\in\mathcal{I}}\alpha^{i}_{t}.$
(44)
for any subset of indices $\mathcal{I}\subseteq[m]$ such that
$\mathcal{I}_{t}\subseteq\mathcal{I}.$
### A.6 Lower bound on $\gamma_{t}$
Here we assume $\underline{\alpha}_{t}^{i}\geq c\eta.$ Recall that
$\displaystyle\gamma_{t}=\min\left\\{\min_{i\in[m]}\left\\{\frac{\underline{\alpha}^{i}_{t}}{2|\hat{\theta}^{i}_{t}|+\sqrt{\underline{\alpha}_{t}^{i}M_{i}}}\right\\}\frac{1}{\|g_{t}\|},\frac{1}{\hat{M}_{2}(x_{t})}\right\\}.$
where
$\displaystyle\hat{M}_{2}(x_{t})=M_{0}+{\color[rgb]{0,0,0}10}\eta\sum_{i=1}^{m}\frac{M_{i}}{\underline{\alpha}^{i}_{t}}+{\color[rgb]{0,0,0}8}\eta\sum_{i=1}^{m}\frac{(\hat{\theta}^{i}_{t})^{2}}{(\underline{\alpha}^{i}_{t})^{2}}.$
We get the lower bound by constructing a bound on both of the terms inside the
minimum.
1) We have
$\mathbb{P}\left\\{\hat{M}_{2}(x_{t})\leq\left(1+{\color[rgb]{0,0,0}10}\frac{m}{c}\right)M+{\color[rgb]{0,0,0}8}\frac{mL^{2}}{\eta
c^{2}}\right\\}\geq 1-\delta$ (Due to Lemma 6, and by definition of
$\hat{M}_{2}(x_{t})$), which implies
$\mathbb{P}\left\\{\frac{1}{\hat{M}_{2}(x_{t})}\geq\eta\left(\frac{1}{\frac{{\color[rgb]{0,0,0}8}m}{c^{2}}L^{2}+\eta(1+{\color[rgb]{0,0,0}10}\frac{m}{c})M}\right)\right\\}\geq
1-\delta.$
2) Using Lemma 6 we get $\mathbb{P}\left\\{\|g_{t}\|\leq
L_{0}+\sum_{i=1}^{m}\frac{L_{i}}{c}\right\\}\geq 1-\delta.$ Hence, we can
bound
$\mathbb{P}\left\\{\min_{i\in[m]}\left\\{\frac{\underline{\alpha}^{i}_{t}}{2|\hat{\theta}^{i}_{t}|+\sqrt{\underline{\alpha}_{t}^{i}M_{i}}}\right\\}\frac{1}{\|g_{t}\|}\geq\frac{c\eta}{(2L+\sqrt{Mc\eta})L(1+\frac{m}{c})}\right\\}\geq
1-\delta.$
Therefore,
$\mathbb{P}\left\\{\gamma_{t}\geq\frac{\eta}{2}\min\left\\{\frac{1}{\frac{{\color[rgb]{0,0,0}4}m}{c^{2}}L^{2}+\eta({\color[rgb]{0,0,0}0.5}+{\color[rgb]{0,0,0}5}\frac{m}{c})M},\frac{1}{L^{2}(\frac{1}{c}+\frac{m}{c^{2}})+0.5\sqrt{\frac{M\eta}{cL^{2}}}L^{2}(1+\frac{m}{c})}\right\\}\right\\}\geq
1-\delta,$
$\mathbb{P}\left\\{\gamma_{t}\geq\frac{\eta}{2L^{2}(1+\frac{m}{c})}\min\left\\{\frac{1}{\frac{{\color[rgb]{0,0,0}4}}{c}+\frac{{\color[rgb]{0,0,0}5}M\eta}{L^{2}}},\frac{1}{\frac{1}{c}+\sqrt{\frac{M\eta}{4cL^{2}}}}\right\\}\right\\}\geq
1-\delta.$
$\mathbb{P}\left\\{\gamma_{t}\geq\frac{c\eta}{2L^{2}(1+\frac{m}{c})}\min\left\\{\frac{1}{{\color[rgb]{0,0,0}4}+\frac{{\color[rgb]{0,0,0}5}Mc\eta}{L^{2}}},\frac{1}{1+\sqrt{\frac{Mc\eta}{4L^{2}}}}\right\\}\right\\}\geq
1-\delta.$
Finally, the bound is
$\mathbb{P}\left\\{\gamma_{t}\geq\eta C\right\\}\geq 1-\delta.$
with
$C:=\frac{c\eta}{2L^{2}(1+\frac{m}{c})}\min\left\\{\frac{1}{{\color[rgb]{0,0,0}4}+\frac{{\color[rgb]{0,0,0}5}Mc\eta}{L^{2}}},\frac{1}{1+\sqrt{\frac{Mc\eta}{4L^{2}}}}\right\\}.$
### A.7 Proof of Lemma 9
Proof From Fact 2 it follows that
$\forall x\in\mathcal{X}\leavevmode\nobreak\ \exists
s_{x}=\frac{x-x_{0}}{\|x-x_{0}\|}\in\mathbb{R}^{d}:\leavevmode\nobreak\
\langle s_{x},\nabla f^{i}(x)\rangle\geq\frac{\beta}{2R}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall
i\in\mathcal{I}_{\beta/2}(x).$
Let $\hat{x}$ be an approximately optimal point for the log barrier:
$B_{\eta}(\hat{x})-B_{\eta}(x^{*}_{\eta})\leq\eta,$ that is equivalent to:
$f^{0}(\hat{x})+\eta\sum_{i=1}^{m}-\log(-f^{i}(\hat{x}))-f^{0}(x^{*}_{\eta})-\eta\sum_{i=1}^{m}-\log(-f^{i}(x^{*}_{\eta}))\leq\eta.$
Then, for the objective function we have the following bound:
$\displaystyle
f^{0}(\hat{x})-f^{0}(x^{*}_{\eta})\leq\eta+\eta\sum_{i=1}^{m}-\log\frac{-f^{i}(x^{*}_{\eta})}{-f^{i}(\hat{x})}.$
(45)
The optimal point for the log barrier $x^{*}_{\eta}$ must satisfy the
stationarity condition
$\nabla B_{\eta}(x^{*}_{\eta})=\nabla
f^{0}(x^{*}_{\eta})+\eta\sum_{i=1}^{m}\frac{\nabla
f^{i}(x^{*}_{\eta})}{-f^{i}(x^{*}_{\eta})}=0.$
By carefully rearranging the above, we obtain
$\sum_{i\in\mathcal{I}_{\beta/2}(x^{*}_{\eta})}\frac{\nabla
f^{i}(x^{*}_{\eta})}{-f^{i}(x^{*}_{\eta})}+\sum_{i\notin\mathcal{I}_{\beta/2}(x^{*}_{\eta})}\frac{\nabla
f^{i}(x^{*}_{\eta})}{-f^{i}(x^{*}_{\eta})}=\frac{-\nabla
f^{0}(x^{*}_{\eta})}{\eta}.$
By taking a dot product of both sides of the above equation with
$s_{x}=\frac{x^{*}_{\eta}-x_{0}}{\|x^{*}_{\eta}-x_{0}\|}$, using the Lipschitz
continuity we get for $x^{*}_{\eta}$:
$\displaystyle\frac{1}{\min_{i}\\{-f^{i}(x^{*}_{\eta})\\}}\sum_{i\in\mathcal{I}_{\beta/2}(x^{*}_{\eta})}\langle\nabla
f^{i}(x^{*}_{\eta}),s_{x}\rangle\frac{\min_{i}\\{-f^{i}(x^{*}_{\eta})\\}}{-f^{i}(x^{*}_{\eta})}$
(46) $\displaystyle=\frac{\langle-\nabla
f^{0}(x^{*}_{\eta}),s_{x}\rangle}{\eta}-\sum_{i\notin\mathcal{I}_{\beta/2}(x^{*}_{\eta})}\frac{\langle\nabla
f^{i}(x^{*}_{\eta}),s_{x}\rangle}{-f^{i}(x^{*}_{\eta})}\leq\frac{mL}{\eta}.$
(47)
From the above, using Fact 2, we get
$\min\\{-f^{i}(x^{*}_{\eta})\\}\geq\frac{\eta\beta}{2mLR}.$
Hence, combining the above with (45) we get the following relation of point
$\hat{x}$ and point $x^{*}_{\eta}$ optimal for the log barrier:
$\displaystyle
f^{0}(\hat{x})-f^{0}(x^{*}_{\eta})\leq\eta+\eta\sum_{i=1}^{m}\log\frac{-f^{i}(\hat{x})}{-f^{i}(x^{*}_{\eta})}\leq\eta\left(1+m\log\left(\frac{2mLR\hat{\beta}}{\eta\beta}\right)\right).$
(48)
Next, note that the Lagrangian $\mathcal{L}(x,\lambda)$ is a convex function
over $x$ and concave over $\lambda$. Hence, for
$(x^{*}_{\eta},\lambda^{*}_{\eta}):=\left(x^{*}_{\eta},\left[\frac{\eta}{-f^{1}(x^{*}_{\eta})},\ldots,\frac{\eta}{-f^{m}(x^{*}_{\eta})}\right]^{T}\right)$
we have
$\displaystyle\mathcal{L}(x^{*}_{\eta},\lambda^{*}_{\eta})-\mathcal{L}(x^{*},\lambda^{*})\leq\mathcal{L}(x^{*}_{\eta},\lambda^{*}_{\eta})-\mathcal{L}(x^{*},\lambda^{*}_{\eta})\leq\langle\nabla_{x}\mathcal{L}(x^{*}_{\eta},\lambda^{*}_{\eta}),\lambda^{*}_{\eta}-x^{*}\rangle\leq
0.$
Expressing $\mathcal{L}(x^{*}_{\eta},\lambda^{*}_{\eta})$ and
$\mathcal{L}(x^{*},\lambda^{*})$ and exploiting the fact that $\nabla
B_{\eta}(x^{*}_{\eta})=\nabla_{x}\mathcal{L}(x^{*}_{\eta},\lambda^{*}_{\eta})=0$,
we obtain
$\mathcal{L}(x^{*}_{\eta},\lambda^{*}_{\eta})-\mathcal{L}(x^{*},\lambda^{*})=f^{0}(x^{*}_{\eta})-f^{0}(x^{*})-m\eta\leq
0.$ Consequently, we have $f^{0}(x^{*}_{\eta})-f^{0}(x^{*})\leq m\eta.$
Combining the above and (48), we get
$f^{0}(\hat{x})-\min_{x\in\mathcal{X}}f^{0}(x)\leq\eta+\eta
m\log\left(\frac{2mLR\hat{\beta}}{\eta\beta}\right)+m\eta.$
### A.8 Zeroth-order estimator properties proof
The deviation of the gradient estimators $G^{i}(x_{t},\nu)-\nabla
f^{i}_{\nu}(x_{t})$, by definition can be expressed as follows for
$i=0,\ldots,m$
$\displaystyle G^{i}(x_{t},\nu)-\nabla
f^{i}_{\nu}(x_{t})=\frac{1}{n_{t}}\sum_{j=1}^{n_{t}}\left[\underbrace{\left(d\frac{f^{i}(x_{k}+\nu
s_{tj})-f^{i}(x_{t})}{\nu}s_{tj}-\nabla
f^{i}_{\nu}(x_{t})\right)}_{v_{j}^{i}}+\underbrace{d\frac{\xi_{tj}^{i+}-\xi_{tj}^{i-}}{\nu}s_{tj}}_{u_{j}^{i}}\right],$
(49)
where the first term under the summation $v_{j}^{i}$ is dependent only on
random $s_{tj}$, however the second term is dependent on both random variables
coming from the noise $\xi^{i\pm}_{tj}$ and from the direction $s_{tj}$.
Then, using the fact that the additive noise $\xi_{tj}^{i\pm}$ is zero-mean
and independent on $s_{tj}$, we get:
$\displaystyle\mathbb{E}\left\|G^{i}_{\nu,n}(x_{t},\xi)-\nabla
f^{i}_{\nu}(x_{t})\right\|^{2}=\mathbb{E}\left\|\frac{1}{n}\sum_{j=1}^{n}v_{j}^{i}\right\|^{2}+\mathbb{E}\left\|\frac{1}{n}\sum_{j=1}^{n}u_{j}^{i}\right\|^{2}$
(50)
Using the result of Lemma 2.10 (Berahas et al., 2021), we can bound the first
part of the above expression
$\mathbb{E}\left\|\frac{1}{n}\sum_{j=1}^{n}v_{j}^{i}\right\|^{2}$:
$\displaystyle\mathbb{E}\left\|\frac{1}{n}\sum_{j=1}^{n}v_{j}^{i}\right\|^{2}\leq\frac{3d^{2}}{n}\left(\frac{\|\nabla
f^{i}(x)\|^{2}}{d}+\frac{M_{i}^{2}\nu^{2}}{4}\right).\leavevmode\nobreak\
\forall i\in\\{0,\ldots,m\\}.$ (51)
The second part $u_{j}^{i}$ is zero-mean, hence does not influence the bias.
Indeed, using the independence of $\xi^{j\pm}_{tj}$ and $s_{tj}$ we derive
$\displaystyle\mathbb{E}\sum_{j=1}^{n_{t}}u_{j}^{i}=\frac{d}{\nu}\mathbb{E}\left(\sum_{j=1}^{n_{t}}(\xi^{i+}_{tj}-\xi^{i-}_{tj})s_{tj}\right)=0.$
(52)
Its variance can be bounded as follows, using $\|s_{tj}\|=1$:
$\displaystyle\mathbb{E}\left\|\frac{1}{n}\sum_{j=1}^{n}u_{j}^{i}\right\|^{2}=\mathbb{E}\frac{d^{2}}{\nu^{2}n^{2}}\left\|\sum_{j=1}^{n}(\xi^{i+}_{tj}-\xi^{i-}_{tj})s_{tj}\right\|^{2}\leq
4\frac{d^{2}}{\nu^{2}n^{2}}\sum_{j=1}^{n}\mathbb{E}\|\xi^{i+}_{tj}\|^{2}\|s_{tj}\|^{2}\leq
4\frac{d^{2}\sigma^{2}}{\nu^{2}n}.$ (53)
From the above, and Lemma 2.10 (Berahas et al., 2021) the statement of the
Lemma follows directly.
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|
# Self-Translate-Train: A Simple but Strong Baseline
for Cross-lingual Transfer of Large Language Models
Ryokan Ri Shun Kiyono Sho Takase
SB Intuitions
<EMAIL_ADDRESS>
###### Abstract
Cross-lingual transfer is a promising technique for utilizing data in a source
language to improve performance in a target language. However, current
techniques often require an external translation system or suffer from
suboptimal performance due to over-reliance on cross-lingual generalization of
multi-lingual pretrained language models. In this study, we propose a simple
yet effective method called Self-Translate-Train. It leverages the translation
capability of a large language model to generate synthetic training data in
the target language and fine-tunes the model with its own generated data. We
evaluate the proposed method on a wide range of tasks and show substantial
performance gains across several non-English languages.
Self-Translate-Train: A Simple but Strong Baseline
for Cross-lingual Transfer of Large Language Models
Ryokan Ri and Shun Kiyono and Sho Takase SB Intuitions
<EMAIL_ADDRESS>
## 1 Introduction
Cross-lingual transfer is a technique to solve tasks in a target language by
leveraging training data from the other languages (Pikuliak et al., 2021).
This has been increasingly feasible with the rise of multilingual pre-trained
models, which are trained on multilingual corpora and capture commonalities
across languages (Conneau et al., 2020; Xue et al., 2021; Scao et al., 2022).
Capable multilingual models can perform tasks in a target language without
being trained on task-specific data in that language, which is known as zero-
shot cross-lingual transfer (Artetxe and Schwenk, 2019; Chen et al., 2021).
This technique is expected to reduce the disparity between high-resource and
low-resource languages.
Cross-lingual transfer is also exhibited in large language models (LLMs),
which refers to auto-regressive language models with billion-scale parameters
that are trained with a massive amount of text data (Brown et al., 2020;
Touvron et al., 2023). A common approach for zero-shot cross-lingual transfer
is fine-tuning the model with supervised training data available in a source
languages, mostly English, and then applying the model to the target language
(Chen et al., 2024; Shaham et al., 2024). However, we argue that this approach
does not fully elicit the model’s cross-lingual capability as the model has no
clue the input language at the test time. To achieve better cross-lingual
performance, we let the model teach itself how to solve the task in the target
language. In our proposed method, Self-Translate-Train, we produce the target
language’s translation of the training data leveraging the strong capability
of the LLM to generate text, and train the LLM with its own generated
translation.
Figure 1: An overview of Self-Translate-Train. An LLM translates training data
to the target language and then fine-tuned on its own generated data.
We evaluate Self-Translate-Train with several tasks including question
answering, text-pair classification, and mathematical reasoning across
multiple languages. Our experiments show that Self-Translate-Train
consistently improves the performance of baselines given a multilingual
capability of the LLMs. Our results indicate that we can achieve better cross-
lingual performance by correctly elicit the model’s translation capability,
which encourages further exploration of how to better utilize the model’s
cross-lingual capability.
## 2 Related Work
### 2.1 Cross-lingual Transfer Learning
There are two main approaches to transfer task knowledge across languages:
data transfer and model transfer (Pikuliak et al., 2021).
Data transfer translates the source language data to the target language. In
the Translate-test approach, models are trained on source language data and at
inference time, the task inputs are translated into the source language
(Conneau et al., 2018; Asai et al., 2018). Although the training stage is
simple, it incurs additional translation costs at inference time. The
Translate-train approach, on the other hand, translates the training data and
the resulting model is used to predict the target language data directly
(Conneau et al., 2018). Data transfer is quite effective in terms of
performance (Hu et al., 2020), but one drawback is its requirement of
additional translation systems.
Model transfer alleviates the need for translation systems by using
multilingual pretrained models, which are trained on a large amount of data
from multiple languages and capture the commonality between languages. These
models can be fine-tuned on task-specific data in a single source language and
generalize to solve the task in other languages (Pires et al., 2019; Mulcaire
et al., 2019; Conneau et al., 2020), eliminating the need for translation
systems.
Our approach, Self-Translate-Train, leverages LLMs’ translation and cross-
lingual generalization capabilities. It combines the advantages of data
transfer and model transfer by using explicit training signals in the target
language while eliminating the need for external translation systems.
### 2.2 Self-Improvement of LLMs
LLMs have demonstrated remarkable text generation capabilities, which has been
leveraged to generate training data for various purposes (Li et al., 2023b;
Lee et al., 2024). The generated data can be used to further specialize the
LLM itself for downstream applications, without requiring an extensive
collection of additional data. This process can be viewed as a form of self-
improvement (Bai et al., 2022; Huang et al., 2023; Sun et al., 2023; Li et
al., 2023a).
Self-Translate-Train is also a self-improvement approach to specialize the LLM
to a target language by translating the source language data to the target
language.
## 3 Self-Translate-Train
Our framework focuses on fine-tuning LLMs on a small amount of data for a
specific task. Let the training corpus in a source language, say English, be
$\mathcal{D}_{\text{src}}=\\{(\mathbf{x}_{\text{src}}^{i},\mathbf{y}_{\text{src}}^{i})\\}_{i=1}^{N}$,
where $\mathbf{x}$ is the input and $\mathbf{y}$ is the output. In a typical
cross-lingual transfer setting, the model is fine-tuned only on
$\mathcal{D}_{\text{src}}$ and expected to generalize to target languages.
### Translated Synthetic Data
Given the LLM’s translation capability, we can let it translate the training
corpus into a synthetic corpus in the target language
$\mathcal{D}_{\text{tgt}}$. The synthetic data can be added to
$\mathcal{D}_{\text{src}}$ to achieve a better generalization to the target
language.
The translation can be performed in various ways depending on the model’s
capabilities or available resources. In this paper, we experiment with the
few-shot prompting technique (Section 4.4).
### Code-switched Synthetic Data
The generated data has an interesting aspect: each synthetic instance has a
corresponding instance in the original dataset with the same semantics. We can
exploit this to further synthesize data by generating code-switched instances
where the input and output are in different languages.
We pair the original and translated instances to construct
$\mathcal{D}_{\text{cs}}=\\{(\mathbf{x}_{\text{src}}^{i},\mathbf{y}_{\text{tgt}}^{i})\\}_{i=1}^{N}\bigcup\\{(\mathbf{x}_{\text{tgt}}^{i},\mathbf{y}_{\text{src}}^{i})\\}_{i=1}^{N}$.
When the task output is natural language, we manually translate the prompt
“Please answer in {{ tgt }}.” into the target language, and add it to the
input $\mathbf{x}$.
## 4 Experimental Setups
To verify the effectiveness of Self-Translate-Train, we conduct extensive
experiments on multiple tasks and languages.
### 4.1 Task and Datasets
We present a list of datasets for experiments in Table 1. For each task, an
English dataset is used for training and a multilingual dataset for
evaluation. To make the computational cost feasible, we use a 10,000-sample
subset of the training data for SQuAD and MultiNLI.
Task | Training | Evaluation
---|---|---
QA | SQuAD (Rajpurkar et al., 2016) | XQuAD (Artetxe et al., 2020)
Classification | MultiNLI (Williams et al., 2018) | XNLI (Conneau et al., 2018)
Math | GSM8k (Cobbe et al., 2021) | MGSM (Shi et al., 2023)
Table 1: List of datasets for experiments. The details are described in
Section A.1.
### 4.2 Languages
We conducted evaluation on four languages: German (de), Russian (ru), Thai
(th), and Chinese (zh). German is a Germanic language, which is
phylogenetically close to English and expected to show better cross-lingual
transfer, while Russian, Thai, and Chinese are from different language
families. In particular, Thai is a low-resource language with a different
script from English, which is expected to be more challenging for cross-
lingual transfer.
### 4.3 Language Models
Our main experiments use Llama2-7B (Touvron et al., 2023), a public LLM.
Although 90% of its pretraining corpus is English, the model has a
multilingual capability (e.g., Table 2) from the remaining fraction of
multilingual data.
### 4.4 Synthetic Data Generation
Recent LLMs are known to exhibit a translation capability without much task-
specific data Briakou et al. (2023). In our experiments, we elicit the
translation capability of the LLMs via few-shot in-context learning (Brown et
al., 2020).
To construct few-shot translation samples, we sample eight pairs from the
train or validation splits of the multilingual datasets, where instances
across languages form parallel data. The translation was performed for each
field individually, e.g., for GSM8k, we translated the question and answer
separately. The prompt template simply alternates the source and target text
prepended with the language tag (Section A.2).
An important step to ensure the quality of the synthetic data is to filter out
the low-quality data (the details in Section A.3). To remove under- or over-
translation (Tu et al., 2016), we filter out texts with an extreme source-
target length ratio. Also, to address the repetition problem (Holtzman et al.,
2020), we set the max number of tokens for generation and filter out the
translation that does not end with the EOS token. With the translations from
Llama2-7B, this process removes around 10% of the data for most languages and
around 50% for Thai due to the model’s limited generation quality.
To provide the sense of the translation quality, we report the BLEU score
(Papineni et al., 2002) measured by the parallel data constructed from
questions in the MGSM test set in Table 2. Overall, the translation quality is
sufficiently high except for Thai. As we will see in Section 5.1, this poses a
challenge for cross-lingual transfer to Thai.
Model | de | ru | th | zh
---|---|---|---|---
Llama2-7B | 37.1 | 27.2 | 1.9 | 29.4
Table 2: BLEU scores from the MGSM test set. The configuration of BLEU is
described in Section A.4.
### 4.5 Fine-tuning
All the tasks are cast as text generation tasks, where the LLM is given the
inputs as a prompt and generate the answer. Fine-tuning is conducted with
causal language modeling loss, computed only for output tokens. We use LoRA
(Hu et al., 2022), a parameter-efficient tuning technique, to reduce
computational cost.
We use AdamW (Loshchilov and Hutter, 2019) and the cosine learning rate
schedule for optimization, training with a batch size of 64 for 1,000 steps.
For each setting, we conduct six runs with two learning rates (5e-5 and 3e-4)
and different random seeds, reporting summarization statistics of the top four
runs based on validation accuracy to remove runs with optimization failure.
See Section A.5 for other hyperparameters.
## 5 Results
### 5.1 Main Results
| MGSM | XQuAD | XNLI
---|---|---|---
| de | ru | th | zh | de | ru | th | zh | de | ru | th | zh
$\mathcal{D}_{\text{src}}$ | | 30.1
---
$\pm$0.4
| 25.0
---
$\pm$0.7
| 8.1
---
$\pm$0.7
| 21.1
---
$\pm$1.7
| 60.3
---
$\pm$0.8
| 49.3
---
$\pm$0.4
| 34.5
---
$\pm$1.0
| 66.3
---
$\pm$0.8
| 79.7
---
$\pm$0.4
| 76.9
---
$\pm$0.1
| 53.7
---
$\pm$0.9
| 74.1
---
$\pm$0.2
$+\mathcal{D}_{\text{tgt}}$ | * | 36.4
---
$\pm$1.3
* * | 34.0
---
$\pm$1.7
* | 7.7
---
$\pm$3.4
* | 27.1
---
$\pm$1.2
* | 61.7
---
$\pm$0.9
* | 57.8
---
$\pm$0.8
* * | 46.4
---
$\pm$1.3
* * | 77.7
---
$\pm$0.4
* * | 81.6
---
$\pm$0.7
* * | 78.5
---
$\pm$0.6
* | 56.3
---
$\pm$1.5
* | 78.5
---
$\pm$0.3
*
$+\mathcal{D}_{\text{tgt}}+\mathcal{D}_{\text{cs}}$ | * | 35.9
---
$\pm$1.3
* * | 34.5
---
$\pm$2.4
* | 10.4
---
$\pm$1.6
* | 28.8
---
$\pm$1.9
* * | 62.2
---
$\pm$0.9
* * | 58.0
---
$\pm$0.6
* * | 46.2
---
$\pm$1.7
* * | 77.3
---
$\pm$0.7
* * | 81.6
---
$\pm$0.5
* | 78.4
---
$\pm$1.2
* | 58.9
---
$\pm$1.5
* * | 77.6
---
$\pm$0.6
*
Table 3: Results on multilingual evaluation datasets. Scores are marked with ∗
if its improvement is statistically significant ($p<0.05$ in Welch’s t-test)
compared to the baseline $\mathcal{D}_{\text{src}}$. The significant and
highest score in each column is marked in bold.
(a) de
(b) th
(c) zh
Figure 2: Accuracy in the MGSM dataset with different model sizes of Llama2.
As the baseline, we fine-tune the LLM with the source language dataset
$\mathcal{D}_{\text{src}}$. To ensure a fair comparison, we augment
$\mathcal{D}_{\text{src}}$ with the eight target language samples used for
few-shot translation (Section 4.4). We then compare the baseline with the
models fine-tuned on the data generated from Self-Translate-Train
($\mathcal{D}_{\text{src}}$ and $\mathcal{D}_{\text{cs}}$) in Table 3.
First, Self-Translate-Train is indeed an effective method;
$+\mathcal{D}_{\text{tgt}}$ almost consistently outperforms the baseline
$\mathcal{D}_{\text{src}}$. The only exception is Thai (th), where there is no
significant improvement. This is likely due to the low translation quality of
the model in Thai (Table 2).
The effectiveness of code-switching dataset is limited. When we add
$\mathcal{D}_{\text{cs}}$ to $\mathcal{D}_{\text{tgt}}$, there is no
significant improvement from adding $\mathcal{D}_{\text{tgt}}$ only ($p<0.05$
in Welch’s t-test). This indicates that the code-switching data does not
provide additional information for the model to generalize in the task.
### 5.2 Does the model size matter?
The size of the language model can influence both its ability to generalize
across languages and the quality of its translations, which in turn may impact
the effectiveness of Self-Translate-Train. We compare the performance of
Llama2 with different sizes, i.e., 7B, 13B, and 65B, on the math task in de,
th, and zh (Figure 2).
The larger model size generally tends to perform better, and the improvement
from Self-Translate-Train remains consistent across different model sizes. In
Thai (th), we did not observe a significant improvement in the 7B model, but
do in the larger models (13B and 65B), likely due to their better translation
quality. The 7B model has a low Thai translation BLEU score of 1.9 (Table 2),
while the 13B and 65B models have BLEU scores of 5.1 and 12.0, respectively.
The improvement in Thai (th) with the 70B model is the most significant (+19.8
average points). This implies that Self-Translate-Train is particularly
effective when the model struggles with generalizing across the source and
target languages but can still generate reasonable translations.
## 6 Conclusion
We introduced Self-Translate-Train, a method to improve cross-lingual transfer
performance by generating synthetic training data in the target language. We
validated its effectiveness on various tasks and languages, demonstrating
substantial performance gains across several non-English languages. Self-
Translate-Train is effective when the zero-shot cross-lingual transfer
performance is suboptimal and the model can generate reasonable translations.
Self-Translate-Train neither requires external translation systems nor
intensive additional data collection, making it a simple yet effective method
for cross-lingual transfer. We encourage practitioners to try this approach as
an improved baseline for cross-lingual transfer of LLMs.
Our research also shows that relying solely on the model’s generalization
capability may be suboptimal, and there is a better way to elicit the cross-
lingual capability of the model. We hope this work encourages further
exploration of how to better utilize the model’s cross-lingual capability.
## 7 Limitations
Our experiments are conducted on a modern type of LLM, an autoregressive
Transformer decoder, and centered around the Llama2 model families (Touvron et
al., 2023). Although we further validate our method in Appendix B, the
effective of our proposed method is uncertain when applied to other types of
LLMs developed in the future.
Our method is based on the assumption that the model can generate reasonable
translations in the target language. This may be challenging when the task
inputs are long or complex. One solution is to split the input into smaller
segments, as we have done with the SQuAD dataset (Section A.2).
Finally, when the task requires generating long and natural text, the quality
of the generated translation matters more. If the translation quality is low,
the model outputs may degrade due to translation errors or unnaturalness. The
application of our method on more challenging tasks requires further
investigation.
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## Appendix A The Details of Experimental Setups
### A.1 Tasks and Datasets
We provide the details of the datasets introduced in Section 4.1.
#### A.1.1 Question Answering (QA)
SQuAD (Rajpurkar et al., 2016) is an English QA dataset created from Wikipedia
articles as training data. Given a question and a passage, the task is to
extract the answer from the passage. Evaluation is conducted with XQuAD
(Artetxe et al., 2020), which consists of translation of SQuAD into multiple
languages.
Context: Architecturally, the school has a Catholic character. Atop the
Main Building’s gold dome is a golden statue of the Virgin Mary. Immediately
in front of the Main Building and facing it, is a copper statue of Christ
with arms upraised with the legend "Venite Ad Me Omnes". Next to the Main
Building is the Basilica of the Sacred Heart. Immediately behind the basilica
is the Grotto, a Marian place of prayer and reflection. It is a replica of the
grotto at Lourdes, France where the Virgin Mary reputedly appeared to Saint
Bernadette Soubirous in 1858. At the end of the main drive (and in a direct line
that connects through 3 statues and the Gold Dome), is a simple, modern stone
statue of Mary.
Question: To whom did the Virgin Mary allegedly appear in 1858 in Lourdes France?
---
Saint Bernadette Soubirous
Figure 3: An input and output example of the SQuAD dataset.
#### A.1.2 Text-Pair Classification
We also evaluate our method on cross-lingual text-pair classification tasks.
The MultiNLI dataset (Williams et al., 2018) involves determining the logical
relationship between a premise sentence and a hypothesis sentence. XNLI
(Conneau et al., 2018) is a multilingual NLI dataset for evaluation.
Premise: Conceptually cream skimming has two basic dimensions - product and geography.
Hypothesis: Product and geography are what make cream skimming work.
What is their logical relation? Entailment, Neutral or Contradition.
---
Neutral
Figure 4: An input and output example of the MultiNLP dataset.
#### A.1.3 Mathematical Reasoning
GSM8k (Cobbe et al., 2021) is an English dataset of 8.5K high-quality grade
school math problems. Each problem is annotated with a solution that shows the
mathematical steps required to reach the final answer. As the evaluation
dataset, we use MGSM (Shi et al., 2023), a multilingual version of the GSM8k
dataset.
The LLM is trained to generate the step-by-step solution to math problems. The
answer is extracted from the LLM output as the final digits, and the accuracy
is calculated based on the exact match of the extracted answer and the ground
truth.
Natalia sold clips to 48 of her friends in April,
and then she sold half as many clips in May.
How many clips did Natalia sell altogether in April and May?
---
Natalia sold 48/2 = clips in May.
Natalia sold 48+24 = 72 clips altogether in April and May.
#### 72
Table 4: An input and output example of the GSM8k dataset.
### A.2 Prompt Format for LLM Translation
To translate training data using a LLM (Section 4.4), we employed the
following prompt template for each task. The template simply consists of the
source text and target text prepended with the language tag. The text is
surrounded by backticks and the LLM starts generating the target text the open
backtick until the close backtick is found.
{% for sample in few_shot_samples %}
en: ‘{{ sample.data_field }}‘
{{ target_language }}: ‘{{ sample.data_field }}‘
{% endfor %}
en: ‘{{ data_field }}‘
{{ target_language }}: ‘
---
Figure 5: Prompt format for LLM translation.
The SQuAD dataset annotates answer spans in the context passages. We translate
the annotations using the mark-then-translate approach (Chen et al., 2023). We
mark the answer span in the context passage with the tokens “<answer>” and
“</answer>”, translate the marked text, and then extract the translated answer
span from the translated context. Note that in this case, the few-shot samples
are also marked with the answer span.
The context passages in the SQuAD dataset are relatively long, and it is
challenging for the LLM with a limited context window to fit the entire few-
shot samples and the source text. To address this issue, we split the context
into sentences using the spaCy library111https://spacy.io/ and translate them
separately, i.e., the few-shot samples and the source text are sentences.
### A.3 Data Filtering for Synthetic Data
We remove pairs where the target length is less than one-third or more than
three times the source length. The text length is heuristically determined to
account for character length differences between languages. For example,
phonogram-based text (e.g., English) has much more characters than ideograph-
based text (e.g., Chinese). We set normalization factors where English,
German, Thai, and Russian characters count as 1, and Chinese characters as 3.
We also filter out incomplete translations which are typically produced by
repetitive generation. We set the maximum number of tokens for generation
(Table 5) and remove the outputs not ending with the token indicating the end
of the translation, in our case, the backtick character used in the prompt
format.
| Data Field | Max Number of Tokens
---|---|---
SQuAD (Rajpurkar et al., 2016) | context | 512
question | 256
MultiNLI (Williams et al., 2018) | premise | 256
hypothesis | 256
GSM8k (Cobbe et al., 2021) | question | 512
answer | 512
Table 5: Maximum number of tokens set for generating translations.
### A.4 Assessing the Translation Quality
To evaluate the translation quality, we use the BLEU score (Papineni et al.,
2002) measured by the parallel data constructed from questions in the MGSM
test set. The translation is performed in few-shot in-context learning with 8
translation samples constructed from the train set of the MGSM dataset. The
BLEU score is calculated using the SacreBLEU library (Post,
2018)222https://github.com/mjpost/sacrebleu. As the tokenizer option, we use
“13a” for de and ru, “flores101” for th, and “zh” for zh.
Table 6 shows the BLEU scores from the LLMs evaluated in this paper. The
result of Qwen1.5-1.8B is discussed in Appendix B, and gpt-3.5-turbo-0125 in
Section C.3.
Model | de | ru | th | zh
---|---|---|---|---
Llama2-7B | 37.1 | 27.2 | 1.9 | 29.4
Llama2-13B | 41.3 | 33.4 | 5.1 | 34.3
Llama2-70B | 45.6 | 41.9 | 12.0 | 42.4
Qwen1.5-1.8B | 21.9 | 11.3 | 1.5 | 41.3
gpt-3.5-turbo-0125 | 48.0 | 44.6 | 23.1 | 47.2
Table 6: BLEU scores from the MGSM test set.
### A.5 Hyper-parameters for Fine-tuning
We provide the hyper-parameters used for fine-tuning the LLMs in Table 7.
Hyper-parameter | Value | Hyper-parameter | Value
---|---|---|---
Batch size | 64 | Adam $\epsilon$ | 1e-8
Number of steps | 1,000 | Adam $\beta_{1}$ | 0.9
Learning rate | [5e-5, 3e-4] | Adam $\beta_{2}$ | 0.999
LR Scheduler | Cosine | Weight decay | 0.1
Warmup ratio | 0.05 | |
Table 7: Hyper-parameters used for fine-tuning the LLMs.
## Appendix B Results from Qwen1.5-1.8B
To increase the robustness of the results, we also conducted experiments with
Qwen1.5-1.8B 333https://qwenlm.github.io/blog/qwen1.5/. While the model is
mainly trained on Chinese and English data, it is also constructed with the
multilingual use cases in mind.
| gsm8k
---|---
| de | ru | th | zh
$\mathcal{D}_{\text{src}}$ | | 8.0
---
$\pm$0.3
| 6.5
---
$\pm$0.4
| 2.8
---
$\pm$0.5
| 23.3
---
$\pm$0.5
$+\mathcal{D}_{\text{tgt}}$ | * | 18.7
---
$\pm$1.8
* * | 15.0
---
$\pm$1.2
* | 3.4
---
$\pm$1.8
| 21.4
---
$\pm$1.2
$+\mathcal{D}_{\text{cs}}$ | * | 17.1
---
$\pm$1.3
* * | 14.9
---
$\pm$0.5
* | 2.7
---
$\pm$0.8
| 23.2
---
$\pm$0.9
Table 8: Results on the MGSM dataset with Qwen1.5-1.8B. Scores are marked with
∗ if its improvement is statistically significant ($p<0.05$ in Welch’s t-test)
compared to the baseline $\mathcal{D}_{\text{src}}$. The significant and
highest score in each column is marked in bold.
We observe that the results are consistent with the main experiments: Self-
Translate-Train is effective when the zero-shot cross-lingual transfer
performance is suboptimal and the model can generate reasonable translations.
The performance is improved by adding the target language data
$+\mathcal{D}_{\text{tgt}}$. However, when the translation quality is poor as
in Thai (1.5 BLEU score in Table 6), the improvement is not observed.
Additionally, Qwen1.5-1.8B seems to have good cross-lingual capability between
English and Chinese, as indicated by the high BLEU score (41.3 in Table 6).
With this, tuning on the source language data alone is sufficient to achieve
high performance.
## Appendix C Frequently Asked Questions
In this section, we discuss questions that are outside the scope of the main
topic of this paper but are somewhat relevant and may be of interest to
readers.
### C.1 Does Self-Translate-Train improve the performance in the source
language?
The performance somtimes improves, given the task is challenging and the
translation quality is sufficiently high.
Table 9 shows the results on the English test set with Llama2-7B. The
performance improves in the MGSM dataset when adding the synthetic data from
de, ru, and zh. The Thai language does not show the improvement possibly due
to the low translation quality.
However, the improvement is not observed in the XQuAD and XNLI datasets. This
might be because the task performance is already high with the source language
data alone, and the synthetic data does not provide additional information to
improve the performance.
| MGSM | XQuAD | XNLI
---|---|---|---
| de | ru | th | zh | de | ru | th | zh | de | ru | th | zh
$\mathcal{D}_{\text{src}}$ | | 37.8
---
$\pm$0.8
| 70.2
---
$\pm$0.5
| 88.2
---
$\pm$1.3
$+\mathcal{D}_{\text{tgt}}$ | * | 42.6
---
$\pm$1.9
* * | 41.8
---
$\pm$1.3
* | 40.0
---
$\pm$1.9
* | 42.7
---
$\pm$1.0
* | 69.1
---
$\pm$0.7
| 69.6
---
$\pm$0.5
| 70.0
---
$\pm$0.5
| 69.9
---
$\pm$0.4
| 89.0
---
$\pm$0.4
| 88.3
---
$\pm$0.7
| 88.6
---
$\pm$1.0
| 89.3
---
$\pm$0.3
$+\mathcal{D}_{\text{cs}}$ | * | 42.7
---
$\pm$0.7
* * | 42.9
---
$\pm$0.7
* | 39.8
---
$\pm$1.6
| 40.8
---
$\pm$2.0
| 69.5
---
$\pm$0.5
| 69.6
---
$\pm$0.3
| 69.8
---
$\pm$0.5
| 68.7
---
$\pm$0.5
| 88.7
---
$\pm$0.2
| 88.5
---
$\pm$1.0
| 88.0
---
$\pm$0.7
| 88.7
---
$\pm$0.6
Table 9: Results on the Englihs test set with Llama2-7B. Scores are marked
with ∗ if its improvement is statistically significant ($p<0.05$ in Welch’s
t-test) compared to the baseline $\mathcal{D}_{\text{src}}$. The significant
and highest score in each column is marked in bold.
### C.2 Does the synthetic data alone improve the performance in the target
language?
Yes, but adding the source language data is more effective. Table 10 shows the
results with Llama2-7B on the multilingual evaluation datasets. Tuning on the
synthetic data alone ($\mathcal{D}_{\text{tgt}}$) improves the performance in
the target language, but the improvement is not as significant as adding the
synthetic data to the source language data ($+\mathcal{D}_{\text{tgt}}$). In
practice, we recommend using the synthetic data in combination with the
original data to achieve the best performance.
| MGSM | XQuAD | XNLI
---|---|---|---
| de | ru | th | zh | de | ru | th | zh | de | ru | th | zh
$\mathcal{D}_{\text{src}}$ | | 30.1
---
$\pm$0.4
| 25.0
---
$\pm$0.7
| 8.1
---
$\pm$0.7
| 21.1
---
$\pm$1.7
| 60.3
---
$\pm$0.8
| 49.3
---
$\pm$0.4
| 34.5
---
$\pm$1.0
| 66.3
---
$\pm$0.8
| 79.7
---
$\pm$0.4
| 76.9
---
$\pm$0.1
| 53.7
---
$\pm$0.9
| 74.1
---
$\pm$0.2
$D_{\text{tgt}}$ | * | 32.1
---
$\pm$0.6
* * | 30.4
---
$\pm$1.8
* | 8.7
---
$\pm$0.6
| 24.6
---
$\pm$2.5
* | 62.5
---
$\pm$0.4
* * | 57.4
---
$\pm$1.1
* * | 44.3
---
$\pm$0.8
* * | 75.4
---
$\pm$1.1
* * | 81.5
---
$\pm$0.6
* | 77.9
---
$\pm$0.7
| 53.8
---
$\pm$1.7
* | 77.1
---
$\pm$0.5
*
$+\mathcal{D}_{\text{tgt}}$ | * | 36.4
---
$\pm$1.3
* * | 34.0
---
$\pm$1.7
* | 7.7
---
$\pm$3.4
* | 27.1
---
$\pm$1.2
* | 61.7
---
$\pm$0.9
* | 57.8
---
$\pm$0.8
* * | 46.4
---
$\pm$1.3
* * | 77.7
---
$\pm$0.4
* * | 81.6
---
$\pm$0.7
* * | 78.5
---
$\pm$0.6
* | 56.3
---
$\pm$1.5
* | 78.5
---
$\pm$0.3
*
Table 10: Results on the Englihs test set with Llama2-7B with the setting of
tuning the synthetic data alone $\mathcal{D}_{\text{tgt}}$.
### C.3 Is tuning on $\mathcal{D}_{\text{tgt}}$ generated by another model
still effective?
Yes, if the other model can generate reasonable translations. Such approach
can be seen as the Translate-train approach (Section 2.1) or sequence
distillation from a teacher model (Kim and Rush, 2016).
As an upper-bound experiment using the math task, we fine-tune Llama2-7B on
the synthetic data generated by gpt-3.5-turbo-0125 from the OpenAI
API444https://openai.com/index/openai-api/, which produces high-quality
translations across the languages explored in this paper (Table 6).
| MGSM
---|---
| de | ru | th | zh
$\mathcal{D}_{\text{src}}$ | | 30.1
---
$\pm$0.4
| 25.0
---
$\pm$0.7
| 8.1
---
$\pm$0.7
| 21.1
---
$\pm$1.7
$+\mathcal{D}_{\text{tgt}}$ | * | 37.4
---
$\pm$1.0
* * | 35.9
---
$\pm$1.2
* * | 28.5
---
$\pm$1.4
* * | 33.2
---
$\pm$1.1
*
$+\mathcal{D}_{\text{cs}}$ | * | 38.6
---
$\pm$1.5
* * | 34.4
---
$\pm$0.5
* * | 28.7
---
$\pm$1.3
* * | 34.8
---
$\pm$1.3
*
Table 11: Results on multilingual evaluation datasets with Llama2-7B tuned on
the synthetic data generated by gpt-3.5-turbo-0125.
Adding the synthetic data generated by gpt-3.5-turbo-0125 improves performance
across languages.
However, the outputs from external models are often restricted in its
usage555For example, Term of use of OpenAI API (January 31, 2024) restricts
the usage of the outputs for training a model that competes with the API
(https://openai.com/policies/terms-of-use/). Meta Llama 3 License (April 18,
2024) prohibits using the outputs to improve any other large language model
(https://llama.meta.com/llama3/license/)., while the method explored in this
paper can be used with the model at hand without other resources.
Additionally, our interest in this paper is rather to explore the cross-
lingual potential of the model itself and how to better utilize it.
|
# Inverse Design of Nonlinear Metasurfaces for Sum Frequency Generation
Neuton Li ARC Centre of Excellence for Transformative Meta-Optical Systems
(TMOS), Department of Electronic Materials Engineering, Research School of
Physics, The Australian National University, Canberra, ACT 2600, Australia,
<EMAIL_ADDRESS>Jihua Zhang ARC Centre of Excellence for Transformative
Meta-Optical Systems (TMOS), Department of Electronic Materials Engineering,
Research School of Physics, The Australian National University, Canberra, ACT
2600, Australia<EMAIL_ADDRESS>Songshan Lake Materials Laboratory,
Dongguan, Guangdong 523808, P. R. China Dragomir N. Neshev ARC Centre of
Excellence for Transformative Meta-Optical Systems (TMOS), Department of
Electronic Materials Engineering, Research School of Physics, The Australian
National University, Canberra, ACT 2600, Australia<EMAIL_ADDRESS>Andrey A. Sukhorukov ARC Centre of Excellence for Transformative Meta-Optical
Systems (TMOS), Department of Electronic Materials Engineering, Research
School of Physics, The Australian National University, Canberra, ACT 2600,
Australia<EMAIL_ADDRESS>
###### Abstract
Sum frequency generation (SFG) has multiple applications, from optical sources
to imaging, where efficient conversion requires either long interaction
distances or large field concentrations in a quadratic nonlinear material.
Metasurfaces provide an essential avenue to enhanced SFG due to resonance with
extreme field enhancements with an integrated ultrathin platform. In this
work, we formulate a general theoretical framework for multi-objective
topology optimization of nanopatterned metasurfaces that facilitate high-
efficiency SFG and simultaneously select the emitted direction and tailor the
metasurface polarization response. Based on this framework, we present novel
metasurface designs showcasing ultimate flexibility in transforming the
outgoing nonlinearly generated light for applications spanning from imaging to
polarimetry. For example, one of our metasurfaces produces highly polarized
and directional SFG emission with an efficiency of over
$0.2\text{\,}{\mathrm{cm}}^{2}\text{\,}{\mathrm{GW}}^{-1}$ in a
$10\text{\,}\mathrm{nm}$ signal operating bandwidth.
## 1 Introduction
Sum-frequency generation (SFG) is a fundamentally important second-order
nonlinear process with many applications ranging from wavelength conversion of
optical sources [1] and infrared imaging [2, 3, 4, 5] to nonlinear polarimetry
[6]. This phenomenon arises from induced polarizations in the medium, and in
general, it can be observed in the presence of strong optical fields.
Efficient second-order nonlinear frequency conversion, such as the SFG,
traditionally requires long interaction lengths in bulky nonlinear crystals.
As a result, only certain crystalline orientations and input polarisations can
satisfy the phase-matching conditions for producing sizable nonlinear effects.
This limits the types of polarization transformations and the directionality
of emission that are possible in nonlinear bulk crystals.
Recent advances in nanotechnologies have facilitated the development of ultra-
thin single-layer dielectric metasurfaces, where optical nano-resonators can
enhance and tailor the nonlinear interactions with functionalities beyond the
capabilities of traditional bulky crystals [7, 8, 9, 10, 11]. To generate
optical resonances, previous metasurface designs have often relied on semi-
analytical approaches in the limiting cases of Mie-type modes for individual
nanoresonators [12, 13, 14, 15, 16, 17, 18], or bound state in the continuum
resonances [19, 20, 21, 22, 23, 24, 25, 26]. The angular-dependent properties
of nonlocal metasurfaces could also be utilized to tune the nonlinear
interactions over a range of wavelengths [27]. There is ongoing research on
the enhancement of SFG in metasurfaces with resonances at non-degenerate
wavelengths [28, 29, 30, 31]. Non-planar structures with broken 3D symmetries
have been identified for designing effective nonlinear susceptibility response
[32, 33]. Furthermore, several studies have considered inverse-design or
machine learning approaches to optimize for strong resonances [34, 35, 36, 37,
38]. The resulting geometries have highly counterintuitive nanostructured
geometries, which prove superior to conventional designs in their respective
applications.
However, these examples fall short of controlling nonlinear generation beyond
the mere enhancement of conversion efficiency, which is only a part of the
advantages that metasurfaces offer. For many practical applications such as
imaging, it is often important to consider the input and output polarizations
as well as the directional distribution of the generated emission. Control of
the nonlinear polarization and the regulation of diffraction orders while
maintaining high overall efficiency still remain a major challenge in the
field. High-efficiency nonlinear conversion generally requires strong field
enhancements in the nonlinear region, which can be characterized by the
Q-factor of the resonance, its matching with the pulse bandwidth, and the
overlap between high Q-factor modes. The task of engineering overlapping high
Q-factor resonances with a prescribed bandwidth in materials at non-degenerate
wavelengths is difficult in itself. The additional objectives of engineering
emission polarization and direction expand the complexity of the task even
further.
In this manuscript, we address the aforementioned engineering challenges by
developing a novel inverse-design framework for optimizing metasurfaces that
enables the simultaneous enhancement of the SFG efficiency over a desired
pulse bandwidth, tailoring the polarization transformation matrix, and
increasing the emission directionality in a multi-objective manner. Our
resulting metasurfaces account for the intrinsic $\hat{\chi}^{(2)}$ of the
material and combine it with the optimized structural geometries to form an
effective $\hat{\chi}^{(2)}_{\text{eff}}$ of the device [Fig. 1(a)], enabling
ultimate control of nonlinear interactions. Thereby, we broaden the
flexibility and increase the benefits of using nonlinear metasurfaces, from
wavelength conversion to imaging and other applications.
Fig. 1: (a) Schematic of the possible processes that can occur through SFG.
The input fields $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ can have arbitrary
polarization states, which then generate the SFG field $\mathbf{E}_{3}$. Each
diffracted order of the SFG field can have an independent effective
$\chi^{(2)}_{\text{eff}}$ associated with it. By optimization of the
structure, each of these $\chi^{(2)}_{\text{eff}}$ can be tailored as desired.
(b) The energy levels of the SFG process with each field's respective
frequencies.
## 2 Methods
### 2.1 Adjoint Optimization of SFG
Beginning from the perspective of classical nonlinear optics, a polarization
field is induced in the medium that has a nonlinear relationship with applied
electric field [39, 40]
$\mathbf{P}=\varepsilon_{0}\quantity[\chi^{(1)}:\mathbf{E}+\chi^{(2)}:\mathbf{EE}+\chi^{(3)}:\mathbf{EEE}+\dots].$
(1)
This polarization then generates a nonlinear current that excites the fields
at harmonic frequencies. Our work focuses on the second-order process, so we
explicitly write this nonlinear current at the sum-frequency $\omega_{3}$ as
$J_{3,i}(\mathbf{r})=-i\omega_{3}\varepsilon_{0}\xi(\mathbf{r})\sum_{jk}\chi^{(2)}_{ijk}E_{1,j}(\mathbf{r})E_{2,k}(\mathbf{r})$
(2)
for input fields $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ at the frequencies
$\omega_{1}$ and $\omega_{2}$, respectively [Fig. 1(b)]. The implementation of
an additional position-dependent parameter $\xi(\mathbf{r})$ allows the
strength of the $\chi^{(2)}$ interaction to be modified simultaneously with
the value of the refractive index in the presence or absence of material [Fig.
2(a)]. We assume the non-depleted pump approximation, where the driving fields
will not suffer a significant loss of intensity as it propagates due to low
enough conversion efficiency. Then the fields $\mathbf{E}_{1}$ and
$\mathbf{E}_{2}$ are found as solutions of linear Maxwell's equations, and
then $\mathbf{E}_{3}$ at $\lambda_{3}$ may be obtained by solving the
inhomogeneous wave equation with the current from Eq. (2).
We aim to maximize the value of an objective function that depends on the
generated sum-frequency field, which in turn depends on the medium parameters
($p$) and on the incident fields:
$T(\mathbf{E}_{3})=T(\mathbf{E}_{3}(p,\mathbf{E}_{1},\mathbf{E}_{2})).$ (3)
We choose the optimization parameter as $p=\xi({\mathbf{r}})$ and define the
permittivity at each position of the design space as
$\varepsilon(\mathbf{r})=\varepsilon_{c}+\xi({\mathbf{r}})(\varepsilon_{d}-\varepsilon_{c}),\leavevmode\nobreak\
0\leq\xi({\mathbf{r}})\leq 1,$ (4)
where $\varepsilon_{d}$ and $\varepsilon_{c}$ are the permittivity for the
patterned and cladding materials, respectively. Thereby, $p=0$ or $p=1$
corresponds to the absence or presence of a nonlinear material at a particular
spatial location according to Eqs. (2) & (4), while the values of $0<p<1$ are
used during the intermediate stages of the optimization process as we discuss
in the following paragraph.
Fig. 2: (a) Depiction of topology optimization. The material derivative at
each discrete point is calculated through a series of forward and adjoint
simulations. (b) Forward simulations of sum-frequency generation, and (c)
corresponding adjoint simulations. The forward fields are labelled
$\mathbf{E}_{i}$, and the adjoint fields are labelled $\mathbf{v}_{j}$ for
wavelength $i,j$.
We formulate the adjoint optimization approach to nonlinear metasurfaces
following the general principles developed in Refs. [41, 42, 43, 44, 45, 46].
Previously, nonlinear cavities [47] and metasurfaces [35, 38] were optimized
to maximize the total second-harmonic generation. Here, we focus on the SFG
process, which includes three fields with different spatial distributions,
adding further complexity to the optimisation problem. Furthermore, we perform
optimization for SFG radiation over targeted diffraction orders, rather than
just the total converted radiation in all directions, as well as the tailored
nonlinear polarization transformations. For this purpose, we aim to
computationally efficiently calculate the derivatives $dT/dp$ simultaneously
for all spatial points in the nonlinear layer, which then allows for the fast
gradient descent optimization of the metasurface patterns as sketched in Fig.
2(a). Instead of repeating the forward calculation for material variations at
every spatial point [Fig. 2(b)], we identify the adjoint linear problems at
the three wavelengths [Fig. 2(c)], which solutions allow the calculation of
objective function derivative at all spatial locations through the overlaps
between the fields at the three non-degenerate wavelengths and each of their
corresponding adjoint fields. Detailed mathematical expressions for the
material derivative and its formulation can be found in Supplementary S1. In
the main manuscript, we summarize the ways in which the necessary adjoint
electric fields are obtained.
For optimization of sum-frequency emission in the outward far-field and on the
surface $\bf{\Omega}$, we can define the objective function through the
complex amplitude $a$ of a particular wave or mode with a field $({\bf
E}_{3,f},{\bf H}_{3,f})$. This mode can be of any form, including plane waves,
Gaussian beams, vortices, and other beam shapes, in combination with any
polarization structure. Then, the mode amplitude can be defined as
$a_{3}=\frac{1}{N_{3}}\iint_{\Omega}{\bf n}\cdot\left[{\bf E_{3}}\times{\bf
H}_{3,b}-{\bf E}_{3,b}\times{\bf H_{3}}\right]d\Omega\,,$ (5)
where n is the unit normal vector outward the surface $\Omega$, $({\bf
E}_{3,b},{\bf H}_{3,b})$ is a direction-reversed wave from $({\bf
E}_{3,f},{\bf H}_{3,f})$, and the normalisation coefficient is
$N_{3}=\iint_{\Omega}{\bf n}\cdot\left[{\bf E}_{3,f}\times{\bf H}_{3,b}-{\bf
E}_{3,b}\times{\bf H}_{3,f}\right]d\Omega$.
We now consider the most common case, in which all materials are reciprocal
with symmetric permittivity and permeability, and we only have the electric
field-induced electric current source. Then, the adjoint field
$\mathbf{v}_{E,3}$ at $\lambda_{3}$ is a result of linear scattering from the
metasurface for a source whose input wave is $({\bf E}_{3,b}/N_{3},{\bf
H}_{3,b}/N_{3})$.
The adjoint fields $\mathbf{v}_{E,1,2}$ at $\lambda_{1,2}$ are obtained by
solving Maxwell's equations with a current source
$\displaystyle{\bf J}_{v,1}$ $\displaystyle=$ $\displaystyle{\bf
L}_{1}^{T}{\bf v}_{E,3}\,,$ (6) $\displaystyle{\bf J}_{v,2}$ $\displaystyle=$
$\displaystyle{\bf L}_{2}^{T}{\bf v}_{E,3}\,,$ (7)
where
$\displaystyle
L_{1,ij}=-i\varepsilon_{0}\omega_{3}\xi(\mathbf{r})\sum_{k}\chi^{(2)}_{ijk}E_{2,k}\,,$
(8) $\displaystyle
L_{2,ik}=-i\varepsilon_{0}\omega_{3}\xi(\mathbf{r})\sum_{j}\chi^{(2)}_{ijk}E_{1,j}\,.$
(9)
These currents only exist where $\xi(\mathbf{r})\chi_{ijk}^{(2)}$ is non-zero,
i.e. in the nonlinear material. We note that these currents have a dependency
on $\mathbf{v}_{E,3}$, and so it must be obtained before solving the
aforementioned equations.
Finally, the objective function gradient is
$\begin{split}\frac{dT}{d\xi}=\frac{da_{3}}{d\xi}=&-i\varepsilon_{0}\omega_{3}\sum_{ijk}\chi^{(2)}_{ijk}E_{1,j}E_{2,k}\text{v}_{E,3,i}\\\
&-i\sum_{q}\omega_{q}{\bf
v}_{E,q}^{T}(\varepsilon_{q,d}-\varepsilon_{q,c}){\bf E}_{q},\end{split}$ (10)
where indices $q=1,2$ are for $\lambda_{1,2}$. This equation allows the
calculation of derivatives for any functions $T(a_{3})$ using a chain rule.
For example, for the maximization of SFG light power into a particular mode,
we can set $T=\absolutevalue{a_{3}}^{2}$, and obtain
$\frac{d|a_{3}|^{2}}{d\xi}=2\,\text{Re}\left\\{a_{3}^{\ast}\frac{da_{3}}{d\xi}\right\\}.$
(11)
In total, a set of six simulations is required to calculate the gradient using
the above equations for an arbitrarily large number of spatial positions
determined by the computational grid: three forward simulations to calculate
${\bf E}_{q}$, and three adjoint simulations for ${\bf v}_{E,q}$ at the three
non-degenerate wavelengths $\lambda_{q}$ with $q=1,2,3$, respectively.
### 2.2 Optimization for Multiple Polarizations in SFG
We now discuss the methodology for multi-objective optimization by extending
the results in the previous section that were formulated assuming fixed input
waves ${\bf E}_{1}$ and ${\bf E}_{2}$. Of particular interest is to
simultaneously tailor the sum-frequency polarization states for multiple
combinations of different input polarizations. Then, we can define a figure of
merit as
${\cal F}\left(\\{a_{3}^{(m)}({\bf E}_{1}^{(m)},{\bf
E}_{2}^{(m)})\\}_{m=1,\ldots,M}\right),$ (12)
where $m$ enumerates different input wave combinations and $a_{3}^{(m)}$ are
the sum-frequency amplitudes of the chosen polarization and spatial mode
profiles. Then, the derivative can be found using a chain rule
$\begin{split}\frac{d{\cal F}}{d\xi}=&\sum_{m=1}^{M}\frac{\partial{\cal
F}}{\partial a_{3}^{(m)}}\frac{da_{3}^{(m)}({\bf E}_{1}^{(m)},{\bf
E}_{2}^{(m)})}{d\xi}\\\ &+\sum_{m=1}^{M}\frac{\partial{\cal F}}{\partial
a_{3}^{\ast(m)}}\frac{da_{3}^{\ast(m)}({\bf E}_{1}^{(m)},{\bf
E}_{2}^{(m)})}{d\xi},\end{split}$ (13)
where the derivatives on the right-hand-side are determined using the adjoint
formulation in Eq. (10).
In the most general case, the input polarizations for $\lambda_{1,2}$ can each
be decomposed into pairs of orthogonal polarization states. Each of $M=4$
combinations of input waves can generate a different nonlinear current. Then,
a total of 14 unique simulations are required for the material derivative to
fully capture all possible input and output polarization combinations,
including 8 forward (2 at $\lambda_{1,2}$ and 4 at $\lambda_{3}$) and 6
adjoint (2 at each wavelength) simulations.
Furthermore, our method can also optimize diffraction outputs by specifying
the adjoint source waves' $k$-vectors. Diffraction into particular orders can
be enhanced (suppressed) by increasing (decreasing) the corresponding mode
overlap $a_{3}$, respectively.
### 2.3 Numerical implementation
We iteratively update the function $\xi(\mathbf{r})$ at all spatial locations
inside the quadratically nonlinear material. At each iteration, the material
permittivity is updated via gradient descent, or another gradient-based
method, at each discretized point in the domain [44, 48]. For a single-layer
metasurface design, we consider the pattern to be independent of the
longitudinal coordinate $z$. The optimization concludes when the FOM no longer
increases from one iteration to the next or the set maximum number of
iterations has been reached.
In our optimization, we employ several different techniques to ensure that the
free-form structures converge to a state that is both physical and realisable.
We introduce an increasing binarisation factor as the optimization progresses
that eventually forces the final pattern to be either material or air [49,
50]. The patterns are also periodically blurred with a Gaussian filter that
ensures the minimum feature is larger than limitations imposed by fabrication
precision [51, 52]. Importantly, an artificial absorption coefficient is added
onto the material, which will prevent convergence to structures that have
arbitrarily large quality factors (Q factor) [47, 53]. This non-radiative
decay rate defines the finite bandwidth of the SFG process, which is a
practical consideration for future experimental verification and distinguishes
our optimisation algorithms from other works purely relying on high-quality
factor resonances. This artificial absorption coefficient is later removed for
the analysis of the actual metasurfaces.
## 3 Results
Fig. 3: (a) Schematic of an SFG polarizing metasurface. The state of
$\mathbf{E}_{2}$ is fixed, while the state for $\mathbf{E}_{1}$ is
unpolarized. The SFG light from the optimized metasurface is polarized in the
$\ket{V}$, with the higher diffraction orders also being suppressed. (b)
optimized pattern of the metasurface. (c) Poincaré sphere representation of
all input states of $\mathbf{E}_{1}$ being transformed into the same SFG
output ($S_{1}$, corresponding to $\ket{H}$) state. (d) SFG efficiency from an
unpolarized source for different diffraction orders. (e) Signal wavelengths
sweep of SFG efficiency spectra for a pump wavelength fixed at
$1350\text{\,}\mathrm{nm}$, and (f) pump wavelengths sweep of SFG efficiency
spectra for a signal wavelength fixed at $1550\text{\,}\mathrm{nm}$.
In general, we can optimize for any combination of polarizations and elements
in the effective $\chi^{(2)}$ tensor, as discussed above. However, in this
work, we tackle a simpler problem that nevertheless still highlights the
strength of our optimization scheme. Specifically, we consider a signal having
any transverse polarization while the pump has a fixed polarization. Such
functionality may be beneficial for the operation of upconversion infrared
imaging, where the image can have any arbitrary polarization and the pump is a
light source of fixed polarization. For our examples, we have a signal
wavelength $\lambda_{1}=$1550\text{\,}\mathrm{nm}$$ and a pump wavelength
$\lambda_{2}=$1350\text{\,}\mathrm{nm}$$ (resulting in $\lambda_{3}=721.6$
nm). The transformation of the signal state into the SFG state can be
expressed in the form
$\mathbf{E}_{3}=\mathbf{M}\quantity(\mathbf{E}_{2},\chi^{(2)},\varepsilon)\cdot\mathbf{E}_{1}.$
(14)
The fields $\mathbf{E}_{q}$ represent the orthogonal complex amplitudes
$\quantity(E_{q}^{x},E_{q}^{y})$ in the transverse components basis. We can
make a connection of Eq. (14) to the Jones vectors in polarization optics in
the following way. Let $\mathbf{M}$ represent the nonlinear scattering matrix
analogous to the Jones matrix in linear materials. Individual elements of
$\mathbf{M}$ can be interpreted as complex scattering amplitudes. For an
unpatterned film, $\mathbf{M}$ is constrained by the crystal orientation and
is set by the elements of $\chi^{(2)}$ and the input state of
$\mathbf{E}_{2}$. One such case is provided in Supplementary S3.1 for
unpatterned nonlinear film. Then $\mathbf{E}_{1}$ and $\mathbf{E}_{3}$ can be
interpreted as the input and output Jones vectors of the system, respectively.
Now, with the ability to pattern structures into the nonlinear film, the
permittivity $\varepsilon$ is no longer uniform across the domain but instead
can be engineered to achieve the desired transformation of $\mathbf{M}$.
In all the examples, the nonlinear material is indium gallium phosphide
(InGaP) that is of (100) crystalline orientation and $300\text{\,}\mathrm{nm}$
thick. The film is resting on the fused silica substrate, with a
$$900\text{\,}\mathrm{nm}$\times$900\text{\,}\mathrm{nm}$$ unit cell. See
Supplementary S2 for the details regarding InGaP parameters. The
electromagnetic simulations are performed using the commercial COMSOL
Multiphysics software package suite. Each simulation takes approximately 100
seconds when using a discretization of $40\text{\,}\mathrm{nm}$. The resulting
fields are exported to the MATLAB programming language, where we implement our
inverse-design optimization.
### 3.1 Polarizing Nonlinear Metasurface
At the SFG wavelength, multiple diffraction orders exist due to the
periodicity of the metasurface. In infrared imaging applications, the higher
diffraction orders should be ideally suppressed so that the majority of the
SFG light is propagating in the zeroth order. In previous works [5, 30], it
has been a challenge to suppress these higher-order propagating modes.
Simultaneously, with our method, we can also optimize diffraction outputs by
specifying the adjoint source waves' $k$-vectors. This is a significant
advance compared to previous inverse-design approaches for quadratically
nonlinear metasurfaces [35, 38] where only total second-harmonic conversion
but not directionality could be optimized.
We denote the zeroth order of the scattering matrix as $\mathbf{M}_{0}$, whose
elements can be optimized individually. In the first example, the signal is an
unpolarized state while the pump at $\lambda_{2}$ is polarized in the
$\ket{V}$ state. We intend for the SFG zeroth order diffraction to also be
polarized in the $\ket{V}$ state [Fig. 3(a)]. In this scheme, the metasurface
can be considered to be a nonlinear polarizer by taking an unpolarized input
and polarizing it into the $\ket{V}$ state at the SFG output. The figure of
merit is formulated as ${\cal
F}=\frac{1}{2}\quantity(\absolutevalue{\bra{V_{3}}\mathbf{M}_{0}\ket{H_{1}}}^{2}+\absolutevalue{\bra{V_{3}}\mathbf{M}_{0}\ket{V_{1}}}^{2})$,
and its derivative is calculated with Eq. (13). We note that unpolarized light
can be decomposed into equal powers of any pair of orthogonal states, which
are the $\ket{V}$ and $\ket{H}$ pair in our modelling. In this notation, the
states $\ket{\psi_{1}}$ denote the input signal state, while $\bra{\phi_{3}}$
denote the outgoing SFG state.
The optimization converges to a design that is highly non-trivial, as shown in
Fig. 3(b). We calculate the expected transformation of input unpolarized light
and find that it is almost fully converted into the $\ket{V}$ state at the SFG
wavelength, as depicted on the Poincaré sphere in Fig. 3(c). The numerical
values of $\mathbf{M}_{0}$ matrix elements can be found in Supplementary S3.2,
where further analysis is also provided. In Fig. 3(d), we show the predicted
SFG conversion efficiency into all the propagating diffraction orders, of
which the zeroth order comprises approximately 80% of the total SFG light (see
Supplementary S4.1 for precise values). Therefore, the optimized metasurface
directs the vast majority of SFG light perpendicular to the surface, which
greatly benefits imaging applications.
We now determine the frequency bandwidth by fixing the wavelength of the pump
at $\lambda_{2}=$1350\text{\,}\mathrm{nm}$$ and calculating the SFG efficiency
as an unpolarized signal wavelength is swept around the operating wavelength
of $\lambda_{1}=$1550\text{\,}\mathrm{nm}$$ [Fig. 3(e)]. The transmission into
the desired $\ket{V}$ state is significantly larger than the $\ket{H}$ state
and reaches a maximum SFG efficiency of
$0.25\text{\,}{\mathrm{cm}}^{2}\text{\,}{\mathrm{GW}}^{-1}$, which is more
than 3 orders of magnitude larger than an unpatterned film of the same
nonlinear material and thickness. This performance is echoed when we fix the
signal $\lambda_{1}=$1550\text{\,}\mathrm{nm}$$ and sweep the pump wavelengths
instead [Fig. 3(f)]. The full-width at half maximum (FWHM) conversion
efficiency for the signal is approximately $20\text{\,}\mathrm{nm}$, while it
is approximately $3\text{\,}\mathrm{nm}$ for the pump, representing a
reasonably large operating bandwidth suitable for efficient conversion with
short optical pulses. From these plots, it is evident that there are
resonances present within the metasurfaces at the input wavelengths
$\lambda_{1}$ and $\lambda_{2}$ that greatly enhance the SFG process. We
provide the field distributions and further analysis in the Supplementary S5.
### 3.2 Polarization Independent Nonlinear Metasurface
Fig. 4: (a) Schematic of an SFG waveplate metasurface. The state of
$\mathbf{E}_{2}$ is fixed, while the state for $\mathbf{E}_{1}$ is
unpolarized. The SFG light from the optimized metasurface is rotated in the
$\ket{V}$, with the higher diffraction orders also being suppressed. (b)
optimized pattern of the metasurface. (c) Poincaré sphere representation of
all input states of $\mathbf{E}_{1}$ being transformed into different SFG
output states while preserving their orthogonality. (d) SFG efficiency from an
unpolarized source for different diffraction orders. (e) Signal wavelengths
sweep of SFG efficiency spectra for a pump wavelength fixed at
$1350\text{\,}\mathrm{nm}$, and (f) pump wavelengths sweep of SFG efficiency
spectra for a signal wavelength fixed at $1550\text{\,}\mathrm{nm}$. The
dashed orange curves show the ratio of singular values (SV) for each plot at
their respective wavelengths.
In this example, we focus on a metasurface whose SFG enhancement is
independent of the polarization of $\lambda_{1}$. A metasurface that has this
property can enhance SFG conversion efficiency for all input polarization
states equally. This is particularly useful for imaging, where the source is
typically unpolarized or partially polarized. Such an upconverted image will
preserve the relative amplitudes of the original image, even if there are
spatial variations in the polarization. This is in contrast to previous
demonstrations of SFG imaging, where the metasurfaces typically rely on
polarization-sensitive resonant modes of bound states in the continuum [54].
Therefore, for this case, the target $\mathbf{M}_{0}$ matrix is close to
unitary after scaling, or its singular values are close to equality. The FOM
for this case is
${\cal F}=s_{2}\,,$ (15)
where $s_{1}$ and $s_{2}$ are the ordered singular values of $\mathbf{M}_{0}$,
and for this demonstration, the pump polarization at $\lambda_{2}$ is fixed at
$\ket{V}$. By maximising $s_{2}$, which is always defined as the smaller of
the two singular values, we are effectively increasing the SFG enhancement of
the worst-performing polarization input state. For a unitary matrix, the ratio
of the singular values must be $s_{2}/s_{1}=1$.
The optimized metasurface is again a non-trivial free-form pattern, as shown
in Fig. 4(b). For this metasurface, we then calculate the transformation of
various input states at the signal wavelength and plot them on a Poincaré
sphere [Fig. 4(c)]. We see that the metasurface imparts a rotation of the
eigenstates to the input states during the SFG process. We provide further
analysis of the matrix $\mathbf{M}_{0}$ in Supplementary S3.3. Importantly,
the eigenstates are nearly orthogonal, which indicates that the transformation
is indeed near unitary after scaling. The SFG is primarily channelled into the
zeroth diffraction order for an unpolarized signal [Fig. 4(d)], with reduced
light leakage into higher order diffraction modes (see Supplementary S4.2 for
numerical values).
We calculate the SFG efficiency for a range of unpolarized input signal
wavelengths and pump wavelength fixed at
$\lambda_{2}=$1350\text{\,}\mathrm{nm}$$ and find a maximum in average
efficiency of
$6\text{\times}{10}^{-3}\text{\,}{\mathrm{cm}}^{2}\text{\,}{\mathrm{GW}}^{-1}$
[Fig. 4(e)]. Again in this example, the FWHM conversion efficiency for the
signal is approximately $20\text{\,}\mathrm{nm}$, while it is approximately
for $10\text{\,}\mathrm{nm}$ for the pump. The SFG efficiency for two
orthogonal polarizations of $\ket{H}$ and $\ket{V}$ are almost equal. On the
right axis, we show the ratio of singular values that reaches a maximum of
0.8, reasonably close to the ratio of 1 for a truly unitary transformation.
This analysis is repeated for a fixed signal ($\lambda_{1}$ =
$1550\text{\,}\mathrm{nm}$) and varying pump wavelengths [Fig. 4(f)], with the
maximum efficiency peaking at $\lambda_{2}$ = $1350\text{\,}\mathrm{nm}$, as
expected. Therefore, the transformation of input light into SFG output from
the metasurface can be considered to be nearly polarization-independent. The
preservation of polarization information leads to the ability to perform
upconversion polarimetric imaging [6] with greater efficiency than previously
reported.
### 3.3 Structure of Resonances in the Optimised Metasurfaces
Finally, we perform linear simulations to inspect the resonances that we
expect to be present in the metasurfaces at
$\lambda_{1},\lambda_{2},\lambda_{3}$ (Supplementary S5.1). Because the two
metasurfaces presented in this work in Sec. 3.1 and 3.2 above are optimized
for different functionalities, they also have different resonant
characteristics. For the nonlinear polarizer metasurface, only $\ket{H}$
produces a resonance at $\lambda_{1}$, while for the polarization-independent
nonlinear metasurface, resonances appear for both $\ket{H}$ and $\ket{V}$
close to $\lambda_{1}$. As expected for both nonlinear metasurfaces, only
$\ket{V}$ produces a resonance at the pump wavelength because the polarization
at $\lambda_{2}$ was fixed. We note that the $Q$-factors are on the order of
70 to 300 for the different wavelengths, which allows for a reasonably broad
imaging bandwidth and for SFG with short optical pulses. We also provide field
enhancement distributions for the metasurfaces at all the interacting
wavelengths in Supplementary S5.2.
## 4 Conclusions
We have developed a novel method of multi-objective optimization of nonlinear
frequency mixing processes in metasurfaces. Our method allows for the
simultaneous control of polarizations and directionality and maximizes
efficiency across a target bandwidth for SFG processes, which is beyond what
is possible with conventional design strategies. We present a computationally
efficient implementation based on adjoint formulation and demonstrate two
essential examples of metasurfaces that enhance SFG either for one signal
input polarization or for all input polarizations. In both cases, the SFG is
emitted in the forward direction while the higher diffraction orders are
suppressed.
This method can be naturally extended to optimize metasurfaces that explore
other nonlinear phenomena such as four-wave mixing, spontaneous parametric
down-conversion, and high harmonic generation with greater efficiency. In the
future, nonlinear metasurfaces that exhibit more complicated characteristics
will be enabled by sophisticated optimization algorithms, widening the
spectrum of potential functionalities.
This work was supported by the Australian Research Council (CE200100010).
All authors have accepted responsibility for the entire content of this
manuscript and approved its submission.
The authors state no conflict of interest.
The datasets generated during and/or analyzed during the current study are
available from the corresponding author upon reasonable request.
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|
# Magnetically Induced Schrödinger Cat States: The Shadow of a Quantum Space
Partha Nandia<EMAIL_ADDRESS>Nandita Debnathb<EMAIL_ADDRESS>Subhajit Kalac<EMAIL_ADDRESS>A. S. Majumdard<EMAIL_ADDRESS>aInstitute
of Theoretical Physics, University of Stellenbosch, Stellenbosch-7600, South
Africa.
bDepartment of Physics, University of Calcutta, Kolkata 700009, India.
cDepartment of Physics, Indian Institute of Technology Guwahati, Guwahati
781039, Assam, India.
dS. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt
Lake, Kolkata 700106, India.
###### Abstract
Schrödinger cat states, which are superpositions of macroscopically distinct
states, are potentially critical resources for upcoming quantum information
technologies. In this paper, we introduce a scheme to generate entangled
Schrödinger cat states in a non-relativistic electric dipole system situated
on a two-dimensional plane, along with an external potential and a uniform
strong magnetic field perpendicular to the plane. Additionally, our findings
demonstrate that this setup can lead to the phenomenon of collapse and revival
of entanglement for a specific range of our model parameters.
## I Introduction
In quantum theory, the transition between the microscopic and macroscopic
worlds is one of the less-understood features Zuk . Such a transition plays a
direct role in the realm of quantum measurements. In an ideal measurement
paradigm, the interaction of macroscopic equipment and a microscopic system
yields entanglement and a superposed quantum state with both macroscopic and
microscopic components pn2 . Schrödinger was the first to highlight the
physical subtleties of this kind of superposition by replacing the macroscopic
part of the system by a “cat”, in order to illustrate a dramatic superposition
of “states” of both alive and dead cats, that should, in practice, be
distinguished macroscopically par . The superposition of macroscopically
different quantum states, generically referred to as non-classical Schrödinger
Cat State (SCS) par ; 2 ; bose , is crucial for understanding the conceptual
underpinnings of quantum physics, especially with reference to wave function
collapse models pnr ; pr ; hr ; hr1 . In recent years, the advancement of
quantum technologies has brought into sharp focus the utility of several
quantum phenomena such as photon anti-bunching Rov , sub-Poissonian statistics
mar and squeezing mtw , along with the dynamics of SCS.
The success of quantum information theory and its potential applications
qinform ; qinf that significantly outperform their classical equivalents have
recently sparked a renewed interest in the generation of non-classical states
such as SCS. Several applications of cat states have been suggested in the
realm of quantum information qinfor , quantum metrology qmeter , quantum
teleportation qteleport , and quantum error correction schemes qerror1 ;
qerror2 . Besides, the concept of decoherence between two superposed quantum
objects, or the quantum-to-classical transition, can be studied using the SCS
as a platform. In quantum optics, a superposition of two diametrically
opposite coherent states $|\pm\alpha>$ with large value of $\mid\alpha\mid$,
can be interpreted as a quantum superposition of two macroscopically distinct
states, i.e., a Schrödinger cat-like state tw ; legg . However, due to decay
of their interference properties, it is extremely difficult to detect such
states in practice egg . Nonetheless, the universality of SCS enables it to be
realized in a wide variety of physical arenas such as nonlinear quantum optics
Sb , quantum dot systems kaku , superconducting cavities kau , Bose Einstein
condensates (BEC) goenner and quantization of weak gravity rbm ; wolf ; sch .
A fascinating direction of research in recent years has been the mechanism for
the natural generation of SCS in some specific condensed matter systems sch1 ;
sch2 .
Schrödinger cat states with entanglement based protocols provide a novel
technique to explore short-distance quantum physics in a non-relativistic
domain when there is a magnetic dipole interaction background ed . At
extremely short distances, the space-time structure needs to be “granular” in
order to account for both gravity and quantum uncertainty ein . A viable
approach towards quantum gravity is through quantizing space-time itself in ,
rather than the construction of an effective field theory of gravity. This
approach is an active area of research on quantum gravity, commonly referred
to as non-commutative geometry go ; pe . The fundamental goal is to derive
classical geometry from a suitable limit of a non-commutative algebra. Though
such a proposal may appear as ad-hoc pek , the physical justification for such
a non-commutative space-time is strong since it provides a solution to the
geometric measurement problem near the Planck scale.
Non-commutative geometry appears naturally in various non-relativistic planar
systems. For instance, it occurs using the lowest Landau-level (LLL)
projection to study the behavior of charged particles in a strong magnetic
field ek . Further, the incompressibility of fractional quantum Hall fluids
qhefluids has a strong connection to the emergence of a non-commutative
geometry in which the fundamental Planck length is substituted by the magnetic
length. Non-commutative space-time forms an alternative paradigm for studying
the behavior of relativistic anyonic systems in interaction with the ambient
electromagnetic field vpn ; Rabin . Additionally, non-commutative properties
of real-space coordinates in the presence of the Berry curvature k produce
skew scattering by a non-magnetic impurity without relativistic spin-orbit
interactions in a condensed matter system. Non-commutative space provides a
paradigm for describing the behavior of the quantum to classical transition
under the influence of decoherence fg1 ; fg2 , which is relevant for
implementation of quantum information protocols. From an experimental
standpoint, there have been efforts in search of evidence of possible non-
commutative effect manifestations in cosmology and high-energy physics kk ; c
; d . A testable framework has been suggested in low-energy experiments in the
arena of quantum Hall effect qhe ; e .
The motivation for the present study is to investigate whether multi-component
entangled non-classical SCS could be produced in deformed quantum space, where
non-commutativity arises naturally in an easily accessibly low energy physical
system. In this article, we investigate the phenomenology of a two-particle
electric dipole model with an additional harmonic interaction and a strong
background magnetic field, with motion constrained to the plane perpendicular
to the field. Such a system may be considered as a toy version of a real
Excitonic dipole set-up exci . By exploring the high magnetic field limit, we
reveal the emergence of planar non-commutative space as a natural consequence.
Furthermore, we establish the deformed Heisenberg algebra as the origin of
multi-component entangled SCS in this system. Moreover, we quantify the degree
of entanglement of our SCS, and show that the phemomenon of collapse and
revival of entanglement R1 ; yueberly ; R2 occurs in this system under the
influence of the harmonic potential.
The organization of our paper is as follows. The interacting two particle
electric dipole system is introduced in Section 2, showing how classical non-
commutative space appears in the presence of a very strong, constant, uniform
magnetic field. Then, in Section 3, we move on to the quantum picture, where
intricacies of the system dynamics are revealed, in context of mapping between
two reference frames. Section 4 discusses how our model with a harmonic
oscillator potential that is dependent only on one spatial variable is able to
generate entangled multi-component Schrödinger cat states. In Section 5, we
compute the degree of entanglement in the generated SCS system and demonstrate
that it exhibits the phenomenon of entanglement collapse and revival. Section
6 is reserved for concluding remarks and discussions.
## II Two-Particle System: Classical picture
We begin by considering a pair of non-relativistic, oppositely charged
particles with equal mass $m$ moving on the plane subjected to a constant
magnetic field $B$ along the $z$ axis (ignoring Coulomb and radiation
effects). In component form, $x_{i}$ and $y_{i}$ $(i=1,2)$ correspondingly
represents the positive and negative charge coordinates. The $z$ coordinate
can be suppressed since the dynamics of the system is confined in a plane.
Standard Lagrangian in C.G.S. units is used to define the system as follows
Dunne ; bag ; pn :
$\displaystyle
L=\frac{1}{2}m(\dot{x}_{i}^{2}+\dot{y}_{i}^{2})+\frac{eB}{2c}\epsilon_{ij}(x_{j}\dot{x}_{i}-y_{j}\dot{y}_{i})$
$\displaystyle-\frac{K_{0}}{2}(x_{i}-y_{i})^{2}-V({x_{1}});~{}~{}~{}i,j=1,2$
(1)
where $c$ is the speed of light in vacuum and $K_{0}$ is the spring constant
corresponding to the harmonic interaction between the two oppositely charged
particles. This model is constructed in the spirit of the “2D excitonic dipole
model” exmol ; emg ; dipex , wherein $m$ can be realized by the effective mass
of the “electron-hole” pair in some specific cases where the magnitude of the
effective mass of electrons and holes can be considered as approximately same
and the Fermi velocity provides an upper bound for its characteristic velocity
in a real physical solid state system. Note that the first term of the above
Lagrangian (1) represents the kinetic term of the charges and the second term
represents their interaction with the external magnetic field $\vec{B}$. We
use a rotationally symmetric gauge to define the vector potential $\vec{A}$
satisfying the equation $\vec{\nabla}\times\vec{A}=B\hat{z}$. The third term
is the harmonic interaction between the two charges, and finally, the fourth
term describes the additional interaction of the positive charge with an
impurity in the $x_{1}$ direction. The limit of a strong magnetic field $B$
and small mass $m$ such as $\frac{m}{eB}\rightarrow 0$ is of interest here, in
which the kinetic energy term becomes negligible in the Lagrangian (1) BD .
Thus, we may approximate the dynamics by the effective Lagrangian,
$L_{0}=\frac{eB}{2c}\epsilon_{ij}(x_{j}\dot{x}_{i}-y_{j}\dot{y}_{i})-V_{0}(x_{i},y_{i})$
(2)
where $V_{0}(x_{i},y_{i})=\frac{K_{0}}{2}(x_{i}-y_{i})^{2}+V({x_{1}})$.
The Lagrangian equations of motion of the co-ordinates of the positive and
negatively charged particles are given by,
$\dot{x}_{i}=\frac{c}{eB}\epsilon_{ij}\frac{\partial V_{0}}{\partial
x_{j}},~{}~{}\dot{y}_{i}=-\frac{c}{eB}\epsilon_{ij}\frac{\partial
V_{0}}{\partial y_{j}}$ (3)
Since our effective Lagrangian (2) is in first-order form, the effective
Hamiltonian of the model is given by
$H=V_{0}(x_{i},y_{i})$ (4)
In order to show the equivalence between the Lagrangian and Hamiltonian
formalism fj ; Rb , we consider Hamilton’s equations of motion:
$\dot{x}_{i}=\\{{x_{i},H}\\}={\\{x_{i},V_{0}(x_{i},y_{i})}\\}$ (5)
$\dot{y}_{i}=\\{{y_{i},H}\\}={\\{y_{i},V_{0}(x_{i},y_{i})}\\}$ (6)
The nontrivial symplectic structure can readily be obtained now by comparing
the Lagrangian equations of motion (3) with the form of Hamilton’s equations
of motion ($\ref{a1},\ref{a2}$) to yield the following brackets:
$\\{x_{i},x_{j}\\}=\frac{c}{eB}\epsilon_{ij};~{}~{}\\{y_{i},y_{j}\\}=-\frac{c}{eB}\epsilon_{ij};~{}~{}\\{y_{i},x_{j}\\}=0$
(7)
The canonical spatial translation generators for individual charged particles
are given by
$P_{x_{i}}=\frac{eB}{c}\epsilon_{ij}x_{j};~{}~{}P_{y_{i}}=-\frac{eB}{c}\epsilon_{ij}y_{j}$
(8)
Using the above expressions and the nontrivial symplectic structures between
the position co-ordinates (7), it can be checked that the momentum co-
ordinates also satisfy a nontrivial symplectic bracket, given by
$\displaystyle\\{P_{x_{i}},P_{x_{j}}\\}=\frac{eB}{c}\epsilon_{ij};~{}~{}\\{P_{y_{i}},P_{y_{j}}\\}=-\frac{eB}{c}\epsilon_{ij};~{}~{}$
$\displaystyle\\{x_{i},P_{x_{j}}\\}=\\{y_{i},P_{y_{j}}\\}=\delta_{ij}$ (9)
## III Quantum dynamics
In this section, we discuss the quantum theory of our non-relativistic two-
particle model at the strong magnetic field limit by elevating the phase space
variables to the level of quantum operators. We obtain the nontrivial or
unusual commutation brackets between the position operators given by:
$[\hat{x}_{i},\hat{x}_{j}]=il^{2}_{B}\epsilon_{ij};~{}~{}[\hat{y}_{i},\hat{y}_{j}]=-il^{2}_{B}\epsilon_{ij};~{}~{}[\hat{x}_{i},\hat{y}_{j}]=0;~{}~{}i,j=1,2$
(10)
with $l_{B}=\sqrt{\frac{\hbar c}{eB}}$ known as the magnetic quantum length
scale. Likewise, the other nontrivial phase space non-commutative algebras are
given as
$[\hat{P}_{x_{i}},\hat{P}_{x_{j}}]=i\frac{\hbar^{2}}{l^{2}_{B}}\epsilon_{ij};~{}~{}[\hat{P}_{y_{i}},\hat{P}_{y_{j}}]=-i\frac{\hbar^{2}}{l^{2}_{B}}\epsilon_{ij},$
(11)
$[\hat{x}_{i},\hat{P}_{x_{j}}]=[\hat{y}_{i},\hat{P}_{y_{j}}]=i\hbar\delta_{ij};$
(12)
It may be observed that in this case, neither the coordinates nor the momentum
operators commute pnb2 . However, the operators
$\hat{P}_{i}=\hat{P}_{x_{i}}+\hat{P}_{y_{i}}=\frac{eB}{c}\epsilon_{ij}(\hat{x}_{j}-\hat{y}_{j}),$
(13)
can act as proper (commutative) translation generators, so that they satisfy
the following commutation relations:
$[\hat{x}_{i},\hat{x}_{j}]=il^{2}_{B}\epsilon_{ij};~{}~{}[\hat{P}_{i},\hat{P}_{j}]=0;~{}~{}[\hat{x}_{i},\hat{P}_{j}]=i\hbar\delta_{ij},$
(14)
which represents a non-commutative Heisenberg algebra (NCHA) in two
dimensions. In this instance, the operator-valued Hamiltonian of the effective
system is given by
$\hat{H}=\frac{K_{0}}{2}(\hat{x}_{i}-\hat{y}_{i})^{2}+V(\hat{x}_{1})$ (15)
A more conventional setting of this Hamiltonian in terms of the commutative
translation generator $\hat{P}_{i}$ is as follows:
$\hat{H}=\frac{1}{2m_{B}}\hat{P}^{2}_{i}+V(\hat{x}_{1});~{}~{}~{}i=1,2$ (16)
where $m_{B}=\frac{e^{2}B^{2}}{c^{2}K_{0}}$ is the effective mass of the
reduced two-particle system. It turns out to be instructive to introduce the
pair of canonical variables:
$\hat{R}_{i}=\frac{\hat{x}_{i}+\hat{y}_{i}}{2};~{}~{}\hat{P}_{i}=\frac{eB}{c}\epsilon_{ij}(\hat{x}_{j}-\hat{y}_{j});~{}~{}i,j=1,2$
(17)
where $\hat{R}_{i}$ is the centre of mass coordinate and $\hat{P}_{i}$ is the
corresponding canonical momentum of our two-particle system. They satisfy the
usual Heisenberg commutation relations as
$[\hat{R}_{i},\hat{R}_{j}]=0;~{}~{}[\hat{P}_{i},\hat{P}_{j}]=0;~{}~{}~{}[\hat{R}_{i},\hat{P}_{j}]=i\hbar\delta_{ij}$
(18)
However, it is worth noting that the centre of mass position coordinates may
also satisfy non-commutative Heisenberg algebra (NCHA) if the two particles
are assumed to have different masses (For further details, see Appendix A).
Even, if the two particles have the same mass, but their position coordinates
satisfy NCHA with different non-commutativity parameters, in that case also,
the centre of mass position coordinates can give rise to a non-commutative
algebra.
Note further, that since the dynamics of the composite system is realized in
terms of the coordinates of the positively charged particle, the information
of the negatively charged particle is completely suppressed in the equations
(14, 16), but it is incorporated into the expression of commuting momentum
operators. The extended Heisenberg algebra of the type as considered in
eq.(14) has the important property: it is realizable in terms of commutative
usual phase space variables (17) as
$\hat{x}_{1}=Ad_{\hat{U}}(\hat{R}_{1});~{}~{}\hat{P}_{1}=Ad_{\hat{U}}(\hat{P}_{1})$
(19)
$\hat{x}_{2}=Ad_{\hat{U}^{\dagger}}(\hat{R}_{2});~{}~{}\hat{P}_{2}=Ad_{\hat{U}^{\dagger}}(\hat{P}_{2})$
(20)
where we have made use of the fact of adjoint action:
$Ad_{\hat{U}}(\hat{A})=\hat{U}\hat{A}\hat{U}^{\dagger}$ with a quasi unitary
operator $\hat{U}$:
$\hat{U}=\exp[(-\frac{il^{2}_{B}}{2\hbar^{2}})\hat{P}_{1}\hat{P}_{2}],$ (21)
as it does not act unitarily on the entirely non-commutative phase space.
We can observe from the aforementioned equations (19, 20) that the non-
commutative phase space commutation algebra (14) can be simulated in terms of
commutative phase space variables (canonical variables) i.e. the centre of
mass coordinates as
$\hat{x}_{i}=\hat{R}_{i}-\frac{c}{2eB}\epsilon_{ij}\hat{P}_{j},~{}~{}~{}~{}i,j=1,2$
(22)
It may be noted that this transformation is not canonical because it changes
the commutation brackets. This transformation has occasionally been called a
Darboux map n2 or Bopp’s shift hm2 which is of relevance in the Bohmian
interpretation of non-commutative quantum mechanics pbom . Furthermore, this
transformation with an explicit dependence on the deformation parameter,
allows us to convert the Hamiltonian in NC space into a modified Hamiltonian
in commutative equivalent space. It follows that if we are able to solve the
spectrum of the system Hamiltonian in commutative equivalent space, we can
also obtain the spectrum of the system in primitive non-commutative space,
though the states in both situations are not the same. We will discuss how the
aforementioned maps aid in the extraction of non-classical cat states in the
next section.
## IV Preparation of Schrödinger Cat states
Using the formalism presented in the previous section, we are now in a
position to investigate the main goal of this work, viz., how we might
naturally prepare Schrödinger’s Cat states. To do so, we first consider a
particular Hamiltonian with a harmonic oscillator potential in the
$\hat{x}_{1}$ direction, given by
$\hat{H}\rightarrow\hat{H}_{NC}=\frac{\hat{P}^{2}_{1}}{2m_{B}}+\frac{\hat{P}^{2}_{2}}{2m_{B}}+V(\hat{x}_{1}),$
(23)
where $V(\hat{x}_{1})=\frac{1}{2}K\hat{x}^{2}_{1}$ and
$m_{B}=\frac{e^{2}B^{2}}{c^{2}K_{0}}$. The corresponding time dependent
Schrödinger equation is:
$i\hbar\frac{\partial}{\partial t}|\psi(t)>_{NC}=\hat{H}_{NC}|\psi(t)>_{NC}$
(24)
Note that, because of the non-commutativity of this theory, it is impossible
to construct simultaneous eigenstates with noncommutative coordinates, which
makes it difficult to define a local probability density for the wave-function
that corresponds to a particular state $|\psi(t)>_{NC}$ pbom1 . However, this
issue can be bypassed by using the interpretation mentioned in pbom1 , or by
using the coherent states formulation of noncommutative quantum mechanics with
the help of the Voros product g .
In our present case, it can be easily observed that the system Hamiltonian
mentioned above can be rewritten as,
$\hat{H}_{NC}=\hat{U}\hat{H}_{CM}\hat{U}^{\dagger},$ (25)
with
$\hat{H}_{CM}=\frac{\hat{P}^{2}_{1}}{2m_{B}}+\frac{\hat{P}^{2}_{2}}{2m_{B}}+V(\hat{R}_{1}),$
(26)
where we have used the fact that
$V(\hat{x}_{1})=V(\hat{U}\hat{R}_{1}\hat{U}^{\dagger})=\hat{U}V(\hat{R}_{1})\hat{U}^{\dagger}$.
Here $\hat{H}_{CM}$ is the unitarily equivalent form of the system Hamiltonian
expressed in terms of the Center of Mass coordinates, whereas the
$\hat{H}_{NC}$ represents the system Hamiltonian written in terms of the
positively charged particle coordinates. We can readily recognize that
$V(\hat{R}_{1})=\frac{1}{2}K\hat{R}^{2}_{1}$, where $K$ is the spring constant
of the impurity interaction faced by the positive charge in the $\hat{x}_{1}$
direction only. Accordingly, the Schrödinger equation (24) transforms as
follows:
$i\hbar\frac{\partial}{\partial t}|\psi(t)>_{CM}=\hat{H}_{CM}|\psi(t)>_{CM}$
(27)
where $|\psi(t)>_{CM}=\hat{U}^{\dagger}|\psi(t)>_{NC}.$ The ground state of
the unitarily equivalent Hamiltonian ($\hat{H}_{CM}$) is now represented as
$|\psi_{0}>_{CM}=|0>\otimes[d_{+}|+k_{2}>+d_{-}|-k_{2}>],$ (28)
where $|d_{+}|^{2}$ and $|d_{-}|^{2}$ denote the probability of finding the
free particle in $|+k_{2}>$ and $|-k_{2}>$ states respectively, $|0>$
represents the ground state of the 1D harmonic oscillator system with
$\hat{a}_{1}$ and ${\hat{a}_{1}}^{\dagger}$ representing the corresponding
annihilation and creation operators respectively, satisfying the following
algebra:
$[\hat{a}_{1},\hat{a}^{\dagger}_{1}]=\mathbb{I};~{}~{}~{}~{}\hat{a}_{1}=\frac{m_{B}\omega_{B}\hat{R}_{1}+i\hat{P}_{1}}{\sqrt{2m_{B}\omega_{B}\hbar}};~{}~{}~{}~{}~{}\hat{a}_{1}|0>=0,$
(29)
with $\omega_{B}=\sqrt{\frac{K}{m_{B}}}$, and $|\pm k_{2}>$ corresponds to the
right and left moving free particle’s momentum state respectively, which
satisfies:
$\hat{P}_{2}|\pm k_{2}>=\pm P_{2}|\pm k_{2}>;~{}~{}~{}~{}P_{2}=\hbar k_{2}$
(30)
The state vector corresponding to the non-commutative phase space (or in terms
of the positively charged particle coordinates) is given by
$|\psi_{0}>_{NC}=\hat{U}|\psi_{0}>_{CM},$ (31)
$|\psi_{0}>_{NC}$ can be expressed as,
$\displaystyle|\psi_{0}>_{NC}=(\exp[(-\frac{il^{2}_{B}}{2\hbar^{2}})\hat{P}_{1}\otimes\hat{P}_{2}])$
$\displaystyle[|0>\otimes(d_{+}|+k_{2}>+d_{-}|-k_{2}>)]$ (32)
which leads to
$\displaystyle|\psi_{0}>_{NC}=d_{+}([\exp(-\frac{il^{2}_{B}k_{2}}{2\hbar}\hat{P}_{1})]|0>)\otimes|+k_{2}>$
$\displaystyle+d_{-}([\exp(\frac{il^{2}_{B}k_{2}}{2\hbar}\hat{P}_{1})]|0>)\otimes|-k_{2}>$
(33)
On substituting $l^{2}_{B}=\frac{\hbar c}{eB}$ in the above equation, we
arrive at-
$\displaystyle|\psi_{0}>_{NC}=d_{+}([\exp[{(-i\frac{ck_{2}}{2eB})}\hat{P}_{1}]]|0>)\otimes|+k_{2}>$
$\displaystyle+d_{-}([\exp[(i\frac{ck_{2}}{2eB})\hat{P}_{1}]]|0>)\otimes|-k_{2}>$
(34)
Now, for a harmonic oscillator potential, the momentum operator $\hat{P}_{1}$
can be written as-
$\hat{P}_{1}=i\sqrt{\frac{m_{B}\omega_{B}\hbar}{2}}({\hat{a}_{1}}^{\dagger}-\hat{a}_{1})$
(35)
Putting the above expression in equation (33), we obtain,
(36)
It follows that the above state vector (36) may also be written in the form of
a superposition of single-component coherent states as
$\displaystyle\scalebox{0.8}{$|\psi_{0}>_{NC}=d_{+}|+\alpha>\otimes|+k_{2}>+d_{-}|-\alpha>\otimes|-k_{2}>$},$
(37)
wherein $|\pm\alpha>=e^{\pm\alpha({\hat{a}_{1}}^{\dagger}-\hat{a}_{1})}|0>$
with $\alpha=\frac{ck_{2}}{2eB}\sqrt{\frac{m_{B}\omega_{B}\hbar}{2}}$ are
real-valued coherent states (or a displacement of the vacuum) that belong to
the subset of the over complete space of usual complex parameter valued
coherent states bom1 .
Here it may be worthwhile to mention a property of the coherent state
$|\pm\alpha>$: the dimensionless parameter $\alpha$ may be rewritten as
$\alpha=\frac{1}{2}P_{2}({\frac{K}{K_{0}}})^{1/4}\sqrt{\frac{c}{2eB\hbar}}=\xi
k_{2}l_{B}$ (38)
with $\xi=\frac{1}{2}(\frac{K}{4K_{0}})^{\frac{1}{4}}$ . A coherent state
$|\alpha>$ can have an arbitrarily large amplitude, and hence, the energy of a
macroscopic harmonic oscillator scv can be approximated by the energy of a
one-dimensional quantum mechanical HO by suitably choosing $\mid\alpha\mid$ to
be arbitrarily large. For large enough $\mid\alpha\mid$ values, $|+\alpha>$
and $|-\alpha>$ correspond to macroscopically distinguishable states and may
be labelled as ‘(+) (alive)’ and ‘(-) (dead)’ sv ; gl . In this sense, we can
regard the above state (37) as an entangled SCS, holding
$\mid\alpha\mid\sqrt{h}$ fixed with finite value in the classical limit v ; s
. Accordingly, one may consider $|\pm\alpha>$ to be “classical-like” states,
but their coherent superposition is endowed with non-classical properties. In
fact, this type of Schrödinger cat states have been generated by pulsed
stimulation of atomic Rydberg wave packets vn .
In the primitive non-commutative phase space, we may rewrite the state vectors
(36) in the following concise way:
$\displaystyle|\psi_{0}>_{NC}=\mathcal{N}[|+\alpha;+k_{2}>+e^{i\phi}|-\alpha;-k_{2}>];~{}~{}$
$\displaystyle|\pm\alpha;\pm k_{2}>=|\pm\alpha>\otimes|\pm k_{2}>,$ (39)
with an arbitrary phase factor ($\phi$) and normalization constant
$\mathcal{N}$. For the aforementioned reason, the states $|\pm\alpha>$ may be
considered to be “macroscopic” like states with the same amplitude but
opposite in phase. (in the present case, the $\mid\alpha\mid$ parameter is not
arbitrary, but is defined in terms of the spring constants, magnetic field and
electric charge). However, their superposition (39) has several non-classical
characteristics op . Particularly, for the relative phase factor
$e^{i\phi}=\pm 1$, we get even and odd cat states that have been well-studied
in the literature 2 ; bose . Moreover, it is evident from (39) that the
coherent states and the free particle states are entangled: when the coherent
state parameter has a positive sign, the free particle state is right-moving.
On the other hand, the free particle state is left-moving when the coherent
state parameter has a negative sign. Therefore, $|\psi_{0}>_{NC}$ is an
entangled Schrödinger cat state containing the coherent superposition cohsup1
; cohsup2 of two states that are diametrically opposite to one another.
Since a momentum eigenstate is an idealization kop , we consider a more
realistic scenario in which the system’s motion in the commutative phase space
is localized within a specific length scale $\sigma$ along the $\hat{R}_{2}$
direction. In this case, we generalize the notion of free particle states to a
propagating Gaussian state given by
$|\psi_{G}>=\sqrt{\frac{\sigma}{\sqrt{\pi}}}\int_{-\infty}^{+\infty}e^{-\frac{\sigma^{2}}{2}(k_{2}-k_{0})^{2}}|k_{2}>dk_{2}$
(40)
where $\sigma$ is the width and $k_{0}$ is the peak momentum of the wave
packet. Now, following the prescription of (28), we can write the composite
state of the particle, when the dynamics of the system are realized in terms
of the centre of mass coordinates, as
$|\psi_{0}>_{CM}=|0>\otimes|\psi_{G}>$ (41)
Accordingly, we can generalize the notion of a two-component cat state (39) to
(42)
which describes a multi-component entangled Schrödinger cat state rop1 where
each component is specified through the momentum eigenvalues. Such a state is
highly non-classical, which can be verified through the corresponding Wigner
function rop1 . Thus, in the presence of a strong magnetic field background,
one may successfully prepare a Schrödinger Cat State utilizing a non-
relativistic electric dipole model, where non-commutativity plays an important
role. It may be reiterated here that we explore the system in terms of the
positively charged particle coordinates.
## V Collapse and revival of entanglement of SCS
In this section, we will begin by investigating the degree of entanglement of
the SCS state $|\psi>_{NC}$. In order to do so, we first write down the
corresponding density matrix given by
$\displaystyle\hat{\rho}_{NC}=(\sqrt{\frac{\sigma}{\sqrt{\pi}}})^{2}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}[|\alpha(k_{2})>_{A}<\alpha(k_{2}^{\prime})|]$
$\displaystyle\otimes[|k_{2}>_{B}<k_{2}^{\prime}|]e^{\frac{-\sigma^{2}}{2}(k_{2}-k_{0})^{2}}e^{\frac{-\sigma^{2}}{2}(k_{2}^{\prime}-k_{0})^{2}}dk_{2}dk_{2}^{\prime}$
(43)
where the subscripts $A$ and $B$ denote two distinct subsections of our
bipartite system, one of which is associated with coherent states and the
other with momentum eigenstates, each of which corresponds to two distinct
degrees of freedom in the non-commutative plane. Since $|\psi>_{NC}$ is a
composite pure state, the entanglement between the coherent states and free
particle states can be quantified in terms of the von-Neumann entropy given by
$S=-Tr_{A}[\hat{\rho}_{red}~{}ln(\hat{\rho}_{red})]$ (44)
where the reduced density matrix is defined as
$\displaystyle\hat{\rho}_{red}=\text{Tr}_{B}[\hat{\rho}_{NC}]$
$\displaystyle=\frac{\sigma}{\sqrt{\pi}}\int_{-\infty}^{+\infty}[|\alpha(k_{2})>_{A}<\alpha(k_{2})|]e^{-\sigma^{2}(k_{2}-k_{0})^{2}}dk_{2}$
(45)
with
$\text{Tr}(\hat{\rho}_{red})=\frac{\sigma}{\sqrt{\pi}}\int_{-\infty}^{+\infty}e^{-\sigma^{2}(k_{2}-k_{0})^{2}}dk_{2}=1$
(46)
For the present purpose, it suffices to compute the purity function purity ,
given by
$\displaystyle\text{P}(\alpha)=\text{Tr}(\hat{\rho}^{2}_{red})=\sum_{n}<n|\hat{\rho}^{2}_{red}|n>$
$\displaystyle=\sum_{m}\sum_{n}<n|\hat{\rho}_{red}|m><m|\hat{\rho}_{red}|n>$
(47)
After a little algebra, one obtains
$\displaystyle<n|\hat{\rho}_{red}|m>=\frac{\sigma}{\sqrt{\xi^{2}l_{B}^{2}+\sigma^{2}}}\frac{1}{\sqrt{n!}\sqrt{m!}}e^{(-\sigma^{2}k_{0}^{2})}$
$\displaystyle(\frac{\xi
l_{B}}{2\sigma^{2}})^{n+m}\frac{\partial^{n+m}}{\partial
k^{n+m}_{0}}(e^{\frac{\sigma^{4}k_{0}^{2}}{\xi^{2}l_{B}^{2}+\sigma^{2}}})$
(48)
By inserting equation (V) into (V) it follows that
$\displaystyle\scalebox{0.9}{$\text{P}(\xi_{0};l_{B})=(\frac{1}{1+\xi^{2}_{0}})e^{(-2\sigma^{2}k_{0}^{2})}[e^{(\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}})}(e^{\frac{\xi^{2}_{0}}{2\sigma^{2}}\overset{\leftarrow}{\frac{\partial}{\partial
k_{0}}}\overset{\rightarrow}{\frac{\partial}{\partial
k_{0}}}})e^{(\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}})}]$};$ (49)
where $\xi_{0}=\frac{\xi l_{B}}{\sigma}$. The above expression can be
rewritten (see Appendix B) as
$\text{P}(\xi_{0};l_{B})=\frac{1}{\sqrt{{1+2\xi^{2}_{0}}}}$ (50)
In Figure 1, we plot the Purity function versus the parameter $\xi_{0}$. It
can be observed that the purity function reduces from unit value (separable or
disentangled state) with increase of the parameter $\xi_{0}$, indicating
increment of entanglement in the system for higher values of $\xi_{0}$ (or
lower values of the width of the wave packet $\sigma$). We consider the
quantum length scale $l_{B}=1.483\times 10^{-8}$m, and vary the width of the
wave-packet in the range of $O(10^{-11}\to 10^{-6})$. Different $l_{B}$ values
displayed in the figure may originate due to the variation of the magnetic
length scale with different accessible magnetic fields in the laboratory.
Figure 1: The Purity function is plotted against the dimensionless factor
$\xi_{0}$ which varies inversely with the width of the wave-packet $\sigma$.
Plots for several choices of the quantum length scale are displayed.
It may be noted that if we just assume $\xi_{0}<<1$ with $\xi\sim 1$ which
implies that $l_{B}<<\sigma$, i.e, the width of the Gaussian packet ($\sigma$)
is large enough compared to the magnetic quantum length scale such that we can
ignore $\xi_{0}$, then it leads to the unit value of the purity function, or
in other words, the collapse of the entanglement in the state. On the other
hand, we can make the states entangled by choosing $\sigma$ comparable to the
magnetic length scale $l_{B}$ where $\text{P}(\xi_{0};B)$ becomes less than
unity. More interestingly, the revival of the entanglement state can occur, if
one considers a time-dependent regime. Let us recall from the definition of
$\xi$, that it basically depends on the coupling strength $K$ of the
“impurity” interaction.
The dynamic behaviour of impurities in materials is known to lead to time-
varying spring interaction Dj1 ; Dj6 . Such dynamical nature of the coupling
has been studied in the literature in the context of several physical systems
such as in optical lattices Dj5 , and extensively in the domain of quantum
electronic transport Dj ; Dj3 ; Dj4 . Let us now, consider that the spring
“constant” $K$ is a slowly varying periodic function of time due to some
external effects, with the time-variation given by
$K(t)=K\text{cos}^{4}\omega_{d}t=K\text{cos}^{4}\theta(t)$ (51)
which clearly indicates
$\xi(t)=\frac{1}{2}(\frac{K\text{cos}^{4}\omega_{d}t}{4K_{0}})^{(1/4)}$ and
$\xi_{0}(t)=\frac{\xi(t)l_{B}}{\sigma}$. Hence, the purity function gets
modified to,
$\text{P}(\xi_{0};l_{B})=\frac{1}{\sqrt{1+2{\xi^{2}_{0}(t)}}}$ (52)
From the above equation, it follows that the Purity function is periodic. It
may be noted that even if the $\sigma$ is comparable to the magnetic length
scale $l_{B}$, disentanglement occurs for
$t_{d}=\frac{\pi}{2\omega_{d}},\frac{3\pi}{2\omega_{d}},\frac{5\pi}{2\omega_{d}},\frac{7\pi}{2\omega_{d}},.....$
with a separation of $\frac{\pi}{\omega_{d}}$ time period between two
successive collapses. For the rest of the time interval, the states are
entangled. This distinguishing feature is known as the collapse and revival of
entanglement in the literature rop2 . In Figure 2, we plot the Purity function
versus the periodic parameter $\theta(t)$ for several different values of the
wavepacket width $\sigma$. It is clearly seen that the magnitude of
entanglement revival increases more for narrower wavepackets.
Figure 2: Time evolution of the Purity function is plotted against the
parameter $\theta(t)$, for various widths of the wavepackets $\sigma$.
Here it needs to be mentioned that in order to observe entanglement revival of
the states, it is required to choose $\sigma$ of the order comparable to that
of the magnetic length scale $l_{B}$ or less, as
$-1\leq\text{cos}\omega_{d}t\leq+1$. On the other hand, if we choose $\sigma$
to be much larger than $l_{B}$, then the additional term in the denominator of
Eq.(52) can be completely negligible which will take us back again to the
situation of the entanglement collapse, viz. $\text{P}(\xi_{0};B)\sim 1$.
Instances of the phenomenon of entanglement collapse and revival have been
pointed out earlier in the literature predominantly in the context of the
Jaynes-Cummings model for optical systems R2 ; rop2 . Here we furnish a
striking example of entanglement collapse and revival in the context of an
excitonic dipole in a condensed matter system.
## VI Conclusions
To summarize, in this work, we have considered a composite two-particle planar
dipole system in the presence of a strong constant and uniform magnetic field,
in which two oppositely charged particles interact via harmonic interaction,
in addition to an impurity interaction experienced by the positively charged
particle. Our system may be regarded as a toy version of excitonic dipole
models that can be realized in some specific direct band gap semiconductors
exciband2 ; exciband3 ; exciband1 having the conduction band minimum for
electrons and the valence band maximum for holes both located at the same
point of the Brillouin zone, where the effective mass of electrons and holes
can be quite similar in magnitude. This typically arises due to specific band
structures and symmetries of materials. The additional interaction could arise
from intrinsic features such as defects or impurities, as well as from
external influences like an external electric field or strain in the material
e4 .
In our analysis, we have first addressed the classical picture in the context
of our system’s Lagrangian formulation which is the most natural in a strong
magnetic field limit. Using symplectic analysis of this first-order
Lagrangian, we have specified the canonical/Weyl-Moyal type deformed NC
classical phase space to be an intrinsic part of our model. Next, we have
explored the quantum mechanical description of our model by elevating all the
phase space variables to the level of Hermitian operators. The spatial and
momentum sectors of individual charged particles obey a non-commutative
deformed algebra. Here, the non-commutativity emerges as a natural consequence
of placing two oppositely charged particles in a strong constant background
magnetic field. The square of the magnetic length scale acts as the effective
non-commutative parameter.
We have presented a physical interpretation of the mapping from the deformed
phase space to the usual commutative phase space. The non-commutative phase
space represents the system Hamiltonian written in terms of the positively
charged particle coordinates, while the standard quantum mechanical phase
space is more suitable for describing our system in terms of the composite
system’s centre of mass coordinates. The dynamics can, therefore, be analyzed
in terms of non-commuting variables or, alternatively, using phase space
transformations, in terms of commuting variables. In literature, non-
commutativity has been often introduced by hand for a single point particle,
thus ruling out any physicality of commutative phase-space variables in such
cases. However, in the present case, non-commutativity emerges naturally,
thereby giving a physical meaning to the commutative phase-space variables.
Determining the Hamiltonian’s ground state in the commutative phase space
allows us to express the quantum state in the non-commutative phase space as a
superposition of two diametrically opposite coherent states, entangled with
momentum eigenstates. This reveals the emergence of entangled and two-
component as well as multi-component Schrödinger Cat States (SCS) in our
system.
Furthermore, we have estimated the magnitude of entanglement in the system of
multicomponent entangled cat states. By utilizing the purity function, we
demonstrate that the effective non-commutative parameter ($l_{B}^{2}$) is
responsible for the entanglement. We show that when the width of the Gaussian
wave packet ($\sigma$) significantly exceeds the minimal length scale
($l_{B}$), the entangled cat states undergo collapse. Conversely, when
$\sigma$ is comparable to the nonzero magnetic length scale $l_{B}$, the
entanglement can be observed. Moreover, we show that if time-dependent
impurity potential is chosen, entanglement revival and collapse occurs
periodically. So notably, within the same formalism, we observe the phenomenon
of collapse and revival of entanglement in the non-commutative plane in the
time-dependent regime with a suitable choice of the $\sigma$ parameter for the
revival case, while the collapse is completely controlled by the nodes of the
periodic function involved in the impurity interaction.
Before concluding, it may be noted that spin-orbit interactions in solid-state
systems introduce electronic band curvature, leading to the emergence of Berry
curvature in momentum space. Such Berry curvature modifies the usual phase
space symplectic structure of Bloch electrons xio ; xio2 . In light of non-
commutative quantum mechanics, our present analysis can be extended to include
investigations on the possible emergence of Schrödinger cat states in solid
state systems involving the 2D excitonic Coulomb problem with the Berry
curvature of the electron’s and the hole’s Bloch states Be1 ; Be2 ; Be3 . This
may open up a new window to experimentally observe quantum superposition for
“macroscopic” states.
## VII Acknowledgements
PN and ND acknowledge support from S.N. Bose National Centre for Basic
Sciences where this work was initiated. PN would also like to thank the
Institute of Theoretical Physics, Stellenbosch University for providing
postdoctoral funds during the period when a major part of this work was
completed. We thank Biswajit Chakraborty, Debasish Chatterjee, Ananda Dasgupta
and Frederik G. Scholtz for some fruitful discussions. ASM acknowledges
support from the Project No. DST/ICPS/QuEST/2018/98 from the Department of
Science and Technology, Government of India.
## VIII Appendix A
Here we present a manifestation of the non-commutativity of the centre of mass
coordinates arising in the case of two oppositely charged particles with
different masses $m_{+}$ and $m_{-}$ representing the masses of positive and
negatively charged particles respectively. The corresponding centre of mass
(CM) coordinates of the above-discussed system is-
$\displaystyle\hat{R}_{i}=\frac{m_{+}\hat{x}_{i}+m_{-}\hat{y}_{i}}{m_{+}+m_{-}};$
$\displaystyle~{}\hat{P}_{i}=\hat{P}_{x_{i}}+\hat{P}_{y_{i}}=\frac{eB}{c}\epsilon_{ij}(\hat{x}_{j}-\hat{y}_{j});~{}~{}i,j=1,2$
(53)
Now, utilizing the results obtained from equation (10), the commutation
brackets between the CM coordinates can be obtained in the following form-
$[\hat{R}_{i},\hat{R}_{j}]=\frac{m_{+}^{2}-m_{-}^{2}}{(m_{+}+m_{-})^{2}}il_{B}^{2}\epsilon_{ij};~{}~{}i,j=1,2$
(54)
clearly indicating the non-commutativity between the CM position coordinates
with
$\theta=\frac{m_{+}^{2}-m_{-}^{2}}{(m_{+}+m_{-})^{2}}il_{B}^{2}\epsilon_{ij}$
being the effective non-commutativity parameter. However, it is
straightforward to check that the other two commutation brackets remain
preserved.
$[\hat{P}_{i},\hat{P}_{j}]=0;~{}~{}[\hat{R}_{i},\hat{P}_{j}]=i\hbar\delta_{ij}$
(55)
It may be noted that the order of magnitude of the non-commutativity between
the CM position coordinates is much lesser compared to that of the position
coordinates of the individual constituent particles. This is simply because
$l_{B}^{2}$ itself is very small due to the strong magnetic field limit, the
presence of the additional mass factor reduces the whole effective non-
commutativity parameter $\theta$ to a much smaller value.
Now, let us introduce the relative coordinate system:
(56)
The commutation relations satisfied by the relative coordinates are given by
(57)
It is evident that the relative position coordinates commute as we have
considered two oppositely charged particles on a non-commutative space (it has
been shown earlier pmho , that the non-commutativity of a charged particle
differs from its antiparticle and also from any other particle of opposite
charge by the sign). On the other hand, the coordinates of relative momenta
give rise to a nontrivial commutation algebra with a reduced order of
magnitude from that of the individual constituent particle’s momentum
coordinates.
It may be further noted that the position coordinates of the centre of mass
and the position coordinates of the relative motion are not independent,
rather they obey the relation given by
$[\hat{R}_{i},\hat{r}_{j}]=-i{l_{B}^{2}}\epsilon_{ij};~{}~{}i,j=1,2$ (58)
So, clearly, there is a connection between the motion of the centre of mass
and the relative motion of the composite system in the non-commutative space.
This helps us to reduce the two-body problem completely to a one-body problem
for the internal motion in non-commutative space using the CM coordinates of
the composite system where the information of the negatively charged particle
is solely hidden/encoded within the CM momenta giving rise to a standard
commutative algebra.
## IX Appendix B
Here we provide a derivation for the expression of the purity function. We
begin with the expression of the reduced density matrix of the equation (V)
and the expression of the coherent state $|\alpha(k_{2})>$ and definition of
the Purity function from the equation(V),
$\text{P}(\alpha)=\sum_{l}\sum_{s}<l|\hat{\rho}_{red}|s><s|\hat{\rho}_{red}|l>$
(59)
The coherent state can be expressed as
$|\alpha(k_{2})>=e^{-\frac{\alpha^{2}}{2}}e^{\alpha\hat{a}_{1}^{\dagger}}e^{-\alpha\hat{a}_{1}}|0>=e^{-\frac{\alpha^{2}}{2}}e^{\alpha\hat{a}_{1}^{\dagger}}|0>$
$<l|\alpha(k_{2})>=<l|e^{-\frac{\alpha^{2}}{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n>=e^{-\frac{\alpha^{2}}{2}}\frac{\alpha^{l}}{\sqrt{l!}}$
(60)
Similarly,
$<\alpha(k_{2})|s>=e^{-\frac{\alpha^{2}}{2}}\frac{\alpha^{s}}{\sqrt{s!}}$.
Plugging this into the equation (59), one gets
$<l|\hat{\rho}_{red}|s>=\frac{\sigma}{\sqrt{\pi}}\int_{-\infty}^{+\infty}e^{-\alpha^{2}}\frac{(\alpha)^{l+s}}{\sqrt{l!}\sqrt{s!}}e^{-\sigma^{2}(k_{2}-k_{0})^{2}}dk_{2}$
(61)
Now substituting, $\alpha(k_{2})=\beta k_{2}$, where $\beta=\xi l_{B}$, we
get-
(62) (63)
$=\frac{\sigma}{\sqrt{\pi}}\frac{\beta^{l+s}}{\sqrt{l!}\sqrt{s!}}e^{-\sigma^{2}k_{0}^{2}}\frac{1}{(2\sigma^{2})^{l+s}}\frac{\partial^{l+s}}{\partial
k_{0}^{l+s}}\\{\sqrt{\frac{\pi}{\beta^{2}+\sigma^{2}}}e^{\frac{\sigma^{4}k_{0}^{2}}{\beta^{2}+\sigma^{2}}}\\}$
(64)
Using the above expressions in the purity function, we get,
(65)
Performing the summations, we are led to
$\text{P}(\alpha(k_{2}))=\frac{\sigma^{2}}{\beta^{2}+\sigma^{2}}e^{(-2\sigma^{2}k_{0}^{2})}[e^{\frac{\sigma^{4}k_{0}^{2}}{\beta^{2}+\sigma^{2}}}(e^{\frac{\beta^{2}}{2\sigma^{4}}\overset{\leftarrow}{\frac{\partial}{\partial
k_{0}}}\overset{\rightarrow}{\frac{\partial}{\partial
k_{0}}}})e^{\frac{\sigma^{4}k_{0}^{2}}{\beta^{2}+\sigma^{2}}}]$ (66)
Now, replacing $\xi_{0}=\frac{\xi l_{B}}{\sigma}$, we arrive at-
$\text{P}(\xi_{0};l_{B})=(\frac{1}{1+\xi^{2}_{0}})e^{(-2\sigma^{2}k_{0}^{2})}[e^{(\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}})}(e^{\frac{\xi^{2}_{0}}{2\sigma^{2}}\overset{\leftarrow}{\frac{\partial}{\partial
k_{0}}}\overset{\rightarrow}{\frac{\partial}{\partial
k_{0}}}})e^{(\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}})}]$ (67)
Next, we obtain a compactified form of
$[e^{\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}}}(e^{\frac{\xi_{0}^{2}}{2\sigma^{2}}\overset{\leftarrow}{\frac{\partial}{\partial
k_{0}}}\overset{\rightarrow}{\frac{\partial}{\partial
k_{0}}}}e^{\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}}}]$. For that, let us
consider the following integral:
$\int_{-\infty}^{+\infty}e^{-bs^{2}+2sk_{0}}ds=e^{\frac{k_{0}^{2}}{b}}\int_{-\infty}^{+\infty}e^{-b(s+\frac{k_{0}}{b})^{2}}ds=e^{\frac{k_{0}^{2}}{b}}\sqrt{\frac{\pi}{b}}$
(68)
From the expression of $e^{\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}}}$, it
follows that-
$e^{\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}}}=\sqrt{\frac{1+\xi_{0}^{2}}{\sigma^{2}\pi}}\int_{-\infty}^{+\infty}e^{-\frac{(1+\xi_{0}^{2})}{\sigma^{2}}s^{2}+2sk_{0}}ds$
(69)
Therefore,
$[e^{\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}}}(e^{\frac{\xi_{0}^{2}}{2\sigma^{2}}\overset{\leftarrow}{\frac{\partial}{\partial
k_{0}}}\overset{\rightarrow}{\frac{\partial}{\partial
k_{0}}}})e^{\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}}}]$
$\displaystyle=\scalebox{0.9}{$\frac{1+\xi_{0}^{2}}{\sigma^{2}\pi}\int_{-\infty}^{+\infty}e^{-\frac{(1+\xi_{0}^{2})}{\sigma^{2}}s^{2}+2sk_{0}}ds(e^{\frac{\xi_{0}^{2}}{2\sigma^{2}}\overset{\leftarrow}{\frac{\partial}{\partial
k_{0}}}\overset{\rightarrow}{\frac{\partial}{\partial
k_{0}}}})\int_{-\infty}^{+\infty}e^{-\frac{(1+\xi_{0}^{2})}{\sigma^{2}}s^{\prime
2}+2s^{\prime}k_{0}}ds^{\prime}$}$
[where we have used the relation ${e^{a{\frac{\partial}{\partial
k_{0}}}}{e^{bk_{0}}}}=e^{ab}e^{bk_{0}}$]
$=\sqrt{\frac{1+\xi_{0}^{2}}{\sigma^{2}\pi}}\int_{-\infty}^{+\infty}e^{-\frac{(1+\xi_{0}^{2})}{\sigma^{2}}s^{2}+2sk_{0}}e^{\frac{(\sigma^{2}k_{0}+\xi_{0}^{2}s)^{2}}{\sigma^{2}(1+\xi_{0}^{2})}}ds$
$=\sqrt{\frac{1+\xi_{0}^{2}}{\sigma^{2}\pi}}e^{\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}}}\int_{-\infty}^{+\infty}e^{-\frac{(1+2\xi_{0}^{2})}{\sigma^{2}(1+\xi_{0}^{2})}s^{2}+2k_{0}\frac{(1+2\xi_{0}^{2})}{(1+\xi_{0}^{2})}s}ds$
After performing some suitable steps, we get the final simplified form as
$[e^{\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}}}(e^{\frac{\xi_{0}^{2}}{2\sigma^{2}}\overset{\leftarrow}{\frac{\partial}{\partial
k_{0}}}\overset{\rightarrow}{\frac{\partial}{\partial
k_{0}}}})e^{\frac{\sigma^{2}k_{0}^{2}}{1+\xi_{0}^{2}}}]=\frac{1+\xi_{0}^{2}}{\sqrt{1+2\xi_{0}^{2}}}e^{2\sigma^{2}k_{0}^{2}}$
(70)
Now after plugging the above result (70) in equation (67), the expression of
the Purity function reduces to
$\text{P}(\xi_{0};l_{B})=\frac{1}{\sqrt{{1+2\xi^{2}_{0}}}}$ (71)
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|
# Discovering Block Structure in Networks
Rudy Arthur
###### Abstract
A generalization of modularity, called block modularity, is defined. This is a
quality function which evaluates a label assignment against an arbitrary block
pattern. Therefore, unlike standard modularity or its variants, arbitrary
network structures can be compared and an optimal block matrix can be
determined. Some simple algorithms for optimising block modularity are
described and applied on networks with planted structure. In many cases the
planted structure is recovered. Cases where it is not are analysed and it is
found that strong degree-correlations explain the planted structure so that
the discovered pattern is more ‘surprising’ than the planted one under the
configuration model. Some well studied networks are analysed with this new
method, which is found to automatically deconstruct the network in a very
useful way for creating a summary of its key features.
[inst1]organization=University of Exeter, Department of Computer
Science,addressline=Stocker Rd, city=Exeter, postcode=EX4 4PY, country=UK
## 1 Introduction
Modularity [1] is a function which takes a label assignment on the nodes of a
network and returns a score evaluating how effectively the label assignment
partitions the network into non-overlapping communities. Given a network with
(weighted) adjacency matrix $A_{ij}$ and the labelling function $c(i)$ mapping
nodes, $i$, to community labels, modularity is defined as
$Q_{\text{Newman}}=\frac{1}{2E}\sum_{ij}\left(A_{ij}-\gamma\frac{k_{i}k_{j}}{2E}\right)\delta(c(i),c(j)).$
(1)
$\sum_{ij}A_{ij}=2E$ is the total number of edges, $k_{i}$ is the degree of
the node $i$ and $\gamma$ is the so-called resolution parameter [2], set to
$1$ throughout this work. The sum over $A_{ij}$ measures the fraction of
within community edges in the observed network and the sum over degrees gives
the fraction of within community edges expected under a degree preserving
randomization of the network, known as the configuration model [3]. Despite
some well-known issues identifying small communities (the so-called resolution
limit [4]) modularity maximization is the basis for a number of very popular
community detection algorithms e.g. [5, 6].
Modularity has been extended in a number of ways to identify communities in
structured networks. Various authors [7, 8, 9] have given a definition of
modularity appropriate for community detection in bipartite networks and [10]
gives a definition of modularity appropriate for community detection in the
unipartite projection of bipartite networks. A definition of modularity
appropriate for finding multi-core-periphery structure is given in [11]. [12]
defines “anti-modularity” for finding sets of unconnected nodes, see also [13]
which uses modularity in a similar way as [12] to find approximately bipartite
node sets. The original paper defining modularity [1] has, at time of writing,
over 11000 citations, so clearly modularity is an important and well used tool
for community detection in networks.
Figure 1: Same network with two different node labellings, indicated by
colours. (a) emphasises community structure and (b) bipartite structure.
Most work identifies a target structure, like non-overlapping communities or
core and periphery sets, and aims to find a label assignment that maximises
modularity or one of its variants. However, consider Figure 1. The same
network is shown with two different labellings to emphasise either its
approximate community structure or its approximate bipartite structure. While
either of these labellings might be useful in different problem settings, this
paper tries to answer the question of which one is ‘best’ in the sense of
‘least expected under the configuration model’.
Section 2 generalises modularity to arbitrary network structures which are
specified by a block matrix $B$. Section 3 describes algorithms to optimise
this generalised modularity at fixed $B$ and then to optimise $B$ itself.
These algorithms are applied to synthethic and real networks in Sections 3 and
4. Section 5 summarises the results and suggests some directions for future
work.
## 2 Block Modularity
Consider a network with $N$ nodes labelled into $N_{B}$ groups or ‘blocks’.
Define the matrix of modularity $Q$ as the $N_{B}\times N_{B}$ matrix with
elements
$Q_{ab}=\sum_{ij}\left(A_{ij}-\gamma\frac{k_{i}k_{j}}{2E}\right)\delta(c(i),a)\delta(c(j),b)$
(2)
The standard modularity, equation 1, is equal to the trace of this matrix
divided by $2E$. Following [5], it is convenient to change the sum over nodes
in equation 2 to a sum over blocks. Defining
$\displaystyle\Sigma_{ab}$ $\displaystyle=\sum_{i\in a,j\in b}A_{ij}$ (3)
$\displaystyle T_{a}$ $\displaystyle=\sum_{i\in a}k_{i}$ (4)
lets us re-write equation 2 as
$Q_{ab}=\Sigma_{ab}-\gamma\frac{T_{a}T_{b}}{2E}$ (5)
When there are more connections between $a$ and $b$ than the configuration
model would predict $Q_{ab}>0$ and when there are fewer $Q_{ab}<0$. Thus, the
sign and magnitude of $Q_{ab}$ is a measure of how ‘surprising’ the edge
density between node sets $a$ and $b$ is, relative to the configuration model.
Large positive values correspond to an unexpected excess and large negative
values to a deficit.
Define block modularity as
$Q(B)=\frac{1}{2E}\sum_{ab}Q_{ab}B_{ab}$ (6)
Here $B$ is a $N_{B}\times N_{B}$ matrix with entries equal to $\pm 1$. To
gain some intuition it is helpful to consider the block matrices
$B_{0}=\begin{pmatrix}1&-1\\\ -1&1\\\ \end{pmatrix}\text{ and
}B_{1}=\begin{pmatrix}-1&1\\\ 1&-1\\\ \end{pmatrix}$
With nodes split into two blocks, with labels 0 and 1, $Q(B_{0})$ will be
large when there is an excess of edges within node sets and a deficit of edges
between them - this is the usual non-overlapping two community structure.
$Q(B_{1})$ is the opposite, large when there is an excess of edges between
node sets and a deficit within them. Therefore $Q(B_{1})$ will be large for
networks which are bipartite or approximately so.
Other modularity formulations can be recovered by taking different values for
the block matrix $B$. The standard equation 1, can be recovered with
$B_{ab}=\delta_{ab}$ for example. Other formulations can be recovered by
substituting the corresponding block pattern. If we split the label $a$ into a
parity label $x_{a}$ and community label $c_{a}$ then a modularity definition
suitable for a bipartite graph made of multiple communities, after [8], can be
obtained using
$B_{x_{a},c_{a};x_{b},c_{b}}=\delta_{c_{a}c_{b}}(1-\delta_{x_{a}x_{b}})$.
Similarly, the multi-core-periphery modularity of [11] can be recovered with
$B_{x_{a},c_{a};x_{b},c_{b}}=\delta_{c_{a}c_{b}}(x_{a}+x_{b}-x_{a}x_{b})$. In
contrast to most other definitions of modularity, we will use $B$ matrices
with values $\\{+1,-1\\}$ rather than $\\{1,0\\}$. This is to give equal
weight to excesses and deficits of connections between blocks.
Figure 2: Allowed $2\times 2$ and $3\times 3$ block patterns, after [14].
The work of [14] is the most similar to the above. They provide limits on the
types of block structure the configuration model can detect. By enforcing
symmetry and the identity
$\sum_{a}\Sigma_{ab}=\sum_{a}\frac{T_{a}T_{b}}{2E}$ (7)
certain types of block matrix are forbidden. For example, it is not possible
to simultaneously have an excess within and between all node sets. Defining
black cells as ones where $Q_{ab}>0$ and white cells where $Q_{ab}<0$,
equation 7 is equivalent to forbidding completely white or black rows or
columns so, in particular, a $2\times 2$ core-periphery block pattern is
forbidden. The allowed $2\times 2$ and $3\times 3$ patterns are shown in
figure 2.
The maximum value of standard modularity is 1 [15]. To understand the maximum
value of $Q(B)$, let us follow [16] and consider a block matrix $B$ with $+1$
on the diagonal and $-1$ elsewhere for a network of $N_{B}$ cliques with the
canonical labelling. The diagonal terms contribute
$\frac{1}{N_{B}}\left(1-\frac{1}{N_{B}}\right)$
and the off-diagonals give
$-\frac{1}{N_{B}^{2}}$
So that
$Q(B)_{max}=\frac{N_{B}}{N_{B}}\left(1-\frac{1}{N_{B}}\right)+\frac{N_{B}(N_{B}-1)}{N_{B}^{2}}=2-\frac{2}{N_{B}}$
(8)
which converges to $2$ for large $N_{B}$. The value is $2$ instead of the
standard modularity bound of $1$ due to counting the deficits as well as the
excesses. Like standard modularity, the upper bound is achieved only in the
limit of very large $N_{B}$ thus, like standard modularity, $Q(B)$ will tend
to be larger for higher $N_{B}$. In this work we compare $Q(B)$ at fixed
$N_{B}$, comparing values at different $N_{B}$ should be done cautiously.
For any fixed $B$, optimising equation 6 will find the label assignment on the
nodes that best matches that structure. For example $B_{6}$ is the block
pattern of 3 isolated communities. $Q(B_{6})$ for the network in Figure 1 will
be large given labels shown in 1(a). $B_{1}$ is the block pattern of a
bipartite network. $Q(B_{1})$ for the same network will be large for the label
assignment shown in 1(b). Given there are a finite number of allowed block
patterns we can find the optimal label assignment for every $B$ and compare
all of the maximised values of $Q(B)$ for different $B$. The block matrix that
gives the maximum $Q(B)$ score is the least expected under the configuration
model and therefore represents the structure in the network which can be least
well explained by degree correlation. In the following sections we will give
some examples that show how this optimal structure matrix can be useful in
characterising networks.
## 3 Algorithms for Finding Block Patterns
Algorithm 1 $\text{Label Swap}(B,T_{0}=0.01,k_{max}=1000)$
$\text{moves}\leftarrow 1$
$k\leftarrow-1$
Let $\vec{n}$ be the list of nodes
Let $c(n)$ be the label of node $n$
while $\text{moves}>0$ or $k<k_{max}$ do
$\text{moves}\leftarrow 0$
$k\leftarrow k+1$
$T=T_{0}\left(\frac{k_{max}-k}{k_{max}}\right)^{2}$
Randomly shuffle the list of nodes $\vec{n}\leftarrow\vec{n}^{\prime}$
for $n$ in $\vec{n}^{\prime}$ do
for $a$ in block labels where $a\neq c(n)$ do
Compute $dQ(a)$, the change in $Q(B)$ when $c(n)\leftarrow a$
end for
$dq\leftarrow\text{max}\left(dQ(a)\right)$
$c_{max}\leftarrow\text{argmax}\left(dQ(a)\right)$
if $dq>0$ then
$\text{moves}\leftarrow\text{moves}+1$
$c(n)\leftarrow c_{max}$
else if $k<k_{max}$ and $r<\exp(dq/T)$ then
$c(n)\leftarrow c_{max}$
end if
end for
end while
return $Q(B)$
For fixed $B$ and some initial labelling of the nodes $c(i)$, the optimisation
algorithm 1 finds a labelling with high $Q(B)$. The algorithm performs
simulated annealing with quadratic cooling, swapping node labels to increase
$Q(B)$, where moves that decrease $Q(B)$ are allowed at higher temperature.
$r$ is a random number and typical parameters are $T_{0}=0.1$, $k_{max}=100$.
The algorithm 1 performs better than a greedy method (setting $T=0$,
$k_{max}=0$) in most cases.
Figure 3: (a) A tripartite network with the optimal labelling under
$Q(B_{8})$, (b) the values of $Q(B_{i})$ for all $3\times 3$ block patterns,
the optimum is achieved at $B_{8}$.
Figure 3 (a) shows a tripartite network. Running algorithm 1 using each of the
allowed $3\times 3$ block patterns in Figure 2 gives the optimal values of
$Q(B_{i})$ shown in Figure 3 (b). The maximum $Q(B)$ is achieved for
$B=B_{8}$, which is the block pattern corresponding to the tripartite
structure of the network, with the optimal labelling for $Q(B_{8})$ indicated
by the colours in Figure 3 (a).
Figure 4: (a) A core-periphery network (pattern $B_{5}$) with the optimal
labelling under $Q(B_{7})$, (b) the values of $Q(B_{i})$ for all $3\times 3$
block patterns.
It is not always the case that the block pattern ‘planted’ in the network is
the one recovered by optimising $Q(B)$. Figure 4 (a) shows a network
consisting a core-periphery with an isolated communtiy, which corresponds to
block pattern $B_{5}$. The same process is applied and the optimal $Q(B)$ is
achieved with $B=B_{7}$ rather than $B_{5}$. This is further analysed in the
next section and A where it is shown that, due to the high degrees of the core
nodes, connections between core nodes are expected under a degree preserving
randomisation. This means that patterns with bipartite structure (as in
$B_{7}$) are favoured since they admit a labelling that is ‘more surprising’.
Figure 5: Left: The network, the block pattern used in its construction and
the optimal labelling under the optimal $B$ identified on the right. Right:
$Q(B)$ for every allowed $4\times 4$ block pattern.
To generate more complicated networks with planted block structures, for
$N_{B}$ blocks, $n$ nodes per block, construct an $N_{B}\times N_{B}$ density
matrix $P$, where $P_{ab}$ is the probability of a link between nodes in block
$a$ and block $b$. Figure 5 shows two networks where $N_{B}=4$, $n=10$ and $P$
is constructed by replacing black elements with $0.8$ and white elements with
$0.1$ in the corresponding block pattern. This is just the Stochastic Block
Model (SBM), see e.g. the review of [17]. In this work the SBM is only used to
generate networks with interesting block structure, but the SBM has a close
relationship with modularity maximisation. It was shown in [18] that
maximising the modularity is equivalent to finding the maximum likelihood
estimate of the parameters of a particular type of SBM called the planted
partition model. The approach taken here of optimising equation 6 likely has
some relationship with maximum likelihood estimation of the parameters of some
SBM, we will discuss connections to the SBM further in Section 5.
The top row of Figure 5 shows a network which consists of an isolated, dense
community, loosely connected to a tripartite network. Running Algorithm 1 for
all possible $4\times 4$ block patterns to find the optimal labelling for each
gives the result on the right. The planted structure is recovered as the
structure corresponding to the maximum $Q(B)$.
It is appropriate at this stage to look at the performance of Algorithm 1.
This algorithm, at any value of $T_{0}$, can get stuck at local maxima, much
like other modularity maximisation algorithms [5]. Slower cooling schedules
(larger $k_{max}$) typically find better maxima. For the network shown in the
bottom of Figure 5 optimising $Q(B)$ with the block partition shown in the
figure on the left, using the greedy algorithm ($T_{0}=0$) we find $31/100$
runs achieve the optimal label assignment. Using the annealing algorithm
($T_{0}=0.01$, $k_{max}=100$) this goes up to $49/100$. These numbers are
broadly representative of other networks and block patterns, with annealing
finding the optimum a factor of $\sim 2-5$ times more often in most cases.
Thus it is recommended (and implemented in Figure 5 and elsewhere) to run
Algorithm 1 a number of times, $N_{r}$, and choose the run with the largest
value of $Q(B)$. $N_{r}=20$ is found to be sufficient in the cases considered
in this work.
Again, it is not always the case that the planted structure is recovered. The
bottom row of Figure 5 shows a network constructed as an isolated community,
loosely connected to a core-periphery network, where the core has two distinct
peripheries. The optimal $B$ and label assignments are not the canonical ones
implied by the block matrix used to construct the network, a number of other
block patterns admit labellings with higher $Q(B)$. The largest $Q(B)$ is
found with the pattern shown the on right of Figure 5. The nearly isolated
community is recovered exactly but instead of the ‘core double periphery’
pattern there is a very small core connected to one periphery which itself
forms one half of a nearly bipartite pair.
To show explicitly how and why core-periphery structure can ‘vanish’ under
block modularity, consider a fully connected clique of $M$ nodes, all sharing
the label $0$, connected to a periphery of $qM$ nodes, labelled $1$. To
satisfy the consistency condition, equation 7, also add a disconnected clique
of $bM$ nodes. In A it is shown that $Q(B_{5})$, the optimal block modularity
for the planted structure is greater than $Q(B_{7})$ only if $b>q^{2}$.
The reqirement $b>q^{2}$ means that if the periphery is large (high $q$) or if
the core-periphery makes up the majority of the network (low $b$) then the
core-periphery block pattern can be a sub-optimal description of the network
structure under the configuration model. Ultimately, this condition derives
from squaring the degree sum of the core nodes. Since these nodes have very
high degree, under a degree preserving randomisation it is not unlikely that
they are connected to each other. In the network since only a subset of these
‘core’ edges are intra-block connections the high connectedness in the core is
expected and doesn’t contribute to the ‘surprise’ measured by $Q(B)$. From
this we can conclude that, as well as the conditions given by equation 7, for
core-periphery structure to exist (under the configuration model) requires
either a relatively small periphery or that the core-periphery only forms a
relatively small part of the overall network. The result is that core-
periphery structure, even if explicitly planted, can give lower $Q(B)$ than
using a block pattern where the core is missing. The result is analogous to
[14], certain block patterns may intuitively appear to be optimal, but under
the configuration model they are not, due to the importance of degree
correlations.
Algorithm 2 Simulated Annealing($N_{B},T_{0}=0.01,k_{max}=100$)
$B=I_{N_{B}\times N_{B}}$
$Q=\text{Label Swap}(B)$
while $T>0$ do
$T=T_{0}\left(\frac{k_{max}-k}{k_{max}}\right)^{2}$
do
For a random element of $B$
$B_{ij}\leftarrow-B_{ij}$
if $i\neq j$ then
$B_{ji}\leftarrow-B_{ji}$
end if
while equation 7 is false
$Q_{n}=\text{Label Swap}(B)$
if $Q_{n}>Q$ or $r<\exp((Q_{n}-Q)/T)$ then
$Q=Q_{n}$
else
Undo flip
end if
end while
Figure 6: Left: The network and the block pattern used in its construction
(which is also the optimal block pattern identified by Algorithm 2). The
network is labelled according to to the block pattern found by Algorithm 2,
shown on the right. Right: Shows the block pattern found by Algorithm 1 and
$Q(B)$ as a function of temperature.
The number of allowed block patterns grows quite rapidly with $N_{B}$ and an
exhaustive search becomes infeasible. Inspired by the similarity of these
block matrices to spin systems, Algorithm 2 is a simulated annealing approach
to finding the optimal block pattern. Figure 6 shows the final block pattern
and label set found by this algorithm for $T_{init}=0.01$ and $1000$ steps of
quadratic cooling. In practice the final results of algorithms 1 and 2 are
fairly insensitive to the exact cooling scheme, starting temperature and
number of cooling steps. Figure 6 shows that algorithm 2 recovers an
equivalent block pattern to the one used to generate the network, where a
permutation of the labels turns the pattern on the right into the one on the
left. The optimal labelling for this block pattern $B$ is the one expected
based on the planted structure. Algorithm 2, like algorithm 1, can become
stuck at local maxima. The same solution - repeated, independent runs - is
used to alleviate this problem. Algorithms 2 and 1 will find optimal label
assignments and block patterns but are somewhat inefficient. In this work the
intention is to understand $Q(B)$ rather than find the best possible algorithm
to optimise it, so algorithmic improvements are left for future work.
## 4 Real Networks
Figure 7: The optimal block pattern, labelling and value of $Q(B)$ for the
‘Southern Women’ network for $N_{B}=\\{2,3,4,5\\}$. The map of colours to
block indices is {0:blue, 1:orange, 2:green, 3:red, 4:purple, 5:brown}
In this section algorithm 2 will be run on some well known empirical networks
obtained from the KONECT database [19]. Since the optimal number of blocks is
unknown, the algorithm is run for a range of $N_{B}$. The first network to be
analysed is the Southern Women network [20], a bipartite network consisting of
18 women (labelled 1 through 18) and 14 events (labelled 19 through 32).
Typical analysis of this network [21] identifies 2 or more communities of
women and 2 or more classes of event. The block optimisation is summarised in
Figure 7. For $N_{B}=2$ the algorithm identifies the (exact) bipartite
structure of the network. $N_{B}=3$ demonstrates some interesting behaviour;
the optimal labelling does not include any labels for the third block. A block
pattern with lower $N_{B}$ can outperform a higher $N_{B}$ pattern. Algorithms
1 and 2 discover this fact by returning an optimal partition that only uses a
subset of the allowed node labels.
$N_{B}=4$ splits the events into two groups (red and orange) and splits the
women into two groups (red and blue) with green events predominantly attended
by red women and blue women attending orange events. $N_{B}=5$ refines this
picture, with 3 classes of event and 2 communities of women. Blue women attend
orange and purple events, while red women attend green and purple events. This
is also shown by the block matrix. The purple events bridge the two separate
bipartite communities. This network has been analysed in great detail, e.g. in
[21], and it is remarkable that this kind of structure can be detected fairly
automatically by optimising $Q(B)$.
Figure 8: The optimal block pattern, labelling and value of $Q(B)$ for the
Karate Club network with $N_{B}=\\{3,6\\}$. The map of colours to block
indices is {0:blue, 1:orange, 2:green, 3:red, 4:purple, 5:brown}. The bottom
shows a representation of the $6\times 6$ block matrix as a network.
Another commonly studied network is Zachary’s karate club [22], a social
network split into two communities due to a dispute between the club’s
instructor (node 33) and administrator (node 1). Using $N_{B}=2$ recovers the
usual two community pattern, Figure 8 shows results for $N_{B}=\\{3,6\\}$.
Higher $Q(B)$ can be obtained for higher $N_{B}$ but there is a balance to
strike between optimisation and interpretability of the block matrix. It is
also the case that optimisation of larger block patterns is slower and more
prone to get stuck in local minima.
The $3\times 3$ pattern is a community centered around the administrator and a
bipartite community of the other members, where the blue nodes have very few
ties with anyone other than the instructor. The optimal $6\times 6$ pattern is
shown in the middle panel. For large $N_{B}$ it can be hard to interpret the
block matrix from the binary pattern alone, so this pattern is visualised as a
network at the bottom of the figure, where a clearer picture emerges. Blocks 2
and 0 are two independent communities which only interact with the
administrator’s faction (block 1). The instructor’s faction, block 5,
interacts with a ‘loyalist’ block (3) and another block (4) which retains some
ties to the administrator. Important to note is the fact that unless a block
has a self loop it should not be considered a ‘community’. For example, the
nodes in block 3 have very few connections to each other, members of this
group are only connected to each other only via the instructor’s faction.
Figure 9: The optimal block pattern, labelling and value of $Q(B)$ for the
Dolphin network with $N_{B}=\\{2,3,4,5\\}$. The map of colours to block
indices is {0:blue, 1:orange, 2:green, 3:red, 4:purple}. Squares are female
dolphins, circles are male and triangles are unknown. The inset figure shows
the network diagram corresponding to the optimal block pattern, see Figure 8.
The final example is Figure 9 which shows the dolphin social network from
[23], which is suggested to be an example of a multiple core and periphery
structure in [14]. The optimal block patterns for $N_{B}\in\\{2,3,4\\}$ are
$N_{B}$ isolated communities with the labelling shown. For $N_{B}=5$ the
pattern is a little more interesting, the all female group 1 when $N_{B}=4$
splits in two and loses some members to the mixed red group. Blocks 1 and 3
now form two cores connected to a shared periphery, block 4. Unlike Figure 4
where a planted core-core-periphery structure could not be detected, as
discussed in 3, if a core-periphery pattern forms a small enough subset of the
total network, as here, it can be detected. This kind of structure is also
seen in Figure 8, where block 1 is the shared periphery of 0 and 2.
## 5 Conclusions
This paper presents a generalization of modularity, called block modularity,
which provides a framework for detecting arbitrary structure in networks.
Because different structures are evaluated with the same quality function,
these structures can be compared and an optimal one identified, meaning fewer
assumptions need to be made about the network. Searching for particular
patterns, like community structure or core-periphery, is still a useful thing
to do, however block modularity and an algorithm like 2 can approach the
network ‘blind’ and discover potentially interesting and unexpected
structures.
Section 4 shows that optimal block patterns can help create a ‘narrative’
account of a complex network. They can also be counter-intuitive, in
particular it seems that core-periphery structure is sometimes elusive. The
configuration model is a fairly powerful null model and degree correlations
‘explain’ a lot of network structure. It would be interesting future research
to explore in general what circumstances core-periphery networks may be better
explained by other block patterns, under the configuration model.
To apply this method on very large networks (web pages, social networks),
faster optimisation algorithms are required. Algorithm 1 requires $O(N^{2})$
operations per temperature increment and both algorithms can become trapped in
local minima. Direct enumeration of all block patterns is preferable where
possible, though seems to only be practical for $N_{B}<5$. Ideally, like the
Louvain and Leiden methods [5, 6], algorithms could be developed which are not
only fast, but which do not require the number of blocks $N_{B}$ to be
specified in advance, allowing a completely blind approach.
This approach shares many of the problems of standard modularity maximisation.
For example the resolution limit identified by [4] as preventing the detection
of small communities. There is also the fact that ‘modular’ partitions can be
found even for random networks [7]. As emphasised in [24] and [25], modularity
maximisation is not an inferential approach, like maximum likelihood
estimation of a SBM, but is descriptive. The label assignment, and here also
the structure matrix $B$, found by modularity maximisation answers the
question of which sets of nodes, in a specific network, have more or fewer
inter-connections than a degree preserving randomisation would predict. When
applied to any network, even a random one, the maximum modularity score may be
low but something will be found, since we are asking for a description of that
network.
The aim of this paper is not to add to the zoo of community detection methods.
[26] has performed a thorough study of numerous different community detection
algorithms and their relative performance. [27] has shown that the very
general SBM framework subsumes many different ‘mesoscopic pattern extraction’
problems, including community detection by modularity maximisation. Following
[18] or [27] it is likely that the maximisation of equation (6) has some
relationship with the SBM. Understanding these connections and comparing the
outputs of the methods described here (or some refinement of them) against
modern SBM techniques would be interesting future work. This paper proposes a
way to unify the many different modularity-like functions in the literature;
demonstrates some unexpected and interesting consequences for multi-core-
periphery networks and shows how block matrices provide a nice ‘summary’ of a
network. Recent critism of modularity based clustering [24, 25] raises many
interesting points about the drawbacks of the method and the misunderstanding
of its results in applications. However, while modularity remains a popular
approach to structure detection I hope that the unifying framework described
here will help researchers and practitioners to better understand these
methods and use them appropriately.
## Appendix A Core-periphery
Consider a fully connected clique of $M$ nodes, all sharing the label $0$,
connected to a periphery, labelled $1$, of $qM$ nodes and a disconnected
clique of $bM$ nodes.
$\Sigma=\begin{pmatrix}M^{2}&qM^{2}&0\\\ qM^{2}&0&0\\\
0&0&bM^{2}\end{pmatrix}$
$T=\begin{pmatrix}(1+q)M^{2}&qM^{2}&bM^{2}\end{pmatrix}$
$2E=M^{2}((1+q)+(b+q))$
Gives
$Q=M^{2}\left(\begin{pmatrix}1&q&0\\\ q&0&0\\\
0&0&b\end{pmatrix}-\begin{pmatrix}(1+q)^{2}&q(1+q)&b(1+q)\\\
q(1+q)&q^{2}&bq\\\ b(1+q)&qp&b^{2}\end{pmatrix}\frac{1}{(b+q)+(1+q)}\right)$
The matrix element $Q_{11}$
$M^{2}\left(-\frac{q^{2}}{(1+q)+(b+q)}\right)$
is always negative. The matrix elements $Q_{01},Q_{10}$
$M^{2}\left(q-\frac{q(1+q)}{(1+q)+(b+q)}\right)$
are always positive, as long as
$(b+q)>0$
which is always true, since $b$ and $q$ are positive and non-zero for non-
trivial networks.
The interesting term is $Q_{00}$ which is
$M^{2}\left(1-\frac{(1+q)^{2}}{(1+q)+(b+q)}\right)$
This is positive if
$b>q^{2}$
and otherwise negative. If $Q_{00}$ is negative then a bipartite block pattern
with
$\begin{pmatrix}B_{00}&B_{01}\\\
B_{10}&B_{11}\end{pmatrix}=\begin{pmatrix}-1&1\\\ 1&-1\end{pmatrix}$
will give higher $Q(B)$ than the ‘core-periphery’ block pattern.
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|
11institutetext: Institute of Mathematics, Pedagogical University of Cracow,
Podchorazych 2, Cracow, Poland
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>
https://matematyka.up.krakow.pl/
# On diagrams accompanying reductio ad absurdum proofs in Euclid’s Elements
book I. Reviewing Hartshorne and Manders
Piotr Błaszczyk 11 0000-0002-3501-3480
Anna Petiurenko 11 0000-0002-0196-6275
###### Abstract
Exploring selected reductio ad absurdum proofs in Book 1 of the Elements, we
show they include figures that are not constructed. It is squarely at odds
with Hartshorne’s claim that “in Euclid’s geometry, only those geometrical
figures exist that can be constructed with ruler and compass”.
We also present diagrams questioning Manders’ distinction between exact and
co-exact attributes of a diagram, specifically, a model of semi-Euclidean
geometry which satisfies straightness of lines and equality of angles and does
not satisfy the parallel postulate.
###### Keywords:
Impossible figuresEuclidean parts Semi-Euclidean plane.
## 1 Introduction
Euclid’s propositions are of two kinds: constructions and demonstrations. I.1
and I.32 are model examples: the first requires the construction of an
equilateral triangle, the second – a demonstration that angles in a triangle
sum up to “two right angles” ($\pi$, in short). Yet, the proof of I.32
includes the construction of a parallel through a point. Indeed, each
proposition includes a construction part (_kataskeuē_) which introduces
auxiliary lines exploited in the proof (_apodeixis_). In this paper, we focus
on diagrams accompanying reductio ad absurdum proofs, as they undermine the
common understanding of Euclid’s diagram.
The 20th-century Euclid scholarship grows in two trends: mathematical and
historical-philological. For diagrams, the first school examines them only as
straightedge and compass constructions, while the second seeks to show they
convey some mathematical information beyond construction requirements. We
challenge both approaches with specific diagrams.
R. Hartshorne [9] develops a coherent reading of the Elements, Books I–IV
focused on tacit axioms, non-defined concepts, or relations and interprets
them in the system of Hilbert axioms. He does not find Euclidean diagrams
problematic, misleading, or competing with a logical account of geometry; on
constructions, though, he writes: “The constructive approach pervades Euclid’s
Elements. There is no figure in the entire work that cannot be constructed
with ruler and compass […] in Euclid’s geometry only those geometrical figures
exists that can be constructed with ruler and compass” ([9], 18–19).
We discuss two examples undermining Hartshorne’s claim: the figure
accompanying proposition I.7 and one implied by the proof of I.27. The first
is not, and indeed, cannot be constructed, as assumptions of the proposition
introduce an inconsistent object. The non-constructive mode of the second
figure is related to the requirement “being produced to infinity” inherent in
the definition of parallel lines. These two figures are by no means
incidental, as the first props the SSS theorem (I.8), and the second brings us
to the core of the Euclid system.
J. Ferreirós provides a concise picture of the second school: “The original
geometry of Euclid lacks the means to derive its theorems by pure logic, but
it presents us with a most interesting and fruitful way of proving results by
diagrams” ([6], 132). In this vein, K. Manders [13] introduced a distinction
between exact and co-exact information inferred from a diagram that got
widespread renown among scholars exercising diagrammatic approach. Respective
definitions read: “Co-exact attributes are those conditions which are
unaffected by some range of every continuous variation of a specified diagram;
paradigmatically, that one region includes another […], or the existence of
intersection points such as those required in Euclid I.1 […]. Exact attributes
are those which, for at least some continuous variation of the diagram, obtain
only in isolated cases; paradigmatically, straightness of lines or equality of
angles […]. Exact attributes […] are unstable under perturbation of a diagram
([13], 92–93).111[13], p. 92 presents parallelism as an exact attribute.
Instead of ‘continuous variation of a diagram’, we introduce a global
perspective, meaning specifications of the plane on which a diagram lies.
Accordingly, we examine Euclid diagram I.1 on various Cartesian planes showing
that the existence of the intersection of circles involved depend on
characteristics of a plane. Regarding exact attributes, we present a model of
a semi-Euclidean plane that does not affect straightness of lines or equality
of angles but affects parallelism (especially I.29). Both counterexamples meet
the scheme: without touching a diagram but changing assumptions on the space
hosting it, we get different results concerning co-exact (intersection of
circles) and exact (parallelism) attributes. Depending on assumptions
concerning space, the same (from the diagrammatic perspective) circles meet or
not, and the same straight lines are parallel or not.
As for co-exact attributes, we provide an analysis of proposition I.6 that
undermines Mandres’ interpretation of inequality in terms of part-whole.
For the most part, our arguments exploit an interpretation of greater-than
relation. It affects our account of the deductive structure of the Elements,
some existential claims, Manders’ interpretation of greater-than in terms of
part-whole, and his claim regarding exact attributes. Euclid’s arguments
exploring that relation proceed reductio ad absurdum mode. Significantly, out
of eleven indirect proofs in Book I, ten, specifically I.6, 7, 27, 29, employ
greater-than relation.222The others are I.14, 19, 25, 26, 39, 40.
## 2 Euclid and Hilbert construction tools
Hartshorne ([9], 102) introduces term Hilbert construction tools, meaning
transportation of line segments and angles. Hilbert axioms justify these
operations; besides, they also state the uniqueness of respective line
segments and angles.333[8], pp. 597–602 provide a concise account of Hilbert
axioms. C1 and C4, respectively, decree construction tools. Diagrams drawn up
with both tools are acquired using the first alone; it suggests Euclid’s
straightedge and compass are more effective. We follow that clue to contrast
Euclid and Hilbert approaches.
### 2.1 Transportation of segments. I.1–3
I.1 To construct an equilateral triangle on the given line AB.444English
translation by Fitzpatrick [7], diagrams after [10].
Given that $AB=a$, point C, the third vertex of the wanted triangle is an
intersection of circles $(A,a)$ and $(B,a)$. In tables like the one below, we
lay out points resulting from intersections of straight lines and
circles.555The idea of such tables originates from [14].
$(A,a),(B,a)$
---
$C$
I.2 To place a straight-line at point A equal to the given straight-line BC
[$b$].
On the line-segment AB, we construct an equilateral triangle ABD with side a;
the below diagram depicts its shadow in grey. Point G is the intersection of
the circle $(B,b)$ and the half-line $DB^{\rightarrow}$. Now, DG represents
the sum of line-segments $a,\,b$. Circle $(D,a+b)$ intersects the half-line
$DA^{\rightarrow}$ at point L. Due to the Common Notions (CN, in short) 3, AL
proves to be equal $b$.
$(B,b),\,\,{DB}^{\rightarrow}$ | $(D,a+b),\,\,{DA}^{\rightarrow}$
---|---
$G$ | $L$
Owning to I.1-2, $b$ is placed at A in a precise position. Drawing circle
(A,b), one can choose any other position at will, and that is the substance of
I.3.
I.3 To cut off a straight-line equal to the lesser C [$b$] from the greater
AB.
Line-segment $b$ is transported to A into position AL; the above diagram
depicts the shadow of that construction; let $Ab$ be its symbolic
representation. The intersection of circle $(A,b)$ and line AB determines E
such that ${AE=b}$.
$Ab$ | $(A,b),\,\,AB$
---|---
| $E$
Summing up, due to I.1–3, one can transport any line segment to any point and
position. An equilateral triangle is a tool to this end, while the existence
of circle-circle and circle-line intersection points are taken for granted.
The Euclid system requires a circle-circle or circle-line axiom, both finding
grounds in Postulates 1–3 that introduce straight-edge and compass. Logically,
these two tools reduce to compass alone (vide Mohr-Mascheroni theorem), yet,
throughout the ages, the economy of diagrams prevailed and no one questioned
the rationale for Euclid’s instruments. There are, however, models of the
Hilbert system that do not satisfy the circle-circle axiom. Moreover,
Hartshorne shows ([9], 373) that I.1 does not hold in the Hilbert system of
absolute geometry. Thus, already at the very first proposition of the
Elements, we observe that Euclid and Hilbert’s systems follow alternative
deductive tracks. Therefore one cannot simply merge Hilbert’s axioms with
Euclid’s arguments.
### 2.2 Transportation of angles. I.22–23
In I.22, Euclid builds a triangle from three given line segments.666[8], p.
173 observes it is equivalent to the circle-circle axiom.
The below table presents $D,E$ as random- and $G,K$ as intersection- points.
| ${Db}$ | $(D,b),\,DE$ | $Da$ | $Gc$ | $(D,a)$, $(G,c)$
---|---|---|---|---|---
$D$, $E$ | | $G$ | | | $K$
I.23, angle transportation, rests on triangle construction as follows: on
sides of the given angle, Euclid builds a triangle, transports its sides to
$A$, $G$, obtaining another triangle. By the SSS, $\triangle CDE=\triangle
AFG$, hence $\angle KCL=\angle FAG$.
${Ab}$ | $(A,b),\,AB$ | $Aa$ | $Gc$ | $(A,a)$, $(G,c)$
---|---|---|---|---
| $G$ | | | $F$
## 3 Euclid’s vs Hilbert’s deduction: SAS to SSS
Throughout propositions I.1–34, equality means congruence, whether applied to
line segments, angles, or triangles. In I.5–8, showing the SSS theorem, Euclid
assumes I.4, Common Notions, and characteristics of the greater-than relation.
The proof of I.4 (SAS criterion) relies on the ad hoc rule: two straight-lines
can not encompass an area. The diagram depicts an area encircled by the base
$EF$ of the triangle and a curve with ends $E,F$. By contrast, Hilbert axioms
guarantee a unique line through points $E,\,F$ and diagram I.4 has no grounds.
I.5 Let ABC be an isosceles triangle. I say that the angle ABC is equal to
ACB.
The construction part is simple: F is taken at random on the half-line
$AB^{\rightarrow}$, then G such that $AF=AG$ is determined on the half-line
$AC^{\rightarrow}$.
$AB^{\rightarrow}$ | $(A,a+b),\,\,AC^{\rightarrow}$
---|---
$F$ | $G$
(i) Now, due to SAS, $\triangle GAB=\triangle FAC$. Thus $FC=BG$ and
$\beta=\angle AGB=\angle AFC=\beta^{\prime},$ $\gamma=\angle ABG=\angle
ACF=\gamma^{\prime}.$
(ii) Again by SAS, $\triangle BFC=\triangle BGC$, and
$\delta=\angle CBG=\angle BCF=\delta^{\prime}.$
(iii) By CN 3, $\gamma-\delta=\gamma^{\prime}-\delta^{\prime}$. Since
$\alpha=\gamma-\delta,\ \ \gamma^{\prime}-\delta^{\prime}=\alpha^{\prime},$
the equality $\alpha=\alpha^{\prime}$ holds. $\Box$
I.6 Let ABC be a triangle having the angle ABC equal to the angle ACB. I say
that side AB is also equal to side AC.
The proof reveals assumptions in no way conveyed through definitions or
axioms. At first, it is the trichotomy law for line segments. Let $AB=b$,
$AC=c$ (see Fig. 1). To reach a contradiction Euclid takes: if $b\neq c$, then
$b<c$ or $b>c$. Tacitly he assumes that exactly one of the conditions holds
$b<c,\ \ \ b=c,\ \ \ b>c.$
Figure 1: Proof of I.6 schematized.
Let $b>c$. Then the construction follows: “let DB, equal to the lesser AC,
have been cut off from the greater AB”. However, given that angles at $B$ and
$C$ are equal, then $AB=c$, and the cutting off “the lesser AC from the
greater AB” cannot be carried out. On the other hand, if $AB=b$ and $b>c$, the
triangle $ABC$ is not isosceles, and angles at $B$, $C$ are not equal.
Throughout the proof, thus, the diagram changes its metrical characteristic
and cannot meet the assumptions of the proposition. Contrary to Euclid’s
claim, $D$ is a random point on $AB$, rather than introduced via the following
construction
$(B,c),AB$
---
$D$
Now, by SAS, the equality $\triangle DBC=\triangle ACB$ holds, and Euclid
concludes the lesser to the greater. The very notion is absurd.
This time, the trichotomy law applies to triangles. The contradiction
$\triangle DBC=\triangle ACB\ \ \&\ \ \triangle DBC<\triangle ACB$
occurs against the rule: For triangles, exactly one of the following
conditions holds
$\triangle_{1}<\triangle_{2},\ \ \triangle_{1}=\triangle_{2},\ \ \
\triangle_{1}>\triangle_{2}.$
Figure 2: Elements, I.7 – letters $a,b$ added
I.7 On the segment-line $AB$, two segment lines cannot meet at a different
point on the same side of AB.
The proof, atypically, includes no construction as the thesis explicitly
states the impossibility of configuration depicted by the accompanying
diagram. To get a contradiction, Euclid assumes there are two points $C,D$
such that $AC=a=AD$ and $BC=b=BD$ (see Fig. 2). Both triangles $\triangle ACD$
and $\triangle BCD$ are isosceles and share the common base CD. Angles at
their bases are equal, $\alpha=\alpha^{\prime}$ and $\beta=\beta^{\prime}$.
Due to visual evidence, at the vertex $C$, the inequality $\alpha>\beta$
holds, while at $D$, $\beta^{\prime}>\alpha^{\prime}$.777[3] expounds the term
visual evidence in a bigger context. Thus, $\beta^{\prime}>\beta$ and, as
stated earlier, $\beta^{\prime}=\beta$. The very thing is impossible –
clearly, because exactly one of the conditions holds
$\beta^{\prime}<\beta,\ \ \beta^{\prime}=\beta,\ \ \beta^{\prime}>\beta.$
That proof assumes the trichotomy rule for angles and transitivity of greater-
than relation. By modern standards, it is, thus, a total order.888Euclid
applies the phrase “is much greater than” when referring to the transitivity.
In I.8, Euclid literally states the SSS criterion. Since the proof relies on a
superposition of triangles, we propose the following paraphrase: If two
triangles share a common side and have other corresponding sides equal, then
their corresponding angles will also be equal. In I.9–12, it is employed in
that form as Euclid considers two equal triangles on both sides of the common
side. Proof of that modification of I.8 effectively reduces to I.7.
### 3.1 Greater-than and Common Notions
Through §§ 10–11 of [9], Hartshorne seeks to prove Euclid’s propositions
I.1–34 within the Hilbert system, except I.1 and I.23, as they rely on the
circle-circle axiom. He observes that “Euclid’s definitions, postulates, and
common notions have been replaced by the undefined notions, definitions, and
axioms” in the Hilbert system. Commenting on Euclid’s proofs of I.5–8,
Hartshorne writes: “Proposition I.5 and its proof is ok as they stand. […]
every step of Euclid’s proof can be justified in a straightforward manner
within the framework of a Hilbert plane. […] Looking at I.6 […] we have not
defined the notion of inequality of triangles. However, a very slight change
will give a satisfactory proof. […] I.7 […] needs some additional
justification […] which can be supplied from our axioms of betweenness […].
For I.8, (SSS), we will need a new proof, since Euclid’s method of
superposition cannot be justified from our axioms” ([9], 97–99).
The above comparison between Euclid’s and Hilbert’s axiomatic approach
simplifies rather than expounds. Euclid implicitly adopts greater-than
relation between line segments, angles, and triangles as primitive concepts;
similarly to addition and subtraction (a lesser from the greater). In the
previous section, we have shown that he takes transitivity and the trichotomy
law being self-evident. Further characteristics one can recover from his
theory of magnitudes developed in Book V – the only part of Euclid’s geometry
hardly discussed by Hartshorne (see [9], 166–167). Here is a brief account.
Euclidean proportion (for which we adopt the 17th-century symbol $::$) is a
relation between two pairs of geometric figures (megethos) of the same kind,
triangles being of one kind, line segments of another kind, angles of yet
another. Magnitudes of the same kind form an ordered additive semi-group
$\mathfrak{M}=(M,+,<)$ characterized by the five axioms given below ([4], §
3).
1. E1
$(\forall{a,b\in M})(\exists{n\in{\mathbb{N}}})(na>b)$.
2. E2
$(\forall{a,b\in M})(\exists{c\in M})(a>b\Rightarrow a=b+c)$.
3. E3
$(\forall{a,b,c\in M})(a>b\Rightarrow{a+c>b+c})$.
4. E4
$(\forall{a\in M})(\forall{n\in{\mathbb{N}}})(\exists{b\in M})(nb=a)$.
5. E5
$(\forall{a,b,c\in M})(\exists{d\in M})(a:b::c:d),\ \ \mbox{where}\ \
na=\underbrace{a+a+...+a}_{n-times}$.
Clearly, E1–E3 provide extra characteristics of the greater-than relation.
A modern interpretation of Common Notions is simple: CN 1 justifies the
transitivity of congruence of line segments, triangles, and angles, CN 2 and 3
– addition and subtraction in the following form
$a=a^{\prime},\ b=b^{\prime}\Rightarrow a+b=a^{\prime}+b^{\prime},\ \
a-a^{\prime}=b-b^{\prime}.$
The famous CN 5, Whole is greater than the part, allows an interpretation by
the formula $a+b>a$ ([3], 73–76).
In the Hilbert system, the greater-than relation is defined through the
concept of betweeness and refers only to line segments and angles ([9], 85,
95); similarly, addition of line segments and angles are introduced by
definitions ([9], 168, 93). Then counterparts of Euclid’s axioms E2, E3, CN
1–3 are proved as theorems.
Here is a sample argument based on inequalities and its Hilbert-style
counterpart. In I.29, Euclid proves the thesis: When a line falls across
parallel lines $l,\,p$, equality of angles obtains $\alpha=\beta$ (see Fig.
3). For, if they are not equal, one of the angles is greater, suppose
$\alpha>\beta$. Then (implicitly by E3),
$\alpha>\beta\Rightarrow\alpha+\alpha^{\prime}>\beta+\alpha^{\prime}.$
Since $\alpha+\alpha^{\prime}=\pi$, angles $\beta,\,\alpha^{\prime}$ satisfy
the requirement of the parallel axiom, i.e., $\beta+\alpha^{\prime}<\pi$ and
straight lines $l,\,p$ meet, contrary to initial assumption.
Figure 3: Elements, I.29 schematized (left). Hartshorne’s version (right)
In contrast, Hartshorne’s proof of I.29 rests on the axiom stating there is
exactly one line through the point $A$ parallel to $p$ (see Fig. 3). Then, if
$\alpha\neq\beta$, a line $l^{\prime}$ through $A$ making angle $\beta$ with
$n$, by I.27, is parallel to $p$ – it contradicts the uniqueness of a parallel
line through $A$. Euclid’s proof, thus, implies an intersection point of $l$
and $p$, Hartshorne’s – a second parallel line to $p$.
## 4 Existence
### 4.1 Existence via Hilbert axioms
Tables in sections § 1–2 expound grounds for introducing points, namely: a
point is (1) an end of a given line segment, (2) a random point on a line
segment, a straight line, a half-line, or a circle, (3) a circle-circle,
circle-line, or line-line intersection point. Their existence is covered by
Hilbert axioms of Incidence, Betweenness, and Pasch, while circle-circle and
circle-line intersection points require the circle-circle axiom. Propositions
I.7 and I.27 bring in other cases: (4) a random point on the plane, (5) a
vertex of a triangle that exists owning to the definition of parallel lines.
Hilbert’s characterization of a plane does not explain case I.7: the axiom on
the existence of three non-colinear points allows to introduce triangle $ABC$
(see Fig. 2), yet, there are no grounds for point $D$ in the Hibert system.
Similarly, Euclid assumes point $D$ while none of the above rules (1)–(3)
guarantee its existence, and indeed, the very proposition does not include a
construction part. Commenting on I.7, Hartshorne points out the implicit
argument on betweenness, but does not report the suspicious status of $D$ (see
[9], 96, 99).
Ad I.27. Until proposition I.29, Euclid’s arguments do not rely on the
parallel postulate, yet, in I.27, aiming to show $AB\parallel CD$, given that
$\angle AEF=\angle EFD$ (see Fig. 4), he invokes definition of parallel lines:
“Parallel lines are straight-lines which, being in the same plane, and being
produced to infinity in each direction, meet with one another in neither” (I
def. 23).
Figure 4: Elements, I.27 (left). Triangle assumed in the proof (right).
The proof proceeds reductio ad absurdum mode and starts with the claim: “if
not, being produced, AB and CD will certainly meet together”. Suppose, thus,
$AB$ and $CD$ are not parallel and meet in $G$, then, in triangle $EFG$, the
external angle $\angle AEF$ is equal to the internal and opposite angle
$\angle EFD$, but by I.16, $\angle AEF$ is also greater than $\angle EFD$.
Hence, $\angle AEF=\angle EFD$ and $\angle AEF>\angle EFD$. The very thing is
impossible.
The rationale for point $G$ lies in the definition of parallel lines rather
than in construction with straightedge and compass. $G$ is not, and cannot be
determined as an intersection of lines $AB$ and $CD$: straight lines $AB$ and
$CD$ do not meet in either absolute or Euclidean geometry.
Summing up, points $D$, in I.7, and $G$, in I.27, constitute impossible
figures that can not be constructed with a straightedge and compass. Similar
arguments apply to the figure I.39 and those in Book III accompanying
propositions 2, 10, 13, 16, 23, 24. Generally, such impossible figures reveal
Euclid’s struggle with foundational problems: I.39 aims to link the theory of
equal figures and parallel lines, those in Book III – making the foundations
of trigonometry.
### 4.2 Existence and co-exact attributes
Euclid proposition I.1 is a model example for the proponents of diagrammatic
thinking. The enunciation of the proposition introduces point $C$ depicted on
the accompanying diagram through the following phrase “the point C, where the
circles cut one another” (see Fig. 5). It is not the case one has to read off
from the diagram that involved circles intersect. The circle-circle axiom is
not explicitly assumed, but it is not the only tacit supposition of the
Elements. Modern geometry studies how it relates to other axioms and how it
affects the characterization of a plane. The key result in that respect
states: if ${\mathbb{F}}\times{\mathbb{F}}$ is a Cartesian plane over an
ordered field $({\mathbb{F}},+,\cdot,0,1,<)$, the field is closed under the
square operation iff the circle-circle axiom is satisfied on
${\mathbb{F}}\times{\mathbb{F}}$ ([9], 144).
Figure 5: Circles on non-Euclidean and Euclidean planes
Let us consider now Manders’ claim on co-exact attribute as it explicitly
refers to I.1. It reads: “Co-exact attributes are those conditions which are
unaffected by some range of every continuous variation of a specified diagram
[…] or the existence of intersection points such as those required in Euclid
I.1 (which is unaffected no matter how the circles are to some extent
deformed)” ([13], 92).
The key concepts here are deformation and continuous variation. Manders
suggest reference to topological deformations, yet does not provide any
details. Taken literally, they require specification of the plane on which a
diagram lays. Instead of any deformation, we consider diagram I.1 on three
planes: ${\mathbb{Q}}\times{\mathbb{Q}}$, ${\mathbb{R}}\times{\mathbb{R}}$,
and ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$ (see Fig.
5).999${\mathbb{R}}^{*}$ is the set of hyperreals (see § 6 below); circles
with radius $1$ have centers at $(0,0)$ and $(0,1)$. The second and third are
models of the Euclidean planes as fields of real and hyperreal numbers are
closed under the square root operation, while the first is not a Euclidean
plane. Due to calculations, we get to know that on the plane
${\mathbb{Q}}\times{\mathbb{Q}}$ the intersection point of circles does not
exist. However, from the cognitive perspective, one and the same diagram
represents circles in these three various mathematical contexts.101010As for
${\mathbb{Q}}\times{\mathbb{Q}}$, we get sure that circles look like
continuous objects due to a technical result such as [15]. Thus, with no
deformation of the diagram, but by switching from one mathematical context to
another we get or not that the intersection point of the circles exists.
## 5 Inequalities and co-exact attributes
In his analysis of proposition I.6, Manders interprets Euclid’s argument
$\triangle DBC$ is smaller than $\triangle ABC$ in terms of part-whole
relation read-off from the diagram ([13], 109–110). On another occasion, he
explicitly writes: “A strict inequality may be reduced […] to a proper-part
relationship in the diagram” ([13], 112).
In [3], we provide a detailed analysis of Common Notion 5 showing it does not
reduce to part-whole relation. In short, logical analysis gives the following
formula for CN5: $a+b>a$, where $a+b$ stands for the whole and $b$ – for its
part, and $a+b,\,a,\,b$ have to be of the same kind (triangles or angles).
Given that interpretation, triangle $ADB$ is not a Euclidean part of triangle
$ACB$; similarly, the gray angle is not a Euclidean part of the angle $CAB$,
as represented in Fig. 6.
Figure 6: Non-Eculidean parts
Euclid’s I.32 sets another challenge for the diagrammatic philosophy. Fereirós
reads it out as follows: “Most of the proof steps involve exact information,
dealing with equalities of angles, and they depend on previous theorems (I.31,
I.29) and common notions 2. Only two steps involve co-exact information,
bringing in attributions read from the diagram”, namely, that $\angle ACD$
covers $\angle AGE$ and $\angle EGD$ “is co-exact because it has to do with
part-whole relation and it is not affected by deformation” ([6], 135–136).
Here is our interpretation (see Fig. 7). Euclid transports angle $\alpha$ to
point $C$, and draws $CE$, which, by I.27, is parallel to $AB$. Hence, by
I.29, $\angle ECD=\beta$, and angles at $C$ sum up to “two right angles”,
$\gamma+\alpha+\beta=\pi$.
Figure 7: Proof of I.32 schematized.
Now, $CE$ lies inside angle $ACD$ because, by I.17, $\gamma+\alpha<\pi$. It is
not the case, thus, that the only way to get that information is to read it
from the diagram.
## 6 Existence meets Inequalities
### 6.1 Semi-Euclidean plane
In this section, we present a model o semi-Euclidean plane, i.e., a plane in
which angles in a triangle sum up to $\pi$ yet the parallel postulate fails.
[9], p. 311, introduces that term, but the very idea originates in Max Dehn’s
1900 [5], § 9, which built such a model owning to a non-Archimedean
Pythagorean field. Dehn explored a non-Euclidean field introduced already in
Hilbert [11], § 12.111111See also [9], § 18. Example 18.4.3 expounds Dehn’s
model. We employ the Euclidean field of hyperreal numbers. On the Cartesian
plane over hyperreals, the circle-circle and circle-line intersection axioms
are satisfied, meaning one can mirror Euclid’s straightedge and compass
constructions. To elaborate, let us start with the introduction of the
hyperreal numbers.
An ordered field $({\mathbb{F}},+,\cdot,0,1,<)$ is a commutative field
together with a total order that is compatible with sums and products. In such
a field, one can define the following subsets of ${\mathbb{F}}$:
1. $\mathbb{L}=\\{x\in{\mathbb{F}}:(\exists n\in{\mathbb{N}})(|x|<n)\\}$,
2. $\Psi=\\{x\in{\mathbb{F}}:(\forall n\in{\mathbb{N}})(|x|>n)\\}$,
3. $\Omega=\\{x\in{\mathbb{F}}:(\forall n\in{\mathbb{N}})(|x|<\tfrac{1}{n})\\}$.
They are called limited, infinite, and infinitely small numbers, respectively.
Here are some relationships helpful to pursue our arguments.
1. $(\forall x,y\in\Omega)(x+y\in\Omega,xy\in\Omega)$,
2. $(\forall x\in\Omega)(\forall y\in\mathbb{L})(xy\in\Omega)$,
3. $(\forall x\neq 0)(x\in\Omega\Leftrightarrow\ x^{-1}\in\Psi)$.
To clarify our account, let us observe the following equality $\Omega=\\{0\\}$
is a version of the well-known Archimedean axiom. Since real numbers form the
biggest Archimedean field, every field extension of
$({\mathbb{R}},+,\cdot,0,1,<)$ includes positive infinitesimals.
Let $\mathcal{U}$ be a non-principal ultrafilter on ${\mathbb{N}}$. The set of
hyperreals is defined as a reduced product
${\mathbb{R}}^{*}={\mathbb{R}}^{{\mathbb{N}}}/{\mathcal{U}}$. Sums, products,
and the ordered are introduced pointwise. A reader can take for granted that
the field of hyperreals $({\mathbb{R}}^{*},+,\cdot,0,1,<)$ extends real
numbers, hence, includes infinitesimals and infinite numbers; moreover, it is
closed under the square root operation (see [2], [1]). Fig. 8 represents in a
schematized way a relationship between ${\mathbb{R}}$ and ${\mathbb{R}}^{*}$,
as well as between $\mathbb{L}$, $\Psi$, and $\Omega$.
Figure 8: The line of real numbers and its extension to hyperreals
Due to the proposition 16.2 ([9], 144), the Cartesian plane over the field of
hyperreals is a model of Euclidean plane, with straight lines and circles
given by equations $ax+by+c=0$, $(x-a)^{2}+(y-b)^{2}=r^{2}$, where
$a,b,c,r\in{\mathbb{R}}^{*}$; angles between straight lines are defined as in
the Cartesian plane over the field of real numbers. Specifically, on the plane
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$, angles in triangles sum up to $\pi$.
Parallel lines are of the form $y=mx+b$ and $y=mx+c$, while a perpendicular to
the line $y=mx+b$ is given by the formula $y=-\frac{1}{m}x+d$.
Now, take us a subspace $\mathbb{L}\times\mathbb{L}$ of the plane
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$. On that plane, circles are defined
by analogous formula, namely $(x-a)^{2}+(y-b)^{2}=r^{2}$, where
$a,b,r\in\mathbb{L}$, while every line in $\mathbb{L}\times\mathbb{L}$ is of
the form $l\cap\mathbb{L}\times\mathbb{L}$, where $l$ is a line in
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$. Since we want plane
$\mathbb{L}\times\mathbb{L}$ include lines such as $y_{1}=\varepsilon x$,
where $\varepsilon\in\Omega$, it has also to include the perpendicular
$y_{2}=\frac{-1}{\varepsilon}x$, but $\frac{-1}{\varepsilon}\notin\mathbb{L}$.
Formula $l\cap\mathbb{L}\times\mathbb{L}$, where $l=ax+by+c$ and
$a,b,c\in{\mathbb{R}}^{*}$ guarantees the existence of the straight line
$y_{2}$ in $\mathbb{L}\times\mathbb{L}$. Finally, the interpretation of angle
is the same as in the model ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$.
Figure 9: Perpendicular lines with infinitesimal and infinitely large slopes
Explicit checking shows that the model characterized above satisfies all
Hilbert axioms of non-Archimdean plane geometry plus the circle-circle and
line-circle axioms, except parallel axiom; the more general theorem concerning
Hilbert planes also justifies our model, namely [9], p. 425, theorem, 43.7
(a).
Figure 10: Non-Euclidean plane $\mathbb{L}\times\mathbb{L}$ (left) vs.
Euclidean plane ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$ (right)
With regard to parallel lines, let us consider the horizontal line $y=1$ and
two specific lines through $(0,0)$, $y_{1}=\varepsilon x,y_{2}=\delta x$,
where $\varepsilon,\delta\in\Omega$. (see Fig. 10). Since
$\Omega\mathbb{L}\subset\Omega$, the following inclusions hold
$y_{1},y_{2}\subset\mathbb{L}\times\Omega$. In other words, values of maps
$y_{1}(x),y_{2}(x)$ are infinitesimals, given that $x\in\mathbb{L}$. The same
obtains for any line of the form $y=\mu x$, with $\mu\in\Omega$. Since there
are infinitely many infinitesimals, there are infinitely many lines through
$(0,0)$ not intersecting the horizontal line $y=1$.
Since every triangle in $\mathbb{L}\times\mathbb{L}$ is a triangle in
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$, it follows that angles in a triangle
on the plane $\mathbb{L}\times\mathbb{L}$ sum up to $\pi$ (see Fig. 11).
Figure 11: Triangles in Euclidean plane
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$ and its subspace
$\mathbb{L}\times\mathbb{L}$
### 6.2 Exact attributes on the semi-Euclidean plane
Let us go back to Manders’ claim “Exact attributes are those which, for at
least some continuous variation of the diagram, obtain only in isolated cases;
paradigmatically, straightness of lines or equality of angles (neither of
which survive any except exceptional types of deformation, no matter how
small)”.
In Fig. 10, points $A=(\frac{1}{\varepsilon},1)$, $B=(\frac{1}{\delta},1)$ are
intersections of $y_{1},y_{2}$ with the horizontal line $y=1$ on the plane
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$. Vertical line $x=0$ falls on $y=1$
and $y_{1}$ making internal angles less than $\pi$. Switching from
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$ to $\mathbb{L}\times\mathbb{L}$, we
do not modify straightness of lines or equality of angles as far as it regards
the diagram on $\mathbb{L}\times\mathbb{L}$, however, there is nothing in the
diagram which informs us whether they intersect or not. Knowing plane
characteristics, we can infer they intersect, given that being on
${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$. The clue is the diagrammatic
philosophy does not impose any restrictions on Euclid’s clause “being produced
to infinity”. As focused on a diagram, it does not have means to decide
whether lines meet beyond a diagram.
Possibly, the claim on straightness of lines or equality of angles is designed
to exclude hyperbolic representations of lines and angles. Indeed, in the
Poincare model, straight lines are circles and angles are determined between
tangents; in the Klein model, straight lines are straight, while angles are
retrieved from the Poincare model. Both models then change the diagrammatic
representation of straight lines and angles. In the plane
$\mathbb{L}\times\mathbb{L}$, straight-lines are usual straight-lines, angles
are usual angles; moreover, triangles are Euclidean, there are also squares
and usual perpendicular lines. Yet, there are no instruments in the
diagrammatic tool-box to reveal a non-Euclidean character of that plane. It is
because diagrammatic perspective is local, while the phenomenon of parallelism
requires a global perspective.
## References
* [1] Błaszczyk, P.: Galileo’s paradox and numerosities. Zagadnienia Filozoficzne w Nauce 70, 73–107 (2021).
* [2] Błaszczyk, P.: A Purely Algebraic Proof of the Fundamental Theorem of Algebra. AUPC 8, 6–22 (2016); https://didacticammath.up.krakow.pl/article/view/3638
* [3] Błaszczyk, P. Mrówka, K., Petiurenko, A.: Decoding Book II of the Elements. AUPC 12, 39–88 (2020); https://didacticammath.up.krakow.pl/article/view/8462
* [4] Błaszczyk, P., Petiurenko, A.: Euclid’s proportion revised. AUPC 11, 37–61 (2019). https://didacticammath.up.krakow.pl/index.php/aupcsdmp/article/view/6901
* [5] Dehn, M.: Legendre’schen Sätze über die Winkelsumme im Dreieck. Mathematische Annalen 53(3), 404–439 (1900).
* [6] Ferreirós, J.: Mathematical Knowledge and the Interplay of Practices. Princeton University Press, Princeton (2016).
* [7] Fitzpatrick, R.: Euclid’s Elements of Geometry translated by R. Fiztpatrick (2007); http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf
* [8] Greenberg, M.: Euclidean and non-Euclidean Geometries. Freeman, N York (2008).
* [9] Hartshorne, R.: Geometry: Euclid and Beyond. Springer, New York (2000).
* [10] Heiberg, J.: Euclidis Elementa. Vol. I. Teubneri, Lipsiae (1883).
* [11] Hilbert, D.: Grundlagen der Geometrie. Festschrift Zur Feier Der Enthüllung Des Gauss-Weber-Denkmals in Göttingen. Teubner, Leipzig (1899), 1–92.
* [12] Hilbert, D. Über den Zahlbegriff. Jahresbericht der Deutschen Mathematiker-Vereinigung 8, 180–184 (1900).
* [13] Manders, K.: The Euclidean Diagram. In: P. Mancosu (ed.): Philosophy of Mathematical Practice, Oxford University Press, Oxford (2008), 80–136.
* [14] Martin, G.: Geometric Constructions. Springer, Berlin (1998).
* [15] Tan, L.: The group of rational points on the unit circle. Mathematics Magazine 69(3), 162–171 (1996).
|
KDK collaboration
# Precision measurement of 65Zn electron-capture decays with the KDK
coincidence setup
L. Hariasz P.C.F. Di Stefano<EMAIL_ADDRESS>M. Stukel Department of
Physics, Engineering Physics & Astronomy, Queen’s University, Kingston,
Ontario K7L 3N6, Canada B.C. Rasco K.P. Rykaczewski Physics Division, Oak
Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA N.T. Brewer
Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831,
USA Joint Institute for Nuclear Physics and Application, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA R.K. Grzywacz Physics Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Joint
Institute for Nuclear Physics and Application, Oak Ridge National Laboratory,
Oak Ridge, Tennessee 37831, USA Department of Physics and Astronomy,
University of Tennessee, Knoxville, Tennessee 37996, USA E.D. Lukosi
Department of Nuclear Engineering, University of Tennessee, Knoxville,
Tennessee 37996, USA Joint Institute for Advanced Materials, University of
Tennessee, Knoxville, Tennessee 37996, USA D.W. Stracener Physics Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA M. Mancuso F.
Petricca Max-Planck-Institut für Physik, Munich D-80805, Germany J. Ninkovic
P. Lechner MPG Semiconductor Laboratory, Munich D-80805, Germany
###### Abstract
65Zn is a common calibration source, moreover used as a radioactive tracer in
medical and biological studies. In many cases, $\gamma$-spectroscopy is a
preferred method of 65Zn standardization, which relies directly on the
branching ratio of $J\pi(\mbox{${}^{65}$Zn})=5/2^{-}\rightarrow
J\pi(\mbox{${}^{65}$Cu})=5/2^{-}$ via electron capture (EC∗). We measure the
relative intensity of this branch to that proceeding directly to the ground
state (EC0) using a novel coincidence technique, finding
$\mbox{$I_{\text{EC}^{0}}$}/\mbox{$I_{\text{EC}^{*}}$}=\mbox{$0.9684\pm
0.0018$}$. Re-evaluating the decay scheme of 65Zn by adopting the commonly
evaluated branching ratio of $\mbox{$I_{\beta^{+}}$}=1.4271(7)\%$ we obtain
$\mbox{$I_{\text{EC}^{*}}$}=\mbox{$(50.08\pm 0.06)\%$}$, and
$\mbox{$I_{\text{EC}^{0}}$}=\mbox{$(48.50\pm 0.06)\%$}$. The associated
$1115\text{\,}\mathrm{keV}$ gamma intensity agrees with the previously
reported NNDC value, and is now accessible with a factor of $\sim$2 increase
in precision. Our re-evaluation removes reliance on the deduction of this
gamma intensity from numerous measurements, some of which disagree and depend
directly on total activity determination. The KDK experimental technique
provides a new avenue for verification or updates to the decay scheme of 65Zn,
and is applicable to other isotopes.
###### Contents
1. I Introduction
2. II Methods
1. II.1 Apparatus and Visible Features
2. II.2 Physical Processes and Likelihood
3. III Results
4. IV Conclusions
5. A Components of the 65Zn Decay Scheme and Re-evaluated Branching Ratios
6. B Likelihood details
1. B.1 Coincidence sorting
2. B.2 Physical quantities
7. C Systematic errors
## I Introduction
65Zn is a common $\gamma$-ray calibration source [1], and for nearly a century
has been applied in the fields of medicine and biology as a radioactive tracer
[2, 3, 4]. It has been applied in various studies [5, 6] including an
investigation of potential orbital-modulation effects on decay constants [7].
In many applications, $\gamma$-ray spectroscopy is a convenient avenue for
activity determination of 65Zn, which is an emitter of essentially
monoenergetic $\gamma$ rays ($1115\text{\,}\mathrm{keV}$) associated with some
of its electron capture (EC) decays. This technique relies on knowledge of the
absolute $1115\text{\,}\mathrm{keV}$ intensity (fraction of decays that emit
$1115\text{\,}\mathrm{keV}$ $\gamma$s), available from existing decay scheme
evaluations such as those by the Laboratoire National Henri Becquerel (LNHB)
[8] or National Nuclear Data Center (NNDC) [9]. Though both evaluations report
$\mbox{$\mathrm{I}($1115\text{\,}\mathrm{keV}$)$}\sim 50\%$ with relative
errors of $\sim 0.2\%$, values (Table 1) deviate by $\sim 0.4\%$ between the
two sources. These evaluations combine dedicated measurements from the
Euromet-721 exercise [10, 11] with other reported values (e.g. from Refs. [12,
13]). All such determinations of absolute intensity are directly reliant on
activity measurements, which may require various corrections; though the
commonly used $4\pi\beta\gamma$ technique is influenced by low-energy X and
Auger radiation when used with EC-decaying nuclides [14], such corrections
have only been applied in some 65Zn studies [11].
| $\mathrm{I}($1115\text{\,}\mathrm{keV}$)$ (%)
---|---
LNHB (2006) [8] | $50.22\pm 0.11$
NNDC (2010) [9] | $50.04\pm 0.10$
Table 1: Absolute intensities of the 65Zn $1115\text{\,}\mathrm{keV}$
$\gamma$-ray, as reported in decay scheme evaluations of the Laboratoire
National Henri Becquerel and National Nuclear Data Center.
In this work, we present a novel measurement and resulting alternate, precise
determination of $\mathrm{I}($1115\text{\,}\mathrm{keV}$)$ through re-
evaluation of the decay scheme of 65Zn. Using data from the KDK experiment
[15, 16], $1115\text{\,}\mathrm{keV}$-producing electron capture decays (EC∗)
of 65Zn are distinguished from those proceeding directly to the ground state
(EC0) of its Cu daughter. The KDK collaboration has recently employed this
technique to obtain the first measurement of the exceedingly rare EC0 decay of
40K [17, 18]. The measurement of this work obtains a ratio of 65Zn
intensities, $\rho=\mbox{$I_{\text{EC}^{0}}$}/\mbox{$I_{\text{EC}^{*}}$}$. The
decay scheme of 65Zn, evaluated with our measurement, is displayed in Fig. 1.
Figure 1: The decay scheme of 65Zn. Branching ratios are calculated combining
our measurement of
$\rho=\mbox{$I_{\text{EC}^{0}}$}/\mbox{$I_{\text{EC}^{*}}$}$ with the shown,
commonly evaluated value of $I_{\beta^{+}}$ [8, 9] and assuming these three
branches complete the decay scheme
($1=\mbox{$I_{\text{EC}^{*}}$}+\mbox{$I_{\text{EC}^{0}}$}+\mbox{$I_{\beta^{+}}$}$).
Gamma energies are from Ref. [19] ($1115.539(2)\text{\,}\mathrm{keV}$), and
adopted $\gamma$ levels [9] ($770.6(2)\text{\,}\mathrm{keV}$). Intermediate
$\gamma$ intensities are from Ref. [10]. Total half-life is from Ref. [9], and
$\text{Q}_{\text{EC}^{0}}$ is from AME 2020 [20].
## II Methods
The KDK apparatus was designed for the discrimination between electron capture
branches proceeding through an excited daughter state to those transitioning
directly to the ground state. Below, we briefly outline the experimental
setup, which has been fully characterized elsewhere [16], with focus on
features visible in the 65Zn data and subsequently all physical processes
relevant for obtaining a measurement of $I_{\text{EC}^{0}}$/
$I_{\text{EC}^{*}}$.
### II.1 Apparatus and Visible Features
An inner Silicon Drift Detector (SDD) tags X-rays accompanying the source EC
decays of interest, as shown in Fig. 2. Due to its excellent resolution (FWHM
of $200\text{\,}\mathrm{eV}$ at $8\text{\,}\mathrm{keV}$), the Cu Kα
($8\text{\,}\mathrm{keV}$) and Kβ ($8.9\text{\,}\mathrm{keV}$) X-rays of
interest are easily distinguishable. In the same region, the majority of
remaining counts are attributed to K Augers from the same electron captures,
and a flat background originating primarily from the source $\beta^{+}$
branch. Below the signal (Cu) peaks, a small contribution from Zn Kα X-ray
fluorescence ($8.6\text{\,}\mathrm{keV}$) is included when fitting SDD
spectra, as shown further. Moreover, due to the low noise threshold of the
detector, L-shell X-rays (Cu, or fluoresced Zn) are visible near
$0.9\text{\,}\mathrm{keV}$.
Figure 2: SDD spectrum obtained with the 65Zn source over 1.4 days. The main
features of interest are Cu Kα and Kβ X-rays at 8 and 9 keV corresponding to
65Zn electron captures. Some L-shell X-rays are visible below
$1\text{\,}\mathrm{keV}$. A fit to SDD data distinguishes components near the
K X-ray peaks in Fig. 5.
The 65Zn source and SDD are centered inside a large, outer Modular Total
Absorption Spectrometer (MTAS) as depicted in Fig. 3. MTAS is an extremely
efficient NaI(Tl) $\gamma$-tagger, originally designed to study
$\beta$-strength distributions of fission products [21, 22]. For the 65Zn
$\gamma$ of interest ($1115\text{\,}\mathrm{keV}$), MTAS boasts a $\sim 98\%$
tagging efficiency [16].
Figure 3: Schematic of the Silicon Drift Detector
($\mathcal{O}(${\mathrm{mm}}^{2}$)$) and source placed inside the Modular
Total Absorption Spectrometer ($\mathcal{O}(${\mathrm{m}}^{2}$)$; cross-
section displayed). The latter contains a central module along with inner,
middle and outer rings, each containing NaI(Tl) volumes coupled to PMTs.
Several processes are visible in 65Zn MTAS spectra, such as those in Fig. 4.
Fully collected $1115\text{\,}\mathrm{keV}$ $\gamma$s from 65Zn EC∗ decays
form the most-prominent peak, though others originating from the natural MTAS
background along with source+background and source+source sum-peaks are
visible. The data is fit with calibrated, simulated spectra of 65Zn $\gamma$s
and $\beta^{+}$ in MTAS, along with a measured MTAS background and
convolutions. The shape of each spectrum is fixed, while the integral is
allowed to vary. Such spectral analysis verifies simulation methods used to
obtain MTAS efficiencies, and has been used to explore the source-SDD geometry
(Ref. [16] and App. B.2).
Figure 4: MTAS events of the 65Zn run in a
$2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ coincidence window with a SDD
trigger. The dominant component is the $1115\text{\,}\mathrm{keV}$ $\gamma$
spectrum, associated with the 65Zn EC decays of interest. The measured MTAS
background, and its convolution with the $1115\text{\,}\mathrm{keV}$ spectrum
are included. The $\beta^{+}$ spectrum and a $\gamma+\gamma$ convolution
provide additional contributions. Simulated $\gamma$ and $\beta^{+}$ spectra
are calibrated to data in energy and resolution.
SDD data, including the K X-ray electron-capture signal of interest, are
categorized by coincidence with MTAS, generally using a
$t^{\prime}=\mbox{$2\text{\,}\mathrm{\SIUnitSymbolMicro s}$}$ nominal
coincidence window (CW). Details of coincidence characterization are available
in our previous work [16]. The sorted SDD data is used to inform the parameter
of interest, $\rho$, as described in the following section.
### II.2 Physical Processes and Likelihood
The main result is obtained through a simultaneous minimization of the Baker-
Cousins likelihood [23] on coincident and anti-coincident SDD spectra,
$-\ln\mathcal{L}=\sum_{i}\left\\{f_{i}(\bm{\theta})-n_{i}+n_{i}\ln\left[\frac{n_{i}}{f_{i}(\bm{\theta})}\right]\right\\},$
(1)
which compares the total observed events ($n_{i}$) in a bin ($i$) to the
corresponding model-predicted events ($f_{i}$). The parameters $\bm{\theta}$
include that of interest, $\rho$, along with various fixed and free terms
pertaining to the spectra.
Such a simultaneous fit to data sorted using a
$2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ CW is shown in Fig. 5. The main
features are the Gaussian Cu Kα and Kβ peaks of fixed means near 8 and
$9\text{\,}\mathrm{keV}$. An additional X-ray peak near
$8.6\text{\,}\mathrm{keV}$ corresponds to fluoresced source Zn Kαs. Lastly, an
ad hoc component consists of a wide Gaussian term of free mean and width
describing Cu K Augers, and a flat term attributed primarily to source
$\beta^{+}$. The shape of all components is shared across coincident and anti-
coincident spectra, whereas the integral of each can vary.
Figure 5: A fit to 65Zn SDD data sorted using a
$2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ coincidence window with MTAS. Cu Kα
and Kβ peaks from 65Zn electron captures inform
$\rho=\mbox{$I_{\text{EC}^{0}}$}/\mbox{$I_{\text{EC}^{*}}$}$. Fluoresced Zn Kα
and a continuous component consisting of Augers and flat background complete
the fit.
To obtain $\rho$, all processes affecting SDD signal detection and coincident
categorization must be considered. To assess the former, we consider expected
true, detected signal counts from the EC∗ branch ($\sigma^{*}$) along with
those from the EC0 branch ($\sigma^{0}$),
$\displaystyle\sigma^{*}$
$\displaystyle=\mathcal{N}\mbox{$I_{\text{EC}^{*}}$}P_{K}^{*}\omega_{K}(1-\eta_{\gamma})\eta$
$\displaystyle\sigma^{0}$
$\displaystyle=\mathcal{N}\mbox{$I_{\text{EC}^{0}}$}P_{K}^{0}\omega_{K}\eta.$
(2)
Both expressions above require the production of a Cu K X-ray and its
successful detection in the SDD. To first order, the total source decays
throughout data collection ($\mathcal{N}$), fluorescence probability
($\omega_{K}$), and SDD tagging efficiency of Cu K X-rays ($\eta$) do not need
to be known to inform $\rho$,
$\displaystyle\rho$
$\displaystyle=\frac{\mbox{$I_{\text{EC}^{0}}$}}{\mbox{$I_{\text{EC}^{*}}$}}$
$\displaystyle=\frac{\sigma^{0}}{\sigma^{*}}\frac{P_{K}^{*}}{P_{K}^{0}}(1-\eta_{\gamma}).$
(3)
However, K-shell capture probabilities ($P_{K}$) differ between the two
electron capture branches, and in the case of EC∗\- originating signal, we
require that the associated de-excitation gamma is not also seen in the SDD
($1-\eta_{\gamma}$). K-capture probabilities are obtained from BetaShape V2.2
[24], and $\eta_{\gamma}$ is obtained combining a measured geometric
efficiency with simulations (App. B.2). Above, internal conversion, internal
pair formation, and internal Bremsstrahlung have been neglected as these sub-
dominant processes are negligible at our precision.
The expected true signal counts are related to the observed, sorted signal
counts (those in the Cu K X-ray peaks of Fig. 5) by accounting for any process
affecting coincidence sorting. One such effect stems from the imperfect MTAS
$\gamma$-tagging efficiency $\epsilon$, which primarily affects sorting of
EC∗-originating signal. This efficiency has been characterized for multiple
coincidence windows by scaling measured 54Mn efficiencies in energy through
simulations, and correcting for deadtime [16], yielding a value of
$97.93(6)\%$ at the $2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ CW.
Moreover, any process detectable in MTAS can occur in spurious coincidence
with the Cu SDD signal. The natural MTAS background rate is accounted for by
considering a Poisson probability $P^{0}_{\mathrm{BG}}=\exp(-\beta\bar{t})$ of
no natural MTAS background events within the effective coincidence window
$\bar{t}$, along with an analagous probability of no MTAS-detected source
$1115\text{\,}\mathrm{keV}$ $\gamma$s ($P^{0}_{\gamma}$). Expected MTAS
background coincidence rates have been obtained elsewhere [16], and
$P^{0}_{\gamma}$ is obtained in App. B. An additional source of coincidences
with source $\beta^{+}$ has been explored and deemed negligible. All spurious
coincidences are proportional to the $\mathcal{O}($\mathrm{\SIUnitSymbolMicro
s}$)$ CW, and, most significantly, place some EC0-originating signal in the
coincident spectrum. Neglecting these terms thus tends to underestimate
$\rho$, with increasingly dramatic effects at larger CWs.
With the above MTAS gamma efficiency and spurious coincidence considerations,
expected coincident ($\Sigma^{*}$) and anti-coincident ($\Sigma^{0}$) signal
counts are obtained,
$\displaystyle\Sigma^{*}$
$\displaystyle=\frac{\nu}{1+\rho^{\prime}}\biggl{[}\epsilon+(1-P^{0}_{\mathrm{BG}}P^{0}_{\gamma})(1-\epsilon+\rho^{\prime})\biggr{]}$
$\displaystyle\Sigma^{0}$
$\displaystyle=\frac{\nu}{1+\rho^{\prime}}P^{0}_{\mathrm{BG}}P^{0}_{\gamma}(1-\epsilon+\rho^{\prime}),$
(4)
with more detail available in App. B. Above, $\nu=\sigma^{*}+\sigma^{0}$ are
total signal events and $\rho^{\prime}=\sigma^{0}/\sigma^{*}$ is introduced
for simplicity ($\rho^{\prime}\propto\rho$ via Eq. (3)). These expressions for
expected signal counts are inserted directly into the model $f$ of Eq. (1) as
the integrals of Gaussian Cu K X-rays, such that $\rho$ is obtained directly
from the fit along with its statistical error.
## III Results
With the likelihood method described above, we obtain
$\rho=\mbox{$I_{\text{EC}^{0}}$}/\mbox{$I_{\text{EC}^{*}}$}=\mbox{$0.9684\stackrel{{\scriptstyle\mathrm{stat}}}{{\pm}}0.0013\stackrel{{\scriptstyle\mathrm{syst}}}{{\pm}}0.0013$},$
(5)
using the $2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ dataset. This result, and
those obtained using 1 and $4\text{\,}\mathrm{\SIUnitSymbolMicro s}$ CWs are
shown in Fig. 6. Our values of $I_{\text{EC}^{0}}$/$I_{\text{EC}^{*}}$ agree
across coincidence windows, and we note that these measurement uncertainties
are correlated. We also report values derived from branching ratios in
existing NNDC [9] and LNHB [8] evaluations.
Figure 6: Measurements of
$\rho=\mbox{$I_{\text{EC}^{0}}$}/\mbox{$I_{\text{EC}^{*}}$}$ obtained at three
different coincidence windows (red points) are compared to expected values
based on most-recent LNHB [8] and NNDC [9] evaluations. The uncertainties on
our measurements are correlated.
The systematic errors considered in this analysis generally fall into two
categories: (1) physical limitations (finitely known quantities such as
tagging efficiency; App. B.2) and (2) spectral characteristics (e.g. fit
range). The former (1) are accounted for analytically using equations of Sec.
II.2. All physical parameters are varied simultaneously to account for
covariances, and are assumed to follow Gaussian distributions. The latter (2)
are gauged by performing numerous fits, such as that in Fig. 5, while randomly
varying the binning and fit range. The final systematic error of $1.3\times
10^{-3}$ sums those of category (1) and (2) in quadrature, and within rounding
happens to be equivalent to the statistical error. We find that the physical
systematics completely dominate any spectral effects, with the leading
contribution stemming from the error on MTAS $\gamma$-tagging efficiency.
Details of these considerations are provided in App. C, and contributions of
individual sources of systematic error are summarized in Table 6.
We re-evaluate the decay scheme of 65Zn (Fig. 1) by combining our result of
$I_{\text{EC}^{0}}$/ $I_{\text{EC}^{*}}$ with the commonly evaluated branching
ratio $\mbox{$I_{\beta^{+}}$}=1.421(7)\%$ [9, 8], and assuming these three
branches complete the decay scheme. Our result is not sensitive to internal
conversion and pair formation, internal Bremsstrahlung, de-excitation through,
or EC to the intermediate $770\text{\,}\mathrm{keV}$ Cu level. The evaluated
$I_{\text{EC}^{*}}$ is thus equivalent to the absolute
$1115\text{\,}\mathrm{keV}$ $\gamma$ intensity.
With our re-evaluation, we find an improvement in sensitivity to the
$1115\text{\,}\mathrm{keV}$ branching ratio of almost a factor of 2 relative
to that obtained by the Euromet exercise [10, 11]. We compare this result of
$\mbox{$\mathrm{I}($1115\text{\,}\mathrm{keV}$)$}=50.08(6)\%$ to various most-
recent measurements in Fig. 7, with values listed in Table 3. An alternate re-
evaluation combining the result of this work with a theoretical
$\mbox{$I_{\text{EC}^{0}}$}/\mbox{$I_{\beta^{+}}$}$ ratio from Betashape V2.2
[24] yields highly consistent results (Table 2).
Figure 7: Selected measured absolute intensities of the 65Zn
$1115\text{\,}\mathrm{keV}$ gamma-ray (blue points) along with the re-
evaluation of this work (dark red line). Individual measurements [25, 13, 12]
and a collective result from Euromet [11, 10] are shown, with values available
in Table 3. Additional measurements are listed in the evaluations of Refs. [8,
9].
## IV Conclusions
We have re-determined the absolute emission intensity of
$1115\text{\,}\mathrm{keV}$ $\gamma$s originating from 65Zn electron-capture
decays, improving the available precision by a factor of $\sim$2, with a
result that agrees with the existing National Nuclear Data Center evaluation.
This improvement stems from precise determination of a ratio of electron-
capture decay intensities, which does not require precise knowledge of source
activity. As in other experiments, the efficiency of our $\gamma$-tagger is a
limiting systematic, though this precisely known quantity yields a systematic
error roughly equivalent to our statistical uncertainty. Additionally, our
determination is the first involving the total EC0 branching ratio of 65Zn.
Our successful measurement is analogous to an existing result for 40K, and may
be applied to other nuclides which decay through multiple modes of electron
capture. The unique detector configuration of the KDK experiment provides a
precise result available for use in the decay scheme evaluation of the
familiar and vastly used 65Zn.
###### Acknowledgements.
We thank Xavier Mougeot and Sylvain Leblond of LNHB for input on 65Zn. Work
was performed at Oak Ridge National Laboratory, managed by UT-Battelle, LLC,
for the U.S. Department of Energy under Contract DE- AC05-00OR22725. The
United States Government retains and the publisher, by accepting the article
for publication, acknowledges that the United States Government retains a non-
exclusive, paid-up, irrevocable, world-wide license to publish or reproduce
the published form of this manuscript, or allow others to do so, for United
States Government purposes. The Department of Energy will provide public
access to these results of federally sponsored research in accordance with the
DOE Public Access Plan (http://energy.gov/downloads/doe- public-access-plan).
US support has also been supplied by the Joint Institute for Nuclear Physics
and Applications, and by NSF grant EAR-2102788. Funding in Canada has been
provided by NSERC through a SAPIN grant, as well as by the Faculty of Arts and
Science of Queen’s University, and by the McDonald Institute.
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## Appendix A Components of the 65Zn Decay Scheme and Re-evaluated Branching
Ratios
The decay scheme of 65Zn (Fig. 1) can be constructed either wholly
empirically, or with a combination of measurements and theoretical values.
Prior to this work, the purely empirical evaluation relied on intensity
measurements of $1115\text{\,}\mathrm{keV}$ and $511\text{\,}\mathrm{keV}$
$\gamma$s associated with 65Zn.
The $1115\text{\,}\mathrm{keV}$ $\gamma$ intensity is equivalent to the
branching ratio to the $1115\text{\,}\mathrm{keV}$ Cu state
($I_{\text{EC}^{*}}$) assuming a simplified scheme where (1) the intermediate
344, 770 keV $\gamma$ emissions are negligible, and (2) internal conversion is
negligible. The order of precision for $I_{\text{EC}^{*}}$ from both previous
evaluations [8, 9] is $\sim\mathcal{O}(10^{-3})$, and the
$\mathcal{O}(10^{-5})$ intensities of intermediate $\gamma$s [10] along with
the $\mathcal{O}(10^{-4})$ internal conversion process (BrIcc program [26])
can be neglected. Radiative electron capture (REC) may accompany both the EC0
and EC∗ 65Zn decays, resulting in internal Bremsstrahlung emission. However,
for these allowed transitions, the frequency of REC relative to non-REC decays
is of the order of $10^{-4}$ [27] and can be neglected in the determination of
$I_{\text{EC}^{*}}$.
The branching ratio through $\beta^{+}$ disintegration ($I_{\beta^{+}}$) is
informed from measured $511\text{\,}\mathrm{keV}$ intensities obtained taking
annihilation-in-flight [28] into account. The decay scheme is then built with
$1=\mbox{$I_{\text{EC}^{*}}$}+\mbox{$I_{\text{EC}^{0}}$}+\mbox{$I_{\beta^{+}}$},$
(6)
which yields the final required branching ratio ($I_{\text{EC}^{0}}$). To re-
evaluate the scheme with the result of this work, the above assumption of
unitarity is maintained and the measured parameter
$\rho=\mbox{$I_{\text{EC}^{0}}$}/\mbox{$I_{\text{EC}^{*}}$}$ is inserted,
$1=\mbox{$I_{\text{EC}^{*}}$}(1+\rho)+\mbox{$I_{\beta^{+}}$}.$ (7)
Above, $\mbox{$I_{\beta^{+}}$}=1.4271(7)\%$ [8, 9] can be fixed to obtain
$I_{\text{EC}^{*}}$ and subsequently $I_{\text{EC}^{0}}$. We assume this
$I_{\beta^{+}}$ value is effectively uncorrelated with the EC branching
ratios. Alternatively, $\rho$ may be combined with a theoretical value of
$\rho_{B}=\mbox{$I_{\text{EC}^{0}}$}/\mbox{$I_{\beta^{+}}$}=33.4(7)$ from
Betashape V2.2 [24],
$1=\mbox{$I_{\text{EC}^{*}}$}\left(1+\rho+\rho\frac{1}{\rho_{B}}\right),$ (8)
which yields consistent branching ratios. These re-evaluations are compared to
existing ones in Table 2. Both the main and alternate methods above reduce the
uncertainty on 65Zn EC branching ratios by a factor of 2 compared to existing
evaluations [8, 9]. The re-evaluated intensity of the
$1115\text{\,}\mathrm{keV}$ $\gamma$ (=$I_{\text{EC}^{*}}$) is compared to
other measurements in Table 3 and Fig. 7. Calculations with our measurement
remain insensitive to internal conversion and intermediate gamma emissions.
Radiative electron capture does not affect our measurement of $10^{-3}$
precision. Internal Bremsstrahlung photons are detectable in both the SDD and
MTAS, and can create false positives and negatives. All such effects are
suppressed by $\mathcal{O}(10^{-4})$ relative to each radiative electron
capture branch. In the next generation of precision measurements, internal
Bremsstrahlung and other low-order effects will contribute.
| LNHB (2006) | NNDC (2010) | This work (main) | This work (alt.)
---|---|---|---|---
$I_{\text{EC}^{*}}$ (%) | $50.23(11)$ | $50.04(10)$ | $50.08(6)$ | $50.06(5)$
$I_{\text{EC}^{0}}$ (%) | $48.35(11)$ | $48.54(7)$ | $48.50(6)$ | $48.48(5)$
$I_{\beta^{+}}$ (%) | $1.421(7)$ | $1.421(7)$ | $1.421(7)$ | $1.45(3)$
Table 2: Branching ratios of 65Zn decays from existing evaluations reported by the LNHB [8] and NNDC [9] along with the re-evaluation of this work. The main re-evaluation combines the listed, adopted value of $I_{\beta^{+}}$ with the KDK measurement of $\mbox{$I_{\text{EC}^{0}}$}/\mbox{$I_{\text{EC}^{*}}$}=\mbox{$0.9684\pm 0.0018$}$. The alternate re-evaluation combines the KDK measurement with the theoretical ratio of $\mbox{$I_{\text{EC}^{0}}$}/\mbox{$I_{\beta^{+}}$}=33.4(7)$ from Betashape V2.2 [24], leading to a larger error on the resulting $I_{\beta^{+}}$ value relative to the $\sim 4$ times more-precise, measured $I_{\beta^{+}}$. Source | $I($1115\text{\,}\mathrm{keV}$)$ (%) | Rel. error (%)
---|---|---
Schötzig (1990) [25] | $50.2$ | $0.8$
Luca _et al._ (2003) [13] | $49.76$ | $0.42$
Iwahara _et al._ (2005) [12] | $49.71$ | $0.33$
Bé (2006) [11, 10] | $50.21$ | $0.20$
This work | $50.08$ | $0.12$
Table 3: Absolute intensities of the $1115\text{\,}\mathrm{keV}$ $\gamma$-ray
of 65Zn from existing measurements along with the value deduced from the decay
scheme re-evaluation of this work (Fig. 1 and App. A).
## Appendix B Likelihood details
The likelihood fit performed in Fig. 5 consists of 4 distinct spectral
components in both the coincident and anti-coincident spectra. Counts in the
Gaussian Cu Kα and Kβ peaks corresponding to 65Zn electron captures are used
to inform $\rho$ as discussed in the main text, with some details available
below. Symbol definitions are retained from the main text, where applicable.
### B.1 Coincidence sorting
Expected coincident signal counts consist of the terms,
$\Sigma^{*}=\overbrace{\sigma^{*}\epsilon}^{\text{\mbox{EC${}^{*}$},
{\hbox{\gamma}}
detected}}+\overbrace{\sigma^{*}(1-\epsilon)(1-P^{0}_{\mathrm{BG}}P^{0}_{\gamma})}^{\text{\mbox{EC${}^{*}$},
{\hbox{\gamma}} missed , BG
coincidence}}+\overbrace{\sigma(1-P^{0}_{\mathrm{BG}}P^{0}_{\gamma})}^{\text{\mbox{EC${}^{0}$},
BG coincidence}},$ (9)
which contain EC∗-originating events whose gamma-ray was successfully tagged
by MTAS, those where the gamma was missed occurring in coincidence with
another event in MTAS, and lastly EC0-originating events which occurred with
an event in MTAS. With the substitution $\sigma^{*}=\nu/(1-\rho^{\prime})$,
this expression yields Eq. (4) in the main text. The expression for expected
anti-coincident signal counts $\Sigma^{0}$ is the complement to the above such
that all events are accounted for:
$\Sigma^{*}+\Sigma=\sigma^{*}+\sigma^{0}=\nu$.
The primary source of spurious coincidences is the natural MTAS background of
rate $\beta$, corrected for via
$P^{0}_{\mathrm{BG}}=e^{-\beta\bar{t}},$ (10)
which is the probability of no occurrences within an average coincidence
window $\bar{t}$. Additionally, there is a rate of EC∗ events which are not
detected in the SDD, though the associated $\gamma$-ray is detected in MTAS in
coincidence with SDD signal. Such events have an expected rate
$\mathcal{R}_{*}=A\mbox{$I_{\text{EC}^{*}}$}(1-\eta_{*})\epsilon,$ (11)
where $A$ is the source activity and a lack of SDD X-ray detection is ensured
via
$(1-\eta_{*})=P_{K}^{*}(1-\omega_{K}\eta)+(1-P_{K}^{*}).$ (12)
At the order of this correction, the $<1\%$ probability of $\gamma$
interaction with the SDD (App. B.2) is negligible. This additional source of
spurious coincidences has a 0-event probability within $\bar{t}$ of
$P^{0}_{\gamma}=\exp(-R_{*}\bar{t})$. Altogether, the probability of any (1+)
event(s) from $\beta$ or $\mathcal{R}_{*}$ within the timescale $\bar{t}$ is
$(1-P^{0}_{\mathrm{BG}}P^{0}_{\gamma})$.
Expected MTAS background counts $\beta\bar{t}$ have been measured directly
[16], and are far larger than those from $\mathcal{R}_{*}$, as shown in Table
4. To obtain $\mathcal{R}_{*}$, known values for K-capture [24] and
fluorescence [29] probabilities are used, along with an assumed partial EC∗
activity $A\mbox{$I_{\text{EC}^{*}}$}$ from calculated source activity [16]
and $\mbox{$I_{\text{EC}^{*}}$}\sim 0.5$ with an assigned error of 10%. MTAS
$\gamma$ and SDD X-ray tagging-efficiencies have been measured elsewhere [16],
wherein average coincidence windows $\bar{t}$ are obtained from reported
quantities $\beta\bar{t},\beta$.
CW ($\mathrm{\SIUnitSymbolMicro s}$) | $\beta\bar{t}$ ($10^{-2}$) | $\mathcal{R}_{*}\bar{t}$ $(10^{-3}$)
---|---|---
1 | $0.74(1)$ | $0.7(1)$
2 | $1.25(1)$ | $1.1(2)$
4 | $2.27(2)$ | $2.1(3)$
Table 4: Expected counts leading to spurious coincidences of SDD signal with
MTAS events for a coincidence window $\bar{t}$. The natural MTAS background
($\beta\bar{t}$) dominates the effect of $\gamma$s from EC∗ events which were
missed by the SDD ($\mathcal{R}_{*}\bar{t}$).
The effect of neglecting spurious coincidences is depicted in Fig. 8. Without
this correction, results for $\rho$ are directly anti-correlated with
coincidence window. Applying the $\mathcal{O}(\text{CW})$ corrections for such
coincidences resolves the unphysical behaviour.
Figure 8: Obtaining
$\rho=\mbox{$I_{\text{EC}^{0}}$}/\mbox{$I_{\text{EC}^{*}}$}$ of 65Zn while
neglecting expected spurious coincidences of signal with background (black
squares) produces an unphysical, linear dependence on coincidence window. The
corrected results (red circles) are consistent across CWs.
### B.2 Physical quantities
Beyond coincidence sorting, any processes affecting relative production or SDD
detection of EC0 and EC∗ events must be accounted for. As per Eq. (3), $\rho$
is directly dependent on relative K-shell capture probabilities
$P_{K}^{*},P_{K}^{0}$, and SDD efficiency of tagging gammas $\eta_{\gamma}$.
K-capture probabilities from Betashape V2.2 [24] yield
$P_{K}^{0}/P_{K}^{*}=1.00690(44)$. Due to the high precision of the
measurement, this seemingly small variation in relative probabilities is the
second-most-dominant source of systematic error, as discussed further. It is
notable that previously reported [8] values from a 1996 evaluation [29] yield
a consistent ratio of $P_{K}^{0}/P_{K}^{*}=1.0067(27)$.
The SDD $\gamma$-tagging efficiency is obtained from a mixture of measurement
and simulations. The SDD K X-ray tagging efficiency obtained elsewhere [16] is
assumed to be equivalent to the geometric efficiency of the SDD; 8-9 keV Cu
X-rays penetrate the dead layers of the SDD without escaping the detector, and
all are likely to escape the source, as same-energy Auger electrons are
readily visible in the SDD (Fig. 5). Two terms are considered,
$\eta_{\gamma}=\eta\eta_{\gamma}^{i},$ (13)
where $\eta=21.5(11)\%$ is the geometric SDD efficiency [16], and
$\eta_{\gamma}^{i}$ is the probability that the $\gamma$ interacts with (does
not escape) the SDD volume.
Simulations are not relied upon for total SDD tagging efficiencies due to the
uncertainty in sub-mm, geometric source-SDD modelling. As such, only the
probability of $\gamma$ interaction with the volume it passes through is
obtained through a simulation of 10 million events, yielding
$\eta_{\gamma}^{i}=1.522(6)\%$. The SDD $\gamma$-tagging efficiency is then
$\eta_{\gamma}=0.327(17)\%$.
## Appendix C Systematic errors
Systematic errors are accounted for in two separate groups: physical sources
(1) discussed in App. B.2, and spectral characteristics (2) chosen prior to
the likelihood fit.
Every physical source of error is included in expressions for expected
coincident and anti-coincident signal counts of Eq. (4). Combined with Eq.
(3), $\rho^{\prime}=\sigma^{0}/\sigma^{*}$ and spurious coincidence
expressions (Eqs. (10)–(12)), $\rho$ is related to all physical parameters.
Values of all such parameters are listed in Table 5.
Parameter | Value | Source
---|---|---
$P_{K}^{*}$ (%) | $87.497(28)$ | Betashape V2.2 [24]
$P_{K}^{0}$ (%) | $88.101(26)$ | Betashape V2.2 [24]
$\omega_{K}$ (%) | $45.4(4)$ | Ref. [29]
$\eta$ (%) | $21.5(11)$ | Ref. [16]
$\epsilon$ (%) | $97.93(6)$ | Ref. [16]†
$\beta\bar{t}$ | $0.0125(1)$ | Ref. [16]†
$\beta$ (kHz) | $2.63951(15)$ | Ref. [16]†
$A\mbox{$I_{\text{EC}^{*}}$}$ (kBq) | $0.268(29)$ | App. B.2
$\eta_{\gamma}^{i}$ (%) | $1.522(6)$ | App. B.2
Table 5: Values and errors of physical parameters which contribute to the
overall systematic error. †Value for $2\text{\,}\mathrm{\SIUnitSymbolMicro s}$
coincidence window, with others available within Ref. [16].
All physical parameters, such as the MTAS $\gamma$-tagging efficiency, are
assumed to follow Gaussian distributions with a width corresponding to their
68% CL error. The effect of this efficiency on $\rho$ is gauged by sampling
1,000 values from the efficiency distribution, and obtaining the resulting
difference in obtained $\rho$ values. The width of the latter distribution
corresponds to the systematic error due to MTAS $\gamma$-tagging efficiency,
which is found to be $1.2\times 10^{-3}$.
To account for covariances, all physical parameters are varied simultaneously
over 10,000 iterations, resulting in a distribution of $\rho$ variation as
shown in Fig. 9. The spread of this distribution, $1.33\times 10^{-3}$, is
equivalent to the total systematic error of category (1).
Figure 9: Variation in $\rho$ induced from 10,000 instances of randomly
varying all physical parameters used to obtain the result. The standard
deviation of this distribution ($1.3\times 10^{-3}$) corresponds to the
systematic error on $\rho$. Binned counts are normalized to total iterations.
Remaining sources of systematics are of category (2), and contain the choice
of binning and fit range. Bin widths were considered between $[10,30]$ eV,
ensuring X-ray peaks remain easily distinguishable. The low end of the fit
range is constrained by the validity of the ad hoc model, as the wide Gaussian
describing primarily Auger counts (of three different energy ranges) is not
sufficient to describe data at much lower energies than pictured in Fig. 5
(data at lower energies is shown in Fig. 2). Both the low and high energy cuts
must encompass the ad hoc background in a region not dominated by X-rays. The
sets of low and high cuts considered are $[7.2,7.5]\text{
keV}\times[10.5,14.0]\text{ keV}$.
The effect of the binning is gauged by performing fits while randomly varying
the bin width from a uniform distribution bounded as described above. The
resulting distribution of $\rho$ values has a width of $\mathcal{O}(10^{-5})$.
The effect of varying the fit range is similarly small. The overall systematic
error of category (2) is equivalent to the width of the $\rho$ distribution
obtained varying all parameters of this category simultaneously. This width is
$3\times 10^{-5}$, which is negligible relative to the systematic error of
category (1). The total systematic error is obtained summing those of both
categories in quadrature to obtain $1.33\times 10^{-3}$.
A summary of systematic errors from individual sources (of both categories) is
displayed in Table 6. The dominant source of error is the MTAS
$\gamma$-tagging efficiency, which defines the order of the systematic error.
Though varying spectral characteristics can affect the ad hoc model in the
fit, any effects on $\rho$ are minimal as the signal Cu peaks of interest
dominate in counts. The statistical error of $1.25\times 10^{-3}$ is of the
same order as the overall systematic error. Added in quadrature, these two
quantities yield the overall error on $\rho$ of $1.83\times 10^{-3}$.
Source | Systematic error
---|---
MTAS $\gamma$-tagging efficiency ($\epsilon$) | $1.2\times 10^{-3}$
K-capture probabilities ($P_{K}^{*},P_{K}^{0}$) | $4.2\times 10^{-4}$
Partial $1115\text{\,}\mathrm{keV}$ activity ($A\mbox{$I_{\text{EC}^{*}}$}$) | $2.5\times 10^{-4}$
Natural MTAS backgrounds ($\beta\bar{t},\beta$) | $2.1\times 10^{-4}$
SDD-source geometric efficiency ($\eta$) | $1.8\times 10^{-4}$
Expected $\gamma$ interaction with SDD ($\eta_{\gamma}^{i}$) | $1.3\times 10^{-5}$
K fluorescence probability ($\omega_{K}$) | $1.8\times 10^{-6}$
|
Fit range | $5.2\times 10^{-5}$
Binning | $1.1\times 10^{-5}$
Table 6: Effects of individual sources of systematic error on $\rho$. The
statistical error is 0.0013. Sources of error stemming from physical
limitations (top group) have the associated symbol(s) in parentheses as used
in the text. Spectral characteristics (bottom group) have a sub-dominant
contribution to the overall systematic error of $1.33\times 10^{-3}$ obtained
in the text.
|
# Improving weakly supervised sound event detection with self-supervised
auxiliary tasks
###### Abstract
While multitask and transfer learning has shown to improve the performance of
neural networks in limited data settings, they require pretraining of the
model on large datasets beforehand. In this paper, we focus on improving the
performance of weakly supervised sound event detection in low data and noisy
settings simultaneously without requiring any pretraining task. To that
extent, we propose a shared encoder architecture with sound event detection as
a primary task and an additional secondary decoder for a self-supervised
auxiliary task. We empirically evaluate the proposed framework for weakly
supervised sound event detection on a remix dataset of the DCASE 2019 task 1
acoustic scene data with DCASE 2018 Task 2 sounds event data under 0, 10 and
20 dB SNR. To ensure we retain the localisation information of multiple sound
events, we propose a two-step attention pooling mechanism that provides a
time-frequency localisation of multiple audio events in the clip. The proposed
framework with two-step attention outperforms existing benchmark models by
22.3 %, 12.8 %, 5.9 % on 0, 10 and 20 dB SNR respectively. We carry out an
ablation study to determine the contribution of the auxiliary task and two-
step attention pooling to the SED performance improvement.111The code is
publicly released..
Index Terms: sound event detection, self-supervised learning, pooling function
## 1 Introduction
Sound Event Detection (SED) aims to determine the presence, nature and
temporal location of sound events in audio signals. Many SED algorithms rely
on strongly labelled data [1, 2, 3] for training to perform accurate event
detection and localisation. However, producing strongly labelled data for SED
is quite expensive in terms of the expertise, time and human resources
required for the annotation. This has led to the creation of weakly labelled
sound event detection dataset like Audioset [4] which contains audio clip
level annotations without the corresponding onset and offset times of the
audio events.
The weakly supervised sound event detection was first formulated as a
Multiple-Instance Learning (MIL) problem [5, 6] with the recent emergence of
Neural MIL. In Neural MIL, the first half of the network (segmentation
network) produces temporal predictions which are then aggregated by the second
half of the network (classification network) usually a pooling operator to
produce audio clip level predictions. The benefit of such formulation is,
along with detecting audio events in the clip, it provides insight into time
level localisation of those sound events in the audio clip. Since then, recent
works have focused on improving the model architecture of the segmentation
network [7, 8, 9] and developing better pooling methods [10, 11, 12, 13, 14].
However, few works have focused on how sound event detection models perform in
either limited data or noisy settings let alone in both of them.
The noisy data also affects the training of networks for sound event
detection. Specifically, the deep CNN architectures [15, 16] currently used to
provide benchmark performance for different speech and audio tasks [17]
require large labelled clean datasets to train on and when considered in a
noisy environment the performance is known to deteriorate [10]. The two
general learning strategies used as solutions are transfer learning and
multitask learning which were recently utilised for sound event detection [18,
19, 20]. However, in the multitask learning setup, it’s assumed you have
richly annotated labels for all the tasks. We investigate a counterpart of
this where only weak labels are available without any labels for the secondary
task. For this setting, we propose a self-supervised auxiliary task that will
be jointly trained with the primary task of sound event detection. The
auxiliary task is chosen to be the reconstruction of log Mel spectrogram of
audio and we show how the auxiliary task denoises internal representations and
improves network performance in noisy settings.
In all, in this paper, we address the challenge of training sound event
detection models in noisy (domestic or environmental) and limited data
settings. To that effort, we make two-fold contributions. First, identify
appropriate self-supervised auxiliary task for sound event detection in noisy
settings and demonstrate performance benefits to the same. Second, develop a
two step attention pooling mechanism that improves time-frequency localisation
of audio events and indirectly improves sound event detection performance in
noisy settings. We perform all the experiments on a standard noisy sound event
detection dataset remix [10] and release the code publicly.
Figure 1: Our proposed self-supervised learning assisted framework for weakly
supervised sound event detection. (A) The general architecture with shared
encoder and multiple decoder branches. Shared encoder, primary decoder,
auxiliary decoder is represented by $g$, $g_{2}$, $g_{4}$ respectively (B)
shows the two step attention pooling function used for primary decoder. (C)
The attention mechanism used for frequency and time attention in two step
attention pooling along different axis. (D) The CNN architecture used for
shared encoder and auxiliary decoder. The last layer is either class or
reverse convolution for encoder and decoder respectively.
## 2 Related work
A prominent recent work [10] analysed the performance of different model
architectures (segmentation network and pooling functions) under different
Signal to Noise Ratio (SNR) for sound event detection and localisation. The
paper showed that the segmentation network of type ‘VGG-like’ CNN performed
best for audio tagging and variability in performance resulted from the choice
of pooling methods with not a clear winning pooling method across different
SNR. Specifically, Global Attention Pooling outperformed other pooling methods
on some SNR and metrics, while Global Weighted Rank Pooling (GWRP) results in
the best performance on others. Still, the work on sound event detection
performance in limited data and noisy settings is sparse.
Though the various type of multitask learning methods have been greatly
explored for vision and natural language processing (NLP) tasks [21], it has
not been utilised by the audio community. Most of the works in multitask
learning for SED focus on jointly training SED with another strongly labelled
task like Sound Source Localisation (SSL) [18] or Acoustic Scene
Classification (ASC) [19, 22, 23]. A combination of multitask learning and
self-supervised learning is shown to improve performance on speech and audio
tasks [20]. However, the work uses large scale speech datasets like
Librispeech [24] as pretasks to pretrain the networks using self-supervised
learning and does not analyse the effect of noise (domestic or environmental)
on sound event detection performance.
## 3 Methodology
This section contains the details of the proposed approach for SED,
segmentation mapping network $g_{1}$, classification mapping network $g_{2}$,
and the auxiliary time-frequency reconstruction auxiliary task. The
architecture is depicted in figure 1
### 3.1 Self-supervised Learning formulation for SED
Let the raw audio be represented by $X=\\{{x_{i}}\\}_{i=1}^{T}$ where each
$x_{i}\in\mathbb{R}$ is a frame in the audio clip. We extract time-frequency
features for each audio, let them be represented by
$\hat{X}=\\{{\hat{x}_{i}}\\}_{i=1}^{T}$ where each
$\hat{x_{i}}\in\mathbb{R}^{d}$, $d\in\mathbb{Z}$ corresponds to frame in the
audio clip. In practice, d are the number of mel bins obtained after computing
the spectrogram. As per MIL formulation, we can represent each sample in
dataset as a bag $B_{j}=(\\{\hat{x}_{i}\\}_{i=1}^{T},y)|_{j=0}^{N}$ where
$y\in\mathbb{R}^{C}$ is the weak label, N are the number of samples and C are
the number of audio events. The primary task in our self-supervised framework
is SED. The segmentation mapping $g_{1}(.)$ of SED also acts as a shared
encoder for the auxiliary task. The shared encoder maps the feature set
$\\{{\hat{x}_{i}}\\}_{i=1}^{T}$ to $Z=\\{z_{i}\\}_{i=1}^{T}$ where
$z_{i}\in\mathbb{R}^{C\times F\times T}$. The second part of SED task is
network which classifies $\\{z_{i}\\}_{i=1}^{T}$ to $P=\\{p_{i}\\}_{i=1}^{C}$
where $P\in\mathbb{R}^{C}$. The network learns a mapping $g_{2}$ which maps
each audio events time-frequency segmentation to corresponding presence
probabilities of $c^{th}$ event known as $p_{k}$
$g_{1}:\hat{X}\mapsto Z\quad\quad g_{2}:Z\mapsto P$ (1)
The auxiliary self-supervised task chosen needs to help in learning robust
representations which generalise to noisy settings without requiring
additional labels. This will impact not only the learned internal
representation but also downstream sound event detection and localisation
performance. Inorder to achieve that we choose auxiliary task as
reconstruction of extracted time-frequency features for audio. By having time-
frequency reconstruction auxiliary task we hypothesise the network will learn
representations which retain audio event information better [25, 26]. We use
an auto-encoder structure for reconstruction where the encoder is shared with
the primary task of SED. If $g_{3}(.)$ is encoder mapping for reconstruction
task, we now represent $g_{1}(.)=g_{3}(.)=g(.)$ as the shared segmentation
mapping function. The second part of auxiliary task, is a decoder network
which learns a mapping $g_{4}$ such that $g_{4}:Z\mapsto\bar{X}$ where
$\bar{X}$ is the reconstructed time-frequency representation. Specifically
$\\{\bar{x_{i}}\\}_{i=1}^{T}=g_{4}(\\{z_{i}\\}_{i=1}^{T})$. Here the learned
mapping function $g_{4}(.)$ should satisfy:
$g_{4}^{-1}(g(.))=g^{-1}(g_{4}(.))=I$ (2)
To learn the function mappings satisfying primary SED task, let the objective
function be $\mathcal{L}_{1}$. To enforce the constraint of auxiliary task,
let the objective function be $\mathcal{L}_{2}$ where the aim is to minimise
the difference between T-F representation $\\{\hat{x_{i}}\\}_{i=1}^{T}$ and
predicted time-frequency representation $\\{\bar{x_{i}}\\}_{i=1}^{T}$ of audio
clip. If the learnable parameters are W = $[w,w_{2},w_{4}]$ and
$w,w_{2},w_{4}$ corresponding to $g(.),g_{2}(.),g_{4}(.)$ respectively, then
the optimisation problem can be framed in terms of these weights W over all
data points as:
$\underset{W}{\text{min}}\;\mathcal{L}_{1}(P,y|w,w_{4})+\alpha\mathcal{L}_{2}(\\{\bar{x_{i}}\\}_{i=1}^{T},\\{\hat{x_{i}}\\}_{i=1}^{T}|w,w_{2})$
(3)
The parameter alpha ($\alpha$) accounts for scale difference between losses
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$. It helps in adjusting the
contribution of auxiliary task relative to the primary task in learning
weights.
### 3.2 Shared encoder and auxiliary task decoder network
The segmentation mapping function (shared encoder) converts the time-frequency
audio input into a T-F representation for each of the audio events. The time-
frequency feature extracted for audio here is log Mel spectrogram as it has
shown to provide better performance [27, 28, 17]. We choose a CNN based
architecture similar to ‘VGG-like’ [10] for both shared encoder and auxiliary
task decoder. The shared encoder has CNN based network consists of 8 blocks of
2D Convolution, BatchNorm and ReLU with an Average Pool after every 2 blocks.
Having a common encoder helps the network to learn a shared representation by
exploiting the similarity across SED and T-F reconstruction and enables the
network to generalise better on our original task. We use a hard parameter
sharing framework to reduce the risk of overfitting [29] to limited samples.
Table 1: Weakly supervised sound event detection performance across different SNR Network | SNR 20 dB | SNR 10 dB | SNR 0 dB
---|---|---|---
encoder | pooling | aux. | micro-p | macro-p | AUC | micro-p | macro-p | AUC | micro-P | macro-p | AUC
VGGish | GAP | ✗ | 0.5067 | 0.6127 | 0.9338 | 0.4291 | 0.5390 | 0.9144 | 0.3295 | 0.4093 | 0.8694
VGGish | GMP | ✗ | 0.5390 | 0.5186 | 0.8497 | 0.5263 | 0.5023 | 0.8422 | 0.4640 | 0.4441 | 0.8189
VGGish | GWRP | ✗ | 0.7018 | 0.7522 | 0.9362 | 0.6538 | 0.7129 | 0.9265 | 0.5285 | 0.6084 | 0.8985
VGGish (dil.) | AP | ✗ | 0.7391 | 0.7586 | 0.9279 | 0.6740 | 0.7404 | 0.9211 | 0.5714 | 0.6341 | 0.9014
VGGish | 2AP | ✓ | 0.7829 | 0.7645 | 0.9390 | 0.7603 | 0.7486 | 0.9343 | 0.6986 | 0.6892 | 0.9177
The decoder of the auxiliary-task takes $Z=\\{z_{i}\\}_{i=1}^{T}$ as input and
reconstructs it to $\\{\bar{x_{i}}\\}_{i=1}^{T}$. The decoder consists of CNN
based network for combining the intermediate time-frequency representations
obtained for each audio event to an audio level time-frequency representation.
The architecture closely follows the common encoder structure in reverse order
consisting of 8 blocks of 2D Convolution, BatchNorm and ReLU with Average Pool
after every two blocks with a decreasing number of filters.
### 3.3 Primary decoder
The primary decoder is not CNN based, instead, it is a pooling operator to
satisfy MIL formulation. The choice of pooling operator has a significant
performance effect on both the SED and each audio events intermediate time-
frequency representation obtained. Global max pooling and global average
pooling results in underestimate and overestimate the audio event’s temporal
presence respectively, and to overcome this problem dynamic poolings were
proposed [10, 12, 14]. However, the developed pooling mechanisms still lacks
the granularity in temporal predictions and does not provide frequency
localisation which might be used to further disambiguate sound events. Also,
the standard attention pooling [14] is known to be unstable with cross-entropy
usually used for multi-class setup in practice.
We propose a two-step attention pooling mechanism to covert each audio events
segmentation maps $\\{z_{i}\\}_{i=1}^{T}$ into audio level predictions $P$.
The first step in the two step attention pooling takes
$Z=\\{z_{i}\\}_{i=1}^{T}$ as input. This undergoes two independent learned
linear transformation to produce classification and attention output
respectively. The attention output is squashed to ensure its valid probability
distribution. Mathematically, the attention output $Z_{a_{1}}$ and
classification output $Z_{c_{1}}$ are:
$Z_{a_{1}}=\frac{e^{\sigma(ZW_{a_{1}}^{T}+b_{a_{1}})}}{\sum_{i=1}^{F}e^{\sigma(ZW_{a_{1}}^{T}+b_{a_{1}})}}\quad
Z_{c_{1}}=(ZW_{c_{1}}^{T}+b_{c_{1}})$ (4)
This is followed by a weighted combination of classification output
$Z_{c_{1}}$ by attention weights $Z_{a_{1}}$:
$Z_{p_{1}}=\sum_{i=0}^{F}Z_{c_{1}}\cdot Z_{a_{1}}$ (5)
The time level attention is similar to frequency (first step) attention except
it operates along time axis:
$Z_{a_{2}}=\frac{e^{\sigma(Z_{p_{1}}W_{a_{2}}^{T}+b_{a_{2}})}}{\sum_{t=1}^{T}e^{\sigma(Z_{p_{1}}W_{a_{2}}^{T}+b_{a_{2}})}}\quad
Z_{p_{1}}=(ZW_{c_{2}}^{T}+b_{c_{2}})$ (6)
$Z_{p_{2}}=\sum_{t=0}^{T}Z_{c_{2}}\cdot Z_{a_{2}}$ (7)
where $Z_{p_{2}}\in[0,1]$ and denotes the presence probability of each sound
event in the audio clip. Figure 1 subsection c, provides an overview of a
single attention step. In relation to figure, $Z_{a}$, $Z_{c}$, $Z_{p}$ are
the outputs after attention matrix, classification matrix and $P(.)$
respectively in the first stage and second stage depending on subscript. By
breaking the attention into two steps, it makes the pooling more interpretable
by answering the questions of what frequency bins and what time steps
contributes to which audio events by visualising normalised attention weights
$Z_{a_{1}},Z_{a_{2}}$ and output $Z_{p_{1}},Z_{p_{2}}$. Also, the sigmoid
($\sigma$) ensures the attention output stays between 0 to 1 and avoids
unstable training for multilabel training with cross-entropy in practice.
## 4 Experiments
### 4.1 Dataset
To study the effect of noise in limited data settings, we form a noisy dataset
by mixing DCASE 2019 Task 1 of Acoustic scene classification [30] and DCASE
2018 Task 2 of General purpose Audio tagging [31]. The DCASE 2019 Task 1
provides background sounds (noise) recorded from a variety of real world
scenes in which the sounds from DCASE 2019 Task 2 are randomly embedded [10].
To ensure the noise conditions are natural, diverse and challenging, we use
the new DCASE 2019 Task 1 instead of DCASE 2018 as used in [10]. The 2019
variant extends the TUT Urban Acoustic Scenes 2018 with the other 6 cities to
a total of 12 large European cities. This results in 32000 audio clips with
8000 audio clips for each 20,10,0 dB SNR where each audio clip is of 10 secs
with background noise and three random audio events (out of total 41) in it.
### 4.2 Set up
The raw data is converted to time-frequency representation by applying FFT
with a window size of 2048 and an overlap of 1024 between windows. This is
followed by applying Mel filter banks with 64 bands and converting them to log
scale to obtain log Mel spectrogram. The network architecture used is
described in section 3.2. The entire network is trained end-to-end with a
batch size of 24 and learning of 1e-3 using Adam optimiser [32]. The code and
setup is publicly
released222https://github.com/soham97/MTL_Weakly_labelled_audio_data.
## 5 Results
### 5.1 Sound event detection
We evaluate our self-supervision assisted architecture and pooling method
against different baselines, benchmark architectures and pooling methods [14,
10]. Table 1 shows weakly supervised sound event detection performance across
different SNR of 20,10, and 0 dB. The important evaluation metric here under
consideration is micro precision (micro-p), as it uses global counts of true
positives, false negatives and false positives for metric computation against
macro precision which does simple unweighted averaging disregarding class-
imbalance. The VGGish (dil.) encoder here indicates VGGish architecture but
with dilated/atrous convolutions known to provide benchmark performance for
sound event detection, [14]. The VGGish encoder with reconstruction based
auxiliary task and two step attention pooling outperforms the existing
benchmark of atrous attention pooling [14] on SNR 20, 10 and 0 dB by 5.9%,
12.8% and 22.3% respectively. Apart from improving performance, by breaking
the attention into two steps, it allows for the intermediate use of sigmoid
which helps in ensuring the outputs don’t overflow above 1 during training.
Table 2: Ablation study to determine auxiliary task contribution auxiliary task | SNR 20 dB | SNR 10 dB | SNR 0 dB
---|---|---|---
$\alpha$ = 0.0 | 0.7772 | 0.7430 | 0.6937
$\alpha$ = 0.001 | 0.7829 | 0.7603 | 0.6986
$\alpha$ = 0.1 | 0.7637 | 0.7428 | 0.6792
### 5.2 Ablation study for auxiliary task contribution
We perform an ablation study to determine the contribution of reconstruction
auxiliary task and two step attention pooling towards the total performance
improvement. As described in Section 3.1, the total loss is:
$\displaystyle\mathcal{L}=\mathcal{L}_{1}(P,y|w,w_{4})+\alpha\mathcal{L}_{2}(\\{\bar{x_{i}}\\}_{i=1}^{T},\\{\hat{x_{i}}\\}_{i=1}^{T}|w,w_{2})$
(8)
By changing the value of $\alpha$ before training, we can adjust the
contribution of the auxiliary task to primary sound event detection. When
$\alpha=0.0$, the network has no contribution from the reconstruction
auxiliary task during training and it can be used to evaluate the performance
of two step attention pooling. In terms of micro-precision, the two step
attention pooling outperforms existing benchmark of atrous AP (row 4) from
table 1 on SNR 20, 10 and 0 dB by 5.2%, 10.2% and 21.4% respectively. By
adding the auxiliary task contribution with a relative weightage of
$\alpha=0.001$, an additional improvement of 0.7%, 2.3% and 0.7% is observed.
This indicates that two step attention has a prominent contribution in
improving the performance of sound event detection in limited data and noisy
settings, with additional performance gains from the auxiliary task. When
$\alpha$ is increased to 0.01, the performance compared to $\alpha=0.001$ is
decreased. This suggests that the auxiliary task’s loss contribution starts to
overpower the primary SED task’s loss contribution rather than improving
generalisation.
Table 3: Two top and worst performing sound events- SNR 0 dB model | aux. | bus | cowbell | gong | meow
---|---|---|---|---|---
Atrous + AP | ✗ | 0.2 | 0.781 | 0.692 | 0.583
VGGish + 2AP | ✗ | 0.572 | 0.921 | 0.643 | 0.483
VGGish + 2AP | ✓ | 0.627 | 0.94 | 0.663 | 0.532
### 5.3 SED performance on specific audio events
For almost all audio events, our proposed architectures have the best
precision scores against GMP, GAP, GWRP, Atrous across all SNR = 0, 10, 20.
Particularly, for audio events like ‘Bass drum’, ‘bus’, ‘double bass’,
‘cowbell’ the architecture outperforms other models by a large margin as shown
in table 3. However, the proposed model struggles in audio events like
‘gong’,‘chime’ and ‘meow’ where the attention pooling with dilated convolution
encoder performs better [14]. This indicates using atrous or dilated
convolutions helps in detecting audio events whose energy is spread wide in
the temporal domain. This can be incorporated into our current architecture by
replacing the linear convolutions in the shared encoder with dilated
convolutions. Further analysis and event-specific results are available in the
long version of paper 333https://arxiv.org/pdf/2008.07085.pdf and skipped due
to space constraints.
Figure 2: Visualisation of two step attention pooling and reconstruction
decoder outputs. Subplot 1 depicts the scaled log Mel spectrogram of an audio
clip. Subplot 2 is the output of the reconstruction auxiliary task. Subplots
3,4 and 5 are attention weights for the three most probable audio events in
the audio clip. Subplot 6 is the output of the first step attention pooling.
Subplot 7 and 8 is the attention weight and output of second step attention
pooling respectively. The y-axis in subplot 1-4 corresponds to Mel-bins and
sound events in subplot 5-6. The x-axis in subplot 1-7 corresponds to time and
sound events in subplot 8
### 5.4 Interpretable visualisation of audio events
Apart from improved performance, using two step attention pooling provides a
way to localise each audio event present in the audio clip along with both the
time and frequency axis. To illustrate this, we pick a random example with SNR
20 dB and show the end to end visualisation of the two step attention pooling
mechanism in figure 2. The audio under consideration has three events
occurring in it: telephone ringing, cello playing and cat meowing, with
outdoor environmental background noise. Subplot 2 in figure 2 depicts the
reconstructed Mel spectrogram of the audio clip. From the subplot, we can see
that the decoder is not only able to reconstruct the audio events clearly but
it is also denoising the log Mel spectrogram retaining the key elements of
three audio events. A future extension of work is to jointly train sound
source separation along with weakly supervised SED by using the auxiliary task
reconstruction output.
## 6 Conclusions
This paper proposes assisted self-supervised task for improving sound event
detection in limited data and noisy settings. The architecture consists of
sound event detection as a primary task with two-step attention pooling as a
primary decoder and time-frequency representation reconstruction as an
auxiliary task. We empirically evaluate the proposed framework for multi-label
weakly supervised sound event detection, on a remix DCASE 2019 and 2018
dataset under 0, 10 and 20 dB SNR. The proposed self-supervised auxiliary task
framework with two-step attention outperforms existing benchmark models by
22.3 %, 12.8 %, 5.9 % on 0, 10 and 20 dB SNR respectively. The ablation study
carried out indicates the majority of performance improvement is associated
with two step attention pooling with secondary performance improvement from
self-supervised auxiliary task. Furthermore, by using two step attention, we
can easily visualise the sound event presence along both time-frequency axis.
The code is publicly released.
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|
# Mixing across stable density interfaces in forced stratified turbulence
Miles M. P. Couchman1<EMAIL_ADDRESS>Stephen M. de Bruyn Kops2 Colm-cille
P. Caulfield1,3 1Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge CB3 0WA, UK 2Department of Mechanical and
Industrial Engineering, University of Massachusetts Amherst, Amherst, MA
01003, USA 3Institute for Energy and Environmental Flows, University of
Cambridge, Cambridge CB3 0EZ, UK
###### Abstract
Understanding how turbulence enhances irreversible scalar mixing in density-
stratified fluids is a central problem in geophysical fluid dynamics. While
isotropic overturning regions are commonly the focus of mixing analyses, we
here investigate whether significant mixing may arise in anisotropic
statically-stable regions of the flow. Focusing on a single forced direct
numerical simulation of stratified turbulence, we analyze spatial correlations
between the vertical density gradient $\partial\rho/\partial z$ and the
dissipation rates of kinetic energy $\epsilon$ and scalar variance $\chi$, the
latter quantifying scalar mixing. The domain is characterized by relatively
well-mixed density layers separated by sharp stable interfaces that are
correlated with high vertical shear. While static instability is most
prevalent within the mixed layers, much of the scalar mixing is localized to
the intervening interfaces, a phenomenon not apparent if considering local
static instability or $\epsilon$ alone. While the majority of the domain is
characterized by the canonical flux coefficient
$\Gamma\equiv\chi/\epsilon=0.2$, often assumed in ocean mixing
parameterizations, extreme values of $\chi$ within the statically-stable
interfaces, associated with elevated $\Gamma$, strongly skew the bulk
statistics. Our findings suggest that current parameterizations of turbulent
mixing may be biased by undersampling, such that the most common, but not
necessarily the most significant, mixing events are overweighted. Having
focused here on a single simulation of stratified turbulence, it is hoped that
our results motivate a broader investigation into the role played by stable
density interfaces in mixing, across a wider range of parameters and forcing
schemes representative of ocean turbulence.
## 1 Introduction
In a density-stratified fluid, turbulence enhances the rate at which scalars
are irreversibly diffused throughout the flow, a process of great interest in
a variety of geophysical, environmental and industrial settings (e.g. Fernando
(1991)). Of particular importance is characterizing the role of turbulence in
the vertical transport of heat within the ocean, a crucial mechanism for
driving the required upwelling of cold bottom waters to maintain the ocean’s
vertical stratification profile and to complete global circulation currents
(Wunsch & Ferrari, 2004). Turbulence in the ocean generates dynamically
relevant motions on the order of millimeters, which cannot currently be
resolved in numerical circulation models and must therefore be parameterized,
with the choice of parameterization found to influence future climate
projections strongly (Whalen et al., 2020). Considerable observational,
numerical and theoretical work has thus been focused on developing more
accurate and universal mixing models which account for the wide range of
turbulent processes observed in different flow regimes within the ocean
(Caulfield, 2020).
The rate at which turbulence mixes a non-uniform density field is often
defined in terms of an appropriately-averaged vertical density flux
$B\equiv\left\langle\rho^{\prime}w^{\prime}\right\rangle$, where
$\rho^{\prime}$ and $w^{\prime}$ denote fluctuations in density and vertical
velocity away from the mean flow, respectively. If $B$ is to be used as a
robust indicator of irreversible mixing, it is critical that measurements of
$B$ are averaged over sufficiently large spatial volumes or time intervals, in
order to isolate irreversible diffusive processes from reversible stirring
motions (Villermaux, 2019). Stirring, occurring on relatively large scales,
may be thought of as the adiabatic rearrangement of fluid parcels of different
density induced by the underlying turbulence, which in principle is
reversible. Hence, a pointwise measurement of $B$ would not be a sufficient
indicator that irreversible mixing had occurred, as the sign of $B$ could
subsequently switch direction yielding a net flux of zero. Thus, we here use
the term mixing to refer specifically to the diffusive transport of density
across gradients that have been enhanced by such macroscopic stirring motions,
irreversibly leading the system toward a state of greater homogenization. In
order to isolate only irreversible contributions to mixing, Lorenz (1955)
introduced the concept of an available potential energy (APE). APE quantifies
the difference between a system’s current potential energy and its minimum
background potential energy (BPE) that could be achieved if fluid parcels were
adiabatically sorted into their most stable configuration. For a Boussinesq
fluid, Winters et al. (1995) demonstrated that irreversible mixing may be
described as the conversion of APE into BPE, with a system’s BPE increasing in
time as it homogenizes. Generalizing this mixing framework to compressible
flows, Tailleux (2009) argued that the mixing of a thermally-stratified fluid
should most rigorously be defined as the conversion of APE into internal
energy, which in the Boussinesq limit then exactly matches the generation of
BPE.
Given a variety of sampling limitations involved with collecting turbulence
data within the ocean, it is exceedingly difficult to perform the averaging
required to extract the irreversible component of density fluxes from direct
observational measurements of $B$. Therefore, a number of indirect methods
have been proposed that infer such fluxes from more readily available
quantities, which may be computed locally (Gregg et al., 2018). Two such
quantities, associated with what is conventionally referred to as turbulent
microstructure, are the dissipation rates of kinetic energy $\epsilon$ and
scalar variance $\chi$,
$\epsilon=\frac{\nu}{2}\left(\frac{\partial u^{\prime}_{i}}{\partial
x_{j}}+\frac{\partial u^{\prime}_{j}}{\partial
x_{i}}\right)^{2},\quad\chi=\frac{g^{2}\kappa}{\rho_{0}^{2}N^{2}}\left(\frac{\partial\rho^{\prime}}{\partial
x_{i}}\frac{\partial\rho^{\prime}}{\partial x_{i}}\right),$ (1)
representing the rates at which viscosity $\nu$ and molecular diffusivity
$\kappa$ smooth gradients in the turbulent velocity $\mathbf{u^{\prime}}$ and
density $\rho^{\prime}$ fields, respectively. In equation (1), $g$ denotes the
gravitational acceleration, $\rho_{0}$ a reference background density and
$N=\sqrt{\left(-g/\rho_{0}\right)\partial\overline{\rho}/\partial z}$ the
buoyancy frequency, defined by an appropriately averaged ambient density
gradient $\partial\overline{\rho}/\partial z$ against which the turbulence
acts. The quantities $\epsilon$ and $\chi$ are intimately related to the
irreversible processes associated with the conversion of kinetic energy and
available potential energy into internal energy, respectively, as is further
described by Caulfield (2021). In particular, for the class of direct
numerical simulation considered here, characterized by an imposed uniform
background stratification $N^{2}_{0}$, Howland et al. (2021) demonstrated that
$\chi$ computed using $N^{2}=N_{0}^{2}$ in equation (1) provides an excellent
approximation to the destruction rate of APE and is therefore a good measure
of local irreversible mixing.
As discussed by Ivey et al. (2018), $\chi$ also arguably provides the most
robust method for estimating irreversible mixing from oceanographic
measurements, since $\left\langle
B\right\rangle\simeq\left\langle\chi\right\rangle$ in steady-state provided
that averaging is performed over sufficiently long times and large volumes so
that reversible processes and transport terms are negligible (Osborn & Cox,
1972). Importantly, $\chi$ is both directly proportional to the scalar
diffusivity $\kappa$ and sign-definite, providing a robust local measure of
the irreversible fluxes associated with molecular diffusion, which does not
require the averaging of the density flux $B$ needed to filter our reversible
local stirring motions in the turbulent flow. Due to a scarcity of $\chi$
measurements, however, $\epsilon$ is more commonly used to infer mixing
following the method of Osborn (1980), which requires the introduction of a
flux coefficient $\Gamma\equiv\chi/\epsilon$ to prescribe the fraction of
turbulent kinetic energy that leads to irreversible mixing, as opposed to
being directly dissipated by viscosity. A constant value $\Gamma=0.2$ is
commonly assumed when estimating global patterns of oceanic mixing (MacKinnon
et al., 2017), which has been found to be in agreement with tracer release
experiments (Gregg et al., 2018). However, there is significant evidence
suggesting that $\Gamma$ varies appreciably in different flow regimes
(Caulfield, 2021) and so a clear physical picture has not yet emerged
explaining why $\Gamma=0.2$ is a reasonable assumption.
In the absence of measurements of $\epsilon$ or $\chi$, mixing locations are
primarily inferred from the presence of unstable overturns in vertical density
profiles, as proposed by Thorpe (1977). Assuming that the vertical extent of
an overturn is correlated with the Ozmidov length
$L_{O}=\sqrt{\epsilon/N^{3}}$, $\epsilon$ may be inferred from the measurement
of overturns which can then be converted into a flux via $\Gamma$. However,
this assumed correlation between the vertical overturning scale and $L_{O}$ is
not always robust, as has recently been discussed, for example, by Ivey et al.
(2018), Ijichi et al. (2020) and Mashayek et al. (2021). Using a forced direct
numerical simulation (DNS) similar to that considered here, Taylor et al.
(2019) quantified the errors associated with the indirect flux estimates of
Osborn & Cox (1972), Osborn (1980) and Thorpe (1977) by sparsely sampling
vertical profiles of the computational domain in order to mimic oceanographic
measurements.
Spatio-temporal intermittency in stratified turbulence greatly reduces the
applicability of classical turbulence modeling assumptions, including the
common assumption of log-normal distributions for $\epsilon$ and $\chi$ (de
Bruyn Kops, 2015). Cael & Mashayek (2021) found that global ocean measurements
of $\epsilon$ were not well approximated by an assumed log-normal distribution
but instead had a skewed right tail, indicating that a small number of extreme
events dominated the bulk statistics. By considering local correlations
between direct ocean measurements of $\epsilon$ and $\chi$, Couchman et al.
(2021) further emphasized the importance of extreme events, finding that while
the majority of the sampled domain was characterized by the canonical flux
coefficient $\Gamma=0.2$, isolated mixing events containing the largest $\chi$
were not reflected by a corresponding local increase in $\epsilon$, yielding a
dramatic increase in $\Gamma$.
Vertical layering is also known to be a canonical feature of stratified
turbulent flows, with the density field often forming ‘staircases’ of deep,
relatively well-mixed layers separated by thin interfaces with strong
gradients (Caulfield, 2021). For sufficiently stratified environments,
vertical shearing induced by the decoupling of horizontal and vertical motions
in such a layered structure becomes an important mechanism for triggering
instability and the ensuing generation of turbulence (Lilly, 1983; Billant &
Chomaz, 2001). Parameterizations of mixing based on simple domain averages are
thus unlikely to be accurate as rare extreme events and spatial heterogeneity
within the flow will be missed, a potential cause of the highly-scattered
mixing statistics currently reported throughout the literature (Gregg et al.,
2018).
In an attempt to classify such intermittency in an automatic, yet robust and
interpretable manner, Portwood et al. (2016) devised an algorithm for
splitting a snapshot from a forced DNS into three dynamically distinct
regions: quiescent regions, intermittent layers and turbulent patches. These
regions were distinguished by an increasing degree of local overturning, as
determined by computing the fraction of unstable points $\partial\rho/\partial
z>0$ within an extended neighbourhood. Local overturning fractions and
dissipation rates $\epsilon$ were found to be strongly correlated, in
agreement with the arguments of Thorpe (1977). For the relatively large filter
sizes used to segment the domain, on the order of a buoyancy length
$L_{B}=2\pi u_{h}/N$ (where $u_{h}$ denotes a characteristic horizontal
velocity scale), distributions of $\chi$ associated with each region were also
found to be correlated with $\epsilon$, although the finer spatial
distributions of $\epsilon$ and $\chi$ within each region, and the resulting
flux coefficient $\Gamma$, were not considered.
Motivated by the automated flow segmentation of Portwood et al. (2016) in
terms of unstable local density gradients $\partial\rho/\partial z>0$, and the
observation of Couchman et al. (2021) that, within the ocean, extreme events
in $\chi$ are not necessarily correlated with those in $\epsilon$, we here
analyze spatial mixing distributions within a computational domain by
considering local correlations between $\epsilon$, $\chi$ and
$\partial\rho/\partial z$. In particular, we wish to probe whether overturning
alone provides a robust indicator for local mixing, or if significant mixing
as revealed by $\chi$ might occur in other regions that would seem
inconspicuous based on consideration of only $\epsilon$ or
$\partial\rho/\partial z$.
In line with the previous investigations of Portwood et al. (2016) and Taylor
et al. (2019), we consider a forced DNS of stratified turbulence using the
methodologies presented in Almalkie & de Bruyn Kops (2012). In $\S$2, we
summarize the DNS dataset considered here and highlight the presence of a
(previously-unreported) robust vertically-aligned vortex generated by the
forcing scheme, that injects energy into the domain at large scales and
induces vertical layering in the surrounding flow. In $\S$3, we then consider
pointwise correlations between $\partial\rho/\partial z$, $\epsilon$, $\chi$
and the flux coefficient $\Gamma$, which suggests that mixing occurs not only
in overturning regions, but also in areas of local static stability. In $\S$4,
we move beyond pointwise statistics to consider extended mixing structures
within the flow, highlighting two ways in which local static instability in
the density gradient fails to be a sufficient indicator of mixing: within the
vortex a lateral density gradient is correlated with the majority of $\chi$,
and outside the vortex extreme values of $\chi$ are localized to relatively
‘sharp’ stable density interfaces at the bounding edges between overturning
layers. Finally, in $\S$5, we summarize our results and discuss implications
for parameterizing turbulent mixing within the ocean.
## 2 Summary of DNS dataset
We consider a statistically-steady, forced DNS of stratified turbulence from
the simulation campaign originally reported by Almalkie & de Bruyn Kops
(2012), and subsequently analyzed by Portwood et al. (2016) and Taylor et al.
(2019). Using a characteristic root-mean-square horizontal velocity scale
$u_{h}$, length scale $L$, and background buoyancy frequency $N_{0}$, the non-
hydrostatic Boussinesq approximation of the Navier-Stokes equations may be
written in the following dimensionless form:
$\nabla\cdot\mathbf{u}=0,\quad\frac{\partial\mathbf{u}}{\partial
t}+\mathbf{u}\cdot\nabla\mathbf{u}=-\left(\frac{2\pi}{Fr}\right)^{2}\rho\hat{\mathbf{z}}-\nabla
p+\frac{\nabla^{2}\mathbf{u}}{Re}+\mathcal{F},\quad\frac{\partial\rho}{\partial
t}+\mathbf{u}\cdot\nabla\rho-w=\frac{\nabla^{2}\rho}{RePr}.$ $None$
The governing equations ($None$) are numerically integrated using a
pseudospectral technique in a triply-periodic domain, as detailed by Almalkie
& de Bruyn Kops (2012). The dimensionless parameters governing the flow are
the Prandtl number $Pr=\nu/\kappa$, Froude number $Fr_{h}=2\pi u_{h}/(N_{0}L)$
and Reynolds number $Re_{h}=u_{h}L/\nu$. The density field satisfies
${\color[rgb]{0,0,0}\rho\left(\boldsymbol{x},t\right)=\rho_{0}(1-N_{0}^{2}z/g)+\rho^{\prime}\left(\boldsymbol{x},t\right)},$
(3)
where $\rho_{0}(1-N_{0}^{2}z/g)$ defines a time-independent, linear background
density gradient characterized by a reference density $\rho_{0}$ and an
imposed constant background buoyancy frequency $N_{0}$. Density perturbations
$\rho^{\prime}$ away from this linear background state satisfy the periodic
boundary conditions and are used to compute $\chi$ in equation (1). The
imposed constant background buoyancy frequency $N_{0}$ is used as the
characteristic ‘appropriately-averaged’ buoyancy frequency $N$ required to
compute $\chi$ in equation (1), as is widely considered the natural choice
when quantifying irreversible mixing in numerical simulations with an imposed
background stratification (see e.g. Shih et al. (2005); Maffioli et al.
(2016); Garanaik & Venayagamoorthy (2019); Portwood et al. (2019)). By
explicitly computing the available potential energy (APE) of a triply-periodic
domain with an imposed uniform background stratification $N_{0}$, Howland et
al. (2021) confirmed that normalizing $\chi$ by $N_{0}$ indeed provides an
excellent approximation to the true irreversible mixing rate as computed
through changes in the system’s APE. The forcing term $\mathcal{F}$ is
governed by the deterministic scheme denoted ‘Rf’ in Rao & de Bruyn Kops
(2011), which adds energy to horizontal motions larger than 1/8th of the
horizontal box size so as to match a target kinetic energy spectrum at small
wavenumbers. We consider a simulation characterized by $Pr=1$, $Fr_{h}=2.23$
and $Re_{h}=1271$, in a domain of size $2\pi\times 2\pi\times\pi$ with
$4096\times 4096\times 2048$ grid points, resulting in a grid spacing of
$\Delta\approx L_{K}/2$, with $L_{K}$ denoting the Kolmogorov length scale.
For reference, the characteristic buoyancy Reynolds number of the simulation
is $Re_{b}=\left\langle\epsilon\right\rangle/\nu N_{0}^{2}=50$. We consider a
single snapshot of the flow in time and all figures are displayed on grids
that have been sparsed by a factor of eight in each dimension.
Figure 1: Large-scale characteristics of the velocity and density fields
within the computational domain. a) The vertical average of the dissipation
rate $\epsilon$ at gridpoint $(x,y)$, normalized by the domain average. The
whole domain is shown, with gridpoints sparsed by a factor of eight in each
dimension. The dashed circle highlights a region of elevated $\epsilon$,
coinciding with a vortex in the velocity field, as is further examined in
Figure 3. A vertical slice of the domain at $y=200$ is considered in Figure 4.
b) The horizontal velocity normal to the plane for a vertical slice at
constant $y$ passing through the center of the dashed circle in panel a),
revealing a vertically-aligned vortex rotating counterclockwise. The grid has
been shifted in $x$ relative to panel a) so as to center the vortex. The
buoyancy length $L_{B}$ and Taylor length $L_{T}\approx 25L_{K}$ (where
$L_{K}$ is the Kolmogorov length) are marked for reference, as were the filter
sizes used in the segmentation analysis of Portwood et al. (2016). Green
contours mark stable interfaces in the density field, as shown in panel d). c)
The radially-averaged angular velocity $u_{\theta}$ and vertical component of
vorticity $\omega_{z}$ as a function of the distance $r$ from the center of
the vortex. The dashed line at $r\approx 80$ corresponds to the dashed circle
plotted in panel a). d) The density field corresponding to the vertical slice
of velocity plotted in panel b). The green lines in panels b) and d)
illustrate contours at the minimum values of the histogram of the density
field in panel e), as are marked with vertical green lines, delineating the
interfaces between relatively well-mixed density layers in the flow
surrounding the vortex.
The main characteristics of the dataset are summarized in Figure 1. The
vertically-averaged dissipation rate $\epsilon$ (Figure 1a) reveals a dominant
patch of elevated turbulence that is generated by a large-scale vertically-
aligned vortex in the velocity field, rotating counter-clockwise (Figure 1b).
Radial averages, centered on the vortex, of the angular velocity $u_{\theta}$
and vertical component of vorticity $\omega_{z}=\partial v/\partial x-\partial
u/\partial y$ are plotted in Figure 1c, indicating a Rankine-type vortex that
is approximately characterized by rigid body rotation at small radii $r$ from
the vortex core, followed by a transition to roughly irrotational flow at
larger $r$. It is important to note that such a description characterizes the
radially-averaged flow, and that smaller-scale vortical motions will still
certainly be present in the turbulent patch surrounding the vortex. A series
of horizontal currents traveling in alternating directions are found to
emanate from the vortex, characterized by a vertical scale on the order of a
buoyancy length $L_{B}$. In Figure 1d, we plot the vertical slice of the
density field that corresponds to the velocity field shown in Figure 1b,
highlighting an analogous vertically-layered structure outside of the vortex,
with relatively well-mixed density layers separated by sharp, stable
interfaces. The approximate locations of these density interfaces (delineated
by green contours) correspond to minima in the histogram of $\rho$ (Figure
1e), which highlights a strong perturbation of the density field away from its
uniform background gradient. Superimposing these density contours on the
velocity field in Figure 1b highlights that the sheared interfaces in the
velocity field are strongly correlated with the stable interfaces in the
density field. This correlation is further demonstrated in Supplementary Video
1, where rotations of the slices in Figures 1b,d around the center of the
vortex are shown. In $\S$4, we demonstrate that these interfaces,
characterized by both high shear and a strong statically-stable density
gradient, are critically important for the mixing generated outside of the
vortex.
The spontaneous formation of a persistent vortex is a key, yet previously
unreported feature of the forcing scheme of Rao & de Bruyn Kops (2011) used to
generate statistically-steady turbulence. In particular, the identification of
the vortex in Figure 1 provides insight into how the segmentation results of
Portwood et al. (2016) (see their Figure 2c), who used an identical forcing
scheme, are related to the background flow field. Specifically, the roughly
cylindrical patch of most vigorous turbulence detected by Portwood et al.
(2016), using a filter of size $L_{B}$, extends across the entire vertical
domain and almost certainly corresponds to an analogous vortical structure in
their DNS. Similarly, their ‘intermittent layers’ are primarily composed of
horizontal offshoots from the central vertically-aligned turbulent patch, and
are shaped by a similar pattern to the sheared velocity interfaces observed in
Figure 1b. As it is now evident that Portwood et al. (2016) have broadly
identified such a vortex to be a turbulent hotspot, a goal of this study is to
perform a finer analysis of mixing patterns both within and outside of the
vortex, in order to determine how patterns in the small-scale turbulent
microstructure, as described by $\epsilon$ and $\chi$, are related to the
larger-scale layered structure of the flow.
## 3 Pointwise statistics conditioned on local density gradient
Motivated by the flow segmentation of Portwood et al. (2016) in terms of the
local fraction of overturning $\partial\rho/\partial z>0$, we first consider
how the pointwise distributions of $\epsilon$, $\chi$ and
$\Gamma=\chi/\epsilon$ depend on the magnitude of $\partial\rho/\partial z$,
for both statically stable and unstable points, as shown in Figure 2. For
illustration, in Figure 2a we split the distribution of $\partial\rho/\partial
z$ into three regions: two tails containing 10$\%$ by volume of the most
stable and unstable points (coloured blue and red, respectively), and the
remaining 80$\%$ of the intermediate values (green). For such a division, we
then consider the distributions of $\epsilon$, $\chi$ and $\Gamma$ within each
region, as shown in Figures 2b-d. Although the distribution characterizing the
bulk of the domain (green) is centered around the canonical flux coefficient
$\Gamma=0.2$ (see Figure 2d), such points contain the lowest $\chi$ (Figure
2c) and are thus not of primary importance for the total mixing arising within
the computational domain. Instead, it is the extreme tails of the
$\partial\rho/\partial z$ distribution that must be considered, containing the
most significant values of $\chi$. While both the blue and red tails contain
elevated but similar distributions of $\epsilon$, they may be distinguished by
their asymmetry in $\chi$; the stable tail (blue) contains disproportionately
elevated $\chi$ as compared to $\epsilon$, and therefore some of the highest
values of $\Gamma$ within the domain.
Figure 2: Pointwise mixing statistics of the DNS data, conditioned by the
local vertical density gradient. a) Histogram of the perturbed density
gradient $\partial\rho^{\prime}/\partial z$ normalized by the magnitude of the
imposed uniform background gradient, with values greater than one indicating
local overturning. The distribution is split into three regions, by assigning
a fixed volume (here $10\%$) to each tail. Panels b)-d) illustrate the
distributions of $\epsilon$, $\chi$ and $\Gamma$, respectively, for the whole
domain (black) and the subdomains encompassed by the coloured regions in panel
a). Circles mark the median values of each distribution, and the dashed line
in panel d) indicates the canonical flux coefficient $\Gamma=0.2$ for
reference. Panels e)-g) illustrate how the medians of the respective
distributions in panels b)-d) vary with the tail volume selected in panel a).
The circles mark the medians for the segmented distributions shown in panels
b)-d) for the case of $10\%$ tail volume. For panels e) and f), the fraction
of each quantity ($\epsilon$, $\chi$) contained within each tail relative to
the entire domain is also indicated. The dashed diagonal lines mark what would
be expected for a uniform distribution of each quantity throughout the domain.
In Figures 2e-g, we then analyze how the medians of the $\epsilon$, $\chi$ and
$\Gamma$ distributions change as a function of the volume contained within the
blue and red tails, and additionally plot the relative contributions of these
tails to the domain total. Comparing the right panels of Figures 2e and 2f
reveals that $\chi$ is far more dominated by extreme events than $\epsilon$,
in agreement with the analysis of oceanographic data by Couchman et al.
(2021). For instance, when each tail contains $10\%$ volume, the stable (blue)
and unstable (red) tails each contain approximately $20\%$ of the total
$\epsilon$ in the domain, but $45\%$ and $30\%$ of the total $\chi$,
respectively. Furthermore, while the contributions to $\epsilon$ from both
tails is roughly equal, the contribution to $\chi$ from the stable tail is
always roughly $50\%$ greater than for the unstable tail. While $\Gamma=0.2$
may thus be a suitable approximation for the bulk of the domain, it may here
not be relied upon for capturing the most extreme events in $\chi$, which
dominate the bulk mixing statistics.
Additionally, the statistics in Figure 2 suggest that local instability may
not be a sufficient indicator for mixing, given the significance of the blue
stable tail. However, we note that such a conclusion cannot definitively be
drawn from the _pointwise_ distributions of $\partial\rho/\partial z$, as such
a distribution provides no information about the extended spatial environment
around each point. For example, in regions of fully-developed turbulence that
might emerge after the collapse of a shear-induced billow, there is likely a
random mixture of neighbouring unstable and stable points in close proximity
(roughly a $50\%$ mixture as identified by Portwood et al. (2016) in their
most turbulent patches), and so points within the red and blue tails of Figure
2a could be direct neighbours in space. Therefore, in $\S$4, we extend our
pointwise analysis by identifying spatially extended and coherent stable
regions, which appear to take the form of ‘interfaces’ with enhanced density
gradients. We then assess the significance of these non-overturning structures
to the overall mixing statistics.
Figure 3: Mixing patterns within the vortex. Vertical averages of the a)
dissipation rate of kinetic energy $\epsilon$, b) dissipation rate of scalar
variance $\chi$, c) vertical velocity $w$, and d) density perturbations
$\rho^{\prime}$, for the vortex region delineated in Figure 1a. The outer
green circle in panels a)-d) coincides with the black circle in Figure 1a. The
gray curve in panel d) delineates the sharp lateral gradient separating
regions of positive and negative $\rho^{\prime}$ and is found to be correlated
with the spiral distribution of $\chi$ observed in panel b). Radial averages
of e) the azimuthal velocity $u_{\theta}$, f) $\epsilon$, g) $\chi$ and h)
$\Gamma$, as a function of the distance $r$ from the center of the vortex,
with shading denoting the standard deviation around the radial mean. In panel
h),
$\left\langle\Gamma\right\rangle_{r}=\left\langle\chi\right\rangle_{r}/\left\langle\epsilon\right\rangle_{r}$
is the ratio of the red lines in panels f) and g). The vertical green dashed
lines in panels e)-h) mark the radial locations of the maximum and first zero
of the radially-averaged azimuthal velocity, and correspond to the radii of
the green dashed circles in panels a)-d).
## 4 Extended mixing structures
We now consider coherent spatial distributions of the microstructure
quantities $\epsilon$ and $\chi$, and their relation to the large-scale flow
patterns observed in Figure 1, by focusing on mixing structures arising both
within and outside of the vortex. We first perform a closer examination of
mixing within the vortex, as shown in Figure 3. Vertical averages of
$\epsilon$, $\chi$, $w$ and $\rho^{\prime}$ are plotted in Figures 3a-d,
respectively, highlighting clear differences in the spatial distributions of
$\epsilon$ and $\chi$. Such differences are further illustrated in Figures
3e-h, which show the respective radial distributions of the azimuthal velocity
$u_{\theta}$, $\epsilon$, $\chi$ and $\Gamma$ with respect to the vortex core.
These radial distributions illustrate that the inner section of the vortex,
characterized by roughly rigid-body rotation, is well-mixed and contains the
largest values of $\epsilon$ despite having minimal scalar diffusion rates
$\chi$. This observation is consistent with the density field shown in Figure
1d, where initially horizontal contours of constant density (green lines) are
strongly deflected toward the vertical before reaching the center of the
vortex, resulting in a vertically-extended core of roughly constant density
(seen predominantly in the vertical interval $25\lesssim z\lesssim 175$).
Conversely, the majority of $\chi$ is found outside the core at radii where
the angular velocity begins to decay, and is distributed in a roughly spiral
pattern (Figure 3b). Examination of the vertically-averaged perturbed density
field $\rho^{\prime}$ (Figure 3d) reveals the presence of a strong lateral
density gradient, induced by the alternating upwelling of dense fluid and
downwelling of lighter fluid within the vortex as a function of $r$.
Superimposing the position of this lateral gradient (gray) onto the
distribution of $\chi$ in Figure 3b reveals that this gradient is strongly
correlated with the spiral distribution of the most intense $\chi$. While the
vortex was identified by Portwood et al. (2016) to be a patch of vigorous
turbulence with elevated $\epsilon$ due to its generation of significant local
vertical overturning, our analysis suggests that much of the mixing within the
vortex, as quantified by $\chi$, instead results from diffusion across a
strong lateral gradient in the perturbed density field.
Outside of the vortex, the vertical homogeneity of the velocity and density
fields collapses, forming a vertically layered structure. In Figure 4, we
consider a vertical $(x,z)$ slice of the domain at position $y=200$ in Figure
1a, in order to understand how this large-scale layering pattern gives rise to
mixing at the microscale. Motivated by the significance of the stable tail
(blue) in the pointwise distribution of $\partial\rho/\partial z$ in Figure
2a, and the observation of horizontally-extended stable filaments of
$\partial\rho/\partial z$ in Figure 4a, we examine whether such structures
contribute substantially to mixing in the layered flow surrounding the vortex.
To isolate these stable filaments, we apply a Gaussian filter to the density
field with standard deviation $\sigma\approx 6L_{K}$ (corresponding to 2 grid
points in Figure 4), where $L_{K}$ denotes the Kolmogorov length scale. The
intent of such a filter is to isolate spatially-coherent stable structures
from patchy overturning regions that would contain a random assortment of
stable and unstable neighbouring points. We note that our filter length is on
the order of $10L_{K}$ as suggested by Kuo & Corrsin (1971) for removing
internal intermittency. Further, it is significantly finer than the Taylor
length $L_{T}\approx 25L_{K}$, which was the smallest filter size considered
by Portwood et al. (2016) in their identification of ‘intermittent layers’,
allowing us to examine the importance of finer-scale structures within the
flow.
Having filtered the density field (Figure 4b), we then extract the most stable
density structures by considering points in the bottom (most stable) $q$
percent of the filtered $\partial\rho/\partial z$ distribution. For
illustration, we here extract structures comprised of the most stable $q=15\%$
of points (Figure 4c), and in Appendix A demonstrate the effect of changing
this percentage. The green contours from Figures 1b,d are overlaid on Figure
4c, demonstrating that the extracted filaments correspond to segments of the
sharp interfaces separating relatively well-mixed layers in the density field.
Importantly, Figure 4d highlights that the concentration of locally-overturned
points (the segmentation indicator used by Portwood et al. (2016)) is greatest
in the regions between these stable interfaces. In Figures 4e,f, we again
highlight that these stable interfaces are also roughly correlated with
regions of high vertical shear in the layered velocity field, as are generated
by the vortex. Corresponding slices of the dissipation rates of kinetic energy
$\epsilon$ and scalar variance $\chi$ are shown in Figures 4g,h, respectively.
The spatial distribution of $\epsilon$ is seen to be much more diffuse than
that of $\chi$, with extreme values of $\chi$ being primarily concentrated
within thin filamentary structures such as those identified in Figure 4c.
Crucially, there are many examples of locations in the flow (see green
crosses, Figures 4g,h) where the stable interfaces identified in Figure 4c
contain highly-elevated local signatures of $\chi$ without a proportional
local increase in $\epsilon$. Such an observation thus raises the question as
to whether these stable interfaces contribute significantly to the total
mixing within the domain, in addition to the mixing occurring in more
conventionally-studied isotropic overturning regions (such as the large
overturn located in the vicinity of $(x,y)=(450,125)$ in Figure 4).
Figure 4: Characteristics of the layered flow outside the vortex. Vertical
slices at $y=200$ in Figure 1a of: a) the density field $\partial\rho/\partial
z$ normalized by the magnitude of the imposed background gradient; b) the
result of applying a Gaussian filter with a standard deviation of two
gridpoints ($\approx 6L_{K}$) to the density field in panel a); c) extracted
stable filaments from panel b) obtained by retaining points in the bottom
(most-stable) $15\%$ of the filtered density distribution; d) the local
fraction of unstable overturned points computed with a filter size
corresponding to the Taylor length ($L_{T}\approx 25L_{K}$), following the
method of Portwood et al. (2016); e-f) the $x$ and $y$ horizontal components
of velocity, respectively; and g-h) the dissipation rates of kinetic energy
$\epsilon$ and scalar variance $\chi$, respectively. The stable filaments from
panel c) are overlaid on panels d)-f) for reference. The contours coloured
green in panels c-f) and white in panels g-h) correspond to those plotted in
Figures 1b,d, marking the stable interfaces separating relatively well-mixed
density layers. Green crosses in panels g-h) indicate examples of locations
where $\chi$ is locally high, due to the presence of a stable density
interface, without a corresponding increase in local $\epsilon$.
We address the question of whether the identified stable filaments contribute
substantially to the total mixing occuring within the domain in Figure 5,
where we consider the relative contributions of both the vortex and the
isolated stable interfaces to the domain totals of $\epsilon$ and $\chi$. In
agreement with Portwood et al. (2016), despite occupying less than $10\%$ of
the domain volume, the vortex contributes approximately a third of the entire
domain’s $\epsilon$ and $\chi$ (red bars, Figure 5a). Outside of the vortex,
however, it is the stable interfaces that play a key role in the overall
mixing, contributing
$\frac{\textrm{(\% in interface) $\cap$ (\% outside vortex)}}{(\textrm{\%
outside vortex) }}=\frac{26\%}{26\%+40\%}=39\%$ (4)
of the total $\chi$ outside the vortex, despite appearing unremarkable based
on their much smaller contribution to $\epsilon$
($11\%/\left(11\%+55\%\right)=17\%$). Figure 4d thus highlights a key
conclusion of this study: while the concentration of overturned points is most
prevalent within the well-mixed density layers, relatively thin stable
interfaces between such relatively deep layers, which are also correlated with
high vertical shear, yield a crucial component of the bulk scalar mixing rate
$\chi$. In particular, Figures 5b,c highlight that while these interfaces may
be strongly distinguished by their distributions of $\chi$, where the median
values differ by almost an order of magnitude, they are virtually
indistinguishable based on their distributions of $\epsilon$. This mismatch
between the spatial distributions of $\epsilon$ and $\chi$ results in
significantly elevated $\Gamma$ within the interfaces, well above the
canonical value $\Gamma=0.2$ (Figure 5d). It thus appears crucial to consider
the independent information provided by the distributions of $\epsilon$ and
$\chi$ within a domain when quantifying mixing, particularly for identifying
the locations of the most extreme scalar mixing events.
Figure 5: Mixing contributions of the vortex and stable interfaces. a)
Fractional contributions to the domain total of the dissipation rates of
kinetic energy $\epsilon$ and scalar variance $\chi$, from within the vortex
(red), stable interfaces outside the vortex (dark blue), as illustrated in
Figure 4c, and the rest of the domain outside both the vortex and interfaces
(light blue). Histograms of b) $\epsilon$, c) $\chi$ and d) $\Gamma$ outside
the vortex (black), further split according to whether points are contained
within a stable interface (dark blue) or not (light blue). Dashed lines
indicate median values of the respective distributions.
## 5 Discussion
We have considered local correlations between the vertical density gradient
$\partial\rho/\partial z$ and the dissipation rates of kinetic energy
$\epsilon$ and scalar variance $\chi$ in order to characterize the spatial
distributions of mixing within a forced direct numerical simulation of
density-stratified turbulence. The forcing scheme is found to generate a
vertically-aligned vortex within the domain, largely explaining the
concentrated ‘patch’ region of vigorous turbulence reported by Portwood et al.
(2016). Outside of the vortex, the flow is characterized by a layered density
profile, with thin, highly stable interfaces separating relatively well-mixed
layers. While a mixing analysis based solely on the identification of local
overturning would deem the well-mixed layers to be of primary importance, as
in the identification of ‘intermittent layers’ with elevated $\epsilon$ by
Portwood et al. (2016), we have demonstrated that a significant fraction of
$\chi$ is localized to the edges of such layers, within the _stable_
intervening interfaces. Notably, these interfaces appear unremarkable if
looking at $\epsilon$ alone (see Figure 5b), emphasizing the importance of
$\chi$ as an independent indicator of local mixing. A number of other studies
have also highlighted that significant mixing rates may be found in regions
devoid of local overturning, emphasizing the importance of considering other
mixing mechanisms present within stratified flows. For instance, by
considering a different class of forced direct numerical simulations to those
analyzed here, Basak & Sarkar (2006) demonstrated that horizontal shear is
able to generate a complex pattern of vorticies which efficiently mix the
density field without local overturning. A striking experimental demonstration
of dye being transported across stationary, highly-stable density interfaces
has been demonstrated by Oglethorpe et al. (2013), where a “scouring” rather
than overturning dynamic generates the mixing. As the flux coefficient
$\Gamma=\chi/\epsilon$ has been found to strongly depend on the time history
of a turbulent event (Mashayek et al., 2021), it would be instructive to now
consider the time evolution and formation of the stable interfaces identified
in our study, characterized by strongly elevated $\Gamma$. For instance, as
the density interfaces are correlated with regions of high vertical shear, it
is conceivable that they might be remnants of the previous collapse of shear-
induced billows that are now only visible in signals of $\chi$ but not
$\epsilon$, as coined ‘fossil turbulence’ by Nasmyth (1970).
Our findings have two potential implications for the parameterization of ocean
mixing. Firstly, our analysis highlights the importance of adequately sampling
rare, yet extreme mixing events in a turbulent flow, as was also recently
discussed by Cael & Mashayek (2021). In agreement with the analysis of
oceanographic data by Couchman et al. (2021), Figure 2d demonstrates that
although the majority of the domain indeed appears to be well characterized by
the canonical flux coefficient $\Gamma=0.2$, significantly elevated $\Gamma$
is associated with the most extreme events in $\chi$, events that are not
reflected by a corresponding local increase in $\epsilon$. Given the current
relative sparsity of measurements within the ocean, mixing parameterizations
may thus be biased toward the most commonly measured events, which are not
necessarily the most significant. Secondly, even with perfect sampling,
different proxies for mixing are likely to yield contrasting predictions for
the amount and spatial distribution of mixing within the highly-anisotropic
layered flow considered here. For example, if measurements of $\chi$ were not
available, the stable filaments at the edges of the overturning layers (Figure
4d) would appear unremarkable, as they appear locally quiescent based on their
density gradient and are not correlated with any discernible increase in
$\epsilon$. Further, given the strong spatial variability of $\Gamma$ within
the vertically-layered flow (see Figure 5d), it is unclear what value of
$\Gamma$ should be used in the method of Osborn (1980) if trying to infer a
flux from values of $\epsilon$ measured directly by a microstructure profiler
or derived from a Thorpe overturning analysis.
As discussed by Caulfield (2021), an accurate parameterization of the flux
coefficient $\Gamma$ is likely to depend on multiple dimensionless groups
characterizing the underlying flow, such as the buoyancy Reynolds number
$Re_{b}$, Froude number $Fr$, and Prandtl number $Pr$. For instance, DNS
studies have demonstrated that bulk-averages of $\Gamma$ decrease with
increasing $Pr$ (Salehipour et al., 2015) and decreasing $Fr$ (Maffioli et
al., 2016). A promising future direction of inquiry would be to try and
rationalize such variations in $\Gamma$ in terms of differences in the
prevalence and structure of smaller-scale extreme events within the flow, such
as analyzing changes in the morphology of the stable filaments considered
here. It would also be instructive to extend our analysis to simulations of
decaying turbulence which also develop layered structures (de Bruyn Kops &
Riley, 2019), in order to establish whether the spatial distribution of mixing
events observed here changes significantly in forced versus unforced
scenarios.
Finally, following Portwood et al. (2016) and typical oceanographic
measurements, we have here primarily relied upon the local density gradient
$\partial\rho/\partial z$ to inform our analysis of spatial mixing patterns.
However, there are likely more optimal flow variables, or linear combinations
thereof, that could lead to a more robust segmentation of the turbulent domain
into distinct regimes. For example, one could imagine constructing more
insightful indicator functions of mixing from components of the velocity
gradient tensor $\partial u_{i}/\partial x_{j}$, as suggested by de Bruyn Kops
et al. (2019). Applying data-driven techniques, such as unsupervised
clustering or dimensionality reduction, to the wealth of observational,
experimental and numerical stratified turbulence data currently available has
the potential to discover automatically optimal mixing indicators free of
human bias. Such an analysis would hopefully further our understanding of the
dominant mixing mechanisms present in different flow regimes, along with their
prevalence, guiding the search for a more universal and accurate mixing
parameterization.
[Acknowledgements] This research used resources of the Oak Ridge Leadership
Computing Facility at the Oak Ridge National Laboratory, which is supported by
the Office of Science of the U.S. Department of Energy under Contract No. DE-
AC05-00OR22725. SdeBK was supported under U.S. Office of Naval Research Grant
number N00014-19-1-2152. For the purpose of open access, the authors have
applied a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising from this submission.
[Declaration of interests]The authors report no conflict of interest.
## Appendix A Thresholding of stable filaments
The stable filaments (black) plotted in Figure 4c were extracted by
identifying points within the bottom (most stable) $q=15\%$, by volume, of the
Gaussian-filtered distribution of the density gradient $\partial\rho/\partial
z$ (Figure 4b). We here briefly consider how changing this thresholding
percentage $q$ influences the characteristics of the extracted stable
structures.
In Figure 6, we plot the stable structures that are identified by varying the
percentage $q$ from $5\%$ to $30\%$. As $q$ is increased, meaning that more
points in the stable tail of the filtered $\partial\rho/\partial z$
distribution are considered, the identified stable structures are found to
grow primarily in the horizontal direction, tracing out more of the stable
interfaces identified by the green contours from Figures 1b,d. Figure 6 thus
highlights that the magnitude of the vertical density gradient along such
stable contours is not uniform, with certain segments having stronger
gradients (as identified by using a smaller $q$) and thus being characterized
by a larger local $\chi$.
It is also natural to consider how the mixing statistics presented in Figure
5a depend on the thresholding percentage $q$. In Figure 7, considering only
the computational domain outside of the vortex, we plot the percent
contribution of the extracted interfaces to $\epsilon$ and $\chi$, as a
function of $q$. The points at $q=15\%$ correspond to the statistics presented
in Figure 5a, noting that in Figure 7 the percent contributions are normalized
by the domain total outside the vortex, and not the entire domain including
the vortex as in Figure 5a. Figure 7 demonstrates that over a wide range of
threshold percentages $q$, the identified stable filaments always contribute
over twice the amount of $\chi$ as compared to $\epsilon$.
Figure 6: Extracted stable filaments as a function of the thresholding
percentage $q$. Filaments (black) are identified as the union of points within
the bottom (most stable) $q\%$ of the filtered density distribution (see
discussion in $\S$4). The filaments detected using values of $q$ between $5\%$
and $30\%$ are shown in panels a)-f), respectively. Panel c) corresponds to
Figure 4c.
Figure 7: The percent contributions of $\epsilon$ and $\chi$ contained within
the stable interfaces identified in Figure 6, normalized by the domain totals
outside of the vortex, as a function of the thresholding percentage $q$. The
statistics at $q=15\%$ correspond to those presented in Figure 5a, noting that
in Figure 5a the contributions are normalized by the total domain including
the vortex.
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|
MRI]Materials Research Institute, The Pennsylvania State University,
University Park, PA 16802 MTSE]Department of Materials Science and
Engineering, The Pennsylvania State University, University Park, PA 16802
MTSE]Department of Materials Science and Engineering, The Pennsylvania State
University, University Park, PA 16802 [ICDS]Institute for Computational and
Data Sciences, The Pennsylvania State University, University Park, PA 16802
# Crystal Growth Characterization of WSe2 Thin Film Using Machine Learning
Isaiah A. Moses [ Chengyin Wu [ Wesley F. Reinhart<EMAIL_ADDRESS>[
###### Abstract
Materials characterization remains a labor-intensive process, with a large
amount of expert time required to post-process and analyze micrographs. As a
result, machine learning has become an essential tool in materials science,
including for materials characterization. In this study, we perform an in-
depth analysis of the prediction of crystal coverage in WSe2 thin film atomic
force microscopy (AFM) height maps with supervised regression and segmentation
models. Regression models were trained from scratch and through transfer
learning from a ResNet pretrained on ImageNet and MicroNet to predict
monolayer crystal coverage. Models trained from scratch outperformed those
using features extracted from pretrained models, but fine-tuning yielded the
best performance, with an impressive 0.99 $R^{2}$ value on a diverse set of
held-out test micrographs. Notably, features extracted from MicroNet showed
significantly better performance than those from ImageNet, but fine-tuning on
ImageNet demonstrated the reverse. As the problem is natively a segmentation
task, the segmentation models excelled in determining crystal coverage on
image patches. However, when applied to full images rather than patches, the
performance of segmentation models degraded considerably, while the regressors
did not, suggesting that regression models may be more robust to scale and
dimension changes compared to segmentation models. Our results demonstrate the
efficacy of computer vision models for automating sample characterization in
2D materials while providing important practical considerations for their use
in the development of chalcogenide thin films.
##### Keywords:
WSe2 thin film, Crystal coverage, Machine learning, Semantic Segmentation,
Transfer learning, Materials characterization
## 1 Introduction
Great advances are being made in the synthesis of two-dimensional materials
(2D)1, 2, 3, since the successful isolation of graphene in 20044. The
transition metal dichalcogenides (TMD) is a major class of 2D materials that
have gained much attention due to their interesting properties and potential
for applications in areas including electric and optoelectronic, energy, and
sensing3, 5. A number of synthesis methods, including mechanical exfoliation3,
powder vaporization6, 7, pulsed laser deposition8, chemical vapor deposition
(CVD), and metal organic chemical vapor deposition (MOCVD)9, 10, 11, 12 are
being deployed in a bid to improve both the quality and scalability of the
grown TMDs. Associated with the materials synthesis is the need for an
efficient characterization technique to determine the various features of the
samples, ranging from the basic crystal qualities to the determination of the
properties and potential applications of the materials13, 14.
Atomic force microscopy (AFM) is a scanning probe microscopy that is widely
applied in 2D materials characterization due to its versatile capability in
electrical, mechanical, chemical, thermal, electrochemical, and topological
characterization of samples15, 16, 17. The topological mode of the AFM is
crucial in determining the quality and properties of a sample as it is used to
produce an AFM image from which several characteristics, including crystal
coverage, domain size, shape and thickness, and nucleation density can be
determined18, 19, 10, 20, 21. Given the fundamental role the information from
the AFM image analysis plays in determining the grown sample’s quality, even
before further characterization to determine their properties and potential
applications, the fidelity and efficiency of the analysis are of major
priority in the workflow to accelerate the 2D materials qualitative and
quantitative synthesis and exploration.
The conventional approach to AFM image analysis, such as with manual image
correction in ImageJ22, is prone to inconsistencies that inherently arise from
human errors. Beyond potential inaccuracies, this laborious and time-consuming
process can become a bottleneck in the materials discovery process. Therefore,
the deployment of a method that minimizes human interference and could be
applied to thousands of images in a matter of seconds is a necessity for a
high throughput synthesis and characterization of TMDs. The application of
machine learning (ML) for AFM image analysis provides an alternative that
eliminates the limitation posted by the manual analysis23.
A number of studies have been reported on the deployment of ML models to the
AFM image analysis. Among them are the segmentation of the molecular resolved
AFM images23, classification of quasi-planar molecules that spans relevant
structural and compositional moieties in organic chemistry based on AFM
images24, identification of self-organized nanostructures25, extraction of
molecule graphs of samples from AFM images26, atomic structure recovery from
AFM images27, and quantitative analysis of MoS2 thin film micrographs.28
Crucial to the determination of the quality of the materials synthesis is the
domain size and thickness, and surface coverage29, 12, 18, 19, an isolation of
the grown crystal from the substrate on which it is grown.
The crystal coverage is a basic metric that indicates the extent to which the
thin film has grown on the substrate. A rapid and automated determination of
the crystal coverage can enhance materials synthesis as the growth parameters
can be optimized based on this figure of merit. In our present study,
convolutional regression models are developed to be deployed in determining
the crystal coverage of 2D WSe2 grown using MOCVD9. Additionally, robust
semantic segmentation models30, 31, 32, 33, 34 which give a pixel-wise
classification of the grown samples AFM images, as either belonging to the
substrate or the crystals, are trained. Our models exhibit excellent results
with $R^{2}$ exceeding 0.99 in quantification of the crystal coverage in held-
out test samples.
Furthermore, we have systematically evaluated the efficacy of different
transfer learning schemes, namely feature extraction and fine-tuning. We also
include the effects of different pretraining domains, specifically materials
micrographs compared to miscellaneous everyday objects. Our results have some
important and counter-intuitive implications on the practical implementation
of these computer vision models in materials characterization workflows.
## 2 Method
### 2.1 Dataset
Figure 1: WSe2 samples used in the study showing the growth parameters space.
$T$ is the growth chamber inner temperature, $P$ is the pressure, and time is
the growth time. Multiple micrographs are obtained for each sample, so there
are fewer unique conditions than images in our study. Bolder circles indicates
more samples at the same point. Some samples in the test set occupy unique
points in the parameter space, such as the samples at the lowest $T$. Figure
2: Sample AFM images of WSe2 thin film in our dataset. Data ingested in our
workflow have already been preprocessed by other software and include
dimensional scale bar, color scale, and text annotations.
The WSe2 AFM data used in this research were grown by Eichfeld et al.9 and
stored in the Lifetime Sample Tracking (LiST), a database hosted by the 2D
Crystal Consortium (2DCC).35 The 52 WSe2 thin film samples were synthesized
using the metal-organic chemical vapor deposition (MOCVD) technique. The
samples were grown at various conditions, including the growth time, chamber
inner temperature, and pressure (Fig. 1), resulting in significant variations
in the morphological features of the AFM micrographs obtained. Additionally,
different imaging conditions were employed for the samples, with
characterization obtained at the centers and edges of the wafer and at
different resolutions. This resulted in a total of 221 micrographs from the 52
grown samples.
The micrographs were preprocessed by another software which performed
flattening, inserted a colorbar, and annotated the images with text labels and
a scale bar. These images were finally stored as TIF files such as those shown
in Fig. 2. One important consequence of this choice is that our models were
trained not on height maps, but on height-normalized images. That is, the
relationship between pixel intensity and the original height measurement was
different within each image. The same was true of the length scale, where
pixels represented different sample area within each image. We believe this
better represents the practical use case for these models compared to
carefully controlled height and length scales.
Figure 3: Sample images with their lightness histograms demonstrating how
segmentation could be performed on the basis of a bimodal lightness
distributions (one for foreground, one for background). This assumption is
often violated due to imperfect flattening, texture, or artifacts.
The figure of merit for these thin film sample is the monolayer coverage,
which can be computed from an AFM height map according to the fraction of
pixels in the foreground compared to the overall image. This essentially
reduces the problem to a segmentation task, which has many possible solutions.
One simple method to perform binary segmentation (i.e., foreground/background
separation) is to define a lightness threshold (corresponding to a height
threshold) based on the assumption of a mixture of approximately Normal
distributions for each height range of interest (such as background and
foreground). Each image was cropped to only the AFM micrograph portion (no
padding, annotations, color bar, scale bar, etc.), a lightness histogram was
prepared, and a threshold value was selected based on an assumed bimodal
distribution, as shown in Fig. 3. Choosing this threshold produces a binary
mask for each image; these thresholds were chosen and masks evaluated manually
for each micrograph. This labeling procedure resulted in 221 image-mask pairs,
from which the monolayer coverage was computed by counting the number of
pixels above the lightness threshold (i.e., masked).
#### 2.1.1 Augmentation
A dataset consisting of only 221 images might be insufficient to effectively
train a robust ML model. Therefore, in this study, we utilized image patching,
a common data augmentation technique to generate additional data points with
greater variance in image characteristics, thus creating a more diverse
dataset for deep learning model training. We utilized the random transforms
implemented in torch-vision from the Pytorch library36 to generate the image
patches, with a final patch height and width of $224\times 224$ for regression
models and $512\times 512$ for the segmentation models. Each patch had an
equal and independent chance of being flipped vertically, horizontally,
$0-360^{\circ}$ rotation, $0.5-2.0\times$ rescale, and random crop within the
rescaled image. An example of this procedure is shown in Fig. 4. Because this
random transformation could result in out of bounds pixels, we rejected any
patch that did not fall entirely within the original image. We repeated this
sampling until 10 valid patches were obtained for each image.
Figure 4: Data augmentation and image patching schematic scheme. The original
AFM image (top left) is thresholded to produce a mask (top right). Random
image patches are jointly taken from both the image and mask to yield new
(image, mask, coverage) sets where all patches are of size $224\times 224$ but
represent different portions of the original image.
### 2.2 Regression Models
Figure 5: Schematic of different transfer learning paradigms. Feature
extraction is a scheme that only modifies the trainable weights in the fully
connected layer (or other shallow model) while leaving the pre-trained weights
in the convolutional layers unchanged. In fine tuning, all the trainable
weights from the pre-trained model are adjusted to improve the model’s fit to
the new task. In end-to-end learning, the entire model is trained from
scratch, without any knowledge transfer.
We consider two variants of the ML task: regression (predicting the coverage
label directly from the image) and segmentation (predicting the binary mask
and then computing the coverage from the mask). Within the regression task, we
further consider three training paradigms: training from scratch using end-to-
end learning (i.e., with randomly initialized weights), transfer learning by
fine-tuning (i.e., intializing the model with pretrained weights), and
transfer learning by feature extraction (i.e., training a shallow model to
predict target label with pretrained convolutional filters).
For all the regression models, Adam optimizer, ReLU activation function, and
mean squared error (MSE) loss functions were used. 10% of the data samples,
grown under different growth parameters than the rest of the data and/or
obtained under different imaging conditions, were held out to determine how
well the models generalize to out of distribution data (Table 1).
Additionally, about 80% and 10% were used for the training and validation,
respectively.
We started by training a small Convolutional Neural Network from scratch
(CNNsc). The architecture of the CNNsc network was optimized using Bayesian
hyperparameter tuning implemented in the ax-platform package37 which leverages
a Gaussian-process-based Bayesian optimization38.. After each of the
convolutional layers, a max pooling and ReLU activation function were applied
to downsize the feature maps and extract the most important features, and
introduce non-linearity, respectively. This network was deliberately
simplified compared to the pretrained models to evaluate whether fewer
trainable weights would be more robust in extrapolating to the test domain.
We also explored the application of pretrained models, specifically ResNet18
architecture pre-trained on ImageNet39 and MicroNet40 datasets, to predict the
coverage of WSe2 thin films. We chose ResNet18 as it is among the shallowest
standard computer vision architectures available today, which we felt was
important given our low data volume. The features were extracted from the
average pool layer of the pretrained models, given 512 features. MLP models
were then built to learn the crystal coverage from the image features obtained
from the ResNet18 pretrained on the ImageNet and MicroNet. The MLP models are
hereafter referred to as MLP-I and MLP-M, respectively. MLP model
hyperparameters were tuned using ax-platform as in the case of the CNNsc.
For completeness, we also employed the fine-tuning paradigm of transfer
learning. This allowed us to assess the performance of these pre-trained
models in our specific context and evaluate their potential for accurate thin
film coverage prediction. The pretrained models’ classifiers were replaced
with 2 FC layers of 512 and 100 neurons and an output layer. Between the 2 FC
layers is a ReLU activation function to introduce non-linearity and a dropout
of 0.25 to minimize over-fitting. Sigmoid activation function was additionally
placed before the output layer to ensure only values between 0.0 and 1.0
(range of coverage values) are predicted. The models were then tuned with our
data to learn the crystal coverage. The fine-tuning were carried out for the
ResNet18 pretrained on the ImageNet (CNN-I) and another on the MicroNet
(CNN-M).
### 2.3 Segmentation Models
Separately from the regression task, we attempt to solve the problem using
segmentation models to work natively with the binary mask. Similar to the
regression models, encoders pretrained on MicroNet by Stuckner et al. 40 were
used. In their report, they found ResNeXT,41 SE,42 Inception,43 and
EfficientNet44 encoder architectures to give better performances.
Additionally, Unet45 and Unet++46 decoders were found to outperform others.
Specifically, SE_ResNeXt-50_32x4d and SE_ResNeXt-101_32x4d encoders pretrained
on MicroNet coupled with Unet++ decoders gave, on the average, the best
intersection over union (IoU) accuracy for models trained on the full sets of
2 different SEM images (nickel-based superalloys and environmental barrier
coatings). We therefore used SE_ResNeXt-50_32x4d and SE_ResNeXt-101_32x4d
encoders pretrained on MicroNet coupled with Unet++ decoders in our study.
These segmentation models are termed SEG50 and SEG101, respectively.
In order for us to compare the performance of the segmentation and regression
models from the same pretrained architectures, we have additionally trained
segmentation models based on the ResNet18 pretrained encoder and using the
Unet++ decoder. Both encoders pretrained on the ImageNet and MicroNet were
used, and termed SEG18-I and SEG18-M, respectively. The Adam optimizer, 1e-4
learning rate, and a batch size of 6 were used on the training. We utilized an
early stopping after 30 epochs of training without further improvement on the
IoU accuracy of validation set, while the loss function was a weighted sum of
balanced cross entropy (BCE) and dice loss with a 70% weighting towards BCE.
## 3 Results and Discussion
### 3.1 Regression Models
#### 3.1.1 Training from Scratch
Figure 6: (a) is the CNN architecture built from scratch (CNNsc) showing the
convolutional (Conv1, Conv2, Conv3, and Conv4), the pooling (MaxPool), and
fully connected layers (FC1, FC2, FC3), as well as the feature maps and
channel sizes for each of the convolution layer and the neurons connecting the
FC layers. (b) is the root mean squared error value (RMSE) on the train and
validation (val) data against the learning iteration (epochs). (c) is the
parity plot of the predicted and target coverage. The $R^{2}$ and RMSE values
in (c) are for the test set.
The architecture of the CNNsc network found by hyperparameter tuning consisted
of four convolutional layers and three fully connected (FC) layers (Fig. 6).
The kernel size was 5 with a stride of 1 and zero padding. This model was
trained to minimize the MSE loss between the target and the predicted
coverage. A stochastic behavior is observed in the learning resulting in the
fluctuation in losses with the training iterations both for the training and
validation set (Fig. 6(b)). The random initialization of the weights might
have resulted in such behavior. To obtain an optimally trained model, the
model was set to stop once the minimal obtainable value of the training and
validation loss was achieved. This results in the model’s performance with
train, validation, and test set RMSE of 0.022, 0.062, and 0.042, respectively
(Fig. 6(c) and Table 1). These correspond to $R^{2}$ values of 0.995, 0.961,
and 0.982 for train, validation, and test, respectively. Only a few scattered
points were observed in the validation and test parity plots, indicating a
minimal over-fitting.
#### 3.1.2 Feature Extraction
Figure 7: (a), (c), and (e) are the root mean squared error value (RMSE) on
the train and validation (val) data against the learning iteration (epochs)
for the multilayer perceptron model (MLP) trained with features extracted
using the ResNet18 pretrained on the ImageNet data (MLP-I), MLP trained with
features extracted using the ResNet18 pretrained on the MicroNet data (MLP-M),
and for the CNN built from scratch without on-the-fly data augmentation
(CNNsc*), respectively. (b), (d), and (f) are the parity plots of the
predicted and target coverage corresponding to (a), (c), and (e),
respectively. The $R^{2}$ and RMSE values in (b), (d), and (f) are for the
test set.
The MLP architectures were tuned (with an objective of minimizing the
validation loss) to yield 2 hidden layers with (120, and 84) neurons in the
MLP-I and MLP-M. The trained MLP-I exhibited an $R^{2}$ value of 0.873 on the
test set (Fig. 7 and Table 1). MLP-M performs better than the MLP-I, though
still slightly worse than the CNNsc. A better performance observed in the
MLP-M than the MLP-I might be due to the proximity of the data for the
pretraining and our data; MicroNet consists of gray scale micrographs while
ImageNet is made up of the macroscale color images of natural objects. The
features extracted from the former may therefore be more relevant in learning
our image features than those from the latter.
The superior performance of the CNNsc may be due to its smaller size or its
on-the-fly data augmentations; random rotations and flips were applied to the
data while training. To verify if the data augmentations applied to the CNNsc
made a significant difference to the model performance, we trained the same
architecture of CNN with the same hyperparameters without the augmentations
(CNNsc*). The result shows that the augmentations indeed significantly enhance
the performance of the CNNsc (Fig. 7 and Table 1). Overfitting is observed to
set in soon after the first few epochs of training on data without
augmentation. The model accurately predicts the coverage for the train set but
a worse performance than both MLP-I and MLP-M is observed in the validation
and test sets.
However, the on-the-fly augmentation cannot be readily applied in the feature
extraction case as data are not seen by the model more than once. The closest
we can get to the on-the-fly augmentation is to obtain different features for
the rotated and horizontal and vertically flipped images, then training the
MLP model on all of these at once. We also tried average pooling on these
variants as input to the model rather than trying to learn a many-to-one
mapping. Both of these approaches gave worse performance compared to the
vanilla MLP models, with the augmentation giving the $R^{2}$ values of 0.86
for the MLP-I and 0.93 for the MLP-M, while the pooling strategy was worse.
These results underscores a fundamental difference in the static augmentation
of the data for the MLP models and the on-the-fly augmentation for the CNN
models.
#### 3.1.3 Fine-Tuning
Figure 8: (a), and (c) are the root mean squared error value (RMSE) on the
train and validation (val) data against the learning iteration (epochs) for
the fine-tuned ResNet18 pretrained on the ImageNet data (CNN-I), the fine-
tuned ResNet18 pretrained on the MicroNet data (CNN-M), respectively. (b) and
(d) are the parity plots of the predicted and target coverage corresponding to
(a) and (c), respectively. The $R^{2}$ and RMSE values in (b) and (d) are for
the test set.
Finally, we examined the fine-tuning of the pretrained model to predict the
crystal coverage. This approach needs to be explored especially because of our
observation of the significant impact data augmentation has on CNN model
performance. Fine-tuning is carried out for the ResNet18 pretrained on the
ImageNet and another on the MicroNet. These models are termed CNN-I and CNN-M,
respectively. As observed in the CNNsc, capturing the grokking effect is
important in obtaining the optimally trained model; the training and
validation losses were closely monitored, and the training halted once the
minimal obtainable validation loss is reached. The validation loss associated
with the grokking point was determined by an initial training for the models
for a few thousands epochs. The performance of the CNN-I and CNN-M are quite
similar, with CNN-I giving a marginally better result. Both have accurate
predictions on the validation and test set with $R^{2}$ value of 0.99 (see
Fig. 8 and Table 1).
Interestingly, while a significantly better performance is observed from
features extracted from the model pretrained on MicroNet than that from the
ImageNet, the fine-tuning shows the reverse. This means that the filters
pretrained on the MicroNet extract much useful features from the AFM than that
pretrained on the ImageNet. However, the latter scenario seems to provide more
generic image features in which case fine-tuning on sufficient target data has
yielded a better result. A nearly non-existent over-fitting, even on the held-
out test data is noteworthy. The excellent performance of CNN-I and CNN-M
underscores the advantage of not just the transfer learning but also the data
augmentations used with CNN at combating the over-fitting and producing models
that have been accurately trained on our target data which shares generic
features learned from larger data sets used for the pretraining.
#### 3.1.4 Summary of Regression Results
The results of all the regression models have been compiled in Table 1. While
comparable performance on training data can be obtained by all three learning
paradigms, their test performance vary substantially. Fine-tuning yielded the
best results in this regard, followed by training from scratch, then feature
extraction. However, this seems to have been largely a result of on-the-fly
data augmentation, as our ablation study showed that removing this from the
trained-from-scratch CNNsc led to a nearly triple test RMSE, making it the
worst model. Unfortunately, this approach could not be applied to the feature
extraction strategy to improve its performance. Between the two pretraining
domains, there was no clear winner; ImageNet gave better performance in fine-
tuning, while MicroNet was superior in feature extraction. This is not an
obvious result and may warrant further investigation regarding the nature of
the pretrained filters.
Table 1: RMSE and $R^{2}$ values for the predicted coverage on the train, validation (val), and test sets for models trained from scratch and through transfer learning. CNNsc and CNNsc* are the CNN trained from scratch with and without on the fly data augmentation, respectively. MLP-I and MLP-M are the MLP trained using the features extracted with ResNet18 architecture pretrained on ImageNet and MicroNet, respectively. CNN-I and CNN-M are the fine-tuning models of the ResNet18 architecture pretrained on ImageNet and MicroNet, respectively. The best performance in each row is shown in bold, including ties and near-ties. | From scratch | Feature extraction | Fine Tuning
---|---|---|---
RMSE | CNNsc | CNNsc* | MLP-I | MLP-M | CNN-I | CNN-M
train | 0.018 | 0.013 | 0.012 | 0.023 | 0.013 | 0.022
val | 0.039 | 0.120 | 0.098 | 0.047 | 0.021 | 0.030
test | 0.041 | 0.121 | 0.101 | 0.054 | 0.029 | 0.035
$R^{2}$ | CNNsc | CNNsc* | MLP-I | MLP-M | CNN-I | CNN-M
train | 0.997 | 0.998 | 0.998 | 0.995 | 0.998 | 0.995
val | 0.984 | 0.855 | 0.904 | 0.978 | 0.995 | 0.991
test | 0.979 | 0.818 | 0.873 | 0.963 | 0.989 | 0.984
### 3.2 Segmentation Models
Figure 9: (a) and (b) are the parity plots of coverage predicted, using the segmentation model pretrained on the MicroNet, and the target coverage. The encoder for the SEG50 and SEG101 models are SE_ResNeXt-50_32x4d and SE_ResNeXt-101_32x4d, respectively. (c) are sample images, the corresponding ground truth mask, and the predicted mask by the SEG50 and SEG101 models. The $R^{2}$ and RMSE values are for the test set. Table 2: The RMSE, $R^{2}$, and IOU values on the train, validation (val), and test data sets for the segmentation models. SEG18-I and SEG18-M uses ResNet18 pretrained on ImageNet and MicroNet, respectively. SE_ResNeXt-50_32x4d (SEG50) and SE_ResNeXt-101_32x4d (SEG101) encoder are pretrained on MicroNet data. | RMSE | $R^{2}$ | Average IoU (%)
---|---|---|---
| train | val | test | train | val | test | train | val | test
SEG18-I | 0.017 | 0.021 | 0.022 | 0.997 | 0.995 | 0.994 | 89$\pm{21}$ | 88$\pm{26}$ | 90$\pm{13}$
SEG18-M | 0.028 | 0.043 | 0.020 | 0.992 | 0.977 | 0.995 | 87$\pm{22}$ | 87$\pm{27}$ | 90$\pm{14}$
SEG50 | 0.007 | 0.024 | 0.020 | 0.999 | 0.993 | 0.995 | 92$\pm{19}$ | 90$\pm{23}$ | 92$\pm{9}$
SEG101 | 0.013 | 0.020 | 0.025 | 0.998 | 0.997 | 0.992 | 90$\pm{18}$ | 89$\pm{25}$ | 90$\pm{13}$
We now reframe the task as a binary segmentation, where crystal (foreground)
is separated from the substrate (background) and then counted to obtain the
crystal coverage. SE_ResNeXt-50_32x4d and SE_ResNeXt-101_32x4d encoders
pretrained on MicroNet coupled with Unet++ decoders are termed SEG50 and
SEG101, respectively. While ResNet18 encoder pretrained on the ImageNet and
another on the MicroNet with both coupled with the Unet++ are termed SEG18-I,
and SEG18-M, respectively. As this is natively a segmentation problem, it is
not surprising that these models can achieve excellent performance; all the
segmentation models all have marginal improvement over the regression models
as shown in Table 2. To be specific, the best model from the regression
models, CNN-I (Fig. 8 and Table 1) exhibits a test RMSE of 0.029, whereas
SEG18-M and SEG50 both obtain 0.020 RMSE.
Based on the patches of the images, it seems that segmentation models provide
higher performance in determining the crystal coverage than regression models.
Additionally, segmentation models offer the advantage of giving impressive
performances even with a much smaller data set for training40, 47, 48 since
each pixel is in effect a training data point. In our present study, the total
image patches used in the segmentation models are half of that used in the
regression models.
In addition to the coverage value determination, segmentation models provide
pixel-wise classification of the image, classifying each pixel in the AFM
images of WSe2 samples as either belonging to the substrate or the crystal.
This has some additional utility in determining not only how much crystal is
present, but its location in the micrograph. The intersect over union (IoU)
metric shows high performance even on the pixel-level classification task,
with 92% (SEG50) and 90% (SEG101) IoU on held-out test images. It is worth
noting that similar performances are observed on both the train and test sets,
indicating low memorization. This level of generalization, despite the held
out test set samples being grown at different conditions and/or obtained at
different imaging conditions, underscores the potential of the models to
produce reliable results in practical applications.
### 3.3 Inference on Full Images
Figure 10: The original (whole) images in the hold-out test set (first rows),
and the pixel-wise classification, as either belonging to crystal or substrate
(second rows) obtained from the SEG50 model. The intersection over union (IoU)
accuracy for each image is given below the classification. Figure 11: Coverage
analysis and segmentation of the original (whole) test images. Results
obtained using the segmentation models, SEG50 and SEG101, and the best
regression models, CNN-I and CNN-M are shown. The S/No. corresponds to the
image # shown in Fig. 10.
The test set discussed in the previous sections is based on patches created
from the full image test set. However, it is important to characterize the
held-out test set in its original full image format, as this is the real
measure of the practical value of our trained models. For this test, we are
using SEG50 and SEG101 and only the best regression models: CNN-I and CNN-M.
While SEG50 gives best performance on the held-out test set among the
segmentation models, SEG101 and SEG18-I give similar results (Table 2).
The full images were padded such that they match the exact multiple of model
training patch size, $224\times 224$ and $512\times 512$, for regression and
segmentation, respectively, or the last row/column is lost. The tiles (with
the same sizes as those used in training the models) are then obtained from
the full images and the coverage and segmentation are predicted using the
trained models. For CNN-I and CNN-M, the predicted coverage for each tiles is
multiplied by the size of the tile in order to obtain the number of pixels
with the value above the threshold for crystal. The pixel values above the
threshold are added for all the tiles from the same full image. The crystal
coverage of a given full image is then obtained by dividing the sum of the
number of pixels above the crystal threshold from all the tiles by the size of
the full image (the total number of pixels in the full image). Meanwhile for
the SEG50 and SEG101, the resulted segmented tiles are concatenated and the
artificial padding added is removed. The coverage label is then obtained based
on the concatenated segmentation mask.
For the 23 held-out test images which were grown with growth parameters and/or
obtained at different imaging conditions than the train and validation sets
(Table 1), the performance of the models are not as good as on the patched
images for any model. The regression models are at least 30% worse while the
segmentation models are at least four times worse – this means that the
regression models outperform the segmentation models in practice despite worse
test performance on image patches (Figures 10 and 11). The results obtained
from SEG50 are mostly consistent with the results on image patches with an
average IoU accuracy of 86% compared to 92%. Except for a few cases such as
the image #6, 15,and 19, less than 10% error are typical for both the coverage
and the IoU.
In contrast, the SEG101 performed quite poorly, despite being a similar
architecture compared to SEG50, which is surprising because both models give
comparable performance on the patched images. The fact that SEG101 gave the
best result on the first 4 images, which are the same size but different from
the rest of the test set, provides the clue as to why the model performs
poorly on most of the images as well as the SEG50’s lower accuracy on the full
images compared to the patches. Creating the tiles for the full image
inference requires processing that could result in the loss of some parts of
the original images. The resizing involved in the patches created for training
the models is also inevitably not exactly the same as that for the tiles. The
sensitivity of the different models to the different image processing and the
image morphological features have therefore resulted in the observed variation
in the model performances. Also worthy of note is the fact that significant
variations in the segmentation model performances have been observed depending
on the encoder and/or decoder architecture.40
Overall, the results on full images show an important distinction between the
training protocol and real-world application of CNNs. Deep CNNs such as SEG101
may not be robust in practical micrograph analysis despite excellent
performance even on held-out test data due to the image augmentation scheme.
Meanwhile, even though the calculation of crystal coverage is natively a
segmentation problem, the regression models perform well on the full images,
suggesting that they may be more robust to changes of scale, dimension, or
other factors compared to the segmentation models.
## 4 Conclusion
In this study, we conduct a comprehensive analysis of crystal coverage (the
proportion of the substrate covered with grown crystal) in WSe2 thin film
atomic force microscopy (AFM) micrographs using regression and segmentation
models. Regression models were trained to predict the monolayer crystal
coverage from image patches. Models were trained from the scratch and using
transfer learning from ResNet pretrained on ImageNet and MicroNet. MicroNet
consists of grayscale micrographs while ImageNet is made up of the macroscale
color images of natural objects. For transfer learning, both feature
extraction and fine-tuning approaches were used.
Our analysis revealed that the CNN models trained from the scratch outperforms
MLP models trained on features extracted from the pretrained models, while
fine-tuning gave the best performance with up to 0.99 $R^{2}$ value on the
held-out test set. Interestingly, while a significantly better performance is
observed from features extraction using MicroNet than that from the ImageNet,
the fine-tuning shows the reverse. This means that the filters pretrained on
the MicroNet extract more useful features from the AFM than that pretrained on
the ImageNet. However, the latter scenario seems to provide more generic image
features in which case fine-tuning on sufficient target data has yielded a
better result.
Beyond the prediction of crystal coverage over entire patches, segmentation
models provide pixel-wise classification of the image, classifying each pixel
in the AFM images of WSe2 samples as either belonging to the substrate or the
crystal. This has some additional utility in determining not only how much
crystal is present, but its location in the micrograph. Based on the patches
of the images, the segmentation models provide higher performance in
determining the crystal coverage than regression models. The intersection over
union (IoU) metric shows high performance even on the pixel-level
classification task, with up to 92% IoU on held-out test images.
The results on full images show an important distinction between the training
protocol and real-world application of the models. Contrary to the results
from image patches, the regression models performed better than the
segmentation models at predicting the monolayer crystal coverage of the full
images of the held-out test set, giving the $R^{2}$ values of 0.98 and 0.90,
respectively, from the best models. The average IoU on the full held-out test
images reduced to 86% from the 92% obtained for the patch images. Our finding
suggests that the regression models may be more robust to changes of scale,
dimension, or other factors compared to the segmentation models. Overall,
these results highlight the efficacy of machine learning for automated, high-
throughput sample characterization, demonstrating its potential for
accelerating the high-throughput development of chalcogenides for
technological applications. At the same time, it provides practical guidelines
for implementing standard computer vision workflows in real-world materials
characterization applications.
## Acknowledgments
This study is based upon research conducted at The Pennsylvania State
University Two-Dimensional Crystal Consortium – Materials Innovation Platform
(2DCC-MIP) which is supported by NSF cooperative agreement DMR-2039351.
## Data Availability
The raw data required to reproduce these findings are available to download
from Ref.35
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|
# Data-driven classification of low-power communication signals by an
unauthenticated user using a software-defined radio
Tarun Rao Keshabhoina1 and Marcos M. Vasconcelos2 1T. R. Keshabhoina is with
the Department of Electrical Engineering and the Commonwealth Cyber
Initiative, Virginia Tech, USA. Email<EMAIL_ADDRESS>M. Vasconcelos is
with the Department of Electrical and Computer Engineering, FAMU-FSU College
of Engineering, Florida State University, USA. Email<EMAIL_ADDRESS>
###### Abstract
Many large-scale distributed multi-agent systems exchange information over
low-power communication networks. In particular, agents intermittently
communicate state and control signals in robotic network applications, often
with limited power over an unlicensed spectrum, prone to eavesdropping and
denial-of-service attacks. In this paper, we argue that a widely popular low-
power communication protocol known as LoRa is vulnerable to denial-of-service
attacks by an unauthenticated attacker if it can successfully identify a
target signal’s bandwidth and spreading factor. Leveraging a structural
pattern in the LoRa signal’s instantaneous frequency representation, we relate
the problem of jointly inferring the two unknown parameters to a
classification problem, which can be efficiently implemented using neural
networks.
## I Introduction
Multi-agent robotic systems are used in various modern applications, including
industrial automation, agriculture, and environmental monitoring [1, 2]. In
these systems, autonomous robots work together to accomplish a common goal,
such as monitoring an environment or cooperatively completing a task. In such
systems, coordination, and communication among the robots are critical to
their success. Each robot must be aware of the state and actions of the other
robots in the system to coordinate their actions and achieve their goals. For
example, in an agricultural monitoring system, each robot may be responsible
for monitoring a different field area, and they must coordinate their
movements to ensure that the entire field is covered. Therefore, communication
among the robots must be reliable, even in challenging scenarios such as
remote or outdoor environments, which are subject to disruption by obstacles
or malicious interference. Protecting such networks against denial-of-service
attacks is of paramount importance to prevent service disruption and economic
loss.
Figure 1: Block diagram for the communication scenario herein: two legitimate
agents communicate a signal represented by $X$, an attacker observes a
correlated signal $\tilde{X}$, with the intent to emit a jamming signal $J$.
A Low Power Wide Area Network (LPWAN) protocol LoRaWAN (Long Range Wide Area
Network) offers long-range and low-power communication capabilities well-
suited to multi-agent robotic systems [3]. Additionally, LoRaWAN supports
creating large-scale networks with multiple nodes, making it an ideal solution
for coordinating the activities of large groups of robots communicating
intermittently. While LoRaWAN is one of the most robust and resilient low-
power communication protocols, it is still vulnerable to a class of denial-of-
service attacks known as jamming.
A jamming attack follows the diagram in Fig. 1: a transmitting agent, Tx,
sends a signal to a receiving agent, Rx; the transmitted signal is intercepted
by an attacker using a software-defined radio unit; The attacker then infers
two private parameters used for communication between Tx and Rx, and
subsequently sends a jamming signal to interfere with the transmitted signal
at the receiver.
### I-A Related Work
Wireless communication protocols transmit over the air, which makes them
vulnerable to interference from any radio transmitter within their vicinity.
This fundamental aspect of shared media in wireless networks has made way for
extensive research in the wireless jamming domain [4, 5, 6]. Energy-
constrained jamming methodologies attempt to block the channel in reaction to
transmission activity to save power. Herein, we discuss such a reactive
jamming strategy for LoRa PHY. Securing communication systems and improving
performance in the presence of intelligent jammers [7, 8] is the motivation to
this work.
Numerous studies have examined the throughput and performance of ultra-narrow
band (UNB) and spread spectrum-based technologies in the unlicensed
Industrial, Scientific, and Medical (ISM) band [9, 10]. Amongst these, a
comprehensive study of PHY layer vulnerabilities, countermeasures and security
features of LoRaWAN are presented in [11], and its authors also provide a
brief overview of jamming methodologies for LoRa. Long-range transmissions on
LoRa are susceptible to several attack strategies such as replay attacks,
wormhole attacks, and compromising network key information, in addition to
jamming [11].
LoRa’s medium access control (MAC) layer design introduces many configurable
parameters that affect its service reliability. An in-depth explanation of
such parameters, and their resulting performance tradeoffs are presented in
[12]. Choices of these parameters, driven by service requirements, also play a
role in the PHY layer encoding of signals, having implications on the
approaches adopted by intelligent jammers.
When signals from one packet are $6\mathrm{dB}$ stronger than another, it goes
on to be demodulated, leaving the weaker packet to be discarded (this is the
so-called channel capture effect) [13]. Building on the concept, the authors
of [14] have shown that LoRa can be jammed using commercially available
hardware. Herein, they induce collisions on the channel, by flooding it with
numerous packets of identical parameter choices. A more advanced technique,
targetting the symbol demodulation process in LoRa was explored in [15],
introducing the idea of jamming chirps. They revealed that LoRa receivers
cannot distinguish between a well-synchronized jamming chirp and a legitmate
chirp.
LoRa was found vulnerable to interference when two packets employ the same
configuration of two parameters known as the Bandwidth ($BW$) and Spreading
Factor ($SF$). The symbol demodulation process in LoRa involves two steps:
first, dechirping, and then, FFT (Fast Fourier Transform). Symbols are
determined by identifying peaks within the FFT. When interfering packets
utilize the same $BW$ and $SF$, this can cause multiple indiscernible peaks in
the FFT, leading to symbol errors [16].
Contemporary work in LoRa jamming exploit this property, and an empirical
analysis of the approach is discussed in [17]. While they prove the
effectiveness of this strategy, they also make a hard assumption.
Particularly, that the jammer has apriori knowledge of the target signal’s
$BW$ and $SF$ choices, neccessary for generating the jamming chirps. However,
these parameters are generally not available to adversarial agents, which are
unathenticated users of the network.
In this paper, we take a step further, exploring how an adversary may employ a
simple neural network implementation to estimate this information and jam LoRa
signals reactively, without such assumed knowledge. We provide numerical
results on the detection and identification of the $BW$ and $SF$ parameters
from observed signals. Then, we quantify the robustness of our model by
evaluation on a wide range of signal-to-noise ratio (SNR) levels of signals.
The rest of this paper is organized as follows. Section II introduces LoRa PHY
and the chirp spread spectrum. Section III describes system architecture.
Section IV describes our proposed feature extraction technique. Section V
describes the architecture of the neural network classifier. Section VI
presents our simulation results and discusses our system’s performance.
Finally, Section VII concludes the paper and outlines future research
directions.
## II Signal description
Figure 2: A chirp signal with BW = 125 KHz, and SF = 7 in continuous time
(left), discrete time (middle), and its spectrogram (right).
LoRa PHY is a pass band modulation technique that uses chirp spread spectrum
(CSS) to modulate digital information onto a carrier wave. In CSS, a chirp is
a signal whose instantaneous frequency increases or decreases linearly as a
function of time.
In LoRa, each transmitted symbol is mapped into a chirp. The bandwidth ($BW$)
and spreading factor ($SF$) are the most critical parameters defining a LoRa
chirp. The $BW$ corresponds to the range of frequencies of the channel
occupied by the chirp, and the $SF$ determines the number of bits transmitted
in a symbol. Each symbol carries $SF$ bits (i.e., values ranging from $0$ to
$2^{SF}-1$). The joint choice of $SF$ and $BW$ determines the data rate of the
communication link. Following [18], in this section, we describe the CSS
modulation.
A fundamental characteristic of the LoRa chirp is its cyclically shifted
frequency. Wherein, the frequency incrementally rises from the initial
frequency in discrete steps. Upon reaching the highest frequency, it wraps
around to the lowest frequency and continues its ascent until it cycles back
to the initial frequency. The chirp encodes information by adjusting its
starting frequency according to its symbol value, $s_{n}$.
Consider the transmission of a sequence of symbols $\mathbf{s}:=\\{s_{n}\\}$.
Each symbol carries $SF$ bits, denoted by a vector
$\mathbf{w}_{n}=(w_{n,0},\ldots,w_{n,SF-1})$, where $w_{n,b}\in\\{0,1\\}$,
$b\in\\{0,\ldots,SF-1\\}$. A new symbol is transmitted every $T_{s}$ seconds,
corresponding to a chirp signal’s duration in time. The value of the symbol
$s_{n}$ is given by
$s_{n}=\sum_{b=0}^{SF-1}w_{n,b}\times 2^{b}.$ (1)
Since $s_{n}$ can take on $2^{SF}$ distinct values, the channel bandwidth is
divided into $2^{SF}$ discrete levels. Each of these levels signifies the
starting frequency for a specific symbol value.
Therefore, the chirp completes $2^{SF}$ discrete steps throughout its
duration, in cycling back to its initial frequency. For a chosen bandwidth,
$BW$, each step lasts for a duration of $T=1/BW$ seconds, adding up to the
entire symbol duration $T_{s}$. Thus, $SF$ determines the number of steps, and
$BW$ determines the time period of each step, collectively defining the symbol
duration, $T_{s}=2^{SF}/BW$.
Let $f_{c}$ denote the channel’s center frequency. The $n$-th transmitted
symbol, $s_{n}$, is mapped into a chirp signal $c_{n}(t)\in\mathbb{C}$ given
by
$c_{n}(t)=\frac{1}{\sqrt{2^{SF}}}\exp\big{\\{}j\big{(}2\pi
f_{n}(t)\big{)}t\big{\\}},\ \ t\in[0,T_{s}]$ (2)
where,
$f_{n}(t)=f_{c}+\mathrm{mod}\big{(}s_{n}+t\times
BW,2^{SF}\big{)}\times\frac{BW}{2^{SF}}-\frac{BW}{2},$ (3)
and $\mathrm{mod}(\xi,2^{SF})$ is the remainder of the division of $\xi$ by
$2^{SF}$.
In LoRa $SF\in\\{7,8,9,10,11,12\\}$. It is customary to represent a chirp in
discrete-time using $2^{SF}\times f_{s}/BW$ samples indexed by $k$, where
$f_{s}$ is the sampling frequency and $f_{s}/BW$ is the oversampling factor.
Letting $t=k/f_{s}$, we obtain:
$c_{n}(k)=\frac{1}{\sqrt{2^{SF}}}\exp\bigg{\\{}j2\pi\times\Big{(}f_{c}+\\\
\mod\big{(}s_{n}+k\times\frac{BW}{f_{s}},2^{SF}\big{)}\times\frac{BW}{2^{SF}}-\frac{BW}{2}\Big{)}\bigg{\\}}\times
k,$ (4)
where $k=\\{0,1,2,\ldots,(2^{SF}\times f_{s}/BW)-1\\}.$ Figure 2 shows a chirp
in continuous time, in discrete time and in its time-frequency representation.
## III System description
Traditionally, jamming in the physical layer corresponds to adding white
Gaussian noise (AWGN) to the transmitted signal. Such naïve strategies are
ineffective in LoRa communications. Due to that resiliency to AWGN, LoRa has
also been referred to as a secure communication protocol. However, it has been
shown by [17] that LoRa is vulnerable to jamming using a chirp-type waveform.
Generating the chirp-type waveform to cause destructive interference requires
the knowledge of $BW$ and $SF$.
The LoRaWAN specification fixes the choice of these parameters to a finite set
of $18$ combinations ($BW\in\\{125\mathrm{kHz},\ 250\mathrm{kHz},\
500\mathrm{kHz}\\}$ and $SF\in\\{7,8,9,10,11,12\\}$). These parameters are
agreed by the legitimate communicating parties, but are not readily available
to a jamming adversary. Hence, the jammer needs to estimate this information
from an observed signal.
Figure 3: Block diagram for a reactive jammer in a communication system that
uses CSS modulation.
Figure 3 shows the block diagram of the data pipeline used by a reactive LoRa
jammer. Each component of this system is described in the following
subsections.
### III-A Data batch preprocessing block
The SDR captures signals in real time and outputs a stream of In-phase and
Quadrature (IQ) samples of indefinite length. On the other hand, our neural
network classifier operates on data batches of finite size. The preprocessor
block collects data flowing in from the SDR into a matrix of appropriate size
for processing in the subsequent blocks.
The SDR is tuned to the channel of interest and configured to a sampling rate
of $1\mathrm{MHz}$. Due to the Shannon-Nyquist Theorem, a minimum sampling
rate of $1\mathrm{MHz}$ is required since the maximum $BW$ in LoRa is
$500\mathrm{KHz}$. A lower sampling rate might result in distortion from
aliasing, and higher rates imply higher demand for computational resources.
Therefore, the SDR generates a noisy IQ stream $\tilde{X}$ of discrete-time
samples to the host PC. The preprocessor block parses this stream of complex
values into smaller signal blocks and reshapes them into a matrix of
dimensions $B\times M$. Where $B$ represents batch size and $M$ represents
length of the signal segment.
Determining the proper block length $M$ is crucial, as it must contain enough
samples to distinguish the LoRa configurations reliably. If the block length
is too small, the signal is truncated and information is lost. If the block
length is too large, the the neural network processing introduces latency.
Hence it must be as small as possible yet carry enough signal information.
We have empirically determined that the ideal block length must span two LoRa
symbols for the longest configuration. The longest configuration in LoRa is
$BW=125\mathrm{KHz}$, and $SF=12$, resulting in a symbol duration of
$T_{s}=2\times 2^{12}/125000$ seconds. For a sampling frequency of
$1\mathrm{MHz}$, we obtain an over-sampling factor of $8$, resulting in
$2\times 2^{12}\times 8=65,536$ of samples. Therefore, we fix the block length
to $M=65,600$.
### III-B Feature Extraction
The feature extraction block employs an algorithm based on the instantaneous
frequency (IF), which leads to a compact representation of LoRa signal
sequences. Such representation accentuates features related to the
identification of $BW$ and $SF$. The algorithm first transforms the signal
vectors from the time domain to the frequency domain and tracks the
instantaneous frequency of the signal over time. In the frequency domain, any
pair of LoRa signals corresponding to different configurations appear
distinctly different. The algorithm takes in a batch of signal blocks from the
preprocessor block, $V$, and applies the algorithm described in Section IV to
produce a matrix $F$ of IF vectors.
Our goal is to infer the parameters $SF$ and $BW$. One influences the duration
of the chirp, and the other affects both the duration and frequency sweep
range in the chirp. Our approach to feature extraction here is to characterize
the instantaneous frequency of the signal, describing the evolution of the
frequency in the signal with time. Through this representation, we can observe
both the range of the frequencies swept and the time elapsed for each sweep,
enabling simultaneous estimation of $SF$ and $BW$.
### III-C Chirp classifier
The chirp classifier block uses a neural network (NN) to identify the
transmitted chirp signal. Our model is trained using a dataset of IF vectors
labeled with their corresponding $BW$ and $SF$ configurations. Once trained,
this block receives an IF vector and performs a soft-decision classification
of $BW$ and $SF$ in a vector $C$ of probabilities for each of the $18$
possible signal configurations. This information passed to the chirp-generator
block.
In the context of classifying LoRa signals based on their features, it is
important to note that the relationship between these features and their
respective classifications is non-linear. NNs can learn complex relationships
and patterns in data, making them suitable for tasks like classifying signals
with intricate or non-linear relationships between their features and
categories. With proper training and a sufficiently rich architecture, NNs can
provide accurate signal classification even at extremely low levels of SNR. We
will discuss the NN architecture in more detail on Section V.
### III-D Chirp generator
The chirp generator block is responsible for utilizing the inferred $BW$ and
$SF$ to generate a stream of discrete-time IQ values for the jamming chirps,
denoted by $J$. The IQ stream should be sent to the SDR, which uses a Digital
to Analog Converter (DAC) that converts them from discrete-time to a
corresponding continuous-time signal. Once converted to analog, the SDR can
adjust the signal to the channel’s center frequency for transmission. The
resulting signal would represent a chirp with the same $BW$ and $SF$ as the
target signal, leading to interference at the receiver.
LoRa uses a two-step demodulation procedure: the first is known as dechirping,
followed by an FFT. The dechirping operation multiplies the sampled signal
with a base down chirp of the same $BW$ and $SF$. The resulting signal has a
constant frequency, which matches the chirp’s initial frequency. Then, from
its FFT, we identify the bin index of this frequency, determining the encoded
symbol’s value. Under this demodulation scheme, when two signals of the same
$BW$ and $SF$ configuration interfere at the receiver, they result in multiple
indiscernible peaks in the FFT step. Such interference deceives the receiver
into misidentifying the original symbol. This misidentification leads to
symbol demodulation errors, resulting in packet drops, effectively jamming the
signal.
With the knowledge of $BW$ and $SF$ we can generate chirp signals using Eq. 4.
However, the chirp’s polarity (upchirp or downchirp), the symbol value, and
the arrival time influence the effectiveness of interference with the target
signal. Considering these factors, the authors of [17] introduced three
effective methods to jam LoRa signals when $BW$ and $SF$ are known, which can
be implemented in the chirp generator block, summarized as follows:
* •
Identical chirps: A simple approach is to continuously repeat the same symbol
in sequence. By transmitting continuously, we avoid sudden shifts across
demodulation windows. Any delays and time offsets only affect the initial
frequency of the chirp and still result in demodulation errors. This method is
lightweight because it does not require strict time synchronization.
* •
Consecutive downchirps: This method targets the Start Frame Delimiter (SFD)
symbol of LoRa packets, which is a base downchirp that marks the beginning of
the packet header. From transmitting base downchirps consecutively, the
receiver is tricked into making errors in identifying the legitimate SFD,
resulting in incorrect packet parsing and leading to packet drops.
* •
Synchronized chirps: This method is considered to be the most effective
jamming strategy in LoRa [15, 17]. It involves transmitting random symbols
that perfectly align with the demodulation window at a receiver. This is made
possible by estimating and compensating the Carrier Frequency Offset (CFO) and
the Sampling Time Offset (STO), as in a legitmate LoRa demodulator. The
synchronized chirps method requires strict synchronization and additional
computing, however, it is the most effective and difficult to detect method
known to date.
In conjunction with the inferred parameters, the chosen method defines the
sequence of jamming chirps to be transmitted. The IQ values corresponding to
this sequence is streamed from the chirp generator block to the SDR at a fixed
rate. Consequently, the SDR transmits this waveform over the air to jam the
target signal at the receiver. This strategy shows that it is possible to jam
LoRa signals of unknown $BW$ and $SF$ configurations by an unauthenticated
agent.
Figure 4: Feature representations for various LoRa Configurations
## IV Feature extraction
In this section, we identify a pattern in the data, also known as feature,
that aids in distinguishing one category from another. To that end, we compute
the instantaneous frequency of the signal. Considering this feature, we can
retain information about the range of frequencies swept, and their sweep rate
simultaneously, which are directly related to our two parameters of interest,
$BW$ and $SF$.
Here, we follow a two-step procedure to computing the instantaneous frequency:
a Short Term Fourier Transform (STFT) followed by Instantaneous Frequency (IF)
estimation.
### IV-A Short Term Fourier Transform (STFT)
Given the inherent time-varying nature of frequency in a chirp signal, we
employ the STFT on each input signal segment [19]. A given signal segment is
further subdivided into overlapping windows, each consisting of $W=128$
samples, with an overlap of $L=64$ samples. Subsequently, an FFT is executed
on these windows. This operation obtains the power distribution across all the
frequencies in the channel bandwidth, $BW$ as the signal evolves in time, as
follows:
$Q[k,m]=\sum_{n=0}^{W-1}x[n+mL]w[n]e^{-j2\pi nk/W},$ (5)
where $Q[k,m]$ is the STFT coefficient at frequency bin $k$ and time index
$m$, $x$ is the input signal segment, and $w[n]$ is the Hann window function
[20].
### IV-B Instantaneous Frequency Estimation
Unlike stationary signals where the spectral properties are constant, the
frequency of a chirp signal varies linearly with time [21]. For such signals,
we must compute the instantaneous frequency instead of frequency. The
instantaneous frequency is a time-varying parameter related to the average of
the frequencies present in the signal as it evolves in time [22].
From the STFT operation in Eq. 5, we obtain the energy distribution over all
frequency bins for every time-step. We use this energy distribution to compute
a weighted average of the frequencies at each time-step, obtaining the
instantaneous frequency of the signal, as follows:
$f_{inst}(m)=\frac{\sum_{k=1}^{K}P(k,m)f(k,m)}{\sum_{k=1}^{K}P(k,m)},$ (6)
where $f_{inst}(m)$ is the instantaneous frequency at the time index $m$,
$f(k,m)$ is the peak frequency at frequency index $k$ and time index $m$, and
$P(k,m)$ is the power spectral density, computed as $P(k,m)=|Q[k,m]|^{2}$.
## V Neural network architecture
LoRa nodes operate under power constraints (typically from $10\mathrm{dBm}$ to
$20\mathrm{dBm}$) and often transmit over long communication distances
(typically from $10^{3}$m to $10^{4}$m). As a result, LoRa signals are often
received at low SNR, sometimes even below the noise floor. Identifying and
distinguishing such signals reliably demand a classifier model with high noise
tolerance and discriminative power.
Neural networks have been extensively used for signal classification in
wireless communications, spanning applications such as channel sensing,
interference detection and spectrum management [23, 24, 25]. Central to their
efficacy in these applications is their inherent ability to model non-linear
relationships between parameters and noisy data [26].
Figure 4 illustrates the feature representations corresponding to different
$BW$ and $SF$ configurations. The first three sub-figures show the case of
fixed $BW$, and the last three figures illustrate the case of fixed $SF$.
These graphs indicate that changes in $BW$ and $SF$ result in clearly distinct
waveforms. Additionally, the characterization based on the IF of these
waveforms makes the task of distinguishing signals of different configurations
much simpler by converting the the problem of estimating $BW$ and $SF$ into a
signal classification problem.
For this classification task, we use a feed-forward neural network as
illustrated in Figure 5. The model features two hidden layers with 16 neurons
each, and an output layer of 18 neurons, as specified in Table I. The input is
the IF vector, where $t$ is the time index. To classify an IF vector into one
of the $18$ categories, we use a softmax function in the output layer to
obtain a probability distribution on the likelihood of each class given the
observed data. Consider an output of the final layer,
$Z=[z_{1},z_{2},\dots,z_{18}]$ of $18$ real numbers, the softmax function,
$S(\cdot)$, is defined as:
$S(z_{i})=\frac{e^{z_{i}}}{\sum_{j=1}^{18}e^{z_{j}}},\ \ i=1,\dots,18.$ (7)
Figure 5: Neural network architecture used in our system. TABLE I: Key Attributes of the Neural Network Architecture Attribute | Description
---|---
Input | Flatten Layer
Hidden Layer 1 | Dense (16 units, tanh activation)
Hidden Layer 2 | Dense (16 units, tanh activation)
Regularization | Dropout (0.5 rate)
Output Layer | Dense (18 units, softmax activation)
Loss Function | Categorical Cross-Entropy
Optimizer | Adam
Evaluation Metric | Classification Accuracy
## VI Simulation Results
We use synthetic datasets of LoRa signals, creating separate datasets for
training and validation.111The data and code for all the simulations and
numerical experiments in this paper are available at https://github.com/MINDS-
code/jammingSDR.git. Here, the noisy signals are generated according to an
Additive White Gaussian Noise (AWGN) model producing signal data at diverse
SNR levels. Our training dataset has $10$ SNR levels, ranging from $0$ to
$20\mathrm{dB}$. For each of the $18$ configurations, we have generated $50$
signal files. Thus, the training dataset contains a total of $9000$ entries.
Our validation dataset has a broader SNR range, from $-15$ to $20\mathrm{dB}$,
leading to $18$ SNR levels in total. Here, we have generated $20$ signal files
for each case, leading to a total of $6480$ entries. We found out that if the
training dataset included signals with SNR below zero, the classification
performance of the of the NN is severely degraded.
Figure 6: Classification accuracies against SNR with a 95% confidence
interval: (1) overall, (2) by spreading factor, and (3) by bandwidth.
Consider a clean signal, denoted by $X$, subjected to AWGN denoted by $Z$ as
follows:
$\tilde{X}=X+Z,$ (8)
where $\tilde{X}$ is the resulting noisy signal. The power level of $Z$ is
determined by the desired SNR level. We obtain confidence intervals on results
by repeating the experiment $30$ times. Addionally, we experiment with fixed
$BW$ and $SF$ choices, observing their influence on classification
performance. Figure 6 (left) illustrates the classifier’s overall accuracy in
relation with SNR. Classification accuracy starts at around 12% for
$-15\mathrm{dB}$ SNR. The accuracy increases sharply and saturates at
$-5\mathrm{dB}$ SNR. Figure 6 (middle) illustrates the classifier’s accuracy
as a function of SNR for the different possible $SF$ configurations. Each
curve differs from the others by their saturation points and the accuracy
levels they can reach. The curve for $SF$ 12 reaches saturation the earliest
and at the highest accuracy level, succeeded by $SF$ 11, with subsequent
configurations following in descending order. We observe that for a fixed SNR
level, higher SF choices yield consistently higher accuracy scores. The mean
classification accuracy improved with an increase in $SF$ from 7 to 12. This
trend results from LoRa’s spreading waveform, where $SF$ determines the sweep
rate of the chirp. A higher $SF$ leads to a longer chirp duration resulting in
a more elongated and discernible frequency trajectory over time. With more
samples constituting the waveform, identification becomes more precise,
improving classification accuracy.
Figure 6 (right) illustrates the classifier’s accuracy as a function of SNR
for three $BW$ configurations: $125\mathrm{KHz}$, $250\mathrm{KHz}$, and
$500\mathrm{KHz}$. Before reaching saturation, the $125\mathrm{KHz}$ curve
shows a higher accuracy compared to the other two. Meanwhile, the classifier’s
accuracy for the $250\mathrm{KHz}$ curve is consistenlty higher than for
$500\mathrm{KHz}$.
After reaching saturation, the $500\mathrm{KHz}$ curve exhibits higher
accuracy over the other two. However, beyond this point, all three
configurations deliver high classification accuracy. Thus, despite the high
accuracy of the $500\mathrm{KHz}$ curve post saturation, the real
differentiator lies in their points of saturation. The earlier the saturation,
the lower the minimum SNR needed to classify the signal reliably. Thus, the
order in which the curves saturate imply that lower $BW$ configurations yield
better detection.
The mean classification accuracy saturates later for higher $BW$ choices from
$125\mathrm{KHz}$ to $500\mathrm{KHz}$. With a wider $BW$, the signal’s
frequency changes on a broader range in a reduced period. This rapid shifting
causes the instantaneous frequency vectors to become too closely spaced,
making it more challenging for the classifier to distinguish them.
The choice of $BW$ and $SF$ in LoRa is motivated by the application’s quality
of service requirements. However, in practice, nodes switch between several
parameter choices to save power and optimize throughput. Therefore, when
jamming or extensive interference is a concern, legitimate nodes must consider
switching to faster $BW$ and $SF$ choices. Our results conclude that, to avoid
detection by unauthorized agents, legitmate LoRa nodes must opt for lower $SF$
choices and higher $BW$ choices whenever possible.
## VII Conclusions and future work
Many large-scale multi-agent systems rely on LPWAN protocols. Amongst these,
LoRaWAN has found widespread adoption, due to its energy efficiency, long
range, and use of unlicensed spectrum. However, it is succeptible to cyber-
attacks, including eavesdropping and jamming. In this paper, we explored the
vulnerability of LoRa to signal jamming.
A survey of related literature revealed that LoRa is vulnerable to jamming
with a particular chirp type signal. However, generating such signals require
the knowledge of the bandwidth, and spreading factor of the target LoRa
signal. We argue that this information is shared amongst legitimate parties
but unavailable to an unautheticated adversarial agent. In this work, we
presented the high-level design of a practical jammer, that makes use of a
neural network classifier for estimating these parameters by eavesdropping and
reactively emits jamming chirps.
Leveraging a structural pattern in LoRa’s signal waveform, we relate the
problem of estimating these parameters to a signal classification task. To
that end, we proposed a feature extraction method that computes the
instantaneous frequency of signals, enhancing features pertinent to
identifying $BW$ and $SF$ configurations. Then we trained a feedforward neural
network classifier on a dataset LoRa signals to learn these characteristics
for predictive analysis. Our results indicate that the classifier begins to
reliably estimate these parameters for signals stronger than $-5\mathrm{dB}$
SNR. Additionally, we analyzed detection performance at various configurations
of $BW$ and $SF$. Ultimately revealing that, to hinder such detection,
legitimate users of LoRa must use lower $SF$ and higher $BW$.
Directions for future work include experimenting this classifier on a dataset
of real signals captured using a software radio to provide a real-world
validation of this analysis, and an end-to-end implementation of the proposed
jammer to explore real-time performance of the design.
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|
# Exceptional points in dielectric spheroid
Evgeny Bulgakov Kirensky Institute of Physics Federal Research Center KSC SB
RAS 660036 Krasnoyarsk Russia Reshetnev Siberian State University of Science
and Technology, 660037, Krasnoyarsk, Russia Konstantin Pichugin Kirensky
Institute of Physics Federal Research Center KSC SB RAS 660036 Krasnoyarsk
Russia Almas Sadreev Kirensky Institute of Physics Federal Research Center
KSC SB RAS 660036 Krasnoyarsk Russia
(August 27, 2024)
###### Abstract
Evolution of resonant frequencies and resonant modes as dependent on the
aspect ratio is considered in a dielectric high index spheroid. Because of
rotational symmetry of the spheroid the solutions are separated by the
azimuthal index $m$. By the two-fold variation of a refractive index and the
aspect ratio we achieve exceptional points (EPs) at which the resonant
frequencies and resonant modes are coalesced in the sectors $m=0$ for both TE
and TM polarizations and $m=1$.
## I Introduction
Optical properties of dielectric particle is described by resonant frequencies
and corresponding resonant modes. The most famous case is a dielectric sphere
whose resonant modes and frequencies were first considered by Stratton [1].
The solutions in the form quasi-normal modes (QNMs) leaking from the sphere
were considered in Refs. [2, 3]. The frequencies of these solutions resonances
are complex because of coupling of the dielectric particle with the radiation
continuum and can be considered as the eigenvalues of the non-Hermitian
Hamiltonian [4, 5, 6, 7]. Material losses as well as thermal fluctuations [8]
of dielectric particle can considerably contribute into the imaginary part of
complex resonant frequencies through complex refractive index. Non-Hermitian
phenomena drastically alters the behavior of a system compared to its
Hermitian counterpart describing the closed system. The best example of such a
difference is the avoided resonant crossing (ARC) because of coupling of a
particle with the radiation continuum [9, 10, 11, 12, 13]. In turn the ARC can
emerge to singularities, bound states in the continuum at which the imaginary
part of resonance turns to zero [14, 15] that gives rise to collapse of Fano
resonance and exceptional points (EPs). The last is remarkable by that complex
frequencies become degenerate and the eigenmodes coalesce [16, 17, 18]. Early
experiments on microwave coupled resonators revealed the peculiar topology of
eigenvalue surfaces near exceptional points for encircling of EP [19].
Many works on EPs and their applications are associated with parity-time (PT)
symmetric optical systems with a balanced gain and loss. In that case, EPs can
be easily found by tuning a single parameter, namely, the amplitude of the
balanced gain and loss [20, 21, 22, 23, 24]. Since it is not always easy or
desirable to keep a balanced gain and loss in an optical system there is of
significant interest to explore EPs and their applications in non-PT-symmetric
optical systems. Currently, there exist studies concerning EPs for resonant
states in extended periodic dielectric structures sandwiched between two
homogeneous half-spaces [25, 26, 27, 28], dual-mode planar optical waveguides
[29] and plasmonic waveguide [30], layered structures [31, 32, 33], two
infinitely long dielectric cylinders [34, 35, 36, 37, 38] and even single rod
with deformed cross-section [39, 40, 41, 36, 42]. As for compact dielectric
resonators we distinguish the only study of EPs in compact coated dielectric
sphere [43].
In the present paper we consider similar compact elementary dielectric
resonator such as a spheroid in which EPs can be achieved by two-fold
variation of aspect ratio and refractive index. Although the spheroid allows
the solution due to separation of variables in spheroidal coordinate system
[44, 45], analytical expressions for solutions are too cumbersome. We use
software package COMSOL Multiphysics which allows to obtain numerically the
complex resonant frequencies and corresponding resonant modes of particle of
arbitrary shape embedded into the radiation continuum by use of perfectly
absorbing boundary conditions.
## II Evolution of resonant frequencies in spheroid
A rotational symmetry of spheroid implies that the azimuthal index $m$ is
preserved. That allows to calculate the resonant frequencies and resonant
eigenmodes separately in each sector $m$ and calculate EM field configurations
as series over the orbital momenta outside spheroid [44]
$\overrightarrow{E}^{(m)}(\overrightarrow{r})=\sum_{l=1}^{\infty}[a_{lm}\overrightarrow{M}_{lm}(\overrightarrow{r})+b_{lm}\overrightarrow{N}_{lm}(\overrightarrow{r})]$
(1)
where $\overrightarrow{M}_{lm}$ and $\overrightarrow{N}_{lm}$ are the
spherical harmonics [1]. In what follows we consider the sectors $m=0$ and
$m=1$.
The sector $m=0$ is simplified compared to the sector $m=1$ because of
separation of TE and TM modes. Figure 1 presents evolution of complex TE
resonant frequencies with variation of the equatorial radius $R_{\bot}$
relative to the polar radius $R_{z}$ from oblate silicon spheroid
$R_{z}=0.4R_{\bot}$ to prolate spheroid $R_{z}=1.6R_{\bot}$. $k$ is the wave
number, and $R=(R_{z}R_{\bot}^{2})^{1/3}$ is the mean radius that equalizes
volumes of sphere and spheroid. For the reader convenience we split the
frequency range in Figure 1 into two parts. The insets show the QNMs of a
sphere.
Figure 1: Evolution of complex TE resonant frequencies in silicon spheroid
with permittivity $\epsilon=12$ for variation of aspect ratio of polar $R_{z}$
and equatorial $R_{\bot}$ radii in the sector $m=0$. Wave patterns show
azimuthal component of electric field $|E_{\phi}|$ of the Mie resonant modes
in sphere at points marked by closed circles where integers above the insets
notify the orbital momentum $l$ and the radial index $n$. ’x’ marks the case
of oblate spheroid with $R_{z}=0.4R_{\bot}$ while ’+’ marks the case of
prolate spheroid with $R_{z}=1.6R_{\bot}$. Figure 2: Evolution of resonant
frequencies and resonant modes labelled as 1 and 2 in Fig. 1 versus ratio of
radii $R_{z}$ and $R_{\bot}$.
In Figure 2 we demonstrate a phenomenon of avoided crossing of resonances
marked as 1 and 2 in Figure 1 which is the result of interaction of the dipole
QNM with the octuple QNM [46]. There is a general belief that a homogeneous
spherical dielectric body represents the ideal case, so that any perturbation
of shape of sphere can only degrade the resonance (the imaginary part
increases or the $Q$-factor decreases). Lai et al [8, 46] have shown this,
however, provided that imaginary part of the spherical QNM is small enough.
For the QNMs with low $Q$-factor their frequencies deviate from the complex
eigenfrequencies of sphere linearly [4].
This anomalous behavior of the low-$Q$ resonances can be comprehend if to
refer to the series over spherical harmonics (1). For the TE polarization we
have
$E_{\phi}=\sum_{l}a_{l0}M_{l0}^{\phi}$ (2)
where $l=1,3,5,\ldots$ if $E_{\phi}$ is even relative to $z\rightarrow-z$ and
$l=2,4,6,\ldots$ if $E_{\phi}$ is odd. Once a sphere transforms into spheroid
the orbital momentum $l$ is not preserved. Figure LABEL:TEm0coef shows as new
multipole radiation channels are opened with this transformation.
Figure 3: Evolution of multipole coefficients in series (1) for evolution of
resonant modes $l,n$ shown in Fig. 1.
Let us consider some of resonances shown Figure 1. For variation of the polar
radius $R_{z}$ the lowest mode shown by black line goes through the Mie dipole
mode $1,0$ of a sphere with the frequency $kR=0.862+0.0414i$. As seen from the
first subplot of Figure 3 at this moment the only radiation channel is given
by the coefficient $a_{10}$. The resonant widths of the Mie resonant modes
fast fall down with the orbital momentum $l$ and grow with the radial index
$n$ [47]. As a result, when a sphere is deformed, the fast decaying dipole
channel is weakening at the cost of linear arising of the next slower decay
octuple channel $l=3$ in accordance to Eq. (2). These comprehensive
considerations were issued by Lai et al [46]. Respectively the resonant width
is decreased as shown in Figure 1 by black line. However, there are exceptions
from this rule, for example, the QNMs $l=2,n=1$ and $l=2,n=0$ (The last column
of subplots in Figure 3). In both cases the same slower decaying radiation
channels with $l=4$ and $l=6$ are attaching to the quadruple channel with
$l=2$ for deviation from a sphere. Nevertheless the behavior of resonant
widths is dramatically different as seen from Figure 1. For the radial quantum
$n=0$ we observe a degradation of the quadruple QNM, while for $n=1$ we
observe the opposite behavior. That shows the importance of the radial indices
for resonant widths [47].
Let us consider also the resonances evolving with the Mie resonances with
higher orbital momentum, octuple resonance $3,0$ with the frequency
$kR=1.629+0.0042i$ shown by green line in Figure 1. Corresponding evolution of
multipole coefficients is shown in Figure 3 in subplot labelled $3,0$. In
contrast to previous dipole and quadruple resonances the high-$Q$ decaying
octuple resonance is substituted by the fast decaying dipole resonance $1,0$.
As a result we observe an increase of resonant width in Figure 1 for
transformation of sphere into spheroid. Other subplot $4,0$ in Figure 3 shows
the same result.
We omit analysis of the TM resonances shown in Figure 4 because of a
similarity with the case of the TE resonances except that the series (1) for
magnetic field are given by the coefficients $b_{l0}$ with the same sequence
for $l=1,3,5,\ldots$ for the even solutions of magnetic field $H_{\phi}$ and
$l=2,4,6,\ldots$ for the odd solutions relative to $z\rightarrow-z$.
Figure 4: Evolution of complex TM resonant frequencies. Wave patterns show
azimuthal component of magnetic field $|H_{\phi}|$ of the Mie resonant modes.
’x’ marks $R_{z}=0.4R_{\bot}$ and ’+’ marks $R_{z}=1.6R_{\bot}$. The inset
shows behavior of multipolar coefficients on the aspect ratio.
As a result we have similar rules for resonant widths. The Mie TM dipole and
quadrupole resonances yield to spheroid resonances in the $Q$ factor in
contrast to the Mie resonances with higher orbital momenta. However there is
an exception for the resonance $3,1$ which have no minimal resonant width at
$R_{z}=R_{\bot}$. The inset in Figure 4 shows that in the prolate spheroid we
have extremely large contribution of the spherical harmonic $l=5$ compared to
the dipole harmonic $l=1$ that suppresses emission from the prolate spheroid.
Figure 5: Evolution of resonant frequencies for traversing from the oblate
spheroid $R_{z}=0.4R_{\bot}$ (pluses) through a sphere (closed circles) to the
prolate spheroid $R_{z}=1.6R_{\bot}$ (crosses) in the sector $m=1$. Titles
above the insets indicate the orbital momentum $l$ and radial index $n$ (the
number of radial nodal circles). The TE/TM modes are presented by the
azimuthal components $|E_{\phi}|$/$|H_{\phi}|$.
The sector $m=1$ is destined to show that the phenomena of ARCs exist in the
other sectors of the azimuthal index $m$, in particular $m=1$ as demonstrated
in Figure 5. Moreover one can observe the same tendency of degradation of the
high-$Q$ QNMs and, visa versa, enhancement of the $Q$-factor for the low-$Q$
QNMs for deformation of sphere.
## III Exceptional points.
The sector $m=0$ demonstrates EPs separately for each polarization. Figure 6
shows numerous examples of avoided crossing of TE modes highlighted by open
circles.
Figure 6: ARCs of TE QNMs for evolution of sphere into spheroid in the sector
$m=0$.
It is interesting that the ARC phenomena are observed only for the oblate
spheroids below $R_{z}/R_{\bot}=1/2$. The behavior of QNMs is presented in
Figure 7 which shows as the modes are exchanging for variation of the aspect
ratio of spheroid.
Figure 7: Evolution of selected resonant frequencies and resonant modes
labelled as 1 and 2 in Fig. 6 vs ratio of radii $R_{z}$ and $R_{\bot}$. The
insets show the azimuthal component $|E_{\phi}|$ of corresponding resonant
modes at points marked by closed circles.
As shown in Figure 6 (b) the ARCs are complemented by strong enhancement of
the $Q$-factor in an agreement with numerous considerations in different
dielectric resonators [11, 48, 49, 50].
What is remarkable, the oblate spheroid demonstrates numerous EPs for the two-
fold variation of the permittivity and the aspect ratio for both sectors $m=0$
and $m=1$. Figure LABEL:fig80 shows the behavior of QNMs with the aspect ratio
at $\epsilon=17.2$ in the sector $m=0$. One can see that inside the areas
highlighted by open circles two QNMs coalesce into the one QNM.
Figure 8: Evolution of resonant frequencies and resonant modes versus
$R_{z}/R_{\bot}$ at $\epsilon=17.2$ in the sector $m=0$. Open circles
highlight EPs. The left one at $R_{z}/R_{\bot}=0.292,\epsilon=17.2$ and the
right one at $R_{z}/R_{\bot}=0.304,\epsilon=18.4$. The insets show the
$|E_{\phi}|$ profiles of TE QNMs at points marked by closed circles.
Such a behavior of resonances close to the EP behavior was observed in
different dielectric structures [40, 29, 36, 37, 43].
Figure 9: Encircling of EPs shown by open circles in Fig. LABEL:fig80. (a) and
(b) Encircling separate EPs. (c) Encircling of both EPs. Insets show the
component $E_{\phi}$ of resonant mode.
In order to be convinced that there are indeed the EPs we encircle the EP
points shown open circles in Figure LABEL:fig80 by three ways. In the first
case the rectangular contour encircles only the left EP at the point
$R_{z}/R_{\bot}=0.292,\epsilon=17.2$ as shown in Figure 9 (a). Respectively,
in the second case the contour encircles the right EP point
$R_{z}/R_{\bot}=0.304,\epsilon=18.4$ as shown in Figure 9 (b). At last, we
present also the case of encircling of both EPs shown in Figure 9 (c). In all
cases we encircle EPs counterclockwise.
Let us consider the first case shown in Figure 9 (a) where encircling starts
with point $R_{z}/R_{\bot}=0.32,\epsilon=17$ marked by open circle in the
inset of Figure. In the first downward path we decrease the aspect ratio at
the same permittivity reaching the point till
$R_{z}/R_{\bot}=0.27,\epsilon=17$ marked by cross. In the complex plane this
path maps into sharp trajectory shown by dot-dashed blue line that features
high response of resonant frequency on shape of spheroid. Respectively the
resonant mode demonstrate sharp change of the resonant mode. In the next
horizontal path we slightly increase the permittivity from $\epsilon=17$ till
$\epsilon=17.8$ of the oblate spheroid with the same shape and reach the point
$R_{z}/R_{\bot}=0.27,\epsilon=17.8$ marked by square in the inset. In the
complex plane this path maps into monotonic descent of resonant frequency by
law $(kR)^{2}\epsilon\approx C$ or $kR\approx\sqrt{C/17}(1-\Delta\epsilon/2)$.
That linear part of trajectory is plotted by solid blue line in Figure 9 (a).
The resonant mode presented by the insets at staring and finishing points also
does not show visible changes. The third upward part of rectangular contour
goes from the point marked by square $R_{z}/R_{\bot}=0.27,\epsilon=17.8$ to
the point marked by star $R_{z}/R_{\bot}=0.32,\epsilon=17.8$ maps into sharp
trajectory shown by blue dash line. However the resonant mode is not changing
that is related to far distance between the left EP and the path as distinct
from the first downward path from circle to cross. By doing so we closed the
rectangular contour however as the resonant frequency as the resonant mode are
interchanged as was first demonstrated by Dembowskii et al in a microwave
metallic resonator [19]. And only the second encircling of the left EP
restores the resonant mode as demonstrated in Figure 9 (a) by red lines.
The right EP $R_{z}/R_{\bot}=0.304,\epsilon=18.4$ is expected to give rise to
the same features. However as shown in Figure 9 (b) counterclockwise
encircling of this EP demonstrates clockwise behavior of the resonant
frequency and mode opposite to the case of counterclockwise encircling of the
left EP. That is related to that the signs of winding numbers of neighboring
EPs arising after crossing of two lines in the complex plane are opposite each
other [51, 52]. Figure 9 (c) presents graphical evidence for that. The one
encircling of both EPs restores the resonant modes of each resonance.
Next, we show an existence of EPs in the sector $m=1$ too in which the QNMs
with mixed polarizations can be excited by plane wave incident along the
z-axis as different from the case $m=0$ [1]. The first example of evolution of
the QNMs (only the component $E_{\phi}$ is presented) and their complex
eigenfrequencies in the sector $m=1$ is presented in Figure 10.
Figure 10: Evolution of resonant frequencies and resonant modes marked as 1
and 2 in Fig. 5 (sector $m=1$) versus ratio of radii $R_{z}$ and $R_{\bot}$
around EPs at $\epsilon=12$. The EP is given by the point
$\epsilon=12,R_{z}/R_{\bot}=0.84,kR=1.25$.
The EPs occur for precise two-fold tuning of the aspect ratio $R_{z}/R_{\bot}$
and the refractive index of spheroid that is challengeable experimentally.
However there is a way to show EPs by encircling the EP through which resonant
eigenmodes are interchanged [19]. Figure 11 demonstrates as for encircling of
the EPs in plane $R_{\bot}/R_{z}$ and $\epsilon$ one of resonant modes
restores only after encircling by $4\pi$. It is clear that the same refers to
the multipole coefficients $a_{l1}$ and $b_{l1}$ as shown in Figure 3.
Figure 11: Evolution of the field patterns $E_{y}$ for encircling the EPs
$\epsilon=12,R_{z}/R_{\bot}=0.84$ marked by star. Figure 12: Evolution of the
expansion coefficients $a_{l1}$ (TE modes) and $b_{l1}$ (TM modes) for
encircling the EP shown in Fig. 11.
There are also many other EPs with higher frequencies. One example of the EP
is presented in Figures 13 and 14.
Figure 13: Evolution of resonant frequencies and resonant modes marked as 3
and 4 in Fig. 5 (sector $m=1$) versus ratio of radii $R_{z}$ and $R_{\bot}$
around EPs at $\epsilon=12$. The EP is given by the point
$\epsilon=12.46,R_{z}/R_{\bot}=1.11,kR=2.56$. Figure 14: Evolution of the
field patterns $E_{y}$ for encircling the EPs highlighted in Fig. 13.
## IV Summary and conclusions
It seems reasonable that resonances of any dielectric particle shaped
differently from a sphere yield to the Mie resonances of sphere by the
$Q$-factors because the surface of sphere is minimal. However as Lai et al [8,
46] have shown that is truth only for those resonances whose imaginary part is
small enough. We present numerous examples which confirm this rule and give
comprehensible insight by demonstration of multipole radiation channels for
evolution of a sphere into spheroid. However we also show exceptions from this
rule.
However the main objective of the present paper was demonstration of EPs in a
spheroid that has fundamental significance because of compactness of these
dielectric resonators. Moreover, evolution of expansion coefficients in Fig6
demonstrate multipole conversion for encircling of EPs and what is the most
remarkable this evolution has a period $4\pi$. In the photonic system, the
appearance of EPs can be exploited to a broad range of interesting
applications, including lasing [53], asymmetric mode switching [29],
nonreciprocal light transmission [54, 55], enhancement of the spontaneous
emission [56] and ultrasensitive sensing [57].
###### Acknowledgements.
The work was supported by Russian Foundation for Basic Research projects No.
19-02-00055.
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|
# Sparse Regression for Extreme Values
Andersen<EMAIL_ADDRESS>[ Minjie<EMAIL_ADDRESS>[
Department of Statistics, Rice University
Genevera I. Allen<EMAIL_ADDRESS>[ Department of Electrical and
Computer Engineering, Rice University,
Department of Computer Science, Rice University,
Department of Statistics, Rice University,
Department of Pediatrics-Neurology, Baylor College of Medicine,
Jan and Dan Duncan Neurological Research Institute, Texas Children’s Hospital
###### Abstract
We study the problem of selecting features associated with extreme values in
high dimensional linear regression. Normally, in linear modeling problems, the
presence of abnormal extreme values or outliers is considered an anomaly which
should either be removed from the data or remedied using robust regression
methods. In many situations, however, the extreme values in regression
modeling are not outliers but rather the signals of interest; consider traces
from spiking neurons, volatility in finance, or extreme events in climate
science, for example. In this paper, we propose a new method for sparse high-
dimensional linear regression for extreme values which is motivated by the
Subbotin, or generalized normal distribution, which we call the extreme value
linear regression model. For our method, we utilize an $\ell_{p}$ norm loss
where $p$ is an even integer greater than two; we demonstrate that this loss
increases the weight on extreme values. We prove consistency and variable
selection consistency for the extreme value linear regression with a Lasso
penalty, which we term the Extreme Lasso, and we also analyze the theoretical
impact of extreme value observations on the model parameter estimates using
the concept of influence functions. Through simulation studies and a real-
world data example, we show that the Extreme Lasso outperforms other methods
currently used in the literature for selecting features of interest associated
with extreme values in high-dimensional regression.
62J05,
62J07,
62P10,
62P05,
linear regression,
sparse modeling,
extreme values,
Subbotin distribution,
generalized normal distribution,
###### keywords:
[class=MSC]
###### keywords:
and
###### Contents
1. 1 Introduction
2. 2 Regression for Extreme Values
1. 2.1 Sparse Extreme Value Regression
3. 3 Theoretical Results
1. 3.1 Consistency of the Extreme Lasso
1. 3.1.1 Sub-Gaussian Errors
2. 3.1.2 Subbotin Error
2. 3.2 Influence of Extreme Values
4. 4 Empirical Investigations
1. 4.1 Linear Model Simulation Study
2. 4.2 Mixture Model Simulation Study
3. 4.3 Real Data Investigation: Calcium Imaging
4. 4.4 Real Data Investigation: Climatology
5. 5 Discussion
6. A Proofs for Section 3
1. A.1 Lemma 3.3
2. A.2 Theorem 3.1
3. A.3 Theorem 3.2
4. A.4 Lemma 3.5
5. A.5 Lemma 3.7
6. A.6 Theorem 3.3
7. A.7 Theorem 3.4
8. A.8 Theorem 3.5
7. B Full Tabular Results
1. B.1 Linear Model Simulation Study
2. B.2 Mixture Model Simulation Study
## 1 Introduction
When applying linear regression models, one often encountered issue is the
presence of rare extreme values, defined here as abnormally large magnitude
observations. This can occur in the form of outliers in the response variable
as well as in the form of highly influential points in the predictor
variables. Historically, statisticians have tried to develop methods to ignore
or dampen the effects of outliers in data sets when doing a linear regression
analysis. Metrics such as residual analysis, Cook’s distance, and DFFIT can be
used to identify and possibly remove outliers from the data set [31]. New
regression methods have also been developed to handle outliers in response
variables as well. For example, robust regression [15] has been used in many
different applications, and much work has been to done to show theoretical
asymptotic performance in the presence of outliers [14, 32]. More recently,
several have studied robust regression procedures for high-dimensional data
[23, 42].
However, in certain contexts, the important information in the response
variable that we want to model or predict is in the rare, abnormally large
magnitude observations. For these types of applications, rather than wanting
to remove outliers or use robust regression methods, we instead want to focus
on these extreme values when fitting models to the data. For example, in
neuroscience, calcium imaging data collected contains measurements of
fluorescence traces of neurons in the imaged brain [43]; the signal that is
important in this situation is the occurrences of neuron firing, indicated by
large positive spikes in the fluorescence trace. Extreme value regression
models are often used as well in climatology to measure the rate and strength
of extreme climate or weather events [21], or in finance to predict periods of
high volatility of asset prices [7]. Their potential usage has also been
studied in spectroscopy analysis and signal processing[25].
Several different possible approaches to the problem of high-dimensional
regression for extreme values have been used in various fields. Sparse
regression methods based on classical extreme value theory utilize a
generalized linear model framework. The extreme values above a predetermined
threshold in a response variable are specified to follow a distribution, such
as the Gumbel, whose parameters are a linear function of the predictor
variables and which determine the frequency and magnitude of the extreme
values [3, 30]. Another regression model commonly applied to model extreme
values in the high-dimensional setting is sparse quantile regression,
specifically applied to a very high or very low quantile [16]. These types of
models use a weighted absolute deviation loss function in order to find the
expected value of a response variable at a particular quantile. Extensions to
high-dimensional sparse $\ell_{1}$ quantile regression have also been studied
extensively [5, 22]. These types of regression methods have shown to be
effective for finding features which are correlated to larger magnitude values
of a response variable when there is ample data to create a reliable model. In
the types of applications we are considering, though, the extreme values tend
to be very rare for a typical set of observations. Because of this, it is
unclear how the desired quantile should be chosen based on the number and
magnitude of the extreme events. The rarity of the extreme values can also
cause the results from the regression model to be numerical unstable due to
the lack of adequate data to get accurate estimates and to sensitivity to
choice of quantile at the extremes. Additionally, quantile regression will not
be as useful in the situation when the response variable of interest has both
positive and negative extreme values, as the model by construction will
upweight the impact of one side of the extreme values while heavily
downweighting the other. Thus, quantile regression potentially restricts us to
focusing only on some of the extreme values while essentially ignoring others.
One other widely-used approach for modeling extreme values involves pre-
processing the data via some type of thresholding algorithm, keeping only the
observed values of each variable which are above either a static or dynamic
threshold and zeroing out the others. Examples of this in different fields
include spike calling or deconvolution in neuroscience [36] or Otsu’s method
in image processing [4]. After these algorithms have been applied to the data,
typical high-dimensional regression methods are then applied to the data. In
general, thresholding data can help in regression analysis for extreme values
by removing any influence from non-extreme values. However, this type of
filtering is not necessarily desirable in all situations. Thresholding
approaches by their nature binarize the observations of a variable in to
extreme and non-extreme categories, whereas in some cases it may make more
sense to smooth the transition from extreme to non-extreme values if it is not
clear where the boundary between the two should lie. Also, the addition of an
extra data pre-processing step can potentially lead to less precise estimates
from the following regression analysis, since any errors made in the former
will propagate to the latter regression step.
In this paper, we explore a different potential approach to tackle the problem
of modeling and predicting extreme values. Our approach to this problem is to
increase the relative weight of larger magnitude losses compared to regular
ordinary least squares. Conceptually, this problem is analogous to increasing
the power of the Gaussian kernel function, which leads to the generalized
normal distribution [35]. Thus, we base our method on $\ell_{p}$ norm
regression, which uses a general $p$ norm for regression rather than the
ordinary $\ell_{2}$ norm. This is a method which has been well-studied as a
whole in the past in the statistics literature [26, 28]. However, much of the
effort in previous research has been focused on showing that $\ell_{p}$-norm
regression can be more robust to outliers [13, 34] by using a norm between 0
and 1. On the other hand, we are interested in using this type of regression
model to create a method which is more sensitive to extreme values in the
response by using norms larger than the squared error loss, i.e. when $p>2$.
By doing this, we skew the regression results toward finding the relationships
with extreme values in a response variable without disregarding potentially
useful observations that could otherwise be ignored by thresholding or
substantially downweighted by quantile regression. We also analyze the
theoretical influence of extreme value observations on our proposed regression
model as well as the finite sample performance guarantees of the estimation
procedure. While general theoretical properties of $\ell_{p}$ norm regression
have been examined in previous literature [18], the performance with respect
to regression for extreme values when $p>2$ for $\ell_{p}$ norm regression has
not been well-studied; this particular situation presents its own unique
theoretical and practical challenges, which we will investigate in this paper.
The rest of the paper is organized as follows. Section 2 introduces and
characterizes the extreme value linear regression method and presents the
algorithm used for parameter estimation. We then prove consistency and
sparsistency results in section 3. Lastly, in section 4, we investigate the
performance of the extreme value linear regression through simulation and real
data studies.
## 2 Regression for Extreme Values
Let $\mathbf{X}\in\mathbb{R}^{n\times p}$ be a data matrix of predictor
variables and $\mathbf{y}\in\mathbb{R}^{n}$ be a corresponding vector of
responses. For our particular problem, we would like to find features in
$\mathbf{X}$ that are correlated with the extreme values of $\mathbf{y}$. (For
simplicity, we presume without loss of generality that each of the variables
are centered and scaled.) In this paper, we consider the context of a linear
data generating model, which will be the focus of the theory presented in
section 3 and the empirical investigations of section 4. Here, we assume that
the data are generated from a simple linear process:
$\mathbf{y}_{i}=\mathbf{X}_{i}\mathbf{\boldsymbol{\beta}}^{*}+\epsilon_{i},\,\epsilon\text{
i.i.d.}.$
In order to produce large magnitude extreme values in the observed response
$\mathbf{y}_{i}$ from this model, either some of the corresponding predictors
at the observed time $\mathbf{X}_{i}$ need to be large in magnitude relative
or some of the parameters in $\mathbf{\boldsymbol{\beta}}^{*}$ need to be
large in magnitude.
Note that we assume that the errors $\epsilon_{i}$ are independently and
identically distributed, but do not necessarily assume that they follow a
Gaussian distribution. In section 3, we will study both the cases where
$\epsilon$ follows a Gaussian distribution and where $\epsilon$ follows a
generalized normal, or Subbotin, distribution [35]. The generalized normal
distribution is defined as
$f(\boldsymbol{\epsilon})=\frac{\gamma}{2\sigma\Gamma(1/\gamma)}e^{-\left(\frac{|\epsilon|}{\sigma}\right)^{\gamma}}$
for scale parameter $\sigma>0$ and shape parameter $\gamma>0$. When
$\gamma=2$, the generalized normal distribution will be equivalent to a
Gaussian distribution, while when $\gamma>2$ the generalized normal
distribution will have a thinner tail compared to a Gaussian. Thus, we are
specifically interested in studying the case where the generalized normal
distribution with $\gamma>2$ as a potential error distribution of the data
generating model, as this relatively discourages the presence of extremely
large residuals in the regression model estimate when compared to a Gaussian
error distribution.
To get estimates of the parameters of the model above, we propose to use the
extreme value linear regression model, which is characterized by the
$\ell_{\gamma}$-norm regression for $\gamma>2$. The foundation for this method
is a generalized linear model applied to the generalized normal distribution
as described above. It follows naturally from the Gaussian case that
estimating the parameters of the generalized normal distribution for a
particular value of $\gamma$ is analogous to minimizing an $\ell_{\gamma}$
norm regression model loss function [26], which is of the form
$\mathcal{L}(\mathbf{y},\mathbf{X},\hat{\mathbf{\boldsymbol{\theta}}})=\frac{1}{\gamma
N}\|\mathbf{y}-\mathbf{X}\hat{\mathbf{\boldsymbol{\theta}}}\|_{\gamma}^{\gamma}$
where $\gamma$ corresponds to the shape parameter in the generalized normal
distribution. As follows from above, we are particularly interested in the
case of $\ell_{\gamma}$ norm regression for $\gamma>2$.
Figure 1: Loss functions for ordinary linear regression, extreme
$\ell_{\gamma}$ norm regression, and quantile regression.
To demonstrate the differences between the different regression methods
discussed in Section 1, we show the respective loss functions for each in
Figure 1. Specifically, we show the extreme linear regression loss function
for $\gamma=4,6,$ and $8$ and the loss for quantile regression at the 0.5 and
0.99 quantiles are shown. Comparing the different methods, we see the
advantage that the $\gamma$-th power error loss has over the other two loss
functions. Relative to the squared error loss function, the extreme value loss
function puts much less weight on very small residuals. However, as the
magnitude of a residual increases, the weight given by the extreme value loss
function grows exponentially compared to the squared error loss function. In
particular, this means that the extreme linear regression will reduce the
presence of abnormally large residuals, which occur when there is an extreme
value in the response variable which is not captured by the estimate from the
model. Thus, the extreme linear regression will find parameter estimates for
the model which better predict the occurrences of the extreme values of a
response variable. Quantile regression, on the other hand, proportionally
increases the relative weight of extreme values by adjusting the weights of an
absolute value loss function using linear constants. However, since the loss
function only grows linearly, the weight of extreme values compared to
relatively large but non-extreme values in the response variable will be small
for the quantile regression loss function compared to the extreme linear
regression loss function. Additionally, quantile regression can only put
increasing weights on either positive or negative residuals in the regression
estimate and not both, meaning that it is not suitable in the case where a
response variable has both positive and negative extreme values.
In terms of the impact to the weight of observations in a regression model,
our extreme linear regression model functions most similarly to the
$\epsilon$-invariant loss used in SVM regression [9] and to the heterogeneous
noise regression models [33]. Both of these methods can also be used to
substantially decrease the weight of smaller magnitude residuals compared to
the larger magnitude ones; this is accomplished by the $\epsilon$-invariant
loss by setting the loss for all residuals below a selected magnitude to be 0,
while the heterogeneous noise regression models can be used to increase the
weight of the observations which are large with respect to either the
predictors or the response variable. However, the extreme linear regression
model has some notable advantages compared to these techniques; it is not
sensitive to the choice of a thresholding hyperparameter as is the case for
SVM regression, and it does not require estimation of extra parameters as in
the heterogeneous noise model in order to achieve the desired effect for this
particular application.
### 2.1 Sparse Extreme Value Regression
In high-dimensional regression problems, automatic feature selection
techniques are used to obtain sparse solutions. In many contexts, this is done
by adding a sparsity-inducing regularization penalty. In the case of ordinary
linear regression, this leads to the penalized squared error loss function.
Applying the same idea to the extreme value linear regression model gives the
loss function:
$\min_{\mathbf{\boldsymbol{\beta}}}\frac{1}{2N}\|\mathbf{y}-\mathbf{X}\mathbf{\boldsymbol{\beta}}\|_{\gamma}^{\gamma}+\lambda\mathcal{P}(\beta).$
The form of the extreme value $\ell_{\gamma}$ norm loss function permits the
usage of any type of regularization penalty that can be applied to the
ordinary linear regression case. For example, one can employ more complex
penalties such as SCAD [39] or MCP [44], or specify a more specific structure
with penalties such as the Fused Lasso [37], Group Lasso [10], or Exclusive
Lasso [8].
Input : $\mathbf{y}\in\mathbb{R}^{N\times 1}\mathbf{X}\in\mathbb{R}^{N\times
p}$, $\lambda\geq 0,\gamma>2,\delta>0,0<\alpha<1.$
Initialize: $\mathbf{\boldsymbol{\beta}}^{(0)}=\boldsymbol{0}_{p}$
while
_$\frac{1}{N}\|\boldsymbol{\beta}^{(r)}-\boldsymbol{\beta}^{(r-1)}\|_{1}\geq\delta$_
do
(1) Find gradient $\nabla g(\mathbf{\boldsymbol{\beta}})$ and optimal step
size $t_{r}$ via backtracking:
(a) Set $t_{r}=1$.
(b) Calculate $\nabla
g^{(r)}(\mathbf{\boldsymbol{\beta}}^{(r)})=-\gamma\mathbf{X}^{T}[|y-\mathbf{X}\boldsymbol{\beta}^{(r)}|^{\circ(\gamma-1)}\circ\text{sgn}(y-\mathbf{X}\boldsymbol{\beta}^{(r)})]$
(c) Repeat:
(i)
$\mathbf{z}=\text{prox}_{\lambda*t_{r}\mathcal{P}}(\boldsymbol{\beta}^{(r)}-t_{r}g^{(r)}(\mathbf{\boldsymbol{\beta}}^{(r)}))$
(ii) $t_{r}=\alpha t_{r}$
until $g(\mathbf{z})\leq g(\boldsymbol{\beta}^{(r)})-\nabla
g(\boldsymbol{\beta}^{(r)})^{T}(\boldsymbol{\beta}^{(r)}-\mathbf{z})+\frac{1}{2t_{r}}\|\mathbf{z}-\boldsymbol{\beta}^{(r)}\|_{2}^{2}$
(2) Update $\mathbf{\boldsymbol{\beta}}^{(r+1)}=\mathbf{z}.$
(3) Update $r=r+1.$
end while
return _$\hat{\mathbf{\boldsymbol{\beta}}}=\boldsymbol{\beta}^{(r)}$_.
Algorithm 1 Regularized Extreme Regression Algorithm with Backtracking
Similar to the Lasso and other penalized ordinary linear regression models,
the objective function for the penalized extreme linear regression can be
decomposed in to the sum of two convex functions, the residual norm and the
penalty terms. Thus, a proximal gradient descent algorithm can be used to
estimate $\hat{\mathbf{\boldsymbol{\beta}}}$. Algorithmic convergence
properties of proximal gradient descent algorithms for penalized linear
regression have been well-studied in recent literature. Notably, it has been
shown that the proximal gradient algorithm is guaranteed to converge to a
minimum. Additionally, because the $\ell_{\gamma}$ loss function is convex for
$\gamma>2$, if the regularization penalty is also convex, then the algorithm
is guaranteed to converge to a global solution [29]. Algorithm 1 gives the
general outline of the computational methodology. As a practical
consideration, one can choose the $\gamma$ parameter for the regression model
either by a priori preference or by using a stability selection criteria,
which chooses the $\gamma$ value for the regression model that provides the
most consistent estimates of the sparse feature set based on a bootstrapping
procedure.
## 3 Theoretical Results
In this section, we present theoretical results for the performance of the
sparse extreme value regression method introduced previously. Specifically, we
focus our studies on the Extreme Lasso, i.e. the extreme value $\ell_{\gamma}$
norm estimator with an $\ell_{1}$ regularization penalty. We note that, for
the following results, we assume $\gamma$ to be a fixed parameter rather than
a parameter to estimate. Our analysis below is separated in to two parts.
First, we derive high-dimensional and finite-sample performance guarantees for
the Extreme Lasso estimator, showing that it is consistent and variable
selection consistent under two different error distributions appropriate for
the generalized normal. We then study our method with respect to the concept
of influence functions, a statistic to measure the effect of infinitesimal,
pointwise contamination of the covariates and response variable on the
resulting regression coefficients. Specifically, we formulate the influence
function of the Extreme Lasso regression model and use this to demonstrate
that the Extreme Lasso method is more heavily skewed toward selecting features
associated with extreme values compared to the ordinary Lasso regression
method. Formal proofs for all of the statements in Section 3 can be found in
the Appendix.
### 3.1 Consistency of the Extreme Lasso
We now present theoretical results for consistency and model selection
consistency of the Extreme Lasso. Our results bear similarity to existing
results for the consistency of Lasso-regularized M-estimators; the main
difference between the results presented here and those in previous works lies
in the distributional assumptions of the errors. Specifically, our
contribution lies in deriving concentration bounds for sub-Weibull and sub-
Gamma random variables. Consider the linear data generating model:
$\displaystyle\mathbf{y}_{i}=\mathbf{x}_{i}^{T}\mathbf{\boldsymbol{\beta}}^{*}+\epsilon_{i},\,\epsilon\text{
i.i.d.}.$
The Extreme Lasso regression thus solves the optimization problem:
$\displaystyle\operatorname*{minimize}_{\beta}\sum_{i=1}^{n}|y_{i}-\mathbf{x}_{i}^{T}\operatorname{\boldsymbol{\beta}}|^{\gamma}+\lambda\|\operatorname{\boldsymbol{\beta}}\|_{1}$
For simplicity, we consider the case when $\gamma$ is an even integer. The
problem can now be written as:
$\displaystyle\operatorname*{minimize}_{\beta}\sum_{i=1}^{n}(y_{i}-\mathbf{x}_{i}^{T}\operatorname{\boldsymbol{\beta}})^{\gamma}+\lambda\|\operatorname{\boldsymbol{\beta}}\|_{1}$
Define
$\mathcal{L}(\operatorname{\boldsymbol{\beta}})=\frac{1}{n}\sum_{i=1}^{n}\ell(\mathbf{x}_{i}^{T}\operatorname{\boldsymbol{\beta}}-y_{i}).$
Clearly, $\mathcal{L}$ belongs to the family of M-estimators, whose properties
have been widely studied in literature; in particular, Negahban et al. [27],
Loh et al. [23], and Loh and Wainwright [24] have established the consistency
of M-estimators in the high-dimensional setting. Thus, we can apply the ideas
and theories for high-dimensional M-estimators from these papers to the
Extreme Lasso case to obtain the results for the regularized extreme value
linear regression.
We first state the previous results regarding the consistency and variable
selection consistency for general robust M-estimators which we use below. In
the literature, Negahban et al. [27] established consistency for high-
dimensional M-estimators:
###### Lemma 3.1 (Estimation Consistency [27]).
Suppose $\mathcal{L}$ satisfies the Restricted Strong Convexity (RSC)
condition with curvature $\kappa_{\mathcal{L}}$ and
$\displaystyle\lambda\geq
2\|\nabla\mathcal{L}(\operatorname{\boldsymbol{\beta}}^{*})\|_{\infty}.$
Then $\hat{\operatorname{\boldsymbol{\beta}}}$ exists and satisfies the
bounds:
$\displaystyle\|\hat{\operatorname{\boldsymbol{\beta}}}-\operatorname{\boldsymbol{\beta}}^{*}\|_{2}$
$\displaystyle\leq\frac{3\sqrt{s}}{\kappa_{\mathcal{L}}}\lambda$
where $s=|\text{supp}(\operatorname{\boldsymbol{\beta}}^{*})|$, i.e.,
$\|\operatorname{\boldsymbol{\beta}}^{*}\|_{0}$.
Note that Lemma 3.1 corresponds to Theorem 1 in Negahban et al. [27] assuming
that the restricted strong convexity (RSC) holds with tolerance parameter
$\tau_{\mathcal{L}}=0$. Also, here we consider $\ell_{1}$ penalty and
$\Psi(\mathcal{M})=\sqrt{s}$. Similarly, Loh et al. [23] established model
selection consistency, also known as sparsistency, for high-dimensional robust
M-estimators:
###### Lemma 3.2 (Model Selection Consistency [19]).
Suppose the following conditions hold:
(1) $\ell$ satisfies RSC.
(2) $\ell$ satisfies irrepresentability.
Let $\kappa_{\text{IC}}$ denote the compatibility constant defined in Lee et
al. [19]. Then, for any
$\frac{4\kappa_{\text{IC}}}{\tau}\|\nabla\mathcal{L}(\operatorname{\boldsymbol{\beta}}^{*})\|_{\infty}<\lambda<\frac{\kappa_{\mathcal{L}}^{2}}{2L}\big{(}2\sqrt{s}+\frac{\sqrt{s}}{\kappa_{\text{IC}}}\frac{\tau}{2}\big{)}^{-2}\frac{\tau}{\kappa_{\text{IC}}}$,
the optimal solution to an M-estimator problem is unique and model selection
consistent: $\hat{\beta}\in M$.
Further, if
$\min_{a\in\mathcal{S}}|\beta_{a}^{*}|>\frac{2}{\kappa_{\mathcal{L}}}\big{(}\sqrt{s}+\frac{\tau}{4}\frac{\sqrt{s}}{\kappa_{\text{IC}}}\big{)}\lambda$,
then the estimator is also sign consistent:
$\text{sign}(\hat{\operatorname{\boldsymbol{\beta}}}_{\mathcal{S}})=\text{sign}(\operatorname{\boldsymbol{\beta}}^{*}_{\mathcal{S}})$.
Lemma 3.2 refers to Theorem 3.4 in Lee et al. [19]. The finite constant
$\kappa_{\text{IC}}$ is the compatibility constant between the irrepresentable
term and $\rho^{*}$. $\tau$ is the constant in the irrepresentable condition.
Since we consider the $\ell_{1}$-norm, i.e., $\rho=\|\cdot\|_{1}$, we have
$k_{\rho}=\sqrt{s}$ and $k_{\rho^{*}}=1$ in the theorem. $L$ is a constant
such that
$\|\nabla^{2}\ell(\operatorname{\boldsymbol{\beta}})-\nabla^{2}\ell(\operatorname{\boldsymbol{\beta}}^{*})\|_{2}\leq
L\|\operatorname{\boldsymbol{\beta}}-\operatorname{\boldsymbol{\beta}}^{*}\|_{2}$.
Note in the Lasso problem, it can be shown that $L=0$; hence there is no upper
bound for $\lambda$. In the Extreme Lasso case, in general we have $L\neq 0$
and there is an upper bound for $\lambda$.
Importantly, the results from both Lemma 3.1 and Lemma 3.2 are entirely
deterministic. Thus, we can guarantee that, under certain conditions, the
extreme value linear regression with the Lasso penalty will provide consistent
estimates of the true parameters of the model. Additionally, both Lemma 3.1
and Lemma 3.2 suggest that the key ingredients for statistical consistency are
the boundedness of $\|\nabla\mathcal{L}(\beta^{*})\|_{\infty}$, which
ultimately determines the rate of convergence of
$\hat{\operatorname{\boldsymbol{\beta}}}$ to
$\operatorname{\boldsymbol{\beta}}^{*}$ and the local RSC condition. Notice
that when $\ell$ is the squared error loss, we get the same consistency and
model selection consistency rate for the Lasso regression problem:
$\displaystyle\|\nabla\mathcal{L}(\beta^{*})\|_{\infty}=\frac{1}{n}\|\mathbf{X}^{T}(y-\mathbf{X}\operatorname{\boldsymbol{\beta}}^{*})\|_{\infty}=\|\mathbf{X}^{T}\epsilon\|_{\infty}/n.$
On the other hand, for the Extreme Lasso case, i.e.
$\ell(\mathbf{x}_{i}^{T}\operatorname{\boldsymbol{\beta}}-y_{i})=(y_{i}-x_{i}^{T}\operatorname{\boldsymbol{\beta}})^{\gamma}$,
we have:
$\displaystyle\|\nabla\mathcal{L}(\beta^{*})\|_{\infty}=\gamma\cdot\frac{1}{n}\|\mathbf{X}^{T}\epsilon^{\circ(\gamma-1)}\|_{\infty}.$
To establish complete results for consistency and model selection consistency
for the Extreme Lasso, we first build a concentration bound for the quantity
$\|\nabla\mathcal{L}(\beta^{*})\|_{\infty}$, i.e.,
$\gamma\cdot\frac{1}{n}\|\mathbf{X}^{T}\epsilon^{\circ(\gamma-1)}\|_{\infty}.$
To do this, we first need to build a tail bound for $\epsilon_{i}^{\gamma-1}$,
which will differ under different distributional assumptions on the covariates
and error terms in the linear model. These assumptions on the distributional
properties will come into play in verifying that the inequality and the RSC
condition hold with high probability under the prescribed sample size scaling.
We can then combine the tail bound results with Lemma 3.1 and Lemma 3.2 to
derive full results. Below, we present tail bounds for
$\frac{1}{n}\|\mathbf{X}^{T}\epsilon^{\circ(\gamma-1)}\|_{\infty}$ under two
different distribution assumptions on the error $\epsilon$.
#### 3.1.1 Sub-Gaussian Errors
We first assume that $\epsilon_{i}$ follows a sub-Gaussian distribution, and
we construct a tail bound for a sub-Gaussian random variable raised to a
power.
###### Lemma 3.3 (Tail Bound for Sub-Gaussian Raised to a Power).
For sub-Gaussian random variable $\mathbf{Q}$, we have
$\displaystyle\mathbb{P}(|\mathbf{Q}|^{\gamma-1}\geq t)\leq
2\exp\bigg{\\{}-\frac{t^{2/(\gamma-1)}}{2\sigma^{2}}\bigg{\\}}.$
Under ordinary least squares, i.e. when $\gamma=2$, we get the usual sub-
Gaussian tail bound; when $\gamma=3$, $\mathbf{Q}^{2}$ follows a sub-
exponential distribution. When $\gamma\geq 4$, as we have for the Extreme
Lasso, $\mathbf{Q}^{\gamma-1}$ is neither sub-Gaussian nor sub-exponential.
Instead, in this situation the tail bound will follow what is known in the
literature as a sub-Weibull distribution [17, 38], which we define below.
###### Definition 3.1 (Sub-Weibull Variables).
A random variable $\mathbf{Z}$ is said to be sub-Weibull of order $\alpha>0,$
denoted as sub-Weibull($\alpha$), if
$\|\mathbf{Z}\|_{\psi_{\alpha}}<\infty,\quad\text{ where
}\psi_{\alpha}(x):=\exp\left(x^{\alpha}\right)-1\quad\text{ for }x\geq 0.$
Based on this definition, it follows that if $\mathbf{Z}$ is sub-Weibull
$(\alpha),$ then
$\mathbb{P}(|\mathbf{Z}|\geq t)\leq
2\exp(-\frac{t^{\alpha}}{\|\mathbf{Z}\|_{\psi_{\alpha}}^{\alpha}}),\text{ for
all }t\geq 0.$
In the Extreme Lasso problem, since $\epsilon_{i}$ is sub-Gaussian, we have
$\mathbb{P}(|\epsilon_{i}|^{\gamma-1}\geq t)\leq
2\exp\bigg{\\{}-\frac{t^{2/(\gamma-1)}}{2\sigma^{2}}\bigg{\\}}$, which means
$\epsilon_{i}^{\gamma-1}$ is sub-Weibull, i.e.,
$\|\epsilon_{i}^{\gamma-1}\|_{\psi_{2/(\gamma-1)}}<\infty$. In the literature,
Kuchibhotla and Chakrabortty [17] established concentration inequalities
related to sub-Weibull random variables. We apply the results and build a tail
bound for
$\|\sum_{i=1}^{n}\mathbf{x}_{i}\epsilon_{i}^{\gamma-1}\|_{\infty}/n$, i.e.,
$\|\mathbf{X}^{T}\epsilon^{\circ(\gamma-1)}\|_{\infty}/n$ by making the
substitution $\mathbf{Z}=\epsilon_{i}^{\gamma-1}$. Note that by Negahban et
al. [27], restricted strong convexity (for M-estimators) with respect to the
$\ell_{2}$-norm is equivalent to the restricted eigenvalues condition (for the
Lasso estimator).
###### Lemma 3.4 (Concentration Bound for Sum of Sub-Weibull Random Variables
[17]).
Consider the Lasso estimator for linear regression case. Suppose there exists
$0<\alpha\leq 2$, and $\gamma,K_{n,p}>0$ such that
$\max\bigg{\\{}\|X_{i}\|_{M,\psi_{\alpha}},\|\epsilon_{i}\|_{\psi_{\gamma}}\bigg{\\}}\leq
K_{n,p}\hskip 14.22636pt\text{for all}\hskip 5.69054pt1\leq i\leq n.$
Also suppose $n\geq 2$, $k\geq 1$ and the covariance matrix $\Sigma_{n}$
satisfies $\lambda_{\min}(\Sigma_{n})\geq K_{n,s}$. Then, with probability at
least $1-3(np)^{-1}$,
$\left\|\frac{1}{n}\sum_{i}^{n}X_{i}\epsilon_{i}\right\|_{\infty}\leq
7\sqrt{2}\sigma_{n,p}\sqrt{\frac{\log(np)}{n}}+\frac{C_{\tau}K_{n,p}^{2}(\log(2n))^{1/\tau}(2\log(np))^{1/\tau}}{n}$
where $\frac{1}{\tau}=\frac{1}{\alpha}+\frac{1}{\gamma}$.
###### Theorem 3.1 (Consistency for Sub-Gaussian Error).
Given the Extreme Lasso program with regularization parameter
$\lambda_{n}=2\gamma\big{(}7\sqrt{2}\sigma_{n,p}\sqrt{\frac{\log(np)}{n}}+$
$\frac{C_{\tau}K_{n,p}^{2}(\log(2n))^{1/\tau}(2\log(np))^{1/\tau}}{n}\bigg{)}$,
then with probability at least $1-3(np)^{-1}$, any optimal solution
$\hat{\beta}$ satisfies the bounds:
$\displaystyle\|\hat{\operatorname{\boldsymbol{\beta}}}-\operatorname{\boldsymbol{\beta}}^{*}\|_{2}$
$\displaystyle\leq\frac{6\sqrt{s}}{\kappa_{\mathcal{L}}}\cdot\gamma\bigg{(}7\sqrt{2}\sigma_{n,p}\sqrt{\frac{\log(np)}{n}}+\frac{C_{\tau}K_{n,p}^{2}(\log(2n))^{1/\tau}(2\log(np))^{1/\tau}}{n}\bigg{)}.$
where $\tau=2/(\gamma-1)$.
###### Theorem 3.2 (Model Selection Consistency for Sub-Gaussian Error).
Consider the Extreme Lasso program with sub-Gaussian error. Assume that the
loss $\ell$ satisfies Restricted Strong Convexity and covariance matrices
satisfy irrepresentability. Consider the family of regularization parameters
$\lambda=\frac{4\kappa_{\text{IC}}}{\tau}\cdot\gamma\bigg{(}7\sqrt{2}\sigma_{n,p}\sqrt{\frac{\log(np)}{n}}+\frac{C_{\tau}K_{n,p}^{2}(\log(2n))^{1/\tau}(2\log(np))^{1/\tau}}{n}\bigg{)},$
then the following properties holds with probability greater than
$1-3(np)^{-1}$:
(i) The Lasso has a unique solution with support contained within $S$, i.e.
$S(\hat{\beta})\subset S(\beta^{*})$.
(ii) If $\min_{a\in
S}|\beta^{*}_{a}|>(\frac{\tau}{\kappa_{\text{IC}}}\cdot\frac{1}{4}+1)\cdot\frac{2\sqrt{s}}{\kappa_{\mathcal{L}}}\cdot\frac{4\kappa_{\text{IC}}}{\tau}\cdot\gamma\bigg{[}7\sqrt{2}\sigma_{n,p}\sqrt{\frac{\log(np)}{n}}+\frac{C_{\tau}K_{n,p}^{2}(\log(2n))^{1/\tau}(2\log(np))^{1/\tau}}{n}\bigg{]}$
, the lasso estimator is also sign consistent:
$\text{sign}(\hat{\beta}_{S})=\text{sign}(\beta^{*}_{S})$.
Applying the result of Theorem 3.1 for $\gamma=2$, we can achieve the usual
consistency rate of $\sqrt{k\log p/n}$ for the ordinary squared error Lasso
loss function under the constraint
$K_{\varepsilon,r}(\log(np))^{-1/2}(\log(2n))^{1/2}=o\left(n^{1/2}\right)$
Note that the probability of the bound being satisfied approaches 1 as
$n\to\infty$, and thus the bound is proportional $\log(np)$ instead of the
usual $\log p$. By setting the probability to be $1-O(p^{-1})$, the usual
Lasso rate $\sqrt{k\log p/n}$ can be recovered.
#### 3.1.2 Subbotin Error
In the following section, we assume that $\epsilon$ follows a Subbotin
distribution, i.e., $\epsilon\sim\text{Subbotin}(\gamma)$. We study this
particular distributional assumption as the Extreme Lasso problem is
equivalent to minimizing the negative log-likelihood of the Subbotin
distribution plus the regularization penalty. To see this, recall the
likelihood of Subbotin distribution:
$\displaystyle f_{Y}(\mathbf{y};\mathbf{X};\operatorname{\boldsymbol{\beta}})$
$\displaystyle=c_{1}\prod_{i=1}^{n}\exp\bigg{[}-|y_{i}-\mathbf{x}_{i}^{T}\operatorname{\boldsymbol{\beta}}|^{\gamma}\bigg{]}$
$\displaystyle=c_{1}\exp\bigg{[}-\sum_{i=1}^{n}|y_{i}-\mathbf{x}_{i}^{T}\operatorname{\boldsymbol{\beta}}|^{\gamma}\bigg{]}$
Thus, the negative log-likelihood,
$\ell(\mathbf{x}_{i}^{T}\operatorname{\boldsymbol{\beta}}-y_{i})\propto\sum_{i=1}|y_{i}-\mathbf{x}_{i}^{T}\operatorname{\boldsymbol{\beta}}|^{\gamma}$,
corresponds to the loss function in the Extreme Lasso problem. Similar to
before, our goal is to build a tail bound for
$\|\mathbf{X}^{T}\epsilon^{\circ(\gamma-1)}\|_{\infty}/n$. To do this, we
first observe that $\epsilon_{i}^{\gamma}$ follows a Gamma distribution.
###### Lemma 3.5 (Change of Variables).
Suppose $\mathbf{Z}\sim$ Subbotin($\alpha$), where $\alpha$ is an even
integer, then
$\mathbf{Y}=\mathbf{Z}^{\alpha}\sim Gamma(\frac{1}{\alpha},1).$
Thus, by Lemma 3.5, we have
$\epsilon_{i}^{\gamma}\sim\text{Gamma}(\frac{1}{\theta},1)$. Hence,
$\epsilon_{i}^{\gamma}$ follows a Gamma distribution and can be bounded by
sub-Gamma tail bounds in literature [6]. These results are stated in Lemma 3.6
and used to derive the results for Theorem 3.3 and Theorem 3.4 below.
###### Lemma 3.6 (Concentration Bound for Sub-Gamma Random Variables [6]).
If $\mathbf{Z}\sim$ Gamma($\alpha,\beta$), then we have:
$\displaystyle\mathbb{P}[\mathbf{Z}-\mathbb{E}\mathbf{Z}]\geq\sqrt{2\gamma
t}+ct]\leq e^{-t}\hskip 14.22636pt$
where $\gamma=\alpha\beta^{2}$, $c=\beta$. We call that $\mathbf{Z}$ is sub-
Gamma with $(\gamma,c)$.
###### Lemma 3.7 (Concentration Bound for Sum of Sub-Gamma Random Variables).
If $\mathbf{Z}\sim$ Gamma($\alpha,\beta$), then with probability at least
$1-c_{1}\exp(-c_{2}\log p)$,
$\left\|\frac{1}{n}\sum_{i}^{n}X_{i}\epsilon_{i}\right\|_{\infty}\leq\sqrt{\frac{\log
p}{n}}\bigg{[}2\sqrt{\frac{2}{\gamma}}+\sqrt{\frac{\log p}{n}}\bigg{]}$
###### Theorem 3.3 (Consistency for Subbotin Error).
Given the Extreme Lasso program with regularization parameter
$\lambda_{n}=2\gamma\sqrt{\frac{\log
p}{n}}\bigg{[}2\sqrt{\frac{2}{\gamma}}+\sqrt{\frac{\log p}{n}}\bigg{]}$, then
with probability at least $1-c_{1}\exp(-c_{2}\log p)$, any optimal solution
$\hat{\beta}$ satisfies the bounds:
$\displaystyle\|\hat{\operatorname{\boldsymbol{\beta}}}-\operatorname{\boldsymbol{\beta}}^{*}\|_{2}$
$\displaystyle\leq\frac{6\sqrt{s}}{\kappa_{\mathcal{L}}}\gamma(\sqrt{\frac{\log
p}{n}}\bigg{[}2\sqrt{\frac{2}{\gamma}}+\sqrt{\frac{\log p}{n}}\bigg{]}).$
###### Theorem 3.4 (Model Selection Consistency for Subbotin Error).
Consider the Extreme Lasso program with Subbotin distributed error. Assume
that the loss $\ell$ satisfies Restricted Strong Convexity and the covariance
matrices satisfy irrepresentability. Consider the family of regularization
parameters $\lambda=\frac{4\kappa_{\text{IC}}}{\tau}\gamma\sqrt{\frac{\log
p}{n}}\bigg{[}2\sqrt{\frac{2}{\gamma}}+\sqrt{\frac{\log p}{n}}\bigg{]}$, then
the following properties holds with probability greater than
$1-c_{1}\exp(-c_{2}\log p)$:
(i) The Lasso has a unique solution with support contained within $S$, i.e.
$S(\hat{\beta})\subset S(\beta^{*})$.
(ii) If $\min_{a\in
S}|\beta^{*}_{a}|>(\frac{\tau}{\kappa_{\text{IC}}}\cdot\frac{1}{4}+1)\cdot\frac{2\sqrt{s}}{\kappa_{\mathcal{L}}}\cdot\frac{4\kappa_{\text{IC}}}{\tau}\gamma\sqrt{\frac{\log
p}{n}}\bigg{[}2\sqrt{\frac{2}{\gamma}}+\sqrt{\frac{\log p}{n}}\bigg{]}$, the
lasso estimator is also sign consistent:
$\text{sign}(\hat{\beta}_{S})=\text{sign}(\beta^{*}_{S})$.
Note that Gaussian distribution is equivalent to the Subbotin distribution
when $\theta=2$. Thus, in the case where $\epsilon_{i}$ is a Gaussian random
variable, we have by Lemma 3.5 that $\epsilon_{i}^{2}$ is
Gamma($\frac{1}{2},1)$. Hence, $\epsilon_{i}^{2}$ is sub-Gamma with
$(\frac{1}{2},1)$. Suppose that $\|\mathbf{X}_{j}\|_{\infty}\leq 1$, we then
have $\mathbf{X}_{j}^{T}\epsilon$ is a sub-Gamma$(n/2,1)$ random variable.
Thus, it follows from Lemma 3.6 that, in this particular case, we have:
$\displaystyle\mathbb{P}\big{(}\mathbf{X}_{j}^{T}\epsilon-\mathbb{E}[\mathbf{X}_{j}^{T}\epsilon]\geq
2\sqrt{nt}+t\big{)}\leq e^{-t}.$
However, if we instead use known Lasso results for $\epsilon$ with sub-
Gaussian tail bounds and set $t=\sigma\sqrt{\frac{c\log p}{n}}$, then we have:
$\displaystyle\mathbb{P}\big{(}|\mathbf{X}_{j}^{T}\epsilon|/n\geq{t}\big{)}\leq
2e^{-\frac{nt^{2}}{2\sigma^{2}}}.$
In effect, the sub-Gamma tail bound has an extra term compared to the sub-
Gaussian bound. This can be seen when comparing the result of Theorem 3.3 and
Theorem 3.4 to the Lasso consistency rate derived using sub-Gaussian tail
bounds. Specifically, there is an extra factor of $\frac{\log p}{n}$ in the
consistency rate result from Theorem 3.3 and Theorem 3.4 compared to the
regular Lasso consistency rate. This is to be expected given that the sub-
Gamma is generally a weaker distributional assumption compared to the sub-
Gaussian. However, this does show that the bound for Theorem 3.3 and Theorem
3.4 is not necessarily tight for any particular values of $\theta$.
### 3.2 Influence of Extreme Values
Here, we demonstrate that our Extreme Lasso estimator is better at selecting
features associated with extreme values than the regular Lasso estimator. We
do this by utilizing the concept of influence functions, which have been
previously proposed in the regression literature as a method for analyzing and
quantifying the effect of outliers in data on statistical estimators [12].
However, in previous works, the influence functions have generally been used
in order to demonstrate the robustness of a regression estimator to the
outlier observations. In our case, we consider the opposite direction, where
we show that the Extreme Lasso estimator is more sensitive to the extreme
values and hence tends to select features associated with extreme values more.
To do this, we show that the value of influence function of the Extreme Lasso
is greater than the Lasso, suggesting that our proposed estimator is affected
more by extreme values.
We follow closely the approach by Wang et al. [40]. Denote as
$\delta_{\mathbf{Z}}$ the point mass probability distribution at a fixed point
$\mathbf{z}=\left(\mathbf{x}_{0},y_{0}\right)^{T}\in\mathbb{R}^{p+1}$. Given
the distribution $F$ of $(\mathbf{x},y)$ in $\mathbb{R}^{p+1}$ and proportion
$\epsilon\in(0,1)$, the mixture distribution of $F$ and $\delta_{\mathbf{Z}}$
is $F_{\epsilon}=(1-\epsilon)F+\epsilon\delta_{\mathbf{Z}}$. Let
$\displaystyle\boldsymbol{\beta}_{0}^{*}$
$\displaystyle=\operatorname*{arg\,min}_{\beta}\left[\left\\{\int\left(\|y-\mathbf{X}^{T}\boldsymbol{\beta}\|^{\gamma}\right)\mathrm{d}F\right\\}+\sum_{j=1}^{p}p_{\lambda_{j}}\left(\left|\beta_{j}\right|\right)\right]$
and
$\displaystyle\boldsymbol{\beta}_{\epsilon}^{*}$
$\displaystyle=\operatorname*{arg\,min}_{\beta}\left[\left\\{\int\left(\|y-\mathbf{X}^{T}\boldsymbol{\beta}\|^{\gamma}\right)\mathrm{d}F_{\epsilon}\right\\}+\sum_{j=1}^{p}p_{\lambda_{j}}\left(\left|\beta_{j}\right|\right)\right].$
For the Lasso and Extreme Lasso,
$p_{\lambda_{j}}\left(\left|\beta_{j}\right|\right)=|\beta_{j}|$. For an
exponential-type estimator, the influence function at a point
$\mathbf{z}\in\mathbb{R}^{p+1}$ is defined as
$\operatorname{IF}\left(\mathbf{z};\boldsymbol{\beta}_{0}^{*}\right)=\lim_{\epsilon\rightarrow
0^{+}}\left(\boldsymbol{\beta}_{\epsilon}^{*}-\boldsymbol{\beta}_{0}^{*}\right)/\epsilon,$
as long as the limit exists. We use this definition to derive the specific
form of the influence function for the Extreme Lasso; the result is shown
below in Theorem 3.5.
###### Theorem 3.5 (Influence Function of Extreme Lasso).
For the penalized extreme value regression estimators with
$\ell_{\gamma}$-norm loss, the $j$th element of the influence function
$\operatorname{IF}_{j}\left(\mathbf{z};\boldsymbol{\beta}_{0}^{*}\right)$ has
the following form:
$\operatorname{IF}_{j}\left(\mathbf{z};\boldsymbol{\beta}_{0}^{*}\right)\\\
=\begin{cases}0,&\text{ if }\beta_{0j}^{*}=0,\\\ -\Gamma_{j}\left\\{-\gamma
r_{0}^{\gamma-1}x_{0}+v_{2}\right\\},&\text{otherwise,}\end{cases}$
where $\Gamma_{j}$ denotes the $j$th row of
$\left\\{A\left(\gamma_{0}\right)-B\right\\}^{-1},r_{0}=y_{0}-$
$\mathbf{x}_{0}^{T}\boldsymbol{\beta}_{0}^{*}$,
$\displaystyle v_{2}$
$\displaystyle=\left\\{p_{\lambda_{1}}^{\prime}\left(\left|\beta_{01}^{*}\right|\right)\operatorname{sign}\left(\beta_{01}^{*}\right),\ldots,p_{\lambda_{d}}^{\prime}\left(\left|\beta_{0d}^{*}\right|\right)\operatorname{sign}\left(\beta_{0d}^{*}\right)\right\\}^{T},$
$\displaystyle B$
$\displaystyle=\operatorname{diag}\left\\{p_{\lambda_{1}}^{\prime\prime}\left(\left|\beta_{01}^{*}\right|\right),\ldots,p_{\lambda_{d}}^{\prime\prime}\left(\left|\beta_{0d}^{*}\right|\right)\right\\},$
and
$A(\gamma)=\int\mathbf{x}\mathbf{x}^{T}\gamma(\gamma-1)\left(y-\mathbf{x}^{T}\boldsymbol{\beta}_{0}^{*}\right)^{\gamma-2}\times\mathrm{d}F(\mathbf{x},y).$
One important implication of this result is that the Extreme Lasso with
$\ell_{\gamma}$ regression is more sensitive to features containing extreme
values, as formally stated below in Corollary 1.
###### Corollary 1.
The influence function of the Extreme Lasso with $\gamma>2$ is greater than
the influence function of Lasso.
Corollary 1 can be shown by using a direct comparison with the Lasso influence
function, i.e. the case where $\gamma=2$. Specifically, we have:
$\displaystyle\frac{\operatorname{IF}\left((x,y);T_{\text{Extreme}},F_{\beta_{j}}\right)}{\operatorname{IF}\left((x,y);T_{\text{Lasso}},F_{\beta_{j}}\right)}$
$\displaystyle=\frac{\gamma
r_{0}^{\gamma-1}x_{0}-v_{2}}{2r_{0}x_{0}-v_{2}}\cdot\frac{A(2)-B_{1}}{A(\gamma)-B_{1}}.$
Recall that $\beta_{0}^{*}$ is the coefficient of fitting the data without
extreme values. Hence, if $x_{0}$ is an influential point,
$r_{0}=y_{0}-\mathbf{x}_{0}^{T}\beta_{0}^{*}$ is sufficiently large, which
means in this case that $r_{0}^{\gamma-1}\gg r_{0}$ for $\gamma>2$. Hence,
$\frac{\operatorname{IF}\left((x,y);T_{\text{Extreme}},F_{\beta_{j}}\right)}{\operatorname{IF}\left((x,y);T_{\text{Lasso}},F_{\beta_{j}}\right)}>1$,
i.e., the influence function evaluated at $\gamma>2$ is greater than evaluated
at $\gamma=2$. Thus, the Extreme Lasso will be more likely to select features
associated with large magnitude values of $\mathbf{x}$ and $y$ compared to the
ordinary Lasso regression.
## 4 Empirical Investigations
Below, we analyze the performance of sparse extreme value linear regression
below on two sets of simulations studies and two real-world case studies.
### 4.1 Linear Model Simulation Study
We first study the performance of our method on a simulation study with data
generated from the linear model as described in section 2, i.e.
$\mathbf{y}_{i}=\mathbf{X}_{i}\mathbf{\boldsymbol{\beta}}^{*}+\epsilon_{i}$.
We let $\boldsymbol{\epsilon}\overset{iid}{\sim}Gamma(\alpha,\beta)$ (using
the rate parameterization) before centering such that $\bar{\epsilon}=0.$ The
predictor matrices $\mathbf{X}$ contain $n=1000$ observations and $p=750$
features. The columns of the matrix are generated as AR(1) processes with
variance 1 and a cross-correlation with one other column of $\rho=0.9$. We
then add large positive extreme values to the columns at known observation
points; these are different for each column. The true parameter vector
$\mathbf{\boldsymbol{\beta}}^{*}$ is set to have 10 randomly selected nonzero
entries. Our goal is to recover the full non-zero support of
$\mathbf{\boldsymbol{\beta}}^{*}$ without recovering false positives. We
analyze four different varying simulation specifications:
1. 1.
The signal to noise ratio of the extreme events relative to baseline noise,
which we denote as $\tau$.
2. 2.
The number of extreme events added to each of the columns of $\mathbf{X}$.
3. 3.
The distribution of the errors $\boldsymbol{\epsilon}$.
4. 4.
The number of dimensions $p$, holding the number of observations and parameter
sparsity level constant.
We compare the Extreme Lasso regression model, as defined in section 3, with
the ordinary Lasso, $\ell_{1}$ quantile regression, and Lasso regression after
preprocessing the data using data-driven thresholding. We fit the Extreme
Lasso regression model, as defined in section 3, using $\gamma=4$ and
$\gamma=6$. For $\ell_{1}$ quantile regression, we find parameter estimates at
the 0.5, 0.9, 0.99, and 0.999 quantiles. Data-driven thresholding is done by
using the adaptive CUSUM method [41] to identify extreme values in the
response variable and removing any data which does not correspond to those
observed extreme values. The number of variables for all methods is selected
via approximate oracle sparsity tuning. We use 4 replications for each
scenario. The results for each of the simulations studies are shown below
using average F-1 scores along with the standard deviations across all
replications. The full results, which include F-1 scores, true positive rates,
and false positive rates for each of the simulations, as well as comparisons
with different regularization penalties for the extreme value linear
regression and ordinary linear regression models, can be found in the
Appendix.
##### Scenario 1: Magnitudes of Extreme Values in Response
Here, we change the size of the signal to noise ratio, comparing
$\tau=6,7,11$, and $15$. The results are shown in Table 1. When the signal to
noise ratio of the extreme values is not sufficiently large, none of the
methods are able to select the correct features. Similarly, if the signal to
noise ratio is large enough, all of the methods except quantile regression are
able to pick out the correct features. However, we see that there is a fairly
large window of $\tau$ values in which the Extreme Lasso is able to find the
correct features while ordinary linear regression and thresholding fail.
Table 1: Average F-1 scores, changing relative extreme value magnitudes for the linear model. | $\tau$ = 6 | $\tau$ = 7 | $\tau$ = 11 | $\tau$ = 15
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.196 (0.1382) | 0.209 (0.1778) | 0.875 (0.05) | 0.938 (0.0481)
ExLasso ($\gamma=6$) | 0.296 (0.1416) | 0.782 (0.0894) | 0.85 (0.0577) | 0.938 (0.0481)
Lasso | 0.2 (0.1414) | 0.225 (0.15) | 0.3 (0) | 0.938 (0.0481)
Median | 0.149 (0.0357) | 0.301 (0.2087) | 0.529 (0.0626) | 0.44 (0.1056)
Q0.9 | 0.149 (0.0357) | 0.127 (0.0429) | 0.185 (0.1239) | 0.147 (0.0508)
Q0.99 | 0.095 (0.0394) | 0.09 (0.0194) | 0.102 (0.0355) | 0.111 (0)
Q0.999 | 0.132 (0.1028) | 0.219 (0.2222) | 0.328 (0.2583) | 0.321 (0.1821)
Threshold | 0.028 (0.0556) | 0 (0) | 0 (0) | 0.893 (0.1056)
##### Scenario 2: Number of Extreme Events in Response
We now vary the number of extreme value events $E$ from 1 to 4, with $\tau=6$.
Results are shown in Table 2. As we observed above, in the case of one extreme
event at $\tau=6$, none of the methods do well. When there is more than one
extreme event though, the Extreme Lasso is able to pick out the correct
features. None of the other methods are able to perform nearly as well when we
increase the number of extreme value events in this case, with only a slight
improvement in performance at $E=4$ compared to $E=1$.
Table 2: Average F-1 scores, changing number of extreme events for the linear model. | E = 1 | E = 2 | E = 3 | E = 4
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.875 (0.05) | 0.225 (0.05) | 0.79 (0.0838) | 0.913 (0.0857)
ExLasso ($\gamma=6$) | 0.85 (0.0577) | 0.788 (0.2022) | 0.779 (0.1447) | 0.85 (0.1291)
Lasso | 0.3 (0) | 0.36 (0.1925) | 0.339 (0.0773) | 0.325 (0.05)
Median | 0.529 (0.0626) | 0.513 (0.059) | 0.472 (0.1155) | 0.457 (0.1337)
Q0.9 | 0.185 (0.1239) | 0.301 (0.0809) | 0.311 (0.157) | 0.414 (0.1092)
Q0.99 | 0.102 (0.0355) | 0.107 (0.0053) | 0.099 (0.0048) | 0.126 (0.0376)
Q0.999 | 0.328 (0.2583) | 0.232 (0.0992) | 0.334 (0.0793) | 0.445 (0.2531)
Threshold | 0 (0) | 0 (0) | 0 (0) | 0.075 (0.15)
##### Scenario 3: Error Distributions
In this scenario, we change the distribution of the added errors by changing
the rate parameter of the pre-centered Gamma distribution from which they are
generated. By decreasing the rate parameter, we increase the variance of the
errors and thus increase the probability of the presence of added errors with
magnitudes that are approximately as large as the true extreme events
themselves. We study the cases where $\beta=0.33,0.2,0.125$, and $0.083$ at
$\tau=11$. We can see from Table 3 that, starting from the baseline scenario
with $\beta=0.33$, the increasing rate parameter significantly affects the
Extreme Lasso in terms of accuracy compared to the other methods. This is not
surprising, since we would expect the Extreme Lasso to be more sensitive to
large errors that are not actually true signal. However, even in the scenario
with the largest error variance, the Extreme Lasso still outperform all of the
others. Thus, even in the presence of potentially large residuals, the Extreme
Lasso is still a preferable method compared to the others.
Table 3: Average F-1 scores, changing error distribution for the linear model. | $\beta$ = 0.33 | $\beta$ = 0.2 | $\beta$ = 0.125 | $\beta$ = 0.083
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.875 (0.05) | 0.8 (0.1155) | 0.625 (0.1258) | 0.275 (0.2217)
ExLasso ($\gamma=6$) | 0.85 (0.0577) | 0.75 (0.1732) | 0.682 (0.1284) | 0.425 (0.15)
Lasso | 0.3 (0) | 0.2 (0.1414) | 0.262 (0.1103) | 0.175 (0.15)
Median | 0.529 (0.0626) | 0.338 (0.1134) | 0.46 (0.1078) | 0.403 (0.1414)
Q-0.9 | 0.185 (0.1239) | 0.122 (0.0465) | 0.121 (0.041) | 0.097 (0.0071)
Q-0.99 | 0.102 (0.0355) | 0.092 (0.0055) | 0.092 (0.0096) | 0.097 (0.0096)
Q-0.999 | 0.328 (0.2583) | 0.202 (0.1506) | 0.093 (0.0087) | 0.093 (0.0105)
Threshold | 0 (0) | 0.05 (0.1) | 0.073 (0.0994) | 0 (0)
##### Scenario 4: Number of Dimensions
We change the number of dimensions of the model matrix to study the
performance of the different methods in relatively higher dimensional
settings. We let $P=750,1500,2250,$ and $3000$, while we hold the number of
true features constant (thus decreasing the sparsity level as we increase
$P$). Table 4 shows the results. All of the approaches do tend to decay in
accuracy. In particular, the least squares and the Extreme Lasso methods tend
to show a relatively larger decline in performance, while quantile regression
at large quantiles and thresholding appear to be more stable. Once again
though, even when performance decays in the higher dimensional settings, the
F-1 scores for the Extreme Lasso still exceeds any of the others.
Table 4: Average F-1 scores, changing number of dimensions of predictor matrix in the linear model. | P = 750 | P = 1500 | P = 2250 | P = 3000
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.875 (0.05) | 0.75 (0.0577) | 0.827 (0.0848) | 0.627 (0.2906)
ExLasso ($\gamma=6$) | 0.85 (0.0577) | 0.65 (0.1) | 0.642 (0.1962) | 0.55 (0.1)
Lasso | 0.3 (0) | 0 (0) | 0 (0) | 0 (0)
Median | 0.529 (0.0626) | 0.249 (0.0671) | 0.247 (0.1579) | 0.103 (0.0269)
Q-0.9 | 0.185 (0.1239) | 0.117 (0.0553) | 0.102 (0.0119) | 0.103 (0.0269)
Q-0.99 | 0.102 (0.0355) | 0.1 (0.0041) | 0.089 (0.0051) | 0.095 (0.0037)
Q-0.999 | 0.328 (0.2583) | 0.117 (0.0495) | 0.279 (0.2265) | 0.089 (0.0051)
Threshold | 0 (0) | 0 (0) | 0 (0) | 0 (0)
### 4.2 Mixture Model Simulation Study
Next, we study a case where the extreme value linear regression model is
misspecified with respect to the data generating model, but where the data
still have extreme values. We generate data from a mixture model of the form:
$\mathbf{y}_{i}=\sum_{k=1}^{K}\mathds{1}_{ik}\mathbf{X}_{i}\beta_{k}+\epsilon_{i}$
$\boldsymbol{\epsilon}\overset{iid}{\sim}Gamma(\alpha,\beta).$
We simulate 4 different sets of predictor variables. The first set contains
features which are generated from a mean 0 Gaussian distribution with added
extreme values at several randomly selected observation points. The second set
contains variables simulated from a Gaussian distribution with no extreme
values but which has a mean shift of $2\sigma^{2}$ for half of the
observations. The third set contains variables which exhibit cross-correlation
($\rho=0.9$) to one of the variables in the first feature set, but with
different extreme value observation points. The fourth set contains
uncorrelated white noise variables. We then create a response variable using
the above mixture model with $K=2$ mixture components, where the first
component is comprised of the first set of the simulated predictor variables
with extreme values, and the second component is comprised of the second set
of the simulated predictor variables with mean shift. The first component
creates extreme values in the response variable because of their presence in
the first set of predictor variables, while the second component will be
correlated with the non-extreme values in the response because of the mean
shift of the corresponding predictors. Our goal is thus to recover as the
support set the variables associated with the first mixture component, i.e.
the ones which generate the extreme values in the response, without selecting
any variables from any of the others.
The predictor matrices $\mathbf{X}$ we simulate contain $n=1000$ observations
and $p=750$ columns; 10 features assigned to the each of the first 3 sets of
predictor variables as described above and the rest designated as part of the
last set. As in the linear regression simulation study, we analyze four
different varying simulation specifications:
1. 1.
The signal to noise ratio of the extreme events relative to baseline noise,
$\tau$.
2. 2.
The number of extreme events added to the variables in the first and third
components.
3. 3.
The distribution of the errors $\boldsymbol{\epsilon}$.
4. 4.
The number of dimensions $p$, holding the number of observations and parameter
sparsity level for each of the mixture components constant.
Again, we compare our method with regularized ordinary least squares
regression, $\ell_{1}$ quantile regression, and Lasso regression after
thresholding; we use 4 replications for each scenario, and we compare results
using average F-1 scores. Full results can be found in the Appendix.
##### Scenario 1: Magnitude of Extreme Values of Response Variable
We first vary the size of the signal to noise ratio between $\tau=6,7,9,$ and
$50.$ The results are shown in Table 5. The extreme value methods are able to
select the true features at a relatively smaller level of $\tau$. The least
squares and thresholding methods are unable to select the features associated
with the extreme values until $\tau$ is astronomically large. Meanwhile, the
quantile regression methods appear to do better than many of the other methods
when $\tau$ is relatively small, but the performance does not improve much
with larger values of $\tau$.
Table 5: Average F-1 scores, changing relative extreme value magnitudes for the mixture model. | $\tau$ = 6 | $\tau$ = 7 | $\tau$ = 9 | $\tau$ = 50
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.128 (0.0986) | 0.175 (0.1708) | 0.9 (0.0816) | 1 (0)
ExLasso ($\gamma=6$) | 0.259 (0.3143) | 0.757 (0.0963) | 1 (0) | 1 (0)
Lasso | 0 (0) | 0.075 (0.15) | 0 (0) | 1 (0)
Median | 0.094 (0.0022) | 0.185 (0.1797) | 0.095 (0) | 0.095 (0)
Q0.9 | 0.094 (0.0022) | 0.14 (0.0887) | 0.095 (0) | 0.095 (0)
Q0.99 | 0.179 (0.0599) | 0.098 (0.0193) | 0.312 (0.1434) | 0.739 (0.1504)
Q0.999 | 0.348 (0.0986) | 0.369 (0.1994) | 0.394 (0.1643) | 0.474 (0.1721)
Threshold | 0 (0) | 0.123 (0.1798) | 0.384 (0.392) | 0.977 (0.0455)
##### Scenario 2: Number of Extreme Events in Response
Here, we change the number of extreme value events $E$ from 1 to 4 for
$\tau=6$. Results are shown in Table 6. As we increase the number of extreme
value events, the performance of the extreme value methods steadily increases.
Thresholding and quantile regression also tend to perform slightly better with
more extreme events, although the improvement is not as drastic. The least
squares regression methods never are able to pick any of the features
associated with the extreme events.
Table 6: Average F-1 scores, changing number of extreme events for the mixture model. | E = 1 | E = 2 | E = 3 | E = 4
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.128 (0.0986) | 0.313 (0.1514) | 0.632 (0.1489) | 0.836 (0.0473)
ExLasso ($\gamma=6$) | 0.259 (0.3143) | 0.52 (0.0869) | 0.795 (0.1527) | 0.908 (0.0789)
Lasso | 0 (0) | 0 (0) | 0 (0) | 0 (0)
Median | 0.094 (0.0022) | 0.094 (0.0022) | 0.095 (0) | 0.095 (0)
Q0.9 | 0.094 (0.0022) | 0.094 (0.0022) | 0.095 (0) | 0.095 (0)
Q0.99 | 0.179 (0.0599) | 0.421 (0.1032) | 0.604 (0.093) | 0.65 (0.1238)
Q0.999 | 0.348 (0.0986) | 0.358 (0.0519) | 0.45 (0.1935) | 0.474 (0.1154)
Threshold | 0 (0) | 0.229 (0.1455) | 0.596 (0.1489) | 0.758 (0.1173)
##### Scenario 3: Error Distributions
In this scenario, we vary the distribution of the added errors by changing the
rate parameter to $\beta=0.33,0.2,0.166$, and $0.125$ at $\tau=9$. Table 7
displays the results. Once again, an increase in the rate parameter
significantly degrades the performance the extreme value methods because of
the increased presence of large magnitude errors, while other methods are not
affected nearly as much. We do eventually see a point where the extreme value
methods perform worse than quantile or thresholding.
Table 7: Average F-1 scores, changing error distribution for the mixture model. | $\beta$ = 0.33 | $\beta$ = 0.2 | $\beta$ = 0.166 | $\beta$ = 0.125
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.9 (0.0816) | 0.875 (0.1258) | 0.816 (0.0526) | 0.278 (0.3587)
ExLasso ($\gamma=6$) | 1 (0) | 1 (0) | 0.582 (0.2852) | 0.184 (0.217)
Lasso | 0 (0) | 0 (0) | 0 (0) | 0 (0)
Median | 0.095 (0) | 0.095 (0) | 0.095 (0) | 0.094 (0.0022)
Q-0.9 | 0.095 (0) | 0.095 (0) | 0.095 (0) | 0.094 (0.0022)
Q-0.99 | 0.312 (0.1434) | 0.14 (0.0494) | 0.249 (0.0897) | 0.24 (0.0465)
Q-0.999 | 0.394 (0.1643) | 0.299 (0.0897) | 0.19 (0.0186) | 0.115 (0.0505)
Threshold | 0.384 (0.392) | 0.05 (0.1) | 0.64 (0.0773) | 0.508 (0.2058)
##### Scenario 4: Number of Dimensions
We change the number of dimensions of the model matrix to $P=750,1500,2250,$
and $3000$, with $\tau=9$ and holding the number of features in components 1,
2, and 3 constant. Results are in Table 8. The performance of the extreme
value and least squares methods do not change much with the increased
dimensionality. The quantile regression methods actually perform slightly
better with more dimensions, while thresholding tends to do worse.
Table 8: Average F-1 scores, changing number of dimensions of predictor matrix in the mixture model. | P = 750 | P = 1500 | P = 2250 | P = 3000
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.9 (0.0816) | 0.816 (0.0526) | 0.922 (0.0673) | 0.838 (0.0062)
ExLasso ($\gamma=6$) | 1 (0) | 0.947 (0) | 0.961 (0.0263) | 0.947 (0)
Lasso | 0 (0) | 0 (0) | 0 (0) | 0 (0)
Median | 0.095 (0) | 0.095 (0) | 0.095 (0) | 0.095 (0)
Q-0.9 | 0.095 (0) | 0.095 (0) | 0.095 (0) | 0.095 (0)
Q-0.99 | 0.312 (0.1434) | 0.238 (0.0794) | 0.189 (0.0744) | 0.093 (0.0262)
Q-0.999 | 0.394 (0.1643) | 0.39 (0.0962) | 0.501 (0.0663) | 0.577 (0.2264)
Threshold | 0.384 (0.392) | 0.32 (0.1879) | 0.316 (0.087) | 0 (0)
### 4.3 Real Data Investigation: Calcium Imaging
We now study the performance of regularized extreme value linear regression on
a calcium imaging study from neuroscience, available from the Allen Brain
Atlas Brain Observatory data repository [20]. The data set contains
fluorescence traces of neuronal activity for 227 simultaneously recorded
neurons in the visual cortex of a mouse brain during periods of controlled
visual stimuli. For this study, we work with the parts of the study associated
with drifting grating movies, i.e. during time periods which the mice are
shown moving black and white gratings of various changing angles and
frequencies. Our objective is to predict the recorded fluorescence traces of
each of the neurons, with a specific focus on the large positive extreme
values that represent neuron firing activity. The predictor variables for the
data set are the visual stimulus information from the drifting grating movie,
namely the angular orientation and frequency of the drifting gratings being
shown to the mouse, as well as other recorded data about the activity of the
mouse including treadmill running speed and pupil size and location. We fit
the Lasso and 8th power Extreme Lasso regressions to each neuron independently
and compare both the chosen stimulus features and the predicted neuron
activity traces from each method. Hyperparameter selection for both methods
are performed via 5-fold cross validation.
Figure 2: Top: True (black) vs. predicted (blue) neuron trace from the Lasso.
Times of chosen angular orientation stimuli are highlighted in color. Bottom:
True (black) vs. predicted (red) neuron trace from the Extreme Lasso. Times of
chosen angular orientation stimuli are highlighted in color.
In Figure 2, we look at the results from one particular neuron. In the top of
Figure 2, we see that the prediction from the Lasso does not include any
spikes; instead, we see that the Lasso is fitting to the random baseline
fluctuations in the fluorescence trace, which are likely to be random noise or
measurement artifacts and are not especially useful for this type of data. We
also see that the times of the angular orientation stimulus feature selected
by the Lasso do not correspond with any spiking activity of the neuron,
meaning that the stimuli selected by the Lasso are not particularly
scientifically meaningful in this specific context. On the other hand, in the
bottom of Figure 2, we see that the Extreme Lasso appears to select an angular
orientation stimulus feature for which spikes in the fluorescence trace occur
during the time periods where the angle is shown in the drifting grating
movie. The predictions from the Extreme Lasso reflect the importance of the
extreme values for the estimation procedure as well, with the estimated value
of the fluorescence trace being much more sensitive to the spikes in the
observed data relative to the Lasso.
### 4.4 Real Data Investigation: Climatology
Our second real data example comes from the field of climatology. The data
used here are available from the US EPA AQS Data Mart [1] and from the MERRA-2
project on NASA MDISC [11]. Our goal is to predict and find features
associated with large spikes in the hourly measurements of total volatile
organic compound (TVOC) concentration in parts per billion (ppb) for a single
outdoor monitoring site in Deer Park, Texas. We use as predictor variables for
modeling the contemporaneous average hourly data of various atmospheric
weather conditions, including temperature, humidity, air pressure, ozone
level, wind speed, water vapor concentration, and dew point. From the raw
weather data, we also create new predictors using 1 day moving averages of all
of the aforementioned variables at time lags ranging from concurrent to 7
days. The data set we look at below contains hourly observations from January
1st, 2015 to December 31st, 2017, totaling approximately 52500 total recorded
measurements. We split this in to a training data set which spans the first
two years of our data, and a test data set which spans the final year. For
this case study, we compare the results from Lasso regression and the 10th
power Extreme Lasso regression models. We first perform feature selection with
these two methods using the training data set; for this step, hyperparameter
values are selected using cross-validation. We then fit the corresponding
unbiased regression models to the test data set. Below, we discuss the
features which were selected using the regularized regression methods on the
training data set, and the model predictions and residuals from the unbiased
models on the test data set.
Lasso | Extreme Lasso
---|---
Concurrent hourly air humidity | Concurrent hourly air pressure
Concurrent hourly vapor volume | 1 day average precipitation, 5 day lag
Concurrent hourly dew point | 1 day average precipitation, 6 day lag
Concurrent hourly wind speed | 1 day average precipitation, 7 day lag
1 day average temperature, 0 day lag | 1 day average wind speed, 6 day lag
1 day average humidity, 0 day lag |
1 day average vapor, 0 day lag |
Table 9: Selected predictors from each regularized regression model.
In Table 9, we show the predictors selected by each of the two methods. As we
can see, the Lasso tends to select predictors which are associated with
concurrent and current 1-day average atmospheric weather conditions, such as
concurrent air humidity and wind speed and 1-day average temperature and water
vapor content. On the other hand, the Extreme Lasso mainly selects features
associated with daily average weather conditions from 5-7 days prior,
particularly with respect to precipitation. Thus, we see that the two methods
pick very different sets of predictors. Scientifically, it seems that the
Lasso is finding predictors that tend to be associated with smaller common
fluctuations in TVOCs, while the Extreme Lasso selects predictors that
indicate occurrences of large rainfall events which have been linked in
previous literature to large spikes in pollutant concentrations [2].
Figure 3: Top left: True (black) vs. predicted (blue) hourly TVOC
concentration from the ordinary linear regression model. Top right: True
(black) vs. predicted (red) hourly TVOC concentration. Bottom Left: True vs.
predicted hourly TVOC concentrations from the ordinary linear regression.
Bottom Right: True vs. predicted hourly TVOC concentrations from the extreme
value linear regression.
In Figure 3, we show the predicted TVOC concentrations from the extreme value
linear regression and ordinary linear regression models with their previous
respective selected features on the test data set; in the top row, we see
these plotted over time, and in the bottom we see the predicted and actual
values plotted against each other. As we can see, the linear regression model
appears to be solely capturing the minor fluctuations which occur regularly
across time, but does not seem to capture any of the large spikes in TVOC
concentration which occur several times over the course of a year. On the
other hand, the extreme value linear regression model, while not always
accurate with respect to the smaller value of TVOC, appears to do a much
better job in predicting the instances of extreme events where TVOC
concentrations spike to irregularly high levels. While neither model is
particularly accurate with respect to predicting all of the observed TVOC
concentration values, the extreme value linear regression model actually does
predict occurrences of extreme value events, whereas the the ordinary linear
regression grossly underestimates the TVOC concentration levels when they are
above a few hundred parts per billion.
Figure 4: Average absolute value of model residuals from the extreme value
linear regression and the ordinary linear regression for values above a
concentration threshold.
We analyze more closely the residuals of the extreme of the regression model
estimate fit by both methods in Figure 4. Here, we show the average magnitude
of the regression residuals from the ordinary least squares regression and
extreme value linear regression models for observations with TVOC
concentrations above a changing threshold value. From Figure 4, we see that
the ordinary linear regression predictions are closer to the actual values on
average when we consider the entire data set, i.e. when the threshold is 0.
However, as we start looking only at data points towards the upper quantiles,
we see that the extreme value linear regression begins to outperform the
ordinary linear regression in terms of prediction accuracy. For this
particular example, the extreme value linear regression model becomes more
accurate on average than the ordinary linear regression for observations with
TVOC concentration values above 262 ppm, and the difference between the
prediction accuracy of the two models gradually increases as we consider
smaller, more extreme subsets of the observations of the response variable.
## 5 Discussion
In this paper, we have introduced the extreme value linear regression model, a
potential new methodological approach to linear regression for extreme values.
Our method is motivated by $\ell_{\gamma}$-norm regression, which gives much
more weight to the loss for large magnitude residuals relative to ordinary
least squares. This concept has several advantages over other methods
currently used in the literature, namely that it does not require using a two-
step pipeline of pre-processing the data before analysis, nor does it force
the data to be binarized as either extreme or non-extreme. Our method also
does not necessitate the a priori choice of certain model hyperparameters that
may be difficult to select. Our simulation studies provide promising results
which demonstrate that, for a response variable with rare extreme values, the
extreme value linear regression model with automatic feature selection
performs better than quantile regression, thresholding, and least squares
penalized regression in terms of selecting predictors which are correlated
with the extreme values in the response. We have also shown deterministic
finite sample performance guarantees for consistency and model selection
consistency of the Extreme Lasso regression model under the assumption of a
linear data generating model with different potential error distributions,
demonstrating that the estimates from the extreme value linear regression
model are reliable. The theoretical results here could also be of use for
other types of similar problems. In particular, the concentration bounds and
theory presented for the case of generalized normal distributed errors for
$\gamma>2$ could be applied to generate new theoretical results for other
mathematical statistics problems.
There are several potential areas for future work for the extreme value linear
regression. Our theoretical work has mainly focused on using a simple
$\ell_{1}$ Lasso penalty for regularization under the linear regression data
generating model; however, the extreme values in a response variable could
come from a variety of different data generating models. Theoretical results
for the variance of estimators in the low-dimensional case have also not been
addressed here, and could be of interest for future study. There remains
potential methodological developments for the extreme value linear regression
to explore as well. Just as ordinary regression methods are insufficient for
fitting a model for the extreme values, traditional model selection methods
may not work particularly well in this context. While we use regular cross-
validation to select $\lambda$ during model fitting in our real data examples,
we recognize that this may not be the optimal method. The model selection
problem may instead require more nuanced treatment, as naive cross-validation
methods may not work well when the extreme values are particularly rare.
Though we have described a general approach for selecting $\gamma$ for an
individual data set, additional empirical investigation may be useful for
gaining a better understanding of what values of $\gamma$ are typically useful
for analyzing real-world data. Also, while we have presented a couple of
potential applications, our method has the potential to be applied broadly to
a variety of fields, such as for signal processing or for spectral domain
data; explorations in to other applications could provide new insights in
these areas. In conclusion, we develop a novel method for extreme value
regression modeling that opens many area for future research.
## Appendix A Proofs for Section 3
### A.1 Lemma 3.3
For $t>0$,
$\displaystyle\mathbb{P}(\mathbf{Q}^{\gamma}\geq t)=\mathbb{P}(\mathbf{Q}\geq
t^{{}^{1/\gamma}})=\mathbb{P}(e^{\lambda\mathbf{Q}}\geq e^{\lambda
t^{{}^{1/\gamma}}})\leq\frac{e^{\sigma^{2}\lambda^{2}/2}}{e^{\lambda
t^{{}^{1/\gamma}}}}=\exp\bigg{\\{}\sigma^{2}\lambda^{2}/2-\lambda
t^{{}^{1/\gamma}}\bigg{\\}}.$
The right hand side is minimized by
$\lambda^{*}=\frac{t^{1/\gamma}}{\sigma^{2}}$. Hence, we have
$\displaystyle\mathbb{P}(\mathbf{Q}^{\gamma}\geq
t)\leq\exp\bigg{\\{}-\frac{t^{2/\gamma}}{2\sigma^{2}}\bigg{\\}}.$
$\square$
### A.2 Theorem 3.1
In the Extreme Lasso problem, by Lemma 3.3,
$\|\epsilon_{i}^{\gamma-1}\|_{\psi_{\gamma}}\leq K_{n,p}$ where
$\gamma=\frac{2}{\gamma-1}$. For fixed design $\mathbf{X}$, $\mathbf{X}_{i}$’s
are marginally sub-Weibull $(\infty)$ and
$\max_{1\leq i\leq n}\left\|X_{i}\right\|_{M,\psi_{2}}\leq\max_{1\leq i\leq
n}\left\|X_{i}\right\|_{M,\psi_{\infty}}=\max_{1\leq i\leq n}\max_{1\leq j\leq
p}\left|X_{i}(j)\right|.$
Applying Lemma 3.4 with $\alpha=\infty$, we have $\tau=2/(\gamma-1)$.
Therefore, by choosing $\lambda_{n}$ to be
$\lambda_{n}=2\gamma\big{(}7\sqrt{2}\sigma_{n,p}\sqrt{\frac{\log(np)}{n}}+\frac{C_{\tau}K_{n,p}^{2}(\log(2n))^{1/\tau}(2\log(np))^{1/\tau}}{n}\bigg{)},$
the Extreme Lasso estimator satisfies
$\displaystyle\|\hat{\operatorname{\boldsymbol{\beta}}}-\operatorname{\boldsymbol{\beta}}^{*}\|_{2}$
$\displaystyle\leq\frac{6\sqrt{s}}{\kappa_{\mathcal{L}}}\cdot\gamma\bigg{(}7\sqrt{2}\sigma_{n,p}\sqrt{\frac{\log(np)}{n}}+\frac{C_{\tau}K_{n,p}^{2}(\log(2n))^{1/\tau}(2\log(np))^{1/\tau}}{n}\bigg{)}$
where $\tau=2/(\gamma-1).$
$\square$
### A.3 Theorem 3.2
Similar to Theorem 3.1, we prove model selection Consistency holds by applying
Lemma 3.2 with the concentration bound demonstrated in Lemma 3.4.
$\square$
### A.4 Lemma 3.5
Suppose $\mathbf{Z}\sim$ Subbotin($\alpha$), i.e.
$\displaystyle
f_{\mathbf{Z}}(z)=\frac{\alpha}{2\Gamma(\frac{1}{\alpha})}\exp\big{[}-|z|^{\alpha}\big{]}.$
Let $\mathbf{Y}=\mathbf{Z}^{\alpha}$, then $z=\pm y^{\frac{1}{\alpha}}$,
$|\frac{dz}{dy}|=\frac{1}{\alpha}y^{\frac{1}{\alpha}-1}$ and
$\displaystyle
f_{\mathbf{Y}}(y)=\frac{\alpha}{\Gamma(\frac{1}{\alpha})}\exp[-y]\frac{1}{\alpha}y^{\frac{1}{\alpha}-1}=\frac{1}{\Gamma(\frac{1}{\alpha})}\exp[-y]y^{\frac{1}{\alpha}-1}.$
Thus, $\mathbf{Y}=\mathbf{Z}^{\alpha}\sim$ Gamma$(\frac{1}{\alpha},1)$.
$\square$
### A.5 Lemma 3.7
By Lemma 3.5, we have $\epsilon_{i}^{\theta}\sim$ Gamma$(1/\theta,1)$. Hence,
Lemma 3.6 suggests
$\displaystyle\mathbb{P}\big{(}\epsilon_{i}^{\theta}-\mathbb{E}[\epsilon_{i}^{\theta}]\geq
2\sqrt{2\frac{1}{\theta}t}+t\big{)}\leq e^{-t}.$
If $\gamma-1\leq\theta$, we can show that $\epsilon_{i}^{\gamma-1}$ is also
sub-Gamma with $(\frac{1}{\theta},1)$ as the latter one has lower tail.
$\displaystyle\mathbb{P}\big{(}\epsilon_{i}^{\gamma-1}-\frac{1}{\theta}\geq
2\sqrt{2\frac{1}{\theta}t}+t\big{)}\leq\mathbb{P}\big{(}\epsilon_{i}^{\theta}-\frac{1}{\theta}\geq
2\sqrt{2\frac{1}{\theta}t}+t\big{)}\leq e^{-t}.$
For $\|\mathbf{X}_{j}\|_{\infty}\leq 1$, we have
$\mathbf{X}_{j}^{T}\epsilon^{\gamma-1}$ is sub-Gamma with $(n/\theta,1)$ since
sum of sub-Gamma is also sub-Gamma. From this, we find
$\displaystyle\mathbb{P}\big{(}\mathbf{X}_{j}^{T}\epsilon^{\gamma-1}-\mathbb{E}[\mathbf{X}_{j}^{T}\epsilon^{\gamma-1}]\geq
2\sqrt{2\frac{n}{\theta}t}+t\big{)}\leq e^{-t}.$
By using union bounds, we thus have
$\displaystyle\mathbb{P}\big{(}\|\mathbf{X}^{T}\epsilon^{\gamma-1}\|_{\infty}\geq
2\sqrt{2\frac{n}{\theta}t}+t\big{)}\leq pe^{-t}.$
Choosing $t=\log p$, we get
$\displaystyle\|\mathbf{X}^{T}\epsilon^{\gamma-1}\|_{\infty}\leq
2\sqrt{\frac{2}{\theta}}\sqrt{n\log p}+\log p$
with probability at least $1-c_{1}\exp(-c_{2}\log p)$. This is equivalent to:
$\displaystyle\|\mathbf{X}^{T}\epsilon^{\gamma-1}\|_{\infty}/n$
$\displaystyle\leq 2\sqrt{\frac{2}{\theta}}\sqrt{\frac{\log p}{n}}+\frac{\log
p}{n}$ $\displaystyle=\sqrt{\frac{\log
p}{n}}\bigg{[}2\sqrt{\frac{2}{\theta}}+\sqrt{\frac{\log p}{n}}\bigg{]}$
$\displaystyle\leq\sqrt{\frac{\log
p}{n}}\bigg{[}2\sqrt{\frac{2}{\gamma}}+\sqrt{\frac{\log p}{n}}\bigg{]}$
with probability at least $1-c_{1}\exp(-c_{2}\log p)$.
$\square$
### A.6 Theorem 3.3
By applying Lemma 3.1 with the concentration bound demonstrated in Lemma 3.7,
we have the consistency result.
$\square$
### A.7 Theorem 3.4
By applying Lemma 3.2 with the concentration bound demonstrated in Lemma 3.7,
we have the model consistency result.
$\square$
### A.8 Theorem 3.5
By the KKT condition as required for optimality of
$\boldsymbol{\beta}_{\epsilon}^{*}$, we have:
$\displaystyle(1-\epsilon)\int\left[-\gamma\left(y-\mathbf{x}^{T}\boldsymbol{\beta}_{\epsilon}^{*}\right)^{\gamma-1}\mathbf{x}\right]\times\mathrm{d}F(\mathbf{x},y)$
$\displaystyle+\epsilon\left(-\gamma\left(y_{0}-\mathbf{x}_{0}^{T}\boldsymbol{\beta}_{\epsilon}^{*}\right)^{\gamma-1}\times\mathbf{x}_{0}\right)-v_{1}(\epsilon)=0,$
(A.1)
where
$v_{1}(\epsilon)=\left(p_{\lambda_{1}}^{\prime}\left(\left|\beta_{\epsilon
1}\right|\right)\operatorname{sign}\left(\beta_{\epsilon
1}\right),\ldots,p_{\lambda_{d}}^{\prime}\left(\left|\beta_{\epsilon
d}\right|\right)\operatorname{sign}\left(\beta_{\epsilon
d}\right)\right)^{T}$. Let $r_{0}=y_{0}-\mathbf{x}_{0}^{T}\beta_{0}^{*}$.
Differentiating with respect to $\epsilon$ in both sides of (A.8) and letting
$\epsilon\rightarrow 0$, we obtain
$\displaystyle\int\left[-\gamma(\gamma-1)\left(y-\mathbf{x}^{T}\boldsymbol{\beta}_{\epsilon}^{*}\right)^{\gamma-2}\times\frac{\partial}{\partial\epsilon}\left(y-\mathbf{x}^{T}\boldsymbol{\beta}_{\epsilon}^{*}\right)\mathbf{x}\right]$
$\displaystyle\left.\mathrm{d}F(\mathbf{x},y)\right|_{\epsilon=0}-\frac{\partial
v_{1}(\epsilon)}{\partial\epsilon}$ $\displaystyle=\gamma
r_{0}^{\gamma-1}x_{0}-v_{2},$ (A.2)
where
$v_{2}=\left(p_{\lambda_{1}}^{\prime}\left(\left|\beta_{01}^{*}\right|\right)\operatorname{sign}\left(\beta_{01}^{*}\right),\ldots,p_{\lambda_{d}}^{\prime}\left(\left|\beta_{0d}^{*}\right|\right)\operatorname{sign}\left(\beta_{0d}^{*}\right)\right)^{T}$.
Using (A.8) and (A.8), it can be shown that
$\left(A(\gamma)-B_{1}\right)\left[\operatorname{IF}\left\\{\left(\mathbf{x}_{0},y_{0}\right),\boldsymbol{\beta}_{0}^{*}\right\\}\right]=\gamma
r_{0}^{\gamma-1}x_{0}-v_{2},$
where
$A(\gamma)=\int\mathbf{x}\mathbf{x}^{T}\gamma(\gamma-1)\left(y-\mathbf{x}^{T}\boldsymbol{\beta}_{0}^{*}\right)^{\gamma-2}\times\mathrm{d}F(\mathbf{x},y),$
$\displaystyle B_{1}=$
$\displaystyle\operatorname{diag}\left\\{p_{\lambda_{1}}^{\prime\prime}\left(\left|\beta_{01}^{*}\right|\right)+p_{\lambda_{1}}^{\prime}\left(\left|\beta_{01}^{*}\right|\right)\delta\left(\beta_{01}^{*}\right),\ldots,p_{\lambda_{d}}^{\prime\prime}\left(\left|\beta_{0d}^{*}\right|\right)+p_{\lambda_{d}}^{\prime}\left(\left|\beta_{0d}^{*}\right|\right)\delta\left(\beta_{0d}^{*}\right)\right\\},$
with
$\delta(x)=\begin{cases}+\infty,&\text{ if }x=0,\\\ 0,&\text{ otherwise.
}\end{cases}$
$\square$
## Appendix B Full Tabular Results
For our full results, we also compare different regularization penalties for
the extreme value linear regression model and the linear regression model.
### B.1 Linear Model Simulation Study
Scenario 1: Changing Magnitude of Extreme Values of Response Variable
Table 10: Average F-1 score for changing extreme value magnitude. | $\tau$ = 6 | $\tau$ = 7 | $\tau$ = 11 | $\tau$ = 15
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.196 (0.1382) | 0.209 (0.1778) | 0.875 (0.05) | 0.938 (0.0481)
ExLasso ($\gamma=6$) | 0.296 (0.1416) | 0.782 (0.0894) | 0.85 (0.0577) | 0.938 (0.0481)
ExSCAD 4th | 0.196 (0.1382) | 0.209 (0.1778) | 0.875 (0.05) | 0.938 (0.0481)
ExSCAD 6th | 0.296 (0.1416) | 0.782 (0.0894) | 0.85 (0.0577) | 0.938 (0.0481)
ExMCP 4th | 0.1 (0.1155) | 0.195 (0.1556) | 0.888 (0.0637) | 0.938 (0.0481)
ExMCP 6th | 0.255 (0.0662) | 0.757 (0.1606) | 0.864 (0.0474) | 0.938 (0.0481)
Lasso | 0.2 (0.1414) | 0.225 (0.15) | 0.3 (0) | 0.938 (0.0481)
SCAD | 0.2 (0.1414) | 0.225 (0.15) | 0.3 (0) | 0.938 (0.0481)
MCP | 0.2 (0.1414) | 0.225 (0.15) | 0.3 (0) | 0.963 (0.0477)
Median | 0.149 (0.0357) | 0.301 (0.2087) | 0.529 (0.0626) | 0.44 (0.1056)
Q0.9 | 0.149 (0.0357) | 0.127 (0.0429) | 0.185 (0.1239) | 0.147 (0.0508)
Q0.99 | 0.095 (0.0394) | 0.09 (0.0194) | 0.102 (0.0355) | 0.111 (0)
Q0.999 | 0.132 (0.1028) | 0.219 (0.2222) | 0.328 (0.2583) | 0.321 (0.1821)
Threshold | 0.028 (0.0556) | 0 (0) | 0 (0) | 0.893 (0.1056)
Table 11: Average true positive rates for changing extreme value magnitude. | $\tau$ = 6 | $\tau$ = 7 | $\tau$ = 11 | $\tau$ = 15
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.193 (0.1355) | 0.196 (0.1571) | 0.875 (0.05) | 0.927 (0.0487)
ExLasso ($\gamma=6$) | 0.293 (0.1421) | 0.767 (0.1054) | 0.85 (0.0577) | 0.927 (0.0487)
ExSCAD 4th | 0.193 (0.1355) | 0.196 (0.1571) | 0.875 (0.05) | 0.927 (0.0487)
ExSCAD 6th | 0.293 (0.1421) | 0.767 (0.1054) | 0.85 (0.0577) | 0.927 (0.0487)
ExMCP 4th | 0.1 (0.1155) | 0.191 (0.1488) | 0.877 (0.0517) | 0.927 (0.0487)
ExMCP 6th | 0.262 (0.0828) | 0.764 (0.1467) | 0.855 (0.053) | 0.927 (0.0487)
Lasso | 0.2 (0.1414) | 0.225 (0.15) | 0.3 (0) | 0.927 (0.0487)
SCAD | 0.2 (0.1414) | 0.225 (0.15) | 0.3 (0) | 0.927 (0.0487)
MCP | 0.2 (0.1414) | 0.225 (0.15) | 0.3 (0) | 0.952 (0.0552)
Median | 0.398 (0.2045) | 0.318 (0.16) | 0.446 (0.041) | 0.435 (0.0842)
Q0.9 | 0.398 (0.2045) | 0.164 (0.1179) | 0.158 (0.1061) | 0.145 (0.0449)
Q0.99 | 0.128 (0.1367) | 0.086 (0.0384) | 0.09 (0.0362) | 0.125 (0)
Q0.999 | 0.175 (0.2165) | 0.196 (0.2072) | 0.303 (0.2673) | 0.352 (0.2432)
Threshold | 0.031 (0.0625) | 0 (0) | 0 (0) | 0.864 (0.1174)
Table 12: Average false positive rates for changing extreme value magnitude. | $\tau$ = 6 | $\tau$ = 7 | $\tau$ = 11 | $\tau$ = 15
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.011 (0.0017) | 0.011 (0.0014) | 0.002 (7e-04) | 0.001 (7e-04)
ExLasso ($\gamma=6$) | 0.01 (0.002) | 0.003 (0.0017) | 0.002 (8e-04) | 0.001 (7e-04)
ExSCAD 4th | 0.011 (0.0017) | 0.011 (0.0014) | 0.002 (7e-04) | 0.001 (7e-04)
ExSCAD 6th | 0.01 (0.002) | 0.003 (0.0017) | 0.002 (8e-04) | 0.001 (7e-04)
ExMCP 4th | 0.011 (0.0014) | 0.01 (7e-04) | 0.002 (7e-04) | 0.001 (7e-04)
ExMCP 6th | 0.01 (0.002) | 0.003 (0.0017) | 0.002 (8e-04) | 0.001 (7e-04)
Lasso | 0.011 (0.0019) | 0.01 (0.002) | 0.009 (0) | 0.001 (7e-04)
SCAD | 0.011 (0.0019) | 0.01 (0.002) | 0.009 (0) | 0.001 (7e-04)
MCP | 0.01 (8e-04) | 0.01 (0.002) | 0.009 (0) | 0.001 (8e-04)
Median | 0.004 (0.0061) | 0.009 (0.0046) | 0.011 (0) | 0.008 (0.0013)
Q0.9 | 0.008 (0.0036) | 0.011 (0.0045) | 0.01 (0.003) | 0.008 (0.0016)
Q0.99 | 0.017 (0.0093) | 0.016 (0.0058) | 0.018 (0.0059) | 0.009 (0)
Q0.999 | 0.015 (0.0083) | 0.016 (0.0055) | 0.013 (0.0078) | 0.008 (0.0039)
Threshold | 0.009 (0.0019) | 0.011 (0.0017) | 0.01 (7e-04) | 0.002 (0.0017)
Scenario 2: Changing Number of Extreme Events in Response
Table 13: Average F-1 scores for changing number of extreme events. | E = 1 | E = 2 | E = 3 | E = 4
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.875 (0.05) | 0.225 (0.05) | 0.79 (0.0838) | 0.913 (0.0857)
ExLasso ($\gamma=6$) | 0.85 (0.0577) | 0.788 (0.2022) | 0.779 (0.1447) | 0.85 (0.1291)
ExSCAD 4th | 0.875 (0.05) | 0.225 (0.05) | 0.79 (0.0838) | 0.913 (0.0857)
ExSCAD 6th | 0.85 (0.0577) | 0.788 (0.2022) | 0.779 (0.1447) | 0.85 (0.1291)
ExMCP 4th | 0.888 (0.0637) | 0.25 (0.1) | 0.788 (0.1192) | 0.913 (0.0857)
ExMCP 6th | 0.864 (0.0474) | 0.813 (0.1555) | 0.779 (0.1447) | 0.89 (0.0978)
Lasso | 0.3 (0) | 0.36 (0.1925) | 0.339 (0.0773) | 0.325 (0.05)
SCAD | 0.3 (0) | 0.36 (0.1925) | 0.339 (0.0773) | 0.325 (0.05)
MCP | 0.3 (0) | 0.295 (0.0741) | 0.339 (0.0773) | 0.325 (0.05)
Median | 0.529 (0.0626) | 0.513 (0.059) | 0.472 (0.1155) | 0.457 (0.1337)
Q0.9 | 0.185 (0.1239) | 0.301 (0.0809) | 0.311 (0.157) | 0.414 (0.1092)
Q0.99 | 0.102 (0.0355) | 0.107 (0.0053) | 0.099 (0.0048) | 0.126 (0.0376)
Q0.999 | 0.328 (0.2583) | 0.232 (0.0992) | 0.334 (0.0793) | 0.445 (0.2531)
Threshold | 0 (0) | 0 (0) | 0 (0) | 0.075 (0.15)
Table 14: Average true positive rates for changing number of extreme events. | E = 1 | E = 2 | E = 3 | E = 4
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.875 (0.05) | 0.225 (0.05) | 0.782 (0.0894) | 0.902 (0.0818)
ExLasso ($\gamma=6$) | 0.85 (0.0577) | 0.777 (0.1913) | 0.759 (0.1297) | 0.85 (0.1291)
ExSCAD 4th | 0.875 (0.05) | 0.225 (0.05) | 0.782 (0.0894) | 0.902 (0.0818)
ExSCAD 6th | 0.85 (0.0577) | 0.777 (0.1913) | 0.759 (0.1297) | 0.85 (0.1291)
ExMCP 4th | 0.877 (0.0517) | 0.25 (0.1) | 0.777 (0.0997) | 0.902 (0.0818)
ExMCP 6th | 0.855 (0.053) | 0.802 (0.1436) | 0.759 (0.1297) | 0.882 (0.1133)
Lasso | 0.3 (0) | 0.333 (0.1414) | 0.329 (0.0583) | 0.325 (0.05)
SCAD | 0.3 (0) | 0.333 (0.1414) | 0.329 (0.0583) | 0.325 (0.05)
MCP | 0.3 (0) | 0.291 (0.0676) | 0.329 (0.0583) | 0.325 (0.05)
Median | 0.446 (0.041) | 0.484 (0.078) | 0.449 (0.0937) | 0.444 (0.1012)
Q0.9 | 0.158 (0.1061) | 0.283 (0.0754) | 0.299 (0.148) | 0.409 (0.1113)
Q0.99 | 0.09 (0.0362) | 0.115 (0.0121) | 0.098 (0.0096) | 0.129 (0.0276)
Q0.999 | 0.303 (0.2673) | 0.239 (0.1036) | 0.322 (0.1004) | 0.441 (0.2547)
Threshold | 0 (0) | 0 (0) | 0 (0) | 0.075 (0.15)
Table 15: Average false positive rates for changing number of extreme events. | E = 1 | E = 2 | E = 3 | E = 4
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.002 (7e-04) | 0.01 (7e-04) | 0.003 (0.0013) | 0.001 (0.0011)
ExLasso ($\gamma=6$) | 0.002 (8e-04) | 0.003 (0.0026) | 0.003 (0.0017) | 0.002 (0.0017)
ExSCAD 4th | 0.002 (7e-04) | 0.01 (7e-04) | 0.003 (0.0013) | 0.001 (0.0011)
ExSCAD 6th | 0.002 (8e-04) | 0.003 (0.0026) | 0.003 (0.0017) | 0.002 (0.0017)
ExMCP 4th | 0.002 (7e-04) | 0.01 (0.0014) | 0.003 (0.0013) | 0.001 (0.0011)
ExMCP 6th | 0.002 (8e-04) | 0.003 (0.0019) | 0.003 (0.0017) | 0.002 (0.0017)
Lasso | 0.009 (0) | 0.01 (7e-04) | 0.009 (0) | 0.009 (7e-04)
SCAD | 0.009 (0) | 0.01 (7e-04) | 0.009 (0) | 0.009 (7e-04)
MCP | 0.009 (0) | 0.01 (7e-04) | 0.009 (0) | 0.009 (7e-04)
Median | 0.011 (0) | 0.008 (0.0022) | 0.008 (0.0011) | 0.008 (0.0013)
Q0.9 | 0.01 (0.003) | 0.011 (0.0016) | 0.012 (0.0017) | 0.009 (0.0011)
Q0.99 | 0.018 (0.0059) | 0.01 (0.0013) | 0.012 (0.0013) | 0.011 (0.002)
Q0.999 | 0.013 (0.0078) | 0.01 (0.0017) | 0.01 (0.003) | 0.008 (0.0036)
Threshold | 0.01 (7e-04) | 0.01 (7e-04) | 0.009 (0) | 0.01 (7e-04)
Scenario 3: Changing Error Distribution
Table 16: Average F-1 scores for changing residual distribution. | $\beta$ = 0.33 | $\beta$ = 0.2 | $\beta$ = 0.125 | $\beta$ = 0.083
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.875 (0.05) | 0.8 (0.1155) | 0.625 (0.1258) | 0.275 (0.2217)
ExLasso ($\gamma=6$) | 0.85 (0.0577) | 0.75 (0.1732) | 0.682 (0.1284) | 0.425 (0.15)
ExSCAD 4th | 0.875 (0.05) | 0.8 (0.1155) | 0.625 (0.1258) | 0.275 (0.2217)
ExSCAD 6th | 0.85 (0.0577) | 0.75 (0.1732) | 0.682 (0.1284) | 0.425 (0.15)
ExMCP 4th | 0.888 (0.0637) | 0.825 (0.0957) | 0.625 (0.1258) | 0.275 (0.2217)
ExMCP 6th | 0.864 (0.0474) | 0.788 (0.166) | 0.575 (0.0957) | 0.425 (0.15)
Lasso | 0.3 (0) | 0.2 (0.1414) | 0.262 (0.1103) | 0.175 (0.15)
SCAD | 0.3 (0) | 0.2 (0.1414) | 0.262 (0.1103) | 0.175 (0.15)
MCP | 0.3 (0) | 0.2 (0.1414) | 0.27 (0.1197) | 0.15 (0.1732)
Median | 0.529 (0.0626) | 0.338 (0.1134) | 0.46 (0.1078) | 0.403 (0.1414)
Q-0.9 | 0.185 (0.1239) | 0.122 (0.0465) | 0.121 (0.041) | 0.097 (0.0071)
Q-0.99 | 0.102 (0.0355) | 0.092 (0.0055) | 0.092 (0.0096) | 0.097 (0.0096)
Q-0.999 | 0.328 (0.2583) | 0.202 (0.1506) | 0.093 (0.0087) | 0.093 (0.0105)
Threshold | 0 (0) | 0.05 (0.1) | 0.073 (0.0994) | 0 (0)
Table 17: Average true positive rates for changing residual distribution. | $\beta$ = 0.33 | $\beta$ = 0.2 | $\beta$ = 0.125 | $\beta$ = 0.083
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.875 (0.05) | 0.8 (0.1155) | 0.625 (0.1258) | 0.275 (0.2217)
ExLasso ($\gamma=6$) | 0.85 (0.0577) | 0.75 (0.1732) | 0.667 (0.1247) | 0.425 (0.15)
ExSCAD 4th | 0.875 (0.05) | 0.8 (0.1155) | 0.625 (0.1258) | 0.275 (0.2217)
ExSCAD 6th | 0.85 (0.0577) | 0.75 (0.1732) | 0.667 (0.1247) | 0.425 (0.15)
ExMCP 4th | 0.877 (0.0517) | 0.825 (0.0957) | 0.625 (0.1258) | 0.275 (0.2217)
ExMCP 6th | 0.855 (0.053) | 0.777 (0.1526) | 0.575 (0.0957) | 0.425 (0.15)
Lasso | 0.3 (0) | 0.2 (0.1414) | 0.252 (0.1013) | 0.175 (0.15)
SCAD | 0.3 (0) | 0.2 (0.1414) | 0.252 (0.1013) | 0.175 (0.15)
MCP | 0.3 (0) | 0.2 (0.1414) | 0.266 (0.1146) | 0.15 (0.1732)
Median | 0.446 (0.041) | 0.33 (0.0991) | 0.449 (0.0937) | 0.385 (0.1168)
Q-0.9 | 0.158 (0.1061) | 0.12 (0.0449) | 0.117 (0.034) | 0.094 (0.0136)
Q-0.99 | 0.09 (0.0362) | 0.086 (0.0099) | 0.086 (0.0176) | 0.096 (0.0199)
Q-0.999 | 0.303 (0.2673) | 0.207 (0.1615) | 0.087 (0.0163) | 0.088 (0.0184)
Threshold | 0 (0) | 0.05 (0.1) | 0.072 (0.1048) | 0 (0)
Table 18: Average false positive rates for changing residual distribution. | $\beta$ = 0.33 | $\beta$ = 0.2 | $\beta$ = 0.125 | $\beta$ = 0.083
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.002 (7e-04) | 0.003 (0.0016) | 0.005 (0.0017) | 0.01 (0.003)
ExLasso ($\gamma=6$) | 0.002 (8e-04) | 0.003 (0.0023) | 0.005 (0.0017) | 0.008 (0.002)
ExSCAD 4th | 0.002 (7e-04) | 0.003 (0.0016) | 0.005 (0.0017) | 0.01 (0.003)
ExSCAD 6th | 0.002 (8e-04) | 0.003 (0.0023) | 0.005 (0.0017) | 0.008 (0.002)
ExMCP 4th | 0.002 (7e-04) | 0.002 (0.0013) | 0.005 (0.0017) | 0.01 (0.003)
ExMCP 6th | 0.002 (8e-04) | 0.003 (0.002) | 0.006 (0.0013) | 0.008 (0.002)
Lasso | 0.009 (0) | 0.012 (0.0039) | 0.011 (0.0016) | 0.011 (0.002)
SCAD | 0.009 (0) | 0.012 (0.0039) | 0.011 (0.0016) | 0.011 (0.002)
MCP | 0.009 (0) | 0.011 (0.0019) | 0.01 (0.0014) | 0.011 (0.0023)
Median | 0.011 (0) | 0.009 (0.0019) | 0.008 (0.0013) | 0.009 (8e-04)
Q-0.9 | 0.01 (0.003) | 0.012 (7e-04) | 0.014 (0.0023) | 0.013 (0.0013)
Q-0.99 | 0.018 (0.0059) | 0.015 (0.0017) | 0.015 (0.0029) | 0.013 (0.0026)
Q-0.999 | 0.013 (0.0078) | 0.011 (0.0033) | 0.015 (0.0026) | 0.015 (0.0034)
Threshold | 0.01 (7e-04) | 0.014 (0.0048) | 0.016 (0.0045) | 0.014 (0.0013)
Scenario 4: Changing Number of Dimensions
Table 19: Average F-1 scores for changing number of dimensions. | P = 750 | P = 1500 | P = 2250 | P = 3000
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.875 (0.05) | 0.75 (0.0577) | 0.827 (0.0848) | 0.627 (0.2906)
ExLasso ($\gamma=6$) | 0.85 (0.0577) | 0.65 (0.1) | 0.642 (0.1962) | 0.55 (0.1)
ExSCAD 4th | 0.875 (0.05) | 0.75 (0.0577) | 0.827 (0.0848) | 0.627 (0.2906)
ExSCAD 6th | 0.85 (0.0577) | 0.65 (0.1) | 0.642 (0.1962) | 0.55 (0.1)
ExMCP 4th | 0.888 (0.0637) | 0.75 (0.0577) | 0.615 (0.2091) | 0.425 (0.2062)
ExMCP 6th | 0.864 (0.0474) | 0.664 (0.1218) | 0.521 (0.1279) | 0.6 (0)
Lasso | 0.3 (0) | 0 (0) | 0 (0) | 0 (0)
SCAD | 0.3 (0) | 0 (0) | 0 (0) | 0 (0)
MCP | 0.3 (0) | 0 (0) | 0 (0) | 0 (0)
Median | 0.529 (0.0626) | 0.249 (0.0671) | 0.247 (0.1579) | 0.103 (0.0269)
Q-0.9 | 0.185 (0.1239) | 0.117 (0.0553) | 0.102 (0.0119) | 0.103 (0.0269)
Q-0.99 | 0.102 (0.0355) | 0.1 (0.0041) | 0.089 (0.0051) | 0.095 (0.0037)
Q-0.999 | 0.328 (0.2583) | 0.117 (0.0495) | 0.279 (0.2265) | 0.089 (0.0051)
Threshold | 0 (0) | 0 (0) | 0 (0) | 0 (0)
Table 20: Average true positive rates for changing number of dimensions. | P = 750 | P = 1500 | P = 2250 | P = 3000
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.875 (0.05) | 0.75 (0.0577) | 0.752 (0.0346) | 0.542 (0.2378)
ExLasso ($\gamma=6$) | 0.85 (0.0577) | 0.65 (0.1) | 0.617 (0.1607) | 0.55 (0.1)
ExSCAD 4th | 0.875 (0.05) | 0.75 (0.0577) | 0.752 (0.0346) | 0.542 (0.2378)
ExSCAD 6th | 0.85 (0.0577) | 0.65 (0.1) | 0.617 (0.1607) | 0.55 (0.1)
ExMCP 4th | 0.877 (0.0517) | 0.75 (0.0577) | 0.663 (0.2358) | 0.425 (0.2062)
ExMCP 6th | 0.855 (0.053) | 0.672 (0.0713) | 0.544 (0.1423) | 0.6 (0)
Lasso | 0.3 (0) | 0 (0) | 0 (0) | 0 (0)
SCAD | 0.3 (0) | 0 (0) | 0 (0) | 0 (0)
MCP | 0.3 (0) | 0 (0) | 0 (0) | 0 (0)
Median | 0.446 (0.041) | 0.229 (0.0473) | 0.247 (0.1426) | 0.124 (0.0845)
Q-0.9 | 0.158 (0.1061) | 0.111 (0.0596) | 0.107 (0.0266) | 0.124 (0.0845)
Q-0.99 | 0.09 (0.0362) | 0.101 (0.0083) | 0.081 (0.0084) | 0.091 (0.0068)
Q-0.999 | 0.303 (0.2673) | 0.112 (0.0494) | 0.267 (0.2336) | 0.081 (0.0084)
Threshold | 0 (0) | 0 (0) | 0 (0) | 0 (0)
Table 21: Average false positive rates for changing number of dimensions. | P = 750 | P = 1500 | P = 2250 | P = 3000
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.002 (7e-04) | 0.003 (8e-04) | 0.004 (0) | 0.008 (0.0028)
ExLasso ($\gamma=6$) | 0.002 (8e-04) | 0.005 (0.0014) | 0.005 (0.0019) | 0.006 (0.0014)
ExSCAD 4th | 0.002 (7e-04) | 0.003 (8e-04) | 0.004 (0) | 0.008 (0.0028)
ExSCAD 6th | 0.002 (8e-04) | 0.005 (0.0014) | 0.005 (0.0019) | 0.006 (0.0014)
ExMCP 4th | 0.002 (7e-04) | 0.003 (8e-04) | 0.004 (0.0029) | 0.008 (0.0028)
ExMCP 6th | 0.002 (8e-04) | 0.004 (0.0013) | 0.006 (0.002) | 0.005 (0)
Lasso | 0.009 (0) | 0.015 (0.002) | 0.014 (0) | 0.015 (0.002)
SCAD | 0.009 (0) | 0.015 (0.002) | 0.014 (0) | 0.015 (0.002)
MCP | 0.009 (0) | 0.014 (0.0022) | 0.013 (0.0014) | 0.014 (0.0014)
Median | 0.011 (0) | 0.012 (0.0011) | 0.01 (0.0026) | 0.012 (0.0058)
Q-0.9 | 0.01 (0.003) | 0.014 (0.0028) | 0.013 (0.0028) | 0.015 (0.0029)
Q-0.99 | 0.018 (0.0059) | 0.012 (0.0011) | 0.016 (0.0017) | 0.014 (0.0011)
Q-0.999 | 0.013 (0.0078) | 0.014 (0.0028) | 0.012 (0.0051) | 0.016 (0.0017)
Threshold | 0.01 (7e-04) | 0.021 (7e-04) | 0.02 (0.0098) | 0.016 (0.0052)
### B.2 Mixture Model Simulation Study
Scenario 1: Changing Magnitude of Extreme Values of Response Variable
Table 22: Average F-1 scores for changing magnitude of extreme value magnitude. | $\tau$ = 6 | $\tau$ = 7 | $\tau$ = 9 | $\tau$ = 50
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.128 (0.0986) | 0.175 (0.1708) | 0.9 (0.0816) | 1 (0)
ExLasso ($\gamma=6$) | 0.259 (0.3143) | 0.757 (0.0963) | 1 (0) | 1 (0)
ExSCAD 4th | 0.128 (0.0986) | 0.175 (0.1708) | 0.9 (0.0816) | 1 (0)
ExSCAD 6th | 0.259 (0.3143) | 0.757 (0.0963) | 1 (0) | 1 (0)
ExMCP 4th | 0.028 (0.0556) | 0.106 (0.1222) | 0.82 (0.0688) | 0.972 (0.0556)
ExMCP 6th | 0.105 (0.2105) | 0.653 (0.1974) | 0.946 (0.0454) | 1 (0)
Lasso | 0 (0) | 0.075 (0.15) | 0 (0) | 1 (0)
SCAD | 0 (0) | 0.075 (0.15) | 0 (0) | 1 (0)
MCP | 0 (0) | 0.075 (0.15) | 0 (0) | 1 (0)
Median | 0.094 (0.0022) | 0.185 (0.1797) | 0.095 (0) | 0.095 (0)
Q0.9 | 0.094 (0.0022) | 0.14 (0.0887) | 0.095 (0) | 0.095 (0)
Q0.99 | 0.179 (0.0599) | 0.098 (0.0193) | 0.312 (0.1434) | 0.739 (0.1504)
Q0.999 | 0.348 (0.0986) | 0.369 (0.1994) | 0.394 (0.1643) | 0.474 (0.1721)
Threshold | 0 (0) | 0.123 (0.1798) | 0.384 (0.392) | 0.977 (0.0455)
Table 23: Average true positive rates for changing magnitude of extreme value magnitude. | $\tau$ = 6 | $\tau$ = 7 | $\tau$ = 9 | $\tau$ = 50
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.131 (0.102) | 0.175 (0.1708) | 0.9 (0.0816) | 1 (0)
ExLasso ($\gamma=6$) | 0.246 (0.2936) | 0.742 (0.1067) | 1 (0) | 1 (0)
ExSCAD 4th | 0.131 (0.102) | 0.175 (0.1708) | 0.9 (0.0816) | 1 (0)
ExSCAD 6th | 0.246 (0.2936) | 0.742 (0.1067) | 1 (0) | 1 (0)
ExMCP 4th | 0.031 (0.0625) | 0.112 (0.1315) | 0.842 (0.0618) | 1 (0)
ExMCP 6th | 0.111 (0.2222) | 0.686 (0.1878) | 1 (0) | 1 (0)
Lasso | 0 (0) | 0.075 (0.15) | 0 (0) | 1 (0)
SCAD | 0 (0) | 0.075 (0.15) | 0 (0) | 1 (0)
MCP | 0 (0) | 0.075 (0.15) | 0 (0) | 1 (0)
Median | 0.089 (0.0038) | 0.172 (0.1629) | 0.091 (0) | 0.091 (0)
Q0.9 | 0.089 (0.0038) | 0.131 (0.0795) | 0.091 (0) | 0.091 (0)
Q0.99 | 0.166 (0.0587) | 0.1 (0.0322) | 0.303 (0.1415) | 0.671 (0.1675)
Q0.999 | 0.389 (0.1361) | 0.346 (0.188) | 0.373 (0.1713) | 0.426 (0.1602)
Threshold | 0 (0) | 0.122 (0.1714) | 0.353 (0.3505) | 0.958 (0.0833)
Table 24: Average false positive rates for changing magnitude of extreme value magnitude. | $\tau$ = 6 | $\tau$ = 7 | $\tau$ = 9 | $\tau$ = 50
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.011 (0.0013) | 0.011 (0.0023) | 0.001 (0.0011) | 0 (0)
ExLasso ($\gamma=6$) | 0.01 (0.0028) | 0.004 (0.0017) | 0 (0) | 0 (0)
ExSCAD 4th | 0.011 (0.0013) | 0.011 (0.0023) | 0.001 (0.0011) | 0 (0)
ExSCAD 6th | 0.01 (0.0028) | 0.004 (0.0017) | 0 (0) | 0 (0)
ExMCP 4th | 0.01 (8e-04) | 0.01 (0.0014) | 0.002 (8e-04) | 0 (0)
ExMCP 6th | 0.01 (0.0023) | 0.004 (0.002) | 0 (0) | 0 (0)
Lasso | 0.013 (7e-04) | 0.012 (0.002) | 0.014 (0) | 0 (0)
SCAD | 0.013 (7e-04) | 0.012 (0.002) | 0.014 (0) | 0 (0)
MCP | 0.013 (7e-04) | 0.012 (0.0019) | 0.013 (0.0014) | 0 (0)
Median | 0.014 (7e-04) | 0.012 (0.002) | 0.014 (0) | 0.014 (0)
Q0.9 | 0.014 (8e-04) | 0.014 (8e-04) | 0.014 (0) | 0.014 (7e-04)
Q0.99 | 0.014 (0.004) | 0.014 (0.007) | 0.01 (0.0026) | 0.006 (0.0032)
Q0.999 | 0.007 (0.0032) | 0.011 (0.0052) | 0.01 (0.0036) | 0.01 (0.0051)
Threshold | 0.009 (0.0023) | 0.011 (0.0019) | 0.008 (0.0034) | 0.001 (0.0014)
Scenario 2: Changing Number of Extreme Events in Response
Table 25: Average F-1 scores for changing number of extreme events. | E = 1 | E = 2 | E = 3 | E = 4
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.128 (0.0986) | 0.313 (0.1514) | 0.632 (0.1489) | 0.836 (0.0473)
ExLasso ($\gamma=6$) | 0.259 (0.3143) | 0.52 (0.0869) | 0.795 (0.1527) | 0.908 (0.0789)
ExSCAD 4th | 0.128 (0.0986) | 0.313 (0.1514) | 0.632 (0.1489) | 0.836 (0.0473)
ExSCAD 6th | 0.259 (0.3143) | 0.52 (0.0869) | 0.795 (0.1527) | 0.908 (0.0789)
ExMCP 4th | 0.028 (0.0556) | 0.164 (0.136) | 0.559 (0.2802) | 0.743 (0.1306)
ExMCP 6th | 0.105 (0.2105) | 0.239 (0.2046) | 0.697 (0.2623) | 0.83 (0.0989)
Lasso | 0 (0) | 0 (0) | 0 (0) | 0 (0)
SCAD | 0 (0) | 0 (0) | 0 (0) | 0 (0)
MCP | 0 (0) | 0 (0) | 0 (0) | 0 (0)
Median | 0.094 (0.0022) | 0.094 (0.0022) | 0.095 (0) | 0.095 (0)
Q0.9 | 0.094 (0.0022) | 0.094 (0.0022) | 0.095 (0) | 0.095 (0)
Q0.99 | 0.179 (0.0599) | 0.421 (0.1032) | 0.604 (0.093) | 0.65 (0.1238)
Q0.999 | 0.348 (0.0986) | 0.358 (0.0519) | 0.45 (0.1935) | 0.474 (0.1154)
Threshold | 0 (0) | 0.229 (0.1455) | 0.596 (0.1489) | 0.758 (0.1173)
Table 26: Average true positive rates for changing number of extreme events. | E = 1 | E = 2 | E = 3 | E = 4
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.131 (0.102) | 0.328 (0.1627) | 0.667 (0.1571) | 0.847 (0.0547)
ExLasso ($\gamma=6$) | 0.246 (0.2936) | 0.542 (0.0949) | 0.817 (0.1599) | 0.944 (0.0642)
ExSCAD 4th | 0.131 (0.102) | 0.328 (0.1627) | 0.667 (0.1571) | 0.847 (0.0547)
ExSCAD 6th | 0.246 (0.2936) | 0.542 (0.0949) | 0.817 (0.1599) | 0.944 (0.0642)
ExMCP 4th | 0.031 (0.0625) | 0.182 (0.144) | 0.599 (0.2817) | 0.837 (0.0834)
ExMCP 6th | 0.111 (0.2222) | 0.259 (0.2049) | 0.761 (0.2223) | 0.937 (0.0745)
Lasso | 0 (0) | 0 (0) | 0 (0) | 0 (0)
SCAD | 0 (0) | 0 (0) | 0 (0) | 0 (0)
MCP | 0 (0) | 0 (0) | 0 (0) | 0 (0)
Median | 0.089 (0.0038) | 0.089 (0.0038) | 0.091 (0) | 0.091 (0)
Q0.9 | 0.089 (0.0038) | 0.089 (0.0038) | 0.091 (0) | 0.091 (0)
Q0.99 | 0.166 (0.0587) | 0.397 (0.0874) | 0.588 (0.1032) | 0.691 (0.0929)
Q0.999 | 0.389 (0.1361) | 0.369 (0.0525) | 0.479 (0.2206) | 0.479 (0.1158)
Threshold | 0 (0) | 0.234 (0.1401) | 0.653 (0.1768) | 0.767 (0.1054)
Table 27: Average false positive rates for changing number of extreme events. | E = 1 | E = 2 | E = 3 | E = 4
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.011 (0.0013) | 0.008 (0.0023) | 0.004 (0.0019) | 0.002 (8e-04)
ExLasso ($\gamma=6$) | 0.01 (0.0028) | 0.006 (0.0013) | 0.002 (0.002) | 0.001 (8e-04)
ExSCAD 4th | 0.011 (0.0013) | 0.008 (0.0023) | 0.004 (0.0019) | 0.002 (8e-04)
ExSCAD 6th | 0.01 (0.0028) | 0.006 (0.0013) | 0.002 (0.002) | 0.001 (8e-04)
ExMCP 4th | 0.01 (8e-04) | 0.008 (7e-04) | 0.004 (0.0028) | 0.002 (7e-04)
ExMCP 6th | 0.01 (0.0023) | 0.008 (0.0013) | 0.002 (0.002) | 0.001 (8e-04)
Lasso | 0.013 (7e-04) | 0.013 (7e-04) | 0.013 (7e-04) | 0.013 (7e-04)
SCAD | 0.013 (7e-04) | 0.013 (7e-04) | 0.013 (7e-04) | 0.013 (7e-04)
MCP | 0.013 (7e-04) | 0.012 (0.0013) | 0.012 (0.0019) | 0.012 (0.0019)
Median | 0.014 (7e-04) | 0.014 (7e-04) | 0.014 (0) | 0.014 (0)
Q0.9 | 0.014 (8e-04) | 0.014 (0) | 0.014 (0.0014) | 0.014 (8e-04)
Q0.99 | 0.014 (0.004) | 0.009 (0.0013) | 0.006 (0.0023) | 0.004 (0.0013)
Q0.999 | 0.007 (0.0032) | 0.008 (0.0011) | 0.007 (0.0038) | 0.007 (0.0026)
Threshold | 0.009 (0.0023) | 0.009 (0.0011) | 0.004 (0.0022) | 0.003 (0.0013)
Scenario 3: Changing Error Distribution
Table 28: Average F-1 scores for changing residual distribution. | $\beta$ = 0.33 | $\beta$ = 0.2 | $\beta$ = 0.166 | $\beta$ = 0.125
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.9 (0.0816) | 0.875 (0.1258) | 0.816 (0.0526) | 0.278 (0.3587)
ExLasso ($\gamma=6$) | 1 (0) | 1 (0) | 0.582 (0.2852) | 0.184 (0.217)
ExSCAD 4th | 0.9 (0.0816) | 0.875 (0.1258) | 0.816 (0.0526) | 0.278 (0.3587)
ExSCAD 6th | 1 (0) | 1 (0) | 0.582 (0.2852) | 0.184 (0.217)
ExMCP 4th | 0.82 (0.0688) | 0.818 (0.0819) | 0.735 (0.1126) | 0.288 (0.3796)
ExMCP 6th | 0.946 (0.0454) | 0.917 (0.0556) | 0.693 (0.1714) | 0.144 (0.1744)
Lasso | 0 (0) | 0 (0) | 0 (0) | 0 (0)
SCAD | 0 (0) | 0 (0) | 0 (0) | 0 (0)
MCP | 0 (0) | 0 (0) | 0 (0) | 0 (0)
Median | 0.095 (0) | 0.095 (0) | 0.095 (0) | 0.094 (0.0022)
Q-0.9 | 0.095 (0) | 0.095 (0) | 0.095 (0) | 0.094 (0.0022)
Q-0.99 | 0.312 (0.1434) | 0.14 (0.0494) | 0.249 (0.0897) | 0.24 (0.0465)
Q-0.999 | 0.394 (0.1643) | 0.299 (0.0897) | 0.19 (0.0186) | 0.115 (0.0505)
Threshold | 0.384 (0.392) | 0.05 (0.1) | 0.64 (0.0773) | 0.508 (0.2058)
Table 29: Average true positive rates for changing residual distribution. | $\beta$ = 0.33 | $\beta$ = 0.2 | $\beta$ = 0.166 | $\beta$ = 0.125
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.9 (0.0816) | 0.875 (0.1258) | 0.861 (0.0556) | 0.281 (0.358)
ExLasso ($\gamma=6$) | 1 (0) | 1 (0) | 0.589 (0.2837) | 0.194 (0.2291)
ExSCAD 4th | 0.9 (0.0816) | 0.875 (0.1258) | 0.861 (0.0556) | 0.281 (0.358)
ExSCAD 6th | 1 (0) | 1 (0) | 0.589 (0.2837) | 0.194 (0.2291)
ExMCP 4th | 0.842 (0.0618) | 0.869 (0.1245) | 0.893 (0.1368) | 0.307 (0.3857)
ExMCP 6th | 1 (0) | 1 (0) | 0.821 (0.1798) | 0.17 (0.209)
Lasso | 0 (0) | 0 (0) | 0 (0) | 0 (0)
SCAD | 0 (0) | 0 (0) | 0 (0) | 0 (0)
MCP | 0 (0) | 0 (0) | 0 (0) | 0 (0)
Median | 0.091 (0) | 0.091 (0) | 0.091 (0) | 0.089 (0.0038)
Q-0.9 | 0.091 (0) | 0.091 (0) | 0.091 (0) | 0.089 (0.0038)
Q-0.99 | 0.303 (0.1415) | 0.133 (0.0438) | 0.251 (0.0829) | 0.232 (0.0391)
Q-0.999 | 0.373 (0.1713) | 0.277 (0.0854) | 0.182 (0.0343) | 0.107 (0.0505)
Threshold | 0.353 (0.3505) | 0.05 (0.1) | 0.689 (0.092) | 0.517 (0.2134)
Table 30: Average false positive rates for changing residual distribution. | $\beta$ = 0.33 | $\beta$ = 0.2 | $\beta$ = 0.166 | $\beta$ = 0.125
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.001 (0.0011) | 0.002 (0.0017) | 0.002 (7e-04) | 0.009 (0.0045)
ExLasso ($\gamma=6$) | 0 (0) | 0 (0) | 0.005 (0.0038) | 0.01 (0.0032)
ExSCAD 4th | 0.001 (0.0011) | 0.002 (0.0017) | 0.002 (7e-04) | 0.009 (0.0045)
ExSCAD 6th | 0 (0) | 0 (0) | 0.005 (0.0038) | 0.01 (0.0032)
ExMCP 4th | 0.002 (8e-04) | 0.002 (0.0017) | 0.001 (0.0013) | 0.007 (0.0033)
ExMCP 6th | 0 (0) | 0 (0) | 0.002 (0.0017) | 0.009 (0.0026)
Lasso | 0.014 (0) | 0.014 (0) | 0.014 (0) | 0.013 (7e-04)
SCAD | 0.014 (0) | 0.014 (0) | 0.014 (0) | 0.013 (7e-04)
MCP | 0.013 (0.0014) | 0.013 (7e-04) | 0.012 (0.0016) | 0.012 (0.0019)
Median | 0.014 (0) | 0.014 (0) | 0.014 (0) | 0.014 (7e-04)
Q-0.9 | 0.014 (0) | 0.015 (7e-04) | 0.013 (0.0014) | 0.013 (0.0023)
Q-0.99 | 0.01 (0.0026) | 0.013 (0.002) | 0.01 (0.0023) | 0.011 (0.0013)
Q-0.999 | 0.01 (0.0036) | 0.011 (0.0017) | 0.012 (0.0028) | 0.015 (0.002)
Threshold | 0.008 (0.0034) | 0.011 (7e-04) | 0.004 (0.0013) | 0.006 (0.003)
Scenario 4: Changing Number of Dimensions
Table 31: Average F-1 scores for changing number of dimensions. | P = 750 | P = 1500 | P = 2250 | P = 3000
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.9 (0.0816) | 0.816 (0.0526) | 0.922 (0.0673) | 0.838 (0.0062)
ExLasso ($\gamma=6$) | 1 (0) | 0.947 (0) | 0.961 (0.0263) | 0.947 (0)
ExSCAD 4th | 0.9 (0.0816) | 0.816 (0.0526) | 0.922 (0.0673) | 0.838 (0.0062)
ExSCAD 6th | 1 (0) | 0.947 (0) | 0.961 (0.0263) | 0.947 (0)
ExMCP 4th | 0.82 (0.0688) | 0.782 (0.0708) | 0.765 (0.0679) | 0.683 (0.134)
ExMCP 6th | 0.946 (0.0454) | 0.854 (0.0619) | 0.961 (0.0263) | 0.885 (0.0876)
Lasso | 0 (0) | 0 (0) | 0 (0) | 0 (0)
SCAD | 0 (0) | 0 (0) | 0 (0) | 0 (0)
MCP | 0 (0) | 0 (0) | 0 (0) | 0 (0)
Median | 0.095 (0) | 0.095 (0) | 0.095 (0) | 0.095 (0)
Q-0.9 | 0.095 (0) | 0.095 (0) | 0.095 (0) | 0.095 (0)
Q-0.99 | 0.312 (0.1434) | 0.238 (0.0794) | 0.189 (0.0744) | 0.093 (0.0262)
Q-0.999 | 0.394 (0.1643) | 0.39 (0.0962) | 0.501 (0.0663) | 0.577 (0.2264)
Threshold | 0.384 (0.392) | 0.32 (0.1879) | 0.316 (0.087) | 0 (0)
Table 32: Average true positive rates for changing number of dimensions. | P = 750 | P = 1500 | P = 2250 | P = 3000
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.9 (0.0816) | 0.861 (0.0556) | 0.947 (0.0611) | 0.802 (0.1235)
ExLasso ($\gamma=6$) | 1 (0) | 1 (0) | 1 (0) | 1 (0)
ExSCAD 4th | 0.9 (0.0816) | 0.861 (0.0556) | 0.947 (0.0611) | 0.802 (0.1235)
ExSCAD 6th | 1 (0) | 1 (0) | 1 (0) | 1 (0)
ExMCP 4th | 0.842 (0.0618) | 0.853 (0.0524) | 0.929 (0.0825) | 0.795 (0.1136)
ExMCP 6th | 1 (0) | 1 (0) | 1 (0) | 1 (0)
Lasso | 0 (0) | 0 (0) | 0 (0) | 0 (0)
SCAD | 0 (0) | 0 (0) | 0 (0) | 0 (0)
MCP | 0 (0) | 0 (0) | 0 (0) | 0 (0)
Median | 0.091 (0) | 0.091 (0) | 0.091 (0) | 0.091 (0)
Q-0.9 | 0.091 (0) | 0.091 (0) | 0.091 (0) | 0.091 (0)
Q-0.99 | 0.303 (0.1415) | 0.207 (0.0762) | 0.18 (0.0684) | 0.092 (0.0468)
Q-0.999 | 0.373 (0.1713) | 0.371 (0.1244) | 0.486 (0.0278) | 0.567 (0.2974)
Threshold | 0.353 (0.3505) | 0.344 (0.1929) | 0.335 (0.0958) | 0 (0)
Table 33: Average false positive rates for changing number of dimensions. | P = 750 | P = 1500 | P = 2250 | P = 3000
---|---|---|---|---
ExLasso ($\gamma=4$) | 0.001 (0.0011) | 0.002 (7e-04) | 0.001 (8e-04) | 0.003 (0.0029)
ExLasso ($\gamma=6$) | 0 (0) | 0 (0) | 0 (0) | 0 (0)
ExSCAD 4th | 0.001 (0.0011) | 0.002 (7e-04) | 0.001 (8e-04) | 0.003 (0.0029)
ExSCAD 6th | 0 (0) | 0 (0) | 0 (0) | 0 (0)
ExMCP 4th | 0.002 (8e-04) | 0.002 (7e-04) | 0.001 (8e-04) | 0.002 (0.001)
ExMCP 6th | 0 (0) | 0 (0) | 0 (0) | 0 (0)
Lasso | 0.014 (0) | 0.013 (8e-04) | 0.013 (8e-04) | 0.013 (0.001)
SCAD | 0.014 (0) | 0.013 (8e-04) | 0.013 (8e-04) | 0.013 (0.001)
MCP | 0.013 (0.0014) | 0.011 (0.0013) | 0.012 (0.0017) | 0.013 (0.001)
Median | 0.014 (0) | 0.014 (0) | 0.014 (0) | 0.014 (0)
Q-0.9 | 0.014 (0) | 0.017 (0.0013) | 0.015 (0.0026) | 0.014 (0.001)
Q-0.99 | 0.01 (0.0026) | 0.017 (0.0078) | 0.012 (0.0019) | 0.016 (0.0086)
Q-0.999 | 0.01 (0.0036) | 0.01 (0.0045) | 0.007 (0.0014) | 0.007 (0.0067)
Threshold | 0.008 (0.0034) | 0.007 (0.0017) | 0.008 (0.0016) | 0.012 (0.0019)
## Acknowledgements
The authors acknowledge support from NSF DMS-1554821 and NSF NeuroNex-1707400.
We also thank Dr. Michael Weylandt for providing useful discussions.
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|
# Magnetic Field at the Galactic Centre from Multi-Wavelength Dust
Polarization
M. S. Akshaya1 and Thiem Hoang1,2
1Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea
2Department of Astronomy and Space Science, University of Science and
Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea
E-mail<EMAIL_ADDRESS>(MSA)
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
We have mapped the magnetic field ($B$-field) for a region of about 30 pc
around the centre of our Galaxy, which encompasses the circumnuclear disk
(CND), the mini-spirals, and the 20 km s-1 and 50 km s-1 molecular clouds,
using thermal dust polarization observations obtained from SOFIA/HAWC+ and
JCMT/SCUPOL. We decompose the spectra of 12CO ($J=3\rightarrow 2$) transition
from this region into individual clustered cloud components and find the
polarization observed at different wavelengths might be tracing completely
different layers of dust along the line-of-sight (LOS). We use modified Davis-
Chandrasekhar-Fermi methods to estimate the $B$-field from the observations.
From our analysis we find the mean strength of the plane-of-sky $B$-field
($B_{{}_{\mathrm{POS}}}$) of the CND and the mini-spirals, probed at 53 µm to
be of the order of $\sim 2$ mG. The magnetic field is lowest close to the
Galactic Centre, in the region of the ionized mini-cavity within the CND with
$B_{{}_{\mathrm{POS}}}<1$ mG, and increases outwards. However, the longer
wavelength polarization at 216 µm appears to come from a dust layer that is
cooler and behind the CND and has a stronger $B$-field of about 6 mG. The
$B$-field has the least strength along the Eastern Arm of the mini-spiral,
which is also the only region with $\mathcal{M}_{\mathrm{A}}>1$ and a mass-to-
flux ratio of $\lambda\gtrsim 1$. The similarity between the
$B_{{}_{\mathrm{POS}}}$ estimates of the 53 µm and 850 µm observations might
indicate them originating from the same depth along the LOS, mostly from the
CND and its foreground cloud features, including the negative-longitude
extension.
###### keywords:
dust, extinction – Galaxy: centre – infrared: ISM – ISM: magnetic fields –
ISM: general – polarization
††pubyear: 2024††pagerange: Magnetic Field at the Galactic Centre from Multi-
Wavelength Dust Polarization–Magnetic Field at the Galactic Centre from Multi-
Wavelength Dust Polarization
## 1 Introduction
The centre of galaxies play a vital role in their evolution, right from star
formation to quenching. A significant fraction of the galactic star formation
might be driven by the inflow of material into the galactic centre, whereas
the massive outflows from these starburst activities as well as from the
supermassive blackhole at the galactic centre (like the Sgr A∗ in case of the
Milky Way) can trigger galactic quenching and hence its evolution (Oort, 1977;
Kormendy & Kennicutt, 2004; Veilleux et al., 2020, and references therenin).
Only in our Galaxy do we have the capability to study this complex region in
great detail at high resolution using various observational techniques like
imaging, spectroscopy, and polarimetry. It is well known that the star
formation rate (SFR) of the Milky Way is far below what is expected from
gravitational collapse of molecular clouds, and the SRF in Galactic Centre
(GC) is even lower than the average SFR in the Milky Way, yet the origin for
the low SRF in the GC remains elusive (Genzel et al., 2010; Barnes et al.,
2017; Bryant & Krabbe, 2021; Henshaw et al., 2022, and references therein).
Magnetic field ($B$-field) is one of the popular candidates that can play a
role in this suppressed star formation. In order to understand the role of
$B$-fields in star formation and evolution of the GC, it is crucial to map its
strength and morphology.
Significant efforts have been made to map the magnetic field of the GC in the
past, with most of these studies focused on the circum-nuclear disk which is
the closest molecular reservoir to Sgr A∗ (CND; Becklin et al., 1982; Guesten
et al., 1987; Jackson et al., 1993). The line-of-sight (LOS) strength of the
$B$-field ($B_{{}_{\mathrm{LOS}}}$) was estimated from the observations of
Zeeman splitting in spectral lines. The average $B_{{}_{\mathrm{LOS}}}$ from
the Zeeman measurements at various locations along the CND was about 3 mG
(Schwarz & Lasenby, 1990; Killeen et al., 1992; Plante et al., 1995; Marshall
et al., 1995; Yusef-Zadeh et al., 1996, 1999). Any observed variation between
the measurements was attributed to possible changes in the magnetic field
orientation.
Aitken et al. (1986) and Aitken et al. (1998) predicted the upper and lower
limits of the magnetic field in the region based on the mid-infrared thermal
dust polarization observations. They assumed paramagnetic relaxation as the
mechanism of grain alignment, and estimate the field to be between 2 – 10 mG
around the GC. However, this assumption no longer holds as recent studies show
that paramagnetic relaxation by itself is not strong enough to drive grain
alignment (Hoang & Lazarian, 2016). Regardless, polarized thermal emission
from aligned dust grains is still a popular tool to map the plane-of-sky (POS)
magnetic field ($B_{{}_{\mathrm{POS}}}$). This technique is based on the fact
that non-spherical dust grains tend to align with their longest axis
perpendicular to the $B$-fields (Lazarian, 2007; Lazarian & Hoang, 2007), so
that the polarization of thermal emission is perpendicular to the $B$-fields
(Hildebrand, 1988). As the result, by observing the thermal dust polarization
and rotating polarization vectors by 90°, we can infer the $B$-field
morphology. The GC makes an ideal target for mapping the magnetic fields using
dust polarization because the dust grains are expected to be efficiently
aligned with the $B$-fields in this environment. Indeed, if the dust grains
are not aligned with the magnetic field, then the dust polarization does not
trace the $B$-field morphology. However, from our previous study (Akshaya &
Hoang, 2023), we have found that, due to the high $B$-field strength observed
in the region by Zeeman measurements, grains could achieve perfect alignment
through Magnetically-Enhanced Radiative Torque Alignment mechanism (MRAT;
Hoang & Lazarian, 2016; Hoang, 2022). Therefore, dust polarization can be a
robust tracer of the $B$-fields in the GC.
The GC is a complex and dynamic environment known to have densities, pressure,
and $B$-field orders of magnitude greater than that observed in the diffuse
interstellar medium (ISM). Molecular spectra of the region also indicate the
presence of complex multi-component structures along the LOS (Sutton et al.,
1990; Henshaw et al., 2016; Eden et al., 2020; Hu et al., 2022). If these dust
components present at various distances along the LOS are emitting radiation
at different wavebands (due to difference in their temperatures), we can use
the different polarization morphology observed at multiple wavelengths to get
an understanding of the change in the $B$-field strength, morphology, and
orientation along the LOS.
The goal of this paper is to measure the strength of the $B$-field within a
region of about 30 pc around the supermassive blackhole Sgr A∗ at the centre
of our Galaxy. We will use polarization observations in three wavebands, the
same which were used in our previous study (Akshaya & Hoang, 2023) at 53, 216,
and 850 µm. The observations focus on the CND, which is a warm torus of gas
and dust orbiting around Sgr A∗, and the mini-spirals which are a set of
ionized gas filaments present within and interacting with the CND (Lo &
Claussen, 1983; Christopher et al., 2005). Each observations probes the region
on different scales allowing us to map the small and large scale magnetic
field in the region. The region of the CND is covered in all the three
observations while part of the 20 km s-1 and 50 km s-1 clouds (Kauffmann et
al., 2017) are seen in the 216 µm observation. The 850 µm data covers the
largest area around Sgr A∗ which includes the CND, mini-spirals, 20 km s-1,
and 50 km s-1 clouds. Recent study of the CND using Stratospheric Observatory
for Infrared Astronomy (SOFIA; Temi et al., 2018) observation by Guerra et al.
(2023) estimate the $B$-field strength of the mini-spirals within the CND to
have median values between 5 – 8 mG on a spacial scale of $\lesssim 1$ pc.
The most commonly used technique for the measurement of the POS magnetic field
strength from thermal dust polarization is the Davis-Chandrasekhar-Fermi (DCF)
method (Davis, 1951; Chandrasekhar & Fermi, 1953). It is based on the
propagation of Alfvén waves through the medium, when there is an energy
balance between the gas kinetic energy and magnetic energy. Assuming the
magnetic field lines to be frozen with the matter in the general ISM
conditions, the method attributes any small scale irregularities observed in
the polarization vectors to turbulence. The net magnetic field in the region
can be assumed to be made up of a regular component ($B_{0}$) and a turbulent
component ($\delta B$). A strong $B_{0}$ resists being perturbed by turbulence
such that $\delta B\ll B_{0}$, thus allowing us to characterise the POS
component of $B_{0}$ ($B_{{}_{\mathrm{POS}}}$) by measuring the net
irregularity of the field lines using the relation,
$B_{{}_{\mathrm{POS}}}=\sqrt{4\pi\rho}\frac{\sigma_{v}}{\sigma_{\phi}},$ (1)
where $\rho$ is the gas mass density, $\sigma_{v}$ is the turbulence-induced
velocity dispersion from non-thermal line-width measurements, and
$\sigma_{\phi}$ is the distortion in the $B$-field measured by the
polarization angle dispersion. Even though the method is known to overestimate
the magnetic field due to its restrictive initial assumptions about the
conditions of the underlying medium (Ostriker et al., 2001; Houde et al.,
2009; Chen et al., 2022; Myers et al., 2023, and references therein), it is a
useful technique to get an upper estimate of the magnetic field in simple
environments without self gravity or sheer motion.
The DCF method has been applied successfully by many earlier polarization
studies (Pillai et al., 2015; Planck Collaboration et al., 2016; Pattle et
al., 2017; Guerra et al., 2021; Ngoc et al., 2021; Hwang et al., 2021; Hoang
et al., 2022b). Some of the physical conditions that contribute to the over-
estimate of the mean $B_{{}_{\mathrm{POS}}}$ while using DCF include;
anisotropic turbulence, failure of equipartition between the magnetic and
kinetic energy, and self gravity. It is difficult to disentangle the
importance of each of them without a detailed understanding of the kinematics
of the region. The overestimation could also be a result of integration
effects from within the beam size of individual observations, as well as along
the LOS. These uncertainties arise from the polarization observation and
impact the parameter $\sigma_{\phi}$ in Equation 1 (Hildebrand et al., 2009;
Houde et al., 2009; Skalidis & Tassis, 2021; Li et al., 2022; Guerra et al.,
2023). However, Chen et al. (2022) found that the hydrodynamic properties of
gas in the region contributes equally if not more to the uncertainty of the
measured field (reflected in the parameters $\rho$ and $\sigma_{v}$).
Several attempts have been made to improve the original DCF technique. The DCF
method is a good approximation when the observed angle dispersion is small
i.e. when $\delta B\ll B_{0}$ (Ostriker et al., 2001). Falceta-Gonçalves et
al. (2008) extended the DCF method to cases where the turbulent component of
the magnetic field is comparable to the mean field, thus resulting in large
angle dispersion. Hildebrand et al. (2009) and Houde et al. (2009) further
improved the DCF estimate by using the turbulent-to-ordered magnetic field
ratio to estimate the angel dispersion, based on the second order structure
function of magnetic field position angles introduced by Falceta-Gonçalves et
al. (2008). This method incorporated the large scale structure of the magnetic
field along with the instrumental effects in the observations. The recent
modifications include those by Cho & Yoo (2016) and Lazarian et al. (2022)
where they address the anisotropic nature of MHD turbulence, by incorporating
structure function in combination with velocity centroids to estimate the
velocity fluctuations in the POS. We will apply the DCF modifications from
Houde et al. (2009) and Lazarian et al. (2022) in our study, with the goal to
understand how multi-wavelength polarization can be used to probe the 3D
strength of the $B$-field. No correction factors are used in our estimates and
they can be considered as an upper limit of the $B$-field in the region
covered by each observation.
Figure 1: The maps of polarization from SOFIA/HAWC+ observations at 53 µm
(top), 216 µm (bottom left), and JCMT/SCUPOL at 850 µm (bottom right). The
colorbars represent the intensity in respective wavebands and the beam size of
each instrument is shown on the bottom right of each figure along with the
representative scale of polarization percentage at the top.
In this work, we will measure the $B_{{}_{\mathrm{POS}}}$ from three
observations of the region around the GC spanning a physical scale of 10 – 30
pc in wavebands centered at 53, 216, and 850 µm. Our initial study of the
thermal dust polarization from these observations is discussed in Akshaya &
Hoang (2023), where we focus on the grain alignment physics. We have used the
same observations in this work to understand how the strength of the magnetic
field varies across different scales and wavelengths along the LOS, with
longer wavelengths probing deeper into the LOS compared to shorter
wavelengths. The region of the Galactic disk in general, and GC in particular
are known to have multiple structures at different distances along the LOS. We
will use recent techniques to isolate the individual components and understand
their effects on the derived magnetic field.
The rest of our paper is structured as follows; Section 2 describes the
polarization observations and their data quality assessment. The estimation of
the gas velocity dispersion is discussed in Section 3. The magnetic field
measured from the DCF modifications by Houde et al. (2009) and Lazarian et al.
(2022) is described in Section 4. A detailed discussion of the estimated
$B$-field and its implications on the kinematics of the regions are presented
in Section 5, followed by a brief summary of our results in Section 6.
Figure 2: The top map shows the HAWC+ polarization observation of the CND at
216 µm with the criteria $\Delta\phi_{\mathrm{ref}}<10\degree$, compared with
the 250 µm polarization from PILOT (red vectors). The background intensity map
is the Stokes I measurements from HAWC+. The PILOT polarization vectors are
scaled to $p=5\%$. The map at the bottom compares the HAWC+ observations of
the same region taken using the CNM (red) and OTFMAP (blue) mapping strategies
respectively. The background intensity in this case is from the Stokes I
measurements of the OTFMAP. The scale of polarization percentage is shown on
each map along with the beam size of the HAWC+ instrument.
## 2 Dust Polarization Observations
We have used the thermal dust polarization observations around the GC at 53
and 216 µm from High-resolution Airborne Wide-band Camera Plus (HAWC+; Harper
et al., 2018), which is a far-infrared imager and polarimeter for NASA’s
Stratospheric Observatory for Infrared Astronomy (SOFIA; Temi et al., 2018)
and the reprocessed data at 850 µm from SCUPOL, which was the polarimeter for
the Submillimeter Common User Bolometer Array (SCUBA) instrument on the James
Clerk Maxwell Telescope (JCMT) presented by Matthews et al. (2009). More
details about the data and the polarization cut off criteria used are
presented in Akshaya & Hoang (2023), where the grain alignment of the region
was studied with the same observations. The SOFIA/HAWC+ observations were
reduced using the latest version of the HAWC+ data reduction pipeline i.e. DRP
v3.2.0 compared to the DRP v1.3.0 used in our previous analysis. The quality
of these datasets are nominal with all the problematic files removed during
the reduction procedure.
We have also used the data from Herschel in five wavebands at 70, 160, 250,
350, and 500 µm to derive the gas column density ($N_{{}_{\mathrm{H}}}$) and
the dust temperature ($T_{\mathrm{d}}$) as described in Akshaya & Hoang
(2023). The polarization maps of each observation is shown in Fig. 1. The
magnetic field strength will be estimated for each of the polarization
observations, which we believe to be probing different components of dust
along the LOS. Throughout this work we assume the distance to the GC to be 8
kpc (Trippe et al., 2008; Genzel et al., 2010).
### 2.1 Data Quality Assessment
The observed polarization from the CND and its surrounding can be subject to
reference beam contamination due to the emission from extended features in
this region. This is more of a problem in the 216 µm observation due to the
temperature of the surrounding material. At 53 µm the dust being probed is
much hotter than its surrounding and hence the contribution from the reference
beams is not as significant, allowing us to use the Level 4 data products as
it is at this wavelength. We have estimated the level of reference beam
contamination in the SOFIA/HAWC+ 216 µm observation using the method described
by Novak et al. (1997) and Chuss et al. (2019). The chop angle and amplitude
of the observation are 60$\degree$ and 240′′, respectively with two chop
images taken symmetrically on either side of the source. Herschel 70, 160,
250, and 350 µm observations of the region were used to model the expected
intensity of the chop beams in the HAWC+ filter band-pass using the relation;
$I=A\nu^{2}B_{\nu}(T_{\mathrm{d}}),$ (2)
where $A$ is the amplitude, $\nu$ the frequency, and $B_{v}(T_{\mathrm{d}})$
is the Planck function corresponding to the dust temperature $T_{\mathrm{d}}$.
The mean intensity of the chop beams was used in combination with the
calibrated Stokes I intensity from HAWC+ to determine the contrast of the
source beam with respect to the chop beams. Due to the brightness of the GC,
the ratio of these intensities (represented by $w\equiv I_{r}/I_{m}$, where
$I_{r}$ is the average intensity from the chop beams and $I_{m}$ is the
measured Stokes I intensity of the polarization observation) was found to be
$w<6$ throughout the region. The established method is to only consider
polarization vectors where the contrast is greater than 10 (Santos et al.,
2019). However, this seems unlikely in the Galactic disk at these temperatures
due to the presence of extended emissions along most of the lines of sights.
The other approach is the quantify the level of contamination in the measured
$p$ and $\phi$. Since our goal is to measure the strength of the $B$-field, we
are only interested in the level of contamination in the polarization angle,
which is a key parameter to estimate the $B_{{}_{\mathrm{POS}}}$.
Novak et al. (1997) derived the maximum error in the measured polarization
angle with only the intensity estimates from the reference beam to be,
$\Delta\phi_{\mathrm{ref}}=\frac{1}{2}\mathrm{tan}^{-1}\left[\frac{p_{r}w}{(p_{m}^{2}-p_{r}^{2}w^{2})^{1/2}}\right],$
(3)
where $p_{m}$ is the measured polarization fraction (without debias) and
$p_{r}$ is the assumed polarization of the reference beam.
From Fig. 1 it can be seen that except the top right region of the 216 µm
polarization map, most of the map has a polarization of roughly $p\sim 1\%$.
This is also the level of the observed polarization in the regions of the chop
beams from $Planck$ observations at 350 µm (Planck Collaboration et al.,
2020a, b). Thus the reference beam polarization was set to $p_{r}=1\%$ based
on these observations to estimate the $\Delta\phi_{\mathrm{ref}}$. Following
the cut off criteria defined in Chuss et al. (2019), we use only the
polarization vectors with $\Delta\phi_{\mathrm{ref}}<10\degree$ and the
resulting polarization map is shown in Fig. 2 (top). Earlier estimates of the
contamination from reference beam use $p_{r}=10\%$ (Chuss et al., 2019; Lee et
al., 2021). However this values seems too high for our observation and results
in the rejection of all the polarization vectors.
Due to the low contrast between the source and reference beams in this region,
we also tested the agreement between the polarization angles observed by HAWC+
and those from PILOT balloon experiment presented by Mangilli et al. (2019),
which measured the polarization from an extended region of the GC at 240 µm.
The cut off based on $\Delta\phi_{\mathrm{ref}}$ removed the vectors from low
intense regions, predominantly in the top right as seen in Fig. 1. From our
previous analytical study presented in Akshaya & Hoang (2023), this was the
region where the environmental conditions favour efficient grain alignment,
and hence where we can expect the highest degree of polarization. The other
removed vectors are from the regions where the observed polarization was very
low ($p<0.1\%$). These seem to be the most affected by the reference beam
contamination. The polarization vectors which did not make it with the cut off
set by $\Delta\phi_{\mathrm{ref}}$ also do not match well with the
polarization vectors from PILOT observations, especially in the regions of low
polarization fraction. We consider the remaining vectors good enough for
further analysis and use them as is for the subsequent discussions presented
in this paper.
We have also compared the HAWC+ 216 µm observation after the quality cuts
mentioned above with the observations from the same instrument but taken with
a different observing strategy as part of the Far-Infrared Polarimetric Large
Area CMZ Exploration survey (FIREPLACE; Butterfield et al., 2023, 2024; Paré
et al., 2024). FIREPLACE survey used the on-the-fly mapping mode (OTFMAP) to
measure the polarization of the Central Molecular Zone (CMZ) to overcome some
of the challenges in observing extended emission features posed by the
standard observing strategy of the HAWC+ instrument which was Chop-Nod-Match
(CNM; Harper et al., 2018). The main difference between the two strategies
lies in the background subtraction procedure. While the CNM observations are
limited by background chop images taken close to and symmetrically on either
side of the source, the OTFMAP can choose a background region completely
devoid of any structures associated with the source, as the scans can begin at
a reasonable distance from the target location. This can be very useful when
dealing with extended emission features like in the GC as the chop images
necessary for the background subtraction tend to have some remnant features of
the extended object, resulting in significant reference beam contamination.
The polarization vectors from both these observation strategies is shown in
the bottom map of Fig. 2, where the OTFMAP map follow the standard cuts to the
data with polarization signal-to-noise $p/\sigma_{p}>3$, polarization degree
$p<50\%$, and intensity signal-to-noise (Stokes I) $I/\sigma_{I}>200$. The
main problematic region is the low intense part in the right of the figure
where both the OTFMAP and the CNM observations do not retain any polarization
vectors. The other regions where we have measurements from both strategies
seem to agree well except in the few parts where the CNM observation show low
polarization fraction. Considering the overall agreement in the cutoff
regions, we will use the CNM observation with the
$\Delta\phi_{\mathrm{ref}}<10\degree$ criteria for further analysis from hence
forth.
The 850 µm observation from SCUPOL used in this work is a mosaic created from
69 original observations from SCUPOL archive (Matthews et al., 2009). An
estimation of possible contamination in this data is beyond the scope of the
current paper. We will use this observation as is for our analysis.
Figure 3: Moment zero map of the CO ($J=3\rightarrow 2$) transition used for
our analysis taken from CHIMPS2. The data was binned to match the resolution
of the SOFIA/HAWC+ 216 µm observation.
Figure 4: Sample spectrum of the multiple components observed along the GC is
shown in the top figure along with the respective Gaussian fits to each
identified component in the bottom figure. The blue lines are the decomposed
Gaussian components from scousepy. The dotted red line indicated the best fit
spectrum from the combination of the decomposed components.
## 3 Gas Velocity Dispersion
Carbon monoxide (CO) is the second most abundant molecule in the interstellar
medium (ISM) after H2 and is ubiquitous in the Galactic plane, making it an
ideal tracer of the velocity structures and morphology of gas along this LOS.
Its rotational transitions can be observed in the submillimeter wavelengths,
with different transitions probing different densities. ${}^{12}{\rm
C}^{16}{\rm O}$ (commonly denoted as CO) is the most common isotope of the CO
and can be used to trace gas densities of the order of
$n_{{}_{\mathrm{H}}}\sim 10^{3}-10^{4}$ cm-3 depending on the transition.
Other isotopes like ${}^{13}{\rm CO}$ and ${\rm C}^{18}{\rm O}$ trace denser
regions but are relatively less abundant than CO. The CO ($J=3\rightarrow 2$)
transition is optically thin compared to its other rotational transitions
($J=1\rightarrow 0$ and $J=2\rightarrow 1$) and also has a higher resolution
due to its high frequency ($\nu=345.8$GHz). Thus CO ($J=3\rightarrow 2$) can
trace warmer and denser environments compared to its lower energy
counterparts. Due to the nature of CO ($J=3\rightarrow 2$) transition, the
observations of this line along the Galactic plane often shows multiple
complex emission features corresponding to various structures at different
distances along the LOS. This can be a drawback in cases where the focus is on
a single filament or star forming region and it might be ideal to use higher
density tracers like ${}^{13}{\rm CO}$ or ${\rm C}^{18}{\rm O}$ for these
cases. For our study we chose to use CO ($J=3\rightarrow 2$) transition
because we are interested in seeing all the components along this LOS which
might be contributing to the observed polarization.
Figure 5: Velocity components observed for the region covered by SOFIA/HAWC+
216 µm observation decomposed using scousepy and grouped using acorns. The
centroid velocity of the velocity components is along the z-axis and the data
points are scaled according to their corresponding velocity dispersion. Each
colour represents individual trees resulting from the clustering analysis.
Only the trees with number of leaves greater than four are displayed here.
We have used CO Heterodyne Inner Milky Way Plane Survey 2 (CHIMPS2) data for
the analysis and estimation of the velocity dispersion along the LOS to the
GC. The aim of the CHIMPS2 survey is to map the Inner Galaxy, the CMZ, and a
section of the Outer Galaxy in the 12CO, 13CO, and C18O ($J=3\rightarrow 2$)
emissions using the Heterodyne Array Receiver Program (HARP) on the JCMT. We
have used the first look data of the CHIMPS2 survey towards the CMZ in CO
transition $J=3\rightarrow 2$ presented by Eden et al. (2020), with a spatial
resolution of 15 arcsec and a velocity resolution of 1 km s-1. We re-binned
the spectral cube to match the resolution of SOFIA/HAWC+ 216 µm and the
JCMT/SCUPOL 850 µm observations. The 53 µm observation from HAWC+ is at a much
higher resolution than the CO data, hence we have used the derived velocity
dispersion values at the resolution of the 216 µm observation as an
approximation for its $B$-filed calculation. The moment zero map of the data
is shown in Fig. 3. From the figure it can be seen that the morphology of the
prominent features observed in the polarization maps are also observed in the
integrated intensity map of the CO spectra. This in general is a good
indicator that the chosen spectra can trace the distribution of matter
corresponding to the observed polarization along the LOS and is the basic test
before choosing any molecular species to measure the turbulence needed for the
DCF technique. The complication arises when there are more than one clearly
distinguishable Gaussian components in the spectra for most of the region,
like in the case of the GC.
This complexity of the GC environment makes it a challenging region to measure
the magnetic field using the DCF method. Recent investigation of the method by
Chen et al. (2022) show that the line-width measurement which is used to
constrain the turbulence along the LOS can be the major source of uncertainty
or overestimation of the $B$-field when using the DCF method. The measured
line-width can be a fairly reliable tracer of turbulence in regions where we
can expect a single structure along the LOS, as in the case of filaments
mostly at high Galactic latitudes. But when we look at the spectrum closer to
the Galactic disk, it becomes evident that this approach becomes insufficient.
A sample spectrum of the GC is shown in Fig. 4. The spectrum indicates matter
distributed in multiple layers along the LOS. Another factor to consider when
using these data to measure the line-width is which component/components along
the LOS contribute to the observed polarization.
### 3.1 Spectral decomposition using scousepy
The decomposition of the CO spectrum was done using the Python implementation
of the Semi-Automated multi-COmponent Universal Spectral-line fitting Engine
(scousepy; Henshaw et al., 2016,
2019)111https://github.com/jdhenshaw/scousepy. This is a routine developed in
Interactive Data Language (IDL) and later implemented in Python and is very
useful to decompose complex 3D spectral cube in a systematic and efficient
way. It is a multi-stage procedure and the important steps can be broken down
as follows: (i) specify the region of interest based on position, velocity, or
noise threshold; (ii) the routine then breaks the region into Spectral
Averaging Areas (SAAs) based on the complexity of the spectrum in the region
and extracts the spatially averaged spectrum from each; (iii) the extracted
spectrum is manually fit interactively using
pyspeckit222https://github.com/pyspeckit/pyspeckit; (iv) the best fit solution
of the extracted spectrum from each SAA is used to fit all the individual
spectra within the SAA using a fully automated fitting procedure. The
tolerance levels used in each step is described in Henshaw et al. (2016,
2019).
In our implementation of scousepy on the data re-binned to the HAWC+ 216 µm
resolution, we masked all the pixels with integrated signal below 1.5 K and
used the width of the SAAs to be 10 pixels. We obtained 258 SAAs which were
manually fit to automate the fitting for the 6362 individual spectra of the
region. Similar procedure was performed to obtain the decomposed velocity
components for the SCUPOL 850 µm observation. At the end of the routine we
obtain the number of velocity components in each pixel, their amplitude,
shift, width, and a few additional useful statistics. A sample of the spectra
decomposed using scousepy is shown in the bottom of Fig. 4. We reject the
components with signal-to-noise less than five and use only the robust
resolved components for further analysis.
Figure 6: Moment zero maps of the sub-slabs of PPV cube based on the
clustering shown in Fig. 5. The spectral cube was re-binned to match the
SOFIA/HAWC+ 216 µm observation. The Stokes I maps of HAWC+ 53 µm and SCUPOL
850 µm observation is also shown to compare the morphology of individual slabs
with the morphology of intensity (Stokes I) observed in the polarization maps.
Figure 7: Same as Fig. 6 but zoomed-in to focus on the region covered by the
HAWC+ 53 µm observation, which mostly contains the CND and the mini-spirals.
### 3.2 Hierarchical Agglomerative Clustering using acorns
In order to characterise the decomposed velocity components from scousepy and
understand its effects on the observed polarization, we have used a recently
developed analysis tool based on hierarchical agglomerative clustering called
acorns (Agglomerative Clustering for ORganising Nested Structures; Henshaw et
al., 2019)333https://github.com/jdhenshaw/acorns. The clustering procedure
here begins with the most significant data point and a hierarchy is
established by the merging of clusters based on user defined criteria. For a
PPV cube like our data set, the clustering is performed using the position
(x,y), intensity, and velocity. The criteria we used for the clustering
analysis are as follows: (i) minimum amplitude or peak intensity for the
components to be considered was five times the typical root-mean-square (rms)
value (Henshaw et al., 2019) (ii) minimum size of clusters had to be 30 arcsec
($\sim 1.5$ times the observation beam size) (iii) the difference in velocity
between linked data points cannot be greater than 15 km s-1. For visualization
purpose, we chose trees from the forest which has a minimum of 4 leaves from
the clustering procedure. This resulted in 8 prominent trees and are shown in
Fig. 5. From the figure it can be seen that though there seems to be a great
number of velocity components towards the GC, they can still be grouped into
meaningful distinct sub-structures.
The structures with negative centroid velocities centred at about -60, -40,
and -20 km s-1 are quite distinct and appear to be isolated structures without
much overlap. These components are clearly seen in most of the spectra as
shown in Fig. 4. The positive velocity region however shows significant
overlap in the observed features except for a distinction at around 30 km s-1.
This can also be seen in Fig. 4, where the peaks in the positive velocity
region are more blended together than the negative velocity peaks which are
distinct. Based on these observed velocity features, we divided the initial
spectral cube into sub-slabs with velocity ranges corresponding to the
distinct trees from acorns. The moment zero maps of these slabs are shown in
Fig. 6. We compared the morphology of the velocity features observed in the
moment zero maps of each sub-slab with the intensity maps of the polarization
observations in all the three wavebands considered. The zoom-in version of the
map for the region covered by the 53 µm observation is shown in Fig. 7.
The key take away from this analysis is the difference in morphologies of the
decomposed velocity structures and how they match the polarization
observations at different wavelengths. Probing a region at different
frequencies results in the observation of dust or material at different
temperatures along the LOS, with shorter wavelengths originating from hot
regions with high optical depth and longer wavelength optically thin emission
originating from cooler deeper dust. When we are looking at regions which are
known to have various independent structures along the LOS, we need to be
careful with our measurement of line width to characterise turbulence as the
chosen gaussian component might be probing a layer quite different from the
source of our measured polarization. This effect is demonstrated in Fig. 6 and
Fig. 7.
When we look at the morphology of the PPV sub-slabs and compare them with the
Stokes I maps of HAWC+ 53 and 216 µm observations, the 216 µm observed
intensity follows closely the morphology of the slab with the velocity range
$0<v<45$ km s-1. Though this velocity sub-slab seems to be the major
contributor, there are still traces of features from other slabs in the
intensity map of 216 µm observation. This could indicate that the net observed
emission at this wavelength might be an integrated effect from all the layers
along the line of sight. However, this does not seem to be the case for the 53
µm observation. When we compare its morphology with those of the moment maps
from individual slaps, it becomes clear that most if not all of the observed
emission at this wavelength is originating from the negative velocity
components of our decomposed spectra, and is clearly shown in Fig. 7. The 53
µm observation is dominated by the emission from the CND and the mini-spirals.
There does not seem to be any trace of the morphology observed from the
positive velocity components in the Stokes I map of the 53 µm observation.
This is very important when we are using the spectra to constrain the
turbulence in the region to measure the $B$-field. From this analysis, we
propose that the right estimate of turbulence for the CND and its streamers
comes from the negative velocity components of the CO spectra and use only the
velocity peaks in this range for the measurement of velocity dispersion for
the 53 µm observation. It is also interesting to note that the three negative
velocity components shown in Fig. 7 closely resemble the model for the
velocity of the streamers in Sgr A∗ by Zhao et al. (2009, Fig. 21).
Figure 8: Normalized auto-correlation function calculated from polarized
intensity for each of the observation. The value of $\Delta^{\prime}$ is
determined at half magnitude and corresponds to $0.47^{\prime},0.81^{\prime},$
and $0.78^{\prime}$ at at 53, 216, and 850 µm respectively.
## 4 Measurements of the plane of sky B-field
We now estimate the $B$-field strength using two modified DCF methods; (1)
based on the angle dispersion structure function by Houde et al. (2009) and
(2) the recently proposed Differential Measure Analysis technique by Lazarian
et al. (2022).
### 4.1 $\mathbf{B_{{}_{\mathrm{POS}}}}$ from Angle Dispersion Function
Houde et al. (2009) improved the original DCF technique by using structure
function of polarization angles to calculate the polarization angle dispersion
such that the dispersion due to the change in field orientation is also
accounted for along with the dispersion due to turbulence. Following Houde et
al. (2009), Equation (1) can be rewritten as;
$B_{{}_{\mathrm{POS}}}=\sqrt{4\pi\rho}\sigma_{v}\left[\frac{\langle
B_{t}^{2}\rangle}{\langle B_{0}^{2}\rangle}\right]^{-1/2},$ (4)
where $\langle B_{t}^{2}\rangle/\langle B_{0}^{2}\rangle$ is the turbulent to
large-scale magnetic field strength ratio and can be approximated as the
dispersion in the polarization vectors ($\sigma_{\phi}=\sqrt{\langle
B_{t}^{2}\rangle/\langle B_{0}^{2}\rangle}$). This quantity can be determined
using the two-point dispersion function proposed by Houde et al. (2009),
modelled as a superposition of the large-scale field structure and the small-
scale turbulence. This method takes care of the spatial changes in the
$B$-field morphology and also incorporates the signal integration within the
cloud thickness and the telescope beam. If $\Delta\phi(\ell)$ is the
difference is polarization angle between two vectors separated by an angle
$\ell$ on the sky, then the dispersion function is defined as;
$1-\langle\mathrm{cos}\left[\Delta\phi(\ell)\right]\rangle=\frac{1-e^{-\ell^{2}/2(\delta^{2}+2W^{2})}}{1+\mathcal{N}\left[\frac{\langle
B_{t}^{2}\rangle}{\langle B_{0}^{2}\rangle}\right]^{-1}}+a_{2}\ell^{2},$ (5)
where $\delta$ is the turbulence correlation length, $W$ is the observations
beam radius,
$\mathcal{N}=\Delta^{\prime}(\delta^{2}+2W^{2})/\sqrt{2\pi}\delta^{3}$ is the
number of turbulent cells along the line of sight, and $\Delta^{\prime}$ is
the effective depth of the cloud. The small-scale $B$-field contribution to
the observed dispersion is quantified by the first term in the equation and
the large scale contribution is described by the second term.
We use this angle dispersion structure function to determine $\sigma_{\phi}$
on a pixel-by-pixel basis for each of the observations we have chosen. Before
we apply the dispersion function to the whole image, we need to determine the
values of $\Delta^{\prime}$, $\delta$, and the kernel size $w$. Depth of the
cloud is determined using the width of the auto-correlation function of the
polarized intensity ($P=\sqrt{Q^{2}+U^{2}}$) given by;
$\langle\overline{P}^{2}(\ell)\rangle\equiv\langle\overline{P}(r)\overline{P}(r+\ell)\rangle,$
(6)
as described in Houde et al. (2009). Based on the assumption that the cloud
has similar characteristics across and along its depth, it is a reasonable
approximation of the effective depth of the cloud even though it is a function
of $\ell$ along the cloud surface. We calculated the normalized auto-
correlation function for each of the observations and the corresponding plot
is shown in Fig. 8. The estimated values of $\Delta^{\prime}$ at 53, 216, and
850 µm are $0.47^{\prime}$, $0.81^{\prime}$, and $0.78^{\prime}$ respectively.
Table 1: DCF method parameters.
Parameter | 53 µm | 216 µm | 850 µm
---|---|---|---
beamsize (arcmin) | $4.84$ | $18.2$ | $20$
pixel size (arcsec) | $2.42$ | $4.55$ | $10$
$\Delta^{\prime}$ (arcmin) | $0.47$ | $0.81$ | $0.78$
$w$ (pixels) | 9 | 7 | 13
$\delta_{\mathrm{fit}}$ (arcsec) | $21$ | $21.69$ | -
$[\langle B_{t}^{2}\rangle/\langle B_{0}^{2}\rangle]_{\mathrm{fit}}$ | $0.16$ | $0.019$ | -
$[\langle B_{t}^{2}\rangle/\langle B_{0}^{2}\rangle]_{\mathrm{mean}}$ | $0.29$ | $0.019$ | $0.41$
$\sigma_{v}$ (km/s) | $5.3(8.77)^{*}$ | $8.68$ | $9.75$
$\sigma_{\phi}$ (rad) | $0.42$ | $0.12$ | $0.64$
$n_{{}_{\mathrm{H}}}^{\mathrm{mean}}$ (cm-3) | $10^{4.2}$ | $10^{3.95}$ | $10^{3.96}$
$B_{{}_{\mathrm{POS}}}^{\mathrm{mean}}$ (mG) | $2.07(3.4)^{*}$ | $6.53$ | $1.39$
∗ estimates from all the velocity components.
In order to estimate the dispersion function locally around each pixel, it is
important to choose a kernel size ($w$) in pixel radius over which
$1-\langle\mathrm{cos}\left[\Delta\phi(\ell)\right]\rangle$ is estimated for
each pair of polarization angles. This symmetric two-dimensional normalized
circular kernel ensures no preferential direction is chosen when estimating
the dispersion function. The size has to be greater than the observation beam
size and the turbulence correlation length so that we have enough vectors
within the kernel to calculate the dispersion function with greater accuracy.
We used the method described by Guerra et al. (2021) to determine the optimal
kernel size for each observation. As each of our polarization maps probe a
different physical scale with quite different resolutions, the corresponding
kernel size could also be different. We fit the dispersion function to every
pixel for a range of kernel sizes ranging from $w=5-15$ pixels and calculated
the Spearman’s correlation coefficient ($\rho_{\mathrm{sp}}$) for each kernel
size. The best size was chosen based on the highest median value of estimated
$\rho_{\mathrm{sp}}$ and was found to be 9, 7, and 13 pixels for the 53, 216,
and 850 µm observations, respectively.
Figure 9: Dispersion function fit to the entire field of view for the 53 µm
(top) and the 216 µm (bottom) observations. The blue dots are the estimates
from the observations and the dashed red line is the best fit model for each
case. The first term of the dispersion function which gives an estimate of the
small-scale turbulence component is shown as the cyan line and the large-scale
mean field component is shown by the orange line. The size of the kernel
chosen ($w=9$ and 7 for 53 and 216 µm observations, respectively) for each of
the maps is shown by the black dotted line in both the plots.
Using the above described values of $\Delta^{\prime}$ and $w$, we fit the
dispersion function for each of the polarization observation to determine the
parameters $\delta$, $a_{2}$, and $\langle B_{t}^{2}\rangle/\langle
B_{0}^{2}\rangle$. However, we found most of the estimated $\delta$ values
($\sim 50$%) for the 216 and 850 µm observations to be $\delta<W\sqrt{2}$
which is the correlated beam size of the observations. This indicates that the
resolution is not good enough to resolve the local gas turbulence and might
lead to incorrect estimates of the $B$-field. Thus we tried another approach
presented in Guerra et al. (2021) where we fix the value of $\delta$ based on
the fit of a single dispersion function for the whole region (global
$\delta$). The best fit $\delta$ was found to be $21.69^{\prime\prime}$ for
the 216 µm observation. We used the same value for the 850 µm observation as
they have relatively the same beam sizes ($18.2^{\prime\prime}$ and
$20^{\prime\prime}$ respectively) and due to the lower data quality of the 850
µm observation.
Figure 10: Maps of the $B_{\mathrm{POS}}$ estimated from HAWC+ 216 µm (left)
and SCUPOL 850 µm (right) observations using the DCF method. The location of
the 50 km s-1 and 20 km s-1 clouds are shown as the red and magenta circles,
respectively. The beam size of the observations is shown at the bottom of each
map.
We tried both the approaches of a fixed $\delta$ of $21^{\prime\prime}$ and
using $\delta$ as a free parameter for the 53 µm observation. When $\delta$ is
a free parameter, as in the previous cases about 40% of the pixels still
showed $\delta<W\sqrt{2}$. Hence we chose to fix the $\delta$ value to
$21^{\prime\prime}$ obtained from a single dispersion function fit to the
whole image. There were some pixels with large error bars due to missing data
in the surrounding region within the kernel size ($w=9$). In these cases we
increased the kernel size to 11 to get better estimates of the parameters. By
using this method we were able to get reasonable values of $\langle
B_{t}^{2}\rangle/\langle B_{0}^{2}\rangle$ for most of the pixels of the
observation ($\sim 90\%$).
The plots of the single dispersion function fits to the 53 µm and 216 µm data
are shown in Fig. 9. The small-scale turbulent (first term of the dispersion
function) and the large-scale mean field (last term) components are also
shown. It can be seen from the plots that the turbulent component dominates
the dispersion when $\ell<44^{\prime\prime}$ and $\ell<1^{\prime}$ in 53 and
216 µm observations, respectively. Beyond this size there is significant
contribution from the large-scale component to the observed dispersion. These
sizes are also around the same as the kernel sizes chosen for each
observation.
The best fit values of $\langle B_{t}^{2}\rangle/\langle B_{0}^{2}\rangle$
from each observation was used to estimate $\sigma_{\phi}$ using the
approximation $\sigma_{\phi}\simeq[\langle B_{t}^{2}\rangle/\langle
B_{0}^{2}\rangle]^{1/2}$. The mean value of $\langle B_{t}^{2}\rangle/\langle
B_{0}^{2}\rangle$ was found to be 0.29, 0.019, and 0.41 for 53, 216 and 850 µm
observations respectively. The 216 µm observation shows the least dispersion
which reflects the rather uniform polarization vectors observed in Fig. 1 & 2.
This might be a result of majority of the dust emission at this wavelength
originating from a region of strong magnetic field, and will be discussed in
the following sections.
Figure 11: Maps of $\sigma_{v}$ from blue-shifted velocity components (left),
all velocity components (middle), and $\sigma_{\phi}$ (right) for the 53 µm
HAWC+ observation used to estimate the magnetic field of the region from DCF
method. Figure 12: $B_{{}_{\mathrm{POS}}}$ for the 53 µm observations
estimated using the DCF method from the blue-shifted velocity components is
shown on the left and from all the velocity components is shown in the middle.
The map on the right shows the $B$-field estimate from the DMA method. The
mean field from each map is 2.07$\pm$0.03 mG, 3.4$\pm$0.04 mG, and
2.85$\pm$0.04 mG respectively. The HAWC+ polarization vectors are overlaid on
all the maps with its scale shown at the top of the map on the left.
Figure 13: Histogram distribution of the estimated $B_{{}_{\mathrm{POS}}}$
from the 53 (left), 216 (middle), and 850 (right) µm observations.
Now, we measure the $B_{{}_{\mathrm{POS}}}$ using the modified DCF method
described in Equation 4. The mass density was determined from the gas volume
density ($n_{{}_{\mathrm{H}}}$) using the relation $\rho=\mu
m_{{}_{\mathrm{H}}}n_{{}_{\mathrm{H}}}$, where $\mu=2.8$ is the mean molecular
weight per unit mass of hydrogen (Kauffmann et al., 2008; Sadavoy et al.,
2013) and $m_{{}_{\mathrm{H}}}$ is the mass of hydrogen (Crutcher, 2004). The
volume densities were determined using the depth of the clouds estimated from
the auto-correlation function and the column density measurements from
Herschel observations. The resulting values match well with the earlier
predictions using molecular line observations ($n_{{}_{\mathrm{H}}}\simeq
10^{4.1}$ cm-3; Oka et al., 2011). More details about the estimation of
temperature and gas column density of the region can be found in Akshaya &
Hoang (2023).
The velocity dispersion described in Section. 3 was corrected for the
contribution from molecular thermal motion to extract only the dispersion due
to turbulence using the relation
$\sigma_{v}^{2}=\sigma_{v0}^{2}-k_{\mathrm{B}}T_{\mathrm{gas}}/m_{\mathrm{CO}}$,
where $\sigma_{v0}$ is the dispersion measured from the molecular
spectroscopic data, $k_{\mathrm{B}}$ is the Boltzmann constant, and
$T_{\mathrm{gas}}$ is the temperature of gas in the region. We use the same
approximation of $T_{\mathrm{gas}}=T_{\mathrm{dust}}$ as in our previous
study. We have also ignored the pixels where the best fit values of $\langle
B_{t}^{2}\rangle/\langle B_{0}^{2}\rangle<0.001$ as these are found mostly at
the boundaries of the polarization observations and might be a result of poor
fitting due to insufficient data.
The $B_{{}_{\mathrm{POS}}}$ of the 216 and 850 µm observations were estimated
using the mean $\sigma_{v}$ from all the resolved components shown in Fig. 5.
The resultant maps of the magnetic field are shown in Fig. 10 and have a mean
value of 6.53$\pm$0.08 mG and 1.39$\pm$0.02 mG for 216 and 850 µm
observations, respectively. The high magnetic field estimate of the 216 µm
observation is due to very low dispersion observed in the polarization
measurement at this wavelength, as can be seen from the rather uniform
polarization vectors in Fig. 1. A histogram of the distribution of the
magnetic field for each wavelength is shown in Fig. 13. It is important to
note that these estimates are from the DCF method without any correction
factor applied to the equation, hence the mean $B$-field strength can be
treated as an upper limit of the field at these wavelengths.
#### 4.1.1 B${}_{{}_{\mathrm{POS}}}$ from resolved velocity components
We have used a different approach to measure the $B$-field of the 53 µm
observation. As discussed in Section. 3, the morphology of the observed
emission at the 53 µm wavelength matches well with the integrated map of the
velocity components from the blue-shifted region of the CO ($J=3\rightarrow
2$) spectrum. Not taking this into account while using the $\sigma_{v}$ value
will lead to an overestimation of the $B$-field. To understand the importance
of resolving and isolating the velocity components in such complex regions we
estimate the $B$-field from two values of $\sigma_{v}$. In one case we
consider the mean $\sigma_{v}$ only from the blue-shifted Gaussian components
of the spectrum while in the other case we take the mean value from all the
resolved components. The maps of $\sigma_{v}$ for the two cases and the
$\sigma_{\phi}$ for this region is shown in Fig. 11. Similar to the previous
case, we have ignored the pixels with $\langle B_{t}^{2}\rangle/\langle
B_{0}^{2}\rangle<0.01$ at the boundaries of the polarization data. The
resulting $B_{{}_{\mathrm{POS}}}$ maps are shown in Fig. 12. There is a clear
difference in the estimated strength especially in the regions away from the
central source Sgr A∗. The mean field of the region estimated from the blue-
shifted components is about $B_{-v}^{\mathrm{DCF}}=2.07\pm 0.03$ mG and the
estimate by considering all the components is $B_{v}^{\mathrm{DCF}}=3.4\pm
0.04$ mG. The distribution of the field from both cases are shown in Fig. 13.
The field from all the components seem to show two peaks in the distribution,
at around 1 mG and 4 mG.
Figure 14: Histogram comparing the $B_{{}_{\mathrm{POS}}}$ measured from DMA
and DCF methods for the 53 µm observation, using correction factor $f=1$.
Figure 15: Correlation between the estimated difference in the
$B_{{}_{\mathrm{POS}}}$ measured from DCF and DMA method with the number of
components resolved along the line of sight (top) and the measured velocity
dispersion (bottom).
### 4.2 $\mathbf{B_{{}_{\mathrm{POS}}}}$ from Differential Measure Analysis
We also measure the $B$-field strength using another recent modification of
the original DCF method, called the Differential Measure Analysis (DMA)
technique proposed by Lazarian et al. (2022). The observed dispersion in the
polarization is an effect of the tug-off between the turbulence induced
randomisation and aligning effect due to tension in the magnetic field.
Stronger the magnetic field, lesser the randomised turbulence. The DCF method,
traditionally used for the measurement of the $B$-field from dust polarization
is based on the assumption that the turbulence in isotropic. However, MHD
turbulence is anisotropic by nature (Beresnyak & Lazarian, 2019, and
references therein) and the DMA technique was introduced by Lazarian et al.
(2022) to address this key problematic assumption of the widely used DCF
technique.
The DCF technique we have employed in the previous sections is a modified
version from the original method proposed by Houde et al. (2009), where they
constrain the dispersion in the polarization angle taking into account the
dispersion induced by the changes in the large scale orientation of the
$B$-field as well as the beam integration effects. However, as discussed in
Lazarian et al. (2022) the method still assumes the turbulence to be
isotropic, where we use the LOS velocity dispersion as an accurate description
of the POS velocity fluctuation and comparable to the observed dispersion in
the polarization angle in the POS. The DMA method corrects this assumption by
using the small scale structure function of the velocity centroids (defined as
$C(r)$) for the estimate of the velocity fluctuations instead of the line
width velocity dispersion, similar to the method proposed by Cho & Yoo (2016).
The centroid velocities give an estimate of the average of the mean velocities
from multiple individual eddies along the LOS and are also not affected by
thermal broadening, which is another parameter that contributes to the error
in the $B$-field estimates from DCF. Using the simplest form of DMA method
from Lazarian et al. (2022) and Hu & Lazarian (2023), the POS $B$-field is
given by;
$B=f\sqrt{4\pi\rho}\sqrt{\frac{D_{v}(\ell)}{D_{\phi}(\ell)}},$ (7)
where $D_{v}$ and $D_{\phi}$ are the structure functions of velocity centroid
$C(r)$ and the polarization angle within scale $\ell$ respectively. The factor
$f$ is similar to the one used in the DCF method and in our case we use $f=1$
for both DCF and DMA estimates of the $B_{{}_{\mathrm{POS}}}$. The structure
functions are estimated as,
$\begin{split}D_{v}=\langle(C(r)-C(r+\ell))^{2}\rangle,\\\
D_{\mathrm{\phi}}=\langle(\phi(r)-\phi(r+\ell))^{2}\rangle,\end{split}$ (8)
where the centroid velocity $C(r)$ is given by,
$C(r)=\frac{\int{T_{\mathrm{mb}}(r,v)vdv}}{\int{T_{\mathrm{mb}}(r,v)dv}}.$ (9)
Here $T_{\mathrm{mb}}(r,v)$ is the brightness temperature of the CO line we
have chosen with $r=(x,y)$ and $v$ its respective plane of sky position and
velocity. We have applied the structure function on the same scale as used in
the dispersion function for the DCF method. Thus we choose a kernel size of
$w=9$, keeping the estimates consistent with the DCF from the previous
section.
As discussed in Cho & Yoo (2016), the DCF method gives appropriate estimate of
the $B_{{}_{\mathrm{POS}}}$ when the number of independent fluctuations
(eddies) along the LOS is small. This is not the case for regions along the
Galactic disk, as can be seen from the spectrum shown in Fig. 4, where we
clearly have multiple features along the LOS. Thus the turbulence injection
and driving scale and the number of independent fluctuations along the LOS
become important for the DCF method, while they are taken care of by the
velocity centroid approach in the DMA method. Our approach of resolving the
different individual components which contribute to the net velocity
fluctuation along the LOS might address this problem with DCF to some degree.
We have estimated the $B_{{}_{\mathrm{POS}}}$ for the 53 µm observation from
the above described DMA method using only the blue shifted part of the
spectral cube (with $v<0$ km s-1) and the corresponding map is shown in Fig.
12. The mean magnetic field estimated from the DMA method is
$B_{\mathrm{POS}}^{\mathrm{DMA}}=2.85\pm 0.04$ mG while that from the DCF
method is $B_{\mathrm{POS}}^{\mathrm{DCF}}=2.07\pm 0.03$ mG. A histogram of
the distribution of $B_{{}_{\mathrm{POS}}}$ from both the methods is shown in
Fig. 14. Though the overall distribution of $B_{{}_{\mathrm{POS}}}$ looks
similar, there are regions where there is a large difference between them.
Fig. 15 shows the relation between the difference in the estimated
$B_{{}_{\mathrm{POS}}}$ and the number of resolved components along the LOS
($N_{\mathrm{comps}}$) and $\sigma_{v}$. The difference seems to be the
highest where the velocity dispersion is high and there was no apparent
relation between the estimated difference and the angle dispersion. The key
difference in the two methods is in the way velocity fluctuations are
measured, with the velocity centroids used in DMA being a better way to
constrain the turbulence driven fluctuations along the LOS. The overall
agreement in the measured $B_{{}_{\mathrm{POS}}}$ is a positive indication
that resolving the individual components can address some of the problems
while implementing DCF for the estimate of the magnetic field. Numerical
modelling of this approach paired with DMA can give further understanding on
how the various velocity fluctuations along the LOS resolved from molecular
spectra affect the observed polarization and will be addressed in our future
work.
## 5 Discussion
### 5.1 Alfvén Mach number ($\mathcal{M_{\mathrm{A}}}$)
We have used the 3D Alfvén Mach number ($\mathcal{M}_{\mathrm{A}}$) defined as
the ratio of the turbulent velocity and the Alfvén speed to understand the
interplay between the turbulence in the region and the magnetic field. The
relation is given by,
$\mathcal{M}_{\mathrm{A}}=\sqrt{3}\frac{\sigma_{v}}{\mathcal{V}_{\mathrm{A}}},$
(10)
where $\sigma_{v}$ is the one-dimensional non-thermal velocity dispersion
which characterises the turbulence and $\mathcal{V}_{\mathrm{A}}$ is the
Alfvén velocity given by $B_{\mathrm{tot}}/\sqrt{4\pi\rho}$, where $\rho$ is
the gas mass density and
$B_{\mathrm{tot}}=B_{{}_{\mathrm{LOS}}}+B_{{}_{\mathrm{POS}}}$ is the total
mean magnetic field which can also be approximated as
$B_{\mathrm{tot}}=(4/\pi)B_{{}_{\mathrm{POS}}}$ (Crutcher et al., 2004). The
$\sqrt{3}$ in the equation takes care of the 3D velocity dispersion
approximation from the one-dimensional estimates, assuming isotropic
turbulence in the region. A value of $\mathcal{M}_{\mathrm{A}}<1$ indicates
sub-Alfvénic condition where the magnetic field dominates the gas motion while
a value of $\mathcal{M}_{\mathrm{A}}>1$ indicates the super-Alfvénic condition
where turbulence pressure dominates over the magnetic pressure.
We have estimated $\mathcal{M}_{\mathrm{A}}$ for the HAWC+ 53 and 216 µm
observations. The JCMT 850 µm observation was not considered due to relative
low quality of the data, though can be used to get a rough estimate of the
field strength in the region are not reliable enough to draw conclusions on
the importance of turbulence on these scales. The $\mathcal{M}_{\mathrm{A}}$
estimate for 216 µm observation is sub-Alfvénic throughout the region due to
its high ordered magnetic field. The distribution of
$\mathcal{M}_{\mathrm{A}}$ is much more distinct in the 53 µm observation and
is shown in Fig. 16. $\mathcal{M}_{\mathrm{A}}>1$ along the Eastern Arm of the
mini-spiral where the $B_{{}_{\mathrm{POS}}}\lesssim 1$mG. Turbulent pressure
clearly dominates the gas motion in this region. The dust components probed by
the 53 µm observation show a distribution of sub- and super-Alfvénic
conditions, in contrast to the 216 µm observation where the regions is mostly
sub-Alfvénic.
Figure 16: Alfvén Mach number ($\mathcal{M}_{\mathrm{A}}$) estimated for the
53 µm observation. The $\mathcal{M}_{\mathrm{A}}>1$ is seen along the Eastern
Arm of the mini-spiral indicating the gas kinematics to be driven by
turbulence in the region. The HAWC+ polarization vectors are overlaid on map
with its scale shown at the top and the beamsize shown at the bottom of the
figure.
### 5.2 Mass-to-Flux ratio ($\lambda$)
The CND is known to be clumpy and is a key region to understand how the gas
mass is being fed to our central supermassive black hole (SMBH) as well as the
star formation in this region (Hsieh et al., 2018, 2021). The mini-spirals
observed within the CND are believed to be molecular gas that are in-flowing
from ambient clouds well outside the CND, and contain compact dense cores of
varies sizes as seen from Atacama Large Millimeter/submillimeter Array (ALMA)
observations (Hsieh et al., 2019). These cores need to be dense enough to
survive the tidal disruption along the path where they become part of the CND
and are eventually fed to the SMBH at the centre. An understanding of whether
these cores can lead to successful star formation in this complex and highly
dynamic environment around our SMBH has great implications for the study of
nuclear star clusters and star formation and evolution of galaxies. One of the
key parameters to probe the possible star formation in the presence of
magnetic field is the mass-to-flux ratio ($M/\phi\equiv\lambda$) which
estimates whether the magnetic field can support the cloud against
gravitational collapse. Following Crutcher et al. (2004), $\lambda$ is defined
in terms of the critical value of mass that can be supported by magnetic flux
($M_{\mathrm{crit}}=\phi_{\mathrm{crit}}/2\pi\sqrt{G}$; Nakano & Nakamura,
1978) as,
$\lambda=\frac{(M/\phi)_{\mathrm{obs}}}{(M/\phi)_{\mathrm{crit}}}=\frac{\mu
m_{{}_{\mathrm{H}}}N(H_{2})/B_{\mathrm{tot}}}{1/2\pi\sqrt{G}}=7.6\times
10^{-21}\frac{N(H_{2})}{B_{\mathrm{tot}}},$ (11)
where $\mu=2.8$ is the mean molecular weight, $m_{\mathrm{H}}$ is the mass of
hydrogen atom, $G$ is the gravitational constant, $N(H_{2}$) is the gas column
density in cm-2, and $B_{\mathrm{tot}}$ is the total magnetic field strength
in $\mu$G. A value of $\lambda>1$ indicates that the magnetic field cannot
prevent gravitational collapse of the cloud and is said to be magnetically
supercritical. If $\lambda<1$ then the cloud is magnetically supported and is
said to be magnetically sub-critical. We estimate the mass-to-flux ratio for
the 53 µm observation focused on the CND and the corresponding map is shown in
Fig. 17. Most of the region is magnetically sub-critical due to the high
magnetic field with $B_{{}_{\mathrm{POS}}}>1$ mG. Only along the Eastern Arm
where we also observe $\mathcal{M}_{\mathrm{A}}>1$ do we see $\lambda>1$
indicating the magnetic field might not be strong enough to support gravity.
The physical scale probed by the HAWC+ observation is not high enough to
resolve the dense cores observed along the CND. However, this gives us an idea
of the region of weak magnetic field, where possible star formation can be
triggered in the CND.
Figure 17: Mass-to-Flux ratio ($\lambda$) of the 53 µm observation.
$\lambda>1$ along the Easter Arm of the mini-spiral within the CND where the
estimated magnetic field is $B_{{}_{\mathrm{POS}}}\lesssim 1$ mG. The HAWC+
polarization vectors are overlaid on map with its scale shown at the top and
the beamsize shown at the bottom of the figure.
### 5.3 Grain alignment and field morphology
From our previous study of the CND and its surroundings in Akshaya & Hoang
(2023), we found that the grain alignment in this region can be perfect if the
dust grains contain even low levels of iron atoms (super-paramagnetic grains
with $\sim 20$ iron atoms per cluster). The magnetic field was assumed to be 5
mG throughout the region for the study. The results still hold with the
current detailed map of the $B_{{}_{\mathrm{POS}}}$. The least magnetic field
observed is within the CND with $B_{{}_{\mathrm{POS}}}\leq 1$ mG, where we
also observe a drop in the column density. Earlier Zeeman measurements of this
region by Plante et al. (1995) estimate similar field strength of
$B_{{}_{\mathrm{LOS}}}<1$ mG. It is interesting to note the agreement between
the two methods of $B$-field estimation. Except in the few regions at the
edges of the CND, the overall magnetic field for the 53 µm observation show
$B_{{}_{\mathrm{POS}}}\leq 5$ mG. Thus, we reinforce that if the dust grains
in the region contain even a small percent of iron atoms, we can expect
perfect alignment of dust with the magnetic field in the region where the
alignment will be driven by Magnetically Enhanced Radiative Alignment (MRAT;
Hoang et al., 2022a). But without iron atoms, the grain alignment is only
driven by radiative torques (Dolginov & Mitrofanov, 1976; Draine &
Weingartner, 1997; Lazarian & Hoang, 2007) as the field in the region is not
strong enough to enhance the alignment via magnetic relaxation. A detailed map
to the metallicity of the region can better help constrain the degree of grain
alignment and aid in the construction of a 3D morphology of the $B$-field in
this region.
Hoang et al. (2023) propose a new way to map the 3D $B$-field from the 2D dust
polarization observations based on the MRAT alignment theory, taking into
account the local physical conditions that affect the net polarization
efficiency of the dust grains. Considering the strong magnetic field in the GC
which can promote greater degree of magnetic relaxation in contrast to the
diffuse ISM, particularly in the presence of iron clusters in the dust grains,
it would be insightful to apply this method to derive a complete picture of
the variation of the $B$-field along the LOS and how it effects the transport
of material in this region and will be addressed in a future study.
### 5.4 3D magnetic field from multi-wavelength thermal dust polarization
The major focus of this work has been to map the magnetic field of the GC in
multi-wavelengths and to test if regions such as the GC with multiple
resolvable velocity fluctuations along the LOS can be used to create a picture
of how the $B_{{}_{\mathrm{POS}}}$ changes at different depths, assuming the
different wavelengths are probing different layers at varied temperatures. We
believe we have achieved this to some extent given the limitations of the DCF
method used to determine the magnetic field. The estimated
$B_{{}_{\mathrm{POS}}}$ is in good agreement with the earlier Zeeman
measurements of $B_{{}_{\mathrm{LOS}}}$ from Killeen et al. (1992) and Plante
et al. (1995). Killeen et al. (1992) obtained $B_{{}_{\mathrm{LOS}}}\sim 2$ mG
in the north and the south of the CND while Plante et al. (1995) suggested the
$B_{{}_{\mathrm{LOS}}}<1$ mG within the CND. Assuming the dust polarization
and Zeeman measurements trace similar regions along the CND, using our POS
magnetic field from DCF of $B_{{}_{\mathrm{POS}}}^{\mathrm{mean}}\sim 2.07$
mG, we can estimate an approximate mean full strength of 3D $B$-field of the
CND as;
$\displaystyle
B_{{}_{\mathrm{3D}}}=\sqrt{B_{{}_{\mathrm{POS}}}^{2}+B_{{}_{\mathrm{LOS}}}^{2}},$
(12)
which yields $B_{{}_{\mathrm{3D}}}\sim 2.87$ mG.
The observed polarization in the 53 µm appears to be independent of the deeper
velocity structures identified as shown in Fig. 5, but the 216 and 850 µm
observations seem to trace the relatively cooler dust from different depths
along the LOS. Numerical simulations can be used to investigate if combining
such multi-wavelength polarization observations with the information of the 3D
distribution of dust along the LOS can be used to create a 3D morphology of
the magnetic field. As an extension of this study, we plan to use the recently
upgraded POLArized RadIation Simulator (POLARIS; Reissl et al., 2016) by Giang
et al. (2023), which incorporates the latest grain alignment theories
discussed in our previous work (Akshaya & Hoang, 2023) to model how the
distribution of independent velocity fluctuations affect the observed
polarized emission at different wavelengths in this region. Combined with the
limited $B_{{}_{\mathrm{LOS}}}$ Zeeman measurements, this might be a step
closer to getting a comprehensive view of the elusive 3D morphology of the
magnetic field in the Galactic disk. Due to the nature of the grain alignment
and how sensitive it is to the local physical conditions like the metallicity,
temperature, dust composition, local density, and the amount of incident
radiation, we need to use the latest knowledge of the dust alignment physics
to get a full picture of the dynamical interaction between the magnetic field
and the material in this complex region.
### 5.5 Implications of B-field strength from multi-wavelength polarization
at the GC
The CND is well known to be in the influence of the gravitational potential of
the Sgr A∗ and also play a role in the accretion of material onto the inner
sub-parsec scale ionized cavity surrounding Sgr A∗ (Solanki et al., 2023).
There is also evidence of collisional interaction between the 20 km s-1 cloud
and the outer edges of the CND, which might aid further mass accretion onto
the CND and the inner cavity (Takekawa et al., 2017). The evolution studies of
the CND suggest its formation due to interaction and break down of giant
molecular clouds with the gravitational potential of the Sgr A∗, with two
molecular clouds (20 km s-1 and 50 km s-1 clouds) ideally located at the
Galactic south of the CND (Sanders, 1998; Oka et al., 2011; Mapelli & Trani,
2016). The CND is also observed to have an extended feature in the negative
Galactic longitude direction called the negative-longitude extension (NLE;
Serabyn & Guesten, 1986; Sutton et al., 1990; Takekawa et al., 2017). This is
a foreground (with respect to the CND) feature and we observe a similar
structure in the velocity range of $-50>v>-165$ km s-1 shown in Fig. 18.
Considering the similar $B_{{}_{\mathrm{POS}}}$ estimates for the 53 µm
observation covering the CND and the 850 µm observation with the CND, NLE, and
the 20 km s-1 features, the 850 µm maybe tracing the CND and its foreground
material at a lower temperature. This cannot be confirmed due to the low data
quality of the SCUPOL observation. However, it is interesting nonetheless to
disentangle the source location of the magnetic field to get a view of how it
varies with the depth along the LOS. The 216 µm observations measure the
highest magnetic field out of the three observations that we have considered.
From Fig. 5 it is evident that there is some degree of interaction between the
cloud clusters in the negative velocity region but there is no interaction
between the $v<0$ km s-1 and $v>0$ km s-1 cloud features. Also by looking at
Fig. 6, majority of the emission in the 216 µm observation matches well with
the component morphologies at $v>0$ km s-1, unlike the 53 µm intensity which
match well with the integrated morphologies from $v<0$ km s-1. If the observed
dust emissions are arising from these proposed components, then the strong
magnetic field of the 216 µm observation can be attributed to the velocity
components in the background with respect to the CND. The uniformity in the
polarization vectors at this wavelength have been noted by earlier observation
at 250 µm by the PILOT experiment as well (Mangilli et al., 2019). Considering
our data quality check and the agreement between the observations taken by
other instruments and in different modes, this uniformity of polarization can
be treated as a real physical effect, arising due to dust grains being
perfectly aligned with a strong magnetic field.
Modelling and numerical simulations are necessary to further confirm this idea
of the different field strength observed from multi-wavelength polarization
could be a result of dust being traced at different depths along the LOS and
will be addressed in our future work. The growth of the CND and subsequent
feeding of material into the central black hole is an ongoing study with most
of the simulation not yet considering the effect of the magnetic field in the
material dynamics due to the complexity of the region. An overview of the
strength and 3D morphology of the $B$-field in the region that can be
estimated from polarization observation, complemented with the current grain
alignment physics is a great way to further our understanding of the material
transport at the centre of our Galaxy.
Figure 18: The NLE (black circle) observed in CO ($J=3\rightarrow 2$) spectra
evident in the region covered by the JCMT/SCUPOL observation, right below the
CND and above the location of the 20 km s-1 cloud (shown as the magenta
circle). The component appears in the blue shifted velocity range of
$-50>v>-165$ km s-1.
## 6 Summary
We have used thermal dust polarization observations at 53, 216, and 850 µm
combined with the spectrum of the CO ($J=3\rightarrow 2$) transition to map
the POS magnetic field for a region of about 30 pc around the centre of our
Galaxy. The main conclusions from our study are as follows;
1. 1.
The velocity dispersion in the region can be decomposed into multiple
physically distinct components with an average dispersion of about $\sim 9$ km
s-1.
2. 2.
The physical morphologies of the negative velocity structures match best with
the observed morphology of the CND and the mini-spirals at 53 µm, indicating
that these structure are at a much higher temperature than those at positive
velocities. This also shows that the velocity dispersion at this wavelength
has the contribution arising only from the negative velocity components.
3. 3.
We used the DCF method to estimate the map of the $B_{{}_{\mathrm{POS}}}$ for
all the three observations and found the mean field to be $2.07\pm 0.03$,
$6.53\pm 0.08$, and $1.39\pm 0.02$ mG at 53, 216, and 850 µm respectively.
4. 4.
Most of the region encompassing the CND and the mini-spiral is sub-Alfvénic
with $\mathcal{M}_{\mathrm{A}}<1$ except along the Eastern Arm of the mini-
spiral where $\mathcal{M}_{\mathrm{A}}>1$, indicating the gas motion being
driven by turbulence.
5. 5.
The same region also has a mass-to-flux ratio of $\lambda>1$ indicating
magnetic field cannot prevent gravitational collapse. The CND is known to have
several clumps with $n_{{}_{\mathrm{H}}}>10^{7}$ cm-3, though these scales are
not probed in the current observation the map of $\lambda$ shows the likely
region for star formation in the CND.
6. 6.
We find good agreement between the $B_{{}_{\mathrm{POS}}}$ estimated from DCF
and DMA method for the 53 µm observation, indicating that the decomposing of
the spectra into its constituent multiple velocity components might overcome
the drawback of the velocity dispersion used in the DCF method, where
determining the turbulence driving scale and the number of independent
fluctuations along the LOS become crucial to overcome the overestimation of
the $B$-field.
7. 7.
The similarity in the estimated $B_{{}_{\mathrm{POS}}}$ for the 53 µm and the
850 µm observation might be due to dust emission in these wavebands tracing
the same components, mostly including the CND and its foreground, where as the
216 µm observation might have majority of its emission coming from a component
in the background with respect to the CND, with a much stronger magnetic
field.
## Acknowledgements
MSA thanks Dr. Lopez-Rodriguez and Dr. G. S. Pillai for the insightful
discussion on the polarization data quality assessment. This work was partly
supported by a grant from the Simons Foundation to IFIRSE, ICISE (916424,
N.H.). This study is based in part on observations made with the NASA/DLR
Stratospheric Observatory for Infrared Astronomy (SOFIA). SOFIA is jointly
operated by the Universities Space Research Association, Inc. (USRA), under
NASA contract NNA17BF53C , and the Deutsches SOFIA Institut (DSI) under DLR
contract 50 OK 2002 to the University of Stuttgart. This work made use of
Astropy:444http://www.astropy.org a community-developed core Python package
and an ecosystem of tools and resources for astronomy (Astropy Collaboration
et al., 2013, 2018, 2022). This research made use of APLpy, an open-source
plotting package for Python (Robitaille & Bressert, 2012).
## Data Availability
The data underlying this article will be shared on reasonable request to the
corresponding author.
## Appendix A Supplementary Figures
Figure 19: Maps of the dispersion in velocity (left) and polarization angle
(right) used for the estimation of the $B_{{}_{\mathrm{POS}}}$ from DCF method
for the HAWC+ 216 µm observation.
Figure 20: Maps of the dispersion in velocity (left) and polarization angle
(right) used for the estimation of the $B_{{}_{\mathrm{POS}}}$ from DCF method
for the SCUPOL 850 µm observation.
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|
# Swarm Learning: A Survey of Concepts, Applications, and Trends
Elham Shammar
School of Cyber Science and Engineering,
Wuhan University, Wuhan, China
<EMAIL_ADDRESS>
Xiaohui Cui
School of Cyber Science and Engineering,
Wuhan University, Wuhan, China
<EMAIL_ADDRESS>
Mohammed A. A. Al-qaness
College of Physics and Electronic
Information Engineering
Zhejiang Normal University
Jinhua 321004, China
<EMAIL_ADDRESS>
###### Abstract
Deep learning models have raised privacy and security concerns due to their
reliance on large datasets on central servers. As the number of Internet of
Things (IoT) devices increases, artificial intelligence (AI) will be crucial
for resource management, data processing, and knowledge acquisition. To
address those issues, federated learning (FL) has introduced a novel approach
to building a versatile, large-scale machine learning framework that operates
in a decentralized and hardware-agnostic manner. However, FL faces network
bandwidth limitations and data breaches. To reduce the central dependency in
FL and increase scalability, swarm learning (SL) has been proposed in
collaboration with Hewlett Packard Enterprise (HPE). SL represents a
decentralized machine learning framework that leverages blockchain technology
for secure, scalable, and private data management. A blockchain-based network
enables the exchange and aggregation of model parameters among participants,
thus mitigating the risk of a single point of failure and eliminating
communication bottlenecks. To the best of our knowledge, this survey is the
first to introduce the principles of Swarm Learning, its architectural design,
and its fields of application. In addition, it highlights numerous research
avenues that require further exploration by academic and industry communities
to unlock the full potential and applications of SL.
_Keywords_ IoT, Blockchain, Swarm Learning; Edge Computing, Security,
Decentralized Machine Learning, Federated Learning, Privacy Preservation
## 1 Introduction
The next five years are expected to witness a significant increase in the
number of IoT devices. In 2019, the healthcare sector utilizes one-third of
all IoT devices, which are expected to climb to 40%, or $6.2 trillion, of the
total global IoT technology market value by 2025 [1]. The global adoption of
IoT devices is expected to reach 29 billion by 2030, covering a wide range of
economic sectors and disciplines [2]. Particularly, IoMT devices are poised to
save $300 billion, predominantly in the chronic illness and telemedicine
sectors. This market is considered attractive for investors, with projections
that estimate revenues of $135 billion by 2025 [3]. Moreover, the global
healthcare market is expected to grow to $6.2 trillion by 2028 [4],
necessitating advancements in AI, resource management, data processing, and
knowledge mining. The rapid advancement of the 5G standard and Multi-Access
Edge Computing (MEC) has markedly improved productivity [5].
Modern deep learning models are raising concerns about privacy and security
due to their reliance on centralized servers to store large datasets [6].
Although cloud-based local learning allows some level of collaboration and
improvement of results, it introduces several inherent challenges to this
centralized approach, such as data redundancy, increased data traffic, and
increased security and privacy risks. Two primary challenges associated with
traditional centralized learning methods are data ownership and privacy [4].
Federated learning (FL) emerges as a viable solution to these challenges,
potentially aligning with data protection standards that could conflict with
traditional centralized learning approaches [7]. FL promises notable
improvements in security, fairness, and transparency, setting a new benchmark
for digital data management and model training [8].
FL facilitates collaborative learning that preserves privacy. It addresses
central data storage issues by allowing the raw data to remain on local
devices at each participating node [5],[9],[10]. However, FL is still
vulnerable to sophisticated cyber threats, including membership inference and
data reconstruction attacks, which pose significant risks of data breach. FL
also has limitations in network bandwidth that cause delays. To mitigate these
vulnerabilities, two approaches are introduced: 1) Distributed FL (DFL [11])
and 2) a novel approach called Swarm Learning (SL) that was developed in
collaboration with Hewlett Packard Enterprise (HPE[6]).
DFL and SL are approaches to machine learning that improve privacy and reduce
reliance on centralized data storage. DFL extends the traditional federated
learning model by allowing multiple nodes to train models collaboratively
without a central server[12],[13], [14], while Swarm Learning uses blockchain
technology to create an autonomous peer-to-peer network without a central
authority. Both approaches aim to decentralize learning and enhance privacy,
but SL employs blockchain for even greater security and decentralization.
SL is a decentralized machine learning framework that combines the principles
of blockchain technology with federated learning. Instead of using a central
server to compile model updates as in standard FL, SL uses a peer-to-peer
network that is managed by blockchain to guarantee member validity, data
integrity, and security. SL trains models locally, and only parameter weights
are transmitted on a network of numerous swarm devices. The integration of
blockchain technology ensures secrecy and security, enabling effective
collaboration among disparate entities. Transactions can only be performed by
preauthorized parties through computationally efficient consensus mechanisms.
SL eliminates the need for a central server, reducing the risk of single
points of failure and centralized data breaches. Unlike FL, which ensures data
privacy through aggregating initial local gradients, SL facilitates data
sharing among registered customers via smart contracts, thus preserving data
privacy. A node in SL must undergo registration, authentication, model
retrieval, local training, gradient sharing, and finally, result aggregation
using the Federated Average method [15].
SL enhances fault tolerance, reduces vulnerability to attacks, and supports
scalability, making it ideal for applications requiring high data privacy and
system robustness, such as healthcare, the automotive industry, financial
services, smart cities, edge computing, IoT, and the metaverse. In healthcare,
SL guarantees the preservation of data privacy by allowing hospitals and
research institutions to train models collaboratively without sharing
sensitive patient data [16]. In the industry, SL enables machines and system
components to act as individual learning agents, allowing real-time decision-
making and adjustments without central oversight. It aligns well with Industry
4.0 principles, supporting advanced manufacturing technologies requiring high
levels of data integrity, flexibility, and automation [17], [18], [19], [20].
In financial services, SL can enhance fraud detection systems by learning
transaction data between different entities without compromising client
confidentiality [21]. In smart cities, SL can optimize traffic flow and public
transport management by allowing multiple sensors and nodes to learn and adapt
to real-time traffic conditions. SL supports data sovereignty and
auditability, ensuring compliance with data protection regulations. It also
offers innovation and competitive advantage, allowing faster time to market
and customization[22].
HAN et al. [23] sought to bridge the gap between the theoretical aspects of SL
and its practical application, providing empirical evidence through
experiments carried out on three public datasets. Their findings have
evidenced that SL is supposed to be suitable for most application scenarios,
no matter whether the dataset is balanced, polluted, or biased over irrelevant
features. However, challenges remain, such as backdoor attacks against SL,
managing blockchain integration complexity, and dealing with computational
overhead.
### 1.1 Paper objectives and contribution
The considerable advantages offered by SL require a detailed examination to
understand its current research landscape and practical applications, as well
as to pinpoint areas requiring further improvement. To this end, this SL
survey aims to investigate its capabilities within decentralized learning
environments. Our objectives are to assess its practical implementation,
identify both technical and operational challenges, and highlight potential
avenues for future innovations. Furthermore, the survey seeks to explore
forward-looking developments, such as the integration of advanced
cryptographic techniques to enhance security and the adaptation of SL to
support emerging technologies such as edge computing and the IoT.
This effort will consolidate existing knowledge, clarify research gaps, and
outline strategic directions to expand the adoption of SL. As a resource, this
survey will be invaluable for scholars, researchers, and practitioners. By
improving academic discourse and guiding practical implementations, it aims to
pave the way for a broader application and optimization of SL in various
industries, thus expanding its impact and utility.
To sum up, the main contribution of this paper can be presented as follows:
* •
We present the first survey paper in the field of swarm learning (SL). To the
best of our knowledge, this literature review is the first review on SL.
* •
We provide a comprehensive overview of the existing literature on Swarm
learning and its current applications to give readers a complete picture of
this new and promising research direction.
* •
We studied and analyzed the current applications of SL. We categorized them
into healthcare, transportation, industry, robotic systems, smart homes,
financial services, multimedia IoT, fake news detection, and Metaverse.
* •
We present an in-depth analysis of the current limitations and challenges
facing SL. We explore how these issues impact their development and
deployment. Additionally, we discuss potential future directions to improve SL
technologies and applications. We suggest paths for advancement and areas ripe
for further research to enhance the effectiveness and applicability of SL
technologies.
The road map of this paper, as shown in Fig.1 is outlined as follows: Section
2 provides an introduction to swarm learning and its fundamental concepts and
components. Sections 3 and 4 explore the applications of swarm learning and
its associated challenges, respectively. Section 5 highlights potential
directions for future research in SL. The paper is concluded in Section 6.
Figure 1: Paper Structure
### 1.2 Paper Selection
We conducted a comprehensive search in six databases, namely IEEE, PubMed,
Science Direct, Scopus, Springer, and Web of Science. Specifically, we
retrieved 25 papers from IEEE, 10 from PubMed, 116 from Science Direct, 72
from Scopus, 28 from Springer, and 43 from Web of Science. Subsequently, we
meticulously screened these papers, focusing on those directly related to
swarm learning, while excluding articles on swarm intelligence and swarm
optimization. Following this screening process, we identified a total of 56
papers that met our inclusion criteria.
The number of research papers has increased each year, as shown in Fig. 2.
Research on SL has steadily increased since its humble beginnings in 2020. By
2024, it experienced a significant increase, indicating the growing importance
of SL in various fields. This surge highlights the growing interest of the
academic community in exploring and maximizing the potential of this advanced
technology. The surge in SL research is driven by advances in computational
power, data availability, the proliferation of IoT devices, privacy-preserving
AI techniques, and the emergence of complex problems such as healthcare,
autonomous driving, and smart cities, which require scalable and decentralized
learning methods.
Figure 2: Annual increase in the number of Swarm Learning research papers.
### 1.3 Research Questions
1. 1.
What are Swarm Learning concepts, architecture, and components?
2. 2.
What is the difference between Swarm Learning and Federated Learning,
Distributed FL/Decentralized Federated Learning, and Swarm Intelligence?
3. 3.
What are the applications of swarm learning?
4. 4.
What challenges do we see in the adoption and implementation of swarm learning
in real-world applications?
## 2 Swarm Learning (SL)
SL is a decentralized machine learning framework that enables the training of
the on-device model without the need to transfer raw data. In the SL model,
the data is kept localized at the data owner’s site, substantially reducing
data traffic by avoiding the transmission of raw data [24]. Using blockchain
technology, SL enhances privacy and security through the exchange of only the
model parameters and weights, not the actual data itself. This approach
incorporates smart contracts to manage the training and updating of the
decentralized machine learning models using local user data, distinguishing it
significantly from traditional centralized systems or even FL frameworks that
rely on a central server for aggregating model updates [21]. Additionally, SL
incorporates advanced data privacy and security mechanisms, making it an
ideal, flexible, and secure solution for content caching within contemporary
network architectures [25].
SL employs a permissioned blockchain network and a decentralized hardware
infrastructure to facilitate secure member onboarding, dynamic leader
election, and efficient merging of model parameters. The system utilizes
standardized AI engines within a distributed machine learning context to
ensure secure and reliable operations. An SL library supports an iterative AI
learning process that leverages decentralized data, adhering rigorously to the
prevailing privacy and security standards [26]. This structured approach
secures data and also streamlines the computational process across diverse
network nodes.
### 2.1 Swarm Learning Architecture
The SL architecture encompasses two primary layers: the application layer and
the infrastructure (or hardware) layer. The application layer includes the
Machine Learning (ML) platform, blockchain, and the Swarm Learning Library
(SLL). The hardware layer consists of data sources and models relevant to
specific domains, such as datasets related to missions or geographic locations
[4].
The SL system consists of two components: Swarm edge nodes and Swarm network
(blockchain) [24]. With blockchain technology, SL has the following
characteristics and advantages: (1) storing vast amounts of data locally; (2)
reducing data traffic by not requiring the exchange of original data; (3) not
requiring a secure central network; (3) offering high-level data security and
shielding the model from attacks; and (5) allowing all members to merge
parameters with equal rights [26].
Fig. 3 [6],[16], [27] depicts the architecture of the swarm learning system.
There are several swarm edge nodes (let us say, M nodes), and each node Ci
uses local private data Di, i = 1, 2, 3,…, M, to train its model Li after
downloading an initial model from the network. Then, every node Ci distributes
its model parameters throughout the network. These nodes are recognized,
permitted, and registered with a smart contract in a peer-to-peer blockchain
network to safeguard network security. Each node Ci has an opportunity to be
chosen as a temporary leading node C for model aggregation in a training cycle
t. When the local model Li is trained to satisfy predetermined synchronization
requirements (such as a predetermined training batch), several chosen nodes
will disclose their model parameters to a storm API. As a result, each chosen
node will get the global model parameters from the leading node C, which will
then use a weighted average approach to aggregate them into a global model G
[6].
Figure 3: Swarm learning system architecture
The Swarm network diagram shows how edge nodes are set up to exchange
parameters to learn, with blockchain technology serving as a facilitator.
Private data is used at each node in combination with models provided by the
Swarm network, guaranteeing a decentralized and secure method of collaborative
learning [16]. To take part in model training, Swarm edge nodes must register
via the blockchain’s smart contract. After registering, every node uniformly
downloads the first global model from the blockchain and trains the local
model using its local data. Swarm edge nodes upload the local model parameters
of the training to the leader via the Swarm network. The smart contract on the
blockchain selects the Swarm edge node leader in real-time. The leader will
average the collected local model parameters. To continue local model
training, each Swarm edge node will download the aggregate model from the
Swarm network until the aggregation model meets the requirements of the
trained aggregation model. If not, the leader in a block generates the
aggregation model [24].
The workflow for updating the model in SL, as shown in Fig.4 [28], consists of
two primary stages. Initially, individual organizations trained their local
models and updated them using their own SL nodes. These updates are then
consolidated on their respective permissioned blockchains. In the subsequent
stage, organizations use a network of multiple blockchains to further refine
their local models and synchronize the global model’s state. This approach of
sharing models across various blockchains fosters a more decentralized SL
process and mitigates security risks from external entities [28].
Figure 4: Workflow of SL with multiple permissioned blockchains. The chains of
different colors belong to different participating organizations
In SL, model sharing is seen as a data-transfer process among participating
blockchains. The challenge lies in creating a method for blockchain data
interplay that remains consistent and secure and is adaptable to various
blockchain types without altering the core operations. The diversity in
blockchain structures and consensus protocols used by different organizations
adds to the complexity of enabling interblockchain interactions. Traditional
methods of cross-chain communication, which often rely on a third-party trust
entity, contradict the decentralized nature of SL and are therefore not
suitable. Solutions such as the Cosmos architecture, which relies on a central
hub for blockchain interoperability, also fall short of the ideal
decentralized approach required for SL [28].
### 2.2 Leader Election Algorithm (LEA)
In SL, the fairness and performance of the network are greatly affected by the
leader election process. Swarm edge nodes in SL are best placed on instances
with plenty of bandwidth and processing power to handle the demands of
decentralized decision-making. However, the unfairness of the leader election
mechanism could cause nodes to use excessive amounts of bandwidth, which would
result in inefficiencies and possibly bottlenecks. Participants may be unhappy
with this discrepancy because they believe it is unfair and because nodes with
higher data traffic may be more easily targeted by attackers[23]
The current LEA speculated to be a Proof of Stake (PoS), relies on leadership
election on nodes’ stakes or account balances. The authors in [23] recommended
switching from PoS to a Proof of Work (PoW) model, in which nodes compete to
solve cryptographic puzzles and leadership is established by meeting
predetermined hash value requirements. By equating the likelihood of becoming
a leader based on processing power, this technique seeks to guarantee a more
fair distribution of network load among nodes. Future efforts will focus on
collaborating with Hewlett Packard Enterprise (HPE) to enhance the fairness
and effectiveness of LEA in SL.
### 2.3 Concept of Swarm Learning
ML, in theory, can be carried out locally if enough data and computing
equipment are available. The data and computation existed at different,
disconnected locations (Fig. 5 (A) [16]). In cloud-based computation, data are
transported centrally (Fig. 5 (B)[16]) so that centralized computing can be
used to perform machine learning. It greatly improves the amount of data
available for training, and thereby improves machine learning outcomes.
However, there are some disadvantages, such as increased data traffic and
duplication, as well as problems with data privacy and security. In FL,
parameter settings are managed by a central parameter server, while data
remain with the data owner/contributor, and computing is performed at the
location of local data storage and availability. Dedicated parameter servers
are in charge of gathering and dispersing local learning in FL (Fig. 5 (C)
[16]). Alternatively, SL eliminates the need for a dedicated server, as shown
in Fig. 5 (D).SL distributes the parameters over the swarm network and
develops the models separately at each location using private data [16].
The integration of ML methods into the SL framework can increase training
rates. SL’s decentralized nature allows local data processing at edge nodes,
reducing latency, and potentially speeding up the training process. It also
leverages the computational power of multiple decentralized nodes, improving
training speed. SL reduces communication overhead by distributing workloads
across multiple nodes, reducing the need for frequent communication between
nodes. The blockchain component in SL manages model updates securely and
efficiently, minimizing delays. Dynamic leader elections optimize the training
process by choosing the most capable nodes for crucial tasks. SL’s approach to
handling non-IID data across different nodes can enhance model robustness and
accuracy faster than centralized approaches. SL’s ability to operate on nodes
with varying computational capacities allows for resource optimization[16].
However, integrating ML methods into SL can introduce complexities, making it
difficult to analyze training rate improvements. Traditional machine learning
methods can vary in architecture and complexity, affecting learning rates,
convergence behaviors, and efficiencies. SL’s decentralized nature and varying
computational resources may affect efficiency and scalability. Blockchain
technology for synchronization may introduce overhead, and adjusting ML
methods to fit SL could complicate performance assessment. Empirical studies
and benchmarking against traditional centralized and federated learning
systems are needed to quantify the benefits of SL in real-world scenarios.
Figure 5: Comparative overview of learning models
### 2.4 Swarm Learning Components
As shown in Fig. 6 [29], the SL framework consists of various nodes:
* •
Swarm Learning (SL) node: SL nodes run the core of SL, sharing learnings and
incorporating insights.
* •
Swarm Network (SN) node: Using the Ethereum blockchain, the SN nodes
communicate with each other to track training progress and save global state
information about the model. Additionally, during initialization, every SL
node registers with an SN node, and each SN node manages the training pipeline
for its corresponding SL nodes. Note that the model parameters are not
recorded by the blockchain; instead, it simply stores metadata such as the
model state and the training progress.
* •
Swarm Operator (SWOP) nodes: SWOP nodes manage SL operations, performing tasks
such as starting and stopping Swarm runs, building and upgrading ML
containers, and sharing models for training.
* •
Swarm Learning Command Interface (SWCI) nodes: SWCI nodes monitor the
framework and can connect to any SN node in a given framework.
* •
Swarm Learning Management User Interface (SLM-UI): SLM-UI nodes are GUI
management tools used to install the framework, deploy Swarm training, monitor
progress, and track past runs[29].
* •
SPIFFE SPIRE Server node: SPIFFE SPIRE Server node ensures the SL framework’s
security. A SPIRE Agent Workload Attestor plugin is included in each SN or SL
node, and it interacts with the SPIRE Server nodes to verify the identities of
each node and to get and maintain an SPIFFE Verifiable Identity Document
(SVID) [23].
* •
License Server (LS) node installs and manages the license to run the SL
framework[23].
SL security and digital identity are handled by X.509 certificates, which can
be generated by users or standard security software like SPIRE. SL components
communicate using TCP/IP ports, and participating nodes must be able to access
each other’s ports[29].
Figure 6: Swarm learning Components
### 2.5 Features of Swarm Learning
Swarm learning encompasses several distinct features that strengthen its
application in decentralized settings:
1. A.
Privacy Preservation: SL keeps data at each node which minimizes the risk of
privacy breaches and confidentiality.
2. B.
Decentralization: SL reduces the risk of a single point of failure or data
monopoly by eliminating the need for a central data storage or authority for
model aggregation.
3. C.
Continuous Learning: Models are continuously updated with new data available
at each node, adapting to new conditions such as emerging diseases.
4. D.
Data Diversity and Volume: SL handles larger and varied datasets from multiple
nodes, enhancing model robustness and generalization.
5. E.
Collaborative Learning: Nodes collaborate to train a shared model, benefiting
from shared insights without actual data transfer, crucial to maintaining
patient confidentiality.
### 2.6 Swarm Learning vs. Federated Learning
SL and FL are two distributed learning techniques that provide aggregation of
cooperative models from numerous participating nodes [30],[31]. Several
training rounds will result in the generation of a global model. Furthermore,
to guarantee equitable and safe model aggregation, these participating nodes
are not required to disclose their proprietary datasets. However, there are
two key distinctions between them [26], [32].
* •
Information transmission: In FL, participating nodes and the central server
exchange local model parameters as well as global model updates. However, in
SL, peer-to-peer networks based on blockchain technology and edge computing
work together to ensure that participating nodes can transmit safely and
fairly without the need for central server coordination [6], [33],[34], [35].
* •
With or without a central server: In FL, a central server is utilized to
collect model parameters from involved nodes and employ model aggregation to
generate a global model. On the other hand, SL does not make use of a central
server. During each training cycle, every participating node has the
opportunity to be randomly selected to serve as a temporary server to compile
model modifications. [6]. Using a blockchain-based Swarm network for safe and
decentralized parameter exchange and aggregation of the model, SL eliminates
the need for a central server [26].
SL addresses several key issues in FL and provides many benefits in security,
privacy, and scalability. SL can envision ways to develop more secure,
private, and faster distributed machine learning applications from different
domains.
To tackle the gradient leakage and data privacy issues in FL, Madni et al [15]
developed a secure, collaborative, and decentralized framework for machine
learning training by combining blockchain technology with SL. SL protects the
privacy of the data and the secrecy of the model parameters without unattended
accesses and guarantees data integrity, since it authenticates only trusted
nodes and deploys blockchain mechanisms. Research has been conducted against
common machine learning approaches for anomaly detection, where it is
demonstrated that the SL method gives better precision than current methods
and addresses gradient leakage, which is the current major limiter of the FL.
In their two articles [36], [37], Xu et al. addressed issues such as data
heterogeneity, security, and communication bottlenecks in FL by creating a
strong edge learning framework for smart IoT devices. They presented a new
technique called Communication-Efficient and Byzantine-Robust Distributed
Swarm Learning (CB-DSL). This work is the first thorough theoretical
examination of FL in conjunction with PSO (particle swarm optimization). It
provides a closed-form formula to assess the projected convergence rate of CB-
DSL, which makes it superior to traditional FL approaches such as Federated
Averaging (FedAvg). It also offers a model divergence analysis to assess the
possible advantages of adopting a globally shared dataset for enhancing
learning outcomes in non-IId. situations.
### 2.7 Swarm learning vs Distributed FL/Decentralized FL
Decentralized FL and SL are two approaches to distributed machine learning
that combine edge computing, blockchain technology, and peer-to-peer
networking [38]. Decentralized FL eliminates the need for a central server,
allowing for peer-to-peer communication and a more structured system, such as
blockchain technology [39]. It also includes a consensus mechanism for
updating the global model [40]. SL, developed by HPE Enterprise, integrates
blockchain technology into its core operation, ensuring data integrity, node
authenticity, and traceability. It also improves data privacy by keeping the
data localized and maintaining security through cryptographic measures.
Decentralized FL involves various nodes working together to train a global
model without a central coordinator, while Swarm Learning uses a leader
election mechanism to aggregate updates and update the blockchain. Both
approaches aim to decentralize the machine learning process and maintain data
localization, but SL incorporates blockchain for security and dynamic network
management. Both approaches are suitable for environments requiring high
levels of data integrity and auditability. SL offers advantages such as
enhanced privacy and security but may face challenges in privacy preservation
and server-centric issues. Future research could explore empirical comparisons
and develop hybrid models that combine the strengths of both SL and FL.
Beltrán et al.[12] explored the evolution of Decentralized Federated Learning
(DFL) compared to Centralized Federated Learning (CFL), highlighting its
benefits like improved fault tolerance and scalability. They compared DFL
frameworks and their implementation in various applications, including
healthcare, Industry 4.0, mobile services, military uses, and vehicular
networks. Hallaji et al.[41] explored the security and privacy of DFL,
highlighting its robustness and potential threats. They emphasized the need
for comprehensive security analyses and ongoing research to mitigate inherent
risks in DFL systems. The integration of blockchain technology with
decentralized federated learning (DFL) has been surveyed by Zhang et al.[11],
[42], highlighting its operational workflow and applications in the IoT and
Internet of Vehicles (IoV) domains. It discusses challenges like communication
overhead and system complexity, recommending further research.
The choice between DFL and SL depends on the application’s specific
requirements, such as security, trust requirements, operational complexity,
regulatory compliance, scalability, flexibility, real-time performance, data
privacy, and cost implications. SL is ideal for fields like healthcare and
finance, where data breaches or tampering could have severe consequences. DFL
is suitable for scenarios where operational complexity and resource
availability are concerns. SL is more suitable for highly regulated
environments and requires strict data provenance and audit trails. DFL offers
better scalability and flexibility, while SL may offer enhanced security
features. However, the implementation and maintenance of a blockchain for SL
can be more costly.
### 2.8 Swarm learning vs swarm intelligence
Swarm intelligence is a branch of artificial intelligence that uses the
principles of basic agent behavior research to provide algorithms for
scheduling, routing, and optimization issues. Particle Swarm Optimization
(PSO), Bee Colony Optimization (BCO), and Ant Colony Optimization (ACO) are
examples of swarm intelligence algorithms. In contrast, SL is a subset of
machine learning that focuses on distributed and dedicated learning without
sharing raw data. SL emphasizes decentralized and collaborative machine
learning in a privacy-preserving manner, while swarm intelligence focuses on
problem solving and process optimization, drawing natural influences from a
variety of systems.
Although SL and Swarm Intelligence have related names and are inspired by
natural swarm behaviors, which may be confusing, it is important to compare
them because, in computational and system design settings, they serve distinct
purposes and operate on different principles within computational and system
design contexts. Exploring the intersections and differences between SL and
swarm intelligence can lead to the development of hybrid approaches that
leverage the strengths of both. By comparing SL and swarm intelligence,
researchers can identify new application areas that may benefit from either
approach or a combination of both, aiding in educational and research
development. Ultimately, comparing SL and swarm intelligence enhances the
deployment of these technologies effectively across various domains.
SL provides an innovative set of effective solutions to the difficulties of
conventional optimization algorithms in swarm intelligence. By addressing
these issues, Swarm Learning overcomes the limitations of traditional
optimization algorithms in swarm intelligence and also opens new possibilities
for solving complex, dynamic, and large-scale optimization problems in a
secure, efficient, and privacy-preserving manner.
The Bacterial Foraging Optimization (BFO) algorithm, introduced by Kevin M.
Passino in 2002, is a nature-inspired optimization technique based on E.
coli’s natural foraging behavior. It has been applied in various fields,
including engineering, control systems, and optimization problems. However,
BFO has limitations depending on the problem’s nature and implementation
details, and its performance may not be ideal in all cases. Gan and Xiao [43]
introduced swarm learning strategies to improve convergence accuracy and
prevent premature convergence in BFO. This includes cooperative communication
with the global best bacteria and competitive learning mechanisms, improving
optimal solutions and swarm diversity, and addressing standard BFO
deficiencies.
Bolshakov et al. [44] have developed a deep reinforcement learning algorithm
called Deep Reinforcement Ant Colony Optimization (DRACO), inspired by
traditional ant colony optimization and designed for cooperative homogeneous
swarm learning. DRACO aims to shape collective behavior in decentralized
systems of independent agents, offering an alternative to centralized
learning. The algorithm’s advantages include natural parallelization, solving
collective tasks beyond the reach of single agents, increased reliability,
faster environmental exploration, and economic and energy efficiency.
### 2.9 Swarm Learning and IoT
In conventional cloud-based structures, IoT devices send data to central
servers for analysis. This approach can lead to potential bottlenecks,
compromise data privacy during transmission, and also increase latency. SL, on
the other hand, facilitates local data processing either on the device itself
or on proximate edge servers, thereby decreasing the necessity to transmit
sensitive data over the network and improving response times. SL enhances data
privacy and security by keeping data localized and using blockchain technology
for secure data sharing. This method ensures that sensitive data remain within
the local environment, complying with data protection regulations such as
GDPR. SL enables IoT devices to continuously learn and adapt in real-time,
providing real-time insights and real-time updates. The distributed nature of
SL provides excellent fault tolerance, making it suitable for IoT applications
such as healthcare monitoring systems and industrial automation. It scales
well without relying on a central server, making it suitable for sprawling IoT
networks. Implementing SL improves IoT networks’ efficiency, security, and
privacy compliance, making them better suited to handle vast amounts of data.
The next section will explore more into the applications of swarm learning in
IoT.
## 3 Applications of SL
SL is used in many fields, such as healthcare, autonomous vehicle systems,
environmental monitoring, and robotics, as shown in Fig.7, to improve
diagnostic accuracy, traffic flow, and safety. SL enables data aggregation
without compromising privacy, allows communication and learning from
experiences, and encourages cooperative robots for complex tasks. Its
potential to revolutionize distributed systems and information processing is
significant. The following subsections discuss the applications of SL in the
reviewed papers.
Figure 7: Swarm learning applications
### 3.1 Healthcare
Modern hospitals collect substantial volumes of private patient information
electronically. These data are extremely private and secret because they
pertain to both national security and individual privacy. The exchange of
medical data between institutions is restricted by legal and privacy concerns,
which impact the effectiveness of AI models trained on small datasets. While
distributed deep learning reduces communication and computing costs by making
optimal use of scattered data, it also poses privacy problems [24],[45]. SL
enables local machine learning model training using data from multiple health
nodes, such as hospitals. To maintain data privacy, the trained model
parameters are then exchanged, combined, and dispersed among nodes without the
requirement of a central collecting entity. By using blockchain, SL ensures
data security and confidentiality [45].
As shown in Fig.8[46], the SLN plays a central role by using its unique
digital identifier to train local models with private data and contribute to a
collective global model. The SNN, pivotal for consensus within the blockchain,
manages communication between the SLN and PBN, overseeing the training
process, and maintaining the model’s status. Lastly, the permissioned
blockchain network underpins the model-sharing aspect of swarm learning,
safeguarding the security and confidentiality of the process, and facilitating
effective collaboration between the SLNs.
Figure 8: The framework of metaverse swarm learning, which enables cross-
domain cooperation between metaverse and the physical world via blockchain
SL has demonstrated better performance in healthcare applications, such as
COVID-19 profiles and chest X-ray images, allowing ongoing learning and
enhancement across many data sources while closely respecting privacy laws
such as the General Data Protection Regulation (GDPR) and the Health Insurance
Portability and Accountability Act (HIPAA). It offers opportunities for the
development of cooperative research and diagnostics across hospitals and
research institutions networks and is flexible enough to fit a variety of
medical data environments. For example, German university hospitals are using
SL to evaluate COVID-19 patient data and create AI-based algorithms for the
detection of novel biomarkers. SL will develop into a crucial tool for
collaborative healthcare research and precision treatment [24],[45],[47].
For example, when hospitals use SL to manage COVID-19 data, they first gather
encrypted and anonymized patient data, including symptoms and treatments.
Every hospital sets up a separate SL node for safe local data processing. By
eliminating raw data exchange, these nodes preserve data privacy by locally
training models and sharing just the model parameters over a blockchain. These
parameters are then combined by a blockchain consensus method to update and
synchronize the global model across all nodes. Real-time deployment of this
continuously improving model enables more effective diagnosis and treatment
plans. SL improves predictive models by integrating diverse datasets from
multiple nodes, improving accuracy and treatment efficacy. It prioritizes
privacy and security by keeping sensitive patient data on-premises, reducing
reliance on central repositories. SL also increases efficiency in hospitals by
implementing personalized treatment plans. It is highly scalable, allowing
easy integration of new nodes without significant infrastructure changes.
Warnat-Herresthal et al. in their novel study [16] use SL to train AI models
on large datasets of histopathology images of more than 5,000 patients. SL was
demonstrated for disease classifier development using distributed data from
COVID-19, tuberculosis, leukemia, and lung pathologies, using over 16,400
blood transcriptomes. The study shows SL’s effectiveness in predicting
molecular alterations in colorectal cancer, demonstrating its potential for
enhancing medical imaging analysis without centralized data collection.
Fan et al. [26] was the first to examine the fairness problem in SL as it
relates to healthcare, mainly in duties related to the class of skin lesions.
They evaluated the fairness of the SL model for medical applications,
comparing performance and fairness with the single, centralized, and SL
models. The results show that SL can achieve better performance than single-
institution training and does not amplify biases. However, the study
acknowledges the high complexity of SL implementation due to the complex
configurations of the blockchain network.
To overcome the difficulties presented by non-independent and identically
distributed (non-IID) data in decentralized machine learning, notably in
clinical contexts, Wang et al. [48] introduced a generative augmentation
framework called SL-GAN, which combines a GAN in a swarm learning network to
augment non-IID data into IID data. The non-IID problem is directly addressed
for the first time in the context of SL in this paper, which is emphasized as
a significant advancement in decentralized clinical machine learning research.
The authors suggested improving synthetic data quality by introducing
differential privacy and studying synthetic data privacy.
DeMed, a decentralized privacy-preserving system for medical image processing
that uses blockchain technology, was proposed by Aggarwal et al. [49]. The
approach aggregates data into a classifier using smart contracts after using
self-supervised learning to create low-dimensional representations of medical
images. This paradigm seeks to address security and resilience concerns in
decentralized learning systems, with a particular focus on preventing
malicious or unintentional data alterations. The efficacy of the system is
demonstrated by independent medical picture classification tasks, such as
chest X-rays and pathological data.
By integrating swarm learning with homomorphic encryption. These papers [24],
[50] addressed a significant gap in distributed machine learning privacy-
preserving techniques. Swarm learning participants can securely share model
updates without disclosing sensitive data by incorporating homomorphic
encryption. To maintain participant privacy, the authors devised a partial
decryption algorithm that only required a fraction of the private key to allow
participants to decrypt aggregated model information locally. This
significantly advances the creation of machine learning applications in
domains where privacy is a concern. They recommended handling offline
participants, guarding against model poisoning, and maximizing encryption
trade-offs as areas of future research.
Gao et al.[8] proposed a unique strategy for SL that gathers local knowledge
from each center to overcome the forgetting of global knowledge during local
training. The proposed methodology demonstrates how utilizing data from
several centers can enhance medical picture segmentation while preserving data
privacy and resolving skew problems with non-IID data. The Label Skew-Aware
Loss (LaSA) is introduced to address label skew, preserving global label
information during local training. LaSA maximizes the forecast for the most
likely class determined by the global model. Feature Skew-Aware Regularization
(FeSA) is used to align local feature distributions with the global model,
mitigating the effects of feature skew caused by different imaging techniques
or demography.
Yuan et al. [51] developed a cooperative deep neural network (DNN)
partitioning system to accelerate disease diagnosis in multi-access edge
computing (MEC) networks. They used Swarm Reinforcement Learning (SRL) to
tackle the optimization problem of DNN partitioning and offloading and
blockchain technology to address challenges such as limited resources and
dynamic network environments. The algorithm allows agents to learn from local
data and generate judicious offloading actions.
A study by Saldanha et al. [33] used SL to identify molecular biomarkers in
gastric cancer from pathological images. They focused on microsatellite
instability (MSI) and Epstein-Barr virus (EBV) status. Patients cohorts from
the UK, USA, Switzerland, and Germany are included in their study. Every
dataset is kept apart from the others. However, the study was constrained by
uneven label classifications and the small number of biomarkers examined.
Future research must use a larger number of biomarkers and larger and more
diverse cohorts. To further enhance model performance and interpretability,
the authors recommended investigating attention-based deep learning
techniques.
Pan et al.[52] made a significant contribution to the field of drug
development. The study presented a "Nanonitrator," a nitrate nanoparticle made
of 3000 chitosan, sodium nitrate, and vitamin C as its main constituents. It
was produced utilizing the microencapsulation technique. The purpose of this
innovative nanoparticle is to improve nitrate’s long-circulating delivery
capability, extending its effects on the body’s duration and potency without
sacrificing safety. The authors described a novel method that uses a
combination drug prediction system driven by SL technology to improve the
bioavailability and protective effects of inorganic nitrate.
To predict molecular changes directly from hematoxylin and eosin (H&E)-stained
pathology slides of colorectal cancer, the study by Saldanha et al. [34] uses
SL to train AI models on large datasets of histopathology images from over
5,000 patients. The study shows the effectiveness of SL in predicting the BRAF
mutational status and microsatellite instability, demonstrating its potential
to improve medical imaging analysis without centralized data collection or
control.
A unique methodology based on SL was presented by Zhang et al. [46] for the
safe and equitable sharing of AI models in metaverse healthcare. The framework
addresses security, fairness, and data quality issues, improving model
accuracy and reliability. A novel parameter merging approach is devised to
maximize local models of SL nodes using lower-quality data. A permission
blockchain is used to incentivize high-quality data resources.
Mohammed et al. [53] developed a system using machine learning and SL to
diagnose diseases from nail images. The system uses transfer learning models,
InceptionV3 and VGG16, with an accuracy rate of 80%. The decentralized
approach eliminates trust and uses blockchain technology for parameter
merging. Despite limited training data, the system achieves an accuracy
comparable to or better than centralized models.
The problem of using vast amounts of medical data for cancer research,
particularly breast cancer, while adhering to privacy rules was the main focus
of the study by Shashank et al. [54]. The primary contribution lies in
showcasing SL, as a productive, privacy-preserving approach to improve
clinical research through the analysis of varied datasets from various
sources. They used 1,300 histochemical pictures of breast cancer tumors and
follow-up records to analyze diverse datasets, demonstrating how SL can
enhance clinical research, improve machine learning models, and maintain data
privacy without compromising quality.
By integrating user feedback into AI model training, Purkayastha et al.[55]
introduced a comprehensive approach that enables a more reliable and efficient
collaboration between radiologists and AI. The system uses few-shot learning
and SL, allowing continuous retraining of AI models based on active learning
strategies. The platform presents new capabilities for human-AI partnerships,
such as SL and few-shot learning methods. These techniques enable AI models
built on active learning algorithms to be continuously retrained. Through the
use of tailored model changes and collective knowledge, this approach
facilitates more accurate and repeatable radiological assessments.
Shriyan et al. [56] introduced a novel method to detect cataracts, which is
one of the most common eye conditions in the modern world, using SL. The
authors highlighted the benefits of SL over conventional FL and centralized
learning, emphasizing its effectiveness in the healthcare industry, especially
when data privacy is crucial. Hospitals can improve early cataract detection
by working together to create a global model while maintaining data privacy
through the use of SL. The method advocates for a scalable paradigm that might
include more nodes for higher data diversity and also proposes possible
applications in identifying other retinal illnesses.
Table 1 summarizes the main contributions of those articles.
Table 1: Swarm Learning in Healthcare Ref | Application | Contributions | Methodology | Used Datasets | Key Findings | Future Work
---|---|---|---|---|---|---
[16] | Medical imaging | Demonstrated the potential of SL for enhancing medical imaging analysis. | SL for large-scale pathology image analysis | Tuberculosis, leukemia, COVID-19, and lung pathologies | SL enhanced medical imaging analysis by facilitating multicentric collaboration and maintaining data privacy | Exploring additional medical fields where large, diverse datasets are crucial, possibly extending the decentralized model to more global collaboration.
[26] | Skin disease | Examined the fairness problem in SL | SL in skin lesion classification | Skin lesions | Investigated the fairness aspect of SL, showing robustness to heterogeneous data distributions and maintaining fairness without degrading performance | Future studies to improve model performance within the SL framework, focusing on managing model fairness and designing bias mitigation strategies for SL
[48] | Clinical settings | Overcame the difficulties presented by non-IID data in decentralized ML. | SL-GAN for non-IID data | Tuberculosis, Leukemia, and COVID-19 datasets | Addressed challenges posed by non-IID data in decentralized machine learning, specifically in clinical environments | Continued research to optimize decentralized clinical ML research, potentially exploring new algorithms and integration methods
[49] | Medical image analysis | A privacy-preserving decentralized framework for medical image analysis using blockchain technology. | Distributing a pre-trained Masked AutoEncoder (MAE) as a feature extractor and aggregating trained weights through smart contracts on the blockchain | Chest X-rays and pathological data | Developed a decentralized framework for medical image analysis, leveraging self-supervised learning and blockchain for privacy-preserving model training | Expanding the framework to include more complex medical imaging tasks, potentially increasing the variety of diseases that can be diagnosed using the system
[24], [50] | Privacy-preserving techniques | Including homomorphic encryption into SL. | Enhancing the Paillier homomorphic encryption using the Chinese Remainder Theorem for efficient operations and integrating a blockchain-based SL architecture for decentralized model aggregation through FedAvg | MNIST dataset | Significantly advanced machine learning applications in privacy-sensitive areas by allowing secure model updates sharing without revealing sensitive data | Enhancing defenses against model poisoning, optimizing encryption trade-offs, and handling offline participants
[8] | Medical imaging | Overcoming forgetting global knowledge during local training. Solves skew issues with Non-IID data. | Local knowledge assembly, LaSA loss, FeSA regularization | FeTS, M&Ms, MSProsMRI, MMWHS datasets | Enhanced medical image segmentation by handling Non-IID data issues, preserving data privacy | Further application to systems with unidirectional input constraints and expanding to other medical imaging tasks
[51] | Disease diagnosis | A cooperative DNN partitioning system for accelerating disease diagnosis in MEC networks. | Swarm Reinforcement Learning (SRL) in MEC networks | VGG16, AlexNet, ResNet18, NiN | Accelerated DNN-based disease diagnosis through cooperative DNN partitioning and offloading, minimizing service latency | Real-world applicability validation in clinical settings with specific constraints.
[33] | Molecular biomarker prediction | Predicting molecular biomarkers in gastric cancer from pathological images. | training MSI and EBV prediction models in individual merged cohorts and SL trained, using statistical analysis to assess prediction accuracy and explainability through pathologist-reviewed visualizations | Datasets from Bern, Leeds, TUM Cohort, TCGA | Improved prediction of molecular biomarkers in gastric cancer using multicentric data without compromising privacy | Expansion to include more biomarkers and larger datasets, exploring attention-based DL methods for improved model performance
[52] | Drug development | SL-based combination drug prediction system that identified vitamin C as the drug of choice to be combined with nitrate | AI-driven drug discovery, "Nanonitrator" nanoparticles | DPN, DDN, DTN from DrugBank, ChEMBL, UniProt | Enhanced bioavailability and therapeutic effects of inorganic nitrate for prolonged efficacy and safety | Not explicitly mentioned, but likely involve further clinical trials and detailed pharmacokinetic studies
[34] | Medical imaging | Predict molecular alterations from H&E-stained slides of colorectal cancer | A retrospective analysis of colorectal cancer patient images from five cohorts, using SL to train and validate ML models for predicting molecular features like MSI and BRAF mutations | Datasets from Northern Ireland, Germany, UK, TCGA, YCR BCIP | Demonstrated feasibility and effectiveness of SL in training AI models to predict molecular alterations in colorectal cancer using large, multicentric datasets | Expanding the SL application to other oncology areas and enhancing scalability and applicability of AI technologies in routine diagnostics.
[46] | Metaverse healthcare | Safe and equitable sharing of AI models in metaverse healthcare. A novel parameter merging approach for SL nodes. | SL nodes that train local models using private data, Swarm Network Nodes (SNN) for blockchain communication and monitoring, and a Permissioned Blockchain Network (PBN) for secure collaboration | COVID-19 dataset, PAMAP dataset | Improved accuracy and reliability of healthcare AI models in metaverse by ensuring security, fairness, and data quality distribution | Enhancing security and fairness in model-sharing processes through further integration of decentralized technologies
[53] | Disease diagnosis | Diagnose diseases from nail images. | Integrating three components: SL Node for managing insights, SNN for blockchain operations, and ML Node for training models using pre-trained bases | Four nail disease classification datasets on Google Cloud Drive | Achieved high diagnostic accuracy with a decentralized approach using transfer learning models, maintaining patient privacy | Expansion to other types of medical data and further enhancement of model training processes to maintain high accuracy with limited data
[54] | Cancer research | Using vast amounts of medical data for cancer research while adhering to privacy rules. | SL for training decentralized cancer diagnosis model across two nodes simulating different medical data sources. Data from the WDBC, WPBC, and BreakHis datasets, featuring both tumor characteristics and images, were split between nodes to reflect diverse medical scenarios | BreakHis, WDBC, WPBC | Utilized large volumes of medical data for cancer research while adhering to privacy norms showing how SL facilitates decentralized learning | Extending the decentralized model training to improve oncology research outcomes, leveraging larger and more diverse datasets
[55] | Radiology | A new capability for Human-AI partnerships. | SL with user feedback in AI model training | WDBC, WPBC, BreakHis | Introduced a system that incorporates user feedback in AI training, promoting personalized and efficient radiological assessments | Further development of Human-AI partnership capabilities, optimizing the interaction between radiologists and AI models
[56] | Eye disease detection | A novel method for detecting cataracts. | Pre-processing and data splitting, model training with the VGG-19 architecture, and Swarm Learning integration. | ODIR dataset, a collection of retinal images | Highlighted the advantages of SL over traditional centralized and federated learning systems in detecting cataracts | Expanding the model to include more diseases and larger networks for richer data diversity
### 3.2 Transportation
Innovations in communication and computing technologies have significantly
advanced the Internet-of-Vehicles (IoV). IoV is crucial to improving traffic
management, emergency responses, flow control, and efficiency in Intelligent
Transportation Systems (ITS). FL and Federated Deep Learning (FDL) have been
introduced to address privacy issues in IoV [57],[58]. Despite the benefits of
SL, there are drawbacks to using SL for collaborative Vehicle Trajectory
Prediction (VTP). For example, the need for global communication across a
large-scale network results in significant communication overhead, and the
cost of blockchain increases with the number of participants, making SL less
effective for large networks [57].
A framework that allows Vehicle Users (VUs) to cooperatively train and
aggregate models without the requirement of a central coordinator was
suggested by Lin et al. in [59]. An important consideration in the IoV
environment is the mobility of VUs, which is taken into account in the
proposed cooperative SL architecture. The authors create an incentive system
based on an iterative double auction to entice VUs to participate in the SL
process. An incentive mechanism and real-time models are included for dynamic
vehicle environments. The authors developed an optimization problem that
maximizes social welfare while achieving market equilibrium.
A novel SL approach for edge IoT contexts, communication-efficient, and
Byzantine-robust distributed swarm learning (CB-DSL). To solve issues like
data heterogeneity, communication constraints, and security concerns, CB-DSL
integrates biological intelligence and AI. To strengthen the local model and
the aggregation mechanism within the Direction Decide as a Service (DDaaS)
scheme, they used a three-layer service architecture to transfer traffic data
and control instructions, boosting forecast accuracy and real-time signal
light switching management. The CB-DSL framework is validated using real-world
healthcare datasets and simulation experiments with SUMO (Simulation of Urban
MObility) to demonstrate its effectiveness in reducing traffic congestion
compared to other existing methods.
IoV-SFDL (Internet of Vehicles-Swarm Federated Deep Learning) is a unique
framework that was presented by Wang et al.[58]. It combines SL into the FDL
framework and is specifically tailored for the IoV scenario. The goal of this
framework is to overcome the drawbacks of FDL in IoV, including significant
communication overhead, risks to data privacy, and difficulties brought on by
vehicle movement, erratic communication, and dynamic settings. The system is
more effective for IoV situations where the model training convergence speed
is accelerated through the use of an algorithm in the framework to anticipate
the credibility of weights.
The directed acyclic graph (DAG)-based Swarm Learning (DSL) framework was
created by Huang et al.[60] to address challenges such as unreliable
communications and vulnerability to malicious attacks in IoV. DSL combined
blockchain, Edge Computing (EC), and FL technologies to provide asynchronous
model training and data sharing in IoV. The authors created a Dynamic Vehicle
Association (DVA) algorithm based on DSL to handle vehicle movement and
enhance model training efficiency by maximizing the links between Vehicle
Nodes (VNs) and Road Side Units (RSUs). The DSL framework uses a method to
detect malicious attacks, ensuring security and resilience. It also introduces
a reward mechanism to encourage honest participation in model training,
promoting a collaborative and trustworthy learning environment.
Hou et al. [57] proposed Hierarchical Swarm Learning (HierSL), a novel edge-
assisted framework for Vehicle Trajectory Prediction (VTP). HierSL is proposed
to improve efficiency and security in the collaborative learning process,
particularly for large-scale edge-assisted IoV systems. HierSL reduces global
communications reliance and blockchain costs. Tests are carried out on an
actual NGSIM US-101 data set, and the outcomes demonstrate that the suggested
approach outperforms vanilla Swarm Learning and as well as centralized
learning.
Yin et al. [22] proposed a Multi-Region Asynchronous Swarm Learning (MASL)
framework. MASL is a hierarchical blockchain-powered framework for large-scale
data exchange in IoV. The blockchain, EC, and FL technologies were all merged
by MASL to ensure the anonymity and security of the sharing process. Secure
asynchronous model training and identity authentication have been accomplished
by coordinating the intra-regional (IR) and cross-regional (CR) sharing and
the non-IID data issues between regions. Furthermore, the DAG-enabled MASL is
a fully asynchronous system that is capable of responding to anomalous
vehicles on the IoVs.
Liu et al. [61] introduced a 6G-driven urban traffic congestion mitigation
solution called DDaaS. DDaaS includes a model layer for data collection,
parameter training, and congestion value prediction, a Swarm Network (SN)
layer for safe parameter transmission, and a decision-making layer for signal
light switching. Based on SUMO, simulation trials demonstrate that DDaaS can
reduce traffic congestion and achieve accurate prediction.
Autonomous driving technology has advanced significantly, but privacy concerns
arise due to the use of sensors and cameras. Mishra et al. [62] proposed an
SL-based training approach to address these concerns. By sharing model
learnings across nodes, SL protects sensitive information and reduces privacy
breaches. SL presents a promising solution to create effective and respectful
autonomous driving systems. This approach offers performance comparable to
traditional methods and outperforms other distributed machine learning
techniques like FL. Table 2 shows the main contributions of these articles.
Table 2: Swarm Learning in Transportation Ref | Application | Contributions | Methodology | Used Datasets | Key Findings | Future Work
---|---|---|---|---|---|---
[59] | IoV | A new framework that allows VUs to cooperatively train and aggregate models without the requirement for a central coordinator. An optimization problem that maximizes social welfare while achieving market equilibrium. | Cooperative SL framework with an incentive mechanism based on the mobility of vehicle users | - | Proposes a more communication-efficient method than FL; enhances social welfare and dynamic adjustment to mobility | Develop the incentive mechanism to ensure fair participation and better model aggregation methods
[63] | Edge IoT | Communication-efficient and Byzantine-robust Distributed Swarm Learning (CB-DSL) combining AI with BI principles | Evaluating the model performance under both i.i.d. and non-i.i.d. conditions and in the presence of Byzantine attacks. | CIFAR-10 and MNIST datasets | Improves local model accuracy and decision-making in traffic management; addresses local optima issues | Validate the framework in real-world settings and address more inherent challenges in edge IoT environments
[58] | IoV | IoV-SFDL: Overcomes the drawbacks of FDL in IoV, including significant communication overhead, risks to data privacy, and challenges caused by vehicle movement, erratic communication, and dynamic settings | Integrates SL into Federated Deep Learning framework | Next-Generation Simulation (NGSIM) dataset | Addresses communication overhead, improves model convergence speed in IoV contexts | Explore additional IoV-specific challenges and expand the framework to more dynamic scenarios
[60] | IoV | Improve data sharing and model training in the context of IoV | Directed Acyclic Graph-based SL (DSL) combining edge computing, FL, and blockchain | Traffic Signs Preprocessed dataset based on GTSRB | Enhances data sharing and model training; Introduces dynamic vehicle association and malicious attack detection | Develop more robust mechanisms for attack detection and introduce more adaptive algorithms for vehicle mobility
[57] | Vehicle Trajectory Prediction (VTP) in IoV | A novel edge-assisted framework for VTP | Hierarchical SL with a two-layer learning framework | NGSIM US-101 dataset | Reduces global communication reliance and blockchain costs; improves security in large-scale IoV systems | Optimize synchronization steps and system topology for better accuracy and efficiency
[22] | IoV | A secure, efficient framework for large-scale data sharing in IoVs | Multi-Region Asynchronous Swarm Learning (MASL) with hierarchical blockchain for parallel execution | Traffic Signs Pre-processed dataset based on GTSRB | Addresses scalability, security, and data heterogeneity; maintains user data privacy in large-scale data sharing | Improve the asynchronous training methods and expand blockchain integration for better data privacy and security
[61] | ITS | Direction Decide as a Service (DDaaS) to Reduce Traffic Congestion in 6G-Driven ITS. A traffic simulation and congestion prediction experiment using SUMO in Beijing, China. | Direction Decide as a Service (DDaaS) with a novel three-layer architecture incorporating SL | - | Facilitates the orderly transmission of traffic data and control instructions; improves traffic management and reduces congestion | Enhance the traffic control algorithm for more adaptive and timely decisions; expand to more complex ITS scenarios
[62] | Autonomous Driving Systems | Training autonomous driving systems | SL-based training method for privacy preservation and performance enhancement | Kitti 3d dataset | Claims superior privacy preservation and potentially better performance over traditional methods | Expand research to compare with more distributed machine learning techniques and validate in practical autonomous driving contexts
### 3.3 Industry
The Industrial Internet of Things (IIoT) is being developed using technologies
such as IoT, big data and digital twin (DT). Combining IIoT with AI algorithms
can improve productivity and interoperability, offering solutions for advanced
manufacturing systems. However, the DT technique faces challenges in capturing
dynamic industrial environments due to its data-driven nature and security and
privacy concerns[64]. SL is revolutionizing manufacturing by providing real-
time intelligent agents that improve operational efficiency by streamlining
manufacturing lines, dynamically allocating resources, and instantly resolving
problems. This approach allows for production line optimization, dynamic
resource allocation, and real-time problem solutions without centralized
control. SL is ideal for companies aiming to use Industry 4.0 and smart
manufacturing, creating more resilient and intelligent factories for the
future [65],[20]. However, there is limited research on integrating SL with
IIoT. Reliability issues in industrial systems are crucial, especially in
emergencies. Industrial environments are complex and subject to high
temperatures and noise, making them more complex than normal environments.
With automation, competition for limited communications resources increases
the unreliability of IIoT systems[64].
Pongfai et al.[17] developed a Dragonfly Swarm Learning Process (D-SLP)
algorithm for nonlinear feedback control systems, improving robustness,
performance, and stability. The D-SLP controller demonstrated superior
performance in simulations of a permanent magnet synchronous motor control
system compared to other control methods. However, the study acknowledges
limitations and suggests future work for unidirectional input constraints and
input dead zones in systems.
Using a deterministic Q-Swarm Learning Process (Q-SLP) algorithm and SL
principles, Pongfai et al.[18] created an enhanced control approach. This
method optimizes proportional integral and derivative (PID) controller
parameters, improving system performance, stability, and convergence. The
approach improves convergence time and performance by addressing shortcomings
in conventional techniques. Simulations showed superior performance and
convergence over traditional SLP, improved particle swarm optimization (IPSO),
and the whale optimization algorithm (WOA).
Pongfai and other authors created an adaptive SLP method in a different work
[66]to create the best PID controller possible for multiple-input/multiple-
output (MIMO) systems. The approach dynamically updates online weights
depending on system failures, improving PID parameter autotuning performance,
stability, and resilience. The authors evaluated the algorithm against
conventional techniques using a two-wheel inverted pendulum system as a case
study. The method could be investigated to approximate discrete-time
responses, predict behavior, and observe systems.
Sun et al.[19] proposed a new diagnostic framework for bearing faults in
rotating machinery, addressing data privacy concerns and insufficient labeled
data in factories. The framework uses convolutional neural networks and
adversarial domain networks to train local diagnostic models without sharing
data. Sun et al. in another paper [20] proposed a framework using SL to
diagnose faults in multiple components of the machinery, addressing data
privacy and insufficient data. The framework uses local diagnosis models like
AlexNet and the Chebyshev filter, enhancing efficiency and accuracy.
Xiang et al.[64] presented a ground-breaking architecture for IIoT that is
enhanced by DT technology and powered by credibility-weighted SL. Their method
tries to solve the privacy risks and significant communications costs. With
the aid of DT, they developed a DRL technique to simultaneously optimize
energy consumption and IIoT system reliability. To address issues with
operational efficiency and sustainability, they also developed and solved an
optimization problem in the recommended DT architecture to optimize system
reliability and minimize energy usage.
Wang et al.[65] have introduced a novel approach that utilizes cooperative
multi-agent SL and DT to optimize robot assembly cells and thus can be adapted
to any manufacturing environment. This model of interaction, where each
element acts as an autonomous agent, permits these agents to respond
instantaneously to issues of mechanical structure, networked software, and
hardware integration. The approach considers each component as an agent,
allowing them to interact dynamically to address mechanical structure,
software, and hardware integration changes. The framework supports dynamic
reconfiguration, ensuring efficient manufacturing systems in response to
varying product demands and production cycles.
Luo and Zhang [67] have developed a blockchain-based data management method to
ensure the integrity of the engine data, preventing tampering and deletion.
The method uses SL to verify the integrity of engine test data and protect
privacy. The integrated approach improves trustworthiness, supports
collaborative learning, and optimizes data usage while protecting privacy.
Table 3 shows the main contributions of these articles.
Table 3: Swarm Learning in Industry Ref | Application | Contributions | Methodology | Used Datasets | Key Findings | Future Work
---|---|---|---|---|---|---
[17] | Nonlinear control systems | A Dragonfly Swarm Learning Process (D-SLP) algorithm for nonlinear feedback control systems | Blendsing dragonfly algorithm behaviors with SL protocols to adaptively tune control parameters amidst system variations; A two-layer blockchain framework to ensure secure and private intra-regional and cross-regional data sharing among vehicles and base stations | - | Superior control performance in nonlinear systems compared to conventional methods | Explore application to systems with specific constraints like PAM
[18] | PID controller optimization | Use a deterministic Q-SLP algorithm to optimize and improve the PID parameter’s autotuning process | A Deterministic Q-SLP Algorithm for optimizing PID controllers, combining swarm and learning to refine control parameters KP, KI, and KD, enhancing system response and stability | CPC system | Improved convergence and performance optimization over traditional methods | Not specified
[19] | Diagnostic frameworks for rotating machinery | A new diagnostic framework for bearing faults in rotating machinery | Integrates adversarial domain networks with CNNs | CRWU, HITsz, XJTU-SY, and SCU | Increased efficiency and accuracy in fault diagnosis without compromising data privacy | Not specified
[64] | Industrial Internet of Things (IIoT) | A revolutionary architecture for IIoT powered by credibility-weighted SL and improved by DT technology | Digital Twin technology with credibility-weighted SL | real-world MNIST dataset | Enhancing IIoT system reliability and reducing energy consumption | Further address practical concerns in IIoT for operational efficiency
[65] | Reconfigurable robotic assembly cells | A method for optimizing the layout of reconfigurable robotic assembly cells in manufacturing environments | Multi-agent cooperative swarm learning with digital twin | - | Improved layout optimization and operational efficiency in manufacturing | Enhance the framework to adapt to rapid changes in manufacturing demands
[67] | Data management in engine lifecycle | A blockchain-based data management method to ensure engine data integrity | Utilizing blockchain for secure data interactions, and employs a trusted application (BCAPP) for data processing and validation | NASA open dataset | Enhanced data integrity and security throughout the engine’s lifecycle | Optimize multi-party collaborative learning and data usage
### 3.4 Robotic systems
Learning processes can be significantly accelerated when multiple robots work
together to form a swarm. Such entities could exchange learned information in
a decentralized or centralized fashion. In SL, nodes in the network pool share
locally learned models among themselves without the need for a central
authority. When using SL in networked robotic applications, a collection of
linked robots must be able to operate together or independently to complete
tasks. Rangu and Nair [68], offered a method that uses mobile agents to
execute SL on a group of robots and each learns a task. The learning process
is distributed, with a mobile agent compiling and disseminating the models
learned locally as it moves seamlessly across the network of both simulated
and actual robots. The authors demonstrated the SL approach using a mixed
group of both simulated and real robots, considering that assembling a swarm
solely of real robots would be cost-prohibitive. The application of
reinforcement learning at the local level to groups consisting of simulated,
real, and combinations of these robots has proven the viability and efficiency
of SL within a diverse network of robots.
### 3.5 Smart home
Edge Intelligence (EI) integrates edge computing and AI in smart homes, real-
time video analysis, and precision agriculture. However, centralized machine
learning models have limitations like data privacy breaches and communication
overhead[69]
SL is transforming smart home ecosystems by enabling decentralized decision-
making processes. This allows smart devices to communicate and learn from each
other’s experiences, optimizing energy consumption, security, and automation.
Smart thermostats, lighting, and appliances can adjust settings based on
occupants’ habits, ensuring comfort and energy efficiency. Swarm learning also
allows security networks to analyze data and adapt without human intervention.
Xu et al. [69] introduced a novel cooperative SL framework to overcome Central
Machine Learning issues by leveraging decentralized SL for the prediction of
thermal comfort. This approach reduces communication overhead and improves
model performance by leveraging real data from all nodes within the edge
computing network. The framework’s effectiveness was demonstrated through an
extensive empirical investigation using a Non-IID thermal comfort dataset.
Liu et al.[70] developed ADONIS, a framework for detecting abnormal behavior
in IoT devices. It uses Swarm Learning, knowledge distillation, and human-
computer interaction (HCI) to improve security and operational efficiency. The
decentralized approach reduces central node failure risk and reduces latency
and energy consumption. ADONIS can be applied to smart cities and IoVs, and
its adaptability makes it suitable for various applications. Future research
includes further enhancements and refinement of parameter aggregation methods.
### 3.6 Financial services field
By using decentralized networks for data analysis, decision-making, and risk
management, SL is completely changing the financial services industry. Swarm
learning’s decentralized nature reshapes data-driven decisions in the complex
financial landscape. Enhance investment recommendations and fraud detection
rates while protecting against single points of failure. By using SL,
financial organizations can modify their strategy in response to current
market conditions and consumer trends. John et al.[21] used SL for credit
scoring in Peer-to-Peer lending on a blockchain platform in the financial
services industry, ensuring user data privacy and secure transactions. The
decentralized model training and credit scoring process eliminate centralized
data storage risks. Future work includes testing with real-time datasets and
improving user experience.
### 3.7 Multimedia Internet of Things
By enabling the processing and dissemination of decentralized content in real-
time in environments containing IoT devices, swarm learning is transforming
the Multimedia IoT ecosystem. This method guarantees that content is
personalized for each user, minimizes latency, and maximizes network capacity
utilization. Additionally, processing data locally on devices improves
security and privacy by lowering the possibility that private information will
be hacked.
Zhang et al.[71] have improved the privacy and security of multimedia IoT
devices using Radio Frequency Fingerprinting (RFF) for identity
authentication. They integrated differential privacy, specifically the
Gaussian mechanism, into SL to protect RFF data. They also proposed a novel
node evaluation mechanism to prevent malicious nodes from affecting the
model’s accuracy and integrity. By guaranteeing the security of the underlying
IoT devices through enhanced privacy protection in SL, the research paves the
way for safe multimedia services.
### 3.8 Fake news detection
Social media has significantly impacted the distribution of information, but
the lack of systematic management has led to the spread of fake news. Machine
learning techniques like convolutional neural networks (CNN) and recurrent
neural networks (RNN)can detect fake news, but centralized detection can
violate user privacy. Decentralized methods like SL offer privacy-preserving
learning on local data, reducing hacking risks and allowing users to maintain
confidentiality without sharing data [72]. Dong et al.[72] developed Human-in-
the-loop Based Swarm Learning (HBSL), a decentralized method for detecting
fake news. HBSL uses SL and human-in-the-loop (HITL) techniques to detect fake
news across nodes, ensuring user privacy. It incorporates user feedback,
allowing models to be continuously updated. The method was validated using a
benchmark dataset (LIAR), showing its superiority over existing methods.
### 3.9 Metaverse
The Metaverse faces challenges in reliable extended reality (XR) data
transmission due to a lack of incentives and untrust among users. To address
these issues, a configurable secure resource trading mechanism based on swarm
learning is proposed in [73]. This framework includes subchains for
decentralized Intelligent Reflecting Surfaces (IRS) resource management and
intelligent allocation, a smart contract-enabled scheme, and a decentralized
federated learning-driven IRS allocation scheme. Experimental results
demonstrate the effectiveness of this configurable SL-based resource trading
for reliable XR communication.
Table 4 shows the main contributions of those articles.
Table 4: Swarm Learning Applications Field | Ref | Application | Contributions | Methodology | Used Datasets | Key Findings | Future Work
---|---|---|---|---|---|---|---
Robotic systems | [68] | Networked robotic applications | A method that uses mobile agents to execute SL on a group of robots | Mobile agents executing SL on a group of robots; Each robot learns individually, and a mobile agent facilitates the aggregation and sharing of locally learned models across the swarm | - | Demonstrated viability and efficiency of SL in a mixed robot swarm; reinforced learning applied locally to enhance task completion | Not specified
Smart home | [69] | Edge intelligent computing networks | Cooperative SL framework with cyclic ring all reduce topology for thermal comfort prediction | utilizing stochastic gradient descent within a cyclic edge intelligent computing network. | Non-IID thermal comfort dataset | Demonstrates reduced communication overhead, enhanced data privacy, and improved model performance by leveraging data from all nodes without sharing it | Extend empirical investigations, optimize model performance and handle real-world applications’ data distribution issues
| [70] | IoT, specifically abnormal behavior detection | ADONIS, a framework for detecting abnormal behavior in IoT devices | SL combined with knowledge distillation and HCI for anomaly detection in IoT devices | Traffic dataset | Enhanced security and operational efficiency in IoT networks by local data fusion and a lightweight model to accommodate resource-constrained environment | Further framework enhancement, increase communication efficiency, and refine parameter aggregation for non-IID data
Financial services field | [21] | Credit scoring | Credit scoring in Peer-to-Peer lending on a blockchain platform in the financial services industry | Lending platform on Web 3.0 that connects lenders and borrowers using blockchain technology to ensure secure, peer-to-peer transactions without intermediaries | Universal Bank dataset | Ensures data privacy and secure transactions, with model performance comparable to centralized approaches | Test model with dynamic datasets, explore other decentralized platforms (Solana, Hyperledger, Corda), and enhance user experience
Multimedia IoT | [71] | Multimedia IoT device security using RFF | Improved the privacy and security of multimedia IoT devices using RFF for identity authentication | Integration of differential privacy and a novel node evaluation mechanism in SL | RFF dataset | Enhancing privacy and security for IoT devices by protecting RFF data and making the system resilient against various cyber attacks | Future research could focus on extending these methodologies to broader IoT applications and further improving the robustness of the security measures
Fake news detection | [72] | Decentralized fake news detection | Human-in-the-loop-based swarm learning (HBSL), a decentralized method that incorporates user feedback for detecting fake news | The methodology involves local learning, collaborative model update and human feedback to enhance detection capabilities across the network through a cyclic process | LIAR dataset | Significantly improves the accuracy of fake news detection using local data and user feedback | Design detection models tailored to specific node features to enhance effectiveness
Metaverse | [73] | 6G-Metaverse XR communication | SL-based secure configurable resource trading mechanism for reliable 6G-Metaverse XR communication. | A decentralized trading framework using SL for resource management in a 6G-Metaverse environment, facilitated by IRS and blockchain technology, and Federated Learning for privacy enhancement. | Custom dataset | Effective in reliable XR communication via decentralized management and smart contract-based resource trading | Investigate customization of SL for more fine-grained communication hardware resource management and scheduling
## 4 Challenges
### 4.1 Non-IID Problem in SL
SL enables participants to register, train models, and exchange parameters
through edge nodes, ensuring data sovereignty and confidentiality. However, SL
performance is significantly affected by non-independent and identically
distributed (Non-IID ) data[48], which can lead to inconsistent model updates
and degraded aggregate performance. When data is dispersed unevenly among
various network nodes or participants, it is called the non-IID problem in SL.
This implies that distinct statistical characteristics, such as mean,
variance, and data distribution patterns, may exist in the dataset at each
node. Several factors, including variations in patient demographics, the type
of medical equipment utilized, or even the particular focus or specialization
of the medical institutes providing the data, might contribute to this
heterogeneity in the data. Non-IID data problems include quantity, label, and
feature skews. Feature skew and label skew are caused by differences in
imaging protocols or demographics, leading to inconsistencies in annotations
and Non-IID label distributions. Various strategies, including elastic weight
consolidation and batch normalization, have been proposed to address feature,
label, and quantity skew in classification tasks. However, these methods do
not fully consider label skew, which could cause suboptimal performance[8],
[74].
Two types of strategies are now being used to tackle the non-IID challenge:
algorithm-based and data-based approaches. Algorithm-based methods align local
models with global models, while data-based methods balance distribution but
require a trusted central coordinator. Furthermore, with non-IID data,
convergence problems may arise when utilizing Generative Adversarial Networks
(GAN) for data augmentation[48].
To address the non-IID problem in SL, methods must be created that can either
reduce the impact of data heterogeneity or take advantage of it to increase
the global model’s resilience and generalizability. Strategies such as
advanced aggregation techniques, personalized models, and data augmentation
can improve the robustness and generalizability of the global model[8].
Currently, effective solutions to address the non-IID problem in SL are yet to
be established[48].
### 4.2 Fairness and bias in SL
Fairness and bias in machine learning models indicate how they could perform
or reflect dominant groupings in the data in an unbalanced way. The impact of
SL on model bias and fairness has not yet been fully assessed, even though
fairness issues have been considered in the context of FL. In[26], the authors
suggested comparing SL with centralized learning and subgroup-specific model
training to investigate the fairness of SL in medical imaging tasks without
the need for additional bias mitigation techniques. To provide insight into
how SL might balance performance and fairness in healthcare applications,
their study seeks to determine if SL’s fairness features are more in line with
centralized learning or subgroup-specific training.
### 4.3 Attacks on swarm learning
SL has the potential to handle distributed large-scale data better than FL,
but it also faces significant security issues that require more scrutiny. In
the stages of SL, as shown in Fig.9 [6], different attacks can occur:
unreliable parties may compromise data during local training and before the
locally trained metadata are secured on the blockchain, it might be vulnerable
to various network attacks like Eclipse and DDoS. Furthermore, malicious
participants could introduce harmful parameters during the merging process,
potentially introducing backdoors into the global model. 1) Data poisoning
might occur in the local training phase; 2) eclipse attacks could occur in the
blockchain P2P network in the metadata upload phase; and 3) the global model
could be hacked by poisoned parameters in the parameter aggregation phase [6].
Figure 9: Attack on swarm learning
#### 4.3.1 Backdoor attacks against distributed swarm learning
Despite its privacy and decentralized training benefits, SL faces significant
security threats, such as backdoor attacks, which need to be addressed to
ensure the integrity and reliability of SL systems. Backdoor attacks in
machine learning, especially SL, manipulate data and training processes to
produce incorrect outputs. In SL, where multiple nodes collaborate, a backdoor
attack could be particularly insidious. Moreover, the decentralized nature of
SL makes detecting such attacks challenging due to the non-IID nature of real-
world data. Addressing backdoor attacks requires technological solutions,
robust security practices, and new collaborative learning approaches to ensure
integrity and trustworthiness in decentralized machine learning
environments[6],[35].
Chen et al.[6] conducted a study on security threats in SL using a pixel
pattern backdoor attack method. Their research consists of a number of studies
that evaluate the effectiveness of backdoor attacks in diverse scenarios
utilizing a variety of datasets (MNIST, CIFAR-10, SVHN). These circumstances
include varied network sizes, different data distributions (IID vs. non-IID),
distinct attack targets (single vs. multitarget), and attack continuity
policies (single-shot vs. multiple-shot). To reduce the effects of backdoor
attacks, they also suggested a number of security strategies, including L2
regularization and the addition of noise. Experimental data verify the
efficiency of these protections.
Yang et al.[35] identified a hybrid vulnerability in SL that uses backdoor and
eclipse attacks to propagate backdoors secretly. They introduced a strategy
called sample-specific eclipse (SSE) to target high data contribution nodes,
reducing attack costs and accelerating backdoor propagation. The study
investigates the use of distributed backdoor poisoning attacks in conjunction
with Eclipse assaults for the first time, showing how they can be used
together to allow backdoors to spread covertly among innocent users on the SL
network. Afterward, they suggested a fresh assault plan that concentrates on
nodes that contribute a lot of data, speeding up the spread of backdoors and
requiring fewer resources overall to be effective.
#### 4.3.2 Poisoning attack
SL faces unique challenges from poisoning attacks. Poisoning can compromise
the collective learning process, affecting model parameters and performance.
The decentralized nature of SL complicates detection, as there is no central
authority to monitor data quality or model updates. Therefore, robust
decentralized consensus mechanisms are needed to detect and mitigate poisoned
inputs [35], [71],[28]. Qi Y. et al.[28] developed strategies to prevent
poisoning attacks and ensure the integrity and security of the SL process.
Rongxuan et al.[75] introduced a Zero Trust Architecture (ZTA)-based defense
scheme for SL to combat poisoning attacks in decentralized learning
environments. It identifies a unique vulnerability where a malicious ’header’
node can compromise the model. The defense mechanism emphasizes continuous
risk calculation and anomaly detection, allowing dynamic responses to threats.
The scheme also uses Manhattan distance and accuracy differences to identify
and mitigate risks from both the header and edge nodes. The effectiveness of
the proposed defense strategy is demonstrated through systematic experiments,
proving its practical applicability in real-world scenarios.
#### 4.3.3 Eclipse attack
An Eclipse attack in SL involves an attacker controlling the network
communication between nodes. This is particularly relevant in peer-to-peer
networks where nodes share information and model updates without a centralized
authority [35]. An attacker can isolate a target node or group of nodes by
monopolizing their network connections, potentially introducing false data or
model updates [76]. This could impact the integrity of the model and degrade
performance. To protect against Eclipse attacks, robust peer discovery and
management mechanisms should be implemented, including diverse peer
connections, validating peer identities, and detecting network patterns that
might indicate control of communication channels[35].
#### 4.3.4 Inference attacks
Inference attacks aim to deduce sensitive information about the training data
used by a model, such as recovering private or sensitive attributes. They can
be used to determine if a specific data record was part of the training set,
infer specific attributes or features of data instances, or attempt to
reconstruct a model’s parameters. Inference attacks focus on extracting
information about the training data or model behavior, such as determining if
specific data were used in training or guessing private attributes based on
model outputs. Decentralized machine learning methods allow multiple nodes to
collaboratively learn a shared model without exchanging local data, typically
through blockchain technology [77]. Inference attacks exploit shared model
updates or the final model to infer properties of the training data or
identify unique characteristics of individual participants’ datasets. To
protect against inference attacks, advanced cryptographic and privacy-
preserving techniques such as homomorphic encryption, secure multi-party
computation, and differential privacy are employed. However, the balance
between privacy protection and model performance is a critical challenge in
SL[71].
#### 4.3.5 Model inversion attacks
Model inversion attacks aim directly at reconstructing the inputs used to
train the model, effectively reversing the model’s computations to approximate
or reveal the actual data. They often target models that provide detailed or
confident predictions, which can inadvertently reveal information about the
training data [78]. While inference attacks often derive indirect information
about the data or its attributes, model inversion attacks engage in a more
direct and complex effort to recreate the original training inputs themselves.
In SL, where nodes collaborate to train a model without sharing their local
datasets. The decentralized nature of SL allows each node to contribute to the
model’s learning by updating it based on local data. However, shared model
updates or predictions can leak information, potentially inferring specific
characteristics or reconstructing aspects of the original training data. To
defend against model inversion attacks, strategies such as output
perturbation, differential privacy mechanisms, access controls, and strict
query limits can be implemented[71].
## 5 Future Research
SL addresses privacy and data integration issues, but research gaps exist,
indicating potential areas for further exploration.
* •
Security and Trust: Although SL uses blockchain technology to ensure security
and trust, more investigation is required to solve potential security flaws,
such as sophisticated cyber threats and insider attacks. It is essential to
have strong trust mechanisms and security measures specifically designed for
SL networks. Swarm-FHE [79] offers a significant advancement in SL security by
integrating fully homomorphic encryption and blockchain technology. This
method ensures that collaborative model training is conducted without
compromising data, even in the presence of compromised or malicious
participants. Blockchain technology and lightweight homomorphic encryption are
also combined in a privacy-preserving SL by Li et al. [44], which promotes
model security, data privacy, and computational performance and offers a
competitive substitute for FL in remote machine learning applications [80].
* •
Dynamic Node Management: Enhancing the robustness and dependability of SL
systems may involve investigating dynamic techniques for node participation
and incentive mechanisms to guarantee nodes’ continued and productive
engagement in the swarm network.
* •
Optimizing Leader Election: The leader election process in SL can lead to
disproportionate bandwidth consumption, inefficiencies, and potential
bottlenecks, causing dissatisfaction among participants and potentially
compromising network security. To address these challenges, [23] suggested
refining the leader election mechanism for more equitable network load
distribution.
* •
Scalability and Efficiency: The ability of SL to expand across a growing
number of nodes and a variety of data formats while maintaining efficiency and
model performance should be investigated. Enhancing model aggregation
techniques and communication protocols could be the main areas of research to
facilitate widespread implementations of SL.
* •
Interoperability and Standards: For SL to succeed, standards compliance and
interoperability amongst various systems are essential. To solve issues with
data format, protocols, and compliance, research could examine methods for SL
to seamlessly integrate into existing IT systems. Qi et al.[28] developed a
blockchain twin mechanism to improve the interoperability and efficiency of SL
on different blockchains, introducing an incentive mechanism for active
participation, thus improving the overall performance and security of the SL
process.
* •
Energy Efficiency: Considering the possible magnitude of SL deployments,
especially in the context of IoT, the development of power-saving learning
algorithms is of the utmost importance. The emphasis of such research would be
on minimizing the energy usage of devices involved in the SL process, a factor
that is particularly critical for devices running on batteries or sensors
located remotely.
* •
Cross-domain Applications: Investigating the potential use of SL in diverse
sectors like healthcare, autonomous vehicles, smart cities, and manufacturing
can be extremely advantageous. Each of these areas poses distinct challenges
and demands, and customized SL approaches could result in significant
advancements in the way these sectors employ decentralized learning.
* •
Data Heterogeneity and Non-IID Data: To efficiently tackle the non-IID issue
in SL, forthcoming studies might concentrate on the creation of a hybrid model
adaptation method that merges both algorithmic innovations and robust data
management strategies. The goal of this method should be to reduce the effects
of data heterogeneity and boost the performance and unification of the global
model in a distributed environment.
* •
Advanced-Data Augmentation Techniques: Investigate the application of advanced
generative models, like variational autoencoders (VAEs) or enhanced GANs, for
the production of synthetic data samples. These samples can efficiently
supplement sparse or imbalanced datasets across different nodes, thereby
addressing the non-IID problem.
* •
Ethical AI and Fairness: As SL models become more widespread, it is crucial to
ensure that these models do not perpetuate or exacerbate biases. Research
could focus on developing fairness-sensitive algorithms that promote ethical
AI practices within SL frameworks.
* •
Resource Management: As mentioned in[23], the impact of adding more Swarm
coordinator nodes on resource overhead is negligible. However, the resource
overhead increases linearly with the number of Swarm edge nodes added,
indicating that scaling these nodes should be done with care. This observation
provides valuable guidance and actionable recommendations for developers and
researchers looking to apply SL effectively in real-world scenarios.
* •
Integrating ML into SL: The integration of ML methods into the SL framework
can introduce challenges in analyzing the specific contributions of SL to
training rate improvements. SL uses blockchain technology to synchronize model
updates amongst nodes. Although confidentiality and integrity are guaranteed,
the overhead resulting from blockchain operations (such as consensus processes
and transaction validations) may outweigh the anticipated gains in training
speed from concurrent decentralized training. Therefore, integrating ML
methods into SL may complicate the assessment of training rate improvements.
Empirical studies and benchmarking against traditional systems are needed to
assess its benefits in real-world scenarios.
## 6 Conclusion
SL is a promising advancement in decentralized machine learning that enables
efficient, secure, and privacy-preserving collaborative learning without
central data storage. This review provides invaluable information on the
advantages of SL and emphasizes how SL can facilitate safe, confidential, and
effective collaborative machine learning across dispersed networks. Highlights
the benefits of SL, such as improved data privacy, reduced risk of centralized
breaches, and the ability to learn from diverse data sources without data
transfer. SL has potential applications in healthcare, IoV, industry, etc.
However, challenges like non-IID problems, fairness, bias, and vulnerability
to attacks need to be addressed. Robust decentralized consensus mechanisms and
advanced cryptographic techniques are essential for the integrity and privacy
of SL. These research gaps offer a wide range of opportunities for researchers
interested in advancing the field of decentralized machine learning.
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# Learning New Tasks from a Few Examples with Soft-Label Prototypes
Avyav Kumar Singh
Department of Informatics
King’s College London
<EMAIL_ADDRESS>&Ekaterina Shutova
ILLC
University of Amsterdam
<EMAIL_ADDRESS>&Helen Yannakoudakis
Department of Informatics
King’s College London
<EMAIL_ADDRESS>
###### Abstract
It has been experimentally demonstrated that humans are able to learn in a
manner that allows them to make predictions on categories for which they have
not seen any examples Malaviya et al. (2022). Sucholutsky and Schonlau (2020)
have recently presented a machine learning approach that aims to do the same.
They utilise synthetically generated data and demonstrate that it is possible
to achieve sub-linear scaling and develop models that can learn to recognise
$N$ classes from $M<N$ training samples – aka less-than-one shot learning.
Their method was, however, defined for univariate or simple multivariate data.
We extend it to work on large, high-dimensional and real-world datasets and
empirically validate it in this new and challenging setting. We apply this
method to learn previously unseen NLP tasks from very few examples (4, 8 or
16). We first generate compact less-than-one shot representations called soft-
label prototypes which are fitted on training data, capturing the distribution
of different classes across the input domain space. We then use a modified
k-Nearest Neighbours classifier to demonstrate that soft-label prototypes can
classify data competitively, even outperforming much more computationally
complex few-shot learning methods.
## 1 Introduction
Humans have the remarkable ability to adapt knowledge gained in one domain for
use in another setting and to identify or disambiguate objects after observing
only a few examples Lake et al. (2015). This has inspired machine learning
researchers to build models able to do the same. In the rapidly growing area
of few-shot learning, researchers aim to enable correct classification in a
new task after exposure to only a few examples per class.
In natural language processing (NLP), few-shot learning techniques first
relied on interventions at the data level, such as dataset augmentation Clark
et al. (2018) or generation of adversarial examples from few-shot datasets
Miyato et al. (2016). The advent of large language models Devlin et al. (2018)
led to fine-tuning models on different target tasks, creating prompt-enhanced
few-shot datasets for training Gao et al. (2020); Schick and Schütze (2020b);
Lester et al. (2021) and the application of meta-learning algorithms Finn et
al. (2017); Snell et al. (2017) which optimise model parameters in a specific
manner so as to adapt quickly to new tasks using past experiences as a
reference.
Meta-learning has been successfully applied across a range of NLP tasks. These
included learning new classes from a few examples, e.g. in relation
classification Obamuyide and Vlachos (2019a, b) or word sense disambiguation
Holla et al. (2020) tasks; learning to quickly adapt to new domains
Nooralahzadeh et al. (2020) and to new languages Langedijk et al. (2021); van
der Heijden et al. (2021); Wang et al. (2020). An ambitious recent effort
focused on learning entirely new, previously unseen NLP tasks from as few as
4, 8 or 16 examples Bansal et al. (2019).
Despite many successes, it is important to note that meta-learning algorithms
come with limitations. While the amount of data required for training a model
for a specific few-shot task can be very little, a large amount of data from
diverse few-shot tasks is still needed to learn to generalise well to new
tasks. In practice, training time for meta-training algorithms can still run
in the order of days as they are computationally complex and resource-heavy.
In this paper, we explore and extend an alternative, more simple and yet
powerful, approach to few-shot learning, proposed by Sucholutsky et al.
(2021), which aims to offset the disadvantages mentioned above. The method
relies on representing input data sub-linearly (i.e.: representing $M$ classes
using $N$ points with $M\geq N$) and using a simple machine learning
classifier to classify data. We first generate compact, sophisticated less-
than-one shot representations called soft-label prototypes which are fitted on
the training data of the task at hand, capturing the distribution of different
classes across the input domain space. While Sucholutsky et al.’s original
approach was designed and applied to simpler univariate and multivariate data,
we extend it to work on real-world, high dimensional data. We apply the method
to learning previously unseen NLP tasks from only a few examples of that task.
We experimentally demonstrate that classification with soft-label prototypes
using a simple, modified version of k-Nearest Neighbours Cunningham and Delany
(2022) outperforms a range of strong baselines as well as the best performing
existing few-shot learning approaches in 9/16 investigated tasks.
## 2 Related Work
### 2.1 Few-shot learning in NLP
Few-shot learning techniques in NLP not exclusively requiring large language
models include data augmentation and semi-supervised learning techniques such
as augmentation with adversarial examples Miyato et al. (2016), interpolation
of training data into hidden space Chen et al. (2020) and consistency training
to make models more resistant to noise Xie et al. (2019). Semi-supervised
learning with cloze-style tasks and further fine-tuning large language models
outperformed Schick and Schütze (2020a, b) fine-tuning a large language model
with few-shot datasets using a classifier head as described by Devlin et al.
(2018). GPT-2 Radford et al. (2019) focused on achieving competitive results
with zero-shot training in multi-task settings, however, it’s successor, GPT-3
Brown et al. (2020), outperformed existing state-of-the-art models on machine
translation, cloze-style tasks and question-answering tasks with few-shot
training. Adding prompts, either in the vocabulary space Gao et al. (2020) or
the embedding space Li and Liang (2021) Lester et al. (2021), resulted in the
formation of masked language modeling tasks over the training dataset and led
to enhanced performance in few-shot settings compared to fine-tuning over a
language model.
### 2.2 Meta-Learning
In the meta-learning paradigm, the training and test sets, referred to as
$\mathcal{D}$meta-train and $\mathcal{D}$meta-test, are split into episodes.
Conceptually, each episode encompasses a task $\mathcal{T}$i and consists of a
support set $\mathcal{D}^{(i)}$support and a query set
$\mathcal{D}^{(i)}$query. Meta-learning algorithms initially fit the model on
the support set of the episode and then achieve generalisation across episodes
by optimising performance on the query sets of the episodes. For evaluation,
the model trains on the support set and checks performance on the query set
for each task $\mathcal{T}$i $\in$ $\mathcal{D}$meta-test. Meta-learning
algorithms are divided into three different areas based on their learning
methodology - model-based, metric-based and optimisation-based meta-learning
methods.
#### 2.2.1 Model Agnostic Meta-Learning
MAML Finn et al. (2017) is an optimisation-based meta-learning approach which
principally focuses on adding a generalisability objective in it’s cost
function. Optimising this generalisability objective results in a model which
learns to combine new information on top of existing knowledge efficiently.
Let us parameterise the model $f_{\theta}$ by $\theta$. As defined before,
each task $\mathcal{T}$i consists of a support set $\mathcal{D}^{(i)}$support
and a query set $\mathcal{D}^{(i)}$query. Initially, $\theta$ in
$f_{\theta}(x)$ is optimised to $\theta^{\prime}_{i}$ for a task
$\mathcal{T}$i. This initial step is referred to as inner-loop optimisation.
Mathematically, for a single step
$\mathbf{\theta^{\prime}_{i}}=\mathbf{\theta}-\alpha\nabla_{\theta}\mathcal{L}^{s}_{\mathcal{T}_{i}}(f_{\theta})$
where $\alpha$ is the learning rate and $\mathcal{L}^{s}_{\mathcal{T}_{i}}$
denotes the loss on $\mathcal{D}^{(i)}$support. As a result, each task
$\mathcal{T}$i results in an episode-specific optimised model
$f_{\theta_{i}^{\prime}}$. We can therefore capture the overall objective,
which is to have the model $f_{\theta_{i}^{\prime}}$ generalise across all
tasks in the distribution.
$\underset{\theta}{m}in\sum_{\mathcal{T}_{i}\sim
p(\mathcal{T})}\mathcal{L}^{q}_{\mathcal{T}_{i}}(f_{\theta_{i}^{\prime}})$
$=\underset{\theta}{m}in\sum_{\mathcal{T}_{i}\sim
p(\mathcal{T})}\mathcal{L}^{q}_{\mathcal{T}_{i}}(f_{\mathbf{\theta}-\alpha\nabla_{\theta}\mathcal{L}^{s}_{\mathcal{T}_{i}}(f_{\theta})})$
The losses $\mathcal{L}^{q}_{\mathcal{T}_{i}}$ are defined by the mis-
classifications on the query examples in the episode. The primary idea is to
optimise with respect to $\theta$ even though the losses are computed with
respect to $\theta_{i}$ for each episode — this enables the model to optimise
the initial parameters $\theta$ in a way as to improve generalisability. This
process is called outer-loop optimisation.
$\mathbf{\theta}\leftarrow\mathbf{\theta}-\beta\>\nabla_{\theta}\sum_{\mathcal{T}_{i}\sim
p(\mathcal{T})}\mathcal{L}^{q}_{\mathcal{T}_{i}}(f_{\theta_{i}^{\prime}})$
#### 2.2.2 Prototypical Networks
Prototypical Networks is an example of metric-based meta-learning methods.
Metric-based methods use an embedding function to get a complex, high-
dimensional representation for each instance in the support set of an episode
and then optimise the embedding function such that query examples are
correctly classified — the goal is to minimise the loss measured on the query
set between the true label and classification based on a similarity metric.
Prototypical networks, initially proposed by Snell et al. (2017), use an
embedding network, given by $f(\theta)$, to output a high-dimensional vector
that is the arithmetic mean of the data points per class in the support set of
an episode.
If we denote the sum of the instances belonging to class $c$ in the support
set of an episode by $S_{c}$ then the prototype $\mu_{c}$ is given by
$\mu_{c}=\frac{1}{|S_{c}|}\sum_{(x_{i},y_{i})\in S_{c}}f_{\theta}(x_{i})$
We can use a distance function as our similarity metric and take a softmax
over that function to get the probability of a test point beloning to a
particular class as
$\displaystyle p(y=c|x)=softmax(-d(f_{\theta}(x),\mu_{c})$
$\displaystyle=\frac{exp(-d(f_{\theta}(x),\mu_{c}))}{\sum_{c^{\prime}\in
C}exp(-d(f_{\theta}(x),\mu_{c}^{\prime}))}$
We further define the loss function as
$J(\mathbf{\theta})=-log(p(y=c^{*}|x,\theta))$
During training, each episode contains a support set and a query set
consisting of instances belonging to the same classes in the meta-training
set.
#### 2.2.3 Meta-Learning in NLP
Meta-learning has been applied in a range of natural language processing tasks
in recent years. Obamuyide and Vlachos (2019a) used meta-learning for relation
extraction in a lifelong learning setting. Obamuyide and Vlachos (2019b) also
used meta-learning for relation classification. Holla et al. (2020) use
various meta-learning algorithms to solve the problem of word-sense
disambiguation. Bansal et al. (2020) use a combination of semi-supervised and
supervised tasks to evaluate performance of their models (using meta-learning
algorithms) on tasks such as relation classification, sentiment
classification, entity-typing and natural language inference. Meta-learning
algorithms like MAML have also been applied to cross-lingual classification
tasks van der Heijden et al. (2021) and dependency parsing Langedijk et al.
(2021).
### 2.3 LEOPARD
LEOPARD Bansal et al. (2019) is a meta-learned large language model built by
producing a task-dependent final layer, typically a softmax, based on the
inputs. Given a meta-training set $\mathcal{D}_{i}^{tr}=\\{(x_{i},y_{i})\\}$
containing a task $\mathcal{T}_{i}$, the algorithm initially divides the data
with the task into it’s respective classes given by
$\mathcal{C}_{i}^{m}=\\{x_{i}|y_{i}=n\\}$ where $n\in N_{i}$ and $N_{i}$
represents the number of classes in the task at hand. It then calculates the
weights and biases of the output softmax layer as
$w_{i}^{n},b_{i}^{n}=\frac{1}{|\mathcal{C}_{i}^{n}|}\sum_{x_{j}\in\mathcal{C}_{i}^{n}}g_{\psi}(f_{\theta}(x_{j}))$
where $g_{\psi}$ is a multi-layer perceptron with two layers using a symmetric
sigmoid activation function for each layer, $w_{i}^{n}$ is a $l$-dimensional
vector and $b_{i}^{n}$ is a scalar. Combining the weights and biases for each
class, we get
$\mathbf{W_{i}}=[w_{i}^{1};...;w_{i}^{N_{i}}]\;and\;\mathbf{b_{i}}=[b_{i}^{1};...;b_{i}^{N_{i}}]$
The final equation which is used for classification is given as
$p(y|x^{*})=softmax(\mathbf{W_{i}}h_{\phi}(f_{\theta}(x^{*}))+\mathbf{b_{i}})$
where $h_{\phi}(.)$ is a separate multi-layer perceptron parameterised by
$\phi$ using all classes, given by $N_{i}$, in the task at hand. The primary
idea, as with most meta-learning approaches, is that one should not introduce
task-dependent parameters during any stage of the training process as the
model can learn task-relevant data and use it for inference rather than learn
a good initialisation point that allows the model to generalise well.
Therefore, the softmax parameters vary episodically and the existing model is
used to generate the initial parameters.
Meta-training with LEOPARD involves sampling multiple episodes at a time,
given by $G>1$, to produce a good initialisation point for the softmax
parameters. The remaining episodes are used for meta-adaptation of the model.
Mathematically, the update happens as follows, starting with
$\phi_{i}^{(0)}\leftarrow\phi_{i}$ for $s\in 0,1,...,G-1$
$\phi_{i}^{s+1}=\phi_{i}^{s}-\alpha_{s}\mathbb{E}_{\mathcal{D}_{i}^{tr}\sim
T_{i}}[\nabla_{\phi}\mathcal{L}_{i}(\\{\Theta,\phi_{i}\\},\mathcal{D}_{i}^{tr})]$
Here, $\Theta$ denote the parameters of the encoder which are fine-tuned on
meta-training, $\phi$ are the parameters of the softmax function previously
generated and $\alpha_{s}$ is the learning rate on the support set of an
episode.
Bansal et al. (2019) fine-tuned few-shot datasets over the meta-trained
LEOPARD model to evaluate performance on test datasets over NLP tasks such as
natural language inference, review classification and entity typing. They
reported that LEOPARD outperformed the baselines used by them which included
non meta-trained models as well as a prototypical network based meta-learning
model.
## 3 Method
We encode training and test instances using a large language model. We then
generate soft-label prototypes as defined by Sucholutsky et al. (2021) and
classify test encodings using those soft-label prototypes as input features.
### 3.1 Encoders
Our default base encoder is a BERT Devlin et al. (2018) model - provided by
the HuggingFace Transformers111https://huggingface.co/docs/transformers/index
library Wolf et al. (2019). We use the base cased BERT model as our default
encoder to generate input features which are used to generate and classify
with soft-label prototypes. To facilitate comparison with LEOPARD, which was
additionally meta-trained using GLUE tasks, in a separate experiment we also
use the MT-DNN model Liu et al. (2019) which has been pretrained in a multi-
task learning setting on the set of GLUE tasks. We use MT-DNN to provide the
input features for classification with soft-label prototypes. We use feature-
based encodings derived from BERT and MT-DNN without further fine-tuning.
### 3.2 Defining Soft-Label Prototypes
Sucholutsky and Schonlau (2020) define a soft label as a vector representing a
point’s "simultaneous membership to several classes". It is used in cases
where we can denote a point’s partial association to different classes. Using
this definition, they further define a soft-label prototype given by
$(\vec{X},Y)$ where $\vec{X}$ is a point in the input space (in machine
learning, an input feature vector) and $Y$ is the corresponding soft-label.
Thus, it is possible to represent $N$ classes using $M$ data points where
$M<N$ and each soft-label prototype contains class information of multiple
classes.
### 3.3 Generating Soft-Label Prototypes
As soft-label prototypes assign soft labels to every point in the input domain
(denoted in Figure 1(a)), a soft-label prototype at point $\vec{X}$
essentially represents the class distribution (determined from the training
data) at $\vec{X}$. The process of generating soft-label prototypes from
training data is a two step process detailed below.
#### 3.3.1 Finding Lines Connecting all Centroids
First, we compute the centroid of each class in the input dataset. The next
step involves finding and fitting class centroids on the minimum number of
lines - with the aim to create soft-label prototypes at the ends of each line
which capture the class distribution of all classes along that line.
Sucholutsky et al. (2021) propose three algorithms to generate the optimal set
of lines. We provide a high level overview of the two methods we used to
generate lines.
Brute Force This method first generates combinations of all possible lines
between class centroids. It then evaluates their quality by generating a score
proportional to the perpendicular distance of all centroids from the nearest
line that do not lie on it and finally uses up to $l$ lines which score the
lowest. The complexity is given by $O(n^{2l})$ where $n$ denotes the total
number of classes and $l$ denotes the total number of lines. Practically, this
method is suitable only in cases where the number of classes is small. As it
is independent of the dimensions of the input space, it also works well for
high-dimensional input space given $n$ is small, typically $\leq$10.
Recursive Regression This method clusters centroids hierarchically to group
similar centroids together. The similarity of centroids within a single
cluster is judged by how well all the centroids fit on a regression line. If
the distance of a particular centroid is beyond a pre-defined tolerance
threshold $\epsilon$ from a line, it is removed from that cluster and assigned
to another cluster. We used this method to run all our experiments. It is
computationally less expensive and performs well for high-dimensional data
spread across many more classes relative to the Brute Force method - thus
being more suitable for large, real-world data.
We depict the process of finding lines connecting all centroids in Figure
1(b).
#### 3.3.2 Defining Constraints
Once we find the lines, we use the endpoints of each line as the location of
soft-label prototypes. Therefore, we would have $2l$ prototypes if there are
$l$ lines fitted on $n$ centroids. Sucholutsky et al. (2021) mathematically
define finding the class distribution at each soft-label prototype as a
constrained optimisation problem.
Each class centroid is assigned an interval on the line segment connecting it
to other centroids on the same line. The start of this interval is marked at
the midpoint between the class centroid and the centroid of the class
preceding it along the line. The end of the interval is marked at the midpoint
between the class centroid and the succeeding class’s centroid. At any point
on the line segment, the class influences at that particular point is the sum
of all soft-label prototype class distributions weighted by the inverse of the
distance from each prototype. For this point to lie in the decision region of
one class, the weighted magnitude of that class’s influence must be higher
than that of other classes. The optimisation problem therefore consists
conceptually of two main constraints (1) the desired class has the maximum
influence amongst all classes and (2) the difference between the influence of
the desired class and the sum of the influences of all other classes is
maximised. Therefore, two soft-label prototypes are opened at the ends of each
line as denoted in Figure 1(c).
### 3.4 Classification with Soft-Label Prototypes
We use k prototypes to classify a point using a modified version of k-Nearest
Neighbours called Soft-Label Prototype k-Nearest Neighbors (SLP) Sucholutsky
and Schonlau (2020). Given $M$ soft-label prototypes which represent the input
distribution of $N$ classes, we define
$S=(X_{1},Y_{1}),...,(X_{M},Y_{M})$
to be the set of prototypes used in training where $X_{i}$ is the location of
the $i^{th}$ prototype in the input feature space and $Y_{i}$ is a matrix of
size $[N\times 1]$ denoting the soft label. We suppose that the location of
the point we want to classify is $x$. We calculate the distances from each
prototype to $x$ and denote them by the set
$D={(X_{i},x)}_{i=1,2...M}$
We then sort $S$ in ascending order of distances using $D$. Finally, we weigh
the probability distribution of the $i^{th}$ nearest prototype inversely by
it’s distance from the point and obtain
$Y^{*}=\sum_{i=1}^{k}\frac{Y_{i}}{d(X_{i},x)}$
$x$ is then assigned to the class
$C^{SLP}(x)=argmax_{j}Y^{*}_{j}$
where $Y^{*}_{j}$ is the jth element of $Y^{*}$. The decision boundaries are
described diagrammatically in Figure 1(d).
(a)
(b)
(c)
(d)
Figure 1: Generating soft-label prototypes and representing the final decision
landscape.
## 4 Tasks and Datasets
For comparison purposes, we use the same set of tasks and datasets as Bansal
et al. (2019). This set covers a variety of text classification tasks with a
large number of classes, ranging from 2 to 13. The tasks include (a) Entity
typing \- CoNLL-2003 Sang and De Meulder (2003) and MIT-Restaurant Liu et al.
(2013) datasets; (b) Review rating classification \- review ratings from
Amazon Reviews Blitzer et al. (2007) and three-way classification on the data;
(c) Text classification \- scraped social media data from
crowdflower222https://www.figure-eight.com/data-for-everyone/ which comprises
sentiment and emotion classification in a range of domains, as well as
political bias detection; and (d) Natural language inference in the scientific
domain – the SciTail dataset. Khot et al. (2018).
We use the same datasets and splits as Bansal et al. to ensure direct
comparability of approaches. At evaluation time, Bansal et al. sample an N-way
k-shot dataset $\forall\;k\in\\{4,8,16\\}$ for few-shot fine-tuning and then
evaluate the model performance on the test set of the respective task. As
performance is dependent on the data used for fine-tuning, ten test episodes
per task are sampled in total and each model is fine-tuned and evaluated on
each episode. We use accuracy as the primary metric to compare models, similar
to Bansal et al. (2019). Final performance per classification task is reported
using the mean accuracy across ten episodes as well as the standard deviation.
We ensure that the test episodes are the same as Bansal et al. by directly
using their few-shot splits available
publicly333https://github.com/iesl/leopard/. For entity typing tasks, rather
than generating a full sentence (or text) encoding as done in ratings and text
classification tasks, we obtain token-level encodings for classification.
## 5 Baselines
Our baseline selection is two-fold — essentially (1) we want to determine how
well the soft-label prototype classifier performs compared to other
classifiers given the same few-shot training data and (2) how classification
with soft-label prototypes compares against other models when both are trained
on a larger training set.
### 5.1 1-Nearest Neighbour Classifier
Our first baseline is the simple 1-Nearest Neighbour classifier for each task.
We first encode all of the data points in the episode using BERT. The test
instances are then assigned the class of their closest support example in this
embedding space. We use Euclidean distance as the distance metric.
### 5.2 Fine-tuning BERT
Another baseline we consider is fine-tuning BERT on the support set of the
test task and then evaluating the performance of that fine-tuned model on the
test set. Since this method relies on fitting a large language model on only a
few support instances, there is a higher risk of overfitting. Since 1-NN and
this baseline do not rely on additional training data, we use them to evaluate
performance in a scenario where fine-tuning is only limited to few-shot
datasets.
### 5.3 Multi-Task Learning on BERT
Bansal et al. (2019) further fine-tune the BERT model on different tasks from
the GLUE benchmark - classification, natural language inference, question
answering etc (Wang et al., 2018) prior to applying it to learn an unseen task
in a few shot fashion. Bansal et al.’s multi-task trained BERT (MT-BERT) is
similar to another implementation of a multi-task trained BERT called MT-DNN
Liu et al. (2019) which has been trained on the same set of GLUE tasks. Since
Bansal et al.’s multitask-trained BERT model is not publicly available, we use
MT-DNN as a baseline. We train a classifier head on top of MT-DNN using the
few-shot episodes and compute accuracy on the test set.
### 5.4 Prototypical Networks
We also compare to Prototypical Networks, a meta-learning algorithm, used to
meta-train the BERT encoder on the set of GLUE tasks. We use Euclidean
distance as the distance metric. Classification with soft-label prototypes
also uses Euclidean distance to weigh soft-label probabilities. This baseline
allows us to compare SLP to learning deterministic class prototypes.
### 5.5 LEOPARD
Bansal et al. (2019) report LEOPARD as the meta-learning algorithm which
presented the highest scores across most tasks for entity typing, ratings
classification as well as text classification. We compare our method to theirs
since (to the best of our knowledge) it represents the current state-of-the-
art in few-shot learning of previously unseen tasks.
## 6 Experimental setup
Sucholutsky et al.’s less-than-one-shot learning framework needed some
additional adaptation to work on real-world, complex datasets. They use the
CVXPY Diamond and Boyd (2016) library to perform computations for generating
the class distribution of a soft-label prototype, however, as their
experiments are low-dimensional ($\leq 5$), they are able to perform their
computations using a lightweight optimisation solver called ECOS Domahidi et
al. (2013). As language models have a higher output dimension (for example
BERT outputs a $768$ dimensional vector for a token) we use a more powerful,
commercially available solver called MOSEK MOSEK ApS (2019) capable of
integration within the CVXPY library to perform the required optimisations for
getting the soft-label prototypes — we found that ECOS was unable to handle
the computational complexity which resulted in our experiment program
terminating erroneously. We share the code used to run our experiments and the
detailed descriptions of our extensions to Sucholutsky et al.’s framework
publicly444https://github.com/avyavkumar/few-shot-learning-notebooks.
### 6.1 Hyperparameters
We used a held-out validation task (sentiment classification in the
electronics domain) to determine the ideal hyperparameters for soft-label
prototype classification. The hyperparameters with a major influence on
accuracy are listed below:
1) $k$ which denotes the number of nearest neighbour prototypes. We searched
among the values $k\in\\{1...10\\}$ and we found that $k=1$ provided the
highest accuracy on the validation set.
2) $\epsilon$ which is a control factor used to denote the maximum tolerance
between a centroid and the line assigned to it. The tolerance is measured as
the Euclidean distance between the line and the centroid. We use a tolerance
value of $1e-1$.
3) $l$ which denotes the maximum number of lines used to connect all
centroids. We experimented with $l\in\\{0.2n,0.4n,0.6n,0.8n,n-1\\}$ where n is
the number of centroids. For cases where it was not possible to connect $n$
centroids with $l$ lines, the optimisation process failed. Experimentally, we
found that $l=n-1$ corresponded to the highest accuracy in the validation set.
The result can also be understood intuitively - more prototype centroids can
be connected with more lines. In cases where $n$ points can be represented
with $l^{\prime}\leq l$ lines such that the tolerance is $\leq\epsilon$,
$l^{\prime}$ lines are used.
When training MT-DNN on the few-shot test episodes, we use a learning rate of
$2e-5$, a batch size of 4 and train for 100, 125 and 150 epochs for
$shots\in\\{4,8,16\\}$ respectively.
## 7 Results and discussion
Category (Classes) | Shot | 1NN | BERTbase* | SLPBERT
---|---|---|---|---
Political Bias (2) | 4 | 52.245 ± 4.348 | 54.57 ± 5.02 | 53.447 ± 3.281
| 8 | 54.568 ± 3.015 | 56.15 ± 3.75 | 55.824 ± 3.725
| 16 | 55.884 ± 2.436 | 60.96 ± 4.25 | 58.277 ± 4.128
Emotion (13) | 4 | 8.474 ± 1.028 | 09.20 ± 3.22 | 9.235 ± 2.169
| 8 | 8.418 ± 0.792 | 08.21 ± 2.12 | 8.423 ± 3.36
| 16 | 8.758 ± 0.854 | 13.43 ± 2.51 | 9.414 ± 3.398
Sentiment Books (2) | 4 | 57.56 ± 4.571 | 54.81 ± 3.75 | 59.89 ± 5.385
| 8 | 60.76 ± 4.177 | 53.54 ± 5.17 | 64.34 ± 2.565
| 16 | 60.98 ± 2.857 | 65.56 ± 4.12 | 66.36 ± 2.183
Rating DVD (3) | 4 | 37.289 ± 6.846 | 32.22 ± 08.72 | 38.158 ± 10.056
| 8 | 37.598 ± 5.364 | 36.35 ± 12.50 | 38.504 ± 9.973
| 16 | 37.461 ± 4.506 | 42.79 ± 10.18 | 36.778 ± 9.852
Rating Electronics (3) | 4 | 37.001 ± 5.164 | 39.27 ± 10.15 | 35.554 ± 9.373
| 8 | 37.646 ± 4.223 | 28.74 ± 08.22 | 43.193 ± 9.391
| 16 | 38.633 ± 3.103 | 45.48 ± 06.13 | 45.133 ± 9.754
Rating Kitchen (3) | 4 | 36.769 ± 7.738 | 34.76 ± 11.20 | 38.671 ± 9.775
| 8 | 37.435 ± 6.347 | 34.49 ± 08.72 | 45.142 ± 11.026
| 16 | 38.047 ± 3.226 | 47.94 ± 08.28 | 45.253 ± 13.455
Political Audience (2) | 4 | 51.827 ± 1.754 | 51.02 ± 1.23 | 51.305 ± 2.68
| 8 | 53.113 ± 2.314 | 50.87 ± 1.88 | 53.104 ± 3.669
| 16 | 53.287 ± 1.874 | 53.09 ± 1.93 | 53.888 ± 3.305
Sentiment Kitchen (2) | 4 | 60.17 ± 3.197 | 56.93 ± 7.10 | 61.96 ± 4.594
| 8 | 59.82 ± 2.703 | 57.13 ± 6.60 | 64.83 ± 3.983
| 16 | 61.85 ± 2.65 | 68.88 ± 3.39 | 68.21 ± 3.298
Disaster (2) | 4 | 53.629 ± 9.391 | 55.73 ± 10.29 | 52.77 ± 10.803
| 8 | 57.625 ± 7.638 | 56.31 ± 09.57 | 56.888 ± 11.139
| 16 | 60.93 ± 4.997 | 64.52 ± 08.93 | 65.907 ± 3.691
Airline (3) | 4 | 46.175 ± 4.922 | 42.76 ± 13.50 | 43.653 ± 17.948
| 8 | 48.039 ± 5.677 | 38.00 ± 17.06 | 36.343 ± 17.598
| 16 | 51.754 ± 2.954 | 58.01 ± 08.23 | 54.806 ± 16.779
Rating Books (3) | 4 | 39.324 ± 4.985 | 39.42 ± 07.22 | 45.588 ± 10.329
| 8 | 38.447 ± 4.07 | 39.55 ± 10.01 | 38.77 ± 12.485
| 16 | 41.107 ± 2.683 | 43.08 ± 11.78 | 42.105 ± 11.289
Political Message (9) | 4 | 15.044 ± 1.556 | 15.64 ± 2.73 | 15.424 ± 1.281
| 8 | 15.666 ± 1.053 | 13.38 ± 1.74 | 16.332 ± 2.056
| 16 | 14.844 ± 1.426 | 20.67 ± 3.89 | 18.185 ± 1.885
Scitail (2) | 4 | 53.66 ± 4.594 | 58.53 ± 09.74 | 52.296 ± 4.366
| 8 | 53.212 ± 3.028 | 57.93 ± 10.70 | 55.964 ± 5.705
| 16 | 54.53 ± 4.399 | 65.66 ± 06.82 | 59.675 ± 4.033
Sentiment DVD (2) | 4 | 53.78 ± 1.316 | 54.98 ± 3.96 | 56.06 ± 2.408
| 8 | 53.8 ± 2.843 | 55.63 ± 4.34 | 56.98 ± 3.299
| 16 | 54.05 ± 1.659 | 58.69 ± 6.08 | 58.95 ± 2.813
Restaurant (8) | 4 | 48.194 ± 4.881 | 49.37 ± 4.28 | 46.407 ± 7.062
| 8 | 57.36 ± 3.681 | 49.38 ± 7.76 | 57.434 ± 7.745
| 16 | 64.706 ± 2.913 | 69.24 ± 3.68 | 59.074 ± 8.34
CoNLL (4) | 4 | 45.104 ± 7.724 | 50.44 ± 08.57 | 39.35 ± 5.798
| 8 | 46.897 ± 4.183 | 50.06 ± 11.30 | 47.892 ± 14.31
| 16 | 53.969 ± 3.252 | 74.47 ± 03.10 | 54.397 ± 8.991
Table 1: Classification using SLP is indicated by SLP indexed by the encoder
used. Entries in bold highlight the highest scores. All models were only fine-
tuned and they did not rely on the meta-training set. * represents the
performance of the baselines as reported by Bansal et al. (2019) in their
paper – note that our experimental setting is identical to theirs.
Category (Classes) | Shot | LEOPARD* | MTLMT-BERT* | Proto-Net* | MTLMT-DNN | SLPMT-DNN
---|---|---|---|---|---|---
Political Bias (2) | 4 | 60.49 ± 6.66 | 54.66 ± 3.74 | 56.33 ± 4.37 | 62.46 ± 4.09 | 60.743 ± 10.656
| 8 | 61.74 ± 6.73 | 54.79 ± 4.19 | 58.87 ± 3.79 | 66.37 ± 0.36 | 65.34 ± 2.886
| 16 | 65.08 ± 2.14 | 60.30 ± 3.26 | 57.01 ± 4.44 | 66.10 ± 3.79 | 66.337 ± 0.552
Emotion (13) | 4 | 11.71 ± 2.16 | 09.84 ± 2.14 | 09.18 ± 3.14 | 12.88 ± 0.73 | 11.503 ± 1.28
| 8 | 12.90 ± 1.63 | 11.21 ± 2.11 | 11.18 ± 2.95 | 15.39 ± 1.37 | 14.151 ± 3.366
| 16 | 13.38 ± 2.20 | 12.75 ± 2.04 | 12.32 ± 3.73 | 17.45 ± 0.56 | 13.654 ± 1.943
Sentiment Books (2) | 4 | 82.54 ± 1.33 | 64.93 ± 8.65 | 73.15 ± 5.85 | 85.53 ± 0.99 | 86.07 ± 0.313
| 8 | 83.03 ± 1.28 | 67.38 ± 9.78 | 75.46 ± 6.87 | 85.43 ± 0.34 | 86.18 ± 0.326
| 16 | 83.33 ± 0.79 | 69.65 ± 8.94 | 77.26 ± 3.27 | 85.73 ± 0.71 | 86.13 ± 0.414
Rating DVD (3) | 4 | 49.76 ± 9.80 | 41.23 ± 10.98 | 47.73 ± 6.20 | 51.26 ± 8.3 | 57.183 ± 15.254
| 8 | 53.28 ± 4.66 | 45.24 ± 9.76 | 47.11 ± 4.00 | 55.39 ± 10.07 | 60.192 ± 13.438
| 16 | 53.52 ± 4.77 | 45.19 ± 11.56 | 48.39 ± 3.74 | 60.02 ± 1.8 | 66.987 ± 0.654
Rating Electronics (3) | 4 | 51.71 ± 7.20 | 41.20 ± 10.69 | 37.40 ± 3.72 | 59.82 ± 4.24 | 61.938 ± 9.047
| 8 | 54.78 ± 6.48 | 45.41 ± 09.49 | 43.64 ± 7.31 | 63.37 ± 0.78 | 65.023 ± 1.013
| 16 | 58.69 ± 2.41 | 47.29 ± 10.55 | 44.83 ± 5.96 | 62.85 ± 0.62 | 64.911 ± 0.969
Rating Kitchen (3) | 4 | 50.21 ± 09.63 | 36.77 ± 10.62 | 44.72 ± 9.13 | 54.80 ± 13.02 | 55.56 ± 18.927
| 8 | 53.72 ± 10.31 | 47.98 ± 09.73 | 46.03 ± 8.57 | 57.08 ± 11.95 | 59.631 ± 17.894
| 16 | 57.00 ± 08.69 | 53.79 ± 09.47 | 49.85 ± 9.31 | 63.95 ± 4.34 | 63.285 ± 15.583
Political Audience (2) | 4 | 52.60 ± 3.51 | 51.53 ± 1.80 | 51.47 ± 3.68 | 51.03 ± 7.22 | 53.85 ± 7.353
| 8 | 54.31 ± 3.95 | 54.34 ± 2.88 | 51.83 ± 3.77 | 56.74 ± 5.30 | 56.759 ± 5.072
| 16 | 57.71 ± 3.52 | 55.14 ± 4.57 | 53.53 ± 3.25 | 58.70 ± 2.15 | 58.529 ± 2.3
Sentiment Kitchen (2) | 4 | 78.35 ± 18.36 | 60.53 ± 9.25 | 62.71 ± 9.53 | 82.50 ± 3.49 | 86.57 ± 0.508
| 8 | 84.88 ± 01.12 | 69.66 ± 8.05 | 70.19 ± 6.42 | 86.67 ± 0.60 | 86.67 ± 0.25
| 16 | 85.27 ± 01.31 | 77.37 ± 6.74 | 71.83 ± 5.94 | 86.63 ± 0.50 | 86.85 ± 0.19
Disaster (2) | 4 | 51.45 ± 4.25 | 50.61 ± 8.33 | 50.87 ± 1.12 | 50.29 ± 3.60 | 52.17 ± 4.258
| 8 | 55.96 ± 3.58 | 54.93 ± 7.88 | 51.30 ± 2.30 | 51.61 ± 5.07 | 53.549 ± 4.306
| 16 | 61.32 ± 2.83 | 60.70 ± 6.05 | 52.76 ± 2.92 | 56.08 ± 4.77 | 54.504 ± 4.544
Airline (3) | 4 | 54.95 ± 11.81 | 46.29 ± 12.26 | 40.27 ± 8.19 | 70.41 ± 2.45 | 61.305 ± 17.572
| 8 | 61.44 ± 03.90 | 49.81 ± 10.86 | 51.16 ± 7.60 | 70.22 ± 2.47 | 70.585 ± 0.813
| 16 | 62.15 ± 05.56 | 57.25 ± 09.90 | 48.73 ± 6.79 | 71.04 ± 0.64 | 70.553 ± 0.695
Rating Books (3) | 4 | 54.92 ± 6.18 | 38.97 ± 13.27 | 48.44 ± 7.43 | 65.52 ± 8.96 | 69.728 ± 12.633
| 8 | 59.16 ± 4.13 | 46.77 ± 14.12 | 52.13 ± 4.79 | 69.28 ± 0.27 | 72.825 ± 9.53
| 16 | 61.02 ± 4.19 | 51.68 ± 11.27 | 57.28 ± 4.57 | 68.94 ± 0.78 | 72.998 ± 9.566
Political Message (9) | 4 | 15.69 ± 1.57 | 14.49 ± 1.75 | 14.22 ± 1.25 | 19.78 ± 1.10 | 20.091 ± 2.605
| 8 | 18.02 ± 2.32 | 15.24 ± 2.81 | 15.67 ± 1.96 | 21.56 ± 0.65 | 21.8 ± 1.272
| 16 | 18.07 ± 2.41 | 19.20 ± 2.20 | 16.49 ± 1.96 | 23.63 ± 2.47 | 23.046 ± 1.706
Scitail (2) | 4 | 69.50 ± 9.56 | 63.97 ± 14.36 | 76.27 ± 4.26 | 56.11 ± 3.52 | 62.546 ± 7.193
| 8 | 75.00 ± 2.42 | 68.24 ± 10.33 | 78.27 ± 0.98 | 69.36 ± 3.41 | 68.726 ± 5.825
| 16 | 77.03 ± 1.82 | 75.35 ± 04.80 | 78.59 ± 0.48 | 74.18 ± 3.38 | 73.909 ± 3.531
Sentiment DVD (2) | 4 | 80.32 ± 1.02 | 66.36 ± 7.46 | 74.38 ± 2.44 | 84.30 ± 0.96 | 84.88 ± 0.492
| 8 | 80.85 ± 1.23 | 68.37 ± 6.51 | 75.19 ± 2.56 | 84.90 ± 0.75 | 85.02 ± 0.312
| 16 | 81.25 ± 1.41 | 70.29 ± 7.40 | 75.26 ± 1.07 | 84.73 ± 0.52 | 85.05 ± 0.276
Restaurant (8) | 4 | 49.84 ± 3.31 | 50.49 ± 4.40 | 17.36 ± 2.75 | 28.37 ± 2.65 | 16.584 ± 2.874
| 8 | 62.99 ± 3.28 | 58.01 ± 3.54 | 18.70 ± 2.38 | 46.65 ± 3.80 | 18.866 ± 2.513
| 16 | 70.44 ± 2.89 | 66.16 ± 3.46 | 16.41 ± 1.87 | 58.66 ± 1.24 | 19.332 ± 2.326
CoNLL (4) | 4 | 54.16 ± 6.32 | 55.63 ± 4.99 | 32.23 ± 5.10 | 35.97 ± 2.38 | 30.434 ± 4.716
| 8 | 67.38 ± 4.33 | 58.32 ± 3.77 | 34.49 ± 5.15 | 49.04 ± 0.63 | 35.458 ± 2.937
| 16 | 76.37 ± 3.08 | 71.29 ± 3.30 | 33.75 ± 6.05 | 64.55 ± 1.14 | 35.599 ± 2.617
Table 2: Classification using soft-label prototypes is indicated by SLP
indexed by the encoder used. Entries in bold highlight the highest scores. All
models were either trained or meta-trained on the set of GLUE tasks. MTL
denotes multi-task models. * represents the performance of the baselines as
reported by Bansal et al. (2019) in their paper — note that our experimental
setting is identical to theirs.
Table 1 shows the performance of our SLP models and the respective baselines
in the setting where only the few-shot training data for the test task is
available. Classification with soft-label prototypes matches or outperforms
other baselines for the majority of the tasks considered - 22/48 tasks across
all categories report the highest performance with SLPBERT, 22/48 tasks for
BERTbase and 4/48 tasks for 1-NN. Table 2 shows that these percentages
increase for SLPMT-DNN when using an encoder trained on GLUE tasks with SLPMT-
DNN outperforming it’s counterparts on 27/48 tasks. Classification with MT-
DNN, referred to as MTLMT-DNN, is second-best, attaining superior performance
on 10/48 tasks. LEOPARD and Prototypical Networks register the highest
accuracy on 8/48 and 3/48 tasks respectively.
The performance of SLPMT-DNN contrasts strongly with that of LEOPARD – SLPMT-
DNN outperforms the latter in 36/48 tasks. SLPMT-DNN also outperforms MTLMT-
DNN in 28/48 tasks. SLPMT-DNN generally performs poorly in scientific natural
language inference tasks (SciTail) and entity typing tasks (Restaurant and
CoNLL), registering the lowest accuracy scores of the three baselines
(LEOPARD, SLPMT-DNN and MTLMT-DNN) in these sets of tasks.
In the highly-limited training data setting (Table 1), it can be observed (in
$7/16$ tasks) that if there is a consistent increase in performance from 4
shots to 8 shots, a comparable increase in performance from 8 shots to 16
shots is, however, not achieved. This effect is even more pronounced after
meta-training or training on GLUE tasks — this phenomenon is observed in
$(13/16)$ categories and a few categories within these even show a slight
decline in performance $(4/13)$.
A point to note is how closely related the performance of SLPMT-DNN and
prototypical networks is for entity typing tasks in Table 2. ProtoNet also
suffers from similar performance lapses in entity typing tasks and the
performance of ProtoNet and SLPMT-DNN is almost the same - which points to a
common factor between the two approaches which makes it unsuitable for entity
typing tasks. Both these methods use Euclidean distance as a metric for
measuring similarity — prototypical networks optimise the model by evaluating
the performance of (deterministic) support prototypes on the query set and use
Euclidean distance as the similarity measure. Classification with soft-label
prototypes similarly weighs the class distribution of the nearest $k$
prototypes by the Euclidean distance between the test point and the
prototypes. This seems to suggest that Euclidean distance is not the best
similarity measure for tasks which involve multi-dimensional natural language
embeddings. One potential reason is that for fine-grained linguistic
encodings, it is possible to find different classes within the locus
characterised by the same Euclidean distance from a point (i.e.: a circle in
two dimensions). As the number of dimensions increase, it becomes more
important to consider projection in individual dimensions as opposed to a
generalised measure of similarity - for example, using cosine distance versus
Euclidean distance for higher dimensions - for entity typing tasks.
(a)
(b)
(c)
Figure 2: Representing three classes using two soft-label prototypes
##### Depicting Three Classes using Soft Labels with Two Points
Soft-label prototypes have the ability to synthesis new data points, based on
the input data distribution, with a probability distribution of the class
labels. For instance, consider the probability distribution of two prototypes
of the category airline (derived from one episode of few-shot training data
using shots=16 using MT-DNN as the pre-trained encoder). Using the constraints
highlighted in previous sections, we generate two soft-label prototypes which
contain information about three classes depicted in Figure 2. A key point to
note from this is that we only use two soft-label prototypes to represent
three classes — therefore, this is an example of less-than-one shot
representation with respect to the class labels.
##### Training Time
Training time reduces to three operations - obtaining training encodings from
few-shot training datasets, composing lines from class centroids and
constructing soft-label prototypes at the ends of those lines. The total times
taken for each operation are different for different categories due to the
varying number of data points (given by $shots*classes$). As an example, we
take the average of time taken in each operation in all episodes of the
category airline and depict our results in Figure 3.
(a)
(b)
(c)
Figure 3: Average time taken for different training operations for the
category airline. We use MT-DNN as the encoder.
It should be noted nonetheless that times taken were very small for processes
involving the generation of lines and obtaining soft-label prototypes.
Training time, as a whole, was exceptionally low compared to meta-learning
approaches. Additionally, we did not use a GPU for our experiments.
##### Limitations
Classification with soft-label prototypes relies on creating centroids based
on the mean of the data points of a class in a training episode. The class
distribution at a particular point in the input feature space is therefore
highly dependant on the quality of encodings per class which makes the
generation of soft-label prototypes particularly susceptible to noise. This
can also lead to overfitting as the arithmetic mean of the class attempts to
incorporate all points per class, including outliers.
## 8 Conclusion
Soft-label prototypes generate a sublinear representation of data and perform
classification using a simple, computationally efficient version of k-nearest
neighbours classifier competitively with state-of-the-art optimisation-based
methods and even in very complex scenarios. Soft-label prototypes are
generated on few-shot datasets and directly utilise the knowledge captured
within a large language model. They scale well with the complexity of the
language encoder and are able to capture knowledge of large language models
and represent it well in a new target setting — providing comparable
performance with state-of-the-art approaches or improving over them. Sub-
linear representation and classification of data is also a further step
towards algorithmic less-than-one shot learning, wherein we disambiguate $M$
classes using only $N$ instances in total where $M\geq N$.
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|
# From Questions to Insightful Answers: Building an Informed Chatbot for
University Resources
Subash Neupane1, Elias Hossain1, Jason Keith2, Himanshu Tripathi1,
Farbod Ghiasi1, Noorbakhsh Amiri Golilarz1Amin Amirlatifi2, Sudip Mittal1,
Shahram Rahimi1 1 Department of Computer Science & Engineering
Mississippi State University
{sn922, mh3511, ht557, fg289}@msstate.edu,{amiri, mittal,
<EMAIL_ADDRESS>2 Dave C. Swalm School of Chemical Engineering
Mississippi State University
{keith<EMAIL_ADDRESS>
###### Abstract
This paper presents BarkPlug v.2, a Large Language Model (LLM)-based chatbot
system built using Retrieval Augmented Generation (RAG) pipelines to enhance
the user experience and access to information within academic settings. The
objective of BarkPlug v.2 is to provide information to users about various
campus resources, including academic departments, programs, campus facilities,
and student resources at a university setting in an interactive fashion. Our
system leverages university data as an external data corpus and ingests it
into our RAG pipelines for domain-specific question-answering tasks. We
evaluate the effectiveness of our system in generating accurate and pertinent
responses for Mississippi State University, as a case study, using
quantitative measures, employing frameworks such as Retrieval Augmented
Generation Assessment (RAGAS). Furthermore, we evaluate the usability of this
system via subjective satisfaction surveys using the System Usability Scale
(SUS). Our system demonstrates impressive quantitative performance, with a
mean RAGAS score of 0.96, and satisfactory user experience, as validated by
usability assessments.
###### Index Terms:
Chatbot, LLM, RAG, University resources, information access
## I Introduction
Colleges and universities invest significant time and resources into enhancing
their websites to effectively communicate crucial information about the
institution and available campus resources. The institutional website serves
as its “virtual face”, the face it has chosen to present to the online world,
including potential and current students, faculty, parents, alumni and general
users [1]. Although these websites offer comprehensive information, they lack
the capability to provide personalized responses to user queries. For
instance, when a prospective student needs details about submitting ACT
scores, wants to know their tuition and fees, or is unsure which parent’s
information to use on their FAFSA application, they must navigate through
multiple webpages to find answers. This process frequently requires a
considerable amount of time. Yet, at times, users’ queries are left unanswered
due to either unclear information or lack of personal interaction.
Various campus resources and services, such as academic departments, career
centers, admissions, registration, scholarships, and financial aid, are
available to assist students with both academic and non-academic queries.
These resources are equipped with dedicated officers who provide guidance to
students. However, they are constrained by service-time limitation (may only
be available during specific working hours) and may require an appointment,
which might not always accommodate busy student schedules. Additionally,
delays in responses and longer wait times, particularly during application
periods, can diminish prospective students’ interest in the institution,
ultimately affecting university revenue.
Figure 1: Comparative example of completion (response generation) without
using the RAG approach versus using the RAG approach for a given user prompt
related to specific individual at Mississippi State University.
To address these challenges, universities are currently employing
conversational agents, also known as chatbots, to offer support to users.
Chatbots are _“software systems that mimic interactions with real people”_[2]
by engaging in conversation through natural language using machine learning
technology, specifically Natural Language Processing (NLP)[3]. For instance,
Arizona State University (ASU) developed a chatbot named _Sunny_ [4] with the
intentional design to offer emotional support to students, alongside providing
information regarding ASU. Sunny efficiently addresses frequently asked
questions such as inquiries about financial aid, academic advisors, and
accessing ASU email accounts. Similarly, Georgia State University (GSU)
introduced a virtual assistant named _Pounce_ [5] to tackle obstacles to
enrollment faced by students transitioning from high school to college. These
obstacles encompassed tasks like financial aid applications, document
submissions, immunization records, placement exams, and class registration.
GSU reported a notable 22% reduction in summer melt due to Pounce’s
assistance. Beyond admissions and enrollment, universities are increasingly
deploying chatbots to aid students in their academic pursuits and campus life.
One notable example is the chatbot _Beacon_ [6] developed by Staffordshire
University. Beacon offers personalized and responsive support, including
information on timetables and answers to frequently asked questions.
Apart from the higher education sector, chatbots are being increasingly
adopted across a diverse range of industries and contexts including healthcare
[7, 8], cybersecurity[9, 10], retail[11], and hospitality[12] among others due
to their ability to emulate human conversations, automate services, and reduce
human workload. The meteoric rise in interest in using chatbots by industries
at present is attributed to the overwhelming success of
ChatGPT.111https://chatgpt.com/. In fact, the global chatbot market size was
valued at 5.39 billion dollars in 2023 which is expected to reach 42.83
billion dollars by 2033, according to a market research report [13] published
by Spherical Insights & Consulting.
In this paper, we introduce BarkPlug v.2\- the second iteration of a chatbot
system built for Mississippi State University (MSU), with an architecture that
can be applied to any university setting. This system serves as an assistive
tool, capable of leveraging all university resources to provide more
intelligent analyses of university related content. It responds interactively,
making access to relevant information easier for users. Compared to the other
educational chatbots, BarkPlug v.2 is more comprehensive and covers several
aspects of university functions and services. The development of our chatbot
utilizes Retrieval Augmented Generation (RAG) [14] techniques for response
generation. RAG pipelines consist of two vital components: a retriever and a
generator based on a Large Language Model (LLM). We opt for the RAG approach
because pretrained LLMs, such as _gpt-3.5-turbo_ , alone cannot adequately
answer domain-specific questions or perform well on data outside their
training dataset, often resulting in hallucinated outputs. Figure 1 provides a
comparative overview of the response generation for a given user prompt
without RAG and with RAG. As is evident from Figure 1, ChatGPT clearly fails
to answer domain-specific questions, while BarkPlug v.2, which uses the RAG
approach, can accurately answer a user prompt. In our pipeline we utilize
various campus resources such as information on _academic departments,
financial aid, admission, scholarships, dining, housing, and health center_ as
a corpus of external data source for retrieval.
BarkPlug v.2 project’s key contributions include:
* •
Design and development of a comprehensive chatbot system proficient in
responding to a wide spectrum of queries pertaining to the diverse array of
campus resources available at Mississippi State University.
* •
Demonstrating the possibility of promptly providing personalized, real-time
information, thereby augmenting user engagement through the continuous
availability of the chatbot.
* •
Showcasing the application’s effectiveness through rigorous evaluation,
validating its performance and user satisfaction.
The rest of this article is divided into five connected sections. In Section
II, we present the background and related work. Following that, in Section
III, we explain the architecture and methodology. Section IV provides a
detailed analysis of experimental results. Moving on to Section V, we provide
implementation details and discuss the limitations and future works. Finally,
we conclude our paper.
## II BACKGROUND AND RELATED WORK
In this section, we briefly look into the pre-requisite background followed by
exploring related research that focuses on development of chatbot applications
in educational context.
### II-A Large Language Models (LLMs)
Large Language Models (LLMs) like GPT-4, LLAMA3, and PaLM are at the forefront
of computational linguistics, powered by Transformer-based architectures [15]
with vast parameter spaces, often exceeding hundreds of billions. These models
rely on the self-attention mechanism within Transformers. LLMs excel in
understanding and generating human language, reshaping the Natural Language
Processing (NLP) landscape. They leverage various Transformer architectures
and pre-training objectives, including decoder-only models (e.g., GPT2, GPT3),
encoder-only models (e.g., BERT, RoBERTa), and encoder-decoder architectures
like BART.
These architectures efficiently process sequential data, capturing intricate
dependencies within text while enabling effective parallelization. LLMs
integrate prompting or in-context learning, enhancing text generation by
incorporating contextual information. This capability facilitates coherent and
contextually relevant responses, fostering interactive question-and-answer
engagements [16].
### II-B Retrieval Augmented Generation (RAG)
Pre-trained Large Language Models (LLMs) are proficient at acquiring extensive
knowledge but lack memory expansion or revision capabilities, leading to
errors like hallucinations. To address this, hybrid approaches like Retrieval
Augmented Generation (RAG) have emerged [17, 18, 14].
RAG integrates input sequences with information retrieved from corpus of an
external data source, enriching context for sequence generation. The retriever
component selects the top $k$ text passages relevant to the input query,
augmenting the model’s understanding and enhancing output sequence generation.
This process is governed by the equation: $p_{n}(z|x)$ where $p_{n}$
represents the retriever component with parameters $n$ (number of documents or
passages a user wants to retrieve), selecting relevant passages $z$ from the
knowledge database given input $x$.
### II-C Related Works
Recent research on educational chatbots explores various areas such as
application fields, objectives, learning experiences, design approaches,
technology, evaluation methods, and challenges. Studies have shown that
educational chatbots are used in health advocacy, language learning, and self-
advocacy. They can be flow-based or powered by AI, facilitating answering
Frequently Asked Questions (FAQs), performing quizzes, recommending
activities, and informing users about various events [19][20]. Chatbots have
been found to improve students’ learning experiences by motivating them,
keeping them engaged, and providing immediate online assistance [21].
Additionally, chatbots make education more accessible and available [20].
Design aspects such as the role and appearance of chatbots are significant
factors that affect their effectiveness as educational tools [22]. Chatbots
are designed using various methods, including flow-based and AI-based
approaches, and can incorporate speech recognition capabilities [23].
Technologies used to implement chatbots include Dialogflow
222https://cloud.google.com/dialogflow and ChatFuel333https://chatfuel.com/
among others. These technologies impact chatbot performance and quality,
necessitating careful selection during design and development [24]. Flow-based
chatbots, such as those powered by Dialogflow, can provide structured
interactions based on predetermined scripts, while AI-based chatbots leverage
machine learning and NLP to offer more flexible and dynamic interactions.
In regards to assessment of the effectiveness of educational chatbots,
evaluation methods such as surveys, experiments, and evaluation studies are
used, measuring acceptance, motivation, and usability [25][24][26]. Surveys
gather feedback from students and educators regarding their experiences with
chatbots, while experiments may involve testing chatbots in controlled
settings to measure their impact on learning outcomes. Evaluation studies
provide deeper insights into how chatbots perform in various educational
scenarios and how users perceive their usefulness. In terms of interaction
styles, research examines whether chatbots are user-driven or chatbot-driven,
depending on who controls the conversation [23][19]. Chatbot-driven
interactions often involve more automated and guided conversations, while
user-driven interactions prioritize user input. Striking a balance between
these approaches can result in more natural and effective communication.
However, it’s important to acknowledge that achieving this balance
necessitates addressing substantive challenges to optimize the chatbot’s
applicability across diverse contexts, including the field of education.
Ethical considerations, such as compliance with educational norms and
safeguarding user data, assume paramount importance [21, 27]. Leveraging novel
methodologies in their development, we aim to navigate these issues more
effectively. Moreover, we confront persistent programming complexities and the
importance of sustaining chatbot utility amidst educational evolution [28,
29]. By harnessing advancements in technology, we endeavor to bolster our
chatbots’ resilience to these challenges. These collaborative endeavors offer
a strategic direction, utilizing technological advancements to refine
educational chatbots. Furthermore, the language model (conversational chatbot)
contends with conceptual challenges essential for its operational efficacy,
requiring careful research focus.
Insights from studies such as [30] reveal how language models such as BERT
establish relationships between expressions and queries, shedding light on
chatbot interaction styles and response quality. This study contributes to
understanding how advanced language models can be integrated into chatbots for
more nuanced and context-aware responses. [31] discusses the gap between
chatbot responses and user intent, which can be more pronounced in complex
university settings. Chatbots in academic environments often encounter
questions that require a deep understanding of the subject matter and context.
This necessitates the use of sophisticated models that can handle intricate
queries and provide accurate and relevant responses. [30, 31] underscores the
importance of understanding and controlling the context of language models,
thereby guiding our efforts to integrate advanced language models into
chatbots for more nuanced and context-aware responses. Their context-aware
approach has been instrumental in shaping our chatbot’s unique capabilities.
The integration of chatbots within university platforms and metaverses offers
promising avenues for enhancing user experience and facilitating learning. For
instance, [32] demonstrate how chatbots in metaverse-based university
platforms offer instant, personalized support for tasks such as course
navigation and answering FAQs, leveraging NLP and machine learning to
streamline information dissemination and reduce administrative burdens. This
kind of integration not only facilitates academic processes but also helps in
addressing students’ concerns promptly, ensuring smoother academic
experiences. In specific university contexts, [33] develops a question-
answering system for an Indonesian university admissions using Sequence-to-
sequence learning. This system demonstrates how chatbots can be employed in
specialized areas to address particular challenges, such as providing guidance
during the admissions process. Similarly, [34] introduce a dynamic chatbot
enhancing student interaction by covering admissions, academic assistance, and
event information, prioritizing user feedback for accuracy, reliability, and
safety. Frequent updates ensure that chatbots maintain relevance and continue
to serve as effective tools for student support. Moreover, [35] presents
TutorBot+, which employs LLMs like ChatGPT to offer feedback in programming
courses. Their quasi-experimental research shows positive impacts on students’
computational reasoning abilities, illustrating the potential of such
interventions in education. TutorBot+ demonstrates the benefits of integrating
advanced AI models to support students in understanding complex programming
concepts, potentially transforming how computational subjects are taught.
## III BarkPlug v.2 Architecture & Methodology
Figure 2: Overview of BarkPlug v.2’s two phase architecture. The first phase
_Context retrieval_ is responsible to retrieve relevant documents based on the
user prompt. The second phase, _Completion_ responsible of generating
personalized responses utlilizing retrieved documents as context along with
user prompt.
This section describes the architecture of BarkPlug v.2 consisting two main
phases: _context retrieval_ and _completion_ as shown in Fig. 2. The first
phase retrieves documents relevant to the user prompt. The second phase
utilizes these retrieved documents and user prompts to generate contextual
responses referred to as completions. The subsequent subsections will provide
a comprehensive breakdown of each phase, discussing their functionalities and
methodologies.
Figure 3: Similarity score threshold retrieval.
### III-A Context Retrieval
Retrieval in BarkPlug v.2 involves obtaining pertinent information from an
external data source to establish context for completions. This phase takes a
prompt (query) as an input and produces chunks of documents relevant to the
prompt. In our context, the external data is the university resources
available through Mississippi State University’s Website
444https://www.msstate.edu/. We curate data of 42 different department within
the university using web crawlers. These include _academic departments,
financial aid, admissions, housing, dinning services, library, health center_
etc. Inclusion of these campus resources as external data source is to ensure
BarkPlug v.2 is comprehensive enough to answer diverse question. For example,
a user might ask a question such as _“What are the funding opportunities
available for graduate students in the CSE department?"_. Followed by the
question _“Who do I contact if I have additional questions about majors or
attending MSU?"_ To answer the first question the system should have
information about funding opportunities within CSE, whereas to answer the
second question, information about _academic counselors_ should be present in
the external data source. Please refer to Section IV-A for detailed
explanation on how we curate these data, prepare and process for this phase.
The first step in this phase is to transform the external data source. This
step relies on two important components: an _embedding model_ and a _vector
database_. Embeddings refers to functions that map or transforms raw input
data to low-dimensional vector representations while retaining important
semantic information about the inputs [36]. On the other hand, the vector
database is a type of database that stores data as high-dimensional vectors
that are usually generated by applying embedding functions to the raw data
[37], such as text in our case. It supports complex and unstructured data and
allows fast and accurate similarity search and retrieval. BarkPlug v.2
utilizes an embedding model to vectorize the external data sources, in
particular, we leverage a _text-embedding-3-large_ model managed through API
calls. These vectors are subsequently stored in Chroma DB [38] an in-memory
vector database.
For efficient context retrieval process we use vector store-backed retriever
technique provided by LangChain [39]. It utilizes vector store to retrieve
documents. In general the vector store retriever uses two types of search
methods including Maximum Marginal Relevancy (MMR) and Similarity Search. In
this work, we have leveraged _similarity score threshold retrieval_ , depicted
in Fig.3 as our searching mechanism. This searching strategy returns all the
possible results to a user prompt based on the minimum similarity threshold we
define. The output of this phase is the relevant documents that serves as
context for subsequent completions phase.
### III-B Completion
The second phase is the _completion_ which is also referred to as _response
generation_. We utilize a gpt based LLM for completions, in particular, we
leverage OpenAI’s _gpt-3.5-turbo_ as our base generator model. The input to
the generator consists of retrieved document chunks and the user prompt. Then,
the generator model, or LLM, uses this information as a guideline to produce
acurate and relevant responses (completions). Response generation in BarkPlug
v.2 is managed through OpenAI’s API calls.
An example of completion for the user prompts _"What are the funding
opportunities available for graduate students in the CSE department?"_ and
_"Who do I contact if I have questions about attending MSU_ can be seen in
Fig. 4. In the first example, a user seeks information about funding
opportunities within the CSE department for graduate studies. The completion,
in this case, is a contextualized response generated by combining relevant
information retrieved from campus resources, such as the academic department
(CSE), with the user prompt. In the second example, the user is seeking
information about admission to Mississippi State University. To answer this,
the system must possess knowledge about campus resources, such as admissions
and scholarships. As mentioned earlier, BarkPlug v.2 not only has knowledge
about academic departments but also other campus resources, including
admissions and scholarships. By utilizing admissions and scholarships as
external data sources, the system returns the relevant document reflecting
information about admissions and scholarships, along with contact details, as
a completion for the user prompt.
Prompt: What are the funding opportunities available for graduate students in
the CSE department?
Completion: Graduate students in the Computer Science and Engineering (CSE)
department at Mississippi State University have access…funding
opportunities…graduate assistantships, research grants, fellowships,
scholarships, and industry partnerships…research grants cover project
expenses.
Prompt: Who do I contact if I have questions about attending MSU?
Completion: For queries on admissions at Mississippi State University, reach
out to the Office of Admissions and Scholarships at 662-325-2224 or
<EMAIL_ADDRESS>
Figure 4: A graphical depiction of response generation. Red highlights
indicate BarkPlug v.2’s ability to contextualize and generate factually
grounded response.
## IV Experiments and Results
To evaluate the performance of our system, we adopt a two-fold approach
including both _quantitative_ and _usability assessment_ methods. For the
quantitative evaluation (See Section IV-B), we utilize the RAGAS [40]
framework, while the SUS is adopted for usability assessment (See Section
IV-C). In the following subsection, we first discuss the dataset and steps we
took to prepossess them and then provide a detailed explanation of our
evaluation approaches.
### IV-A Dataset Description & Preparation
To ensure a comprehensive chatbot system capable of answering diverse
questions—whether academic or non-academic—we initially developed a web
scraper to gather information on various campus resources at Mississippi State
University. This collection would then serve as an external data source in our
pipeline. We scraped various campus resources including academic departments,
financial aid, scholarships, housing, dining, parking, and police. In total,
we scraped 42 campus resources into a JSON file. Each JSON file includes the
following information: the URL, title, and content of the scraped webpage, all
wrapped into a JSON object. We consolidated the individual files into a master
JSON file which serves as an external data source and is ingested into our RAG
pipeline. A subset of the data utilized by BarkPlug v.2 can be observed in
Table I.
To enhance retrieval accuracy, we first preprocess the JSON file. This
preprocessing step involves removing noise, such as undesirable Unicode
characters, redundant, and unnecessary information. We then implement a
recursive chunking strategy, with a chunk size of 8000 and an overlap of 1200
characters. This step is crucial for optimizing the performance of RAG chatbot
systems with the objective of ensuring that our chatbot generates an accurate
response that is contextually appropriate. Subsequently, we transformed the
textual data into vectorized representations utilizing an _embedding model_
(Refer to Section III-A to learn for more details on embedding models.).
TABLE I: A subset of an external data source containing campus resources, including both academic and non-academic departments, indicating the total number of tokens associated with each. | Departments | # of Tokens
---|---|---
Campus Resources | Computer Science and Engineering | 200623
Chemical Engineering | 118271
Electrical and Computer Engineering | 328558
Industrial and Systems Engineering | 22390
Agricultural and Biological Engineering | 79978
Civil and Environmental Engineering | 61071
Aerospace Engineering | 37812
Biomedical Engineering | 256761
Housing | 132193
Admission | 276972
MSU Police | 16629
### IV-B Quantitative Evaluation
To evaluate BarkPlug v.2’s ability to produce contextually appropriate
responses, we utilize the RAGAS framework [40]. We choose this framework
because it is specifically designed to assess RAG pipelines. Other popular
evaluation metrics such as ROUGE [41] and BLEU [42] are not suitable in our
context. This is because ROUGE is generally used to evaluate summarization
tasks, while BLEU is designed to evaluate language translation tasks.
TABLE II: Overview of results: Retrieval scores pertain to the _context retrieval_ phase of the architecture, where _prec._ refers to context precision, and recall refers to context recall. Generation scores pertain to the _completion phase_ , where _faith_ stands for faithfulness and _rel._ for answer relevancy. The end-to-end evaluation showcases BarkPlug v.2’s efficiency in generating contextually relevant and accurate answers through metrics such as answer similarity and answer correctness. | Retrieval | Generation | RAGAS Score | End-to-End Evaluation
---|---|---|---|---
Category | Prec. | Recall | Faith. | Rel. | Harmonic Mean | Answer Similarity | Answer Correctness
Engineering Programs | 0.98 | 0.96 | 0.99 | 0.97 | 0.97 | 0.8434 | 0.8620
General Inquiry | 0.95 | 0.97 | 0.98 | 0.96 | 0.96 | 0.7764 | 0.8123
Research Opportunities | 0.97 | 0.98 | 0.96 | 0.99 | 0.97 | 0.8245 | 0.8841
University Resources | 0.96 | 0.99 | 0.97 | 0.98 | 0.97 | 0.8317 | 0.8923
We evaluate both phase of BarkPlug v.2 architecture (See section III) i.e.
_context retrieval_ and _completion_. To evaluate the retrieval, we employ two
metrics such as _context precision and context recall_. The first metric
represents the Signal-to-Noise Ratio (SNR) of retrieved context, while the
second metric evaluates whether the retriever has the ability to retrieve all
the relevant evidence to answer a question. Similarly, to evalaute _completion
or generation_ we employ _faithfullness_ and _answer relevance_ metrics.
Faithfulness evaluates how factually accurate the generated answer is while
answer relevance evaluates how relevant the generated answer is to the
question. The final RAGAS score, representing the harmonic mean of these four
metrics, falls within a range of 0 to 1, with 1 denoting optimal generation.
This score serves as a singular measure of a QA system’s performance.
Therefore, the RAGAS score is essential for assessing the overall performance
and relevance of BarkPlug v.2 in its targeted educational environments.
To conduct phase wise evaluation, we first crafted a set of questions and
their ground truth pertaining to _engineering programs, general inquiries,
research opportunities_ , and _other university resources_. We report RAGAS
score of 0.97, 0.96, 0.97 and 0.97 for these categories respectively in Table
II. These score underlines both retrieval and completion component are
efficient.
We also conduct end-to-end evaluation to measure overall performance of
BarkPlug v.2, as it directly affects the user experience. Metrics such as
_answer similarity_ and _answer correctness_ are employed to assess the
overall performance, ensuring a comprehensive evaluation. In particular,
_answer similarity_ scores that reflect strong alignment with ideal responses
are reported to be high in cases when questions about engineering programs and
research opportunities are asked, with scores of 0.8434 and 0.8317
respectively. Moreover, _answer correctness_ , which indicates high factual
accuracy, is reported to be high when the system is asked questions about
university resources and research opportunities, at 0.8923 and 0.8841
respectively. Overall, these metrics suggest that BarkPlug v.2 effectively
retrieves relevant and accurate answers.
### IV-C Usability Assessment
To further understand the user experience when using BarkPlug v.2, we perform
a subjective satisfaction survey using the System Usability Scale (SUS) [43]
\- a widely reliable method that accesses systems usability through set of
questionnaire. Given the expensive nature of this evaluation we engage a panel
of 50 graduate and undergraduate students undertaking CSE8011 (Seminar course)
at Mississippi State University. The participants were tasked to answer set of
10 questions as depicted in 5, each offering five response options ranging
from “strongly agree” to “strongly disagree”. We then collected their feedback
and calculated an average SUS score of 67.75. The feedback results indicated
satisfactory usability with a room for improvement for future iterations of
our system.
Figure 5: Distribution of average System Usability Scale (SUS) scores.
## V Detailed Analysis and Insights
In this section, we discuss implementation details, where we explain the
technical process behind developing BarkPlug v.2. Then, we discuss constraints
and shortcomings encountered, and provide our plan for the future.
### V-A Implementation Detail
For data curation we employed a multi-thread web crawler with the Scrapy
Python library to collect data from over 42 campus resources (See Section IV-A
for details). We carefully selected important HTML div tags that comprised of
relevant information about a topic. This process was semi-automatic in nature
because every HTML pages were differently formatted with different div ids.
Manual div selection also allowed us to remove noise to some extent. The data
was exported to JSON file format with url, topic and content. Individual JSON
files for each of the campus resources was then consolidated into a master
JSON file for comprehensive retrieval.
We predominantly use LangChain framework to develop BarkPlug v.2. First, we
preprocess master JSON into smaller chunks using Recursive Character Text
Splitter splitting strategy. Given the nature of our data we opted for 8000
chunk size with 1200 overlap. We then apply an embedding function on these
chunks utilizing OpenAI’s _text-embedding-3-large_ model and store the vectors
in Chroma DB. This step allowed us to retrieve documents relevant to specific
user prompts. In our case, we utilize _vectorstore_ for _context retrieval_
with a similarity search threshold as our search strategy (See Section III-A
for more details). For completion or response generation we leverage OpenAI’s
_gpt-3.5-turbo_ model. Both embedding and response generation is managed
through API calls.
BarkPlug v.2 is built with Django framework using python. For front-end we
utilize HTML, CSS and Javascript. The current version of our system has not
only question-answering functionality but also user sign up and log in
feature. Once a user is registered they can ask queries, they can see previous
conversations, delete conversations, and email conversations. Our application
is deployed through a third-party cloud service for accessibility.
### V-B Limitations & Future Direction
Despite the achievements in developing our educational chatbot, several
significant challenges currently limit its broader application.BarkPlug v.2
does not currently have Automatic Speech Recognition (ASR) capability, which
might hinder its use among visually impaired, disabled, or elderly users.
Additionally, given that Mississippi State University hosts a number of
international students annually from non-English speaking countries, it
currently lacks multi-lingual support. In terms of technical limitations, our
retrieval system sometimes fails to provide accurate or relevant results,
occasionally producing incorrect information, a phenomenon known as
‘hallucinations’. We are also limited by a maximum number of output tokens,
which is 4096, and a context window of 16k. This sometimes hinders system’s
ability to capture the full length of the conversation in the memory buffer.
To address the limitations discussed above and enhance BarkPlug v.2’s
functionality and usability, we are planning several key upgrades. These
include adding support for multiple languages to cater to a diverse user base,
integrating ASR and text conversion features to enable various interaction
modes, and improving the retrieval algorithms to boost the accuracy and
relevance of the information provided. Moreover, in response to the token
limitations of the OpenAI API, we aim to apply the map-reduced document chain
approach from LangChain. Through these improvements, we aim to transform
BarkPlug v.2 into a more reliable and accessible educational tool.
## VI Conclusion
This study highlights the significant potential of AI-based chat systems in
improving communication and access to information regarding university
resources. Our system, BarkPlug v.2 integrates large amounts of university
data, including academic programs, campus facilities, student service as
external data corpus into its RAG pipelines for domain-specific question and
answering tasks. By incorporating this external data corpus, our system
ensures the delivery of precise and contextually relevant responses to both
academic and non-academic user inquiries. The comprehensive end-to-end
evaluation process demonstrated BarkPlug v.2’s efficiency in generating
contextually relevant and accurate answers as measured by metrics such as
answer similarity and correctness. Furthermore, system usability experiments
employing the SUS indicated that BarkPlug v.2 is practical and effective for
real-world usage, affirming its reliability and the positive user experience
it offers. The positive outcomes of using BarkPlug v.2 at Mississippi State
University suggest promising opportunities for broader implementation. This
system could be adapted for use in other universities or different sectors and
can be viewed as enterprise document retrieval systems that enhance user
engagement and information access.
## Declaration of Competing Interest
The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.
## Acknowledgement
This work was supported by the PATENT Lab (Predictive Analytics and Technology
Integration Laboratory) at the Department of Computer Science and Engineering,
Mississippi State University.
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|
mythm[theorem]Theorem mylmm[theorem]Lemma University of Southern California,
USA and https://jaredraycoleman.com
[email protected]://orcid.org/0000-0003-1227-2962 Carleton University,
Ottawa, Ontario, Canada and https://people.scs.carleton.ca/~kranakis/
[email protected]://orcid.org/0000-0002-8959-4428Research
supported in part by NSERC Discovery grant. Wesleyan University, Middletown
CT, USA and http://dkrizanc.web.wesleyan.edu/<EMAIL_ADDRESS>California
State University, Long Beach, CA, USA and https://home.csulb.edu/~omorales/
[email protected]://orcid.org/0000-0002-9645-1257 Jared
Coleman and Evangelos Kranakis and Danny Krizanc and Oscar Morales Ponce
<ccs2012> <concept> <concept_id>10003752.10003809.10010047</concept_id>
<concept_desc>Theory of computation Online algorithms</concept_desc>
<concept_significance>300</concept_significance> </concept> <concept>
<concept_id>10003752.10003809.10010047.10010051</concept_id>
<concept_desc>Theory of computation Adversary models</concept_desc>
<concept_significance>300</concept_significance> </concept> </ccs2012>
[300]Theory of computation Online algorithms [300]Theory of computation
Adversary models John Q. Open and Joan R. Access 2 42nd Conference on Very
Important Topics (CVIT 2016) CVIT 2016 CVIT 2016 December 24–27, 2016 Little
Whinging, United Kingdom 42 23
# Line Search for an Oblivious Moving Target
Jared Coleman Evangelos Kranakis Danny Krizanc Oscar Morales-Ponce
###### Abstract
Consider search on an infinite line involving an autonomous robot starting at
the origin of the line and an oblivious moving target at initial distance
$d\geq 1$ from it. The robot can change direction and move anywhere on the
line with constant maximum speed $1$ while the target is also moving on the
line with constant speed $v>0$ but is unable to change its speed or direction.
The goal is for the robot to catch up to the target in as little time as
possible.
The classic case where $v=0$ and the target’s initial distance $d$ is unknown
to the robot is the well-studied “cow-path problem”. Alpert and Gal
[alpern2003theory] gave an optimal algorithm for the case where a target with
unknown initial distance $d$ is moving away from the robot with a known speed
$v<1$. In this paper we design and analyze search algorithms for the remaining
possible knowledge situations, namely, when $d$ and $v$ are known, when $v$ is
known but $d$ is unknown, when $d$ is known but $v$ is unknown, and when both
$v$ and $d$ are unknown. Furthermore, for each of these knowledge models we
consider separately the case where the target is moving away from the origin
and the case where it is moving toward the origin. We design algorithms and
analyze competitive ratios for all eight cases above. The resulting
competitive ratios are shown to be optimal when the target is moving towards
the origin as well as when $v$ is known and the target is moving away from the
origin.
###### keywords:
Infinite Line, Knowledge, Oblivious, Robot, Search, Search-Time, Speed, Target
## 1 Introduction
Search is important to many areas of computer science and mathematics and has
received the attention of numerous studies. In the simplest search scenario,
one is interested in the optimal trajectory of a single autonomous mobile
agent (also referred to simply as a robot) tasked with finding a target placed
at an unknown location on the infinite line. The line search problem is to
give an algorithm for the agent so as to minimize the competitive ratio
defined as the supremum over all possible target locations of the ratio of the
time the agent takes to find the target and the time it would take if the
target’s initial position was known to the robot ahead of time. This classic
problem has led to many variations (see [alpern2003theory] for more on its
history).
In this paper we consider an extension of the line search problem involving an
autonomous robot and an oblivious moving target. The search is again performed
on an infinite line and concerns an autonomous robot starting at the origin of
the line but differs from the previously studied case in that the search is
for a moving target whose speed and direction are not necessarily known to the
searching robot. The robot starts at the origin and the target at an arbitrary
distance $d$ from the origin. The target is moving with constant speed and is
oblivious in that it cannot change its speed and/or direction of movement. We
consider and analyze several alternative knowledge-based scenarios in which
the target’s speed and initial distance from the origin may be known or
unknown to the searching robot. The case where a target with unknown initial
distance from the origin is moving away from the origin was solved by Alpern
and Gal [alpern2003theory]. As far as we are aware, these are the first
results for the remaining cases.
### 1.1 Notation and terminology
On the infinite real line, consider an autonomous robot which is initially
placed at the origin whose maximum speed is $1$ and an oblivious robot (also
referred to as the moving target) initially placed at a distance $d$ to the
right or left of the origin and moving with constant speed $v>0$. As is
usually done in linear search and in order to avoid trivial considerations on
the competitive ratio by adversarially placing the target very close to the
robot, we assume that $d$ is not smaller than the unit distance, i.e., $d\geq
1$.
The target may be moving away from or toward the origin. If it is moving away,
we assume its speed is strictly less than $1$ as otherwise the problem can not
be solved. Further, we assume that the autonomous robot knows the direction
the target is moving (away from or toward the origin). The search is completed
as soon as the robot and target are co-located.
The movement of the autonomous robot is determined by a trajectory which is
defined as a continuous function $t\to f(t)$, with $f(t)$ denoting the
location of the robot at time $t$. Moreover, it is true that
$|f(t)-f(t^{\prime})|\leq u|t-t^{\prime}|$, for all $t,t^{\prime}$, where $u$
is the speed of the agent (be that the searching robot or the oblivious
target). The autonomous robot can move with its own constant speed and during
the traversal of its trajectory it may stop and/or change direction
instantaneously and at any time as specified by the search algorithm.
A search strategy is a sequence of movements followed by the robot. The
competitive ratio of a search strategy $X$, denoted $CR_{X}$, is defined as
the supremum over all possible initial target locations and speeds of the
ratio of the time the agent takes to find the target and the time it would
take if the target’s initial position was known to the robot ahead of time.
The competitive ratio of a certain type of search problem is the infimum of
$CR_{X}$ taken over all possible strategies $X$ for this problem. By abuse of
notation we may drop mention of $X$ when this is easily implied from the
context.
Our goal in this paper is to prove bounds on the competitive ratios of
algorithms under four different knowledge models:
1. 1.
FullKnowledge: The robot knows both the target’s speed $v$ and its initial
distance $d$.
2. 2.
NoDistance: The robot knows the target’s speed $v$ but not its initial
distance $d$.
3. 3.
NoSpeed: The robot knows the target’s initial distance $d$, but not its speed
$v$.
4. 4.
NoKnowledge: The robot knows neither the target’s speed $v$ nor its initial
distance $d$.
For all knowledge models, the robot does not know the target’s initial
position. We study each of the above knowledge models for the case where the
target is moving toward the origin (Toward) and where it is moving away (Away)
from the origin. In each case, we assume the robot knows the direction of
travel of the target.
### 1.2 Related Work
Several research papers have considered the search problem for a robot
searching for a static (fixed) target placed at an unknown location on the
real line, see [BCR93, schuierer2001lower]. The problem was first
independently considered in a stochastic setting by Bellman and Beck in the
1960’s (cf. [beck1964linear, bellman1963optimal] as well as [BCR93,
schuierer2001lower]). In a deterministic setting it is now well known that the
optimal trajectory for this single agent search uses a doubling strategy whose
trajectory attains a competitive ratio of $9$. Linear search has attracted
much attention and been the focus of books including [ahlswede1987search,
alpern2003theory, stone1975theory].
The case of a moving target appears to have been first considered by McCabe
[mccabe1974]. In that paper, the problem of searching for an oblivious target
that follows a Bernoulli random walk on the integers is considered. For the
case of a deterministic oblivious searcher, the only result we are aware of us
is found in Alpern and Gal [alpern2003theory]. There they consider the case
where the target is moving away from the origin at a constant speed $v<1$
which is known to the searching robot. Only the initial distance of the target
is unknown. They give an algorithm with optimal competitive ratio for this
case.
Our problem is reminiscent of the problem of catching a fugitive in a given
domain which is generally referred to as the cops and robbers problem
[anthony2011game]. The main difference is that in those problems, the target
(robber) is itself an autonomous agent. As a result, the techniques considered
there do not apply to our case.
Our problem is also related to rendezvous (of two robots) on an infinite line
but it differs because in our case only one of the robots is autonomous while
the other is oblivious. Related studies on the infinite line include
rendezvous with asymmetric clocks [czyzowicz2018linear] and asynchrnous
deterministic rendezvous [de2006asynchronous]. More recent work on linear
search concerns searching for a static target by a group of cooperating
robots, some of which may have suffered either crash
[czyzowicz2019searchcrash] or Byzantine [czyzowicz2021searchbyz] faults.
### 1.3 Results of the paper
In all situations considered it is unknown to the robot whether the target is
initially to the left or to the right of the origin. We analyze the
competitive ratio in four situations which reflect what knowledge the robot
has about the target. We present results on the FullKnowledge model (the robot
knows $v$ and $d$) in Section 2, the NoDistance model (the robot knows $v$ but
not $d$) in Section 3, the NoSpeed model (the robot knows $d$ but not $v$) in
Section 4, and the NoKnowledge model (the robot knows neither $v$ nor $d$) in
Section 5. For each of these models we study separately the case when the
target is moving away or toward the origin (this knowledge being available to
the robot). The results are summarized in Table 1. We conclude with a summary
and additional open problems.
Knowledge | Movement | Competitive Ratio | Section
---|---|---|---
$v,d$ | Away | $CR=1+\frac{2}{1-v}$ | 2.1
Toward | | $CR=1+\frac{2}{1+v}$ if $v<1$
---
$CR=1+\frac{1}{v}$ otherwise
2.2
$v$ | Away | $CR=1+8\frac{1+v}{(1-v)^{2}}$ | 3.1 [alpern2003theory]
Toward | | $CR=1+\frac{1}{v}$ if $v\geq\frac{1}{3}$
---
$CR=1+8\frac{1-v}{(1+v)^{2}}$ otherwise
3.2
$d$ | Away | | $CR\leq 5$ if $v\leq\frac{1}{2}$
---
$CR\leq 1+16\frac{\left(\log\frac{1}{1-v}\right)^{2}}{(1-v)^{4}}$ otherwise
4.1
Toward | $CR=3$ | 4.2
$\emptyset$ | Away | | $CR\leq 1+\frac{16}{d}\left[\log\log\left(\max\left(d,\frac{1}{1-v}\right)\right)+3\right]$
---
$\cdot\max\left(d,\frac{1}{1-v}\right)^{8}\cdot\log^{2}\left[\max\left(d,\frac{1}{1-v}\right)\right]$
5.1
Toward | $CR=1+\frac{1}{v}$ | 5.2
Table 1: Table of competitive ratio bounds proven for each knowledge model for
cases with the target moving away from or towards the origin with speed $v$
and initial distance $d$ from the robot which is moving with speed $1$.
Equalities indicate that tight upper and lower bounds are proven.
## 2 The FullKnowledge Model
We first study the model where the robot knows the target’s speed $v$ and its
initial distance from the origin $d$.
### 2.1 The FullKnowledge/Away Model
For the case when the target is moving away from the origin, clearly if $v\geq
1$ then the robot can never catch the target. Thus, for this model (and all
other Away models), we assume $v<1$. In this section, we will analyze an
algorithm where the robot chooses a direction and moves for time
$\frac{d}{1-v}$. If the robot does not find the target after moving for time
$\frac{d}{1-v}$ in one direction, then it changes direction and continues
moving until it does.
Algorithm 1 Online Algorithm for FullKnowledge/Away Model
1:input: target speed $v$ and initial distance $d$
2:choose any direction and go for time $\frac{d}{1-v}$
3:if target not found then
4: change direction and go until target is found
###### Theorem 2.1.
For the FullKnowledge/Away model, Algorithm 1 has an optimal competitive ratio
of
$1+\frac{2}{1-v}.$ (1)
###### Proof 2.2.
By Algorithm 1, the robot goes in one direction for a time $\frac{d}{1-v}$.
Observe that if the robot does not encounter the target after this amount of
time, it must be on the opposite side of the origin (in the other direction).
At the time the robot changes direction, its distance to the target will be
equal to $\frac{d}{1-v}+d+\frac{dv}{1-v}=\frac{2d}{1-v}$. Thus, the total time
required until the robot catches up to the target is at most
$\frac{d}{1-v}+\frac{2d}{(1-v)^{2}}.$
Clearly then, the competitive ratio is at most
$\frac{\frac{d}{1-v}+\frac{2d}{(1-v)^{2}}}{\frac{d}{1-v}}=1+\frac{2}{1-v}$
which is as claimed in Equation (1) above.
Optimality follows from the fact that regardless of which direction the robot
chooses to travel, the adversary can place the target in the opposite
direction. Moreover, for the robot to catch up to the target it must visit one
of the points $\pm\frac{d}{1-v}$. If the robot visits location $\frac{d}{1-v}$
to the right (resp. left) the adversary places the target on the left (resp.
right). Therefore the completion time will be at least
$\frac{d}{1-v}+\frac{2d}{(1-v)^{2}}$. This shows the upper bound is tight and
completes the proof of Theorem 2.1.
### 2.2 The FullKnowledge/Toward Model
Consider the following algorithm which is similar to Algorithm 1.
Algorithm 2 Online Algorithm for the FullKnowledge/Toward Model
1:input: target speed $v$ and initial distance $d$
2:choose any direction and go for time $\frac{d}{1+v}$
3:if target not found then
4: change direction and go until target is found
###### Theorem 2.3.
For the FullKnowledge/Toward model, Algorithm 2 has competitive ratio at most
$1+\frac{2}{1+v}.$ (2)
###### Proof 2.4.
The robot goes in one direction for a time $\frac{d}{1+v}$. If the robot finds
the target in this time, the algorithm is clearly optimal. If, however, the
robot does not find the target, then it must be on the opposite side of the
origin (in the other direction). If this is the case, then by time
$\frac{d}{1+v}$ the target has moved a distance $\frac{dv}{1+v}$ and is at
distance $d-\frac{dv}{1+v}=\frac{d}{1+v}$ from the origin. Therefore at the
time the robot changes direction, the distance between robot and target is
$\frac{2d}{1+v}$. Thus, the robot will encounter the target in additional time
$\frac{2d}{(1+v)^{2}}$. It follows that the total time required for the robot
to meet the target is $\frac{d}{1+v}+\frac{2d}{(1+v)^{2}}$ and the resulting
competitive ratio satisfies
$CR\leq\frac{\frac{d}{1+v}+\frac{2d}{(1+v)^{2}}}{\frac{d}{v+1}}=1+\frac{2}{1+v}.$
This completes the proof of Theorem 2.3.
###### Theorem 2.5.
For the FullKnowledge/Toward model, the competitive ratio of any online
algorithm is at least $1+\frac{2}{1+v}$, provided that $v<1$. In particular,
Algorithm 2 is optimal for $v<1$.
###### Proof 2.6.
Consider any algorithm for a robot starting at the origin to meet a target
initially placed at an unknown location distance $d$ from the origin. For any
point at distance $a$ from the origin, the target takes exactly
$\frac{d-a}{v}$ time to reach $a$. Then, let $t$ denote the time the robot
first passes through a point at distance $a$ from the origin. If
$t<\frac{d-a}{v}$, then the robot cannot know whether the target is on the
same or opposite side of the origin. On the other hand, if
$t\geq\frac{d-a}{v}$ and it has not encountered the target, then the target
must be on the opposite side of the origin. Thus, given a trajectory, let $\pm
a$ be the first point such that the robot is at position $\pm a$ at time
exactly $\frac{d-a}{v}$. Clearly such a point must exist for any trajectory
since the target is moving toward the origin. Then whichever side of the
origin the robot is on, consider the instance where the target started on the
opposite side. Clearly then, the robot takes an additional time at least
$\frac{2a}{1+v}$ to reach the target. Thus, the competitive ratio is given by:
$\displaystyle\frac{\frac{d-a}{v}+\frac{2a}{1+v}}{\frac{d}{1+v}}$
$\displaystyle=\frac{(d-a)(1+v)+2av}{vd}=1+\frac{1}{v}+\frac{a(v-1)}{dv}.$ (3)
Observe that whenever $v<1$, the right-hand side of Equation (3) satisfies
$1+\frac{1}{v}+\frac{a(v-1)}{dv}\geq
1+\frac{1}{v}+\frac{\frac{d}{1+v}(v-1)}{dv}=1+\frac{2}{1+v}$
which completes the proof of Theorem 2.5.
With Theorem 2.5 proved, we know Algorithm 2 is optimal for any value of $v$
between $0$ and $1$, but what about when $v>1$? In this case, we’ll prove the
following algorithm is optimal: the robot waits at the origin forever. We call
this algorithm “the waiting algorithm”.
###### Theorem 2.7.
Whenever the target is moving toward the origin with speed $v\geq 1$, the
waiting algorithm has an optimal competitive ratio of $1+\frac{1}{v}$.
###### Proof 2.8.
Clearly the algorithm takes exactly time $d/v$ to complete and so the upper
bound follows trivially. For the lower bound, we build upon the proof of
Theorem 2.5. It simply remains to consider Equation (3) for $v>1$. In this
case, the right-hand side of Equation (3) is increasing with respect to $a\geq
0$, so
$1+\frac{1}{v}+\frac{a(v-1)}{dv}\geq 1+\frac{1}{v}.$
This completes the proof of Theorem 2.7.
###### Remark 2.9.
Observe that the waiting algorithm makes no use of the target’s speed or
initial distance and therefore, as long as the target is moving toward the
origin, applies directly to the other knowledge models.
## 3 The NoDistance Model
In this section we assume that the robot knows $v$ but not $d$. Consider the
following zig-zag algorithm with “expansion ratio” $a>0$ (with the value of
$a$ to be determined).
Algorithm 3 Online Algorithm for NoDistance/Away and NoDistance/Toward Models
1:input: target speed $v$ and expansion ratio $a$
2:$i\leftarrow 0$
3:while target not found do
4: if at origin then
5: $d\leftarrow(-a)^{i}$
6: $i\leftarrow i+1$
7: else if at $d$ then
8: $d\leftarrow 0$
9: move toward $d$
### 3.1 The NoDistance/Away Model
The following result was shown by Alpern and Gal [alpern2003theory].
###### Theorem 3.1.
For the NoDistance/Away model, Algorithm 3 with $a=2\frac{1+v}{1-v}$ has an
optimal competitive ratio of
$\displaystyle 1+8\frac{1+v}{(1-v)^{2}}.$
### 3.2 The NoDistance/Toward Model
Recall first the statement made in Remark 2.9, that the optimality of the
waiting algorithm (which makes no use of any knowledge of $d$) holds for any
$d$ as long as $v\geq 1$. Thus, we need only consider scenarios where $0\leq
v<1$. As we will see, however, when the target is moving toward the origin,
the waiting algorithm is optimal for far slower targets! In general, since the
target is moving toward the origin, the robot need not search ever-increasing
distances away from the origin (i.e. execute Algorithm 3 with an expansion
ratio $a>1$). We call any algorithm which involves the robot never traveling
further than some finite distance from the origin (in one or both directions)
a contracting algorithm. Note that Algorithm 3 for $0<a\leq 1$ is a
contracting algorithm and $a=0$ is exactly the waiting algorithm. We’ll start
by showing that any contracting algorithm cannot have a better competitive
ratio than the waiting algorithm:
###### Theorem 3.2.
The competitive ratio of Algorithm 3 for any $0\leq a\leq 1$ is
$1+\frac{1}{v}$.
###### Proof 3.3.
Let $d^{\prime}$ be the finite distance further than which the robot will
never travel in at least one direction. Then consider the scenario where the
target is initially a distance $d=c\cdot d^{\prime}>>d^{\prime}$ from the
origin in the same direction. Then the competitive ratio is at least
$\displaystyle\sup_{c}\frac{\frac{cd^{\prime}-d^{\prime}}{v}}{\frac{cd^{\prime}}{v+1}}$
$\displaystyle=\sup_{c}\frac{c-1}{c}\frac{1+v}{v}=\lim_{c\rightarrow\infty}\frac{c-1}{c}\left(1+\frac{1}{v}\right)=1+\frac{1}{v}$
which proves Theorem 3.2.
By Theorem 3.2, any algorithm which hopes to out-perform the waiting algorithm
must be expanding. Now we show that the following hybrid algorithm, Algorithm
4, is optimal.
Algorithm 4 Wait or Zig-Zag Search Algorithm for NoDistance/Toward model
1:input: target speed $v$
2:if $v\geq\frac{1}{3}$ then
3: execute waiting algorithm
4:else
5: execute Algorithm 3 with $a=2\frac{1-v}{1+v}$
###### Theorem 3.4.
For the NoDistance/Toward model, the competitive ratio of Algorithm 4 is at
most
$\displaystyle\begin{cases}1+\frac{1}{v}&\mbox{if $v\geq\frac{1}{3}$}\\\
1+8\frac{1-v}{(1+v)^{2}}&\mbox{if $v<\frac{1}{3}$}\end{cases}$ (4)
###### Proof 3.5.
The first case is trivial: the competitive ratio of the waiting algorithm is
exactly $1+\frac{1}{v}$ by Theorem 2.7. The second case, however, is a bit
more complicated. First, observe that if $v<\frac{1}{3}$ then $a$ must be less
than $3$. Indeed, consider the scenario where the robot “just misses” the
target on the very first round of the algorithm (after traveling a distance
$1$ in some direction and then turning around). Then the competitive ratio of
the algorithm is
$\displaystyle 1+\frac{2a+2}{1+v}$
which is greater than $1+8\frac{1-v}{(1+v)^{2}}$ for any $a>3$ and $v>0$:
$\displaystyle 1+\frac{2a+2}{1+v}$
$\displaystyle>1+\frac{8}{1+v}>1+\frac{8}{1+v}\cdot\frac{1-v}{1+v}$
since $\frac{1-v}{1+v}<1$.
Now, consider the round $k$ when the robot catches up to the target and
observe that
$\displaystyle a^{k-2}$
$\displaystyle<d-\left(2\sum_{i=0}^{k-3}a^{i}+a^{k-2}\right)v=d-\left(2\frac{a^{k-2}-1}{a-1}+a^{k-2}\right)v$
since otherwise, the robot would have caught up to the target in round $k-2$.
This yields the following inequality which will prove useful in analyzing the
competitive ratio below:
$\displaystyle a^{k-2}$
$\displaystyle<d-\left(2\frac{a^{k-2}}{a-1}+a^{k-2}\right)v$
$\displaystyle\leq d\frac{a-1}{a-1+v(a+1)}+\frac{v}{a-1+v(a+1)}$
$\displaystyle\leq d\frac{a-1}{a-1+v(a+1)}+\frac{1}{4a-2}$ (5)
Observe the worst competitive ratio, then, is given by the situation where the
robot “just misses” the target on the $(k-2)^{\text{th}}$ round and catches up
to it only on round $k$. It follows the competitive ratio of Algorithm 4 is
$\displaystyle\frac{2\sum_{i=0}^{k-3}a^{i}+a^{k-2}+\frac{2(a^{k-2}+a^{k-1})}{1+v}}{\frac{d}{1-v}}\leq\frac{2\frac{a^{k-2}-1}{a-1}+a^{k-2}+\frac{2(a^{k-2}+a^{k-1})}{1+v}}{\frac{d}{1-v}}$
which, by Inequality (5) (and by substituting each $a^{k-2}$ with the right-
hand side of Inequality (5)), is less than or equal to
$\displaystyle CR$ $\displaystyle\leq
1+\frac{1}{2}\left[\frac{1}{d}\left(\frac{a-3}{a-1}\cdot\frac{v(5-7a)}{1-3a+2a^{2}}\right)+\frac{4a^{2}}{(a-1)+v(a+1)}\right]$
(6)
$\displaystyle\leq\lim_{d\rightarrow\infty}1+\frac{1}{2}\left[\frac{1}{d}\left(\frac{a-3}{a-1}\cdot\frac{v(5-7a)}{1-3a+2a^{2}}\right)+\frac{4a^{2}}{(a-1)+v(a+1)}\right]$
$\displaystyle=1+\frac{2a^{2}}{(a-1)+v(a+1)}$ (7)
which follows since the right-hand side of Inequality (6) is increasing with
respect to $d$ on $1<a\leq 3$. Finally, the right-hand side of Inequality (7)
is minimized at $a=2\frac{v-1}{v+1}$ with a value of
$1+8\frac{1-v}{(1+v)^{2}}$, which proves Theorem 3.4.
Now we show that Algorithm 4 is optimal by proving a tight lower bound on the
competitive ratio for any online algorithm. Our proof is based on techniques
developed in [killick2022cone]. Let $X(t)$ denote be the robot’s position at
time $t$ according to a given strategy.
###### Theorem 3.6.
For the NoDistance/Toward model, any strategy $X$ has a competitive ratio of
at least
$\displaystyle\begin{cases}1+\frac{1}{v}&\mbox{if $v\geq\frac{1}{3}$}\\\
1+8\frac{1-v}{(1+v)^{2}}&\mbox{otherwise}\end{cases}$
###### Proof 3.7.
Let $\beta_{t}=\inf_{t^{\prime}>t}\frac{t^{\prime}}{|X(t^{\prime})|}$.
Clearly, then, if $t_{1}\leq t_{2}$ then $\beta_{t_{1}}\leq\beta_{t_{2}}$.
Furthermore, $\beta_{t}\geq 1$ for all $t$ since the maximum speed of the
robot is $1$. Now let $\beta=\lim_{t\rightarrow\infty}\beta_{t}$. By
definition of the limit infimum, there must exist a finite time $t$ such that
$\beta\leq\frac{t^{\prime}}{|X(t^{\prime})|}$ for all $t^{\prime}\geq t$ and
thus there must exist a time $t_{1}>\frac{\beta(\beta+1)}{\beta-1}t$ such that
the robot reaches a point (without loss of generality, on the right side of
the origin) $X(t_{1})=\frac{t_{1}}{\beta+\epsilon_{1}}$ for any arbitrarily
small $\epsilon_{1}>0$. Consider such a time and observe that, by
construction, the robot could not have reached any point to the left of
$x_{0}=-\frac{t_{1}-X(t_{1})}{1+\beta}$ after time
$t_{0}=\frac{\beta(t_{1}-X(t_{1}))}{1+\beta}$ since $x_{0}\leq-t$ and
$t_{0}>t$ (see Figure 1).
Figure 1: The cone-bounded trajectory of the robot and worst-case placement
$p$ of the target. The small gray triangle is to remind the reader that, by
the definition of $\beta$, the robot trajectory is only guaranteed to be
contained by the cone after some finite time $t$. Thus, in order to maximize
the competitive ratio, we (as the adversary) should place the target so that
its trajectory does not intersect $(x_{0},t_{0})$ or the gray triangle.
Now, consider a target starting at initial positon $p$ (to be determined)
moving at speed $v>0$ toward a robot which starts at the origin and has a
speed of $1$. Thus, by placing the target at a starting location so that the
farthest right the robot could have reached is $x_{0}-\epsilon_{0}$ for any
arbitrarily small $\epsilon_{0}$, the robot can not have reached the target by
time $t_{1}$. Such a target has an initial position of
$\displaystyle p=-\frac{(1+\beta v)(t_{1}-X(t_{1}))}{1+\beta}-\epsilon_{0}$
and follows the trajectory
$\displaystyle X_{\text{target}}(t)=vt+p$ (8)
where $X_{\text{target}}(t)$ denotes the robot’s position at time $t$. Observe
also, if the robot moves directly toward the target after $t_{1}$, then its
trajectory after time $t_{1}$ is given by
$\displaystyle X(t)=X(t_{1})+t_{1}-t$ (9)
Thus, the earliest time the robot could possibly encounter the target can be
computed by finding the intersection between the robot trajectory (Equation
(9)) and the target’s trajectory (Equation (8)) and solving for $t$:
$\displaystyle vt+p$ $\displaystyle=X(t_{1})+t_{1}-t$ $\displaystyle t$
$\displaystyle=\frac{X(t_{1})+t_{1}-p}{1+v}.$ (10)
Then the competitive ratio (Equation (10) divided by $-p/(1+v)$, the optimal
search time) is
$\displaystyle CR$
$\displaystyle\geq\sup_{\epsilon_{0},\epsilon_{1}}\frac{(X(t_{1})+t_{1}-p)/(1+v)}{-p/(1+v)}=\sup_{\epsilon_{0},\epsilon_{1}}\frac{X(t_{1})+t_{1}-p}{-p}=\sup_{\epsilon_{0},\epsilon_{1}}\left[1-\frac{X(t_{1})+t_{1}}{p}\right]$
$\displaystyle=\sup_{\epsilon_{1}}\left[1+\frac{(1+\beta)(t_{1}+X(t_{1}))}{(1+\beta
v)(t_{1}-X(t_{1}))}\right]$ (11)
$\displaystyle=1+\frac{(1+\beta)^{2}}{(1+\beta v)(\beta-1)}$ (12)
where Inequality (11) follows since $p=-\frac{(1+\beta
v)(t_{1}-X(t_{1}))}{1+\beta}-\epsilon_{0}$ for arbitrarily small
$\epsilon_{0}>0$ and Inequality (12) follows since
$X(t_{1})=\frac{t_{1}}{\beta+\epsilon_{1}}$ for arbitrarily small
$\epsilon_{1}>0$. Finally, observe that if $v<\frac{1}{3}$, then the right-
hand side of Equality (12) has a single minimum of $1+8\frac{1-v}{(1+v)^{2}}$
at $\beta=\frac{v-3}{3v-1}$. On the other hand, if $v\geq\frac{1}{3}$, then
the right-hand side of Equality (12) is decreasing with respect to $\beta$ and
thus the competitive ratio satisfies
$\displaystyle
CR\geq\lim_{\beta\rightarrow\infty}\left[1+\frac{(1+\beta)^{2}}{(1+\beta
v)(\beta-1)}\right]=1+\frac{1}{v}.$
## 4 The NoSpeed Model
In this section we assume that the robot knows $d$ but not $v$.
### 4.1 The NoSpeed/Away
For this model, it is clear that the robot cannot execute an algorithm like
Algorithm 1 since no upper bound on the target’s speed is known. Note that, if
any upper bound $\hat{v}<1$ on the target’s speed were known, the robot could
execute Algorithm 1 by assuming the target speed to be equal to $\hat{v}$,
resulting in a competitive ratio of at most $1+\frac{2}{1-\hat{v}}$. Since the
target speed $v$ is unknown (and potentially very close to $1$), however, we
propose another strategy. Consider a monotone increasing non-negative integer
sequence $\\{f_{i}:i\geq 0\\}$ such that $f_{0}=1$ and $f_{i}<f_{i+1}$, for
all $i\geq 0$. The idea of the algorithm is to search for the target by making
a guess about its speed in rounds as follows. We start from the origin and
alternate searching right and left. On the $i$-th round, we use the guess
$v_{i}=1-2^{-f_{i}}$ and search the necessary distance away from the origin
such that, if the target’s speed is less than or equal to $v_{i}$ and the
target’s initial position is in the same direction from the origin that the
robot moves in round $i$, then the target will be found in round $i$.
Otherwise, we can conclude that either the target is moving with a speed
greater than $v_{i}$ or else it is on the opposite side of the origin. In this
case the robot returns to the origin and repeats the algorithm in the opposite
direction.
Later in the analysis we will show how to select the integer sequence
$\\{f_{i}:i\geq 0\\}$ so as to obtain bounds on the competitive ratio. The
algorithm explained above is formalized as Algorithm 5.
Algorithm 5 Online Algorithm for NoSpeed/Away Model
1:input: target initial distance $d$
2: integer sequence $\\{f_{i}:i\geq 0\\}$ such that $f_{i}<f_{i+1}$, for
$i\geq 0$ and $f_{0}=1$;
3:$t\leftarrow 0$
4:for $i\leftarrow 0,1,2,\ldots$ until target found do
5: $v_{i}\leftarrow 1-2^{-f_{i}}$
6: $x_{i}\leftarrow(-1)^{i}\cdot\frac{d+tv_{i}}{1-v_{i}}$
7: move to $x_{i}$ and back to the origin
8: $t\leftarrow t+|x_{i}|$
To compensate for the fact that the starting speed of the robot in the
algorithm is $v_{0}=1-2^{-1}=1/2$ we first need to consider the case
$v\leq\frac{1}{2}$.
###### Lemma 4.1.
For the NoSpeed/Away model, if the unknown speed $v$ of the target is less
than or equal to $\frac{1}{2}$ then the competitive ratio of Algorithm 5 is at
most $5$.
###### Proof 4.2.
According to Algorithm 5 and since $v<1/2$, the robot will find the target
either on its first trip away from the origin, after time at most $2d$, or
after the first time it changes direction of movement. In the worst case it
will spend time $2d$ in one direction and then additional time
$\frac{2d+d+2dv}{1-v}$. It follows that the competitive ratio is at most
$\displaystyle\frac{2d+\frac{2d+d+2dv}{1-v}}{\frac{d}{1-v}}=5$
which proves Lemma 4.1.
Next we analyze the competitive ratio of the algorithm for $v>\frac{1}{2}$.
###### Lemma 4.3.
For the NoSpeed/Away model, if the unknown speed $v$ of the target is greater
than $\frac{1}{2}$ then the competitive ratio of Algorithm 5 is at most
$1+2^{1+\sum_{j=0}^{k}f_{j}}\cdot 4^{k+1}$ where $k$ is the first $k$ such
that $v_{k}\geq v$.
###### Proof 4.4.
Let $d_{i}$ be the distance from the origin the target would be if its speed
was equal to $v_{i}$, where $v_{i}=1-2^{-f_{i}}$ at time
$\sum_{j=0}^{i-1}\left(1-2^{-f_{i}}\right)$. In other words, if $v_{i}\geq v$,
then $d_{i}$ is the maximum distance of the target from the origin (and thus,
the robot) at the beginning of round $i$ of the algorithm. Thus, if the speed
of the target is less than or equal to $v_{i}$ and the robot moves toward it
in round $i$, then it would take at most
$x_{i}=\frac{d_{i}}{1-v_{i}}=2^{f_{i}}d_{i}$ additional time for the robot to
catch up to the target, for $i\geq 0$. Recall the algorithm involves the robot
moving a distance $x_{i}$ (in time $x_{i}$, since the robot’s speed is $1$)
away from the origin and back in round $i$. Observe then that $d_{0}=d$,
$v_{0}=1/2$, and
$\displaystyle d_{i}=d+2v_{i}\sum_{j=0}^{i-1}x_{j}.$ (13)
Therefore, it follows from the definition of $x_{i}$ that
$\displaystyle x_{i}=2^{f_{i}}\left(d+2v_{i}\sum_{j=0}^{i-1}x_{j}\right).$
(14)
As a consequence
$\displaystyle\sum_{j=0}^{i-1}x_{j}=\frac{x_{i}-2^{f_{i}}d}{2^{f_{i}}\cdot
2\cdot v_{i}}$ (15)
Similarly, if we replace $i$ with $i+1$ we have that
$\displaystyle\sum_{j=0}^{i}x_{j}=\frac{x_{i+1}-2^{f_{i+1}}d}{2^{f_{i+1}}\cdot
2\cdot v_{i+1}}$ (16)
Subtracting Equation (15) from Equation (16), we derive the recurrence
$\displaystyle x_{i}=\frac{x_{i+1}-2^{f_{i+1}}d}{2^{f_{i+1}}\cdot 2\cdot
v_{i+1}}-\frac{x_{i}-2^{f_{i}}d}{2^{f_{i}}\cdot 2\cdot v_{i}}$ (17)
Collecting similar terms and simplifying Equation (17) yields
$\displaystyle\frac{x_{i+1}}{2^{f_{i+1}}\cdot 2\cdot v_{i+1}}$
$\displaystyle=\left(1+\frac{1}{2^{f_{i}}\cdot 2\cdot
v_{i}}\right)x_{i}+\left(\frac{2^{f_{i+1}}}{2^{f_{i+1}}\cdot 2\cdot
v_{i+1}}-\frac{2^{f_{i}}}{2^{f_{i}}\cdot 2\cdot v_{i}}\right)d$
$\displaystyle=x_{i}\left(1+\frac{1}{2^{f}_{i}\cdot 2\cdot
v_{i}}\right)+\frac{d}{2v_{i+1}}-\frac{d}{2v_{i}}$ (18)
$\displaystyle\leq\left(1+\frac{1}{2^{f_{i}}\cdot 2\cdot v_{i}}\right)x_{i}$
(19)
following from the fact the sum of the last two terms in Inequality 18 is less
than or equal to $0$.
If we simplify the right-hand side of Equation (19), we derive the following
recursive inequalities
$\displaystyle x_{i+1}$ $\displaystyle\leq 2^{f_{i+1}}\cdot 2\cdot
v_{i+1}\left(1+\frac{1}{2^{f_{i}}\cdot 2\cdot v_{i}}\right)x_{i}$
$\displaystyle\leq 2^{f_{i+1}}\cdot
2\cdot\left(1+\frac{1}{2^{f_{i}}}\right)x_{i}$ $\displaystyle\leq\left(2\cdot
2^{f_{i+1}}+2\cdot 2^{f_{i+1}-f_{i}}\right)x_{i}$ $\displaystyle\leq
2^{f_{i+1}}\cdot 4\cdot x_{i},$ (20)
which follows since $\frac{1}{2}\leq v_{i}<1$ and $f_{i}<f_{i+1}$ for all $i$.
By repeated application of the last Recurrence (20) above and using the fact
that by definition $x_{0}=2^{f_{0}}d$, it follows easily by induction that
$\displaystyle x_{i+1}$ $\displaystyle\leq 2^{f_{i+1}}\cdot 4\cdot x_{i}$
$\displaystyle\leq 2^{f_{i+1}+f_{i}}\cdot 4^{2}\cdot x_{i-1}$
$\displaystyle\leavevmode\nobreak\ \vdots$ $\displaystyle\leq
2^{f_{i+1}+f_{i}+f_{i-1}+\cdots+f_{1}}\cdot 4^{i+1}\cdot x_{0}$
$\displaystyle\leq 2^{\sum_{j=0}^{i+1}f_{j}}\cdot 4^{i+1}\cdot d$ (21)
Consider the first $i$ such that $v_{i}\geq v$. It follows that and
$v_{i-1}<v$ which yields $1-v<1-v_{i-1}=2^{-f_{i-1}}$ and implies that
$2^{f_{i-1}}<\frac{1}{1-v}$. Note that although $v_{i}\geq v$, the robot may
not find the target in round $i$ because it is located in the opposite
direction. It is guaranteed, however, to find the target by round $i+1$.
Moreover the total time that has elapsed from the start until round $i$ is
$2\sum_{j=0}^{i}x_{j}$ at which time the target is at distance
$d+v2\sum_{j=0}^{i}x_{j}$ from the origin.
As a consequence the competitive ratio of Algorithm 5 is at most
$\displaystyle\frac{2\sum_{j=0}^{i}x_{j}+\frac{d+2v\sum_{j=0}^{i}x_{j}}{1-v}}{\frac{d}{1-v}}$
$\displaystyle=1+\frac{2(v+1-v)}{d}\sum_{j=0}^{i}x_{j}$
$\displaystyle=1+\frac{x_{i+1}-2^{f_{i+1}}d}{d}\cdot\frac{2}{2^{f_{i+1}}\cdot
2\cdot v_{i+1}}$ (By (16)) $\displaystyle\leq
1+\frac{x_{i+1}}{d}\cdot\frac{2}{2^{f_{i+1}}\cdot 2\cdot v_{i+1}}$
$\displaystyle\leq 1+\frac{1}{v_{i+1}}2^{\sum_{j=0}^{i}f_{j}}\cdot 4^{i+1}$
(By (21))
Since $v_{i+1}\geq 1/2$ we conclude with an upper bound on the competitive
ratio of Algorithm 5 of
$\displaystyle 1+2^{1+\sum_{j=0}^{i}f_{j}}\cdot 4^{i+1}$ (22)
which proves Lemma 4.3.
We are now ready to prove the main theorem about the competitive ratio of
Algorithm 5.
###### Theorem 4.5.
For the NoSpeed/Away model, the competitive ratio of Algorithm 5 when applied
to the sequence $f_{j}=2^{j}$, for all $j\geq 0$, is at most
$\displaystyle\begin{cases}5&\text{if }v\leq\frac{1}{2}\\\
1+\frac{16\left(\log\frac{1}{1-v}\right)^{2}}{(1-v)^{4}}&\text{otherwise}\end{cases}$
where $\log$ is the base-2 logarithm.
###### Proof 4.6.
Consider the first index $i$ such that $v_{i}\geq v$. It follows that and
$v_{i-1}<v$, and so
$\displaystyle 1-2^{-2^{i-1}}<v\Rightarrow 2^{2^{i-1}}<\frac{1}{1-v}.$
Then, by Lemma 4.3, the competitive ratio of is at most
$\displaystyle 1+2^{1+\sum_{j=0}^{i}f_{j}}\cdot 4^{i+1}$
$\displaystyle=1+2^{1+\sum_{j=0}^{i}2^{j}}\cdot 4^{i+1}$
$\displaystyle=1+2^{2^{i+1}}\cdot 4^{i+1}\leq
1+\left(\frac{1}{1-v}\right)^{4}\cdot\left(4\cdot\log\left(\frac{1}{1-v}\right)\right)^{2}$
$\displaystyle\leq 1+\frac{16\left(\log\frac{1}{1-v}\right)^{2}}{(1-v)^{4}}$
This proves Theorem 4.5.
###### Remark 4.7.
By Theorem 2.1, $1+\frac{2}{1-v}$ is a lower bound on any algorithm when both
$v,d$ are known. As a consequence it must also be a lower bound when $d$ is
known but $v$ is not.
### 4.2 The NoSpeed/Toward Model
Now we consider the case where the target is moving toward the origin.
Algorithm 6 Online Algorithm for NoSpeed/Toward Model
1:input: target initial distance $d$
2:choose any direction and go for time $d$
3:if target not found then
4: change direction and go until target is found
###### Theorem 4.8.
For the NoSpeed/Toward model, Algorithm 6 achieves an optimal competitive
ratio at most $3$.
###### Proof 4.9.
The robot chooses a direction (without loss of generality, say to the right)
and goes for a time $d$ (this is where the robot makes use of its knowledge of
the distance $d$). If it does not find the target it changes direction. In the
meantime the target has moved for a distance $dv$ and now must be at location
$-d+dv$. Therefore at the time the robot changes direction the distance
between robot and target is equal to $d-(-d+dv)=2d-dv$, and hence the meeting
will take place in additional time $\frac{2d-dv}{1+v}$. It follows that the
total time required for the robot to meet the target must be equal to
$d+\frac{2d-dv}{1+v}$. The resulting competitive ratio satisfies
$\displaystyle CR\leq\frac{d+\frac{2d-dv}{1+v}}{\frac{d}{v+1}}=3.$
This proves the upper bound.
To prove the lower bound we argue as follows. If the searcher never visits
either of the points $\pm d$ then the competitive ratio is arbitrarily large
for very small values of $v$. Let $\epsilon>0$ be sufficiently small and let
the speed of the target be $v=\epsilon/3$. Consider the first time, say $t$,
that the robot reaches one of the points $\pm(d-\epsilon)$. Without loss of
generality let this point be $d-\epsilon$ and suppose the target is
adversarially placed at $-d$. Then at time $t$ it will be located at $-d+tv$.
Therefore the distance between the robot and the target at time $t$ will be
$d-\epsilon-(-d+tv)=2d-tv-\epsilon$. The time it takes for robot to find the
target, then, is at least $d-\epsilon+\frac{2d-tv-\epsilon}{1+v}$ and the
competitive ratio is at least
$\displaystyle\frac{d-\epsilon+\frac{2d-tv-\epsilon}{1+v}}{\frac{d}{1+v}}\geq
3-\frac{2\epsilon+(t+\epsilon)v}{d}$
It follows easily that if $t\geq 3d-\epsilon$ then
$CR\geq\frac{t}{d/(1+v)}\geq 3-3\epsilon$. However, if $t\leq 3d-\epsilon$
then $\frac{2\epsilon+(t+\epsilon)v}{d}\leq 2\epsilon+3v\leq 3\epsilon$, since
by assumption $v=\epsilon/3$. Therefore again $CR\geq 3-3\epsilon$. This
completes the proof of Theorem 4.8.
## 5 The NoKnowledge Model
In this section we assume that neither the initial distance $d$ nor the speed
$v$ of the target is known to the robot.
### 5.1 The NoKnowledge/Away Model
We now describe an approximation strategy resembling that described in Section
4.1. For this strategy though, the robot will need to guess both the target’s
speed and its initial distance.
Consider the situation where neither the distance $d$ to the target nor its
speed $v<1$ is known to the robot. Also consider two monotone increasing non-
negative integer sequences $\\{f_{i},g_{i}:i\geq 0\\}$ such that $f_{0}=1$ and
$g_{0}=0$ and $f_{i}<f_{i+1}$ and $g_{i}<g_{i+1}$, for all $i\geq 0$. The idea
of the algorithm is to search for the target by making a guess for its speed
and starting distance in rounds as follows. The robot, starting from the
origin, alternates searching to the right and left. On the $i$-th round, it
guesses that the target’s speed does not exceed $v_{i}=1-2^{-f_{i}}$ and that
it’s initial distance from the origin does not exceed $2^{g_{i}}$. Using these
guesses, the robot searches exactly the distance required (which we will later
denote $d_{i}$) to catch the target, given its guesses are correct and that
the target is in the direction the robot searches in round $i$. If robot does
not find the target after searching this distance, it returns to the origin
and begins the next round. Later in the analysis we will show how to select
the integer sequences $\\{f_{i},g_{i}:i\geq 0\\}$ so as to obtain bounds on
the competitive ratio. We formalize the algorithm described above as Algorithm
7.
Algorithm 7 Online Algorithm for NoKnowledge/Away Model
1:Inputs; Integer sequences $\\{f_{i},g_{i}:i\geq 0\\}$ such that
$f_{i}<f_{i+1}$ and $g_{i}<g_{i+1}$, for $i\geq 0$ and $f_{0}=1$ and
$g_{0}=0$;
2:$t\leftarrow 0$
3:for $i\leftarrow 0,1,2,\ldots$ until target found do
4: $d_{i}\leftarrow 2^{g_{i}}$
5: $v_{i}\leftarrow 1-2^{-f_{i}}$
6: $x_{i}\leftarrow(-1)^{i}\cdot\frac{d_{i}+tv_{i}}{1-v_{i}}$
7: move to $x_{i}$ and back to the origin
8: $t\leftarrow t+|x_{i}|$
Since there is always an integer $i\ \geq 1$ such that both
$v_{i}=1-2^{-f_{i}}\geq v$ and $2^{g_{i}}\geq d$, it is clear that the robot
will eventually succeed in catching the target. Next we analyze the
competitive ratio of the algorithm.
For the NoKnowledge/Away model, if Algorithm 7 terminates successfully in
round $i+1$ then its competitive ratio must satisfy
$CR\leq 1+\frac{2(i+2)}{d}\cdot 2^{g_{i+1}}\cdot 2^{\sum_{j=0}^{i}f_{j}}\cdot
4^{i+1}.$ (23)
###### Proof 5.1.
We call each iteration of the loop in Algorithm 7 a round. For any round $i$,
let $d_{i}$ be the distance from the origin to where the target would be if
its speed was equal to $v_{i}=1-2^{-f_{i}}$ and its starting position
$2^{g_{i}}$. Recall that during the first $i-1$ unsuccessful rounds, the taret
is moving further and further away from the origin. If the robot is at the
origin and the speed of the target is $v_{i}$ then it takes time at most
$x_{i}=\frac{d_{i}}{1-v_{i}}=2^{f_{i}}d_{i}$ for the robot to catch up to the
target, for $i\geq 0$. Observe from the algorithm that $d_{0}=1$ and
$v_{0}=1/2$ and
$\displaystyle d_{i}=2^{g_{i}}+2v_{i}\sum_{j=0}^{i-1}x_{j}.$ (24)
Therefore, it follows from the definition of $x_{i}$ that
$\displaystyle x_{i}$
$\displaystyle=2^{f_{i}}\left(2^{g_{i}}+2v_{i}\sum_{j=9}^{i-1}x_{j}\right).$
(25)
As a consequence
$\displaystyle\sum_{j=0}^{i-1}x_{j}$
$\displaystyle=\frac{x_{i}-2^{f_{i}+g_{i}}}{2^{f_{i}}\cdot 2\cdot v_{i}}$ (26)
Similarly, if we replace $i$ with $i+1$ we have that
$\displaystyle\sum_{j=0}^{i}x_{j}$
$\displaystyle=\frac{x_{i+1}-2^{f_{i+1}+g_{i+1}}}{2^{f_{i+1}}\cdot 2\cdot
v_{i+1}}.$ (27)
Subtracting Equation (26) from Equation (27) we derive the recurrence
$\displaystyle x_{i}$
$\displaystyle=\frac{x_{i+1}-2^{f_{i+1}+g_{i+1}}}{2^{f_{i+1}}\cdot 2\cdot
v_{i+1}}-\frac{x_{i}-2^{f_{i}+g_{i}}}{2^{f_{i}}\cdot 2\cdot v_{i}}$ (28)
Collecting similar terms and simplifying Equation (28) yields
$\displaystyle\frac{x_{i+1}}{2^{f_{i+1}}\cdot 2\cdot v_{i+1}}$
$\displaystyle=\left(1+\frac{1}{2^{f_{i}}\cdot 2\cdot
v_{i}}\right)x_{i}+\left(\frac{2^{f_{i+1}+g_{i+1}}}{2^{f_{i+1}}\cdot 2\cdot
v_{i+1}}-\frac{2^{f_{i}+g_{i}}}{2^{f_{i}}\cdot 2\cdot v_{i}}\right)$
$\displaystyle=\left(1+\frac{1}{2^{f_{i}}\cdot 2\cdot
v_{i}}\right)x_{i}+2^{g_{i+1}-1}\left(\frac{1}{v_{i+1}}-\frac{2^{g_{i}}}{2^{g_{i+1}}\cdot
v_{i}}\right)$ $\displaystyle\leq\left(1+\frac{1}{2^{f_{i}}\cdot 2\cdot
v_{i}}\right)x_{i}+2^{g_{i+1}-1}$ (29)
where Inequality (29) follows since
$\frac{1}{v_{i+1}}-\frac{2^{g_{i}}}{2^{g_{i+1}}\cdot v_{i}}\leq 1$.
If we multiply out with the denominator in the lefthand side of Inequality
(29) and simplify the righthand side we derive the following recursive
inequalities
$\displaystyle x_{i+1}$ $\displaystyle\leq 2^{f_{i+1}}\cdot 2\cdot
v_{i+1}\left(1+\frac{1}{2^{f_{i}}\cdot 2\cdot
v_{i}}\right)x_{i}+2^{f_{i+1}+g_{i+1}}v_{i+1}$
$\displaystyle\leq\left(2(2^{f_{i+1}}-1)+2^{f_{i+1}-f_{i}}\cdot\frac{v_{i+1}}{v_{i}}\right)x_{i}+2^{f_{i+1}+g_{i+1}}$
(30) $\displaystyle\leq 2^{f_{i+1}}\cdot 4\cdot x_{i}+2^{f_{i+1}+g_{i+1}},$
(31)
where in the derivation of Inequality (31) from the previous Inequality (30)
we used the fact that $\frac{v_{i+1}}{v_{i}}\leq 2$.
By repeated application of the last Recurrence (31) above and using the fact
that by definition $x_{0}=2^{f_{0}}d$, it follows easily by induction that
$\displaystyle x_{i+1}$ $\displaystyle\leq 2^{f_{i+1}}\cdot 4\cdot
x_{i}+2^{f_{i+1}+g_{i+1}}$ $\displaystyle\leq 2^{f_{i+1}+f_{i}}\cdot
4^{2}\cdot x_{i-1}+2^{f_{i+1}+f_{i}+g_{i}}\cdot 4^{1}+2^{f_{i+1}+g_{i+1}}$
$\displaystyle\leavevmode\nobreak\ \vdots$ $\displaystyle\leq
2^{g_{0}+\sum_{j=0}^{i+1}f_{j}}\cdot 4^{i+1}\cdot
x_{0}+2^{g_{1}+\sum_{j=1}^{i+1}f_{j}}\cdot
4^{i}+2^{g_{2}+\sum_{j=2}^{i+1}f_{j}}\cdot 4^{i-1}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\cdots+2^{f_{i+1}+g_{i+1}}$
$\displaystyle=\sum_{k=0}^{i+1}2^{g_{k}+\sum_{j=k}^{i+1}f_{j}}\cdot 4^{i-k+1}$
(32)
since $x_{0}=1$.
The total time that has elapsed from the start until the beginning of last
round $i$ (when the robot visits the origin for the last time before catching
the target) will be $\sum_{j=0}^{i}2x_{j}$ at which time the target is at
distance $d+v\sum_{j=0}^{i}2x_{j}$ from the origin. As a consequence the
competitive ratio of Algorithm 7 must satisfy the inequality
$\displaystyle CR$
$\displaystyle\leq\frac{2\sum_{j=0}^{i}x_{j}+\frac{d+2v\sum_{j=0}^{i}x_{j}}{1-v}}{\frac{d}{1-v}}.$
(33)
Simplifying the righthand side of Inequality (33) and using Identity (26)
yields
$\displaystyle CR$ $\displaystyle\leq 1+\frac{2}{d}\sum_{j=0}^{i}x_{j}$
$\displaystyle\leq 1+\frac{x_{i+1}}{d}\cdot\frac{1}{2^{f_{i+1}}\cdot
v_{i+1}}\mbox{\leavevmode\nobreak\ (Use Equation\leavevmode\nobreak\
\eqref{eq:no_knowledge__upper_guess_3})}$ $\displaystyle\leq
1+\frac{1}{v_{i+1}d2^{f_{i+1}}}\sum_{k=0}^{i+1}2^{g_{k}+\sum_{j=k}^{i+1}f_{j}}\cdot
4^{i-k+1}\mbox{\leavevmode\nobreak\ (Use Equation\leavevmode\nobreak\
\eqref{eq:no_knowledge_guess})}$ $\displaystyle\leq
1+\frac{1}{v_{i+1}d}\sum_{k=0}^{i+1}2^{g_{k}+\sum_{j=k}^{i}f_{j}}\cdot
4^{i-k+1}.$
Since $v_{i+1}\geq 1/2$ we conclude with
$\displaystyle CR$ $\displaystyle\leq
1+\frac{2}{d}\sum_{k=0}^{i+1}2^{g_{k}+\sum_{j=k}^{i}f_{j}}\cdot 4^{i-k+1}$
$\displaystyle\leq
1+\frac{2}{d}\sum_{k=0}^{i+1}2^{g_{k}+\sum_{j=k}^{i}f_{j}}\cdot 4^{i-k+1}$
$\displaystyle\leq 1+\frac{2(i+2)}{d}\cdot 2^{g_{i+1}}\cdot
2^{\sum_{j=0}^{i}f_{j}}\cdot 4^{i+1}$ (34)
This completes the proof of Lemma 5.1.
We now prove the following theorem.
###### Theorem 5.2.
For the NoKnowledge/Away model, Algorithm 7 with the sequences
$g_{i}=f_{i}=2^{i}$ has a competitive ratio of at most
$\displaystyle
1+\frac{16}{d}\left[\log\log\max\left(d,\frac{1}{1-v}\right)+3\right]\cdot\max\left(d,\frac{1}{1-v}\right)^{8}\cdot\log^{2}\max\left(d,\frac{1}{1-v}\right)$
where $\log$ is the base-2 logarithm.
###### Proof 5.3.
Observe that if the robot finds the target in round $i+1$, then by design, one
or both of the robot’s round $i-1$ guesses for the target’s speed
($1-2^{-2^{i-1}}$) or initial distance ($2^{2^{i-1}}$) must have been too low,
otherwise the robot would have found the target in an earlier round. In other
words, either $1-2^{-2^{i-1}}<v$ or $2^{2^{i-1}}<d$. It follows, then that
$i-1<\log\log\max\left(d,\frac{1}{1-v}\right)$. Then by Lemma 5.1, an upper
bound on the competitive ratio is given by
$\displaystyle CR$ $\displaystyle\leq 1+\frac{2(i+2)}{d}\cdot 2^{g_{i+1}}\cdot
2^{\sum_{j=0}^{i}f_{j}}\cdot 4^{i+1}$ $\displaystyle=1+\frac{2(i+2)}{d}\cdot
2^{2^{i+1}}\cdot 2^{2^{i+1}-1}\cdot 4^{i+1}$
$\displaystyle=1+\frac{(i-1)+3}{d}\cdot\left(2^{2^{i-1}}\right)^{8}\cdot
16\left(2^{i-1}\right)^{2}$
$\displaystyle=1+\frac{16}{d}\left[\log\log\max\left(d,\frac{1}{1-v}\right)+3\right]\cdot\max\left(d,\frac{1}{1-v}\right)^{8}\cdot\log^{2}\max\left(d,\frac{1}{1-v}\right)$
which proves Theorem 5.2.
###### Remark 5.4.
Observe that a lower bound of $1+8\frac{1+v}{(1-v)^{2}}$ follows directly from
the NoDistance/Away model.
### 5.2 The NoKnowledge/Toward Model
We can prove the following theorem.
The optimal competitive ratio is $1+\frac{1}{v}$ and is given by the waiting
Algorithm.
###### Proof 5.5.
The upper bound follows directly from Theorem 2.7. For the lower bound,
consider an algorithm where the robot does not wait forever and instead moves
a distance $d^{\prime}>0$ to the right (without loss of generality – a
symmetric argument for the case where the robot moves to the left follows
trivially) after waiting at the origin for time $t\geq 0$. Then consider the
scenario where the target with speed $v=\frac{d}{t+d^{\prime}}$ is initially
at $-d$ for any distance $d\geq 1$. Thus, the target reaches the origin at
exactly the time the robot reaches $d^{\prime}$ and so their earliest possible
meeting time is
$\displaystyle
t+d^{\prime}+\frac{d^{\prime}}{1+v}=\frac{d}{v}+\frac{d^{\prime}}{1+v}\geq\frac{d}{v}$
Thus, the competitive ratio is at least
$\displaystyle\frac{d/v}{d/(1+v)}=1+\frac{1}{v}$
This proves Theorem 5.2.
## 6 Conclusion
We considered linear search for an autonomous robot searching for an oblivious
moving target on an infinite line. Two scenarios were analyzed depending on
whether the target is moving towards or away from the origin (and this is
known to the robot). In either of these two scenarios we considered the
knowledge the robot has about the speed and starting distance of the target.
For each scenario we gave search algorithms and analyzed their competitive
ratio for the four possible cases arising. Our bounds are tight in all cases
when the target is moving towards the origin. They are also shown to be tight
when the target is moving away from the origin and its speed $0<v<1$ is known
to the robot; for this scenario we also obtain upper bounds when $v$ is not
known to the robot. It remains an open problem to prove tight bounds for the
case when $v$ is unknown to the robot and the target is moving away from the
origin. It also remains open to find tight bounds for the case where the
direction of movement of the target is unknown.
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|
goodness | $\chi^{2}_{\mathrm{cp}}$ | goodness of fit for closure phases | 1.8
of fit | $\chi^{2}_{\mathrm{lca}}$ | goodness of fit for log closure amplitudes | 2.3
| $\mathcal{A}^{\mathrm{(sc)}}_{\mathrm{intra}}$ | mean self-calibration gain for co-located stations | 0.98
| $\mathcal{A}^{\mathrm{(sc)}}_{\mathrm{\tiny{\acs{lm}}}}$ | LMT mean self-calibration gain | 1.13
| $\mathcal{A}^{\mathrm{(sc)}}_{\mathrm{\tiny{\acs{az}}}}$ | SMT mean self-calibration gain | 1.01
| $\mathcal{A}^{\mathrm{(sc)}}_{\mathrm{\tiny{\acs{sp}}}}$ | SPT mean self-calibration gain | 1.03
EHT-HOPS image | $\chi^{2}_{\mathrm{amp}}$ | goodness of fit for amplitudes | 1.2
goodness | $\chi^{2}_{\mathrm{cp}}$ | goodness of fit for closure phases | 2.1
of fit | $\chi^{2}_{\mathrm{lca}}$ | goodness of fit for log closure amplitudes | 1.2
| $\mathcal{A}^{\mathrm{(sc)}}_{\mathrm{intra}}$ | mean self-calibration gain for co-located stations | 0.98
| $\mathcal{A}^{\mathrm{(sc)}}_{\mathrm{\tiny{\acs{lm}}}}$ | LMT mean self-calibration gain | 1.15
| $\mathcal{A}^{\mathrm{(sc)}}_{\mathrm{\tiny{\acs{az}}}}$ | SMT mean self-calibration gain | 1.01
| $\mathcal{A}^{\mathrm{(sc)}}_{\mathrm{\tiny{\acs{sp}}}}$ | SPT mean self-calibration gain | 1.00
Note – $\mathcal{A}^{\mathrm{(sc)}}_{\mathrm{intra}}$ corresponds to the mean
gain of ALMA, APEX, JCMT, and SMA.
Figure 7: Cen A data properties from April 2017. The top left panel shows the
$(u,v)$ coverage. A priori calibrated amplitude (before self-calibration) and
closure phase data points are shown in the top right and bottom panel,
respectively, overplotted with lines from the final image model as a function
of $(u,v)$ distances. The error bars indicate thermal noise and $5\,\%$ non-
closing error uncertainties added in quadrature, which are smaller than the
plotted symbols in some cases. The color-coding shows different baselines.
Amplitudes projected along and perpendicular to the jet position angle are
given in Supplementary Fig. 8.
Figure 8: Source structure along specific position angles on the sky. A priori
calibrated amplitudes are shown projected along the jet position angle (PA) on
the sky in the left panel and perpendicular to the PA in the right panel. The
color coding and error bars shown are the same as in Supplementary Fig. 7.
Figure 9: Determination of the jet apex location. A zoomed-in version of the
final image model is shown. The solid blue lines show simple linear
extrapolations of the inner NW and SE jet arms, which would place the jet apex
well within the counterjet region. The dashed white lines mark the certain
edges of the approaching jet and the counterjet. The quadrangle enclosed by
the solid and dashed lines is the region where the jet apex is located. Inside
this quadrangle, a tentative convergence of the two streamlines can be seen.
The apex position assumed in this work is indicated with a white cross. The
surrounding blue dashed circle corresponds to the
$z_{\mathrm{col}}=32\,\acs{muas}$ distance. Vertical black bars mark the
brightest regions along each jet arm, which correspond to the assumed location
of the radio core.
*[VLBI]: very long baseline interferometry
*[SMBH]: supermassive black hole
*[Cen A]: Centaurus A
*[EHT]: Event Horizon Telescope
*[$\mu$as]: microarcseconds
*[GRMHD]: general relativistic magnetohydrodynamics
*[Jy]: jansky
*[PA]: position angle
*[LLAGN]: low-luminosity AGN
*[AGN]: active galactic nuclei
*[pc]: parsec
*[ALMA]: Atacama Large Millimeter/submillimeter Array
*[APEX]: Atacama Pathfinder Experiment
*[JCMT]: James Clerk Maxwell Telescope
*[LMT]: Large Millimeter Telescope Alfonso Serrano
*[SPT]: South Pole Telescope
*[SMA]: Submillimeter Array
*[SMT]: Submillimeter Telescope
*[S/N]: signal-to-noise ratio
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Subsets and Splits
Filtered Text Samples
Retrieves 100 samples of text containing the specific phrase "You are a helpful assistant", providing limited insight into the dataset.
Helpful Assistant Text Samples
Returns a limited set of rows containing the phrase 'helpful assistant' in the text, providing basic filtering of relevant entries.